Kant's Philosophy of Mathematics: Modern Essays (Synthese Library, 219)

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KANT'S PHILOSOPHY OF MATHEMATICS

SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE [I

MODERN ESSAYS

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1 Managing Editor:

Edited by

CARLI. POSY JAAKKO HINTIKKA, Boston University

Duke University, North Carolina, U_SA

Editors: DONALD DAVIDSON, University of California, Berkeley GABRIEL NUCHELMANS, University ofLeyden WESLEY C. SALMON, University of Pittsburgh

KLUWER ACADEMIC PUBLISHERS VOLUME 219

DORDRECHT I BOSTON I LONDON

Library of Congress Cataloging-in-Publication Data

TABLE OF CONTENTS Kant's philosophy of mathemat1cs modern essays I edtted by Carl J. Posy. p. cm. -- (Synthese library; v. 219) ISBN 0-7923-1495-6 (HB : aCid-free paper) 1. Mathematics--Phllosophy. 2. Kant. Immanuel. 1724-1804. I. Posy. Carl J. II. Series. QAB.4.K36 1992 91-86207 501--dc20

ISBN 0-7923-1495-6

PREFACE

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ACKNOWLEDGEMENTS

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CARL J. POSY -

Introduction: Mathematics in Kant's Critique of Pure Reason

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I. CLASSIC PAPERS OF THE 1960'S AND 1970'S

Published by Kluwer Academic Publishers P.O. BOX 17, 3300 AA Dordrecht. The Netherlands. I I

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Kluwer Academic Publishers incorporates the publishing programmes ofD. Reidel, Martinus Nijhoff. Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publisher. 101 Philip Drive. Norwell, MA 02061, U.S.A. In all other countries, sold and distributed

by Kluwer Academic Publishers Group, P.O. BOX 322, 3300 AH Dordrecht, The Netherlands. Printed on acid-free paper

All Rights Reserved © 1992 by Kluwer Academic Publishers

No part of the material protected by this copyright notice may be reproduced or utilized in any fonn or by any means, electronic or mechanical, including photocopying, recording or by any infonnation storage and retrieval system, without written permission from the copyright owner.

JAAKKO HINTIKKA - Kant on the Mathematical Method CHARLES PARSONS - Kant's Philosophy of Arithmetic - Postscript MANLEY THOMPSON - Singular Tenns and Inmitions in Kant's Epistemology - Postscript PHILIP KITCHER - Kant and the Foundations of Mathematics

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69 81

102 109

II. RECENT WORK CHARLES PARSONS - Arithmetic and the Categories J. MICHAEL YOUNG - Construction, Schematism and Imagination MICHAEL FRIEDMAN - Kant's Theory of Geometry STEPHEN BARKER - Kant's View of Geometry: a Partial Defense ARTHUR MELNICK - The Geometry of a Fonn of Inmition WILLIAM HARPER - Kant on Space, Empirical Realism, and the Foundations of Geometry CARL J. POSY - Kant's Mathematical Realism GORDON G. BRITIAN, JR. - Algebra and Inmition JAAKKO HINTIKKA - Kant's Transcendental Method and His Theory of Mathematics

Printed in the Netherlands

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221 245 v 257 293 315 341

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TABLE OF CONTENTS

CONTRIBUTORS

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INDEX

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PREFACE

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For the bulk of the twentieth century, Kant's philosophy of mathematics fell into low disrepute. His views suffered partly from the nineteenth and twentieth century challenges to the hegemony of Euclidean geometry. And their reputation declined partly because the great thinkers - Russell, Hilbert, Brouwer and others - who shaped the twentieth century's philosophical approach to mathematics each took aim in his own way at central Kantian doctrines. To be sure, the early years of the twentieth century saw lively debates about Kant's mathematical views. (The famous interchanges between Russell and Couterat are but one instance of this.) But the generally negative assessment of Kant's philosophy of mathematics led naturally to general neglect. So, with scant exception, the middle of the century saw little creative work on this topic. The one main exception is the late Gottfried Martin's work. Martin's Arithmetik und Kombinatoric bei Kant appeared in 1938. Though the book's positive thesis - that Kant viewed arithmetic as a fonnal axiomatic science is still highly controversial, its scholarship and its treasure trove of insights about Kant's way of thinking remain unsurpassed. It is indeed still the best starting place for anyone who wishes to do scholarly work on reactions to Kant's philosophy of mathematics. But for a long time Martin's work had sadly small impact on the wider philosophical community, and exploration of Kant's views about mathematics remained a philosophical backwater. All of that changed in the mid-1960's with the publication of a lively debate about issues in Kantian mathematics between Charles Parsons and J aakko Hintikka. The debate, which was fascinating in its own right, - each of these authors is an accomplished logician and a solid Kantian scholar had the added effect of reigniting philosophical interest in Kant's );houghts about mathematics. A trickle of fascinating studies - inspired by the issues that Parsons and Hintikka raised - followed in the 1970's. Then in the next decade the movement grew into a full-fledged renaissance of interest in Kant's philosophy of mathematics. This was further sparked by a 1983 conference on the topic at Duke University. vii Carl J. Posy (ed.), Kant's Philosophy of Mathematics, vii- viii.

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The present anthology is designed first of all to chronicle and publicize this renaissance. But hopefully it will also show the importance of Kantian thinking for general issues in the philosophy of mathematics. Part I includes a pair of the seminal papers by Hintikka and Parsons. It also includes influential papers by Manley Thompson and Philip Kitcher which followed up on this initial work and which appeared in the ensuing decade. Parsons and Thompson have added more recent postscripts to their papers. Part II contains a selection of papers that represent the flowering of interest in Kant's philosophy of mathematics during the 1980's and the beginning of the 1990's. It includes new work by both Hintikka and Parsons. I hope that the essays in this volume will demonstrate that Kant's views about mathematics are interesting, that they are defensible (or at least respectable in the light of our current knowledge) and that they are inseparable from his overall philosophical system. Indeed, a leisurely thumb through the volume will readily testify to the fascination that many modem Kant scholars and philosophers of mathematics have for Kant's mathematical views. As for defensibility, you will find in the pages that follow quite a few attempts to vindicate or reconstruct Kantian doctrines and arguments from a

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modem perspective. In fact, far from viewing Kant's work on mathematics as dead, the volume contains several attempts to connect his views with quite recent findings in physical science, cognitive science and mathematical logic. Finally, you will find the volume rich and diverse with suggestions connecting Kant's mathematical views with his overall philosophy. This collection contains a mixture of scholarship and specUlation, reconS!rUction and close textual analysis.1t represents it fair sample of the current state of thinking on Kant's philosophy of mathematics. To be sure, you will find no unanimity on any single point; and it is hard to tell which, if any, of the perspectives and interpretations presented here will prevail. This very

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diversity and vigorous debate, however, are themselves the signs of a vibrant

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ACKNOWLEDGMENTS

Part of the editorial and clerical costs for this anthology were underwritten by a generous grant from the Duke University Research Council. The editor is also indebted to Tod Davis and Marianne Grochowski for editorial assistance. The editor gratefully acknowledges the permission given by the authors, editors and copyright holders of the following publications to reproduce them in original or revised form in this anthology: "Kant on the Mathematical Method" was first published in The Monist, volume 51, 1967, pages 352-375. It is copyright © 1967 The Monist, and is reproduced here by kind permission of Jaakko Hintikka and the publishers of The Monist, La Salle, Illinois, 61301. "Kant's Philosophy of Arithmetic" was first published in Sidney Morgenbesser, Patrick Suppes, and Morton White, eds., Philosophy, Science and Method: Essays in Honor of Ernest Nagel, 1969, St. Martin's Press. It is reproduced here by kind permission of Charles Parsons and the editors. The Postscript to "Kant's Philosophy of Arithmetic" was first published in Charles Parsons, Mathematics in Philosophy: Selected Essays, copyright 1983, Cornell University Press. It is reproduced here by kind permission of· Charles Parsons and Cornell University Press.

field of study.

Durham, 1991

"Singnlar Terms and Intuitions in Kant's Epistemology" was originally published in the Review of Metaphysics, Volume 26, copyright 1972, it is repnnted here by kind permission of Manley Thompson and the Review of Metaphysics. "Kant and the Foundations of Mathematics" was first published in The Philosophical Review, Volume 84, copyright 1975. It is reprinted here by kind permission of Philip Kitcher and The Philosophical Review.

ix Carl J. Posy (ed.), Kant's Philosophy of Mathematics. ix-x.

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ACKNOWLEDGEMENTS

"Arithmetic and the Categories" was first published in Topoi volume 3, copyright 1984, It is reprinted here by kind permission of Charles Parsons and of Topoi. "Construction, Schematism, and Imagination" was first published in Topoi volume 3, copyright 1984. It is reprinted here by kind permission of J. Michael Young and of Topoi. "Kant's Theory of Geometry," was first published in The Philosophical Review, Volume 94, copyright 1985. It is reproduced here by kind permission of Michael Friedman and of The Philosophical Review. "The Geometry of a Form of Intuition" was first published in Topoi volume 3, copyright 1984. It is reprinted here by kind permission of Arthur Melnick and of Topoi. "Kant on Space, Empirical Realism and the Foundations of Geometry" was first published in Topoi volume 3, copyright 1984. It is reprinted here by kind permission of William Harper and of Topoi. "Kant's Mathematical Realism," was first published in The Monist, volume 67, pages 115-134. It is copyright © 1984 The Monist, and is reproduced here by kind permission of the publishers of The Monist, La Salle, minois, 61301. "Kant's Transcendental Method and His Theory of Mathematics," was first published in Topoi volume 3, copyright 1984. It is reprinted here by kind permission of Jaakko Hintikka and of Topoi.

CARL

J.

POSY

INTRODUCTION: MATHEMATICS IN KANT'S CRITIQUE OF PURE REASON

Kant's views about mathematics have had eloquent and influential detractors among the very thinkers who shaped our present philosophical world view: Frege ridiculed Kant's claims about the role of pure intuition in arithmetic.! Russel denounced the Kantian theory of geometry.2 And Brouwer, who built his intuitionistic mathematics on a Kantian premise about the a priority of time, explicity rejected the a priority of space.3 Moreover these critics follow generations of "mainstream" dissenters going back at least to Eberhard.4 So even sympathetic readers of Kant are tempted to isolate his philosophy of mathematics as a small, separable, outmoded and embarrasing imperfection in the full critical framework. 5 But nothing could be further from the truth. The essays in this volume will, I believe, serve to affirm that Kant's views about mathematics are interesting, and that where they may be ntistaken the mistakes are explainable. Indeed, though the authors of these essays are not mindlessly uncritical of Kant, you will find in the pages that follow quite a few attempts to vindicate or reconstruct Kant's mathematical views from a modem perspective and to connect those views with quite recent work in empirical science. in mathematics and in logic. In this introduction, however, I want to address the question of separability. References to and even sustained discussions of mathematics abound in all of Kant's philosophy from the early writings through the great Critiques and their collateral works. But Kant's pivotal book is clearly the Critique of Pure Reason; so in this essay I shall introduce the papers that follow by showing how some of the issues that they raise bear on the central themes of that work. My point is not merely that the first Critique touches on mathematical issues. Rather, in introducing these papers I want to show you that these mathematical issues are bound intimately into the fabric of epistemology, metaphysics and philosophy of science that is woven by the Critique. I would like you to see that the effort extended in the present collection of essays to understand Kant's views about mathematics will also necessarily illuminate his most basic philosophical doctrines. I Carl J. Posy (ed.), Kant's Philosophy afMathematics, 1-17. tel 1QQ?' T(/lIwpr Academic Publishers. Printed in the Netherlands.

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I have fu-ranged these introductory remarks according to the major divisions of the Critique of Pure Reason. And for readers who may not be familiar with its general outline, I have also included brief sketches of the strocture of the Critique and of its main sections. I. THE" AESTHETIC"

1. The Structure of the first Critique and of the "Aesthetic" Kant divided the Critique of Pure Reason into a "Doctrine of Elements" and a "Doctrine of Method." The Doctrine of Elements studies the scope and limitations of empirical knowledge. It focuses on the grounds of a priori knowledge (specifically on the possibility of synthetic a priori knowledge, which is Kant's response to Hume's skepticism) and argues that we can discover these possibilities by examining the presuppositions of unproblematic empirical knowledge. The "Transcendental Aesthetic," the first main division of the Doctrine of Elements, isolates the faculty of sensibility and explores its contribution to a priori knowledge. The second main division, the "Translendental Logic," (which concentrates on t"e intellect) is itself divided into two parts: The "Analytic" concerns the role of the intellect in empirically (and therefore sensorily) based knowledge. (The intellect thns limited is what Kant calls the faculty of understanding.) The "Dialectic" demonstrates that this faculty cannot be employed to provide nontrivial knowledge which is not sensorily based. ("Reason" is Kant's name for the intellect when it is not limited to sensorily based knowledge.) Central to Kant's arguments in all three of these divisions is his so called "critical" move (his analysis of the distinct epistemological, semalltic and ontological contributions of these separate faculties) and his doctrine of transcendental idealism (the view that sensibility and understanding do provide their own fully adequate theory of truth and their own ontology, quite separate from the semantics and ontology of reason)~ After having established these points to his own satisfaction, Kant devotes the "Doctrine of Method" to comparing the methodological techniques appropriate to the practice of empirical science, philosophy and mathematics. Turning to the "Aesthetic," three of the issues raised by this part of the Critique play central roles in several of the papers in this anthology: (a) Kant's notion of intuitions and of their distinctive epistemological roles; (b) Kant's exposition of space as an a priori form of intuition; "transcendentally ideal" and (e) Kant's claims that space and time "empirically real."

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INTRODUCTION

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Each of these themes actually also reverberates throughout the rest of the Critique. I shall discuss them in order.

2. The Nature of Intuitions According to Kant all synthetic judgements (and hence all nontrivial elements of knowledge) rest ultimately on intuitions. Iiltuition is thus for him a fundamental epistetuic notion. (The German word is "Anschauung.") Nevertheless, there is no clear consensus about what Kant means by Anschauung, and the opening essays by Hintikka and Parsons contain a debate about this basic notion. The debate is relevant to the philosophy of mathematics because Kant insists that mathematics contains synthetic knowledge. (Mathematics is, in fact, Kant's paradigm of synthetic a priori knowledge.) And so he must define "intuition" in 'a way that includes the evidence for mathematical jndgments. Indeed, Kant's argument for the intuitivity of geometric space provides one clear criterion for intuitions. This is the criterion of singularity. He says (at A25!B39) that "we can represent to ourselves only one space; and if we speak of diverse spaces, we mean thereby only parts of one and the same unique space."6 The opening passage of the "Aesthetic" appears, however, to add a second criterion, immediacy. "In whatever manner and by whatever means a mode of knowledge may relate to objects, intuition is that through which it is in immediate relation to them .. ."(AI9!B33) These dual characteristics, singularity and immediacy, are further reinforced at A320!B376-77, and they form the basis of Parsons' view in "Kant's Philosophy of Arithmetic." Thongh immediacy entails singularity, the reverse, says Parsons, need not hold. There are singular concepts, whose relation to their objects is presumably mediated by the more general concepts of which they are composed. Parsons also assumes that "immediate" awareness must be quite literally perceptual. This makes Kant's claim that geometry is synthetic quite plausible. For Kantian geometry, on Parsons' interpretation, is about ordinary spatial objects; and geometric reasoning nses standardly perceivable objects for its constructions. This reasoning is a priori because it abstracts from the accidental features of the particular object used and treats the object as a representative for all relevantly similar spatial figures. These same considerations, however, open a hard question about arithmetic. For numbers and numerical concepts are not concretely instantiated

the way geometric concepts are.

CARL J. POSY

INTRODUCTION

That is a question which Hintikka's interpretation is well suited to address. Hintikka claims that immediacy is not a separate criterion for intuitions, but rather a consequence of singularity. Moreover, Hintikka (building on E. W. Beth's work) suggests that the syntheticity of a mathematical judgment derives not from perceptual evidence for the judgment but rather from the role of singular representations in the proof of the judgment. Hintikka agrees that geometric proofs are a priori because of the arbitrariness of their figures. But he claims that this a priority is best expressed by the restrictions on the free variables used in the instances of universal quantifier introdnction and existential quantifier elimination that occur in formalizing these proofs. This approach clearly applies to number theoretic arguments as well. Indeed, existentialquantifier elimination is especially useful in arithmetic proofs (e.g., proofs by mathematical induction). That, he says, is the source of pure intuition for arithmetic. In "Kant on the Mathematical Method" Hintikka goes on to suggest that this reduction of intuitivity to the use of singular terms in formal proofs corresponds to the Euclidian notion of eethesis. That in tum, says Hintikka, is the model for the centrality of "construction" in Kant's theory of mathematics. Parsons, however, insists, as I said, that intuitivity requires something phenomenologically like perception. So to account for the syntheticity of arithmetic he finds its intuitive base in the system of numerals, a system which is indispensible for arithmetic calculation. He claims indeed that calculation plays the role arithmetic that spatial constructions play in geometry: Calculation goes beyond conceptual analysis, engages perceivable individuals and thus grounds arithmetic truths in intuition. Finally, Mauley Thompson, in "Singular Terms and Intuitions in Kant's EpistemOlOgy" builds on this dispute between Hintikka and Parsons to provide bis own intermediate view. He agrees with Hintikka that singularity and imuiediacy always coincide for Kant. But unlike Hintikka, he denies that linguistic singular terms generally represent Kantian intuitions. Intuitions, according to Thompson, almost never have direct linguistic, counterparts. Indeed, he says, Kant's philosophy is virtually without singular terms under-

3. Space as A Priori

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stood in the modem sense. "Space," "time," and numerals, to be sure, come

close to being singular terms. But (citing A7l9/B747) Thompson insists that the mathematical objects they denote do not exist. He accepts Parsons' point that mathematical construction - be it "ostensive" as in geometry or "symbolic" as in algebra-demonstrates possibility and not existence.

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Kant's "Metaphysical" Exposition" of the concept of space provides four arguments, the first two of which are designed to show that the representation of space is an a priori representation. That is to say, judgments whose justifications rest solely on this representation (especially the judgments of geometry) are judgments wbich can be known a priori. This claim of the a priority of geometry is perhaps the most frequently attacked claim of the Critique of Pure Reason. For it involves two subsidiary claims that have not fared well in modem science. It follows first of all from Kant's notion of a priority that the substantive claims of geometry are directly about empirical objects. There is thus no legitimate division in the Kantian scheme between abstract and applied geometry. Secondly, there is the !,nfamous Kantian belief that that these substantive a priori geometric judgments are just the theorems of old-fashioned Euclidean geometry. Objections to Kant's conflation of abstract and applied geometry (which stem from formalism in modern pure mathematics) strike at the heart of

Kant's transcendental idealism and his theory of empirical objects. I will discuss them fully when treating Kant's general notion of "objective Validity" in section 10 below. But for now let me tum to Stephen Barker's "Kant's View of Geometry: A Partial Defense," which addresses both sets of complaints in the setting of empirical science. In our own century, Barker notes, the conception of space that was prompted by relativity theory and then philosophically developed by thinkers like Reichenbach, Camap, Hempel and Nagel, has led to the separation of physical from purely mathematical geometry. In particular, physical science has established tests for the concept of straight line and for other geometric concepts. It is now a matter of empirical testing to determine which (if any) general laws are true of these concepts. And as a matter of fact, these laws are not those predicted by Euclidean geometry. So geometry is neither a priori nor Euclidean. 7 Barker also notes, however, that there has been a tradition of dissent from this standard modern view, a tradition wbich goes back at least to Poincare. Barker himself suggests that Euclidean geometry is consistent with our "ordinary conception of space." And he argues (along with Poincare) that the mode of thought wbich finds Euclidean geometry in conflict with empirical theories depends itself on some controversial assumptions about the criteria

for accepting scientific theories.

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INTRODUCTION

Barker doesn't intend these considerations as a vindication of Kant. Indeed he points out that to be defensible Kant's geometric claims must be interpreted modestly. But he sees this fact itself as a ''partial defense," which shows that the attack on Kant's Euclideanism is itself dependent on sweeping and problematic theses in general philosophy of science.

Melnick contrasts this active spatializing behavior with the passive (indeed "reactive'') behavior associated with sensory stimuli. He uses this contrast to interpret Kant's matter/form distinction and to explain how space can be an intUition in its own right. (Spatial activity can, after all, be directed without attending to any accompanying sensations.) This sort of operational view of space gives poiguancy to problems about how subjectively generated activity (in this case spatializing activity) can dictate the forms of physical objects. Problems of this sort are problems of "objective validity," in Kant's terms; and Melnick responds to these problems by suggesting an operational account of Kant's general metaphysics. In particular he interprets Kant's transcendental idealism as an operational semantics for empirical language, and contrasts that with a purely descriptive language favored by transcendental realism. 8 The second ~argument for the claim that space is an intuition and not a concept rests on the fact that spatial regions extend infinitely. Kant points out that though a concept can have infinitely many things which "fall under" it (i.e., an infinite extension), no concept can comprehend infinitely many things "within it." Michael Friedman analyzes this argument by adapting Bertrand Russell's criticism that Kant's philosophy of mathematics rests on the outmoded constraints of Aristotelian logic. Without the Aristotelian limitation to monadic predicates, says Friedman, Kant would have been able to express the infinity of space in a purely conceptiIal fashion. For while a sentence in the language of monadic predicate logic ntight have (per accidens) an infinite model, no such sentence can force its model to be infinite. Thus, in particular, if the language is purely monadic, no simple collection of axioms for the concept of space can force the infinity of space. RusselJ employs this sort of logical criticism to undermine not only Kant's view of geometry but his entire theory of synthetic a prion· judgments. But Friedman, by contrast, simply uses our historical hindsight to "deepen our understanding of the difficult logical problems" with which Kant's theory of space is struggling. In particular, Friedman emphasizes certain topological and analytic properties of the Kantian conception of space (e.g., denseness and continnity) whose expression we now know requires a polyadic relational language and multiple quantifiers. Since, according to Friedman, Kant cannot form concepts which force these properties to hold of space, he must rest the

4. Space as an Intuih·on The latter two arguments of the "Metaphysical Exposition of Space" are each designed to shows that the original representation of space is an intuition and not a concept. That conclusion is, of course, needed in order to establish the syntheticity of geometry. The first of these paragraphs argues, as we have seen, that the representation of space is necessarily singular. The representation of space, says Kant, encompasses individual spatial regions as parts and not as subsidiary elements which fall under a more general concept. The second paragraph attempts to prove the intuitivity of space by pointing to its infinite extent and claiming that no concept could by itself establish the infinity of space.

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CARL L POSY

These arguments provide the material respectively for Arthur Melnick's "The Geometry of a Form of IntUition" and Michael Friedman's "Kant's Theory of Geometry." I wilJ discuss them in that order. At the heart of Kant's first argument for the intuitivity of space in his observation that an individual spatial region is derived from the original representation of total space by the imposition of lintitations. This is designed to show that the representation of total space could not be derived by abstraction from inferior representations in the way a general concept is derived from the elements of its extension. Melnick, however, elegantly develops the positive riotion that a spatial region is actively delimited by a specifically "spatial" behavior. He does this by outlining "spatial" behavior. He does this by outlin~ ing an "operational" interpretation of Kantian geometry. Melnick's central notion is that of "ostending or delimiting". a spatial region. The subject ostends a region in his inunediate vicinity by pointing and tracing out its boundaries. The ostension can be an active finger pointing; it can be a sweep of the gaze, or even a simple mental shift of attention. Melnick's interpretation captUres the global aspect of space because this regional ostending is a special case of the larger spatializing activity which includes moving one's entire body (e.g., by walking) and delimiting regions along the way.

properties on a non-concepttIal basis: pure intuition and construction in

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INTRODUCTION

5. Space as Empirically Real

The "Metaphysical Deduction" ptovides an elaborate derivation of Kant's list of these "categories" from a specially constructed table of the forms of judgment. And the "Transcendental Deduction" attempts to prove the objective validity of the categories in a most abstract way. This is then followed by the "Schematism" chapter (which uses the faculty of imagination to connect the abstract versions of the categories to concrete empirical concepts), the fourpart "Analytic of Principles" (which shows how these empirical versions of the categories actually structure our scientific knowledge), and a pair of sections relating Kant's views to more traditional historical and metaphysical issues. A number of the papers in this anthology touch on the role of the categories of quantity in the "Analytic", both in their abstract, transcendentaldeduction form and in their spatio-temponJ.! versions in the "Schematism" and in the "Analytic of Principles." The anthology also deals in some detail with the genenJ.! metaphysical issue of objective validity.

If we assume that the syntheticity and a priority of geometry both stem from the special nature of the reptesentation of space, then we must inevitably wonder how we can be certain that geometry holds true of physical objects. Kant's answer is the doctrine of transcendental idealism. I mentioned -Melnick's interpretation of this doctrine in Section (3) above, and I will discuss some other interpretations below. But there is a concomitant doctrine which deserves independent mention here. That is Kant's doctrine of "empirical realism." which is the subject of WJ.!liam Harper's "Kant on Space, Empirical Realism and the Foundations of Geometry." Empirical realism is the doctrine that space and the objects in space (though "transcendentally ideal") are perfectly real when viewed from the perspective of empiricru science. Harper interprets this doctrine in a strong way. Kant is arguing, he says. for the conclusion that the representation of spatial objects presupposes the presence of non-mental objects. Thus he views empirical realism as Kant's attempt to refute Berkeley's phenomenalism. The heart of Kant's argument, as Harper sees it, is the observation that ordinary perception must contain in it more than unadorned sensation. Perception must, in particular, contain a counterfactual element, a projection of "what would happen if ... ,. This element will include the constraints of geometry, constraints which detennine, for instance, how the object will appear when viewed from other positions. But counterfactuals of this sort are the rock on which a Berkeleyan phenomenalism must founder. For no phenomenalism can ultimately support such counterfactuals. Indeed, they rest, Harper insists, on the presupposition that there exists a non-mental object which is an "inexhaustably rich source of additional perceptible features". Thus you can see that the mathematical issues taken up by these papers issues conceming the syntheticity of arithmetic and geometry and of their physical applicability - are inseparable from themes like transcendental idealism and empirical realism which are at the heart of Kant's metaphysics, and general questions about the nature of intuitions and other basic elements of Kant's philosophy of mind. Let's turn next to the "Analytic." II. THE" AN ALYTIC"

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6. The Structure of the "AnoJytic" The "Analytic" investigates the grounds for and the objective validity of synthetic a priori judgments based on the formal concepts of the understanding.

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7. The Categories of Quantity Charles Parsons, in "Arithmetic and the Categories," points out that the categories of quantity (unity, plurality and totality) depend more on the part/whole relation than on their official derivation from universal, particular and singular judgments. He laments this because modem quantification theory (together with set theory, some of which may already be present in the Critique) can link the official derivation of these categories to the concept of number. This, in turn, would have provided a direct path to the principle of extensive magnitude stated in the "Axioms of Intuition." Drawing on sevenJ.! published and unpublished texts, Parsons then shows that Kant was struggling with two very delicate issues about number and quantity. Kant was first of all concerned with the distinction between continu_ ous and discretely ordered aggregates. He handled this, as Parsons demonstrates, by distinguishing between those aggregates whose "unifying concept" dictates a unique method for distinguishing its parts and those whose concept leaves the nature of the parts undetermined. The former are discrete (and therefore numberable), the latter are continuous. Parsons observes that this Kantian distinction does not require any element of intuition, even for the notion. of number. However, the second Kantian concern is precisely whether or not the notion of number per se must have an intuitive element. The "Schematism" clearly implies that "number" is an intuition based concept. But Parsons also brings passages that tell the other way.

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INTRODUCTION

He suggests that we can adapt the modem distinction between structural and constructive treatments of the numbers to interpret the Kantian alternative positions. But he also suggests that Kant himself may never have come to a full conclusion on this.

9. The "Axioms of Intuition" and the "First Analogy"

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8. The "Schematism"

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The schema of a concept is a rule which guides the imagination to produce a paradigmatic representation of those objects which fall under the concept. The categories, of course, are so general that their schemata could not be actual images. So Kant argues that their schemata are general rules for the temporal ordering of appearances by the imagination. These schemata are importantly like mathematical objects: Both exemplify the general structure of intuition with particular concrete objects. To fully appreciate this similarity, and indeed to understand Kant's point in the "Schematism" chapter, we must understand Kant's notion of imagination. This is among the main tasks of J. M. Young's "Construction, Schematism and Imagination." The "construction" in Young's title refers to "ostensive construction" (Le., the activity of intuitively exemplifying mathematical concepts), an activity which lies at the heart of Kant's theory of mathematical proof. Like Parsons (in "Kant's Philosophy of Arithmetic") Young insists that the exemplifying intuition must be sensible. Indeed, again with Parsons, he suggests that a numeral system provides the best exemplifying instances for the concepts of arithemetic. Specifically, a numeral for n indicates the number n precisely because it is "the last thing that would be generated in a standard collection of n sensible things." This sort of collection (e.g., the first n numerals in some numeral system) is a concrete instance of the concept corresponding to the number n, and its "standardness" provides the generality needed for mathematical thought. But here is where imagination comes in. Using a numeral or a standardized collectiou to represent a particular number requires that we "construe" that collection as falling under the corresponding numerical concept. Kantian imagination, says Young, is the central element of the process of "construing." For construing an empirical object as falling under a concept involves imagining how that thing might present itself in other contexts. That in turn, Young claims, requires imaginative construction.

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William Harper, expanding his analysis of geometry, argues that the "Axioms of Intuition" support a sophisticated empirical realism by requiring every dimension of any empirical object to have a precise real-valued maguitude. Though Harper does not do this, we can view his argument as responding to a criticism made in Philip Kitcher's "Kant and the Foundations of Mathematics. "9 Kitcher's complaint rests on the fact that we humans have perceptual thresholds and aims to assail the Kantian claim that space is infiuitely divisible (a claim crucial to the assigument of real-valued maguitudes). The usual argument for infinite divisibility rests on the possibility of iterating the division of any given spatial region. Kitcher points out that this possibility must not be purely logical (or conceptual), for then the claim of infinite divisibility would be anaJytic. However, he insists, we can have no observational ground for this possibility. For there will be observable magnitudes whose bisection we cannot observe. These will be maguitudes at or near the minimal threshold. And we caunot assume that the divisibility of more familiar spatial regions is a general feature of space. Harper responds to this by claiming once again that Kant's empirical realism supplies principles which outstrip our ordinary perceptual abilities. Thus, in particular, he points out that the result of any measurement can be located within observable intervals. Then he suggests that, even though an exact real number caunot be determined by human measurement, a device like van Fraassen's supervaluations can assure the empirical truth of the claim asserting the existence of a precise real number for the measurement. Harper argues that this approach will allow Kant a full empirical realism for mathematics. Indeed, he goes on to claim that the "First Analogy" (in which Kant establishes the permanence of physical substance) similarly establishes a general empirical realism even for unobservable physical objects. 10. Objective Validity At the end of the "Analytic" (at A239-40/B298-99) Kant suggests that were it not for the fact that space and time are the forms of empirical objects, mathematics would be a "mere play of imagination or of understanding." This observation has two contemporary upshots. The first is that - uulike the modern study of formal systems - Kant caunot

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CARL J. POSY

INTRODUCTION

allow the mathematical importance of a pure, uuinterpreted calculus. Modem abstract treatments of geometry, for instance, allow multiple interpretations of the axioms. And, similarly, alternative standard and non-standard realizations of Peano's axioms for arithmetic abound. Kant, however, seems to miss this altogether in his monolithic attachment to observable applications. This upshot is controversial. On the one hand nothiog Kant has said precludes the logical possibility of constructing abstract nninterpreted systems. Indeed, Stephen Barker points out in his essay that Kant explicitly allowed the consistency of non-Euclidean geometries. If we follow Barker's view, we can read the passage just quoted from the end of the "Analytic" as considering the possibility that mathematics be practiced abstractly without application to empirical objects. The main thmst of the passage is then that mathematics, thus considered, would not count as empirical knowledge. Michael Friedman, on the other hand, rejects the plausibility that Kant held any such a proto-modem view. Kant, he argues, "has no notion of possibility on which both Enclidean and non-Euclidean geometries are possible." The second upshot concerns the general ontology of mathematics. For it appears from the quoted Kantian remarks that the proper objects of mathematics are just the full-blooded objects of the empirical world. These, at any rate, are the only objects of which mathematics (or any science) can provide knowledge. But we must then ask whether Kant would admit a separable realm of purely mathematical objects. Here too the commentators will differ. Certaiuly Kant's remarks about numbers indicated that he thought of these as legitimate objects. On the other hand his remark at A719!B747 that in mathematics "there is no question of existence at all" seems to indicate that purely mathematical objects do not exist. Manley Thompson, in his essay, understands this last passage in that way. He suggests as a consequence that there are no objects with which to instantiate the quantified statements of a purely mathematical language. And thus he proposes to interpret generality in such a language ouly by free variables. Charles Parsons suggests in his "Kant's Philosophy of Arithmetic" a less austere reading accordiiig to which the quantifiers would be interpreted in a modal sense. Thus, in particular, existence in a mathematical language would connote the possibility of constructing an appropriate empirical object. These then are some of the central questions of the "Analytic": the ways in which concepts unify the representations of objects, the nature of imagination and categorial schemata, and the whole issue of objective Validity. Once again it should be clear that as you read these essays about quantity,

numerals, arithmetic and geometry, you will inevitably encounter the broader Kantian questions. Let us ruin finally to the essays which touch on the "Dialectic" and the "Doctrine of Method."

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III. THE "DIALECTIC" AND THE "DOl;TRINE OF METHOD"

11. The Structure of the "Dialectic" Following a well established practice, Kant inserts a division of the Critique to diagnose the misuses of the concepts and principles detailed in the "Analytic." He arranges these "mistakes" systematically: They alI derive from natural but improper syllogistic uses of the categorial principles. HO'Yever, the real thrust of the "Dialectic" is Kant's sustained attack on traditional metaphysics (specifically, rational psychology, cosmology and theology). The "Dialectic" also contains his most forceful attack on transcendental realism. For the realist, according to Kant, is trapped into the natural mistakes that lead to the problems of traditional metaphysics. The transcendental idealist, by contrast, allegedly has the tools to avoid these mistakes.

12. The "First Antinomy" Kant's attack on transcendental realism is especially potent in the "Antinomy" section of the "Dialectic", where he attempts to show that the realist (hut not the transcendental idealist) is committed to four pairs of mutually contradictory propositions. The initial two antinomies are of special interest, since these are directly mathematical: The first commits the realist to both the finite and infinite extent of the world in space and in past time. The second commits him to the infinite divisibility of matter and to the opposite atomism as welL The first of these is the focus of my own essay on "Kant's Mathematical Realism." The essay interprets Kant's transcendental idealism as if it were the modem linguistic anti-realism (or "assertabilism") that has come to be associated with intuitionistic logic. Transcendental realism, by contrast, maintains the standard "classical" logic. This allows me to illuminate several features of the "Antinomy" argument. Thus, for instance, the conflicting claims about the age of the universe come as a pair of statements which are classically contradictory. If the realist must accept them both (as Kant claims) then transcendental realism is indeed inconsistent. These statements are not intuitionistically contradictory - I

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INTRODUCTION

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display a Kripke model which falsifies them both - and so transcendental idealism escapes the reductio. Indeed, that same Kripke structure models Kant's claim that, for the idealist, the series of past moments may be continued indefinitely but not infinitely. I show, moreover, how to modify the philosophical side of assertabilism in order to explain the actual arguments that Kant. attributes to the realist and idealist. Finally I suggest that all of this offers an insight into Kant's view of the logic of mathematics. The idealist rejects classical logic for empirical judgments because there will always be observationally undecidable empirical claims. Undecidability plus assertabilism yield a non-classical logic. But, I point out, the "purity" of mathematical intuition leads Kant to believe that all mathematical judgments are decidable. So I conclude that though Kant advo-

only spatio-temporal intuition can ground the complete determination of mathematical objects. In particnlar Brittan uses Kant's views on algebra to dispute those who rest the intuitive nature of mathematics on the central role of calculation. The difference between algebra and arithmetic, he notes, is a difference of generality and not a difference in kind or in subject matter. And so, elaborating a point derived from Gottfried Martin, Brittan insists that algebraic reasoning is just as synthetic and intuitive as the other more specific branches of mathematics. Even in algebra, according to Brittan, this intuitivity rests on the possibility of complete detennination. But, in a bold proposal, Brittan suggests that it is relational structures whose properties are fully detertuinable by algebraic reasoning and not individual mathematical objects.

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cates an intuitionistic logic for empirical discourse, he favors a classical logic

for mathematics.

15. Mathematics and Transcendental Philosophy 13. The "Doctrine of Method"

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This last logical theme is echoed in the "Doctrine of Method," where Kant observes (at A792/B820) that "apagogical" (i.e., indirect) proofs have their "'true place" in mathematics but not in empirical science. For these proofs are classically but not intuitionistically valid. But the "Doctrine of Method" compares mathematics with empirical science (and indeed with philosophy) on more than just the question of their respective proof strategies. It analyzes the roles of axioms, hypotheses and definitions in each of these disciplines as well; and it diagnoses the possibility and methodological sources of synthetic knowledge in each of them. 14. Kant on Algebra

The "Doctrine of Method" contains Kant's only sustained discussion of algebra in the Critique, and this is the focus of Gordon Brittan's "Algebra and

Intuition." Brittan - who, unlike Thompson, does allow Kant a separable realm of mathematical objects - associates Kant's claim that mathematics requires

intuition with the view that these objects are "fully detennined." For Brittan the full detennination of a mathematical object means that all singular mathematical propositions which refer to that object are semantically bivalent. He notes that conceptual analysis alone cannot provide this bivalence, and so

In a long series of papers Jaakko Hintikka has produced perhaps the most comprehensive discussion of Kant's views on mathematical method. And indeed he has generally made great strides towards integrating considerations of mathematical method with the large systematic themes in Kant's speculative philosophY. This is apparent, for instance, in his "Kant on the Mathematical Method," which I discussed in section 2. In "Kant's Transcendental Method and His Theory of Mathematics" Hintikka connects his interpretation of Kant's views on mathematical method with the broadest of Kantian themes, the question of the objects and procedures of "transcendental philosophy." Specifically, according to Hintikka, Kant's notion of "transcendental knowledge" is not confined to knowledge about one particular conceptual system, but rather it refers to knowledge about the human activities which "create and maintain" that conceptual system. Kant's view of mathematical knowledge, says Hintikka, gives a prime illustration of this point. Recall that, according to Hintikka, Kantian mathematical method rests on the introduction of instantiating singnlar tenns in mathematical proofs. These tenns, which play the role of pure intuitions, ordinarily come without ties to specific empirical objects. So Kant is faced with the problem of how intuitions like these - nnconnected as they are to actual objects - can yield empirical knowledge a priori. And his answer, according to Hintikka, is that the method of instantiation anticipates the properties which we ourselves introduce "in the processes through which we come to know individuals (particulars)." Of course, those processes themselves are, for Kant, the processes involved in sense perception.

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To be sure, Hintikka criticizes Kant for basing knowledge of particulars on sense perception. (He calls this Kant's "Aristotelian premise.") For Hintikka views sense perception (even Kantian sense perception) as entirely passive. And he contrasts this with the most general process of gathering information abont individuals, a process which includes the activities of seeking and finding. However, when we correct Kant, and view knowledge gathering in this more active way, we produce, Hintikka believes, a more consistently Kantian ("transcendental") epistemology. We also get a revised Kantian view of logic. This will center on instantiation rules and will found these rules in tnm on a general theory of seeking and finding. Moreover, Hintikka argues, the revision itself will produce improved readings of Kant's notion of a thingin-itself and of his general theory of identity.

REFERENCES

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Duke University NOTES 1 See Frege's Foundations of Arithmetic, especially sections 12 ff. 2 Russell's most sustained criticism of Kant's philosophy of geometry is contained in his An

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Essay on the Foundations of Geometry. 3 See chapter II of Brouwer's dissertation, Over de grondslagen del" wiskunde, his lecture "Het wezen der meetkunde," and his essay "Intuitionism and Formalism." 4 See the articles by J. A. Eberhard in Philosophischen Magazin (1788-1792) and Philosophischen Archiv (1793-1794). An excellent summary of Eberhard's criticisms and those of others together with a translation of Kant's response in the essay "On a Discovery According to which any New Critique of Pure Reason has been made Superfluous by an Earlier One" is fOWld in H. Allison's The Kant Eberhard Controversy. 5 See for instance Norman Kemp Smith's A Commentary on Kant's Critique of Pure Reason, (especially his treatment of the "Introduction" and of the concept of space in the "Aesthetic") and Jonathan Bennett's Kant's Analytic. ~ Making the same point about time, Kant says a' bit later on, that ''the representation which can be give only through a single object is intuition." (A32/B47). 7 It is worth noting that Kant was an explicit foil for the early nineteenth century mathematicians who challenged both the a priority and Euclidean natu~ of space, though these challenges weren't fully accepted, until much later. See M. Greenberg's Euclidean and Non-Euclidean Geometries: Development and History, Chapter 6. g Melnick has expanded this theme into a full-blown interpretation of the central parts of the Critique of Pure Reason in his book, Space, Time and Thought in Kant. 9 Actually, Harper is responding to a point in Charles Parsons' "Infinity and Kant's Conception of the 'Possibility of Experience'," a paper which influenced Kitcher's criticism.

Allison, H.: 1973, The Kant-Eberhard Controversy, The Johns Hopkins University Press, Baltimore. Bermett, J.: 1966, Kant's Analytic, Cambridge University Press, Cambridge. Brouwer, L.EJ.: 1907, Over de Grondslagen der Wiskunde. (Dissertation, University of Amsterdam; translated as 'On the Foundations of Mathematics,'), in Brouwer (1975). Brouwer, L.E.J.: 1909, 'Het Wezen der Meetkunde', (Translated as The Nature of Geometry'), in Brouwer (1975). Brouwer, L.E.J.: 1913, 'Intuitionism and Formalism', in Bull. Amer. Math. Soc. 20; reprinted in Brouwer (1975). Brouwer, L.EJ.: 1975, Collected Works, v. I. (A. Heyting, ed.), North Holland, Amsterdam. Frege, G.: 1955, The Foundations of Arithmetic. 0. L. Austin. trants.). Oxford University Press. OxfQfd. "Greenberg. M.: 1974. Euclidean and Non-Euclidean Geometries: Development and History (Second Edition), W. H. Freeman and Co., New York. Melnick, A.: 1989, Space, Time and Thought in Kant, Kluwer Academic Publishers, Dordrecht. Farsons, c.: 1964, "Infinity and Kant's conception of 'the 'Possibility of Experience"', Philosophical Review, 73. Russell, B.: 1956, An Essay on the Foundations of Geometry, Dover, New York. Smith, N. K.: 1929, A Commentary on Karrt's Critique of Pure Reason, Macmillan, London.

JAAKKO HINTIKKA

KANT ON THE MATHEMATICAL METHOD

I. MATHEMATICAL METHOD TURNS ON CONSTRUCTIONS

According to Kant, "mathematical knowledge is the knowledge gained by reason from the construction of concepts," In this paper, I shall make a few suggestions as to how this characterization of the mathematical method is to be under,stood, The characterization is given at the end of the Critique of Pure Reason in the first chapter of the Transcendental Doctrine of Method (A 713 = B 741),1 In this chapter Kant proffers a number of further observations on the subject of the mathematical method. These remarks have not been examined very intensively by most students of Kant's writings. Usually they have been dealt

with as a sort of appendix to Kant's better-known views on space and time, presented in the Transcendental Aesthetic. In this paper, I also want to call attention to the fact that the relation of the two parts of the first Critique is to a considerable extent quite different from the usual conception of it. To come back to Kant's characterization: the first important term it contains is the word 'construction'. This term is explained by Kant by saying that to construct a concept is the same as to exhibit, a priori, an intuition which corresponds to the concept.2 Construction, in other words, is tantamount to

the transition from a general concept to an intuition which represents the concept, provided that this is done without recourse to experience. 2. A VULGAR INTERPRETATION OF KANTIAN CONSTRUCTIONS

How is this tenn 'construction' to be understood? It is not surprising to meet

it in a theory of mathematics, for it had in Kant's time an established use in at least one part of mathematics, viz. in geometry. It is therefore natural to assume that what Kant primarily has in mind in the passage just quoted are the constructions of geometers. And it may also seem plausible to say that the reference to intuition in the definition of construction is calculated to prepare

21 Carl J. Posy (ed.), Kant's Philosophy of Mathematics, 21-42.

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KANT ON THE MATHEMATICAL METHOD

the ground for the justification of the use of such constructions which Kant gives in the Transcendental Aesthetic. What guarantee. if any. is there to make sure that the geometrical constructions are always possible? Newton had seen the only foundation of geometrical constructions in what he called 'mechanical practice' (see the preface to Principia). But if this is so, then the certainty of geometry is no greater than the certainty of more or less crude 'mechanical practice'. It may seem natural that Kant's appeal to inmition is designed to fumish a better foundation to the geometrical constructions. There is no need to construct a figure on a piece of paper or on the blackboard, Kant may seem to be saying. All we have to do is to represent the required figure by means of imagination. This procedure would be justified by the outcome of the Transcendenral Aesthetic, if this can be accepted. For what is allegedly shown there is that all the geometrical relations are due to the structure of our sensibility (our percepmal apparams, if you prefer the term); for this reason they can be represented completely-in imagination without any help of senseimpressions. This interpretation is the basis of a frequent criticism of Kant's theory of mathematics. It is said, or taken for granted, that constructions in the geometrical sense of the word can be dispensed with in mathematics. All we have to do there is to carry out certain logical argnments which may be completely formalized in terms of modem logic. The only reason why Kant thought that mathematics is based on the use of constructions was that constructions were necessary in the elementary geometry of his day, derived in most cases almost directly from Euclid's Elementa. But this was only an accidental peculiarity of that system of geometry. It was dne to the fact that Euclid's set of axioms and postulates was incomplete. In order to prove all the theorems he wanted to prove, it was therefore not sufficient for Euclid to carry out a logical argnment. He had to set out a diagram or figure so that he could tacitly appeal to our geometrical intuition which in this way could supply the missing assumptions which he had omitted. Kant's theory of mathematics, it is thus alleged, arose by taking as an essential feamre of all mathematics something which only was a consequence of a defect in Euclid's parricnlar axiomatization of geometry.3 This interpretation, and the criticism based on it, is not without relevance as an objection to Kant's fnIl-fiedged theory of space, time, and mathematics as it appears in the Transcendental Aesthetic. It seems to me, however, that it does less than justice to the way in which Kant acmally arrived at this theory.

It does not take a sufficient accOlmt of Kant's precritical views on mathematics, and it even seems to fail to make sense of the arguments by means of which Kant tried to prove his theory. Therefore it does not give us a chance of expounding fully Kant's real arguments for his views on space, time and mathematics, or of criticizing them fairly. It is not so much false, however, as

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too narrow. 3. KANT'S NOTION OF INTUITION

We begin to become aware of the insufficiency of the above interpretation when we examine the notion of construction somewhat more closely. The defiuition of this term makes use of the notion of intuition. We have to ask, therefore: What did Kant mean by the term 'inmition'? How did he define the teIID? What is the relation of his notion of intuition to what we are accustomed to associate with the term? The interpretation which I briefly sketched above assimilates Kant's notion 'I of an a priori intuition to what we may call mental pictures. Intuition is something you can put before your mind's eye, something you can visualize, \ something you c:an represent to yo.ur imagination. This is not at all .the basic meanmg Kant hImself wanted to gIve to the word, however. According to hlS"j definition, presented in the first paragnaph of his lectures on logic, every particnlar idea as distingnished from general concepts is an inmition. Everything, in other words, which in the human mind represents an individual is an intuition. There is, we may say, nothing 'intuitive' about intuitions so defined. Inmitivity means simply individuality.' Of course, it remains true that later in his system Kant came to make intuitions intuitive again, viz. by arguing that all our human intuitions are bound up with our sensibility, Le., with our faculty of sensible perception. But we have to keep in mind that this connection between intuitions and sensibility was never taken by Kant as a mere logical consequence of the definition of intuition: On the contrary, Kant insists all through the Critique of Pure Reason that it is not incomprehensible that other beings might have intuitions by means other,-than senses.5 The connection between sensibility and inmition was for Kant something to be proved, 'not something to be assumed.6 The proofs he gave for assuming the connection (in the case of human beings) are presented in the Transcendenral Aesthetic. Therefore, we are entitled to assume the connection between sensibility and intuitions only in those parts of Kant's system which are logically posterior to the Transcendenral Aesthetic.

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JAAKKO H1NTIKKA 4. THE SYSTEMATIC PRIMACY OF KANT'S THEORY OF THE MATHEMATICAL METHOD

My main suggestion towards an interpretation of Kant's theory of the mathematical method, as presented at the end ofthe first Critique, is that this theory is not posterior but rather systematically prior to the Transcendental Aesthetic. If so, it follows that, within this theory, the term 'intuition' should be taken in the 'unintuitive' sense which Kant gave to it in his definition of the notioTI. In particular, Kant's characterization of mathematics as based on the use of constructions has to be taken to mean merely that, in mathematics, one is all the time introducing particular representatives of general concepts

and carrying out arguments in terms of such particular representatives, arguments which cannot be carried out by the sole means of general concepts. For if Kant's methodology of mathematics is independent of his proofs for connecting intuitions and sensibility in the Aesthetic and even prior to it, then we have, within Kant's theory of the method of mathematics, no justification whatsoever for assuming such a connection, i.e., no justification for giving

the notion of intuition any meaning other than the one given to it by Kant's own definitions. There are, in fact, very good reasons for concluding that the discussion of

the mathematical method in the Doctrine of Method is prior to, and presnpposed by, Kant's typically critical discussion of space and time in the Transcendental Aesthetic. One of them should be enough: in the Prolegomena, in the work in which Kant wanted to make clear the structure

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of his argument, he explicitly appeals to his discussions of the methodology of mathematics at the end of the Critique of Pure Reason in the beginning and dnring the argument which corresponds to the Transcendental Aesthetic, thus making the dependence of the latter on the former explicit. This happens both when Kant discusses the syntheticity of mathematics (Academy edition of Kant's works, Vol. 4, p. 272) and when he discusses its intuitivity (ibid., p. 281; cf. p. 266). Another persuasive reason is that at critical junctures Kant in the Transcendental Aesthetic means by intuitions precisely what his own definitions tell us. For instance, he argues abnut space as follows: "Space is not a' ... general concept of relations of things in general, but a pure intuition. For ... we can represent to ourselves only one space.... Space is essentially one; the manifold in it, and therefore the general concept of spaces, depends solely on the introduction of lintitations. Hence it follows that an ... intuition underlies all' concepts of space" (A 24-25 = B 39). Here intuitivity is

KANT ON THE MATHEMATICAL METHOD

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inferred directly from individuality, and clearly means nothing more than the latter. 5. THE HISTORICAL PRIMACY OF KANT'S THEORY OF THE MATHEMATICAL METHOD

But I am afraid that, however excellent reasons there may be for reversing the order of Kant's exposition in the first Critique and for putting the discussion of mathematics in the Methodenlehre before the Transcendental Aesthetic, my readers are still likely to be incredulous. Could Kant really have meant nothing more than this by his characterization of the mathematical method? Could he have thought that it is an important peculiarity of the method of mathematicians as distinguished from the method of philosophers that the mathematicians make use of special cases of general concepts while philosophers do not? Isn't suggesting this to press Kant's definition of intuition too far? The answer to this is, I think, that there was a time when Kant did believe that one of the main peculiarities of the mathematical method is to consider particular representatives of general concepts.7 This view was presented in the precritical prize essay of the year 1764. Its interpretation is quite independent of the interpretation of Kant's critical writings. In particular, the formulation of this precritical theory of Kant's does not turn on the notion of intuition at all. It follows, therefore, that the idea of the mathematical method as being based on the use of general concepts in concreto, Le., in the fonn of individual instances, was the starting-point of Kant's more elaborate views on

mathematics. Whether or not my suggested reading of Kant's characterization of mathematics is exhaustive or not, that is, whether or not intuition there

means something more than a particular idea, in any case this reading is the one which we have to start from in trying to understand Kant's views on mathematics. It is useful to observe at this point that the reading of Kant which I am suggesting is not entirely incompatible with the other, more traditional, interpretation. On one hand, a fully concrete mental picture represents a particular, and therefore an intuition in the sense of the wider definition. On the other hand, particular instances of general concepts are usually much easier to deal with than general concepts themselves; they are much more intuitive in the ordinary sense of the word than. general concepts. The two interpretations therefore don't disagree as widely as may first seem. What really makes the difference between the two is whether Kant sometimes had in mind, in

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KANT ON THE MATHEMATICAL METHOD

addition to 'usual' intuitions in the sense of mental pictures or images, some other individuals that are actually used in mathematical arguments. This, I

the use of intuitions, i.e., on the use of representatives of individuals as distinguished from general concepts. After all, the variables of elementary algebra range over numbers and don't take predicates of numbers as their substitution values as the variables of a formalized syllogistic may do. Then we can also understand what Kant had in mind when he called algebraic operations, such as addition, multiplication, and division, constructions. For what happens when we combine in algebra two letters, say a and b, with a functional sign, be this f or g or + or . Or:, obtaining an expression like f(a, b) or g(a, b) or a + b or a . b or a: b? These expressions, obviously, stand for indiVidUal) numbers or, more generally, for individual magnitudes, usually for individuals different from those for which a and b stood for. What has happened, therefore, is that we have introduced a representative for a new individual. And such an introduction of representatives for new individuals, i.e., new intuitions, was just what according to Kant's definition happens when we construct something. The new individuals may be said to represent the concepts 'the sum of a and b', 'the product of a and b', etc. Kant's remarks on algebra therefore receive a natural meaning under my interpretation, quite apart from the question whether this meaning is ultimately reconcilable with what Kant says in the Transcendental Aesthetic. We might say that the purpose of Kant's use of the term 'intuition' here is to say that algebra is nominalistic in Quine's sense: the only acceptable values of variables are individuals.

think, is something we must make an allowance for. 6. KANT ON ALGEBRA

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In fact, if we have a closer look at Kant's actual theory of mathematics as presented at the end of the Critique of Pure Reason, we shall see that many things become natural if we keep in mind the notion of intuition as a particular idea in contra-distinction to general concepts. Usually, people read Kant's theory of the mathematical method in the light of what he says in the Transcendental Aesthetic. In other words, they read 'intuition' as if it meant 'mental picture' or 'an image before our mind's eye' or something of that sort. But then it becomes very difficult to understand why Kant refers to algebra and to arithmetic as being based on the use of intuitions. The point of using algebraic symbols is certainly not to furnish ourselves with intuitions in the ordinary sense of the word, that is, its purpose is not to furnish ourselves with more vivid images or mental pictures. Scholars have tried to reconcile Kant's remarks on algebra and arithmetic with his critical doctrines as they are presented in the Transcendental Aesthetic. The outcome of these attempts is aptly summed up, I think, by Professor C. D. Broad in a well-known essay on 'Kant's Theory of Mathematical and Philosophical Reasoning', where he says that "Kant has provided no theory whatsoever of algebraic reasoning.'" This is in my opinion quite correct if we read Kant's descriprion of the mathematical method in the light of what he says in the Transcendental Aesthetic. But then Broad's view becomes, it seems to me, almost a reductio ad absurdum of the assumption that the Transcendental Aesthetic is, in Kant's mind, logically prior to the discussion of mathematics at the end of the first Critique. For on this assumption the statements Kant makes on arithmetic and algebra are not only deprived of their truth but also of their meaning. If the Transcendental Aesthetic were logically prior to Kant's methodology of mathematics, it would become entirely incomprehensible what on earth Kant could have meant by his remarks on arithmetic and algebra which so obviLously are at variance with his professed theories. On the other hand, if we assume' that by 'intuition' Kant only meant any representative of an individual when he commented on arithmetic and algebra, a number of things, although not necessarily everything, become natural. If we can assume that the symbols we use in algebra stand for individual numbers, then it becomes trivially true to say that algebra is based on

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7. KANT ON ARITHMETICAL EQUATIONS

Kant's remarks on arithmetic present a somewhat more complicated problem. I shall not deal with them fully here, although they can be shown to square with the view I am suggesting. There is only one point that I want to make here. In the case of the arithmetic of small numbers, such as 7, 5, and 12, the ordinary reading of Kant's remarks is not without plausibility. What Kant seems to' be saying is that in order to establish that 7 + 5 = 12 we have to visualize the numbers 7, 5, and 12 by means of points, fingers, or some other suitable illustrations so that we can immediately perceive the desired equation. He goes as far as to say that equations like 7 + 5 = 12 are immediate and indemonstrable (A 164 = B 204). This is not easy to reconcile with the fact that Kant nevertheless described a procedure which serves, whether we call it a proof or not, to establish the truth of the equation in question and that he said that his view is more natural as applied to large numbers (B 16). I hope

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to be able to show later what Kant meant by saying that equations like 7 + 5 = 12 are 'immediate' and 'indemonstrable'. He did not mean that the equation can be established without an argument which we are likely to call a proof. 'Immediate' and 'indemonstrable' did not serve to distinguish immediate perception from an articulated argument, but to distioguish a certain subclass of particularly straightforward arguments from other kinds of proofs. The ordinary interpretation of Kant's theory therefore fails here too. Of the correct view I shall try to give a glimpse later.

he used the term exposition for a process analogous to that of mathematical

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8. EUCLID AS A PARADIGM FOR KANT

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construction.

The setting-out or ecthesis is closely related to the following or third part of a Euclidean proposition, the auxiliary construction. This part was often called the preparation or machinery (Ka~acrKE1Jti). It consisted in stating that the figure constructed in the setting-out was to be completed by drawing certain additional lines, points, and circles. In our example, the preparation reads as follows: "For let BA be drawn through the point D, let DA be made equal to CA, and let DC be joined." The construction was followed by the apodeixis or proof proper (i'm60E1~1~). In the proof, no further constructions were carried out. What

One good way of coming to understand Kant's theory of mathematics is to ask: What were the paradigms on which this theory was modelled? The most obvious paradigm, and in fact a paradigm recognized by Kant himself, was Euclid's system of elementary geometry.9 In the beginning of this paper, we

happened was that a series of inferences were drawn concerning the figure

saw that a usual criticism of Kant's theory of mathematics is based on a comparison between Kant's theory and Euclid's system. It seems to me, however, that it is not enough to make a vague general comparison. It is much more

way in which it was constructed.

useful to ask exactly what fearores of Euclid's presentation Kant was thinking of in his theory. In view of the interpretation of Kant's notion of intuition that I have suggested, the question becomes: Is there anything particular in Euclid's procedure which encourages the idea that mathematics is based on the use of particular instances of general concepts? It is easy to see that there is. For what is the structure of a proposition in Euclid? Usually, a proposition consists of five (or sometimes six) partslO First, there is an enunciation of a general proposition. For instance, in proposition 20 of the Elementa he says: "In any triangle two sides taken together in any manner are greater than the remaining one." This part of the proposition was called the 1tp01:a<Jl~. But Euclid never does anything on the basis of the enunciation alone. In every proposition, he first applies the content of the enunciation to a particular figure which he assumes to be drawn. For instance, after having enunciated proposition 20, Euclid goes on to say: "For let ABC be a triangle. I say that in the triangle ABC two sides taken together in any manner are greater than the remaining one, namely, BA, AC greater than BC; AB, BC greater than CA; BC, CA greater than AB." This part of a Euclidean proposition was called the setting-out or ecthesis (IlK9E<Jl<;' in Latin expositio). It is perhaps no accident that Kant used the German equivalent for setting-out (darstellen) in explaining his notion of construction, and that

which had been introduced in the selring-out and completed in the auxiliary construction. These inferences made use (I) of the axioms, (2) of the earlier propositions, and (3) of the properties of the figure which follow from the After having reached the desired conclusion about the particular figure, Euclid returned to the general enunciation again, saying, e.g., 'Therefore, in any triangle, etc.' 9. 'ECTHESIS' AS A PARADIGM OF KANTIAN CONSTRUCTIONS

When this structure of a Euclidean proposition is compared with Kant's account of the mathematical method, the agreement is obvious. Kant's idea of geometry was, it may be said, Euclidean in more than one sense of the word. When Kant says that it is the method of mathematicians always to consider general concepts in concreto, in a particular application, he has in mind the setting-out Of ecthesis of a Euclidean proposition where a general geometrical

propOSItion is 'exhibited' or 'set out' by means of a particular figure. This is borne out by the examples by means of which Kant explains his theory of the mathematical method. He says that the superiority of the mathematical method over the philosophical one in geometry lies in the fact that the mathematician can draw actual figures and carry out proofs in terms of such figures. For instance, if a philosopher (qua philosopher) tries to prove that the sum of the internal angles of every triangle is equal to two right angles, he is reduced, Kant says, to analyzing the concepts 'straight line', 'angle', and 'three', and is unable to get anywhere. A mathematician, in contrast, can draw a figure of a triarigle, complete it by means of suitable additional constructions (i.e., intro-

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duce suitable uew lines, circles, etc., into the argument) and thereby make the proposition to be proved obvious. (See A 716-717 = B 744-745.)11 This example shows that, in addition to the seuing-out or ecthesis of a Euclidean proposition, Kant also had in mind the part of the proposition which follows the ecthesis, viz. the preparation or 'machinery'. Setting-out and preparation were the two parts of a Euclidean proposition where constructions in the usual sense of the word were made; and we have seen that these two parts were also the ones in which constructions in Kant's abstract sense of the word were needed, i.e., where new individual points, lines, etc., were introduced. This, then, means that within geometry Kant's notion of construction coincides with the ordinary usage of the term 'construction'. This outcome of our comparison between Kant and Euclid supports what was said earlier. It shows that Kant's notion of a construction accommodates as a special case the usual geometrical notion of construction. Now the constructions of the geomenical kind need not take place in the human mind. More often than not, they are carried out on a piece of paper or on the blackboard. What is common to all such constructions is that some new lines. points, or circles are introduced. If these geometrical entities are conceived of as individuals, they fit into Kant's general definition of an intuition. There is no need, therefore, to assume that the constructions of geometry mean for Kant something else than what we are prepared to call constructions. But this is not all we can get out of the comparison. If we have a somewhat closer look at the relation between Kant's theory of the mathematical method and Euclid's practice, the relation serves to suggest several insights into Kant's theory. Here I shall only mention a few of them.

to make a desired construction, by actually carrying out constructions. What distinguishes the two methods, therefore, is broadly speaking the fact that in the analytic method no constructions are made while the synthetic method is

30

based on the use of actual constructions. 12

Kant indicates that what makes mathematics in general and geometry in particular synthetic is the use of inmitions, i.e., the use of constructions. We have seen that his notion of construction coincides, in geometry, with the ordinary mathematical usage of the teoo 'construction'. What this means, then, is that Kant's distinction between analytic and synthetic is modelled, within mathematics at least, on a usage of mathematicians which was current at his time. (Mathematicians to-day still speak of synthetic geometry, meaning geometry which turns on the use and study of geomettical constructions). This suggestion is supported by Kant's own comments on the subject, which serve to narrow down his sense of synthetic so as to connect it explicitly with constructions in an almost geomettical sense. The distinction between analytic and synthetic in geometry was earlier often used to separate two methods of finding a desired proof or construction, or, in some cases, to separate two methods of exposition. What Kant needed was a distinction between two different means of carrying out a proof. For him, the paradigm of synthesis was precisely synthesis in the geometrical sense of the word, i.e., the completion of a figure by means of the introduction of new geomenical entities. This he distinguished from the other usage which was based on the paradigm of proceeding 'inversely' from a ground to a consequence. This difference is stated by Kant, if not in so many words, in a footuote to the first paragraph of his Dissertation of the year 1770. I3

10. ANALYTIC AND SYNTHETIC METHODS

11. KANT AND ANALYTIC GEOMETRY. 'INDEMONSTRABLE' EQUATIONS

There is in geometry an ancient distinction between two kinds of methods. There is, on one hand, the method of assuming a desired result to be achieved, for instance, of assuming that we have succeeded in making a desired construction, in the ordinary sense of 'construction'. From these assumptions one then argues 'backwards', so to speak, to the conditions on which this construction is possible and to the ways in which it can be effected. This is called the analytic method. It was sometimes ascribed to Plato, but it was not to be employed explicitly and systematically in a large scale until the analytic geometry of Descartes, the very name of which is derived from the 'analytic' method in question. The other method was the synthetic one. In applying it one nies to effect the desired result, for instance,

There is another way in which an awareness of the respective geometries of Euclid and Descartes helps us to understand Kant. We can make a particularly interesting observation if we compare Euclid's geometry with Descartes'. According to Descartes, the main idea of the analytic geometry was a correlation or analogy between algebraic and geomenical operations. Just as all we need in arithmetic are the four or five basic operations of addition, subtraction, multiplication, division, and the extraction of roots, exactly in the same way we need in geometry only a few basic constructions, Descartes says. 14 What we are interested in here is the analogy between algebralc and geomenical operations, in parricular the fact that algebraic opera-

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tions correspond to certain geometrical constructions. This gives, I think, the key to what Kant means by saying that simple arithmetical equations, such as 7 + 5 = 12 are 'immediate' and 'indemonstrable'. We see this if we try to cast the argument by means of which 7 + 5 = 12 is verified into the form of a Euclidean proposition. Because of the analogy between algebraic operations and geometrical constructions, the actual addition of 7 and 5 corresponds to the third stage, i.e., the preparation or 'machinery', of a Euclidean proposition. Kant's explanations also show that, according to him, the numbers 7 and 5 must somehow be 'set out' or 'exhibited' before the actual operation of addition, in analogy to the 'ecthesis' of a Euclidean proposition. (This is what his remarks on "points or fingers" illustrate.) But what, then, corresponds to the proof proper, to the apodeixis? Obviously, all that we have to do in order to show that 7 '+ 5 = 12 is to carry out the operation of addition; the proof proper is reduced to a mere minimum, to the mere observation that the result of the addition equals the desired result 12. In a perfectly good sense, therefore, one can say that no proof (proper), no apodeixis is needed to establish that 7 + 5 = 12. This equation is 'immediate' and 'indemonstrable' in the precise sense that it can be established by the mere auxiliary construction or kataskeue of a Euclidean proof. This is all that Kant's statement amounts to. And the fact that this really was Kant's idea is shown by a letter of his to Schultz, dated November 25, 1788. The main difference is that, instead of using the terminology which pertains to the theorems of the Euclidean geometry, Kant in this letter makes use of the parallel terminology pertaining to geometrical problems. r This is important, I think, over and above the interpretation of particular passages, for it shows how Kant intended the intuitivity of arithmetic to be understood. The inunediacy of arithmetical truths is not due to the fact that simple equations like 7 + 5 = 12 are perceived to be true and not argued for, but to the fact that the only thing we have to do in order to establish such equations is to carry out the computation. This serves to explain why Kant

contradiction "which the nature of all apodeictic certainty requires." This passage becomes very natural if we take Kant for his word and understand him as referring solely to the apodeictic or 'proof proper' part of a Euclidean proposition. Taken literally, the proof proper or apodeixis is after all the only part of a Euclidean proposition where inferences are drawn. Taken in this way Kant's statement expresses precisely what he would be expected to hold on my interpretation, viz. that the distinction between on one hand apodeixis and on the other hand ecthesis and the auxiliary construction separates the analytic and the synthetic parts of a mathematical argoment.

said his account of the equations is more readily understood in the connection Loflarge numbers (B 16; cf. A 78 = B 104). 12. APODEICTIC INFERENCES ANALYTIC

I snspect that a particularly perplexing passage in the first Critique receives a natural explanation pretty much in the same way as the remarks on arithmetic. I mean the statement Kant makes in B 14 to the effect that all the inferences (SchlUsse) of mathematicians are based on the principle of

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13. INTUITIONS MADE INTUITIVE

What have we accomplished so far? We have seen that in Kant's theory of the mathematical method, presented towards the end of the first Critique, one has to keep in mind the possibility that by intuitions Kant means particular representatives of general concepts. We have seen that a number of things about Kant's theory of algebra, arithmetic, and geometry become natural from this point of view. But, it may be said, the possibility of intuitions which are not sensible is ruled out in the Transcendental Aesthetic. Kant argues there that all the use of intuitions in mathematics is based on the intuitions of space and time, and that these intuitions are due to the structure of our sensibility. There is, therefore, no room left in mathematics for intuitions that are not connected with sensibility. I have no desire to deny that this is what Kant says. But I want to point out that the disagreement between the above interpretation of Kant's methodology of mathematics and his theory of space and time in the Transcendental Aesthetic does not disprove my interpretation. The discrepancy between the two parts of Kant's system belies my reading of Kant ouly if the account of mathematics given in the Transcendental Aesthetic is correct. Kant claims there that the use of intuitions in mathematics can only be understood if we assume that all these intuitions are due to our sensibility. If there now are intuitions, say the individual variables or 'intuitions' of algebra, which have no relation to our sensibility, then the only possible conclusion is not that these alleged intuitions are not intuitions at all in Kant's sense. The other possibility is to say that they are genuine intuitions but that Kant just was wrong in saying that all the intuitions used in mathematics are sinnlich, i.e., due to onr sensibility. But then it remains to be explained how Kant came to entertain the mistaken doctrine. I have implied that the notion of the mathematical method as

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being based on the use of individual instances was the starting-point of Kant's better-known theory that all the intuitions we use in mathematics are due to our sensibility. What is there in the notion of an intuition as an individual instance which made Kant think that this conclusion is inevitable? We have discussed the role of intuitions, in the sense of representatives for individuals, in algebra, in arithmetic, and in geometry. What is the common feature of these uses which can only be explained, according to Kant, by assuming that the mathematical intuitions are sensible? What is the common

logic in the way Aristotle did. For this reason, Kant could not reduce all the syllogistic modes to the two modes of Barbara and Celarent which he recognized as the basic ones, and was bound to reject all the others as being

denominator of all the mathematical 'constructions' we have discussed? 14. 'ECTHESIS' IN LOGIC

It seems to me that a natural generalization is virtually contained in the above analysis of Euclid's propositions. The most important part of a Euclidean proposition which is intuitive in Kant's sense is the setting-out, the ecthesis. Now this notion of eethesis occurs not only in Greek geometry. It also occurs in the Aristotelian logic. Aristotle never explains explicitly what the procedure called eethesis is, but we can perhaps say that it was a step in which Aristotle moved from considerations pertaining to a general term over to considerations pertaining to a particular representative of this general term. Fat instance, in Analytiea Priara I, 2, 25a15, Aristotle seems to argue as follows: If no A is a B, then no B is an A. For if not, then some B's are A's. Ta..i<:e a particular b of this kind. This particular b has both the property B and the property A and shows, therefore, that it is impossible that none of the A's is a B as we assu.riJ.ed. This contradiction proves the conclusion. A later passage (Analytiea Priara I, 41, 49b33ff.) seems to indicate that Aristotle took the logical eethesis to be essentially the same as the geometrical one. 15 I suggest that this notion of eethesis offers a very good reconstruction of Kant's notion of construction, i.e., of the notion of the exhibition of a general concept by means of particular representatives. It agrees, as we see, very well with the way in which Kant defines the notion of construction. Its use in the Aristotelian logic may perhaps explain why Kant criticized (in the essay on the 'false subtlety of the four syllogistic figures') certain parts of this logic. He went as far as to reject, in effect, all the syllogistic modes except the first two modes of the first figure. The explanation may perhaps lie in the fact that the particular application of eethesis I just outlined was calculated to prove one of the rules of conversion which Aristotle needed in order to reduce all the syllogistic modes to the first two modes. Since the use of eethesis was for Kant a typically mathematical method of reasoning, he conld not use it in

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~impure'

and 'confusing'. The notion of ecthesis can be made precise in tenns of modern logiC. 16 It becomes, in effect, identical with one of the most important rules of inference

of quantification theory (existential instantiation). And, in terms of the notion of eethesis so reconstrued, we can see in what sense the equation 7 + 5 = 12 can be said to be based on the use of particular representatives of general

concepts, i.e., on the use of eethesis. It would take us too far, however, to go into this question here. 17 15. PARTICULARS PARTICULARLY INTUITIVE?

I shall conclude this paper by sketching very briefly and in un-Kantian terms how the reconstruction of Kant's notion of construction in terms of ecthesis or in some similar way makes sense of his attempt to connect the mathemati-

cal method with sensibility. It was already suggested that the notion of construction may perhaps be identified with certain methods of proof in modern logic. If this is so, then Kant's problem of the justification of constructions in mathematics is not made obsolete by the formalization of geometry and other branches of mathematics. The distinction between intuitive and nonintuitive

methods of argnment then reappears in the formalization of mathematical reasoning as a distinction between two different means of logical proof. But does there remain any sense in which the use of such . intuitive' methods is

problematic? Would Kant have accepted such a reconstruction of the notion of intuition as a premise of his argument that all intuitions are due to our sensibility? The answer to the questions is, I think, yes. We can see why it was natural for Kant to make the transition from the use of individual instances of any kind to their connection with sensibility. I shall briefly outline two explanations.

Historically, it may be said, nothing was more natural for Kant than to connect individuals with the use of our senses. Aristotle already held that "it is sense-perception alone which is adequate for grasping the particulars" (Analytiea Posteriora I, 18, 8Ib6). All knowledge, therefore, which is obtained by means of particulars, must be perceptual. How natural the application of this general Aristotelian idea to the case of constructions in Kant's sense was, is perhaps shown by the fact that Alexander the Commentator

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already applied Aristotle's idea to the process of eethesis, Alexander held that the singular tenn introduced in the ecthesis is given by perception, and that the proof by eethesis therefore consists in a sort of perceptual evidence. I8 And the general Aristotelian assumption about individuals and senses was echoed by Kant's Gennan predecessors. 16. CONSTRUCTIONS AS ANTICIPATIONS OF EXISTENCE

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Another, and perhaps a more important, way of making Kant's ideas plausible may be derived from the division of the Euclidean propositions into parts. We have seen that for him, the use of constructions took place in the second and the third parts of a Euclidean proposition, while in the fourth part the argumentation was purely nonconstructive or, which amounts to the same, purely analytic. Now the distinction between these parts of a Euclidean proposition corresponds, according to a widespread view of which Kant seems to have accepted, to a distinction between two kinds of principles of Euclid's system. The principles of construction are the so-called postulates, while the principles of proof proper are called axioms (common notions). It is significant that the examples Kant gives of analytic principles used in geometry (B 17) obviously fall into the second category. (This shows, incidentally, that Kant's notion of construction in geometry was not, as sometimes had been suggested, sometlting alien to the axiomatic treatment of geometry. The very examples Kant gives of geometrical constructions are based either directly on Euclid's posrulates, or else on explicit propositions Euclid had proved earlier; a fact of which Kant . scarcely could have been unaware. In point of fact, the main construction needed in Kant's favourite example, the theorem about the internal angles of a triangle, is based on the postulate of parallels which Kant himself had tried to prove.) Hence, the distinction between intuitive and logical ways of reasoning was for Kant, within geometry at least, equivalent with the distinction between the use of posrulates, i.e., principles of construction, and the use of axioms, i.e., principles of proof. What, then, constitutes the latter distinction? According to a widespread view which may be traced back to Aristotle and certaiuly back to the Greeks, posrulates are assumptions of existence. Kant's problem of the justification of constructions, therefore, amounts to the problem of justifying the use of existential assumptions in mathematics.

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17. HOW CAN CONSTRUCTIONS YIELD A PRIORI KNOWLEDGE?

Stated in this form the whole problem may seem spurious. There is certainly nothing that could prevent a mathematician from studying axiom systems which incorporate general existential assumptions. The problem only makes sense if we are concerned with the applicability of mathematical reasoning to reality. But this certainly is something Kant was concerned with in the Transcendental Exposition, in spite of the fact that he insists that he is speaking of pure mathematics ouly. (This appears particularly clearly from paragraphs 8-9 of the Prolegomena; cf. Vaihinger's discussion of these paragraphs.) We may ask: What happens when we apply to reality a particular mathematical argument in the course of which a postulate, i.e., a general existential assumption, has been used? In applying it, we have to introduce a representative for a new individual, as Kant puts it "without any object being present, either previously or now, to which it could refer." The introduction of the new representative for an individual is carried out a priori. The existence of the individual object in question, in other words, is not given by experience. Kant describes the situation by saying that the intuition or, in O.!IT tenns, the representative for an individual object precedes its object. The only thing to make sure that there is any object at all corresponding to the representative is the general existential assumption. But it may seem as if there is no general justification for the application of existential assumptions at all unless we are in fact acquainted with the objects that are assumed to exist, which simply is not the case with applications of our a priori knowledge. It seems, as Kant puts it, impossible to intuit anything a priori. For in the absence of actual acquaintance there is in reality nothing to make sure that we can always find objects which the representatives we have introdnced really stand for or that they have the properties we expect them to have. 19

Kant's solution of this (real or apparent) problem consists in saying that there is one and only one case in which we can be sure that the individuals we have assumed to exist really do so and have the desired properties. This is the case in which we have ourselves created the objects in question or our-

selves put the desired properties and relations into them.20 And he seems to tltink that there is only one stage of our coming to be aware of objects in which this kind of 'putring properties into objects' can take place. Or, rather, there is ouly one-stage in which we can 'put properties' into all (individual) objects. This stage is sensible perception. For sensible perception is the only

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way in which an individual object can 'make its way' into OUT consciousness. Outer sense is the only way in which we can become aware of external objects. For this reason, it is the only stage of our coming to know objects at which we can ourselves give spatial relations to all external objects. Therefore, the spatial relations postulated in geometry must be due to the

(4) The mutual relations of the individuals with which mathematical reasoning is concerned is due to the process by means of which we come to know the existence of individuals. These systems of mutual relations may be expected to be reflected by the structure of mathematical reasoning. Now Kant has been seen to assume that (5) the process by means of which we come to know the existence of individuals in general is perception (sensation). From (4) and (5) it follows that (6) the structure of mathematical reasoning is due to the structure of our apparatus of perception. Now (6) is in effect a basic feature of Kant's full and final doctrine of the mathematical method, as complemented by the results he thought he had achieved in the Transcendental Aesthetic.

structure of our outer sense.

I am putting forth this partial reconstruction only as a first approximation to what Kant had in mind in the Transcendental Exposition. This reconstruction is related fairly closely to Kant's 'transcendental argument' for his theory of space and time especially as it is presented in the Prolegomena. I have merely tried to fill in those steps which Kant does not himself emphasize in the light of his general assumptions. The relation of my partial reconstruction to Kant's other arguments for his views is more complicated, and requires a

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longer discussion than I can undertake here. I want to emphasize that I am not at all clainting that Kant's argument is correct. The main purpose which the reconstruction serves here is to suggest that Kant's problem of the possibility of constructions in mathematics, and ( his attempted solution to the problem, makes perfectly good sense even when i by 'construction' one only means 'the introduction of a new individual repre; sentative for a general concept'. 18. THE STRUCTURE OF KANT'S ARGUMENT

The structure of Kant's argument in the form presented here is nevertheless worth a closer look. Its several stages may be represented in the light of what has been said somewhat as follows: (I) Mathematical reasoning is principally concerned with the existence of individuals. (2) The results of mathematical reasoning are applicable to all experience a priori. In virtue of Kant's general 'Copernican' assumptions ("we can know a priori of things only what we ourselves put into them") (I) and (2) force us to conclude:

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(3) The existence of the individuals with which mathemarical reasoning is concerned is due to the process by means of which we come to know the existence of individuals in general. Of course, what really matters is not the existence of the individuals as such (there are plenty of individuals existing in the world) but the existence of individuals having the appropriate relations to each other. Hence we may perhaps paraphrase (3) as follows:

19. ARE INDIVIDUALS 'GIVEN TO US'?

This line of thought (1)-(6) is not without interest and even without certain plausibility. Since we have seen that Kant's point can be translated so as to apply to modem logic, we are therefore led to ask what the corresponding argument will look like as applied to symbolic logic. Steps (1)-(2) and (4) do not seem to me completely implausible as applied to logic instead of mathematics. It is in (5) that Kant really goes wrong. It is simply not true that we usually or always come to know the existence of individuals in the world by means of perception in the sense that perception is the whole of the process involved. It may even be asked whether any perception at all need be involved. When we come to establish the existence of a number of a certain kind, it is ntistaken to assume that perception is always involved. (But is a number really an individual? Maybe not; but certainly a number was an individual for" Kant when he called the symbols of algebra inntitions, i.e., representatives of individuals. Kant's account of algebra stands or falls with the assumption that 'individuals' of the sort represented by the variables of algebra are also known by the sale means of perception.) The concept of an inner sense to which Kant resorts here is one of the weakest points of his system. To think of all knowledge of individual objects as being due to perception is to succumb to a temptation which for Kant may have been very real but which it is important to get rid of. This is the temptation to think that the basic materials of human knowledge are given to us passive receivers who do not have to actively search for these materials. On tltis fallacious idea the human mind, often conceived of as a disembodied spirit inhabiting an

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alien machine, has to wait until the signals from the outside strike its receptors. (It is interesting in this connection to observe the way in which Kant stressed the passive nature of perception, speaking, e.g., of how objects are given to us in perception.) The fact that the mind can indirectly spur the machine into a movement is not thought to alter the situation materially. Nor is the situation essentially changed by the fact that according to Kant the human mind can in many ways actively organize the raw materials thus obtained, add to them and perhaps even modify them. I hope that I do not have to argue here that this picture is grundfalsch, thoroughly false. It is more interesting to ask for a better account. If perception is not the general concept which covers all that we want, what is? It seems to me that insofar as we can give a general name to all the processes by means of which we come to know the existence of individuals, they may rather be called processes of searching for and finding than acts of perception, albeit we have to accommodate the accidental perception of an object as well as the deliberate construction of an object as special cases of 'searching' and 'finding' in this broad (broadest possible) sense. Hence we have instead of (5):

(5)* The process by means of which we come to know the existence of individuals is that of searching for them. Instead of (6) we thus have to conclude: (6)* The structure of a logical argument is due to the structure of the processes of searching for and finding. My attempted partial reconstruction of the main point of Kant's philosophy of mathematics as applied to modern symbolic logic instead of mathematics thus gives rise to an interesting suggestion for our present-day philosophy of logic. The suggestion is to consider the logic of quantification as being essentially the logic of the notions of searching for and finding (suitably generalized). It seems to me that this suggestion is likely to give rise to interesting and important considerations, if cartied out systematically. Boston University NOTES 1 In referring to the Critique of Pure Reason, I shall use the standard conventions A = first edition (1781), B = second edition (1787). All decent editions and translations give the pagination of one or both of these editions. In rendering passages of the first Critique in English, I shall normally follow Norman Kemp Smith's translation (Macmillan, London and New York, 1933). 2 Loc. cit.

KANT ON THE MATHEMATICAL METHOD

41

3 A paradigmatic statement of this view occurs in Bertrand Russell's Introduction to Mathematical Philosophy (George Allen and Unwin. London. 1919). p. 145: "Kant, having observed that the geometers of his day could not prove their theorems by unaided arguments. but required an appeal to the figure. invented a theory of mathematical reasoning according to which the inference is never strictly logical. but always requires the support of what is called 'intuition'." Needless to say, there does not seem to be a scrap of evidence for attributing to Kant the _'observation' Russell mentions. 4 See, e.g., Kant's Dissertation of 1770, section 2, § 10; Critique of Pure Reason A 320 = B 376377; Prolegomena §8. Further references- are given by H. Vaihinger in his Commentar zu Kants Kritik der reinen Vernunft CW. Spemann, Stuttgart, 1881-1892), Vol. 2, pp. 3, 24. Cf. also C. C. E. Schmid, Worterbucn zum leichteren Gebrauch der Kantischen Schriften (4th ed., Croker, Jena, 1798) on Anschauung. 5 "We cannot assert of sensibility that it is the sole possible kind of intuition" (A 254 = B 310). Cf., e.g., A 27 = B 43, A 34-35 = B 51. A 42 = B 59, A 51 = B 75 and the characteristic phrase 'uns Menschen wenigstens' at B 33. 6 The opening remarks of the Transcendental Aesthetic seem to envisage a hard-and-fast connection between all intuitions and sensibility. As Paton points out, however. they have to be taken partly as a statement of what Kant wants to prove. See H. J. Paton. Kant's Metaphysic of Experience (George Allen and Unwin, London, 1936), Vol. I, pp. 93-94. 7 This has been brought out clearly and forcefully by E. W. Beth, to whose writings on Kant I am greatly indebted, although I do not fully share Beth's philosophical evaluation of Kant's theories. See 'Kants Einteilung der Urteile in analytische und synthetische', Algemeen Nederlands Tijdschrift voor Wijsbegeerte en Psychologie 46 (1953-54) 253-264; La crise de la raison et la logique (Gauthier-Villars, Paris. 1957); The Foundations of Mathematics (North-Holland Publishing Company, Amsterdam, 1959), pp. 41-47. 8 Proceedings of the Aristotelian Society 42 (1941-42) 1-24. 9 See the Academy Edition of Kant's works, Vol. 2, p. 307. Concerning the Elementa. see Sir Thomas Heath's translation and commentary The Thirteen Books of Euclid's Elements (Cambridge University Press, Cambridge, 1926). 10 Heath, op. cit., Vol. I, W. 129-131. II We can see here that according to Kant the peculiarity of mathematics does not lie in the axioms anl.!postulates of the different branches of mathematics, but in the mathematical mode of argumentation and demonstration. 12 We haye to realize, however. that the mere difference of the directions in which one is proceeding in an analysis and in a synthesis, respectively, was sometimes emphasized at the expense of the questions whether constructions are used or not. One could thus distinguish between a 'directional' and a 'constructional' (or- 'problematic') sense of analysis and synthesis. ct. my paper, 'Kant and the Tradition of Analysis', in Deskriprion, Existenz und Analytizitiit, ed. by P. Weingartner (Pustel, Munich. 1966), reprinted as Chapter 9 of Jaakko Hintikka, Logic, Language-Games, and Information (Clarendon Press, Oxford, 1973). 13 Cf. also, Prolegomena, "§5 (Academy Edition, Vol. 4, p. 276, note). We can also say that Kant's remarks in'~effect serve to distinguish between the directional and the constructional (problematic) sense of analysis and synthesis, and to indicate that Kant opts for the latter. (See the preceding note at:l-d the article mentioned there.) 14 See La Geometrie, the first few statements (pp. 297-298 of the first edition). J5 Concerning the notion of ecthesis in Aristotle, see W. D. Ross, Aristotle's Prior and Posterior Analytics: A Revised Text with Introduction and Commentary (Clarendon Press, Oxford. 1949),

42

JAAKKO HINTIKKA

pp. 32-33, 412-414; Jan Lukasiewicz, Aristotle's Syllogistic from the Standpoint of Modern Formal Logic (Clarendon Press, Oxford, 1951), pp. 59-67; GUnther Patzig, pie Aristotelische Syllogistik (Vandenhoeck und Ruprecht, GOrtingen, 1959), pp. 166-178; B. Einarson, 'On Certain Mathematical Terms in Aristotle's Logic" American Journal of Philology, 57 (1936) 34--54, 151-172, esp. p. 161. As will be seen from these discussions, the precise interpretation of the Aristotelian notion of ecthesis (as used in his logic) is a controversial problem to which no unambiguous solution may be available. The interpretation which I prefer (and which I shan rely on here) assimilates logical ecthesis to the 'existential instantiation' of modem logic. I cannot argue for this interpretation as fully here as it deserves. For Aristotle's use of the term ecthesis in geometry, which seems to me to be closely related to the logical ecthesis, cf., e.g., Analytica Priora I, 41 49b30-50a4 and Analytica Posteriora I, 1O,76b39-77a2. 16 I am here presupposing the interpretation mentioned in the preceding note. For further remarks on this interpretation, cf. my paper, 'Are Logical Truths Analytic?', Philosophical Review 74 (1965) 178-203, reprinted my Knowledge and the Known (Reidel, Dordrecht, 1974) and E. W. Beth's discussion of the relation of ecthesis and modem logic in 'Semantic Entailment and Fonnal Derivability', Mededelingen van de Koninklijke Nederlandse Akademie van Werenschappen,Afd. Letterkunde, N. R., 18, no. 13 (Amsterdam, 1955), pp. 309-342. 17 Some remarks on these points are contained in my paper, 'Kant Vindicated', in Deskription. Existenz und Analydzittit, ed. by P. Weingartner (Pustet, Munich, 1966), reprinted as Chapter 8 qf Logic, Language-Games, and Information (note 12 above). 18 Alexander of Aphrodisias,In Aristotelis Analyticorum Priorum LibnmlI Commentarium, ed. by M. Wallies, in Commentaria in Aristotelem Graeca, Vol. 2(a) (Berlin 1883), p. 32, cf. pp. 32-33, 99-100, 104; lukasiewicz, op. cit., pp. 60-67. An attempt to explain and to justify the mathematical ecthesis from an Aristotelian point of view also easily gives rise to striking anticipations of Kantian doctrines. Thus we find, for instance, that according to Theophrastus mathematical objects "seem to have been, as it were, devised by us in the act of investing things with figures and shapes and ratios, and to have no nature in and of themselves ... " (Theophrasrus, Metaphysica 4a18ff., pp. 308-309 Brandisii). Cf. also Anders Wedberg, Plato's Philosophy of Mathematics (Almqvist and Wiksell, Stockholm, 1955), p. 89, who emphasizes that Aristotle likewise seems to anticipate some of the most salient features of Kant's theory of mathematics. 19 This difficulty was emphasized by Kant's early critics. For instance, J. G. E. Maas writes in his long paper, 'Ueber die transcendentale Aesthetik', Philosophisches Magazin 1 (1788) 117-149, as follows, apropos Kant's notion of an a priori intuition: "Hierbey karm ich (I) die Bemerkung Dicht vorbeilassen, dass eine Anschauung a priori ... nach Kants eigenen ErkHirungen Dicht denkbar sey. Eine Anschauung ist eine Vorstellung. Sollte sie a priori seyn, so miisste sie schlechterdings nicht vom Objecte hergenommen werden, und eine Anschauung ist doch nur moglich, sofem uns der Gegenstand gegeben wird, dieses aber ist widerum nur dadurch moglich, dass er das Gemiith auf gewisse Weise afficiere. Eine Anschauung a priori ist demnach unmoglich, und kann mithin auch in Ansehung des Raumes nicht zum Grunde liegen" (pp. 134 -135). Maas does not realize, however, that the possibility of a successful use of a priori intuitions is precisely the problem Kant was trying to solve in the Transcendental Aesthetic. 20 In B xviii Kant says that he is "adopting as our new method of thought ... the principle that we can know a priori of things only what we ourselves put into them." Cf. also B xii-xiv: I have commented briefly on the historical background of this Kantian assumption in 'Kant's "New Method of Thought" and his Theory of Mathematics', Ajatus 27 (1965) 37-47, reprinted in Knowledge and the Known (note 16), and in 'Tieto on valtaa', Valvoja 84 (1964) 185-196.

CHARLES PARSONS

KANT'S PHILOSOPHY OF ARITHMETIC

The interest and inflnence of Kant's philosophy as a whole have certainly been great enough so that this by itself would be enough to make Kant's philosophy of arithmetic of interest to historical scholars, It is also possible to show the influence of Kant on a number of important later writers on the foundations of mathematics, so that Kant has importance specifically as a figure in the history of the philosophy of mathematics, However, my own interest in this subject has been animated by the conviction that even today what Kant has to say about mathematics, and arithmetic in particular, is of interest to the philosopher and not merely to the historian of philosophy. However, I do not know how much of an argument the following will be for this. Kant does not discuss the philosophy of arithmetic at any great length, so that it is virtually impossible to understand him without making use of other material. What I have used consists mainly of two considerations: the integration of Kant's theoretical philosophy as a whole, and modem knowledge on the foundations of logic and mathematics. The justification for using the second is twofold; first, I think experience shows that one does not get far in understanding a philosopher unless one tries to think through the problems on their own merits, and in this one must use what one knows; second, if one is today to take Kant seriously as a philosopher of mathematics, one must confront him with this modem knowledge, which after all in major respects shows immense progress from the situation in his lifetime. I shall be concentrating mainly on one question, which I think must be answered before one goes farther with the subject: Why did Kant hold that arithmetic depends on sensible intuition, indeed that arithmetical propositions in some. way refer to sensible intuition? 1bis is, of course, closely related to the question of why he regarded such propositions as synthetic rather than analytic. In considering this question, one must very soon consider Kant's views on logic and its relation to arithmetic. Also since the answer to the above question is much clearer if "arithmetic" is replaced by "geometry", we shall also give some consideration to Kant's views on geometry. 43 Carl J. Posy (ed.), Kant's Philosophy afMathematics, 43-79 . • _ .... ,.".,_.

n._' .. ~.J' ••• '._IlT_~I.

__ ' __ J_

,r

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CHARLES PARSONS

In order to clarify our problem, let us first briefly consider Kant's concept of intuition. Intuition is a species of representation (Vorstellung) or, in the language of Descartes and Locke, idea. Having intuitions is one of the primary ways in which the mind can relate to or be conscious of objects. The nearest thing to a definition in the Critique of Pure Reason occurs in a classification of representations: This [knowledge] is either intuition or concept (intuitus vel conceptus). The former relates immediately to the object and is singular [einzeln], the latter refers to it mediately by means of a feature which several things may have in common. [A 320 = B 376-7]2

In the opening sentence of the Transcendental Aesthetic, Kant says: In whatever manner and by whatever means a mode of knowledge may relate to objects, intuition is that through which it is in immediate relation to them, and to which all thought .as a means is

I:

directed. [A 16 ~ B 33]

Ii

A passage in §l of Jasche's edition of Kant's lectures on logic reads:

I

I

KANT'S PHILOSOPHY OF ARITHMETIC

45

justificatory argwnent which the Principles do (A 87 = B 120). By the immediacy criterion Kant's conception of intuition resembles Descartes's, while by the singularity criterion and his insistence on a nonintuitive conceptual factor in all knowledge, Kant's theory of intuition differs from that of Descartes. That what is immediately present to the mind are individual objects seems to be an axiom of Kant's epistemology, or one might also say metaphysics, since it goes with the conviction that objects, the primary existences, are in

the first instance individual objects. Thus what satisfies the immediacy criterion of intuition will also satisfy the singularity criterion. It does not seem that the converse must be true. The idea of a singular rep-

resentation fonned from concepts seems quite natural to us. Such a representation would relate to a single object if to any at all, but it hardly seems immediately. By associating it with a definite description rather than with a general tenn, we wonld distinguish it from a concept under which exactly one object falls (even if necessarily). For Kant, however, the passage from A 320 = B 376-7 seems to allow such a representation to be a concept; this might also be suggested by the fact that the idea of God is called a concept; it is nowhere suggested that it is an intuition. However, Kant never remarks, so

All modes of knowledge, that is. all representations related to an object with consciousness are either intuitions or concepts. The intuition is a singular representation (repraesentatio singularis), the concept a general (representatio per nofas communes) or reflected representation (representatio discursiva).3

Intuitions are thus contrasted with concepts, which relate to objects only

mediately, by way of certain properties and by way of intuitions which instantiate them and which relate indifferently to all the objects which possess the required properties. What is meant by calling an intuition a singular representation seems quite clear. It can have only one individual object. The objects to which a concept "relates" are evidently those which fall under it, and these can be any which have the property which the concept represents, so that a concept will only in exceptional cases have a single object. Thus far, the distinction corresponds to that between singular and general terms.

One might think that the criterion of "immediate relation to objects" for being an intuition is just an obscure fonnnlation of the singularity condition. But it evidently means that the object of an intuition is in some way directly present to the mind, as in perception, and that intuition is thus a source, ulti-

mately the only source, of immediate knowledge of objects. Thus the fact that mathematics is based on intuition implies that it is immediate knowledge and thus, even though synthetic a priori, does not require the elaborate

far as I know, on the implications of the possibility of nonimmediate singular representations for the concept of intuition. This omission may give support to a theory which has been advanced by Jaakko Hintikka according to which the singularity criterion is the sole defining criterion: An intuition is simply an individual representation. In Kant and his immediate predecessors, the teon "intuition" did not necessarily have anything to do with appeal to imagination or to direct perceptual evidence. In the fonn of a paradox, we may perhaps say that the "intuitions" Kant contemplated were not necessarily very intuitive. For Kant, an intuition is simply anything which represents or stands for an individual object as distinguished from general concepts.4

Many of the passages Hintikka cites also mention the immediacy criterion, and it is not clear why Hintikka thinks it nonessential. The main reason, which we shall consider later, is that this assumption supports a theory of Beth and Hintikka to explain Kant's notion of "construction of concepts in intuition" and the resnlting analysis of mathematical demonstration. Another seems to be the absence of the immediacy criterion in the Logic and the fact that Kant makes remarks on concepts which seem to exclude essentially singular concepts and thus to imply that all singular representations are intuitions. s Hintikka also points out that the part of the Transcendental Aesthetic

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KANT'S PHILOSOPHY OF ARITHMETIC

where Kant argues that space is an intuition argues essentially that the representation of space is singular. However, he has opened the Aesthetic by stating the immediacy criterion (A 16 = B 33, cited above) and in the proof of intuitivity he does say that space is given (B 39, also A 25). Moreover, in arguments for the same thesis in the Inaugural Dissertation of 1770, Kant does appeal to immediacy: Immediately after arguing that space is a pure intuition because it is a "singular concept," Kant says of geometrical propositions that they "cannot be derived from any universal notion of space but only as it were seen in space itself as if in something concrete" (§ 15c, emphasis Kant's). Later he says, "Geometry makes use of principles which fall under the gaze of the mind." It seems to me that the textual evidence for Hintikka' s view is not sufficient to outweigh the clear statements and emphases on the immediacy criterion, even though the alternative view must assume that Kant in dis-

resulting exposure to contingency and the necessity of concepts in order to represent objects not present. Thus not only sensibility but also thinking, or consciousness through concepts (knowledge through concepts, B 94), are characteristics of finite intelligences. The alternative is an "intuitive understanding" whose activity would create the objects of its awareness. Its awareness would be only intuition; it is called "intellectual intuition" because it has the spontaneity- which for us is characteristic of thought and because the unity which with us is the result of synthesis of the given is for it already present in the intuition. It seems clear that intellectual intuition wonld satisfy the immediacy criterion.

cussing these matters did not keep in mind the possibility of nonimmediate

modem training most forcefully in considering Kant's outlook on logic is the limitation of his knowledge of and conception of it. Kant learned and taught the established logical lore at a very uncreative time in the history of the subject. Thus the formal logical analysis he undertakes is pretty well limited to the categorical proposition-forms of the theory of the syllogism, with gestures toward hypothetical and disjunctive propositions. The inferences which are covered are the syllogisms and immediate inferences of the Aristotelian theory and a few propositional inferences such as modus ponens. Of propositional logic as an additional developed theory, or of the additional possibilities of quantification theory, Kant had no idea. Kant not only had very limited technical resources at his command; what is more striking and more damaging to his standing as a philosopher, he was largely satisfied with logic as he found it. Technically he could hardly in any case have gone very far beyond the state of the science in his own time, and

singular representations. 6 But Hintikka' s theory really stands or falls on the interpretation of the role of intuition in mathematics. A thesis about intuition which is of great importance for Kant is that our mind can acquire intuitions of actual objects only by being affected by them. Just what this "affection" is I shall not venture to say, but it involves for the subject a certain passivity, so that our perceptions are not on the face of it brought about by our own mental activity, and also a certain exposure to con-

tingency in our relations with objects. Thus we do not perceive objects unless they physically affect our sense organs. A particular and highly important twist of Kant's philosophy is that the nature of our capacity to be affected by objects, our sensibility, already determines certain characteristics of our intuitions. These are said to be the form of our intuition in generaL Among them is spatiotemporality. This must be

understood to mean that the nature of the mind detennines that the objects we intuit should be spatial and temporal, and indeed intuited as such. The intuition which plays a role in mathematics, which is not the direct result of the affection of our tuind by objects, expresses an intuitive insight which we have into our forms of intuition and is in that sense still an intuition of sensibility. It is apparently also sensible intuition in the sense of being intuition of inner sense.

As Hintikka rightly emphasizes, this intrinsic connection between intuition and sensibility does not come directly from the concept of intuition but represents a characteristic of man, or more generally of finite intelligences. Such an intelligence derives the content of its consciousness from outside with the

47

II

Let us now turn to Kant's views on logic. What must strike a person with

he was not a creative mathematician. But what would have been needed for

Kant to be dissatisfied with "traditional logic" might only have been more insight into his own discoveries. As is well known, Kant attributed the lack of progress in logic to the absence of any need for it. He held that logic was established as a science and then finished off once and for all by Aristotle. This is a false view not only of the possibilities of discovery in logic but also of the history of the subject, which, far from not being "able to advance a single step" nor "required to retrace a single step" since Aristotle, had done both more than once. Kant's opinion was also influential and served to create resistance to more reason-

able views both of logic itself and of its history.

f

-'t

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CHARLES PARSONS

Why Kant should have thought the science of logic both completable and completed is a question which I shall not attempt to answer. here. I do not know whether a serious effort to answer it would uncover interesting ideas of Kant, which as it is we do not understand. In general, it can be said that the view harmonized exttemely well with the more rationalistic side of Kant's way of thinking and with the belief, which he was not the ouly great philosopher to hold, that his own work finished off an important part of philosophy. Kant certainly thought that there were inexhaustible sources of problems, even philosophical problems, for the human intellect to wrestle with. But he held that this inexhaustibility lay within limits fixed by a form, the basic properties of which could be exhaustively described. This form would belong to the human faculty of thought itself, which so long as it was dealing with "itself and its form" and not with objects given from outside or with the manner in which they might be given from outside, was bound to be capable not only of being on sure ground but of uncovering and analyzing every relevant factor. Reason and the understanding are "perfect unities" (A xiii, A 67 = B 92). We also find an echo of the Cartesian idea that the self is better known than objects: "I have to deal with nothing save reason itself and its pure thinking; and to obtain complete knowledge of these, there is no need to go far afield, since I come upon them in my own self' (A xiv). Logic is, according to Kant, the most general of all divisions of knowledge. It applies to all objects of our thought in general, and all ttue statements and sound inferences must conform to it. In particular and especially important, logical possibility is the most inclusive kind of possibility. If something is possible in any respect whatsoever, it is logically possible; its concept does not involve a contradiction. In particular, at least as far as Kant's explicit statements are concerned, the applicability of logic is not limited by the forms of our sensibility. The relation between logic and the forms of intuition can best be seen by contrast with geometry: The forms of intuition provide the basis for certain necessary ttuths, in particular those of geometry, in the sense that if the forms of intuition were not as they are the ttuths in question would not hold, and if we did not have a certain insight into our forms of intuition, we would not know them. The application of these ttuths, however, is limited to the objects which affect our senses. Moreover, the principles are ttue of these objects only as they appear and not as they are in themselves. These limitations do not obtain for logic. In particular, there are states of affairs which are logically possible but which are excluded by the forms of intuition, such as the existence of spatial configurations conttary to the theo-

KANT'S PHILOSOPHY OF ARITHMETIC

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rems of Euclidean geometry; so that geometry is a more special theory than logic, not ouly in the sense that it deals with a more restticted type of object but also in the sense that it makes statements about these objects which are not logically necessary, although they are necessary in another way. Logic is also not subject to the great lituitation of knowledge based on intuition, that of applying ouly to appearances. When Kant says that it mnst be possible to think of things in themselves, he implies first that such a conception does not contradict the laws of logic, and second that in the statements we make about them, the logical laws are still a negative criterion of ttuth. If he could not trust logic in this realm, Kant's metaphysics of morals would not be able to get off the ground. Already on this level, it is possible to see quite clearly some reasons why Kant should have regarded geometry as synthetic a priori and used an idea such as that of a form of intuition in order to explain how snch a science was possible. Geometry is a more special theory than logic first in the sense that it contains nonlogical primitives, second in that its theorems cannot in general be proved merely by means of definitions and logic, as Leibniz apparently thonght. Indeed this is much more obvious to us than in Kant's time, given that we have non-Euclidean geometry and are in general less tempted to overestimate the power of logic, especially traditional logic. It is worth pointing out that Euclid's postulates are what are in effect existence assumptions, so that here Kant's general views abont existence would imply that they could not be analytic. That Kant should then found geometry on the form of our sensible intuition is not difficult to understand. On the one hand spatiotemporality is a . characteristic property of the objects given to the senses. Moreover, Kant emphasized that space was an individual the notion of which was understood in a way analogous to ostension, and the sarne ostensive understanding would be necessary for the particular prituitives of geometry. On the other hand, Kant started from the idea that geometry was a body of necessary ttuths with evident foundations. That the axioms of geometry should be empirically verified directly is conttary to their necessity; that they should be some sort of high-level hypotheses is conttary to their evidence. The second observation to make about Kant's views on logic is that he never suggests a conventionalist account of logical validity. It is ttue that the very general character of logical and analytical ttuths goes with their uninformativeness. They reflect the nature of the mind and of certain particular concepts, and apparently not at all how the world is otherwise. But this nature and the manner in which particular concepts give rise to the analytic ttuths

so

KANT'S PHILOSOPHY OF ARITHMETIC

that they do seem to be something given, which will be in fact the same for all discursive intelligences, even if their forms of intuition are. quite different from ours, Kant does not give much explanation of how this is, and perhaps he felt some doubt as to the possibility of giving such an explanation. If we tty to apply the insight which we might get from the Transcendental Deduction to this question, we get into a very difficult dilemma. Namely the essential activity of the understanding seems to be in relation to material given in intuition, to bring it to the unity expressed in an objective judgment. In other words, the notions of object, concept, judgment get their whole sense from their application to experience. Nonetheless .the understanding has a greater generality than intuition: The forms of intuition are not logically necessary; and in operating logically with a given notion, it is not necessary to appeal to intuition or even to suppose that the notion has an intuition corresponding to it. It is possible in some way for us to recognize that what can be given in experience is not the whole of possible reality, and even to recognize that with the help of inmition we can know objects only in a relative way, as they appear. All this points, even apart from the requirements of Kant's moral philosophy, to the presence in us of more general conceptions of object, concept, judgment, and a fortiori inference. This dilemma will occupy us again later, since it has an application to the problems of arithmetic.

subject, the situation is analogous to that in which the subject concept is defined by the conjunction of the predicate concept with perhaps certain others. This would be a paradigm case where the connection of subject and predicate is "thought through identity" (A 7 ~ B II). An idealized version of an analytic judgment would be one of the form' All AB are A', Or 'All C are A', where 'C' is defined as 'A and B'. This is idealized because, according to Kant, outside mathematics concepts do not in general have definitions in the proper sense. It seems certain that a number of other forms would have to be admitted as analytic, e.g., 'No AB are not A' or the propositional 'If p and q, then p'.7 But there is no particular reason why '7 + S ~ 12' should be. Kant says (B IS) that for '7 + S ~ 12' to be analytic, it would have to follow from the concept of a sum of 7 and S by the law of contradiction. This would be as if it were provable from definitions by a very restricted logic, probably included in the limited traditional apparatus at Kant's command, and it is hard to see how it could be true otherwise. However, it is one thing to say there is no reason to expect this and another to understand Kant's specific reason for thinking it false. Kant indicates that the way you find out that 7 + S ~ 12 is by a process like counting, of progressing from 7 to 12 by successive additions of I, in which one must operate with a particular instance of a groop of objects, which can ouly be given in intuition.

III

II

SI

CHARLES PARSONS

With respect to Kant's philosophy of geometty, the difficulties do not concern why Kant thought geometty to be a priori intuitive knowledge, but rather whether this is true and what precisely the theory was by which he proposed to explain how it could be true. When we tum to Kant's philosophy of arithmetic, there is even less difficulty as to why he should have thought arithmetical propositions a priori. But it is already by no means easy to see why Kant regarded them as synthetic, as based in some way on our forms of intuition, in particular on the form of inner intuition, time, and as limited in their application to appearances. It will become clearer why Kant regarded arithmetical propositions as synthetic if we observe that Kant's concept of analytic proposition most likely had a much narrower extension than the corresponding concept in more recent philosophy, e.g., in Frege and logical positivism. Kant does not formulate his concept with enough precision so that we can be altogether sure about this. But it seems rather clear from the examples that when Kant speaks of the concept of the predicate of an analytic judgment as contained in that of the

We have to go outside these concepts. and call in the aid of the intuition which corresponds to

one of them. our five fingers, for instance, or, as Segner does in his Arithmetic, five points. adding to the concept of 7, unit by unit, the five given in intuition. For starting with the nwnber 7, and for the concept of 5 calling in the aid of the fingers of my hand as intuition, I now add one by one to the number 7 the units which I previously took together to fonn the number 5, and with the aid of that figure [the hand1 see the number 12 come into being. [B 15-161

It is, ~owever, still not clear why that process cannot be either itself put in the form of a purely logical argnment or replaced by something quite different which can. There was an attempt to do just this with which Kant was in a position to be familiar, by Leibniz in the Nouveaux Essais. 8 Leibniz worked with '2 + 2 ~ 4', but the type of argument suffices for any addition formula. He assumed as an axiom the substitutivity of identity, which Kant would in all probability have regarded as analytic. Leibniz took as definitions

I + I, 3 ~ 2 + I, 4 ~ 3 + I, 2~

CHARLES PARSONS

52

which is approximately what is done in modem formalizations. Then the proof goes as follows: 2=2+1+1 =3+1 =4

(def. of "2") (def.of"3") (def. of "4")

The standard modem objection to this argument is that Leibniz should have inserted brackets, so that it goes 2 + 2 = 2 + (I + 1) = (2 + 1) + 1 = 3 + 1

and therefore assumes an instance of associativity. We cannot exclude the possibility that this was known to Kant when he was working on the Critique of Pure Reason, since it occurs in effect in the book Prufung der kantischen Kritik der reinen Vernunft, vol. I (Konigsberg, 1789), by Kant's pupil Johann Schultz, professor of mathematics in Konigsberg. Putting great weight on the evidence of writings by Schultz and other disciples of Kant, Gottfried Martin has put forth the hypothesis that Kant envisaged an axiomatic foundation of arithmetic similar to the classical axiomatizations of geometry.9 He sees the claim that arithmetic is synthetic as resting on the first instance on the logical point that arithmetical propositions such as '7 + 5 = 12' cannot be proved by mere logic from definitions such as those Leibniz uses. An axiomatic foundation of the sort which would answer to Mattin's ideas is given in Schultz's Prufung. Without explicitly mentioning Leibniz, Schultz points out that the sort of proof of an arithmetical identity that Leibniz gives rests on the assumption of associativity. He gives, for '7 + 5 = 12', an argument which also rests on commutativity, and seems, wrongly, to think this assumption unavoidable. But of course commutativity has to be used sooner or later in arithmetic. 10 Schultz gives two axioms, the commutativity and associativity of addition. He neither asserts nor denies the independence of the corresponding laws of multiplication and of the distributive law. He also gives two "postulates" which are worth quoting in full: 1. From several given homogenous quanta, to generate the concept of one quantum by their successive connection, Le .. to transform them into one whole. 2. To increase and to diminish any given quantum by as much as one wants, that is, to infinity. I I

The second postulate implies that Schultz is not thinking specifically of the arithmetic of integers but also of continuous quantities. In any case, the first

KANT'S PHILOSOPHY OF ARITHMETIC

53

postulate is the basis for the supposition that the function of addition is

defined; i.e., given numbers m, n. there actually exists a number m + n. If we accepted this as actually giving Kant's con ception, there would still remain the question how intuition enters into the foundation of these axioms and postulates. About this Schultz has in fact something to say. But in transferring the conception to Kant we are faced immediately with the difficulty that he explicitly says that arithmetic does not have axioms. As regards magnitudes (quantitas), that is, as regards the answer to be given to the question, "What is the magnitude of a thing?" there are no axioms in the strict meaning of the term, although there are a number of propositions which are synthetic and immediately certain (indemonstrabilia)_ [A 163-4 = B 204]

He considers two possibilities, rules of equality, which he asserts to be analytic (a proper axiom must be synthetic), and the elementary arithmetical identities, such as '7 + 5 = 12', which are what he seems to be referring to at the end of our quotation, which are indeed synthetic and indemonstrable, but which he declines to call axioms because they are singular. This position is reaffirmed in a letter from Kant to Schultz dated November 25, 1788,12 in which he comments on the manuscript of volume I of the Prufung. There he gives a reason, which I shall mention later, why arithmetic should not have axioms_ He does say that arithmetic has postulates, "immediately certain practical judgments." The general tone of his discussion suggests that he might regard the general directive to carry out addition, on the presupposition that this can always be done, as a postulate, i.e., that he might accept Schultz's first postulate. But what he seems to have specifically in mind is what he elsewhere calls numerical formulae, i.e., 7 + 5 = 12. We cannot be certain, however, that the mathematical material of the published version of the Priifung was present in the manuscript that Kant was commenting on. For it seems from the letter, as Martin points outY that the manuscript maintained that arithmetical propositions were analytic, and thus it is clear that it was considerably revised after Schultz received Kant's letter. The fact that in the published version the axiomatic analysis is used to snpport the conclusion that arithmetic is synthetic does not prove that it was not present in the manuscript, although the supposition that the postulates were there is a bit strained. But that Schultz might have argued that the commutative and associative laws were analytic is not at all impossible. (Leibniz argued this at least for commutativity. 14) Even so, unless one accepts Mattin's rather unlikely idea that the axiomatic analysis was contributed by Kant to Schultz after the letter, it is

I .

54

KANT'S PHILOSOPHY OF ARITHMETIC

hard to escape the conclusion that Schultz understood the mathematical issue in at least one respect better than Kant himself: Kant does not seem to have had an alternative view of the status of such propositions as the commutative and associative laws of addition. He can hardly have denied their truth, and it seems that if they are indemonstrable, they must be axioms; if they are demonstrable, they must have a proof of which he gives no indication. If when speaking of the axiomatic character of arithmetic, Martin means that according to Kant arithmetic must make use of propositions which cannot be deduced by logic and definitions, then there can be no disagreeing with him. But if he means that Kant had in mind setting up arithmetic as an axiomatic system of which Schultz's is a very primitive instance and that it is

didates for such arguments are arguments involving singnlar tenns. For Beth the fonn of argument involved is illustrated by the proof that the base angles

in the verification of such laws as the commutative and associative that the

primary application of inmition in a..-ithmetic is to be found, then Kant's actual words go against him. Even if Martin's view of the matter is quite correct as far as it goes, it cannot satisfy us. In the first place, it does not answer the question why arithmetic should depend on intuition, except in the sense, entirely bound to the primitive level ofaxiomatics in Kant's time, that so far as one can see the obvious alternative is insufficient. In the second place, it carries over to arithmetic the considerations which were at work in geometry while our original

I!i'

sense of difficulty arose from the difference between the two. And there are many indications, in particular some remarks in the letter to Schultz, which I shall discuss, that he saw some of this difference and did not intend to give an entirely symmetrical account.

I

il III.

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CHARLES PARSONS

of an isosceles triangle are equal: We proceed, as is well known, as a rule as follows: first we consider a particular triangle, say ABC, and suppose that AB = AC; then we show that 4. ABC = 4. ABC and have thus proved that the assertion holds in the particular case in question. Then one observes that the proof is correct for an arbitrary triangle, and therefore that the assertion must hold in general. 18

The general fonn of the argument is as follows: We want to prove '(x) (Fx::oGx),. We assume a particular a such that Fa. We then deduce 'Ga'. We then have 'Fa::oGa' independently of the hypothesis. But since a was arbitrary, '(x) (Fx::oGx), follows. This- form of argument, as for example in Beth's case, is the characteristic fonn of a proof in Euclid. In the "Discipline of Pure Reason in Its Dogmatic Employment," the section of the Critique where Kant sets forth his view of mathematical proof as proceeding by "construction of concepts in pure intuition," this fonn of argument appears clearly in the geometrical example (A 716-7 = B 744-5). The geometer "at once begins by constructing a triangle." By a series of constructions on this triangle and applications of general theorems to it "through a chain of inferences guided throughout by intuition he arrives at a fully evident and universally valid solution of the problem." Hintikka concentrates attention rather on the rule of existential instantiation, that is on arguments of the form (3x)Fx Fa

IV

I

II

,.

i

d

Ii'I

"~'I

The problem of the asymmetry of arithmetic and geometry could be solved by an interpretation suggested by E. W. Beth ls and developed by Hintikka. 16 From their interpretation it seems to follow that if a proposition B of geometry is proved by a proof which appeals to axioms A I ... An (I here include postulates),17 then in general the conditional Al & ... & An '::0 B is synthetic; at any rate an appeal to intuition is made over and above any which is made in verifying the axioms. One could then argue that since arithmetic according to Kant does not have axioms, only the first type of appeal to pure intuition occurs in arithmetic. Beth's and Hintikka's hypothesis is that for Kant certain arguments which can nowadays be fonnulated in first-order predicate logic involve an appeal to intuition. In view of the singularity criterion for intuition, the natural can- .

p where a is introduced to indicate an F, in view of the fact that the previous line affinns that there are F' s. 19 Both of these arguments have in common that they tum on the use of a free variable which indicates anyone of a given class of objects, so that an argument concerning it is valid for all objects of the class. They thus have a fonnal analogy with the appeal to pure intuition, in that a singular tenn is used in such a way that what is proved of it can be presumed generally valid. Moreover, the manner in which this generality is assured, namely by not allowing anything to be assumed about a except what is explicitly stated in

CHARLES PARSONS

KANT'S PHILOSOPHY OF ARITHMETIC

premises, is reminiscent of a statement of Kant about the role of a constructed figure in a proof: "If he is to know anything with a priori certainty he must not ascribe to the figure anything save what necessarily follows from what he has himself set into it in accordance with his concept." (Bxi;) It is noteworthy that in traditional algebra calculations are carried out on terms and formulae with free variables, where the derivation of such an equation serves to prove a general proposition. Hintikka interprets the rather obscure remarks about "symbolic construction" in algebra in this sense. 20 It would naturally follow from the conception of intuition as simply individual representation that the mere form of these arguments is such that they involve intuition. Of course, it would not give any plausibility to Kant's more far-reaching philosophical theses which tum on the connection of mathematics with the form of sensibility. Thus the philosophically interesting aspects of the concept of pure intuition seem to lose their point when it is pointed out that these arguments can be formalized in pure quantification theory. This is exactly the conclusion which Beth draws. One might object that this seems to presuppose that logic itself does not pose philosophical problems which the notion of pure intuition might be needed to answer, but on this at least Beth is in agreement with Kant in most of his utterances. But anyway it seems unlikely that the break between arguments which tum on the generality interl'retation of free variables and logical arguments which do not is the philosophically most siguificant break within mathematical proof,21 One could wish more clear-cut evidence for the attribution of such a view to Kant or even for the most modest thesis that he started with this idea in developing his philosophy of mathematics. If it was his mature view, Kant's mathematically astute pupil Schultz seems not to have suspected it since there is no suggestion of it in the Prilfung. Schultz took for granted that an adequate axiomatization would be such that if the axioms were analytic so would be all the theorems. Mathematics fails to be analytic just because in its deductive development synthetic premises must be used. The same view is expressed by Kant when he says:

Against this, it is pointed oui" that Kant says of a geometric proof that it proceeds "through a chain of inferences guided throughout by intuition" (A 716-7 = B 744-5). In view of the description Kant gives of the proof, this could easily mean that in the course of the proof one is constantly appealing to the evidences formulated in the axioms and postulates. It would obviously be anachronistic to attribute to Kant a picture of proof modeled on a formal deduction where the axioms are stated at the beginning and everything else is logic and where the purpose is not to show the truth of the proposition proved but merely that it follows from the axioms. On the contrary, for Kant a Euclidean proof is convincing because on each particular application of an axiom or postulate the correctness of what it claims in this particular case is evident. It must be conceded that it might be true that inference by certain rules from analytic premises might yield analytic conclusions while inference according to the same rules from synthetic premises could lead to conclusions which are not only themselves synthetic but such that the conditional of premises and conclusions is also synthetic. In particular, the rule of existential instantiation can only come into play in the presence of an existential quantifier, and it is not clear that, for Kant, a statement in which an existential quantifier occurs essentially can be analytic. I can ouly say that in such cases the text of Kant does not clearly indicate that the necessity of an appeal to intuition arises for the inference and not merely for the verification of the

56

For as it was found that all mathematical inferences proceed in accordance with [nach] the principle of contradiction (which the nature of all apodicitic certainty requires). it was supposed that the flmdamentaI propositions of the science can themselves be known to be true through that principle. This is an erroneous view_For though a synthetic proposition can indeed be discerned in accordance with the principle of contradiction. this can only be if another synthetic proposition is presupposed, and if it can then be apprehended as following from this other proposition. [B 14]

57

premise.

If Hintikka were right, one could expect that in the passages on algebra the role of variables would be emphasized. It is possible to find this emphasis in the passage on A 717 = B 745, but it is not really explicit. The emphasis of A 734 = B 762 seems different, where Kant says, "The concepts attached to the symbols, especially concerning the relations of maguitudes, are presented in intuition" (emphasis mine). The relations would seem to be expressed by algebraic function signs. Although the passages on algebra offer some support for Hintikka' s theory, it is less than decisive. I shall show that there are other possible ways of looking at these passages. The direct evidence thus seems to me on the whole opposed to the BethHintikka theory. However, it would have strong indirect support if there were not other ways to explain how arithmetic can require pure intuition and -to

interpret the notion of "construction of concepts," especially in algebra. To this end we now return to the problem of the difference between arithmetic and geometry.

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v

Both to set forth what we need of this construction for our purposes and to indicate how far one can go without using set-theoretic devices, I shall discuss a logical truth which is closely related in meaning to '2 + 2 = 4' and provides the key to the proof of '2 + 2 = 4' in more extended formalisms. This example will help to indicate how far the cases of arithmetic and geometry are symmetrical. Consider the following schema of the first-order predicate calculus with identity:

The difficulty can be put in this way: The synthetic and intuitive character of geometry gets a considerable plausibility from the fact that geometry can naturally be viewed as a theory about actual space and figures constructed in it. This space is related to the senses by being a field in which the objects given to the senses appear, and geometry seems to give quite substantial information about this space which from the point of view of abstract thought might be false. The content of arithmetic does not immediately suggest such a special character or such a connection with sensibility. Of course in the first instance it speaks of numbers and purely abstract operations and relations - equality, addition, subtraction, etc. Then the question is - what is the field of application of numbers? That is, what sorts of things can be counted, assigned cardi-

II

nal or ordinal numbers, or measured and thus aSsigned continuous quantities? On the face of it, there is no reason to believe that the application of arithmetic need be to objects in space and time. Although this has certainly become more evident since th~ rise of abstract mathematics, that mathematical objects themselves could be numbered was something which Kant was certainly in a position to be aware of. If the application of arithmetic is to be limited to appearances, this limitation has to be understood rather broadly in order to reconcile it with obvious facts. In the case of geometry, it was possible to mention logical possibilities which the concepts allowed but which did not exist according to the mathematical theory; Kant gives the example of a two-sided plane fignre, and many more such possibilities were opened up in the development of non-Euclidean geometry. It was probably impossible in Kant's time to be clear about whether such a possibility exists in arithmetic. If it did, it would give rise to a clear separation of arithmetical from logical truth. This sort of argnment was not available to Kant. The difficulty is made more acute, some would say insoluble, by subsequent developments in logic, particularly the efforts of Frege and others to do just what Kant thought impossible - to reduce arithmetic to logic, to deduce arithmetical propositions from definitions and propositions of pure logic. Of course the extent of what counts as "logic" here is considerably wider than what Kant regarded as such. At the very least, we need for this type of construction to incorporate some of the theory of classes into logic; not just the notion of class and some elementary operations concerning them, but also at least some modest axioms of class existence - how modest depending on

how much arithmetic one wants to deduce.

(3 x)Fx· (3 x)Gx· (x) - (Fx· Gx) 2 2

(I)

'::0

59

(3 x) (Fx V Gx) 4

where '(3 x) Fx' is an abbreviation for '- (3 x) Fx' and '(3 x) Fx' for o n+1 (3 x)[Fx· (3 y) (Fy . y "" x)]. n so that '(3 x) Fx' can be expanded as 2 (2)

(3 x) (3 y) [Fx· Fy ·x""Y· (u) (Fu::o· u=xV u= y)]

and '(3 x) (Fx v Gx)' as

4 (3)

(3 x) (3 y) (3 z) (3 w) [Fxv Gx· Fy V Gy· Fz v Gz· Fw V Gw· x:;t:y 'x*,z ·x:;t:w· y=t:.z· y,,*w· z:;t:w. (u) (Fu vGu·::o· u=xV u = yv u =zV u=w)].

Intuitively, the proof of this schema goes like this: Suppose (3 x) Fx . (3 x) Gx-

2

2

Then in view of (2) and its counterpart for '(3 x)Gx' there are x, y, z, and w such that 2 Fx· Fy ·x""Y· (u) (Fu::o· u =xV u=y) Gz· Gw· z""w' (u) (Gu::o· u=zV u=w).

We then go out to argue, with the help of '(x) - (Fx . Gx)" that x, y, z, w satisfy the condition in the scope of (3), and so we infer that there are x, y, z, w such that this condition holds.

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CHARLES PARSONS

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KANT'S PHILOSOPHY OF ARITHMETIC

This schema requires for its formulation ouly predicate letters, variables, quantifiers, identity, and logical connectives. The only notion involved which could possibly be different in principle from what Kant regarded as general logic is identity, and since that is used in application to quite arbitrary objects, it does not immediately suggest a restriction as to application as the geometrical concepts do. Moreover, the schema is proved without the application of existence axioms: The range of values of the variables can be any 23 universe whatsoever, even the empty one. Frege and his twentieth-century followers certainly thought that by their construction they had refuted the view that arithmetic depends in any way on "pure intuition," sensibility, or time. Thus the temporal notion of the successive addition of units, or the even more concrete one of combining groups of objects, is replaced in Frege's construction by the timeless relation of one class being the union of two others, which can be defined in teTIIlS of the logical connective alternation as it occurs in (1). Moreover, the construction provides a framework for the application of the concept of number far beyond the scope of concrete appearances, in particular, in the elaboration of set theory. Analogous to a non-Euclidean space would be a possible world in which the arithmetical identities turned out differently, for example, in which 2 + 2 = 5. But would that not be a world in which there was a counterinstance to our schema, and therefore in conflict with logic? Ouly, of course, if the connection of meaning between '2 + 2 =4' and the schema (or '2 + 2 =5' and a similar schema) is preserved. I am inclined to regard the breaking of this connection as a change in the meaning of addition. There is, however, one way out of this dilemma. With '2 + 2 = 5' we would associate the schema

(4)

(:l x)Fx . (3 x)Gx . (x) - (Fx) . Gx) . => (3 x)(Fx V Gx)

2

2

5

Now suppose we had a universe U in which for any choice of extensions of 'F' and 'G' this schema came out true. Even according to our notions of logic, there is a possible case in which this happens, and in which (since (1) is valid) there is also no conflict with (1), namely in which U contains fewer than four elements. Io that event the antecedent of the above would always be false?4 If one considers the minimal existence axioms which would be needed to prove the categorical '2 + 2 = 4' in modern set theory, we find that again they require the universe to contain at least four elements, which can be identified with the numbers 1,2,3,4.

61

If we accept first order quantification theory with identity as a logical framework, then it seems that we can maintain the symmetry of arithmetic and geometry in a weak sense, that such propositions as '2 + 2 = 4' imply or presuppose existence assumptions which it is logically possible to deny. To draw the line at this point and to declare thus that set theory is not logic seems to me eminently reasonable; but I shall not argue for this now, particularly since I have done so elsewhere.25 I think the presence of existence propositions in mathematics one of the considerations at stake in Kant's views on mathematics, but it is not clearly differentiated from others. His general views on existence imply that existential propositions are synthetic, but he never applies this doctrine directly to the existence of abstract entities. In the letter to Schultz cited above, Kant says that arithmetic, although it does not have axioms, does have postulates. Postulates as to the possibility of certain constructions, for Kant constructions in intuition, played the role of existence assumptions in Euclidean geometry. Schultz states as a postulate in the Priifung essentially that addition is defined. This factor is also present in Kant's remarks about "construction of con-

cepts in pure intuition," which he regards as the distinguishing feature of mathematical method. If the geometer wants to prove that the sum of the angles of a triangle is two right angles, he begins by constructing a triangle (A 716 = B 744). This triangle, as we indicated above, can serve as a paradigm of all triangles; althongh it is itself an individual triangle, nothing is used about it in the proof which is not also true of all triangles. The proof consists of a sequence of constructions and operations on the triangle.

Kant's view was that it is by this construction that the concepts involved are developed and the existence of mathematical objects falling under them is shown. Although we need not regard this theorem as implying or presupposing that there are triangles, Kant regarded a general proposition as empty, as not genuine knowledge, if there are no objects to which it applies. In this instance only the construction of a triangle can assure us of this. Apart from that, further existeIlce assumptions are used in the course of the proof, in the example of A 716 = B 744 of extensions oflines and of parallels. The same factor is also suggested in the rather puzzling passage in which Kant says that the operation with variables, function symbols, and identity in ttaditional algebraic calculation involves "exhibiting in intuition" the operations involved, which he calls "symbolic construction." Io fact, such operation presupposes that the functions involved are defined for the arguments we permit ourselves to substitute for the variables. Moreover, the construction of an algebraic expression for an object to satisfy a certain condition is the very

KANT'S PHILOSOPHY OF ARITHMETIC

paradigm of a constructive proof of the existence of such an object. However, I think there is something else at stake in this passage, which I shall come to.

instead of the successive addition of "units" we have a timeless relation, for example, that one set is the union of two others; but also with the application of these notions within modem mathematics, in which arithmetical statements can be made about structures which are entirely timeless, and in reference to which any talk of "successive addition" is in on the face of it entirely metaphorical. In the lener to Schultz, Kant qualifies his position in a way which does more justice to this more general character of arithmetic:

VI

It is by no means obvious that the existence assumptions which must be made in the deductive development of mathematics have any connection with sensibility and its alleged form. Frege for one was quite convinced that they did not. What Kant says that bears on this point is not completely clear, partly because in the nature of the case it is bound up with some difficult notions in his philosophy, partly because again he did not disengage this issue from some others. As a preliminary remark, we must observe that Kant certainly did not regard arithmetic as a special theory of, say, time, in the sense in which he regarded geometry as a special theory of space. It does not tum up in this connection in the proofs of the apriority of time in either the Aesthetic or the corresponding discussion in the Inaugural Dissertation (§ 12, § 14 no. 5). Nevertheless it is clear that according to Kant, the dependence of arithmetic on the forms of our intuition is in the first instance only on time. I should venture to say that space enters the picture only through the general manner in which inner sense, and thus time, depends on outer sense, and thus space. We shall be clear about the intuitive character of arithmetic when we are clear about the manner in which it depends upon time. Whenever Kant speaks about this subject, he claims that number, and therefore arithmetic, involves succession in a crucial way. Thus in arguing that intuition is necessary to see that 7 + 5 = 12: For starting with the number 7. and for the number 5 calling in the aid of the fingers of my hand as intuition. I now add one by one to the number 7 the units which I previously took together to form the nwnber 5, and with the aid of that figure [the hand] see the number 12 come into being. [B 15-16; emphasis mine]

When he gives a general characterization of number in the Schematism, the reference to succession occurs essentially:

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CHARLES PARSONS

62

The pure image of all magnitudes (quantorum) for outer sense is space; that of all objects of the senses in general is time. But the pure schema of magnitude (quantitatis), as a concept of the understanding, is number, a representation which comprises the successive addition of homogeneous units. [A 142 = B 182]

As I said, this seems to conflict not only with the interpretation which number and addition acquire in such constructions as Frege' s, in which

Time, as you quite rightly remark, has no influence on the properties of numbers (as pure determinations of magnitude), as it does on the property of any alteration (as a quantum), which itself is possible only relative to a specific condition of inner sense and its fann (time); and the science of number, in spite of the succession, which every construction of magnitude [Grosse] requires, is a"pure intellectual synthesis which we represent to ourselves in our thoughts.

Earlier in the letter he writes: Arithmetic, to be sure, has no axioms, because it actually does not have a quantum, i.e., an object of intuition as magnitude, for its object, but merely quantity, i.e., a concept of a thing in general by detennination of magnitude.

Kant is here in fact reaffirming a position affirmed in the Dissertation: To these there is added a certain concept which, though itself indeed intellectual, yet demands for its actualization in the concrete the auxiliary notions of time and space (in the successive addition and simultaneous juxtaposition of a plurality), namely, the concept of number, treated of by arithmetic. [§12J

These remarks place arithmetic less on the intuitive and more on the conceptual side of our knowledge. If arithmetic had for its object "an object of intuition as magnitude," i.e., forms such as the points, lines, and figures of geometry, then it would refer quite directly to a form of intuition. But instead it refers to "a concept of a thing in general"; the science of number is a "pure intellectual synthesis." This laner phrase especially suggests that arithmetical notions ntight be definable in terms of the pure categories and thus be associated with logical forms which do not refer at all to conditions of sensibility. Such a view would seem to conflict with the statement of the Schematism that number is a schema. The refere,!ce to "a concept of a thing in general" is no doubt to be meant in the same sense as that in which the categories are said to specify the concept of an object in general, and the pure intellectual synthesis is no doubt that of the second edition transcendental deduction, which is the synthesis of a manifold of intuition in general, which is for us realized so as to yield

'1-

i

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CHARLES PARSONS

KANT'S PHILOSOPHY OF ARITHMETIC

knowledge only in application to intuitions according to our forms of inruition. Thus the "concept of an object in general" could give rise to actual knowledge of objects only if these objects can be given according to our forms of intuition. But does this merely mean that objects in space and time provide the only concrete application of these concepts which we can know to exist, as one might expect from the absence of special reference to inruition? Whether it means this or something more drastic is, I think, a special case of the general dilemma about the understanding which I mentioned in the beginning. In either case, however, it would be a plausible interpretation of Kant to say that the forms of intuition must be appealed to in order to verify the existence

logic, for example where the iuitial element is '0' and the (n + \)st numeral is obtained by prefixing'S' to the nth numeral, have the further property that each numeral contains within itself all the previous ones so that the nth numeral is itself a model of the numbers from 0 to n). The basis for the use of a concrete perception of a sequence of n tenus in verifying general propositions is that, since it serves as a representative of a strucUlre, the same purpose could be served by any other instance of the same strucrure, that in any other perceptible sequence which can be placed in a one-one- correspondence with the given one so as to preserve the successor relation. This might justify us in calling such a perception a "formal inruition." We might note that the physical existence of the objects is not directly necessary, so that we can abstract also from that "material" factor. An empirical intuition functions. we might say. as a pure intuition if it is taken as a representative of an abstract strucrure. Such a perception provides the fullest possible realization before the mind of an abstract concept. One of the important questions about Kant's philosophy of arithmetic is whether a comparable realization exists beyond the limits of scale of concrete perception. Before we can enter into this question, let me point out another closely related reason in Kant's mind for regarding mathematics as dependent on intuition. This. comes out in particular in the concept of "symbolic construction." The algebraist, according to Kant, is getting resnlts by manipulating symbols according to certain rnIes, which he wonld not be able to get without an analogous inruitive representation of his concepts. The "symbolic construction" is essentially a construction with symbols as objects of inruition:

assumptions of mathematics. However, it is not very clear how to apply the general conceptions derived from the Aesthetic and the Transcendental Deduction to the case at hand. The direct existence propositions in pure mathematices are of abstract entities, and it is only in the geometric case that they can be said to be in space and time. I do think that the objects considered in arithmetic and predicative set theory can be construed as forms of spatiotemporal objects. Fnll set theory wonld of course not be accommodated in this way, but it is not reasonable to expect that from a Kantian point of view impredicative set theory should be intuitive knowledge or indeed genuine knowledge at all. It could legitimately be said to posruIate entities beyond the field of possible experience. 26 VII

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It is narural to think of the natural numbers as represented to the senses (and of course in space and time) by numerals. This does not mean mainly that numerals function as names of numbers, although of course they do, but that they provide instances of the strucrure of the naruraI numbers. In the algebraic sense, the set of numerals generated by some procedure is isomorphic to the narural numbers in that it has an iuitial element (e.g., '0') and a successor relation which the notion of naruraI number requires. In this sense, of course, the numerals are abstract mathematical objects; they can be taken as geometric figures. But of course concrete tokens of the first n numerals are likewise a model of the numbers from \ to n or from 0 to n - 1. A set of objects has n elements if it can be brought into one-to-one correspondence with the numbers from I to n; a standard way of doing this is by bringing them in some order into correspondence with certain numerals representing these numbers, that is by counting. (The numerals used in work in formal

65

Once it [mathematics] has adopted a notation for the general concept of magnitudes so far as their different relations are concerned, it exhibits in intuition, in accordance with certain universal rules, __ all the various operations through which the magnitudes are produced and modified. When, for instance, one magnitude is to be divided by another, their symbols are placed together, in accordance with the sign for division, and similarly in the other processes; and thus in algebra by means of a symbolic construction, just as in geometry by means of an ostensive construction (the geometrical construction of the objects themselves) we succeed in arriving at results which discursive knowledge could never have reached by means of mere concepts. [A 717 = B 745]

That· this is a source of the clarity and evidence of mathematics and provides a connection of mathematics with sensibility is indicated by the following remark: "This method, in addition to its heuristic advantages, secures all inferences against error by setring each one before our eyes" (A 734 ; B 762). A connection of mathematics and the senses by way of symbolic opera-

:;-""

Iii

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CHARLES PARSONS

tions is already claimed in Kant's prize essay of 1764, Untersuchung iiber die Deutlichkeit der Grundsiitze der natiirlichen Theologie und der Moral,2? which presents a prototype of the theory of mathematical and philosophical method of the Discipline of Pure Reason in its Dogmatic Employment For example, consider the statement of the latter: Thus philosophical knowledge considers the particular only in the universal, mathematical knowledge the universal in the particular, or even in the single instance, although still always a prioli and by means of reason. [A 714 = B 742]

This distinction corresponds in the prize essay to the following, where the distinctive role of signs in mathematics is explicitly emphasized: "Mathematics considers in its solutions proofs, and inferences the universal

in [unter] the signs in concreto, philosophy the universal through [durch] the signs in abstracto."2' The certainty of mathematics is connected with the fact that the signs are sensible: Since the signs of mathematics are sensible means of knowledge, one can know with the same confidence with which one is assured of what one sees with one's own eyes that one has not left any concept out of account, that every equation has been derived by easy rules. etc.; thereby attention is made much easier in that it must take account only of the signs as they are known individually, not the things as they are represented generally.29

The prize essay suggests a position incompatible with the Critique of Pure Reason, namely that since in mathematics signs are manipulated according to

Illi

rules which we have laid down (in contrast to philosophy, where the value of any definition turns on its having a certain degree of faithfulness to preanalytic usage), operation with signs according to the rules, without attention to what they signify, is in itself a sufficient gnarantee of correctness.30 . These passages show that a connection between sensibility and the intuitive character of mathematics existed in Kant's mind before he developed the theory of space and time of the Aesthetic. However, unlike in the later work, no inference is drawn at this stage from this connection to a limitation

of the application of mathematics to sensible objects. The general point behind the observations on symbolic construction can be put in the following way: In general, a mathematical proposition can be verified only on the basis of a proof or calcnlation, which is itself, a construction in intuition. But in view of the remarks about '7 + 5 = 12', a more special fact may have influenced Kant. Certain "symbolic constructions"

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'2 + (1 + 1)' and the two 1's as it were added on to the '2'. A corresponding proof of '7 + 5 = 12' wonld involve five such steps instead of two. A similar observation concerning the schema (1) has been made by a number of writers. Although the schema does not imply that the universe contains any elements or that any construction can be carried out, the proof of it involves writing down a group of two symbols representing the F's, another such group representing the G's, and putting them together to get four symbols. So that it is not at all clear that '2 + 2 = 4' interpreted as a proposition about the combinations of symbols is not more elementary than the logically valid schema (1). I have already suggested that the "symbolic" construction in generating numerals is already enough to settle the question of their reference. In the same way the actna! carrying out of the calculations shows the well-defined character for individual argnments of recursively defined functions. However, induction, which I have wanted to leave out of account here, is involved in seeing that they are defined for all argnments. Maybe Kant ought to have said that apart from intuition I do not even know that there is such a number as '7 + 5'. And it seems that one could not see by a particnlar construction that there is such a number without also seeing it to be 12. This is in agreement with Hintikka's statement that the sense of Kant's statement that numerical formulae are indemonstrable is that the construction required for their proof is already sufficient. The considerations about the role of symbolic operations apply equally to logic and therefore undermine Kant's apparent wish to distingnish them on this basis. This appears more forcefully in modem logic, where instead of a short list of forms of valid iuference one has an infinite list which must be specified by some inductive condition. In my opinion this is a consequence to

be accepted and is even in general accord with Kant's statements that synthesis underlies even the possibility of analytic judgments.3 ! The special connection of arithmetic and time can, I think, be explained as follows: If one constructs in some way, such as on paper or in one's head, such a sequence of symbols as the first n numerals, the structure is already represented in the sequence of operations and more generally in the succession of mental acts of running through a group of n objects, as in counting. Thus time enters in through the succession of acts involved in construction or in successive apprehension. This connects with Kant's remark about number

associated with propositions about number actually involve constructions iso-

in the Schematism. In the operations involved in representing a number to the

morphic to the numbers themselves and their relations, or at least an aspect of them. Thus in Leibniz's proof that 2 + 2 = 4, '2 + 2' must be written out as

Time provides a universal source of models for the numbers. In particnlar,

senses, we also generate a structure in time which represents the number.

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Kant held that it is only by way of successively perceiving different aspects of a manifold and yet keeping them in mind as aspects of one intuition that we can have a clear conception of a plurality. For quite small numbers this seems doubtful although not for larger ones. Nonetheless the element of succession appears even for the smaller ones in the comparison involved in generating or perceiving them in order, and the order is certainly part of our concept of number. What would give time a special role in our concept of number which it does not have in general is not its necessity, since time is in

according to a rule. To speak of a peculiar kind of intuition in the second case seems qnite tuisleading. The mathematical knowledge involved has a highly complex relation to "intuition" in the more specifically Kantian case.

some way or other necessary for all concepts, nor an explicit reference to

time in numerical statements, which does not exist, but its sufficiency, because the temporal order provides a representative of the number which is present to our consciousness if any is present at all. Of course it is one thing to speak of representation in space and time and another to speak of representation to the senses. What is represented to the senses is presumably represented in space and time, but maybe not vice versa. To establish a link of these two Kant would appeal to his theory of space and time as forms of sensibility. The relevant part of this theory is that the structures which can be represented in space and time are structures of possible objects of perception. The kind of possibility at stake here must be essentially mathematical and go beyond "practical" or physical possibility.32 Consider once again a procedure for generating numerals, say by starting with '0' and prefixing occurrences of'S'. The actual use of these as symbols requires that they be perceptible objects. Nonetheless we say it is possible to iterate the procedure indefinltely and therefore to construct indefinitely many numerals. Thus it is clear that the numerals (numeral types)33 which it is in this sense possible to construct extend far beyond the numeral-tokens which have ever been produced in history or which conld in any concrete sense actually be used as symbols.34 This possibility of iteration is necessary for the constructibility of indefinitely many numerals and therefore for the infinity of natural numbers to be given by intuitive construction. Moreover, some insight into such iteration seems necessary for mathematical induction. Insofar as the appeal to pure intuition for the evidence of mathematical statements is supposed to be an analogy of mathematical and perceptual knowledge, it holds less well for propositions involving the concept of indefinite iteration, such as these proved by induction, than for propositions such as 2 + 2 = 4. There seem to be two independent types of insight into our forms of intuition which a Kantian view requires us to have, that which allows a particular perception to function as a "formal intuition" and that which we have into the possible progression of the generation of intuitions

The complexity must be in some way present in the "intuitions" of space

and time since space is an individual which is given, but its structure also detertuines the limits of possible experience and contains various infinite aspects. No doubt the plausibility of the idea that space is present in immediate experience made it more difficult for Kant to appreciate the differences of the kinds of evidences covered by his notions of pure intuition. I am sure that more could be done to explicate the Kantian view of their connection.

In our discussion of intuition, we have somewhat lost sight of the view of logic which at the start we attributed to Kant, which except for the question of existence resembles the modern views called Platonist. Although Kant's view of intuition fits better with the modem tendencies called constructivist or intuitionist, it seems certain that the concept of pure inruition was meant to go with this view of logic and not to replace it. Without using notion like "concepts" and "object" in a quite general way, it is probably not possible to describe it. It wonld be hasty for that reason to identify Kant's conception of intuition with that of Dutch intuitionists, although Brouwer's undoubtedly shows some affinity. It would also be hasty to regard Brouwer's critique of classical mathematics as altogether in accord with Kantianism. POSTSCRIPT

The remarks in this paper about Kant's conception of intuition provoked some controversy_ Hintikka's reply to my criticisms- of his views concentrated on this issue. 34 It is agreed that Kant's basic conception of an intuition

is ofa representation of an individual object that relates to its object inunediately. The dispute concerned what I called the "immediacy criterion" and how it is related to the individuality or singularity criterion. 35 According to Hintikka, an intuition is simply a singular representation, the analogue among Kantian Vorstellungen of a singular term. In reply to my question why he thought the immediacy criterion nonessential, Hintikka says that it is "simply a corollary of the individuality criterion" (p. 342). He cited the followiog well-known passage: "This knowledge is either intuition or concept (intuitus vel conceptus). The former relates immediately to the object and is singular [einzeln],36 the latter refers to it mediately by means of a feature which several things may have in common" (A 320 = B 376-7). What is not immediate about concepts, as Hintikka reads this passage, is that

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they refer to their objects "only through the mediation of a characteristic which several objects may share" (p. 342). Thus the immediacy of intuitions consists in their not representing their objects by way of properties that they may share with other objects. This is, so far, a defensible reading of the passage. To support Hintikka's thesis, however, it would have to be stretched to say that no representation that is mediate in this sense could be singular. Hintikka does not explicitly argne for this consequence, and strong textual arguments against it were subsequently offered by Robert, Howell.37 An important difficulty is that it would make singnlar judgments an exception to Kant's view that a judgment is a combination of two concepts. This would make nonsense of Kant's assituilation of singular judgments to universal. 38 Kant does not, to be sure, say that

ceptual, content of the representation.44 Nonetheless, the controversy about the immediacy criterion convinced me that my interpretation of its meaning was not so evident as I had thought. Howell draws a reasonable line between what is plain from the text and what has the character of a reconstruction. Howell's own further interpretation is certainly an interesting line of reconstrucrion and may be fruitful. As an interpretation of Kant's intentions, it has the difficulty of relyiug on ideas developed much later in response to problems Kant did not consider. Howell's view, like Hintikka's, attempts to make the distinction between intuitions and concepts entirely within general logic. Kant followed a logiciu tradition that neglected the distinction between singular and general terms. Without breaking with this tradition more clearly than he did, Kant could not give a clear account of how singular representations could enter into propositions and inferences. My strategy was not even to try to do this on Kant's behalf. In understanding immediacy as some kind of direct presence, I was treating the concept of intuition as from the beginning epistemic. Even if Howell is right about the strict definition of the immediacy of intuition, there is undoubtedly an epistemic sense of immediacy in Kant's writings, at stake when he says that mathematical axioms are immediately certain (A 732 = B 760).45 I do not see how to get around regarding some link between the immediacy of intuition and this epistemic sense as an assumption of Kant's, whether or not it was directly embodied in the way he understood the word "immediate" in the definition of intuition. Without some such interpretative hypothesis, I did not see how to make sense of Kant"s theory of geometty. It gives a straightforward explanation of how Kant could think that mathematics depends on sensibility, and extends it from the easier case of geometty to the harder case of arithmetic. I should point out that I intended "present" in a phenomenological sense; in imagination as well as perception an object is "present" in the relevant sense. It follows that intuition does not necessarily involve the existence of the object intuited. It is not clear to me how the direct-reference view achieves the same result, which seems necessary for an account of nonveridical perception.46 At this point I should mention a misunderstanding to which my view can give rise, which is expressed most clearly in the following remarks by Gordon Brittan:

there are singular concepts. It is not concepts themselves but only their use

that can be classified as universal, particular, and singular. 39 But there is no indication that the singular use of a concept makes it an intuition.

A more principled difficulty with Hintikka's interpretation is how it could be, on his account, that all our intuitions are sensible. A definite description is a singular term that refers to its object by means of concepts; quite apart from the passage where he seems pretty clearly to say the contrary,40 I do not see how even after Kant's logical analysis of mathematics (however that is to be interpreted) and the argumeut of the Aesthetic, that can be taken to be a representation of sensibility. Rejection of Hintikka' s view of the immediacy criterion does not of itself

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imply acceptance of my own. A via media is proposed by Howell. He agrees with Hintikka that the above-cited passage gives the strict definition of immediacy; it is simply the absence of "mediation" by marks or characteristics. So, far from agreeing with Hintikka that it is merely a corollary of singularity, he goes on to nnderstand it as direct reference in something'like the sense of

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empirical intuitions at least are analogues of demonstratives.42 Hintikka himself reads immediacy as "direct reference to objects" (p. 342) and may have adopted a view like Howell's, although then he surely should not continue to hold that immediacy is a consequence of singularity.43 Howell's view of the strict definition of immediacy has the great advantage of putting into relief the somewhat hypothetical character of all three of the developed views we are considering, Hintikka's, Howell's, and mine. In my view it also shows the necessity of some further assumption, since it makes the bare-bones notion of immediacy very uninfonnative; indeed, to one trained in modern logic, it appears circular, for what could mediation by marks or characteristics be but some predicative, in Kantian tertuinology con-

If we were to accept Parsons' interpretation - that it is part of the meaning of "intuition" that intuitions are quasi-perceptual (and thus that the "immediacy criterion"" is independent of the singularity criterion and has epistemological import) - then how would we be able to understand Kant's claim (at B 146) that "as the Aesthetic has shown, the only intuition possible to us is sen· sible"? On Parsons' interpretation, that (human) intuition is sensible follows as a trivial conse· quence of the definition and should not require the extended argument of the Aesthetic. 47

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It cannot be my view (or anyone else's) that it is a trivial consequence of the definition that intuition is sensible; as I made clear above (section I), an intellectual intuition would be a counterexample. I do not see why Brittan should think that the more restricted thesis that human intuition is sensible should be immediate from the definition as I read it, since it makes no explicit mention of distinctively human (or even fiuite) intellectual capacities. It may be that he takes it to be obvious from Kant's point of view that it is ouly by sensibility that individuals are immediately present to us. However, this could be true ouly if "individual" is taken to mean concrete object. Since clearly only logical singularity is at issue in the definition of intuition, one cannot derive the sensible character of all human intuition in this way. There is an underlying reason why Brittan's misunderstanding should be natural. I did say that in intuition the object "is in some way directly present to the mind" (p. 112), and that word "present" does suggest that the object is given in the sense that Kant regards as characteristic of the intuition of finite intelligences, that is, that the tuind is to some degree passive and is apprehending an object that is "there" prior to its apprehension. Perhaps we do have to cancel such a suggestion in understanding the idea of an intellectual intuition (to the extent that we can understand it). If so, this reflects the intrinsic difficulty of fortuing a conception of a mode of intuition different from

here expressing a common conception of intuition, not necessarily his own, and that in the end his view is that it fits ouly empirical intuition. In fact it is quite clear (especially from §9) that when Kant in this passage talks of "objects," he means actual concrete objects. A priori intuition of such objects has to be prior to their being given. This, he says in §9, is possible ouly "if my intuition contains nothing but the form of sensibility, antedating in my mind all the actual impressions through which I am affected by objects."49 Kant's problem in this passage is thus how intuition can be "of' an object not yet given. I am not sure why, in Hintikka's view, there should be a problem. Hintikka grants that on my view there is but holds it to be insoluble. I hold that the claim that a priori intuition "contains nothing but the form of sensibility" is the main idea of Kant's solution. In §9 of the Prolegomena, as in many other passages, his stress is on the thesis that this form's being a condition of my tuind on the intuition of objects makes a priori knowledge of such objects possible. The question how knowledge resting on the form of sensibility is intuitive is prior. My own writing on mathematical intuition undertakes to offer a model of how forms may be given in intuition which are yet the forms of objects not yet given. Kant, in giving geometrical examples, must have thought this tolerably clear. - An obstacle to complete clarity is the absence in Kant's philosophy of a theory of mathematical objects. Of course Kant's writing on mathematics abounds in what would ordinarily be read as references to mathematical objects. Sometimes he seems to commit himself to them in a more philosophical way, as when he says that '7 + 5 = 12' is a singular proposition (A 164 = B 205), and that we can "give it [the concept of a triangle1 an object wholly a priori, that is, construct it" (A 223 = B 271). In sections V and VI of this essay I assumed that Kant was concerned with mathematical objects of the usual kind. This was an incautious assumption. The concept of object in tenns of which construction gives the concept of triangle an object is not Kant's primary one, and indeed in that passage Kant partly takes away what he has just given in saying that the triangle is "only the form of an object." Kant never talks explicitly of the existence of mathematical objects; existence for hini ~eems to be concrete existence; this is quite explicit in its schematization as actuality. He seems to decline to attribute existence to mathematical objects at all.50 But what, then, are a priori intuitions, as singular representations, intuitions of! Mathematics contains a priori knowledge, which is knowledge of objects in the full-blooded sense, that is of the objects given in empirical intuition. Sometimes, as in Prolegomena §8, Kant talks as if these objects were the

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Howell's view of Kant's conception of intuition would serve to save Hintikka's analysis of Kant's philosophy of mathematics as a whole. But my own criticisms in section IV above of the rest of Hintikka's analysis would not be affected by the assumption that Kant's basic conception of intuition is as Howell claims. However, Hintikka directly criticizes my statement that "intuition is thus a source, ultimately the only source, of immediate knowledge of objects" (p. 112 above). Appealing to remarks of Kant in §§8-9 of the Prolegomena, he argues (p. 343) that my view would lead to a perverse conception of Kant's problem concerning mathematics. His view appears to be that my reading of the immediacy condition would make a priori intuitions "tuisnomers." He relies here on the puzzle expressed in §8 of the Prolegomena: For the question now is, "How is it possible to intuit anything a priori?" An intuition is such a representation as would immediately depend upon the presence of the object. Hence it seems impossible to intuit spontaneously a priori, because intuition would in that event have to take place without either a fonner or a present object to refer to, and in consequence could not be intuition. 48

I should remark that the claim abont intuition made in the second sentence fits Hintikka's conception even less well than mine. But it is clear that Kant is

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objects of a priori intuition. But how can a priori representations have such reference and still be singular? This is a difficulty conunon to my own and to Hintikka's view of intuition. A picture common to us is of pure intuitions as analogous to free variables, with predicates attached to them representing the concepts they "construct" If we are not to import into Kant the "mathematical-objects picture," then it seems we have to take the range of these variables to be empirical objects. Then a mathematical argument cannot, strictly speaking, establish existence, What pays the role of mathematical existence in Kant is constructibility. The most plausible reconstruction of Kant would be, in my view, to take constructibility of a concept to be a kind of possible existence of a (nonabstract) object falling under the concept Kant's view would then be in line with the modal interpretation of quantifiers discussed elsewhere in this volume [i.e. Mathematics in Philosophy], in particular, in connection with intuition, in Essay 1, section III, However, I say "a kind of possible existence" because it cannot be possible existence in the precise sense Kant gives to those words, The difficulty is not with existence but with possibility, which for Kant is what we might call real possibility, The fact that the concept of triangle can be constructed makes it appear that we can see the possibility of a triangle "from its concept in itself' (A 223 = B 271). What makes the construction itself fall short of showing possibility is that possibility involves agreement with the formal conditions of experience with respect to intuition and concepts, that is, not ouly the forms of intuition but the categories. 51 Thus in order to see the possibility of a triangle, we have to observe that space is a formal condition of outer experience and that "the formative synthesis through which we construct a triangle in imagination is precisely the same 'as that which we exercise in the apprehension of an appearance, in making for ourselves an empirical concept of it" (A 224 = B 271). This, of course, repeats the considerations advanced in the Axioms of Intuition. The "objective reality" in the full sense of mathematical concepts seems to be a proposition not of mathematics but of philosophy, To return to the original issue about the immediacy criterion for being an intuition: I would say that Kant as I interpreted him is of interest to the philosopher today, My line of interpretation parallels Kant's most important influence on rwentieth-century foundational research, through Brouwer and Hilbert. A concept of intuition like that I attributed to Kant is of interest in its own right 52 Hintikka, too, could defend his interpretations partly on the grounds of philosophical interest I must also grant a certain justice to the closing remarks of his conunent on my paper (pp, 344-345). Indeed, a central

problem for Kant was "why the knowledge so obtained [in mathematics] can be applied to all experience a priori and with certainty." There is an important aspect of Kant's answer to that question that I hardly touched on, namely the argument in the Analytic for the claim that mathematics necessarily applies to the objects of empirical intuition. However, I do not find an analysis of that argument in Hintikka's writings either. I do not think that either of us has undertaken the task of constructing a truly Kantian explanation of the a priori character of mathematics. In my own case, I doubt that such an explanation could be given without appealing to Kantian transcendental psychology.53

75

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NOTES 1 An earlier version of this paper was written while the author was George Santayana Fellow in Philosophy, Harvard University, and presented' in lectures in 1964 to the University of Amsterdam and the Netherlands Society for Logic and the Philosophy of Science. I am indebted to 1. J. de longh, J. F. Staal, and G. A. van der Wal for helpful comments. I am also grateful to -Jaakko Hintikka for sending me two unpublished papers on the subject of this paper. 2 I.e., 1st edition, p. 320, 2nd edition, pp. 376-377. All passages are quoted in the translation of Norman Kemp Smith (London, 1929) with slight modifications. Other translations from German are my own. Translations of Kant's Inaugural Dissertation are by John Handyside, in Kanf s Inaugural Dissertation and Early Writings on Space, Chicago and London, 1929. 3 Kants Gesammelte Schriften, ed. by the Prussian Academy of Sciences, Berlin, 1902-1956, IX, 91. This edition will be referred to as "Ale" 4 "Kant's 'new method of thought' and his theory of mathematics", p. 130. Hintikka argues in detail for this thesis in a paper, "On Kant's notion of intuition (Anschauwzg)," in Terence Penelhum and J. H. MacIntosh (eds), Kant's First Critique (Bemont, Calif.. 1969). The same idea seems to underlie the analysis of Kant's theory of mathematical proof in E. W. Beth, "Uber Lockes 'allgemeines Dreieck'" in Kant-Studien 48 (1956-1957), 361-380. 5 "It is a mere tautology to speak of general or common concepts" (Logic 1, Ak. IX 91). 6 One might attribute to Kant the view that there are no such representations. The classification Kant makes in A 320 = B 376 and Logic §1 is of Erkenntnisse, which Kemp Smith translates as "modes of knowledge" but which in many contexts would be more accurately though inelegantly translated as "pieces of knowledge." Then the relation of a representation to its object is that thrOQgh which one can know its object. and it might be held that intuition in the full sense is the only singular representation which can p~vide such knowledge. This view would have the perhaps embarrassing consequence that an object which is not in some way perceived is not really known as an individual. 7 Cf. the examples of "truths of reason" given by Leibniz, Nouveaux Essais, IV, ii, § 1. 8 Ibid. IV, vii, §IO. 9 Arirhmetlk und Kombinatorik bel Kant (Diss. Freiburg 1934), eniarged ed. Berlin 1972; Kant's

76 Metaphysics and Theory

CHARLES PARSONS

of Science.

Manchester, 1953. ch. i; Klassische Ontologie der Zahl,

Kant-Studien Erganzungsheft 70, KGln. 1956, § 12. 10 Neither Leibniz nor Schultz seems to mention the fact that in order to prove fonnulae involv-

ing multiplication, such as '2 . 3 = 6', one also needs instances of the distributive law. 11 1. Aus mehrern gegebenen gleichartigem Quantis dUTCh ihre successive Verknupfung den Begriff von einem Quanta zu erzeugen, d. i. sie in ein Ganzes zu verwandeln. 2. Ein jedes gegebenes Quantum, urn so viet, als man will, d.i. sie ins Unendliche zu vergrossern, und zu vermindern (Prufung, I, 221). 12 Ak, X 554-558. 13 Arithmetik und Kombinatorik bel Kant, p. 64-5. 14 C. J. Gerhardt (ed.), Leibnizens mathematissche Schriften, Halle, 1849-1963, vn 78. Leibniz gives a definition of addition from which he claims commutativity follows immediately. One could read his argument as deriving the commutativity of addition from the commutativity of settheoretic union. 15 "UberLockes 'allgemeines Dreieck'." 16 "Kant's 'new method of thought'," ''On Kant's concept of intuition," also "Are logical truths analytic?" Philosophical Review 74 (1965), 178-203, "Kant on the mathematical method," This volume, 21-42. J7 It ought to be remarked that while no doubt the distinction which Kant makes between axioms and postulates derives historically from that of "common notions" and postulates in Euclid, Kant's distinction does not correspond exactly to Euclid's. Euclid's division is between more general principles and specifically geometrical ones. For Kant postulates are "immediately certain practical judgments," the action involved is construction, and their purport is that a construction of a certain kind can be carned out. The role they play is thus that of existence axioms. Euclid's common notions are all of a type which Kant asserted to be analytic propositions (A 164 = B 204, B 17), while axioms proper must be synthetic. 18 op. cit. p. 365. 19 Cf. W. V. Quine, Methods of Logic, revised ed., New York, 1959, §28. 20 "Kant's 'new method of thought' ," p. 130, also "Kant on the mathematical method." The texts are A 717-B 745, A 734-B 762. 21 In "Are logical truths analytic?" Hintikka develops a distinction between analytic and synthetic according to which some logical truths are synthetic. He suggests that the logical truths which are analytic according to this criterion are roughly those which Kant would have regarded as analytic. It follows, however, that in some of the arguments which according to Beth and Hintikka involve for Kant an appeal to intuition, the conditional of their premises and conclusion is analytic. In particular, this is true of the example that Beth works out in detail in Uber Lockes 'allgemeines Dreieck'." §7. In order to be applied to mathematical examples like Kant's, Hintikka's criterion would have to be extended to languages containing function symbols. The way: 'of doing this which seems to me most in the spirit of Hintikka's definition has some anomalous consequences. See also Logic. Language-Games. and Information, Oxford, 1973, chs. 6-9. 22 Beth, op. cit. p. 363. 23 In fact, (1) is analytic according to the criterion of "Are logical truths analytic?" (see note 21 above). However, according to another criterion which might be more in the spirit of Kant, to consider as synthetic a conditional whose proof involves formulae of degree higher than its antecedent, (1) is synthetic. Hintikka takes account of this in "Are logical truths tautologies?" by making an additional distinction between analytic and synthetic arguments, such that in the rele-

KANT'S PHILOSOPHY OF ARITHMETIC

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vant sense the argument from the conjuncts of the antecedent of (1) as premises to its consequent as conclusion is synthetic. 24 Cf. Hao Wang, "Process and existence in mathematics," in Y. Bar-Hillel, E. I. J. Poznanski, M. O. Rabin, A. Robinson (eds.), Essays in the Foundations of Mathematics, dedicated to A. A. FraenkeI. Jerusalem, 1961,328-351, p. 335. 25 "Frege's theory of number" (1965), in Mathematics in Philosophy, N.Y. 1983. 26 An interesting intermediate case is how constructive proofs as the object of intuitionist mathematics could be interpreted from a Kantian point of view. According to Kant as I interpret him, certain empirical constructions can function as paradigms so as to establish necessary truths because of the intention or meaning associated with them. Intuitionism would require that our insight into th~se meanings be sufficient not only to establish laws directly relating to objects in space and tme but also to establish laws concerning the intentions as ''mental constructions." I leave open the question of whether this is possible from Kant's point of view or not. 27 Ak.1I272-301. 28 Die Mathematik betrachtet in ihren Aufiosungen, Beweisen, und Folgerungen das Allgemeine unter den Zeichen in concreto, die Weltweisheit das Allgemeine durch die Zeichen in abstracto (Erste Betrachtung, §2. heading, Ak. IT 278). 29 Denn da die Zeichen der Mathematik sinliche Erkenntnismittel sind, so kann man mit derselben Zuversicht, wie man dessen, was man mit Augen sieht, versichert ist, auch wissen, class man keinen Begriff aus der Acht gelassen, class erne jede einzelne Vergleichung nach leichten Regeln geschehen sei U.S.w. Wobei die Aufmerk-samkeit dadurch sehr erleichterr wird, dass sie nicht die Sachen selbst in ihrer allgemeine Vorstellung, sondern die Zeichen in ihrer einzelnen Erkenntnis, die da sinnlich ist, zu gedenken hat. (Dritte Betrachtung, § I, Ak. II 291). 30 But cf. the following: in der Geometrie, wo die Zeichen mit den bezeichneten Sachen iiberden eine Ahulichkeit haben, ist daher diese Evidenz noch grosser, obgleich in der Buchstabenrechnung die Gewissheit evenso zuwerliissig ist. (Ibid., 292). 3'1 [This "general accord" now seems to me quite tenuous, and Manley Thompson is probably right in saying that the synthesis required for analytic judgments is clearly distinguishable from that in mathematical judgments ("Singular Tenns and Intuitions in Kant's Epistemology,» p. 342, n, 23). Nonetheless, a reply to the main point, that logic is not entirely independent of intuitive construction, would demand a lot of Kant's distinction between intuitive and discursive proofs, as is clear from Thompson's interesting discussion Qfthis distinction (ibid., pp. 340-342). His interpretation implies rather extreme limits on the role of logic in mathematics. This raises a doubt whether Kant's distinction is in the end tenable.] 32 One might say that it is possible to construct tokens. The sense of possibility in which this is possible is, however, derivative from the mathematical possibility of constructing types (or mathematical existence of the types). For we declare that the tokens are possible either directly on the basis of the mathematical construction, or physically on the basis of a theory in which a mathematical space which is in some way infinite is an ingredient. 33 Cf. my «Infinity and Kant's conception of the 'possibility of experience'" in Mathematics in Philosophy (1964). 34 This does not imply that there is an upper limit on the numbers which can be individually represented, once we admit notations for faster-growing functions than the successor function. This happens already in Arabic numeral notation. The number-"I,OOO,OOO,OOO,OOO, if written in '0' and'S' notation with four symbols per centimeter, would extend from the earth to the mOOIL That there is such an upper limit follows, of course, from the assumption that human history

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must come to an end after a finite time.

35 "Kantian Intuitions", Inquiry 15 (1972) 341-345. In this Postscript this paper is cited merely by page number. 36 Particularly since the singularity criterion itself is not in dispute, I should emphasize that my own understanding of its importance owes much to Hintikka's writings and conversation. I should belatedly thank him for explaining his ideas to me some years before they were published in English.

37 Kemp Smith translates einzeln in this passage as "single." The translation "singular" fits its use in Logic, §l (see above, p. 112), where it is paired with the Latin singularis. Thompson suggests that in the Critique passage it may mean that an intuition is a single occurrence. ("Singular Tenns and Intuitions, in Kant's Epistemology" this volume. p. 105, n. 13.) If intuitions are thus in effect events. that would rule out Hintikka's interpretation (though not the view of Robert Howell discussed below). Though this seems to me to agree with Kant's characteristic way of speaking about institutions, the point is not so clear as to be a serious argument in the present dispute. 38 "Institution. Synthesis, and Individuation in the Critique of Pure Reason," NaUs 7 (1973), 207-232, p. 210. 39 A 71 = B 96; Logic, §21, n. I (Ak., IX, 102). 40 LogiC §l, Note 2 (Ak., IX, 91). This point is discussed at some length by Thompson, "Singular Terms and Intuitions," pp. 83-84. 41 In the discussion of "the black man" in a letter to J. S. Beck, July 3, 1792 (Ak., XI, 347); cf. Howell, "Intuition, Synthesis, and Individuation," p. 210. Other examples that can be given, such as the Idea of God, involve either Ideas of Reason or mathematics and might therefore be regarded as exceptional. 42 "Intuition, Synthesis, and Individuation," pp. 210-211. The distinction between what he takes to be the definition and his further interpretation is not explicit in the paper; here I rely on his clarification of his views in a recent letter. 43 Ibid., p. 215. A somewhat similar picture is presented in Thompson, "Singular Terms and Intuitions." However, Thompson rejects Howell's view that certain demonstratives can be the linguistic expression of intuitions; see Thompson, pp. 91-92, and Howell's reply, p. 232. Thompson holds that a Kantian "canonical language" would be virtually without singular tenns (ibid., p. 10 I). 44 In a discussion in March 1983, after this Postscript has been written; Hintikka stated that Howell's interpretation of immediacy was the view Hintikka had maintained all along. 45 But obviously one needs to look carefully at how Kant and his contemporaries actually viewed the relation of concepts and their "marks." This matter was explored by my student Alan Shamoon in his dissertation, "Kant's Logic," Columbia University 1979. 46 Cf. the contrast on the following page (A 733 = B 761) between discursive and intuitive principles. 47 lowe this last observation to Manley Thompson. However, the direct-reference view might itself suggest an assumption such as I attribute to Kant; compare the connection between direct reference and sense-perception in the philosophy of Bertrand Russell. 48 Kant's Theory of Science, p. 50, n. 15. Chapter 2 of this book is a very clear and instructive discussion of Kant's philosophy of mathematics, with expositions both of Hintikka's and my own views. Howell seems not to be free of the same misunderstanding; see "Intuition, Synthesis, and . Individuation," pp. 210-211.

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49 Ak., IV, 281-282, translation from Beck's edition. 50 Ak., IV, 282, emphasis Kant's. 51 "But in mathematical problems there is no question of this Ithe conditions Wider which the perception of a thing can belong to possible experience], nor indeed of existence at all, but only of the properties of the objects in themselves, solely in so far as these properties are connected with the concept of the objects" (A 719 = B 747). This passage is instructively discussed by Thompson, "Singular Tenus and Intuitions," pp. 98-101. It is largely through this paper that I became aware of the difficulties faced by my own views about mathematical objects and existence in Kant. Brittan comments on the same passage (Kant's Theory of Science, p. 66), but he seems to me to misread it in saying that "mathematical problems" have to do with real possibility. That seems to me to neglect the clear statement of A 223-4 = B 271 that construction does not establish such possibility, and even the remark in the present passage that in mathematical problems there is "no question" of the conditions under which perception of a thing can belong to possible experience. The point is subtle because Kant holds that mathematics is about really possible objects and that this can be established. But it is not mathematics that establishes it. 52 A 218 = B 265. Cf. the whole discussion of possibility this statement introduces. Thompson (p. 106, n. 21) sees a difficulty with a modalist interpretation of how Kant might deal with reference to mathematical objects in Kant's distinction between demonstrations and discursive proofs. I am not sure I understand what the difficulty is, but it is not evident that the attenuated version of modalism suggested for Kant in the text is more exempt from it than the direct version. But cf. note 30 above. 5~ See my «Ontology and Mathematics" in Mathematics in Philosophy (Ithaca, Cornell University Press, 1983), section III; also "Mathematical Intuition, Proceedings of the Aristotelian Society N.S. (1979-1980)." 54 I wish to thank Robert Howell and Manley Thompson for their helpful comments on an earlier version of this Postscript.

MANLEY THOMPSON

SINGULAR TERMS AND INTUITIONS IN KANT'S EPISTEMOLOGY

As Kant explains his concept of intuition, it seems clear that a representation must satisfy two conditions in order to be an intuition: it must be singular and it must relate immediately to its object. Charles Parsons has referred to these as "the singularity condition" and "the immediacy condition," and has doubts that within Kant's philosophy they boil down to the same thing.! Jaakko Hintikka, on the contrary, maintains that in Kant the immediacy condition is only the singularity condition stated in another way, so that "Kant's notion of intuition is not vety far from what we would call a singular term."2 Both Parsons and Hintikka focus their attention primarily on Kant's philosophy of mathematics, and Parsons holds that "Hintikka's theory really stands or falls on the interpretation of the role of intuition in mathematics" (p. 46). In this paper I want to emphasize Kant's treattnent of empirical judgments and. the role that intuition plays in them, as I believe (and will tty to show in the course of my discussion) that this context was the primary one for Kant. As textual evidence for his theory, Hintikka cites a passage from the first Critique in the first section of the Dialectic where Kant gives a classification of representations . ... an objective perception is knowledge (cognitio). This is either intuition or concept (intuitus vel conceptus). The former relates immediately to the object and is single, the latter refers to

it

mediately by means of a feature which several things may have in common [A 320 B 376-77]3

=

After quoting this passage along with a remark from the Prolegomena §8, "Intuition is a representation such as would depend directly on the presence of the object," Hintikka concludes: These quotations . .. show what for Kant was the alternative to an immediate relation to objects. It was a reference to objects by means of certain marks or characteristics which may be shared by several objects, i.e., a reference to objects by means of general concepts. Hence, another way of saying that Anschauungen have an immediate relation to their objects is to say that they are particular ideas or
81 Carl J. Posy (ed.), Kant's Philosophy of Mathematics. 81-107. "'" , nn'} Vl ..... n. A~~rln_;,.. 17 .. 1-.'; .. 1."". .. 17".;",uul;", th", Nuthurlnnrh

r

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MANLEY THOMPSON

To refer immediately, then, is to refer to an object by means of marks or characteristics that it alone possesses. The reference, in other words, is wrique, On the other hand, to refer mediately is to refer to an object along with any other object with which it may share certain marks or characteristics, Hintikka does not explicitly rephrase his interpretation in this way, but it seems to me clearly a correct restatement of what he has said. The trouble is that as thus restated it comes into conflict with other passages in the first Critique. Hintikka is well aware of the conflict and offers an explanation for it. We must, he says, "distinguish two different, though not unrelated, levels of Kant's philosophy of mathematics" (p. 49). He calls the first level "the preliminary theory" and claims it is historically prior. It survives in the Critique mostly in the Methodology. The second level comprises ''the full theory" and the groundwork for it is given in the Aesthetic. The "main difference" between the two levels is "that in the preliminary theory no connection is assumed between intuitions and sensibility while in the full theory Kant tries to show that all intuitions are sensible" (pp. 49-50). If any trace of the view that there is no connection between intuitions and sensibility influences Kant's philosophy of mathematics as he expounds it at places in the Critique, one tuight expect to find a similar influence in some of his remarks about empirical judgments. The passage Hintikka quotes from the first section of the Dialectic seems to be a case in point. This passage as quoted above begins with the second clause of a sentence. The full sentence is: "A perception which relates solely to the subject as the modification of its state is sensation (sensatio), an objective perception is knowledge (cognitio)." Intuitions are then placed with concepts as a species of objective perception in opposition to sensations, which by implication are subjective perceptions. The next sentence after those quoted is, 'The concept is either an empirical or a pure concept." The distinction between pure and empirical intuitions is not mentioned, but there is no reason to assume that its introduction would in any way qualify the separation of empirical intuitions from sensations. One may thus interpret the passage as holding that empirical intuitions differ from empirical concepts, not in their connection with sensations, but only in their mode of reference to an object. They refer to an object immediately because they somehow mark characteristics peculiar to that object alone, while empirical concepts can refer to the same object only mediately because they mark only characteristics that the object may share with indefinitely many other objects. But intuitions then appear to be simply concepts of a special sort - individual or singular rather than general concepts. While Kant defines concepts as general representations his practice does

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not always, as Hintikka observes, seem to conform to this definition. The concepts of space and time and concepts in arithmetic, like the number 12 and the sum of 7 and 5, appear to be obvious exceptions. But these concepts involve the notion of pure intuition and Kant's philosophy of mathematics, and 1, want to postpone consideration of them until later. Another apparent exception in the Critique occurs in remarks Kant makes on the singular judgment, just after he has presented his table of judgments. He speaks of "the concept of the subject" of a singular judgment and contrasts it with a general concept (A 7 I = B 96). It is clear that he is thinking of a concept in this case that is represented in language by a proper narue.' In the Dialectic (A 322 = B 378) he gives 'Caius is mortal' as an exaruple of a syllogistic premise, and iu his Logic (§21),5 he uses the sarue sentence as an exaruple of a singular judgment. There is no doubt that the judgment in this case is empirical. The question is whether the subject term 'Caius' represents an empirical intuition = an empirical concept, or only one of these regarded as distinct from the other. The answer afforded by Kant's Logic is clearly the second alternative. There is no such thing as an intuition = a concept, although the denial here is made without reference to the claim that all intuition is sensuous. Intuitions contrast with concepts as singular with general or discursive representations (§ I), and again as completely determinate coguitions (Erkenntnisse) with cognitions whose logical determination can never be taken as final (§ 15, Note). It would seem to follow that a proper narue as the subject of a singular judgment must represent an intuition '# a concept. but this conclusion is rendered doubtful by other remarks Kant makes in the Logic. He declares, "It is a mere tautology to speak of concepts as general or universal; - a mistake6 that rests on an improper division of concepts into general, particular, and singular. Not concepts themselves but only their use [Gebrauch] can be so divided" (§I, Note 2). This stat~ment suggests that the subject term in 'Caius is mortal' may not represent an intuition but rather a concept used to form the subject of a singular judgment. Unfortunately, Kant never makes clear exactly what he means by giving a concept a singular use. As we noted above, in the passage in the Critique where he speaks of "the concept of the subject" of a singular judgment he contrasts it with a general concept. The contrast, which is repeated in the Logic (§21), is that a "singular" concept has "no extension at all" (applies to at most one object). The contrast is mentioned only to show that the difference in the concepts does not affect the logical point that singular and universal judgments are to be treated alike "in their use according to logical form"

I,'I!

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1.11

(der logischen Form nach im Gebrauche) (§21). The predicate concept of a singular judgment applies without exception to everything the subject concept applies to, just as if the latter had an extension and were thus a general concept. In the passage in the Critique (A 71 = B 96) Kant says only that singular and universal judgments can be treated alike "in the employment of judgments in syllogisms" rather than simply that they "are to be treated alike in their use according to logical fonn." Even this restricted claim is not entirely true of the traditional logic Kant professed to accept,7 and both claims concern the use of judgments, not concepts. It remains unclear how a general representation becomes singular and lacks extension in virtue of its use as the subject concept of a singular judgment. On this point, it seems to me that Kant's failure to say much about the contrast between the singular and the universal use of concepts is offset by' the great deal he says in the Critique about the contrast between intuitions and concepts. From what he says about this second contrast it becomes clear, I think, what he must say about the singular use of concepts, although he could not say it with the meager fonnal logic at his disposaL I want first to propose an interpretation of what he means by 'intuition' before I mm to the question of the singular use of concepts, though I will begin with a general remark about concepts in this use. II

Ii

Iii II II

II

When a concept is used as the subject of a singular judgment it purports to represent exactly one object. But then in order to accomplish what it purports to accomplish in this use it must satisfy two conditions: it must represent an object and do so by means of characteristics that this object alone possesses. In other words, it must satisfy an existence condition and a uniqueness condition. It is of course fundamental to Kant's doctrine in the Critique that satisfaction of the existence condition can never be achieved through concepts alone. For it is never self-contradictory to deny that a concept represents an object, and all that can be achieved through concepts alone is the establishment of claims that cannot be denied without self-contradiction. Intuitions then enter the picture in connection with the conditions through which concepts are given objects and become capable in their use of satisfying the existence condition. Intuitions "without concepts," Kant says, "are blind." It is necessary "to make our intuitions intelligible, that is, to bring them under concepts" (A 51 = B 75). These remarks of course presuppose that all intuitions are sensuous,

and prima facie they seem in conflict with the passage from A 320 = B 376-77 quoted by Hintikka, according to which both intuitions and concepts are species of objective perception, knowledge or cognition. 8 It would seem that as forms of cognition intuitions should be intelligible in themselves, independently of other fonns of cognition. This way of regarding the passage seems all the more plausible in light of the fact that the first sentence begins with a clause in which sensation is separated from intuition and characterized as a perception "which relates solely to the subject as a modification of its state." Another way of regarding the passage emerges when we consider its context. Kant is making a plea for the preservation of the "original [Platonic] meaning" of the word 'idea', and he is addressing philosophers in general, or at least "those who have the interests of philosophy at heart" He urges that 'idea' not become "one of those expressions which are commonly used to indicate any and every species of representation." He continues that there is "no lack of terms suitable for each kind of representation, that we should thus needlessly encroach upon the province of any of them." He then offers a "serial arrangement" (Stu/enleiter) of the various species of the genus representation. But then the fact that his arrangement shows only a separation and , no connection between sensation and intuition is not to be taken as implying that no such connection is recognized in his own philosophy. His aim is

simply to urge acceptance of a common philosophical vocabulary, and how his own philosophy is expressed in terms of this vocabulary is irrelevant. As Hintikka points out, "there was no inseparable connection between the notion of intuition and the [sic] sensibility in the writings of most seventeenth- and eighteenth-century philosophers" (pp. 40-41), and Kant himself "insists that the connection between the two is not a mere logical consequence of the

definition of intuition" (p. 45), It is just a fact about us human beings (uns Menschen) that we can know individual objects immediately only with the aid of sense perception and not by intellect alone. This fact does not affect the definition of intuition as a coguitive representation satisfying an immediacy and a singularity condition. It is thus not a fact that would be reflected in an arrangement of common philosophical terms interrelated solely by their definitions. This second interpretation of the passage seems to me more plausible but it leaves us with a problem of how, when the tenns are used to express Kant's philosophy, intuitions can be cognitive representations and yet blind without concepts, It is clear, I think, that the problem should be approached via Kant's distinction between pure and empirical intuitions. It is siguificant that

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Ii,l

I

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this distinction does not occur in his Stu/enleiter, although the distinction between pure and empirical concepts does. A tenn that could be used appropriately to distinguish pure from empirical concepts existed in the common philosophical vocabulary, viz., notion; but there was no such tenn for distinguishing pure from empirical intuitions. The latter distinction was original with Kant and could not be clarified by an appeal to existing terminology. Actually, the existing tenn that comes closest to Kant's 'pure intuition' is again the tenn 'notion'. This was Berkeley's tenn for the intellectual apprehension of active spirits and relations. Kant notes in the Appendix to the Prolegomena, in a footnote criticizing Berkeley, that "idealism, .. (as can already be seen from Plato) inferred from our knowledge a priori (even that of geometry) another intuition than that of the senses (namely intellectual intuition), because it never occurred to anyone that the senses should also inIDit a priori.,$ Kant does not mention that Berkeley's term for this intellectual intuition was 'notion', since presumably he still felt that associating the tenn with his own 'pure concept' and 'idea' was more appropriate than with 'pure intuition' . One way, then, of reconciling Kant's remark that intuitions without concepts are blind with his remark that intuitions are a species of cognitive representation is to take the one remark as applying to empirical and the other to pure intuitions. An empirical intuition by itself is blind insofar as it is simply an amorphous sensory manifold requiring unification through concepts in order to become intelligible. A pure intuition, on the other hand, is quite different, As Kant says in the Aesthetic, space "is not a concept .. , but a pure intuition"; it "is essentially one"; and "the original representation of space is an a priori intuition, not a concept" (A 24-25 = B 39-40), But then this original representation cannot be blind in the sense requiring unification through concepts to make it intelligible. It would seem to be intelligible by itself as the representation of a single unified space, and therefore to be a coguitive representation. Whatever may be said for this reconciliation, it hardly fits the Stu/enleiter passage. Kant there speaks of intuitions and concepts as cognitive representations of the same object; an intuition represents an object immediately as a single object, and a concept represents the same object mediately by means of features it may share with other objects. There is no indication that the object in this case must be an object of pure intuition like space rather than an ordinary empirical object. A more protuising line of reconciliation suggests itself when we note that the uuification of a sensory manifold through concepts is possible ouly

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because the manifold is given under the fonns of sensibility, space and time. In other words, the manifold is originally intuited as a spatiotemporal something. While this means that there is originally a pure as well as an empirical intu~tion, it does not mean that there is an intuition of space and time themselves as fonns of sensibility. It means rather that it is only to the extent that the original empirical intuition is accompanied by a pure intuition of spatiotemporal unity that a subsequent pure intuition of space and time as the fonns of empirical intuition is possible. An empirical intuition, then, is not merely an immediate apprehension of an amorphous sensory manifold. It is also an immediate apprehension of this manifold as a spatiotemporal something, and in this respect it is a coguitive representation, an objective perception, even though it is blind in the sense that it is not an apprehension of an object characterized by sensory qualities. JO The distinction Kant draws between sensation and intuition in presenting his Stufenldter may thus be interpreted as follows. As relating "solely to the subject as a modification of its state" a perception is sensation. Insofar as it also constitutes an apprehension by the subject of something given in space and time - of a spatiotemporal something - it is an empirical intuition. an objective perception, and a species of cognitive representation. The unification of the sensory manifold through concepts so that the objective perception becomes the perception of an object characterized by sensory qualities involves Kant's difficult notion of synthesis. He says in his remarks introducing his table of categories that "Synthesis in general ... is the mere result of the power of imagination, a blind but indispensable function of the soul" (A 78 = B 103). It is only when this synthesis is brought to concepts that "we first obtain knowledge [cognition] properly so called" (die Erkenntnis in eigentlicher Bedeutung). I do not propose to consider Kant's acc9unt of synthesis. My only concern here is to note that his phrase ''first obtain knowledge properly so called" suggests that he was prepared to recoguize something which occurs prior to conceptual synthesis and which may be called "knowledge" in some sense. It is thus possible to regard an empirical intuition as in a sense knowledge, and to speak of it as blind without concepts because ouly as uuified through concepts can it become knowledge properly so called. The other half of the sentence in which Kant proclaims that intuitions without concepts are blind, declares that concepts without intuitions are empty. Intuitions thus supply objects for concepts, and without objects concepts cannot perfonn their epistemological function, whether the function is that which arises from their general, particular, or singular use.

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An obvious suggestion at this point is to regard concepts for Kant as corresponding to open sentences and their different uses as corresponding to different quantifications of open sentences. I want to explore this suggestion in the light of what we have just noted about concepts in relation to intuitions. III

In the opening section (§ 15) of the second-edition Transcendental Deduction Kant introduces the notion of unity of concept. He says we must look for this unity "in that which itself contains the ground of the unity of diverse concepts in judgment, and therefore of the possibility of the understanding, even as regards its logical employment" (B 131). It is "qualitative" unity and is prior to the unity identified by the category of unity. It is the unity produced by the formation of a single concept as opposed to the unity produced by subsuming many objects under a single concept - in other words, the unity of a concept as opposed to the uniting of objects by subsumption under a concept. Unity of concept is determined simply by logical possibility, by the absence of contradiction. Two concepts F and G combine to form a single concept F that is G if, and only if, it is logically possible that something is both F and G. Kant adds in a footnote on B 131 that it makes no difference here whether or not one concept "can be analytically thought through the other." To illustrate by an example he uses elsewhere, "body" and "extended," no less than "body" and "heavy," combine to form a single concept. That "body" also combines with "weightless" but not with "exten-

sionless" distinguishes "All bodies are heavy" from "All bodies are extended" as a synthetic from an analytic judgment. But this distinction makes no difference in the conceptual unity of "heavy body" and "extended body." Both alike are concepts of a logically possible object. We might try to sum up Kant's position here with the statement that a concept is never just a thought but always a thought of a logically possible object, and while it is logically possible to think contradictory thoughts, it is not logically possible to have the concept of a logically impossible object. The trouble with this statement is that it leaves qnite unclear the relation between a concept and a judgment. In the Logic Kant says, "A judgment is the representation of the unity of the consciousness of diverse representations, or the representation of the interrelation of the same insofar as they

constitute a concept" (§ 17). It would thus seem that a self-contradictory judgment is not properly a judgment at all, any more than a concept of a logically impossible object is properly a concept. In the Critique at the end of the

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Analytic Kant pairs self-contradictory concepts with what he calls "Empty object without concept, nihil negativum" (A 292 = B 348). But then in the two sentences immediately following he still speaks of a self-contradictory copcept as a kind of concept - an empty one that "cancels itself." By analogy, a self-contradictory judgment would be a false one that cancels itself. When the pSyChological and logical factors here are disentangled, concepts and judgments in their logical meaning seem very much like open and closed sentences, respectively.l! Unity of concept seems to have its logical counter-

part in the consistency of an open sentence, and likewise unity of judgment in the consistency of a closed sentence. Yet a good deal that Kant says fails to fit this view. He speaks repeatedly of predication as a relation between two concepts rather than between a concept and an object subsumed under it. While he speaks of concepts as "predicates of possible judgments" (A 69 = B 94), it is clear from his example that he thinks they become predicates of actual judgments by being predicated of other concepts. Yet he does not speak of concepts also as subjects of possible judgments. A reason for his not doing so appears in the Logic (§ II) under the heading "Highest Genus and Lowest Species." When concepts are arranged in a series

in the order of their generality, the series terminates in a highest or most generic concept. But there is no termination at the other end, since a lower

and more specific concept than any already given is always logically possible. Thus no concept is ever an ultimate subject - a representation that relates

immediately to objects and can never be a predicate of further representations. 12

Kant's position here preserves only half of the traditional Aristotelian doctrine according to which the series terminates in an infima species as well as a summum genus. With no infima species it would seem a fortiori that there are

no singular concepts and that Kant's position requires him to regard intuitions, which are singular representations by definition, as the subjects of singular judgments. But I do not think that this is a plausible interpretation of Kant. I want to argue, first, that his doctrine of intuitions precludes his taking them as what the subject terms of singular judgments represent, and, secondly, that his denial of infima species is not incompatible with his accepting definite descriptions as the subjects of sentences representing singular judgments and thus accounting for a singular use of concepts. In the course of

arguing for these two points I will be presenting a case for the interpretation of Kant that regards concepts as corresponding to open sentences and judgments to closed sentences.

(i) That intuitions are the subjects of singular judgments is suggested by

~'I

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the passage I mentioned in Section I above from the Logic (§ IS, Note), in which Kant contrasts intuitions with concepts as completely determinate cognitions with cognitions whose logical detennination can never be taken as final. The force of this suggestion, however, is considerably weakened when we note that in the passage Kant says "completely determinate cognitions can be given only as intuitions, not as concepts" (my italics). He does not say that

temporally distinct from that of every other and is thus, simply as intuition without regard for concepts, different from every other intuition,13 although as falling under a certain concept it is, along with any other intuition that falls under the concept, that concept's intuition. But then by 'an object's intuition' Kant has to mean any intuition falling under a concept that applies to that object. For apart from concepts an object is merely a spatiotemporal some-

such cognitions ever are given. and according to the Critique they never are

thing correlated with a given intuition, so that restricted to our intuitive cogni-

because "to know a thing completely, we must know every possible [predicate], and must determine it thereby, either affirmatively or negatively. The complete determination is thus a concept, which, in its totality, can never be exhibited in concreto" (A 573 = B 601). In other words, a completely determinate cognition of an object x is not merely knowledge that enables us to affirm every pair of contradictory predicates disjunctively of x, but also knowledge that enables us to say which of the disjuncts is true of x. We know in abstracto (and trivially) that any object is either P or not-P, but we have knowledge of a particular object only to the extent that we know which of these is the case. And we can have knowledge of this second sort only when an object is given to us in conctreto, in intuition. According to Kant's doctrine in the Critique, an object thus given is always given under the forms of our intuition, space and time. The object is cognized intuitively as a spa-

tiotemporal something and is determined as an individual only by its spatiotemporal location. The further determination that yields knowledge "properly so called" is conceptual and is achieved as the manifold of intuition is synthesised under concepts. Since this intellectual (conceptual) synthesis is limited by the forms of space and time under which the manifold is given, the determination achieved is never complete. For in order to have a completely determinate cognition of an object we must detennine it in concreto in rela-

tion to every logically possible predicate (e.g" having an immortal soul) and not merely to those an object can have as spatiotemporal. Completely determinate cognitions can be given only as intuitions, but the intuitions must be

tions we cannot speak of different intuitions of the sarne object. Hence, if we take proper names as linguistic representations solely of our intuitions we

cannot speak of applying and reapplying the sarne narne. When we speak thus we treat names as conceptual rather than intuitive representations. Names can

be applied, reapplied, and misapplied; so can concepts, but not intuitions. Demonstrative pronouns in contrast to proper names seem to be more plausible candidates for linguistic representations of Kantian intuitions. We do

not ask what 'this' narnes apart from any context of its use, and when 'this' is used to refer to the sarne object on different occasions we usually do not say that the sarne narne has been applied and reapplied. It hardly seems outlandish to say that 'this' in a sentence of the form 'This is F' represents merely a spatio-temporal something to which the concept F is applied. Of - course, in its normal use such a sentence may be said to have in effect the form 'This G is F', since a further predicate is Wlderstood from the context as

accompanying the pronoun. But then 'this' in the phrase 'this G' seems still to represent a spatio-temporal something to which a concept is applied, and the phrase itself, moreover, seems to represent a singular use of the concept. We thus have the suggestion that a concept acquires a singular use when it is

represented by a general term used as a substantive and preceded by a qemonstrative adjective. But this suggestion at best covers only some of the cases. The function of

'this' can be performed by a demonstrative adverb when the definite article precedes the noun, as in 'The G here is F'. The important question for our purposes i,s._ then whether 'this' in 'this G' represents an intuitive cognition not represented in 'the G here'. If the cognition is not the sarne it must be

intellectual - they must yield a determination of an object in relation to every logically possible predicate. But such intuitions, according to Kant, are impossible for us. It should now be clear that we cannot take proper narnes as linguistic rep-

cursive) apprehension of existence and uniqueness. But I see no way to

resentations solely of our intuitions. We can speak of having the same intuition on different occasions only with respect to the concepts an intuition falls Wlder. When Kant speaks of a concept as having "its empirical intuition in experience" (B 129), he must mean that any intuition falling under the concept is that concept's intuition. For each intuition is given on an occasion

defend this distinction. In both cases, the existence and uniqueness required are for the predicate substituted in the expression, and the uniqueness is limited in the sarne way. What is claimed is not that there is one and only one G absolutely, but that there is one and only one where the speaker points (or is to be understood as pointing) when he utrers 'this' just before he utters 'G'

because 'this ... ' as opposed to 'the ... here' represents an intuitive (nondis-

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or when he comes to 'here' in his ntterance of 'the G here'. Any gesture of pointing appropriate in one case is appropriate in the other. 'This G is F' and 'The G here is F' are thus both equivalent in quantificationallogic to 'There is at least and at most one object x such that x is G and is here, and x is F'. Demonstrative pronouns are then no more and no less likely than demonstrative adverbs as candidates for linguistic representations of Kantian intuitions. But demonstratives in their adverbial fonn are conspicuously representations of concepts. As such, they represent an object of reference in a specific spatial and/or temporal relation to the speaker. With 'the G here' a speaker represents an object as being G, as spatially andlor temporally in a certain relation to himself, and as being the only G standing in that relation. While Kant holds that we must intuit all "outer" objects under the forms of space and time, he does not hold that we must represent them conceptually in any specific spatiotemporal relation to ourselves beyond that indicated by the word 'outer'. The specific relations we represent by demonstratives are conceptual determinations of what we intuit, just as much as any other relation to which we give linguistic representation. For a plausible interpretation of Kant's position on the subjects of singnlar judgments we must tum to the way he can treat definite descriptions. His doctrine of intuitions precludes his taking them as the subjects and as being represented by either proper names or demonstrative pronouns. (ii) An infima species and a completely determinate cognition, as Kant considers them, differ in that with the former the only determination claimed is for objects of possible experience, and hence for spatiotemporal objects. We noted above that a completely detenninate cognition is impossible for us because it requires determination of an object with respect to every logically

possible predicate and not merely to those an object can have as spatiotemporaJ. The impossibility of archieving an infima species, on the other hand, results, not from the limitations imposed by our forms of intuition, but from the discursive character of our empirical knowledge. Two objects falling under the same infima species would be distinguishable only as objects of intuitive cognition, Le., only spatiotemporally. But since we can detennine a

species only by forming a general (discursive) representation which, as general, may always contain less general representations under it, we are

"I'

never entitled to say we have reached an infima species - a limit to all differences except those of spatiotemporallocation. In the Logic (§ 11, Note), Kant says that in practice we establish infima species by convention when we agree to ignore the possibiliry of further differences (nickt tiefer zu gehen). In the Critique, he emphasizes the other side and proclaims "a transcendental law of

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specification, which, as a principle of reason, leads us to seek always for further -differences, and to suspect their existence even when the senses are unable to disclose them" (A 657 ~ B 685). This principle of reason is purely regulative and in no sense constitutive. It does not give us knowledge that there are further differences, as the causal principle, in contrast, gives us knowledge that there is a cause for every event. The law of specification is compatible with there being no further differences and with our actually achieving infima species. The important point is not that the achievement is contingent but that its existential presupposition is also contingent. While any empirical determination of the cause of a given event is contingent, the existential presupposition of this achievement, viz.,

that the cause exists, is not (Kant thinks) also contingent. The achievement of an infima species F presupposes that there is at least one object x such that x is an F, and that there is no object y such that y is an F and not specifically identical with x where 'specifically identical with x' is short for 'differing from x at most in spatiotemporallocation'. In the special case where x and y are numerically and not just specifically identical, F constitutes a definite description rather than an infima species. A uniqueness claim is no less contingent than a claim of specific identity, and neither is less contingent than an existence claim. But there is the difference

that while we know that an object for a concept exists when we are given an appropriate empirical intuition, we know that an object is specifically or uniquely identified by a concept only to the extent that we are not given an intuition incompatible with the assumption of such identification. An existenc:e, claim thus differs from a claim of specific identity or uniqueness as a positive from a negative existence condition. We can never finally decide the identification or uniqueness claims by the intuitions we are given; in practice we resort to convention, but in theory we must acknowledge the regulative

principle that the possibility of further differences should always be kept open. Kant's position is thus compatible with giving concepts a singular use

through the addition of a uniqueness claim. It is incompatible only with the assumption of intellectual representations that refer to objects immediately rather than mediately through intuitions. That singnlar terms satisfy a uniqueness condition only by convention may seem incompatible with any claim to knowledge of an objective spatiotemporal world of individuals. But we must

remember that for Kant we know an existing object as singular, as one individual object, not through a concept that represents it immediately as unique, but through an intuition that represents it under the forms of space and time.

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Its singularity consists in its being intnitively cognized as a spatiotemporal something, and not in its being conceptually represented as an object uniquely characterized by a certain concept. We cannot combine the two by forming the concept of being at such and such a specified place at a certain time, and then subsuming an object under this concept. Such a procedure would merely locate an object spatiotemporally relative to other objects, and knowing it as singular would then depend on knowing these other objects as singular. Either they are thus known through intnitions and not concepts or knowledge of them as singular depends on knowledge of still other spatiotemporal objects as singular. Ultimately there is no escape from the conclusion that singularity is known intuitively and not conceptually. Kant's position here, of course, depends on his doctrine that space and time are a priori forms of sensibility and that all intuition is sensuous. To know the singUlarity of a spatiotemporal object intnitively is not the same as to know its location in an empirically determined co-ordinate system. Knowledge of the latter sort requires empirical concepts that prescribe how the origin of the system and distances in space and time are to be detertnined. The system as thus determined empirically is not known as unique; it is always logically possible that the system and everything in it is duplicated elsewhere in space. Yet the system itself as a spatiotemporal object is known intnitively as singular, and one knows a priori that its duplication elsewhere in space would be a different spatiotemporal object, which would itself be singular. There is no need to assume reidentifiable particulars that preserve the unity of space and time. The assumption of such particulars is needed for the unity of empirically determined systems but not for the unity of space and time themselves. The latter as a priori forms of sensibility are intuited as singular by pure intuition,14 and their singularity logically entails the singularity of every object intnited as spatiotemporal. For an object so intuited not to be singular it would have to be intuited as spatiotemporal only in the sense of being something related to different parts of a single space and time without itself having any location in this single space and time, just as a concept (general term) is understood to be true of objects in different parts of a single space and time although it has no spatiotemporallocation itself. But it is not logically possible to intuit an object in this way, given Kant's initial premise that space and time are a priori forms of sensibility and that all intuition is sensuous. A singularity and a uniqueness condition differ, then, in that the former is satisfied through intnitions alone while the latter requires concepts. But Kant's immediacy condition and his singularity condition tum out to be

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equivalent - whenever one is satisfied so is the other. The objects immediately represented by intuitions are also always represented as singular. While this conclusion agrees with Hintikka's that the immediacy condition is only the singularity condition stated differently, it disagrees with his conclusion in taking it as essential to Kant's position that all intuition is senSllOUS and in denying that a singular term constitutes a linguistic representation of an intuition. If we ask what does constitute a linguistic representation of an intuition,

the answer, I think, is simply that for Kant an intnitive representation has no place in language, where all representation is discursive. In language we presuppose intuitive and create discursive representations. When I judge that an

object before me is such and such, I presuppose an intuitive representation (cognition) that I subsume under certain empirical concepts. I cannot give my intuition linguistic representation through a phrase such as 'spatiotemporal

something'. Though I may say that my intuition is the innnediate apprehension of an object simply as a spatiotemporal something, I cannot take the phrase 'spatiotemporal something' as representing my intuition. Such a phrase applies to the object of any intuition and is thus a general (conceptual, discursive) representation. No discursive representation can have the immedi-

. acy and singUlarity of an intuition. There are only discursive representations of existence and Wliqueness claims, both of which claims, as we have noted, are contingent and subject to correction by further experience. The contingency of existence claims runs even deeper in Kant's epistemology. Not only is the nonemptiness of concepts contingent on the occurrence of intuitions, but also whether or not a given intuition is a veridical perception is contingent on the occurrence of other intuitions. Not "every intuitive

representation [anschauliche VorstellungJ of outer things involves the existence of these things, for their representation can very well be merely the product of imagination (as in dreams and delusions)" (B 278). Empirical knowledge thus remains discursive even while it appeals to intuitions.

"Whether this or that supposed experience be not purely imaginary, must be ascertained from its special determinations, and through its congruence with the criteria of all real experience" (B 279). An intuitive representation as cognition or knowledge is contingent on the truth of an existence claim and is thus never incorrigible. Some intuitions are "merely the reproduction of pre-

vious outer perceptions." Concepts are related to objects mediately through intuitions, but the immediacy of the latter in relation to objects is subject to correction by concepts in the form of discursively represented criteria of empirical knowledge (the analogies of experience).

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The form of predication in Kant's general logic is represented by'S is P', and, as we have noted, he regarded predication as a relation between concepts and not between a concept and an object. One is tempted to think that had Kant known quantificational logic he would have recognized 'Fx' as representing the form of predication and would have seen that the corresponding relation in his transcendental logic (epistemology) is that of a concept to an object. It may be objected that 'Fx' does not provide a subject-predicate relation that fits Kant's category of substance and accident, which arises from the relation of'S' to 'P'. But as Kant himself observed, since all affirmative categorical propositions convert, the traditional'S' is 'P' leaves it logically undetermined which concept has the role of subject (B 128-29). His reply that the role is determined when one of the concepts is "brought under the category of substance" is prima facie circular, since the category is supposed to be obtained in the first place by a metaphysical deduction from the subjectpredicate form of judgment. His contention that the "function" of judgments with this form is "that of the relation of subject to predicate" is not enough to remove the circularity if one must still apply the category to detennine which concept is the subject. The concept Kant wants, as he phrases it, is the concept of "that which must always be considered as subject and never as mere predicate"; but this is the concept represented by 'x' in 'Fx' rather than 'S' in'S is P'. The general logic required by Kant's transcendental logic is thus at least first order quantificationallogic plus identity but minus proper names or other singular terms that are in principle eliminable. 15 A proper name represents an empirical concept used with an existence and a uniqueness claim and is hence eliminable in favor of a predicate expression. 'Caius is marta!' has the force of 'The man who ... is mortal' just as 'This is F' has the force of 'The ... that is here is F'. Since a uniqueness claim as well as an existence claim is always contingent, its addition can never ttansforrn a predicate expression into a logical subject - into a representation of a concept that must always be thought as subject. I want to conclude with a few observations relating this interpretation of Kant's transcendental logic to his doctrine of pure intuition and his philosophy of mathematics. IV

Kant's concepts of space and time must, it seems to me, on any interpretation of the Critique be recognized as unique. The words 'space' and 'time' are as close as anything in his philosophy to genuinely singular terms, since they are

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expressions that represent exactly one object, in each case a particular form of sensibility, immediately and not through some further representation required to satisfy an existence claim. I 6 Yet the two are nonetheless essentially different. Tiroe is the form of inner sense, and since "inner sense yields no shape [Gestalt], we endeavor to make up for this want by analogies." From the properties of a line taken as representing the sequence of time, we reason to "all the properties of time, with this one exception, that while the parts of a line are simultaneous, the parts of time are always successive" (A 33 = B 50). It would thus seem that 'time' is not quite a qenuinely singular term for Kant since it can represent its object ouly with the aid of spatial (outer) intuition, 17 although in this case there is hardly appeal to afurther representation. The appeal to spatial representations seems necessary for a repre-

sentation of time at all; it is not an appeal to an intuitive representation which is required in order to satisfy an existence claim for what has already been represented. (Can we be said to have a "gestaltless" intuition of time that is not a representation of it?) I am inclined to suspect that this difference in the representation of space and time remarked in the Aesthetic may underlie the distinction Kant draws in the Methodology between "ostensive" and "symbolic" constructions. He says that "in algebra by means of a symbolic construction, just as in geometry by means of an ostensive construction (the geometrical construction of the objects themselves), we succeed in arriving at results which discursive knowledge could never have reached by means of mere concepts" (A 717 = B 745, my italics). The basic point here, I take it, is that in geometry we actually construct geometrical objects, triangles, circles, etc., while in algebra we do not actually construct the objects themselves, numerical relations and numbers, but only spatial representations of them, written fonnulas

and numerals. What is missing in the latter that makes them ouly symbolic and not ostensive constructions is also what is missing in the representation of the sequence of time by a line, viz., the successive existence of the parts. In the Analytic Kant speaks of number as "simply the unity of the synthesis of the manifold of a homogeneous intuition in general, a uuity due to my generating time itself in the apprehension of the intuition" (A 143 = B 182). But as this generation of time itself is missing and must be supplied by us in the representation of the sequence of time by a line, so likewise in the representation of a number by a numeral and in the representation of a numerical relation by an algebraic formula containing only function signs and schematic letrers 18 representing numerals. I will not attempt to argue for this suggestion of a connection between

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symbolic construction and the representation of time. But it should be noted that the snggestion, if sound, provides evidence against Hintikka' s contention that the philosophy of mathematics expressed in the Methodology reflects a preliminary theory in which intuition is not associated with sensibility. I want now to consider briefly how the suggested connection would affect the role of expressions like 'the number 12' and 'the sum of 7 and 5', and how it bears on the question of the existence of numbers. The numeral '12' is the twelfth alter '0' in our decimal system; also, the second in the second decade, the second alter '10'. The numeral possesses these ordinal properties in virtue of our system of numerals, but the properties also correspond to (represent) arithmetical properties of the number 12. This number is the twelfth after 0 and is the sum of 10 and 2. A numeral is thus a symbolic construction of a number, and in arithmetic by further symbolic constructions from numerals (by calculation) we discover further arithmetical properties of numbers, just as in geometry by further ostensive constructions from figures we discover further geometrical properties of figures. The further symbolic constructions in arithmetic require function signs like 'sum of' and 'product of'. These signs do not represent spatial relations, like the descriptive signs 'diagonal of' and 'radius of'; they represent rather a certain operation to be performed. 'The sum of 7 and 5' represents the operation of sucessively adding seven units to 0, and then five more; 'the product of 7 and 5' represents the operation of adding seven units to ofive times. In his letter to Johann Schultz, November 25, 1788, Kant speaks as though the above account is purely subjective, applying to the way we actually operate in arithmetic and not to the way we think of numbers objectively. He writes, "The science of numbers, notwithstanding the succession that every

construction of quantity requires, is a pure intellectual synthesis, which we represent to ourselves in thought."19 But when we do arithmetic we do not represent numbers in this way; we employ symbolic constructions and "succeed in arriving at results which discursive knowledge could never have reached by mere concepts" (A 717 = B 745). Numbers represented only in thought by pure intellectual synthesis would be numbers Platonistically conceived as objects existing prior to and independently of our symbolic constructions of them. But according to the doctrine of the Critique, we can never be justified in saying that such numbers exist, since existence requires that an object be given in intuition. While mathematical constructions, whether ostensive or symbolic, provide objects for mathematical concepts and thus answer existence questions within

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mathematics, they do not answer existence questions absolutely. What appear as existence questions in mathematics are really questions of constructibility and not existence. Such at least seems to be the point of the following remarks in the Methodology. There is indeed a transcendental synthesis [framed] from concepts alone, a synthesis with which the philosopher is alone competent to deal; but it relates only to a thing in general, as defining the conditions under which the perception of it can belong to possible experience. But in mathematical problems there is no question of this, nor indeed of existence at all. but only of the properties of the objects in themselves. [that is to say], solely in so far as these properties are connected with the concepts of the objects [A 719 = B 747. my italics].20

What exists is thus ouly what is given through perception (empirical intuition) and hence in space and time; what is given a priori through pure intuition is either a form of intuition or the construction under such a form of a

mathematical (quantitative) concept, and in neither case does what is thus given exist. "Mathematical concepts are not, therefore, by themselves knowledge, except on the supposition that there are things which allow of being presented to us only in accordance with the form of that pure sensible intuition" (B 147; if. A 223-24 = B 271-72). That mathematics yields knowledge of existence is thus established in transcendental philosophy and not in mathematics proper. It should be noted that in the last quotation from the Methodology, mathematical problems are said to concern properties of objects in themselves (an sich selbst) and not of objects as they appear to us. If mathematical objects were themselves existing objects rather than particular modifications of the forms under which existing objects can be intuited, the rationalist's dream of a priori knowledge of existences in themselves would be realized. Construction in pure intuition resembles intellectual intuition, which Kant characterizes in the Aesthetic as intuition that is "selfactivity" (B 68); the difference that still defeats the rationalist's claim is that the constructions are not themselves existing objects.

In sentences representing empirical judgments, variables of quantification would for Kant range over objects of possible experience; and it would seem that with his view of existence it is only when variables are thus restricted that the particular quantifier should be read as 'there exists'. If quantifiers are

used in mathematics the parricular quantifier tuight be given the force of 'there is constructible', but it is hard to see what use this phrase would have for Kant. Constructibility in the physical sense of 'not too long to write down' is obviously irrelevant. One must see (intuit) the constructibility, as one sees that a straight line can be prolonged indefinitely and that the number of digits in an Arabic numeral can be indefinitely large. There is no need for a

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special symbol by which one represents discursively (asserts that there is) the coustructibility one must intuit. The situation may seem different in the negative case, where a proposed construction is impossible. Kant remarks in the Methodology, with regard to an apagogical proof, "In respect of convincing power, it has ... this advantage over direct proofs, that contradiction always caries with it more clearness of representation than the best connection, and so approximates to the intuitional certainty of a demonstration" (A 789 = B 817, my italics). He remarks three pages later that "In mathematics ... apagogical proofs have their true place" (A 792 = B 820). His remarks here presuppose his distinction earlier in the Methodology between demonstrations and acroamatic (discursive) proofs. Demonstrations are possible only in mathematics, and, "as the term indicates," they "proceed in and through the intuition of the object" (A 735 = B 763). Their representation is thus diagrammatic, rather than verbal as in discursive proofs. In an apagogical proof showing the impossibility of a proposed construction, it is shown that if the construction is possible (e.g., the construction of a rational number = -.12), the construction of a contradiction is also possible (e.g., the construction of an integer both odd and even). The proof thus "approximates to the intuitioual certainty of a demonstration" because its only discursive element is an intellectual grasp of two concepts as contradictory. There is no further question about COflstructibility, and no need to represent discursively (assert) that there is no such construction. Variables of quantification thus seem to have no place in mathematical constructions. Since an ostensive construction in geometry represents a par-

1,1

II~

:

ticular ouly as falling nnder a concept and thus as in no way differing from any other such particular, it "must in its representation express universal validity for all possible intuitions which fall under the same concept" (A 713 = B 741). But the formulas representing symbolic constructions in algebra are said to be true of all nnmbers, of some, or of exactly one, and they are thns open sentences and contain quantifiable variables. I am not sure how we

should express the Kantian position here, though it seems clear he would want to say that the truth of the formulas is to be grasped with the intuitional certainty of a demonstration and not the conceptual certainty of a discursive proof. But then the formulas have to be taken as diagrams rather than open sentences, and it is for this reason that I spoke of them above as containing schematic letters rather than variables. Also, a use of quantifiers. in this case would strongly suggest that numbers exist and thus tend to confuse constructibility with existence.2 !

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This confusion becomes inevitable if numerals and other expressions in

demonstrations such as '7 + 5', are taken as singular terms representing numbers. A numeral is a symbolic construction of a number and not a name of an existing object. If we ask what is a number for Kant, the correct answer,

I think, is essentially the view expressed recently by R.L. Goodstein: "the natural numbers are roles in arithmetic, the number two for instance being the role which arithmetic assigns to the numeral 2, and so on."22 In the letter to

Schultz, Kant speaks of '3 + 4' as the "expression of a problem" (Ausdruck eines Problems), although when "objectively regarded" it represents a concept "objectively identical" with the concept of the number 7. But such objective (discursive) representations have no place in the couduct of arithmetic. They represent only concepts (thoughts), and to take these concepts as singular representations of existing objects would be to revert to Platonism

and intellectual intuition. The pure intuition by which mathematical demonstration proceeds, while a form of self-activity, issues in an immediate apprehension, not of existing objects, but of constructibility under forms of sensibility. That it also leads to mediate knowledge of existing objects can ouly be proved in transcendental philosophy by proving that all existing objects are given nnder these forms of sensibility. In sum, Kant's philosophy is virtually one that is technically without singular terms. The closest thing to such a term is the word 'space', and to a lesser extent 'time', although neither of these, any more than numerals and other apparently singular terms in mathematics, represent existing objects. Proper names as subjects of sentences representing empirical judgments are them-

selves representations of empirical concepts which, as concepts, are general, although their use is singular. Adequate representation of the universal, particular, and singular use of coucepts in empirical judgments requires a general logic that contains at least first order quantificational logic plus identity, a logic in which the form of predication is 'Fx' and not'S is r.23 Empirical intuitions are the data for knowledge of existing objects but they have, in separation from concepts, no proper discursive (linguistic) representation. When this interpretation of Kant is applied to his philosophy of mathematics, the result makes this part of his philosophy seem more restricted and more intuitionist than Hintikka and even Parsons want to make it.

But however his philosophy of mathematics is interpreted, I am convinced that the interpretation must take into acconnt the role of singular terms and intuitions in his general epistemology. In ignoring this role one can readily come to vety nnKantian conclusions about the existence of mathematical objects. 24

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concept. A determination is possible only in transcendental logic where concepts are related to objects of experience; a concept is then determined as

Of the many remarks in this paper that I now find in serious need of revision and further development perhaps the most important is my "remark at the close of section III that in the general logic required by Kant's transcendental logic the form of predication is that represented by the 'Fx' of quantificationallogic rather than the'S is P' of Aristotelian logic. The remark as it stands conflicts with Kant's distinction between general and transcendental logic. General logic as Kant viewed it "abstracts from all content of knowledge, that is, from all relation of knowledge to the object, and considers only the relation of any knowledge to other knowledge; that is, it treats of the form of thought in general" (A55 = B79). In first order quantificational logic 'x' in 'Fx' represents any individual object to which the predicate expression 'F' applies. But if general logic abstracts from all relation of knowledge (cognition) to object, 'Fx' cannot represent the form of predication in general logic. It is only in transcendental logic, where inmitions are introduced to supply content to concepts (to relate knowledge to objects), that 'Fx' can be taken as representing the form of predication. Instead of saying that in the general logic required by Kant's transcendental logic the form of predication is that represented by 'Fx', I would now say that this becomes the form when general logic is reconsidered and developed from Kant's transcendental logic rather than taken as developed in abstraction from any relation of knowledge to object. There is a relation of knowledge to object for Kant only as sensible intuitions are synthesized under categories -

"concepts of an object in general, by means of which the intuition of an " object is regarded as determined in respect of one of the logical functions of judgment" (B 128). These logical functions, as distinguished in general logic in abstraction from any relation of knowledge to object, provide the "clue" to the discovery of all categories, given the thesis of transcendental logic that the synthesis of concept and intuition yielding a concept of an object in general is always an act corresponding to a logical function of judgment (cf. A66 = B91; A79 = Bl04-5). But if the distinction of these functions in general logic requires a form of predication that presupposes relation of knowledge to object, there is no essential difference between distinguishing logical functions and distinguishing categories and the claim that the functions provide only the clue to the discovery of categories is undermined. The form'S is P' represents predication as a relation between two concepts, a substance and an accident. But as the conceptual relation is convertible it provides only a clue and not the determination of a definite categoreal

substance when "its empirical intuition in experience must always be consid-

ered as subject and never as mere predicate" (BI29). With this detenuination the form of predication becomes 'Fx', and when general logic is reconsidered with this form the procedure leads to quantificationallogic as developed since Kant. This fact, however, does not warrant the conclusion that the general logic required by Kant's transcendental logic is at least first order quantificationallogic plus identiry. With general logic restricted to the concepmal relations Kant distinguished as determiuing logical functions of judgment, it is by no means always clear how these functions correspond to the categories he lists in his transcendental logic. The correspondence of subject to substance for example, requires his

distinction in transcendental logic between mathematical and dynamical categories, and his correlation of the universal judgment with the category of unity and the singular with totality is plausible at all only with a restricted view of the extensions of concepts in general logic. I discuss these issues in my "Unity, Plurality, and Totality as Kantian Categories," The Monist 72, 1989. The University a/Chicago NOTES 1 "Kant's Philosophy of Arithmetic," Philosophy, Science, and Method, eds. S. Morgenbesse:r, P. Suppes, M. White. New York: St. Martin's Press, 1969; reprinted in this volume pp. 43-79. All page references to Parsons are to this volume. 2 "On Kant's Notion of Intuition (Anschauung)," The First Critique: Reflections on Kant's Critique of Pure Reason, eds. T. Penelhum, J. MacIntosh. Belmont. Calif.: Wadsworth Publishing Co., 1969, pp. 38-53. All page references to Hintikka are to this work. But see below, end of footnote 23. .3 I follow the usual practice in giving references to the Critique of Pure Reason: A for the 1st ed. and B for the 2nd. The numbers are always page numbers in the respective editions. When only one reference is given, the passage does not appear in the other edition. All quotations from the Critique are from the translation by Norman Kemp Smith, London, 1929. 4 That is, when one expresses the singular judgment in language one uses a sentence with a proper name as subject. The name then represents the singular concept in that it does the job in the sentence, viz., the job of representing exactly one object, that the concept does in the judgment. While the name thus represents both a concept and an object, this point should occasion no confusion if one remembers that the name represents the concept only in the sense that it represents in the sentence what the concept represents in the judgment. Whether or not one can make judgments without sentences to express them is beside the point, since I am concerned only with

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judgments that are so expressed.

5 References to Kant's Logic are to lasche's edition of Immanuel Kant's Logile Ein Handhuch zu Vorlesungen, 1800. The translations are mine. 6 The mistake (Fehler) Kant is referring to must be that of failing to see that it is a mere tautology. I make considerable use of this passage in my interpretation of Kant. Parsons (n. 5) quotes only the first half of the sentence and does not mention the mistake or the distinction between a division of concepts themselves and a division of their use. 7 The difference between singular and universal judgments can be ignored only with syllogisms in the four direct moods of the first figure. The difference must be recognized in the conversions

required for the reduction of indirect moods and figures. 'Some mortal is Caius' converts to 'Caius is mortal', but an I does not convert to an A proposition. Of course, Kant regarded the doctrine of indirect moods and figures as false sophistry or a needless subtlety to be removed from traditional logic. Cf his Die falsche Spitzjindigkeit der vier syllogistischen Figuren elwiesen, 1762; also, B viii, B 141 n. 8 Intuitions are also referred to as species of knowledge or cognition (Erkenntnisse) in the passage from the Logic (§ 15, Note) mentioned above. I will tum to this passage in Section m. 9 Kant: Prolegomena, tr. by P. G. Lucas, Manchester: Manchester University Press, 1953, p.146, n. \0 In a footnote in the B version of the transcendental deduction (B 160), Kant seems to acknowledge this point. The note begins: Space, represented as object (as we are required to do in geometry), contains more than mere form of intuition; it also contains combination of the manifold, given according to the form of sensibility, in an intuitive representation, so that the form of intUition gives only a manifold, theformal intuition gives unity of representation [italics in Smith's tr.J. By "space represented as object" Kant must mean space as determined by geometrical figures and not space as a form of sensibility. Through the accompanying formal (pure) intuition the manifold of empirical intuition is unified as a spatial something identified by a particular geometrical figure. That this fonnal intuition is a cognitive representation prior to conceptual synthesis is brought out by the rest of the note, which continues: In the Aesthetic I have treated this unity as belonging merely to sensibility, simply in order to emphasize that it precedes any concept, although, as a matter of fact, it presupposes a synthesis which does not belong to the senses but through which all concepts of space and time are possible. For since by its means (in that the understanding detennines the sensibility) space and time are first given as intuitions, the unity of this a priori intuiiion belongs to space and time, and not to the concept of the understanding (cf. § 24) [first italics mine1· From the reference to § 24, we must conclude that the "presupposed" synthesis here is the figurative synthesis explained in that section and distinguished from the "intellectual synthesis" which "is thought in the mere category in respect of the manifold of an intuition in general" (B 151). See, note 20. II In this remark I pass over the complications introduced by Kant's view of modalities. "In a judgment [Urtheil] the relation of different representations in the unity of consciousness is thought merely as problematic; in a proposition [San], on the contrary, it is thought as asserroric. A problematic proposition is a contradictio in adjecto" (Logic, § 3D, Note 3). It might seem that a closed sentence corresponds more to a Kantian proposition, while a judgment is formed from a proposition by the addition of a modal operator. But Kant continues in the next sentence, "Before

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I have a proposition I must first make a judgment:; and I judge concerning many things about which I have not reached a decision, which I must do as soon as I have determined a judgment as a proposition." This suggests that a proposition is formed from a judgment, which in itself is always problematic. 12 Kant admitted that the second of these conditions is satisfied in the case of one, but only one, concept - that of an ens realissimum. "For only in this one case is a concept of a thing - a concept which is in itself universal - completely detennined in and through itself, and known as the representation of an individual" (A 576 = B 604). But this concept is an idea of reason and Kant argues at length that it cannot represent an existing object. 13 In the sentence from A 320 = B 376-77 quoted by Hintikka, Smith's translation of einzeln as "single" seems plausible, although in the Logic, § I the word is paired with the Latin singularis and must mean "singular." An intuition thus differs from a concept both in being single (a single occurrence) and in being a singular representation (a representation of but one object). 14 P. F. Strawson chooses to ignore this point when he claims that for Kant "the preservation of the unity of space from judgment to judgment requires the persistence and re-identifiability of occupants of space." The Bounds of Sense, London: Methuen, 1966, p. 83. By taking reidentifiable parriculars as basic, Strawson is forced to maintain an identificatory as distinct . from a referential function for singular terms that is foreign to Kant. See Note 15. 1:5 Kant is thus in agreement with Quine and opposed to Strawson on the issue of whether singular terms have an identificatory function that cannot be dispensed with in favor of general tenus and variables of quantification. "In Word and Object a conspicuous effect of regimentation is that a predication of the form 'Pa', with identificatory singular tenn in the 'a' place, goes over ipto the symmetrical fonn '(3x) (Px· Ax)'. A uniqueness clause regarding 'A' may still be added, but the identificatory work of singular tenns has lapsed." CW. V. Quine, "Replies: To Strawson," Synthese 19 (1968-69), p. 293; reprinted in Words and Objections, Donald Davidson and Jaakko Hintikka, eds. Dordrecht: D. Reidel Publishing Co., 1969, p. 321. Strawson needs the separateness of the identificatory function because he assumes that reidentifiable particulars are required to preserve the unity of space and time. (See, note 14.) But then does Quine's position require a Kantian-like assumption that the unity of space and time (or space-time) is independent of the ontological unity of particular spatiotemporal objects? 16 There is of course no existence claim to be satisfied since space and time are not existing objects but the forms of intuition under which all existing objects are experienced. 17 This point, implied in the Aesthetic, is stated explicitly in the Analytic. We "cannot obtain for ourselves a representation of time, which is not an object of outer intuition, except under the image of a line, which we draw" (B 156). 18 I come later to my reason for saying "schematic letters" here rather than ''variables.'' Por this use of 'schematic letter', cf W. V. Quine, From a Logical Point of View, Cambridge, Mass.: Harvard University Press, 1953 and 1961; Set Theory and Its Logic, Cambridge, Mass.: Harvard University Press, 1963, revised ed., 1969. 19 Kant: Philosophical Correspondence, ed. & trans. by A. Zweig, Chicago: University of Chicago Press, 1967, p. 130. The letter appears in Kant's Gesammelte Schriften, ed. by the Prussian Academy of Sciences, vol. X. 20 The transcendental synthesis framed from concepts alone referred to in this passage is undoubtedly the "intellectual synthesis" referred to in § 24 of the B version of the transcendental deduction (B 151; See, note 10). This intellectual synthesis relates to "objects of intuition in general, whether that intuition be our own or any other, provided only it be sensible" (B 150). The forms of our intuition, space and time, are required for a preconceptual representation of an

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existing object unified merely as a spatiotemporal something, and this representation is achieved by the figurative synthesis of § 24. The intellectual synthesis provides a necessary but not a sufficient condition for a priori knowledge of existing objects, i.e., the knowledge that the object is singular, the figurative synthesis, which is simply the former carried out in productive imagination rather than concepts, provides a necessary and sufficient condition, i.e., the knowledge that the object as singular is a spatiotemporal something. The philosopher alone is competent to deal with the intellectual synthesis, since qua philosopher he is concerned with a priori knowledge of existence and is thus led to see the necessity of the figurative synthesis. Qua mathematician, one is concerned only with a priori knowledge of the constructibility of concepts and not with a priori knowledge of the application of concepts to existence. If as mathematician one begins with the intellectual synthesis, one can succeed only in thinking about numbers as objects, and this is not according to Kant, the way one does arithmetic or the way one relates arithmetic to reality. 21 Hilary Putnam in "Mathematics Without Foundations" (Journal of Philosophy, LXN (1967), pp. 5-22) distinguishes two views of mathematics, an "object" view, according to which "mathematics is wholly extensional, but presupposes a vast totality of eternal objects"; and a "modal" view, according to which "mathematics has no special objects of its own, but simply tells us what follows from what" (p. 11). The second view seems Kantian in spirit in that it affords a way of interpreting quantifiers in mathematical fonnulations that does not require the existence of mathematical objects. But the extent to which this view can actually be accommodated in Kant's philosophy of mathematics depends on what we make of Kant's distinction between demonstrations and discursive proofs and his claim that logical necessity (what follows from what) is grasped intuitively in the one case and conceptually in the other. I do not see how the side of Kant's philosophy of mathematics can be squared with Putnam's modal view, and I am inclined to think that there is no way in which one can subscribe to much of Kant's position without restricting arithmetic to what can be expressed in free-variable fonnulas. 22 R.L. Goodstein, Essays in the Philosophy of Mathematics. Leicester: Leicester Vniv. Press, 1965. p. 72. 23 Since general logic so conceived contains symbolic constructions and demonstrations, it would seem at least in this sense to be something Kant would have to regard as a branch of mathematics. One may be tempted to take Kant's reconstructed position to be that general (fonnal) logic uses symbolic constructions and demonstrations to determine valid forms of discursive proof, and is thus a special case of mathematics. This is essentially the position of C.S. Peirce. But this position conflicts with Kant's view that logical possibility is purely conceptual and constructibility intuitive. Parsons holds that the priority of constructibility "is a consequence to be accepted and is even in general accord with Kant's statements that synthesis Wlderlies even the possibility of analytic judgments" (p. 67). But as I read Kant the underlying synthesis in this case is purely conceptual and not intuitive. The shift from'S is P' to 'Fx' as the fonn of predication achieves conceptual clarity, not intuitive certainty; and the role of quantifiers is no less discursive than the role Kant assigned to the copula. The use of symbolic constructions and demonstrations in quantificationallogic, though far more extensive, is not essentially different from the constructions Kant recognized in general logic - the square of opposition, the use of circles and squares to diagram the relation of Sand P in categorical judgments (cf- Logic §§ 21, 29), and the four figures of the syllogism (though he regarded the latter as needless subtleties). These remarks on the relation between mathematics and logic, and indeed much of what I have said in section IV, are contrary to the reconstruction of Kant's position proposed by Hintikka in papers I have not mentioned, since I did not have space here to comment on them. I

SINGULAR TERMS AND INTUITIONS IN KANT'S EPISTEMOLOGY 107 have thus not done full justice to his position. Cf in particular his «Kant on the Mathematical Method," The Monist. 51 (1967). pp. 352-375; reprinted in this volume, pp.21-42. 24 I am indebted to Charles Parsons for helpful criticisms of an earlier version of this paper.

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l

The heart of Kant's views on the nature of mathematics is his thesis that the

judgments of pure mathematics are synthetic a priori. Kant usually offers this as one thesis, but it is fruitful to regard it as consisting of two separate claims, a metaphysical subthesis and an epistemological subthesis. (KM) The truths of pure mathematics are necessary, although they do (KE)

not owe their truth to the nature of our concepts. The truths of pure mathematics can be known independently of particular bits of experience, although one cannot come to know them through conceptual analysis alone.

For Kant, pure mathematics includes geometry, arithmetic, algebra, kinematics' "pure mechanics," and, I think, analysis. On the basis of the above subtheses Kant proposes to establish a general theory which will provide particular theories for each of the pure disciplines. His aim is to reveal the nature of the propositions of the disciplines and the nature of our knowledge of those propositions.

To understand Kant's theory we shall need to untangle parts of the Aesthetic and reconstruct some of Kant's arguments and theses. Only then shall-we be in a position to see how Kant's view of mathematics errs. I. KANT'S BASIC NOTIONS

I propose to explicate Kant's conception of necessity by appeal to the device of possible worlds, Bizarre as this approach may seem, it will, nonetheless, prove its worth in understanding Kant's views on mathematics. Let us use the tenn "proposition" to convey the sense in which Kant uses "judgment" when

he is interested in the object of judgment rather than the act of jUdging. Propositions may be regarded as what are expressed by suitable declarative sentences and we shall also take them to be truth-bearers. For Kant, every proposition ascribes a property to a subject. If that subject has that property in a world w then the proposition is true in world w. Conversely, if the subject 109 Carl J. Posy (ed.). Kant's Philosophy of Mathematics, 109-13 I.

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lacks that property in w, or if it does not exist in w, then the proposition is false in w, Necessary truths are propositions true in all possible worlds. If a proposition is true in world w, it is true because of a particular fearure of w - namely, the subject's having the appropriate property in w. Thus a proposition which is necessarily true is necessarily true because all possible

priori knowledge is taken for granted and left unexplained. Similarly, Kant feels that the fact that we have knowledge a posteriori of propositions is uncontroversial. In this case an intuition of the world tells us that the predicate applies to the subject. Kant would not deny that most propositions which we know are known on the basis of inference. The two modes of knowledge just discussed concern how we can reach the starting points for our inferences. But if there is no way to generate synthetic conclusions from analytic premises, then some further factor must be brought in to account for a priori knowledge of synthetic propositions. Kant believes that there are only two routes to knowledge. Either Our recognition of the connection between subject and predicate is brought about by unveiling the structure of our concepts, or our recognition requires the aid of intuition. Kant introduces the notion of pure intuition as that inmition which is involved in a priori knowledge of synthetic proposi_tions. We shall examine the various guises of this notion in later sections. Let us now look at a class of propositions whose importance Kant tended to stress. Assume that the proposition that all A's are B's is synthetic a priori. Since Kant believes that the epistemological notion of apriority is coextensive with the metaphysical concept of necessity, he holds the propOSition to be necessary.- Because the proposition is synthetic it is logically possible that there be an A which is not a B. But since it is necessary we cannot experience such A's. Synthetic a priori propositions thus state nonlogical constraints on what we can experience. Kant's theses (KM) and (KE) can now be stated more clearly. (KM) tells us that mathematical truths state nonlogical conditions on our experience. (KE) contends that the principles stating these conditions can be known a priori and Kant wonld also contend that we do know some of them a priori. Since they are not analytic, they cannot be known just by conceptual analysis, but must be known by means of pure intuition. Kant argues for his theses in Section V of the introduction to the Critique. He takes it for granted that the truths of mathematics are necessary.

worlds have a particular feature in common. The connection between true

propositions and features of worlds is crucial for Kant's explanations of versions of (KM). To fix the concept of necessity is to specify the domain of possible worlds. For Kant a possible world is a totality of possible appearances - that is, experiences which could be experiences for us, constituted as we are. (It is assumed that we all have the same constitution). We shall follow Kant in taking the concept of a possible experience to be primitive. 2 Kant has a broad notion of necessity in that some propositions which are logically possible fail to hold in any Kantian possible world. What fearures a world may have are limited by the strucrure of our concepts. Some propositions are true in each world in virtue of this limitation. In a derivative sense, these propositions can be said to be true in virtue of the

structure of our concepts because they owe their truth to particular fearures of that structure. Kant calls these truths "analytic." Kant's second subthesis uses the notion of apriority. Kant believes that we can know some propositions independently of experience. By this he does not mean that we can know these propositions before we have any experience at

all but that, no matter what experiences we have had, provided that those experiences suffice for our acquisition of the concepts involved in an a priori proposition, then we can still know the proposition. 3 Th~ particular stream of perceptions of the world which I have in fact had is quite irtelevant to my a priori knowledge except insofar as it plays a role in acquainting me with the appropriate concepts. The grounds of that knowledge lie elsewhere. We can use Kant's contention that all propositions are of subject-predicate form to sharpen the question of how we know propositions and so to clarify the notion of a priori knowledge. In coming to know a proposition we recognize a connection between subject and predicate. This can happen in various ways. Analytic propositions can be known by uncovering the constitution of the subject and predicate concepts. Kant is confident of our ability to do this. Although he adds the clause that the predicate may have been "thought confusedly" in the subject of an analytic proposition (A7; BIl), he assumes that we can normally analyze our concepts quite easily. He expects, for example, that his reader will quickly agree with his diagnosis of what is and is not contained in

OUT

mathematical concepts. Concepmal analysis as an avenue to a

First of all, it has to be noted that mathematical propositions, strictly so called, are always judgments a priori, not empirical; because they carry with them necesSity, which carmot be derived from experience. If this be demurred to, I am willing to limit my statement to pure mathematics, the very concept of which implies that it does not contain empirical, but only pure a priori knowledge [B15].

The second sentence adds nothing to Kant's argument. If he takes the apriority of pure mathematics as following from the concept of pure mathematics, he should not assume (as he does) that geometry, arithmetic, algebra, and so

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forth fall under this concept. To make that assumption is to pull a substantive thesis out of thin air by a method akin to that of double definition.5 Either we can take pure mathematics to be (by definition) that part of mathematics consisting of necessary truths; or we can take it to be (by definition) geometry, arithmetic, algebra, and so forth. Depending on how we choose our definition we should argue for and justify the other proposition. Kant tries to have it both ways at once and because of this, while the first sentence of the quoted passage merely states the thesis, the second fails to prove it. Clearly Kant did not take the idea that mathematical propositions ntight be falsified by experience at all seriously, and he expected his readers to agree with him. Kant did anticipate opposition to his thesis that truths of mathematics are nonanalytic and offered an argrnnent to meet it. If we take a true proposition . _of ari_thmetic or geometry, subjecting the subject concept and the predicate concept to close scrutiny, we shall not be able to find the latter contained in the former. In "thinking" the sum of seven and five, for example, we do not, according to Kant, already "think" the number twelve. But since Kant claims that we can always eventually find the constituents of a concept, be concludes that our inability to uncover the predicate concept in the subject concept shows that it is not contained therein. Kant also appeals to the nature of our mathematical knowledge to show that the truths of mathematics are not analytic. Only if mathematical truths were nonanalytic would we need the aid of intuition to convince ourselves of them. Yet if we reflect on the way in which we do recognize propositions of mathematics as true, we shall find that we always require intuition. This fact supports the thesis that intuition is necessary for mathematical knowledge and hence the thesis that mathematical truths are synthetic. These arguments which Kant offers in support of (KM) and (KE) are not equal to the task. Aside from the fact that he has not faced squarely the problem of establishing the necessity of mathematics, Kant's optimism about our powers of conceptual analysis must also be questioned. Frege, for example, would reply that the exhibition of the analyticity of the truths of arithmetic is a long and difficult affair. Where Kant has erred is in supposing that our ability to uncover the constituents is such that we can decide what concepts contain by casual reflection. Hence Kant's argrnnent would fail to show that the propositions of, for example, arithmetic are nonanalytic and would merely indicate a case where naIve reflection is an untrustworthy gnide to conceptual structure. Alternatively, if Kant were to stick with the idea that analytic truth can be revealed so easily, he would be trivializing his sense of analyticity in playing down deep and exciting conceptual relations. In this

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trivial sense of analyticity his conclusion might follow, but it would be uninteresting and would not eliminate the possibility of our learning arithmetic truths by probing our concepts more deeply. But exhibiting the shakiness of Kant's argument does not present the real difficulty with (KM) and (KE). That is revealed in Kant's efforts to explain . these theses. II. THE EXPLANATION OF GEOMETRY

Since Kant's theory of geometry is much clearer than his view of other parts of pure mathematics, it is advisable to start with it. The core of Kant's geometrical docttine is the restrictions of (KM) and (KE) to the case of geometry, to wit: (GM) The truths of geometry are necessary, although they do not owe

their truth just to the nature of our concepts. (GE) The truths of geometry can be known independently of particular

bits of experience, although we cannot know them through conceptual analysis alone.

We can understand these theses by reference to the discussion in section 1. (GM) and (GE) are to be explained by the thesis of the transcendental ideality of space. This thesis asserts that space is an a priori form of intuition. It consists of the following two claims: (SM) All possible intuitions of what we normally take to be the exter-

(SE)

nal world are subject to conditions imposed by space, which can therefore be said to be the form of outer intuition. We can know the principles which state these conditions and which thus describe space. We can know them a priori by means of a pure intuition of space.

(SM) is supposed to be the only explanation for (GM) and (SE) the ouly explanation for (GE). Since Kant regards the truth of (GM) and (GE) as established, by showing that (SM) and (SE) are the ouly explanations for these truths, he takes himself to have demonsttated the truth of (SM) and (SE).6 We begin with the argrnnent for (SM). (GM) is our premise. It asserts that

the truths of geometry are synthetic and necessary - that is, that they state nonlogical conditions on what we can experience. Further, by an assumption ascribed to Kant above, geometrical truths must either be about some particu-

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lar feature of the world - that feature in virtue of which they are true - or they must state some particular property of our concepts. Since they are not analytic. the latter cannot be the case. So geometric truths are true because of some facet of the world. But Euclidean geometry is true in virtue of the fact that space is Euclidean. Geometry thus describes the structure of space? Geometrical truths are true in every possible world. Hence every possible world has the same spatial structure - namely, that described by geometry. Thus there are laws which describe the spatial structure of any possible world

priori. Without recourse to experience, we can construct geometrical figures in thought. In doing so, we bring object-space into being as the object of a pure intuition. Ouly by means of our construction do we have an object at all. For form-space is only the form of intuition, providing for intuition once an object meets its conditions. By drawing geometrical diagrams "in the mind's eye" we are able to construct determinate object-space with its metric and projective properties. Kant sums this up as follows:

- of any world, that is, of which we can have experience. We can explain this

conclusion only by supposing that space imposes conditions on what we are able to experience. These conditions are not logical conditions. Hence they are conditions not on what we can understand but on what we can intuit. Space may therefore properly be called the form of intuition. This establishes (SM).

To construct the argument for (SE), we shall have to tackle the notion of pure intuition. To do so we must begin with the particular case of geometry. For, despite the fact that Kant does introduce the key concept of pure intuition in a quite general theory, his use of it is quite hard to understand except by reference to the geometrical version of it. The notion of pure intuition is obscured through the treatment of (KM) and (KE) together and (SM) and (SE) together. At the beginning of the Aesthetic Kant tells us that intuition is a mode of knowledge "in immediate relation" to objects and that pure intuitions are intuitions in which everything belonging to experience is subtracted (A20-21; B24-25). But he goes on at once to equate pure intuition with the fOml, or faculty;' of intuition. The situation is all the more complicated in that we are supposed to know the features of pure intuition (the faculty or form) through pure intuitions (representations without empirical content). Intuitions must have an object. Kant tries to provide an object for the intuitions which yield our geometrical knowledge by showing how space can be the object of intuitions. I shall refer to the form of intuition as form-space and the object of appropriate pure intuitions as object-space. 8 Kant's idea is that by having an intuition of object-space we come to know the properties of form-space. What needs explaining is how object-space can be intuited, how intuitions of object-space can be pure, and how they can give knowledge of the properties ofform-space. Kant's explanation centers on the notion that we can construct object-space a priori with the help of our geometrical concepts. He claims that we can exhibit such concepts as "line," "point," "circle," and so forth, to ourselves a

To know anything in space (for instance, a line) I must draw it, and thus synthetically bring into being a detenninate combination of the given manifold. so that the unity of this act is at the same time the unity of consciousness (as in the concept of a line); and it is through this unity of consciousness that an object (a determinate space) is first known [B 138].

Kant's idea may be clarified by an analogy. Let us imagine that we are condemned to look toward a surface, normally unlit, onto which pictures are periodically flashed. We can discern some order and pattern in the pictures by learning the geometrical properties of the surface. There is a way to do this without attending carefully to the pictures. We are able to draw luminous figures on the surface. We do so by following rules. To draw a triangle I follow the rule for triangles. The appearance of the resulting figure is determined partly by the rule, partly by the surface, and is, perhaps, partly due to free choices which I have made. Because of the determining role of the surface, the drawn figure can reveal properties of the surface. In Kant's terms, constructing the figure makes the surface an object of intuition. We shall discuss the nature of the rules below. Similarly, Kant would contend that the drawing of geometrical figures . reveals the properties of space. The constructed triangle yields a representation of object-space. Form-space partly detertuines the appearance of the triangle and so the representation discloses properties of form-space. These properties are to be learned from inspection of the constructed figure for whose appearance they are partly responsible. Kant claims that the construc· tion of object-space can be carried out without recourse to experience. For we can follow the rules for representing mathematical concepts no matter what

our experiences of the world have been. We are now ready for Kant's argument for (SE). By (GE), truths of geometry are known a priori. Although we know most of them by following proofs, some of them must be known immediately a priori. The basic truths of geometry cannot be known by deriving them from other truths. Nor are they knowable by analyzing our concepts; that would make them analytic. Hence we must know them through intuition. Because we can, and do, know them a

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priori, they must be known through a nonempirical kind of intuition. Kant now offers his construction of object-space and the notion of pure intuition as described above, as the only explanation for our ability to know the properties of space in nonempirical intuition. In accepting (GE) we are forced to accept (SF) and the account of the construction of object-space, for, it is claimed, (SF) explains (GF) and there just is no alternative. (SM) and (SF) are related. For without presupposing (SM), we could not set up the account of the construction of object-space as we did. The story of our mental picturings is saved from immediate collapse into the empiricist description of conceptual analysis by the determining role that form-space is supposed to play in it. Despite this connection, Kant's tendency to conllate the theses (GM) and (GF) and the theses (SM) and (SF) in the Aesthetic and his ambiguous use of the term "pure intuition" render his theory of space all the more obscure. 9 For example, in the section on the "Transcendental Exposition of the Concept of Space," given in the second edition, Kant argues from (GE) to (SF) and, without a break and without disambiguating his term, slides into the argument from (GM) to (SM) (compare B40-41). In this and similar passages he fails to made it clear that there are two distinct parts to his ideality thesis. At times, however, Kant insists that his theory of space solves two problems which baffled his predecessors. He points out that Leibniz' relational theory of space "can neither account for the possibility of a priori mathematical knowledge, nor bring the propositions of experience into necessary agreement with it" (A40-41, B57-58). Kant's attack on Leibniz in the Inaugural Dissertation also prods this double weakness.lO l.eibniz is supposedly neither able to explain why geometrical truths have the necessity they do have nor equipped to show adequately how we have a priori knowledge of these propositions. Kaut regards his theory of space as clearing up both difficulties. III. OTHER PARTS OF PURE MATHEMATICS

The reconstruction of the Aesthetic enables us to bring some clarity to Kant's views on other parts of mathematics. Since propositions of aritlunetic, algebra, and so forth are synthetic a priori, they must state nonlogical conditions on our experience. Appropriate restrictions of (KM) would be explained by theses akin to (SM). We shall have to connect arithmetic, algebra, and so on with features of the forms of intuition, space, and time. Furthennore, by an argument parallel to that just rehearsed for the geometrical case, we must be able to know the truths of

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these parts of pure mathematics with the aid of pure intuitions (representations without empirical content). So far we have made sense of the notion of pure intuition only in the context of our geometrical knowledge. Two tasks must be completed for each discipline. We should associate arithmetic, algebra, and so forth with aspects of space and time, and describe for each the nature of appropriate kinds of pure intuition. Arithmetic is the easiest case. Kant did not believe, as is often supposed, that arithmetic stands to time as geometry does to space. In the transcendental exposition of the concept of time, Kant does not mention arithmetic but refers, somewhat vaguely, to the "general doctrine of motion" (B49). Kant" would not have been content with this wave of the hand if he could have suited his theory by offering the much more obvious case of arithmetic as an example. Furthennore, his use of an arithmetical example in the argument for the synthetic status of mathematical truths describes an intuition through which the cited arithmetic truth is known, and it is hard to understand this intuition as a pure intuition of time alone. Finally, we have the word of the Inaugural Dissertation: Hence PURE MATHEMATICS deals with space in GEOMETRY, and time in PURE MECHANICS. In addition to these concepts there is a certain concept which in itself indeed is intellectual, but whose actuation in the concrete requires the assisting notions of time and space (by successively adding a number of things and setting them simultaneously beside one another). This is the concept of number. which is the concept treated in ARITHMETIC [PC, p. 621.

But if arithmetic does not state properties of time, what is it about? The truths of geometry are true in virtue of particular features of space. That does not mean that geometry exhausts the properties of space. The following possibility remains open. Arithmetical propositions are true in virtue of certain structural features of space and of time. We can refer to these properties collectively as "combinatorial" features of space-time. The same combinatorial feature can be observed to hold of space and of time; for example, a unit length added to a two-unit length makes a three-unit length whether we think in terms of space-units or time-units. Arithmetical truths may, perhaps, portray such common combinatorial features. This is somewhat vague, but is, nonetheless. an answer to the problem of associating arithmetic with the forms of intuition. Kant is more specific about what pure arithmetical intuition (the nonempirical representation) is like. Considering our knowledge that 7 + 5 = 12, he claims that we cannot know the proposition by concepts alone. Instead, "starting with the number 7, and for the concept of 5 calling in the aid of the fingers of my hand as intuition, I

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now add one by one to the number 7 the units which I previously took together to form the number 5, and with the aid of that figure (the hand) see the number 12 come into being" (BI6). If we take this as our model, we can suppose that what makes arithmetic possible is a pure intuition of space and time together, which spells out the hint given in the Dissertation. We know that 7 + 5 = 12 by instantiating our concepts of 7 and 5, using stroke symbols, for example, and by successively juxtaposing a stroke to the block of seven for every stroke in the block of five. To put the example graphically: we draw strokes counting from one to seven; we then continue "one-eight" (stroke), "two-nine" (stroke), until twelve "comes into being" with "five-twelve" (stroke). There is one obvious difficulty with this idea. In the case of geometry we could find a role for the structure of space in giving a partial determination to our representations. Using the analogy of the surface we were able to give content to the notion that the representation was a representation of space. A similar suggestive analogy is harder to find for the case of arithmetic. 11 As a result of our difficulty in finding a role for the structure of space-time in the determination of our representation the threat of collapse into empiricist-style conceptual analysis gains new vigor for this case. Kant does not seem to favor the option that intuition of stroke symbols leads to knowledge of general propositions of arithmetic. 12 This introduces a problem when we turn to algebra which Kant understands as a generalized arithmetic dealing with "quantity as such."!3 That problem emerges only with the epistemological thesis. Using Kant's hint of the relation between algebra and arithmetic, we can gloss the metaphysical· thesis as claiming that, while arithmetical propositions owe their truth to relatively simple and concrete features of space and time, algebraic truths are true in virme of more abstract and, perhaps, more fundamental features of the forms of intuition. (Such features would be reflected in laws like the law of commutativity of addition.) Combinatorial features of space-time would be partitioned into two classes, the specific and the more general, the former accounting for arithmetical truth and the latter for algebraic truth. Worse than the difficulty of rendering this distinction (or, indeed, the notion of "combinatorial property") clear and precise is the problem of explaining the algebraic version of pure intuition. Kant has two approaches to this problem. The first is an obscure doctrine which claims that algebra is intuitive because it uses "symbolic construction" (cf. A717, B745). What Kant means is that algebra uses symbols and proceeds by manipulating these symbols. (The term "symbol" is loaded, as we shall see.) His early essay on

the principles of natural theology makes the point quite clearly. Mathematics has an advantage over philosophy in its ability to use symbolism and in its power of ignoring the things symbolized. Kant assumes, significantly, that there could not be a philosophical symbolism which could produce the same benefits. He remarks:

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The, signs used in the philosophical way of thinking are never anything other than words, which can neither show, in their composition, the parts of the concepts out of which the whole idea, indicated by the word, consists; nor can they show in their combinations the relations of philosophical thoughts [PC, p. 91-

Kant's subsequent remarks indicate how he thinks that mathematical symbols can do better. Mathematical signs reveal properties of the objects symbolized which are not contained in the concepts of these objects. This is not altogether absurd. Kant clearly thinks that a particular diagram of a circle can be the symbol for all circles and that the stroke symbol "III" is the symbol for 3. Thus the cases of geometry and arithmetic demonstrate how the use of symbols (in this loaded sense) can be especially useful. We can see, by their aid, that circles intersect in at most two points and that 3 is greater than 2. But Kant cannot extend his conclusions to algebra where the symbolism is different. Perhaps because algebra deals with such colorless objects as magnitudesin-general, its signs cannot serve as pictures in the way in which signs of geometry and arithmetic can. 14 No matter how long we stare at the sign design ra + b = b + al we shall not discover, by means of our scrutiny of these signs, the truth of the law of commutativity of addition. Kant's stress on the mathematician's use of symbolism fails to tie algebra to geometry and arithmetic; instead, it reveals that there are important differences between the type of arithmetical symbolism which interests him and ordinary algebraic symbolism. The theory of "symbolic construction" for algebra only amounts to the weak claim that algebra is "intuitive" in being able to operate with signs. It does not divorce algebra from branches of analytic knowledge, which manipulate signs in the same way. So we are still left with the problem of finding a way in which we can have a priori knowledge of basic algebraic truths. We can take the law of commutativity of addition as an example of a fundamental algebraic law (it is clearly a basic principle of "the general arithmetic of indetenninate magnitudes") (PC, p. 8). In the last paragraph we concluded that it could not be known just by presenting the signs to ourselves. Kant does not leave the issue with the shuffle around "symbolic construction."

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Even the method of algebra with its equations, from which the correct answer, together with its proof. is deduced by reduction, is not indeed geometrical in nature, but is still constructive in a way characteristic of the science. The concepts attached to the symbols, especially concerning the relations of magnitudes, are presented in intuition; ... [A734, B762; my emphasis].

Applying this to our example, Kant would account for our knowledge that a + b = b + a by showing how we construct two magnitudes - both of which can stand for all magnitudes - exhibit the concept of addition (the relation of magnitudes in which we are interested here) and so grasp the commutativity principle. But there is a problem with the idea that the magnitudes exhibited can stand for alL Kant believes that mathematics proceeds to universal conclusions from intuitions of particular objects. By examining a diagram of one particular triangle we come to know properties common to all triangles. 15 Similarly, Kant could suppose us to proceed to knowledge of properties shared by all magnitudes by intuiting particular magnitudes. Since he speaks of algebra as generalized arithmetic, we might think that the appropriate magnitudes to be intuited would be numbers. Kant's remarks about axioms for arithmetic suggest, however, that intuition of stroke symbols is not intended to lead us to general conclusions, that knowledge of algebra is not to be founded on intuitions of indeterminate stroke symbols. In any case, he has an alternative at hand in the finite line ~ segment. Geometrical picturing can be used to reveal the principles of algebra. The line segment which we intuit can stand for all the line segments or for all magnitudes. If one use of the particular figure is legitimate the other will be, too. Kant can thus find a way of accounting for our knowledge of principles of algebra. Significantly, that account explains our knowledge of these principles in their geometrical instantiations. Thus while Kant's metaphysical thesis concerning algebra construes algebra as generalized arithmetic, algebraic knowledge would be gleaned in a way similar to that followed in knowing geometry. IV. PROOFS AND KANT'S FOUNDATIONAL PROGRAM

We have been focusing on immediate knowledge. Kant's theory of pure mathematics distinguishes those truths of pure mathematics which are apprehended immediately from those which are inferred. Thus, in the Methodology, Kant characterizes the general form of mathematical disciplines. The mathematician "constructs his concepts in a priori intuition," and by doing so he is able to "combine the predicates of the object both a priori and immediately" (A732, B760), thus obtaining starting points for proofs;

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then "through a chain of inferences guided throughout by intuition, he artives at a fully evident and universally valid solution of the problem" (A717, B745). Perhaps all that Kant has in mind when he describes proofs as "guided by intuition" is the notion that mathematical inferences are intuitive because they consist of transitions from one set of symbols to another. (As if we first exhibited concepts to ourselves to teach ourselves the proper ways of manipulating signs and then inspected the signs themselves, scrutinizing them to ensure that each move accorded with the established rules.) A more substantial reading can be given for the case of geometry. Various of Kant's examples of geometrical proof indicate that he regards proofs as sequences of mental constructions on figures already constructed. Having drawn a figure in thought, embellishing it reveals successively more complex and recondite features of space. So we can be guided to recognize properties which we cannot learn "all at once." It is hard to envisage, however, how we can construe this substantive use of pure intuition in proof for the cases of arithmetic and algebra. But whether we take proofs to be intuitive in the sense that the constructed objects are kept in view throughout, or whether we suppose that one inspects constructed objects jllst at the beginning of proofs, the rest being surveillance of signs, there is a theoretical challenge for Kant to face. If we grant, for the moment, that Kant can claim that some traditional parts of pure mathematics fit the pattern that he sees in all parts of pure mathematics, there remains the task of showing that the rest of pure mathematics can be accommodated. In particular, he would have to show that the new eighteenth-century disciplines ~ in pure mathematics match his ideal - or else they would have to be dismissed as not belonging to pure mathematics. Kant would have to exhibit the intuitive foundations of complex algebra and of analysis if he wishes to maintain that those subjects belong to pure mathematics. To justify the use of complex numbers one must demonstrate that the practice of applying the usual algebraic operations to those numbers is legitimate. Now we have seen above that Kant's problem with our basic knowledge of algebra can be solved by using geometrical picturing. Let us suppose that some way of representing the (linear) multiplication of line segments has been established and that we use direction to indicate sign. Then, when we are justifying the basic laws of algebra, our pictures will reveal to us that the multiplication of a quantity by itself always yields a positive quantity. These pictures cannot therefore justify us in extending the scope of our laws by enlarging the domain of quantities to include complex numbers. The pictures

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we use may even tempt us to take the nonexistence of square roots of positive quantities to be a law of algebra. And it might be hard to argue that this is temptation. 16 An alternative would be to find a new, more general way to represent magnitudes to ourselves in pure intuition. The emphasis on geometrical construction defines the task further. We need a geometrical model for the complex numbers. Using this model we could exhibit to ourselves the general laws of algebra (taken now as the laws of complex algebra), and the use of the dubious imaginary numbers could be placed on a firm foundation. Mathematicians contemporary with Kant took the task of making complex numbers familiar to be siguificant. Kant's geometrical bias was reflected in the way men like Wechsel, Argand, and Gauss completed the task. Kant's problem with analysis does not concern the exhibition of an unfamiliar concept but the derivation of the theorems of analysis from basic principles apprehended immediately in pure intuition. As in the case of algebra, analysis would be founded upon geometrical constructions, and, although Kant could not have developed nineteenth-century function theory on that basis, he could have exhibited the foundations of the analysis he knew. 17 For, in accordance with the geometrical spirit of the seventeenth century, the originators of the calculus often tended to regard it as an offshoot of geometry. Newton even attempted to show, using his method of first and last ratios, that his calculus could be grounded in ''the geometry of the Ancients."l8 That attempt could be adapted as a Kantian answer to the problem of founding analysis. Newton's assumptions could be justifled by appeal to pure intuition, and the kinematic conception of geomotry which Newton used (an approach which regards figures as generated by the motion of points) is re-echoed in Kant's constructive geometrical acts (the "drawing of the line in thought"). The Newtonian idea that "continuity" is an unproblematic notion could also be defended in Kant's terms. The Anticipations of Perception even suggest the line of defense; and it is noteworthy that, in this passage, Kant uses Newton's own favored terminology and speaks of ''flowing magnitudes" (Al70). Insofar as Kant could contend that any knowledge of mathematics can be gleaned from pure intuitions, he could carry that contention through for all parts of pure mathematics that he knew. The strength of his position lies in its seeming ability to account for everything. When mathematics attempted to go beyond concepts which were intuitively accessible (as in the development of the function concept), the currency of ideas similar to Kant's can be seen in the strength of the protests. And when mathematics did forsake intuitive

geometry for good, the mathematical community abandoned Kant's ideal. The death blow was not struck by Bolyai, Lobatschevsky, and Klein but by the men in the tradition which led to Weierstrass's function, continuous everywhere but differentiable nowhere. But Kant's theory was wrong from the beginning. His attempt at explaining mathematical knowledge gives no explanation at all.

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V. CIRCLES OF MATHEMATICAL KNOWLEDGE

Kant contends that he has explained how we can know the basic propositions of geometry a priori. Pure intuitions are supposed to teach us general truths which describe the structure of space exactly; such are the axioms from which geometry begins. I shall show that they cannot do this. Kant's account of our geometrical knowledge is circular in two different ways. My objections will be developed against the case of geometry, but they apply equally to the cases of arithmetic and algebra; I have chosen to advance them against Kant's account of geometry because that is where his explanation of mathematical knowledge appears most cogent. I think it is easy to see that the criticism I raise could be applied with equal force in the cases of other disciplines. The first circle arises as follows. We begin with the criticism which Berkeley leveled against Locke's theory of abstract ideas. One cannot draw a triangle which has only the properties commOn to all triangles. So, if Kant supposes that we learn general propositions about triangles by drawing particular figures to ourselves in thought, he will either have to show that Berkeley's point is wrong or else explain how we carefully refrain from generalizing over the pecnIiarities of the figure. Let us take a sitnple example. Suppose that I construct a scalene triangle. From my figure I can generalize that all triangles have the side-sum property (the property that the sum of the lengths of any two sides is greater than the length of the third); but I must not infer that all triangles are scalene. Why is the one inference legitimate and not the other? Kant answers this question in a cryptic passage in the Methodology: mathematics can achieve nothing by concepts alone but hastens at once to intuition, in which it considers the concept in concreto, though not empirically. but only in an intuition which it presents a priori, that is, which it has constructed, and in which whatever follows from the universal conditions of the construction must be universally valid of the object of the concept constructed [A715-716, B743-744].

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This is best Wlderstood by means of our surface analogy. We imagined ourselves revealing the contours of a surface by drawing figures on it, and envisaged the drawing as rule-governed. We did not make it clear at that stage what the significance of the rules was to be. Kant calls the set of rules which we follow to produce the object of a concept the schema of that concept, noting. apropos of a discussion of Berkeley's attack on Locke that "it is schemata, not images of objects, which underlie our pure sensible concepts. No image could ever be adequate to the concept of a triangle in general" (AI40-141, BI80). Kant's solution to the problem is thus to claim that we can draw general conclusions using only those features of the image on which the rule has pronounced. In the above example, my production of a scalene triangle was brought about by a free decision of mine over and above my application ofthe rule. It is therefore illegitimate to use the scalene peculiarity to draw the conclusion that all triangles are scalene. The trouble with this reply is that it seems to make the exhibition of a particular triangle in intuition quite unnecessary. For if all that we are allowed to do is to draw out features of triangles prescribed by the schema of the concept "triangle," then we can do this by conceptual analysis alone. We shall arrive only at analytic propositions in this way but, given Kant's above reply, it is not clear that we are entitled to learn more anyway. By resisting generalization over accidental features of the drawn figure, we seem to restrict our ability to generalize to properties which the schema demands be exhibited in all triangles. So we need only look to the schema and not to the constructed object. The way to answer this is to hearken back to our surface analogy. We can divide into three types the properties which a figure drawn on the surface possesses. Some properties belong to the figure just because it has been drawn in accordance with a parricular rule; we shall call these R-properties. Others belong to it in virtue of the application of the rule on the structure of the surface; these will be termed S-properties. Finally, there will just be the accidental properties of the figure which result from free choice and are not determined at all by the application of the rule; let us refer to these as A-properties. If we now revert to Kant's theory of the construction of objectspace and to the example of the triangle which we draw to represent objectspace to ourselves, we can set up parallel categories. R-properties are just the properties which the schema alone determines; for the triangle an example of an R-property would be the property of having three sides. S-properties are those properties which the schema and the structure of space together determine; the side-sum property and the property of having the internal angle-

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sum equal to 180 degrees are both supposed to be S-properties. Finally, there " are the A -properties, pecnliarities of the particular figure drawn, such as the scaleneness of the triangle. Now we can know that all triangles have the R-properties which they do have merely by analyzing our concepts. Again, since none of the A-properties of the particnlar triangle we construct is shared by all triangles we must not conclude that all triangles have an A -property just because we notice that our particnlar triangle has that property. Where pure intuition is supposed to help is in leading us to the S-properties which are shared by all triangles. By this means we arrive at propositions which are synthetic a priori and are basic to geometry. Kant is, however, still in difficnlties. Let us consider three geometrical proposItions. (a) All triangles have three sides. (b) All triangles have the sidesum property. (c) All triangles are scalene. (a) is analytic and knowable by conceptual analysis. (b) is assumed to be synthetic a priori and is just the kind of proposition which we are supposed to know a priori through pure intuition. We now imagine ourselves coming to know (b) in the way Kant suggests. We draw a scalene triangle and see that this triangle has the side-sum property. We also see that it is scalene. If we are now to conclude that all triangles have the side-sum property but recognize that we cannot conclude that all triangles are scalene, then we must be able to distingnish S-properties from A-properties. It is not enough for Kant to provide the distinction between these types of properties. He has to show that we can use the distinction to make the right moves and avoid the wrong ones. Unfortunately, it is difficult to see how we can distingnish S-properties from A-properties without already knowing the properties of space. For there is nothing intrinsic to a property which makes it an S-property rather than an A-property. Suppose that we do not take (b) to be analytic (that is, we do not take it to provide a partial explication of the concept of distance). Then there can be spaces and metric relations on them such that (b) is false. More straightforward is the case of the angle-sum property of a triangle. Kant assumes that we can recognize that this is an S-property of a triangle. In a universe like that described by Reichenbach, however, where a cross-section of the space is a plane with a protruding hemisphere,19 having the sum of its angles equal to 180 degrees would be an-A-property of a triangle. Conversely, it does not seem impossible that what Kant takes to be an A-property might turn out to be an S-property in some spaces. Perhaps there are spaces in which all triangles are scalene. The upshot of this is that, to recognize something as an S-property we

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already have to know what the properties of space are. Without knowing that we were not confronting the Reichenbachian space we could not take the angle-sum property to be ao S-property. The intuition is supposed, however, to show us that we are experiencing Euclidean space. But we cannot draw this conclusion until we have distinguished the S-properties. Clearly the account is turning in circIes.20 In fact we can make the point without the appeal to bizarre spaces and we can make it for any basic general proposition of geometry. Let G be a basic geometrical truth. G is supposed to be synthetic a priori. Its synthetic status arises because its truth value is, partially, determined by the structure of space. It is logically possible that space have a structure such that G be false. (Otherwise, G would be analytic.) Further, it is logically possible that G might have been false in such a way that maoy figures actually had the property ascribed to them by G. How would we have determined from inspection of such a figure that the property was only an A-property and that we should not therefore generalize over it? We can answer this question only if we can decide what counts as the application of a rule on the structure of space and what was our free decision in drawing the figure. Yet to distinguish S-properties from A-properties is just to recognize the structure of space. We could not therefore come to know G in the way which Kaot describes. 2l Kant's explanation of our geometrical knowledge is also trapped in a second circle. So far we have been supposing that it is only with general propositions that problems atise. We have focused on the difficulty of deciding how to get nonanalytic general conclusions from particular figures. We could also have asked for the justification of the basis for generalization. Has Kant really explained how we know that a particular figure has a particular property? We shall use two examples to answer the question. The first is a very clear case due to Charles Parsons. 22 We consider the proposition that all line segments are infinitely divisible. One thing is obvious. We cannot come to this knowledge by observing a line segment infinitely divided. So how do we describe a pure intuition - or sequence of such intuitions which will lead us to knowledge? We may follow Parsons in supposing that there is one appropriate form for the description. We can represent to ourselves the line segment bisected. From this representation we can proceed to another in which we represent (say) the left-hand half of the bisected segment in all the detail in which we formerly represented the whole segment. Now we can bisect this segment aod again represent the left-haod part of the segment in enough detail. Continuing this process, we divide the segment as many times as we wish. Given any number n we see that we can

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divide the segment more thao n times. Hence we conclude that the segment is infinitely divisible. There are two possibilities for the way in which the increase of detail is accomplished. The first is to suppose, as Parsons does, that we increase the acuity of our vision, bringing the leftmost parts of the segment under ever more intense scrutiny. Now there is obviously a physical limit to our ability to do this, a threshhold length beyond which we caonot increase the detail of the leftmost segment sufficiently to bisect it. Let us refer to the leftmost segment at this stage as L. The claim of the Kantian is that, although our myopia prevents us from bisecting L, this is only a physical disability; "in principle" we can bisect L. Because we see that we can bisect L and that we can bisect the descendaot segments L', L", aod so on indefinitely, we see that the original segment is infinitely divisible. There are now two questions: first, what kind of possibility is being invoked here? aod second, how do we know that, in the appropriate sense of possibility, we cao continue to bisect L, L', and so forth? By assumption, it is not practically possible for us to bisect L. Let us then suppose that we know only that it is logically possible for us to bisect L, L', aod so forth. We caonot conclude from this that it is possible in Kant's seuse that L be divided as maoy times as we like. What is logically possible may not be possible according to the intuitability criterion. The only way to find out if the logically possible is indeed possible is to give oneself an appropriate intuition. For the case in haod that course has already been rejected. Hence if we read the principle that ensures the bisectability of L, L', aod so forth as guaraoteeing only logical possibility of bisection, it is too weak to lead to the conclusion we want. Kant might reply to this by describing some way in which we cao show that logical possibility aod his kind of possibility coincide in certain cases, or perhaps even in certain families of cases. By so doing he would be able to conclude that the logical possibility of further bisection guaraotees the possibility of such division according to the stronger notion of possibility which he uses. Natura1ly, the way in which we could come to know results about the equivalence of the two kinds of possibility would bave to be explained. It is clear that Kaot cannot suppose that these results are known through analysis of the concept of humao experience. That would be to undercut the significaoce of pure intuition altogether, by making the propositions of geometry knowable througb the aoalysis of concepts. It is not obvious how the machinery which Kaot develops cao be adapted to aoy other means by which we could know a priori that logical possibility aod Kaotiao possibility coincide for certain cases.

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Indeed there are dangers that in trying to escape the conclusiou that only intuition can show us if the logically possible is really possible Kant would have to 23 sacrifice some theses which are closest to his heart. So we must conclude that the proposition which is supposed to be known is that it is possible to bisect L, L' and so on indefinitely, where we are to understand possibility in Kant's sense. How is this to be known? The proposition cannot be known through conceptual analysis (that would render what we are taking to be a truth of geometry analytic) and it must be knowable a priori. Thus it has to be knowable through pure inntition. But now we are back with the problem from which we began and which the process of bisection was supposed to clear up for us. It is clear that no progress has been

The inadequacy of pure intuition does not arise only in connection with the notion of infinity. It stems immediately from the idea that we can, on inspection, determine the exact nature of a figure, whether physical or "drawn in thought." We can, according to Kant, know by means of pure inntition that there is one and only one straight line joining two given points. At first we might think that we understand what he means by this. We construct the points and draw the line between them. If, however, we were to see that this line is unique, then we should have to be able to distingnish it from any other line which we are able to draw between the two points. Now there are cases in which we cannot distinguish the one straight line from very slightly curved ones, even on close inspection. Confronted with fignres in which this is the case we "see" that only one line is straight - but that is ouly because background geometrical knowledge is available to us. Kant is supposing that we are in the process of gaining this knowledge. Hence he must think that we can distinguish the one straight line from the curves which are "nearly straight." But we cannot. And although this may come about from physical limitations of ours, until we have learned our geometry we are in no position to know that our failure is a medical accident - or indeed that it is failure. For all we know, it could just as well be success in discerning a property of space. The problem lies with the picture behind Kant's theory. That picture presents the mind bringing forth its own creations and the naIve eye of the mind scanning those creations and detecting their properties with absolute accuracy. Kant attempts to derive a clear theory of mathematical knowledge from that picture, the theory described above. Whether that picture has been abandoned in the less clear theories of his constructivist successors is a further question.

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made. Nor is there any solution in the idea that we can, successively, replace our pictures of the whole line with pictures of the half-line. This idea is simpler. We construct the original line segment and bisect it. We now replace this picture with a more detailed picture of the left-hand part of the bisected segment and bisect that. So we continue as long as need be. Here we are immediately in difficulties. Apart from the need for justification of our ability to continue the process indefinitely, what is reqnired is an account of how we know that the successive pictures really do represent the lefunost segments of the previous pictures in more detail. Intuition cannot reveal this to us; fOf we have no way of intuiting that the fine structure of segments is preserved from picture to picture. Nor is it a conceptual truth that this procedure of "eulargement" really is just a method of presenting segments in greater detail. What is wrong in both cases is that, to draw from the sequence of intuitions the conclusion that the original segment is infinitely divisible, we need to know something equivalent to that conclusion. We set out to investigate the properties of space which, on Kant's theory, determine what we can and cannot intuit. We try to learn these properties through a series of representations of space. The series of representations which we can in practice give ourselves does not suffice to show the truth of the conclusion. Hence we suppose that the series can be extended to a series of representations which would show the conclusion. In making this supposition we commit ourselves to the notion that such a series is possible, but this possibility in turn involves the property we were supposed to be discovering. Once again we have a circle. We can know that space has a property ouly by knowing that a series of intuitions is possible. But we can know that that series of inntitions is possible only if we know that space has the original property. We begin by trying to discover the limitations of experience; we end up by assuming them.

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University of California, San Diego NOTES I r would like to thank Paul Benacerraf and Patricia Kitcher for all the patient advice and suggestions they have given me. The criticisms and encouragement of Michael Mahoney, P. F. Strawson, and Margaret Wilson have also been very helpfuL 2 Were we to analyze the concept of a possible experience as that of a logically possible experience we should be able to draw conclusions contrary to Kant's general view. It would be analytic that all human experiences are experiences of a Euclidean world and we should be able to know mathematical truths without intuition, merely by analyzing the concept of human experience. 3 See Ch. I of my dissertation Mathematics and Certainty [Princeton, 1973, unpublished} for a

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more detailed account of the general form of theories of a priori knowledge. It is assumed that we are ail on a par as possible knowers - i.e.• that anything that can be known a priori by one person can be known a priori by any other. 4 See Critique of Pure Reason (B4). 5 See P. Geach and M. Black, Translations from the Philosophical Writings of Gottlob Frege, (Oxford, 1952), pp, 159-170. 6 Kant does not present the argument in a way which makes it clear that (GM) and (GE) are distinct. Despite this, we shall see below that certain passages do seem to indicate his awareness of the distinction between them. 7 As an anachronistic argument on Kant's behalf we might point out that different geometries ascribe different spatial structures to the world. Replying to that argument by insisting that one can talk only about a geometry and a physics together applying to the world would, however, undermine Kant's assumption of an intimate tie between propositions and features of worlds. Without that assumption, Kant's whole line of reasoning would collapse and, worse still, (SM), (GM), and (KM) would all need to be refurbished. Yet perhaps one might try, as Poincare did, to maintain the special status of Euclidean geometry. 8 My reason for using this distinction is that it may be a conceptual error, for Kant, to identify space (the form of our perceptions) with space (the object of our perceptions when we do geometry). Even if no such error is involved, the gain in clarity which the distinction brings is obvious. 9 I think that Kant's obscurity here misleads Jaakko Hintikka In his paper "On Kant's Notion of Intuition" (printed in T. Penelhwn and J. J. Macintosh [eds.], The First Critique [Belmont, Calif., 1969]), Hintikka glosses Kant's task as proving "that the ideas of space and time are inseparably tied up to human sensibility" (ibid., p. 45). But, on the view advanced)n the present paper, there are i'l1'0 tasks which can be characterized by this ambiguous phrase. Hintikka's main discussion focuses on the metaphysical task - the move from (GM) ~o (SM) - without dealing with Kant's attempts to explain mathematical knowledge. It is thus not surprising that Hintikka should conclude by divorcing "intuition" from its epistemological role. 10 See G. Kerferd and D. Walford (eds.), Selected Pre-Critical Writings, (New York, 1968), p. 71. I shall refer to thjs volume as PC. I! Although one can imagine a discrete space-time governed l?y a modular arithmetic Oetting the modulus be, for example, 1,000). If we then suppose ourselves to be drawing stroke symbols on an imaginary surface we shall reveal that there are only 1,000 "places" on that swface. The details resist elaboration, but I think this sketch suggests a way to adapt our gloss of pure intuition to the arithmetical case. 12 See the "Axioms of Intuition" (AI64-165; B205-206). Kant did not avail himself of an option here which was later taken by Hilbert. One can suppose that general laws of arithmetic are known by intuition of indeterminate stroke symbols. 13 See Sec. 2 of the Enquiry on the Clarity of the Principles of Natural Theology and Ethics, esp. PC, p. 8. 14 When we reflect we see that there are two different types of sign used in geometry and arithmetic. There are the "revealing" signs which Kant talks about and also signs like "AB," "MJEF," "3," "7 x 9," which are just as incapable of showing us mathematical truths as the signs used in algebra. 15 This will be discussed in detail below. 16 See below, p. 129. 17 That is, Kant could definitely have reconstructed the analysis developed by Newton and his successors. Whether or not Euler's algebraic analysis had already left the province of geometri-

ca1 intuition is a matter for historical speculation. Of Course, Eulerian analysis does not look much like geometry - but then neither does much of Cauchy's treatment. Yet Kant's geometrical approach might even be adequate to the intuitive concept of the continuum which Cauchy employed.

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18 See my "Fluxions, Limits and Infinite Littlenesse," (Isis, March 1973) for the details of Newton's program. 19 See Hans Reichenbach, The Philosophy of Space and Time, (New York. 1958), p. II. Kant might, perhaps, just deny that there is any need for explanation of how we recognize the S-properties. To do so would be to tum his theory of Constructions into an irrelevant piece of window dressing. For the issue with which he is grappling is the issue of how we know geometrical truths and, unless we explain how we know the S-properties, to answer that we know geometry through knowing which properties are S-properties is like saying that we know geometrical propositions because we know them. If the original demand for explanation needs to be taken seriously, so does the new question. But, as I have shown, the theory of constructions is quite helpless here. 20

21 Whether or not Kant has any other theory as to how the properties of space are known, he certainly has no other clear theory. It may be that one can dredge up from the Aesthetic hints of an alternative approach to mathematical knowledge, but I have preferred to concentrate on the more interesting'and detailed approach which Kant favors throughout the Critique. 22 See Infinity and Kant's Conception of the "Possibility of Experience," Philosophical Review, LXXIII (1964), 182-197; reprinted in R. P. Wolff (ed.), Kant: A Collection of Critical Essays (Notre Dame, Ind, 1968). 23 This claim depends on my view that Kant is engaged in an epistemological as well as a metaphysical enterprise. I have taken him to offer an aCCOllnt of mathematical knowledge and construed "intuition" as a sensuous route to such knowledge ("the science of things sensual"; cf PC, p. 62). To retreat to vagueness at this point is to give up the attempt at explanation, and appeals to "intellectual intuition" are the counsel of vagueness. I hope that the investigations of Sees. III and IV show the power of sensuous intuition to account for everything Kant took to be pure mathematics if it can account for anything. Given this, one may admire Kant's attempt at a thorough and detailed theory of mathematical knowledge while admitting his failure. One does not save him by blurring the central concept.

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On its conceptual side, mathematics as Kant understands it involves in an essential way the categories of quantity. This much should be obvious to readers of the Critique of Pure Reason. To trace this connection in more detail, however, has not been a main concern of interpreters of Kant's philos-

ophy of mathematics, at least recent ones. No doubt it has been thought that the connection is bound up with traditional logic and with a conception of mathematics more restrictive than what has come to prevail since the rise of

set theory and abstract mathematics. The questions concerning Kant's conception of intuition and of construction of concepts that have dominated the literature on Kant's philosophy of mathematics are more directly connected with philosophical debates of recent times. Nonetheless, as investigation of the relation of arithmetic at least to the categories of quantity might promise to be instructive for several reasons. First of all, it should clarify how Kant understands the basic concepts of arithmetic, that of nwnber in particular. Secondly, Kant's conception of nwnber and therefore of arithmetic is bound up with the schematism of the categories, since he describes number as the schema of quantity (A 142; B 182), and thus with problems in Kant's philosophy that go beyond his philosophy of mathematics. Thirdly, on just the point of the relation of nwnber and schematism, Kant appears to have changed his view after the first edition of the Critique, as we shall see below. The purpose of the present paper is to explicate Kant's understanding of arithmetical concepts and their relation to the categories of quantity. This will require some exposition of Kant's conceptions of quantity, for which we have to rely on Reflections and on his lectures on Metaphysics. With this background we will address Kant's view of nwnber and compare what is said in the first e/dition of the Critique with some texts from 1788-1970. This comparison yields some puzzles of interpretation having to do with the place of number with respect to the pure and schematized categories. I will preface this whole discussion with some remarks about Kant's view of mathematical

objects in general. 135 Carll. Posy (ed.), Kanf s Philosophy of Mathematics, 135-158.

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This story has no overwhelming moral. It does show that however different his picture of the basic concepts of mathematics was from our own. however confused it may have been when measured against what we can now do with the help of set theory and modern logic, Kant had more to say about the concept of number and related concepts than has been appreciated.

not a form of actuality. There are indications rather that Kant thought of it under the category of possibility. This is said quite explicitly by Kant's disciple Johann Schultz, in criticizing Eberhard for interpreting Kant's concept of the objective reality of a concept as meaning the actual existence of objects falling under it instead of their possibility.

From our modern point of view, a noteworthy feature of Kant's philosophy of mathematics is the absence of an articulated account of mathematical objects. Kant does talk in a highly general way about objects, in particular in saying that the categories spell out "the concept of an object in general". But even the pure categories, once they are distinguished from the forms of judgment, envisage concrete objects, since they include substance, causality, and com-

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munity. Kant's full-blooded notion of object is that of an object of experience, that is a spatio-tempnral object. 1 Thus Kant rarely expresses a philosophical commitment to specifically mathematical objects, although passages that we would read as involving reference to such objects abound in his writings. Exceptions are the statement that '7 + 5 = 12' i(f a singular proposition (A 164 = B 205) and the statement that "we can give it [the concept of a triangle] an object wholly a priori, that is, cpnstruct it" (A 223 = B 271). In another passage Kant's language is even stronger: As regards the fonnal element, we can detennine our concepts in a priori intuition, inasmuch as

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we create for ourselves, in space and time, through a homogeneous synthesis, the objects themselves - these objects being viewed simply as quanta (A 723 = B 751).

In the second of these tbree places, Kant partly takes away what he has given in saying that the triangle is "ouly the form of an object", thus apparently shifting from a use of 'object' that would comprehend mathematical objects to one that does not.2

Even when he is most explicit about mathematical objects, Kant does not attribute existence to them. In fact he seems to reject such an attribution in

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saying that "in mathematical problems there is no question of ... existence at all" (A 719 = B 747).3 The pure category of existence is schematized as existence at a definite time (A 145 = B 184); it implies actual existence (Wirklichkeit). To know the actual existence of something requires connection with an actual perception by means of the analogies of experience (A 225 = B 272). For this reason it seems clear that mathematical existence is

But unfortunately the example from pure mathematics does not fit, for in mathematics possibility and actuality are one, and the geometer says there are (es gibt) conic sections, as soon as he has shown their possibility a priori, without inquiring as to the actual drawing or making of them from material.4

What plays the role of mathematical existence in Kant's usage is con"Structibility. It is tempting to regard this as possible existence: the construction of a concept shows the possible existence of an object whose form is given by the construction. Given Kant's understanding of possibility, however. construction in pure intuition is not sufficient to show such possible existence witl!.QJl!.the aid of cJ;rtain..l?l!iljJ,,1(IJzbicaL.c.ODS.id6:.ati~:-tobe possI-: ble is. to agree "with the formal conditions of experience, that is, with the conditions of intuition and of concepts" (A 218 = B 266, emphasis mine). The latter conditions are of course the categories. In his discussion, Kant is quite explicit about the relevance to the mathematical realm of this conception of pnssibility. When he says, as we noted above, that a constructed triangle is "only the form of an object", he goes on to say that to "deterntine" the possibility of an object of which it is the form, it must be the case thar such a figure is "thought under no conditions save those upon which all objects of experience rest" (A 224 = B 271). These are not only space, as a condition of outer appearance, but that "the formative synthesis through which we construct a triangle in imagination is precisely the sarne as that which we exercise in the apprehension of an appearance, in making for ourselves an empirical concept of it". These are just the considerations advanced in the Axioms of Intuition. But the consequence seems to be that knowledge of the objective reality of mathematical concepts, that is the possible existence of instances of them, is philosophical rather than purely mathematical knowledge. 5 This state of affairs poses a dilemma for Kant's philosophy with regard to the status of mathematical knowledge. Kant's conception of mathematical knowledge as resting on demonstrative proof in which the essentially mathematical element is construction in pure intuition makes it of a quite different character from philosophical; of course that contrast is the main theme of the Discipline of Pure Reason in its Dogmatic Employment. It seems quite clear

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that Kant thinks of such knowledge as independent of philosophy. But mathematical demonstration seems not to yield knowledge of objects in the genuine sense, unless it is supplemented by some philosophical reflection. A much cited remark in the second edition Transcendental Deduction illustrates the difficulty:

use of the understanding" seems to make this acknowledgment already. If, with Schultz, he were to read the particular quantifier as "es gibt", this would not connote Dasein or Wirklichkeit, but that would not commit Kant to a real theory of "nonexistent objects" of the sort that is attributed to Meinong or inspired by him. 8 A possibly more serious question that would arise for a Kantian conception of mathematical objects, and of mathematics as knowledge of such objects, comes from his view that knowledge of objects requires intuition. When Kant speaks in this vein, he does regard construction of concepts in pure intuition as yielding such objects; in that sense, there would be intuition of them. But strictly speaking, this probably applies ouly to what Kant calls ostensive construction, which is characteristic of geometry, as contrasted with symbolic construction, characteristic of algebra (A 717; B 745). It is the former that is said to be "of the objects themselves". This leaves somewhat unclear in what sense it would be open to Kant to say that construction gives the objects of arithmetic and algebra. J. Michael Young seems to me reasonable in describ-

Through the determination of pure intuition we can acquire a priori knowledge of objects. as in mathematics. but only in regard to their fonn, as appearances; whether there can be things which must be intuited in this form, is still left undecided. Mathematical concepts are not, thereforeyf. by themselves knowledge, except on the supposition that there are things which allow of being presented to us only in accordance with the form of that pure sensible intuition (B 147).

One possible resolution would be to admit an ambiguity in the phrase 'knowledge of objects'; mathematical knowledge, unaided by philosophy, is lmowledge of objects in a weaker sense, in which the objects known are forms such that so far as mathematics is concerned it is "left undecided" whether they are the forms of real objects. That suggestion seems to commit Kant to mathematical objects. But other versions of this resolution might make Kant something like what is nowadays called a modalist; that is, constructibility of a concept would entail the possible existence of a (Physical) . object of the form involved, but the notion of possibility would have to be atrenuated when compared with that explicated in the Postulates; it would be a version of what recent writers have called mathematical possibility. 6

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The resolution most immediately suggested by this passage, however, would still leave mathematical knowledge as knowledge of objects in the full-blooded sense. But although mathematical demonstration would yield knowledge of such objects (since the supposition Kant mentions is true), it would not establish that the concepts involved are objectively real. This suggestion still leaves open the interpretation of quantifiers in mathematics, and thus seems to require either one of the two other solutions mentioned above,

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or the more extreme view of Thompson that a Kantian canonical language for mathematics would not contain quantifiers ar all and would express generality only by free variables. 7 Something like the second of these three solutions may be read into the above-cited remark by Schultz, but more direct evidence that Kant faced the issue of the "ontology" of mathematics is lacking. It is instructive to ask, however, whether Kant could have adopted the first solution and accepted mathematical objects, as he indeed seems to do in some passages cited above. He would have to acknowledge a use of quanti.fiers wider than over 'objects' in his full sense of objects of experience, but his conception of the "logical

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ing the construction involved in the intuitive verification of '7 + 5 = 12' in

the second edition/Introduction as ostensive. 9 But although Kant does speak of seeing the number 12 come into being (B 16), what is constructed is clearly a set or configuration of twelve objects. In passing in this passage, and more explicitly in the Schematism (A 140 ; B 179), he refers to such a configuration as an image of the number (see below). We shall see below that Kant's remarks about number frequently sbow a conflation of the notions of a, particular number n and of a set of n objects. This may have prevented him! from facing the question whether numbers, strictly speaking, can be constructed in intuition. It is noteworthy that both in the Introduction and in the Axioms of Intuition Kant focuses on singular propositions about numbers, so that the question how to interpret generalizations about them is not raised. It is at the latrer point that we ourselves are inclined to see the problem of "ontological commitment to numbers" as arising. Young suggests that Kant tuight regard statements about numbers as statements about finite sets, but he considers only a singular example. 10 In one way or another, Kant must regard some objects of arithmetic and algebra as at a conceptual remove from the intuitions that found statements about them. This, rather than his conception of existence, seems to me to be the most principled difficulty in the way of Kant's adopting the "mathematical-objects picture". In some cases, such as rational nwnbers, it seems that

Kant would fall back on the notion of symbolic construction. Positive and negative rational numbers are talked of in the context of a calculus, in which

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there are definite rules for manipulating expressions of the fonn ± ~. where m and n range over natural numbers. By adding symbols for roots, we can similarly accommodate algebraic real numbers. Kant did, however, make a distinction of status between rational and irrational numbers. When, in a letter of September 1790, August Wilhelm Rehberg asked why the understanding cannot ''think..J2 in numbers", II Kant does not challenge the formulation; for him "number" meant primarily "whole number". It is geometric construction that shows that the concept of ..J2 is not empty, but such a root is "not a number, but ouly the rule of approximating of it".12 But Kant's remark that such a quantity "can never be completely thought in numbers" suggests that the representation of a rational number as 7 does allow us to think it completely in numbers. However this may be, the geometric construction once again yields not..J2 "itself' but rather a representative of..J2, in the form of a pair of lines whose length has ratio ..J2. But then the question arises what, if anything, it means to speak of ..J2 "itself'; this question would lead us into issues about mathematical objects that Kant did not consider.

passage, that a judgment is singular, and its subject concept has singular use, if it has in the subject a demonstrative or the definite article. l6 Thus singular judgments, like universal and particular ones, would have an expression of quantity, in effect a quantifier. But Kant does not offer a theory of proper names. The above passage indicates a clear enough distinction of a formal-logical kind between singular and other judgments. By comparison the justification in the Critique of Pure Reason for including singular judgments in the table is unclear and appeals to epistemological considerations (A 71 = B 96). At all events singular judgments are not at center stage in Kant's logic. Where the singular/general distinction is fundamental in Kant is not in formal logic but _ in the distinction between intuitions and concepts. I will not venture to explicate the category of unity as Kant understands it. In the Table of Categories, we already find notions of whole and part. The categories of quantity are unity, plurality, and totality (A 80 = B 106). In Kant's explanations of these categories, in the Critique and elsewhere, the whole/part notions dominate over those of logical quantity. In view of the fact that number arises from these categories according to Kant, it is disappointing that their connection with logical quantity is not more clear, although Kant is explicit enough about the connection of totality with universality, as we shall see. 17 Modern analyses of number have connected it closely with quantification, but this is not a matter about which Kant achieves much clarity. It requires some explanation to see how the unity, plurality, and totality of the Table are related to the notions of quantity explored in the Axioms of Intuition, which officially presents the principle governing the schematized categories of quantity. Indeed, how Kant understands the categories of quantity as pure categories is not entirely clear. Although Kant's explanations are often obscure and sometimes inconsistent with one another, the issues involved in both these matters concern a subject of modern discussion, namely the relation of the set/element relation to the wholelpart relation. To leam more about how Kant understood notions of quantity, whole, and part, we will turn to Kant's Reflections attached to the sections of Baumgarten's Metaphysica dealing with whole and part l8 and to the notes from Kant's lectures on Metaphysics, which generally contain a section corresponding to the same place in Baumgarten. 19 In the Axioms, Kant tells us that an extensive quantity is one "in which the representation of the parts makes possible that of the whole (and therefore necessarily precedes it)" (A 162 = B 203). "All appearances are intuited as

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Kant's conception of the categories of quantity combines two kinds of notions: "quantity" as understood in logic in his time, and conceptions of whole and part. The connection between these two kinds of ideas is not very clearly made. The first is reflected in the Table of Judgments, in which judgments are classified with respect to quantity as universal. particular, or singular (A 70 = B 95). In the universal and particular cases, quantity is what we would express by the quantifiers 'all' and 'some'. Kant's conception of a singular judgment is less clear. It would be most natural to us to count as a singular judgment one of the form 'a is B', where a is a singular term, and indeed Kant gives such examples. 13 But in the language of concepts, that would suggest that a singular judgment is one in which a concept of a different type (or perhaps even not a concept at all, but an intuition) is the subject. Kant repudiates this suggestion in saying that it is not concepts but their use that can be singular. l4 Kant gives his most explicit explanation when, after talking of the use of the concept house in universal and particular judgments, he remarks: Or I use the concept only for a single thing, for example: This house is cleaned in such and such a way. It is not concepts but judgments that we divide into universal, particular, and singular. IS

This would suggest, as Alan Sharnoon remarks in commenting on this

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aggregates (multiplicities) of previously given parts" (A 163 = B 204). The term translated 'multiplicity' is Menge, later used by Cantor and now the standard German term for set. How it should be translated in Kant is a problem; 'plurality', 'collection', and 'multitude' are also possibilities; my choice of 'multiplicity' is somewhat arbitrary.20 It is suggested by the fact that in one place Kant equates Menge with the Latin multitudo.21 In the Axioms. where what is primarily at issue is the schematized categories of

quantity, Kant is talking of the relation of extended objects to their spatial parts. What Kant calls an aggregate or multiplicity is therefore closer to a mereological sum. This spatial model is evidently not conceived by Kant to be the only form taken by the schematized categories of quantity; indeed he generally, though not always, regards time as more fundamental than space. We shall return to this question. Let us turn now to the pure categories of quantity. Kant says that totality "is nothing but plurality considered as unity" (B I II). This should remind you of Cantor's explanations of the notion of set. 22 This is reinforced by the following remarks from lectures on metaphysics:

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Vieles. insofem es Eins ist. ist die Allbe-it. Id, in quo est omnitudo plurium. est totum. 23

Kant does not distinguish very clearly between the whole/part and the set/element relation. I will show, however, that there is some basis, even though not clearly articulated, for Kant to make such a distinction. Something like the latter relation is needed to make sense of the relation of the categories

to the concept of number. Kant's most elementary notion concerning whole and part is that of a compositum, which seems to be simply an object in which parts can be distinguished. He is concerned to distinguish compositum from quantum, in which the parts must be homogeneous,24 but also from totum. 25 The latter distinction is not too clear. Two distinguishable ideas are that a totum is not part of

something futther, or at least not represented as such?6 and that the concept of a tatum involves unity of the plurality of parts. 27 The former idea seems to dominate in the Reflections on these matters, the latter in the explanation of the category of totality in the Critique (B I II; but cf. B II 4). It is the former

idea that has an obvious connection with the universal form of judgment. Nonetheless it seems to me that it is the latter idea that is the more interesting one, and the more relevant to the concept of number. It is the idea that is found later in Cantor. If a whole of parts is thought of as one object, to which, however, a definite conception belongs of what the parts are, then a set of parts is at least determined. Where the objects are spatio-temporal, what

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distinguishes a sum from a set is precisely that the latter has definite elements. One Kantian manner in which what we would call the elements might be given is by a concept. In fact we find Kant saying: A thing can be seen as a compositum (in a series) but without totality (of aggregate). Therefore the concept of the compositum is not yet that of a totum. To be a quantum requires homogeneity, to be a compositum not. The totum is always considered as a quantum according to a certain concept. Totality belongs to the concept of a compositum as homogeneous, that is as quantum (Refl. 5843).

What this passage suggests is that the "homogeneity" of parts that will make a compositum a quantum is their falling under a common concept; then that concept impatts unity to the plurality of parts, so that they constitute a tatum. On this reading, the tatum is determined by the set of parts falling under the concept in question. But now we would distinguish a whole that has a certain set of parts from the set of parts itself; indeed, the concept defining the set might naturally allow the whole as an "improper" part, so that it will be an element of the set. But the notion of a whole also suggests a different role for a concept, namely a sortal concept that the whole object falls under. Then there will be derivative concepts applying to the parts, marking them as parts of this whole, or of a whole of this kind. It is not easy to see how the parts can be "homogeneous" except by falling under some such derivative concept, which returns us to the first reading of the above passage.28 We have been considering remarks of Kant that are on an abstract level and could plausibly be taken to be explicating pure categories. In the case of spatio-temporal objects, however. Kant evidently thinks that spatio-temporal extension itself constitutes the basic form of division into homogeneous parts. To that extent, the parts of an object are homogeneous simply by virtue of being parts of that object, and clearly there is a deeper homogeneity of the spaces occupied by the parts; because the representation of a spatial whole is a result of synthesis, the synthesis is of the sort Kant calls "mathematical" (B 201 n.). Both of these forms of homogeneity will be bound up with a third. there being some concept that offers unification of the second of the two types mentioned above. 29 Kant evidently intended his definitions concerning quantity to cover both discrete and continuous quantity, and the distinction seems still to be defined in abstraction from space and time:

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A quantum by whose magnitude the multiplicity of its parts is undetermined, is called a continit consists of as many parts as I wish to give it; it does not consist of individual parts. On the other hand, every quantum through whose magnitude I wish to represent the multiplicity of its parts is discrete. 30 A quantum through whose concept the multiplicity of its parts is deteIlIlined, is discrete; one through whose concept of quantity the multiplicity of the parts is in itself undetermined, is a continuum. 3l

uum;

Note that Kant says that in the case of a continuwn the multiplicity (Menge) of parts is undetermined. Kant certainly held the pre-Cantorian view that "number' means finite number. 32 If a quantum were a continuum if its

concept did not determine the number of parts, that would then make every quantwn with infinitely many parts a continuwn. That does not follow from what Kant says. What he evidently means is that the concept of a continuum does not determine what the parts are. Although these definitions are abstract, space and time are of course the paradigms of continua. Kant considers the parts of space and time to be spaces and times, rather than points. It follows that they do not have simple parts; presumably, since they can be divided in arbitrary ways, neither has a definite set of parts. 33 In his theory of matter, Kant in effect holds that objects in space are similarly continuous. Of course the application of arithmetic, and even the development of the mathematics of continuity, require that some quantities be identified as discrete. Evidently Kant acconunodates this by making what are the parts of a quantum depend on how it is conceived, as for example in the above quotation from Reflection 5844. If, as Reflection 5847 has it (see note 33) all real quanta are indefinitely divisible, it must be that the concept that "determines" the parts of a discrete quantity does not stop further division; that is, further division is possible, although the resulting parts would not any longer fall under the concept. That must be the situation if we are to make sense in these terms of attributions of cardinal nwnber. Kant sometimes regarded this

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mereological swn, conceived as having individual people as parts (as opposed to some other conceivable division). Elsewhere Kant describes a "discrete quantity per se" as one "in which the number of parts is detennined arbitrarily by us".35 The text goes on: Number is therefore called quantum discretum. Through number we represent every quantwn as discrete.

This situation evidently results from combining the dependence on a concept, of a division into parts that gives a definite number and the taking of this concept as not intrinsic to the quantum. In fact Kant goes further in treating number as dependent on our representation. But some backtracking will

be necessary before we can go into this. Up to now we have concentrated on Kant's purely abstract discussions of part, whole, and quantity; to all appearances these notions belong to the pure categories. Some considerations concerning space and time have, however,

crept in. When Kant begins to talk of number, the amount that can be said on the pnre categorial level seems to be very limited. Already in the Inaugural Dissertation (§ 0, Kant finds an abstract intellectual conception of the composition of a whole of parts to be possible, but to "follow up" such a conception and represent it in the concrete involves temporal conditions. Thus it is one thing, given the parts, to conceive for oneself the composition of the whole, by means of an abstract notion of the intellect; and it is another thing to follow up this general notion, as one might do with some problem of reason, through the sensitive faculty of knowledge, that is to represent the same notion to oneself in the concrete by a distinct intuition. The former is done by means of the concept of composition in general, insofar as a number of things are contained under it (in mutual relations to each other), and so by means of ideas of the intellect which are universal. The second case rests upon temporal conditions, insofar as it is possible by the successive addition of part to part to arrive genetically, that is by SYNTHESIS, at the concept of a composite, and in this case falls under the laws of intuition. 36

concept as not intrinsic to the quantum, so that a quantum that is continuous if one considers its possible divisions into parts, can be considered as, dis-

The same duality arises again when, in § 12 of the Dissertation, Kant refers to the concept of number:

crete:

In addition to these concepts there is a certain concept which in itself indeed is intellectual, but whose actuation in the concrete (actuatio in concreto) requires the assisting notions of time and space (by successively adding a number of things and setting them simultaneously beside one another). This is the concept of number, which is the concept treated in ARITHMETIc.J7

Quantum discretum is that whose parts are considered as units; that whose parts are considered as multiplicities is called a continuum. We can also consider a continuum as discrete; for example. I can consider the minute as unit of the hour, but also as set which itself contains units. namely 60 seconds. 34

If, stretching Kant's explicit formulations, we allow non-connected "objects"

to count as wholes, we can accommodate the assignment of cardinalities in the physical realm: the nwnber of people in this room would attach to their

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I shall not to sort out what, at this stage, belongs to the abstract concept and what to its "actuation in the concrete". From Kant's later critical standpoint, any construction that would yield models of mathematical notions such as that of number will involve the fonns of intuition; this seems to be true

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even of the most basic notion of a compositum. In the Critique of Pure Reason, the starns of the pure categoriaI notions, and in particular the relation of number to the pure categories, is obscured by Kant's characterizing number as the schema of quantity (A 142 = B 182) and by the fact that most of Kant's explanation of notions of quantity occurs in the Axioms, where he is principally concerned with the schematized categories. Later texts rerum to a position closer to that of the Dissertation, as we shall see. The problem Kant faces is how much beyond some basic definitions he can develop without construction~ which on his own account W1l1 involve intuition. With respect to number, a further factor is that he tends not to distinguish a multiplicity's having a certain number from our knowledge of that fact; indeed from the point of view of transcendental idealism the two should

In the Schematism, Kant uses a numerical example in the course of explaining the notion of schema and distinguishing it from that of image (Bild). If I put five points one after another, he tells us,

be essentially connected. He tends even to characterize number in epistemic tenus: To know a multiplicity distinctly by adding of unit to unit is to count. A number is a multiplicity known distinctly by counting. 38

Very often, when Kant talks of the relation of number and arithmetic to time, time seems to play the role of a subjective condition of apprehension. Needless to say, this does not strengthen Kant's case for the view that arithmetic is synthetic and dependent on inmition. On this matter, I have already written elsewhere.39 The above citation illustrates another phenomenon that is frequent in Kant's remarks about number. That is that he tends not to distinguish, for a given number n, between a "multiplicity" with cardinal number n and the number n itself.4Q This conflation illustrates the lack, discussed above, of an articulated theory of mathematical objects in Kant, and with respect to the idea of ostensive construction of numbers may have contributed to it. Note also that by 'number' Kant evidently means primarily cardinal or ordinal number, at all events whole number as opposed to what we would call rational, real, or complex number. III

I shall now mrn to the discussion of number in the Schematism and to the texts of 1788-90 that seem to be inconsistent with it. Kant appears in the Schematism to reject the idea expressed in the Dissertation and implicit, though not consistently held to, in the Metaphysics lecrnres, of describing the concept of number in terms of the pure categories.

this is an image of the number five (A 140 = B 179). Its relation to its object will seem to us qnite different from that in the other cases he mentions, such as the concepts of triangle and dog (A 14 I = B 180). At all events he continues: But if, on the other hand. I think only a number in general, whether it be five or a hundred. this thought is rather a representation of a method whereby a multiplicity, for instance a thousand, may be represented in an image in confonnity with a certain concept, than the image itself.

It is not entirely clear whether he is here describing the thought of number in general, that is the entertaining of the general notion of namraI number or giving a general description of the thought of a particular number (so that it is the description, rather than the thought described, that is general over the natural numbers). The former reading seems to me slightly more likely. However, even the thought of a particular number will have to be distinguished from an image of it; moreover, the thought of a number as large as 1000 will in practice have to involve general operations on numbers. However, even for a number like 5, for which there is no difficulty in obtaining the sort of thing Kant calls an image, we do not have "a method of representing a multiplicity in an image in confonnity with a certain concept", unless the multiplic~ itself is detennined by a concept, in the example at hand something like dot _the page. This is just Frege's point that a number attaches to a concept.41 We have already seen Kant wrestling with this issue and attempting to fit it into a conception of "multiplicities" based on whole/part ideas. It is curious that when Kant comes to enumerate the schemata of the individual categories, it is only for the categories of quantity that he describes an "image", and what!)e says does not exactly fit what he has said previously: The pure image of all magnitudes (quantorum) before outer sense is space; that of all objects of the senses in general is rime. But the pure schema of magnitude (quantitatis), as a concept of the understanding, is number, a representation which comprises the successive addition of homogeneous units. Number is therefore simply the unity of the synthesis of the manifold of a homogeneous intuition in general, a unity due [0 my generating time itself in the apprehension of the intuition (A 142 - 3 = B 182).

No doubt what is meant by calling space and time "pure images" of quanta is that their strucrnre relevant to the application of the categories of quantity can

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be represented by spatial or temporal strucmre. In particular, the image of a number in the sense of the previous passage will be spatio-temporal. Indeed, Kant's emphasis on successive addition in deSCriptions of the concept of number makes it possible that here he conceives the image to be essentially temporal: the points are an image of the number five by being put one after the other (hintereinander): thus, they constitute an image of a number by virtue of being generated in succession.42 Kant at this time seems to have rejected the distinction of the Dissertation between the "intellectual concept" of number and its "actuation in the concrete". The abstract conception of whole, part, and quantity is little in evidence in the Critique, in particular not where number is discussed. Nonetheless, the identification of number as a schema would have its difficulties, for it attributes a temporal content to the notion of number itself. Kant may have been prepared to accept this consequence, for more than one possible reason: any construction that would give rise to the series of numbers would generate them successively, each one by addition of one more from the previous ones. In particular, coming to know the number of a multiplicity by counting involves the generation of a sequence (of acts or tokens) isomorphic to the numbers up to a given one. On transcendental idealist grounds, Kant might have resisted the distinction between a multiplicity's having a certain number and the condition being fulfilled for our knowing this in a canonical way. But the strongest reason would probably have been his conviction of the necessity of construction for arithmetic.43 There is also a conceptual gap which, whether or not Kant was conscious of it, makes his definitions of discrete quantity fall short of capturing the notion of finite quantity, which he would need for his own conception of number. A discrete quantity, as Kant defines it, will have a definite number of parts, but there is no necessity that this number should be finite; in fact, on this level Kant does not offer much of a conceptual basis for comparing magnitudes and for formulating answers to questions about the magnitude of particular quanta.44 Kant's appeal to "successive repetition" was possibly an attempt to capture the notion of finiteness. Consider:

number, even though other remarks of his reject it. But he does not take the key step taken by Cantor: giving a general definition of when two sets have the same cardinal number, and what he says about greater and less is somewhat crude.45 But even when all this has been done, two further steps need to be taken for a set-theoretic theory of cardinal number: the notion of cardinal has to be related to that of ordinal; from Cantor on it has been accepted that an informative answer to the question of the cardinality of a given set will place it in the sequence of ordinals.46 Second, finiteness has to be characterized. The finite ordinals and ordinals in general are often explained in terms of different notions of iteration; finite iteration is an abstract counterpart of the notion of successive repetition. But to describe it in abstract tenns was quite beyond the logical and mathematical resources of Kant and his contemporaries; the task was first accomplished in the 1880's by Frege and Dedekind. Whatever considerations may have motivated Kant's position of 1781, in some later texts he returns to a view close to that of the Dissertation, and holds that at least some essentials of the concept of number are intellectual and presumably derive from the pure categories. This may have been made possible for him by his reworking of the Transcendental Deduction for the second edition of the Critique, with its distinction between a more abstract level of the argument, presented in §§ 15-20, which considers the synthesis of a given manifold of intuition in general, without making any assumptions about our particular forms of intuition, and the application of these abstract considerations to our forms of intuition, in the argnment of §§ 24-26. In particular, Kant distingnishes in this context between intellectual and figurative synthesis (B lSI). The former is that "which is thought in the mere category in respect of the manifold of an intrtition in general". How this new formulation works out for the categories of quantity and the notion of number is not very explicit in the second edition of the Critique. It is reasonable to conjecture, however, that Kant saw in the notion of intellectual synthesis a framework into which to fit the abstract conceptions of quantity developed in his lectures. Note that he characterizes the concept of a quantum as "the consciousness of the manifold [and] homogeneous in intuition in general" (B 203).47 So far, Kant has not pnt the concept of number into his framework. But that is just what he seems to do in his letter to Johann Schultz of November 25, 1788. There he says that arithmetic has for its object ''merely quantity (Quantitiit), i.e. a concept of a thing in general by determination of magnitude",48 and he goes on to say:

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The concept of magnitude in general can never be explained except by saying that it is that determination of a thing whereby we are enabled to say how many times a Wlit is posited in it. But this how-many-times is based on successive repetition, and therefore on time and the synthesis of the homogeneous in time (A 242 = B 300).

We might compare the situation with that obtaining once we have the settheoretic notion of cardinality. In his definition of discrete quantity and identification of number with it, Kant leaves open the possibility of infinite

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Time, as you quite rightly remark, has no influence on the properties of numbers (as pure determinations of magnitude), ... and the science of number, in spite of the succession, which every construction of magnitude requires, is a pure intellectual synthesis which we represent to ourselves in our thoughts. 49

understanding of a number" and that the understanding "makes for itself the concept of;/2 arbitrarily".53 No synthesis in time is required for the mere concept of the square root of a positive quantity; even the impossibility of a

Kant might seem to be responding to the point, later much emphasized by Frege, that the concept of number applies to objects in general, independently of such conditions as those Kant associates with sensibility. But althongh, according to Kant, we may have such an intellectual concept of number, it is applicable only to sensible things (sensibilia). This much would, however, be to be expected if what is at issue is application to yield knowledge of objects in the full sense. But what Kant says by way of argument for it may just as well include pure mathematics:

quantity".

Insofar as quantities are to be determined in accordance with it [the science of number]. they must be given to us in such a way that we can take up their intuition successively, and so this

taking up must be subjected to the condition of time, so that we can still subject no object to our 5o estimation of quantity by numbers except that of our possible sensible intuition.

Kant thus leaves doubt how much of a "science of number" there can be without intuition and time; it is not entirely clear that the difference between his position here and that of 1781 is more than terminological. Kant's response to Rehberg seems, however, to be more emphatic.

Rehberg challenges the formulations of the Schematism. He admits that the application of arithmetical truths to sensible appearances would be subject to the condition of time, but he claims that to see the "truth of the arithmetical propositions themselves" no intuiting of the form of sensibility is necessary

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since no intuiting of time is required, in order to carry out arithmetical and algebraic proofs, which are rather immediately evident from the concepts of numbers, and only require sensible signs, from which the concepts are recognized during and after the operation of the understanding. SI

He expresses puzzlement as to why "the understanding, in the generation of numbers, which are a pure act of its spontaneity, is bound by the synthetic propositions of arithmetic and algebra". In particular, the form of our sensibility does not prevent us from "thinking ;/2 in numbers" in the way in which the nature of space prevents us from thinking "straight lines that would be equal to certain curved ones". The problem Rehberg raises is, in effect, that of the difference between geometry as a theory of space, and arithmetic, whose relation to space and time must on any account be more indirect, a

perennial problem for the interpretation of Kant's philosophy of arithmetic.52 In reply, Kant seems to concede the existence of a "mere concept of the

square root of a negative quantity can be known "from mere concepts of As soon as, however, instead of a,54 the nwnber of which it is the sign is given. in order not merely to designate its root, as in algebra, but to find it. as in arithmetic, the condition of all generation of numbers, namely time, is unavoidably presupposed.55

This remark expresses a constant view of Kant, that time is involved necessarily in mathematical construction. at least ostensive construction. This

holds for geometry as well as arithmetic, as is indicated by remarks to the effect that thinking of a line involves "drawing it in thought" (B 154). However, one might find in it the startling view that algebra, and therefore presumably symbolic construction, is independent of conditions of time, at

least as regards its objects. Could we go on to say that the "science of number" which in 1788 was said to be a "pure intellectual synthesis" is in fact just algebra, where one crosses the line from algebra to arithmetic and

constrains one's objects by the forms of intuition, as soon as one undertakes to calculate actual values of algebraic expressions for particular given arguments? If so, Kant missed an opportunity to say so in the letter to Schultz, where in fact there is no word of symbolic constr:uction; instead he says that

this pure intellectual synthesis is one which "we represent to ourselves in thoughts". Further doubt on such an interpretation is cast by one of Kant's prelimi-

nary sketches for his reply to Rehberg. There he says that although the objects of arithmetic and algebra are "with respect to their possibility not under conditions of timen, such conditions do govern the construction of the concept of quantity in their [the objects'J representation through the synthesis of imagination, namely composition. without which no object of mathematics can be given.56

So far, the force of Kant's remark could be limited to ostensive construction. But he goes on to characterize algebra as the art of bringing under a rule the generation of an unknown quantity through numeration (Zlihlen), independently of every actual number, only through the given relations of the quantities. This quantity to be generated is always a rule of numeration . .. 57.

Since they differ in emphasis from the actual letter, these remarks do not necessarily represent Kant's considered position. But it is hard to imagine his

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ARITHMETIC AND THE CATEGORIES

having written them if he had consistently in this period thought of algebra as containing only purely intellectual concepts. Kant evidently found suggestive the fact that a geometric construction was needed to give an adequate intuitive representation of an irrational quantity; it fit in neatly with the view of the Refutation of Idealism that space and time are interconnected in such a way that consciousness of things in space is necessary for me to locate myself in objective time. Both Reflection 13 and the last paragraph of Kant's letter argue that space and time are interconnected in mathematical construction. With respect to the concept of number, Kant in one text argues that both space and time are necessary to the determinate representation of a number:

foundations of mathematics, with logicism and set-theoretic realism emphasizing the former, and the different forms of constructivism the latter. The fact that these differences persist should remind us that the task of striking the right balance in describing this duality has not become an easy one even after the great advances in the foundations of mathematics since Kant's time.

152

We cannot represent to ourselves any number other than by successive enumeration in time and then taking together this mUltiplicity in the unity of a number. This laner, however, cannot occur otherwise than by my putting them next to one another, for they must be thought as given at the same time, that is, as taken together into a single representation; otherwise this representation fonus no quantity (nwnber).58

That by 'next to one another' he means next to one another in space is clear from the context. The conclusion to be drawn from examining these texts, in my opinion, is that Kant did not reach a stable position on the place of the concept of number in relation to the categories and the forms of intuition. One could find connections between this difficulty and other problems in Kant's philosophy, for example that concerning the status of the "intellectual" representation '" think" (B 423 n.). As regards arithmetic, one might take Kant's problem to be solved by a modem distinction between, on the one hand, characterizing the natural numbers as an abstract structure and developing "arithmetic" as the theory of what must be true in such a structure and, on the other hand, actually constructing an instance of the structure (or some initial part of it). The former would belong to the realm of "mere concepts", and neither time nor anything else Kant would regard as involving intuition would be part of its content. Time would enter as a condition of construction, for example, such that models for the numbers can be constructed in it if any can be constructed at all. 59 In its general lines, this seems tome a defensible position about the relation of the intuitive and the abstract with respect to arithmetic. 60 But there is no moat division of labor, as is shown by the role of calculation in developing the consequences of an abstract characterization of the structure of numbers.61 More generally, the duality of abstract conceptualization and intuition in mathematical thought is exhibited in the philosophical differences about the

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HaJi1ard University NOTES

* Earlier versions of this paper were presented at the Robert Leer Patterson conference on Kant's philosophy of mathematics at Duke University in March 1983 and at colloquia at Columbja University and the Graduate Center of the City University of New York in May 1983 and March 1984 respectively_ r am indebted for their comments to all three audiences. The questioning of Jerrold J. Katz and others at CUNY concerning the relation of ideas of whole and part to space and time influenced the final version considerably. I wish to thank Dieter Henrich for a helpful conversation, and lowe a special debt to Carl Posy, without Whom the paper would not have been written. The Critique of Pure Reason is cited according to the pagination of the original editions, in the standard way. Quotations are in Kemp Smith's translation, sometimes modified. Other references to Kantian texts are to Kants Gesammelte Schriften, edited by the Prussian Academy of Sciences and its successors (Reimer, later de Gmyter, Berlin, 1900- ), abbreviated as Ak. Translations, unless otherwise stated, are my own. My own 1969 paper, 'Kant's philosophy of Arithmetic', is cited according to the reprint (with Postscript) in Mathematics. in Philosophy (Cornell University Press, Ithaca, N.Y., 1983). 1 On this point see § 2 of my 'Objects and Logic', The Monist 65 (1982),491-516. 2 Kant's notion of an object of experience as explicated by the schematized categories does give place to one type of "object" that is at least not a spatia-temporal thing, namely the accidents or states of substances. This would license an analogous shift in the use of 'object'. Probably Kant thought of the "forms" of object as quanta as similarly provided for by the categories of quantity. 3 See Manley Thompson, 'Singular Terms and Intuitions in Kant's Epistemology', this volwne, 81-107, pp. 99 -101, also 'Kant's Philosophy of Arithmetic" Postscript, p. 74. 4 Review of Vol. IT of Eberhard's Philosophisches Magazin (1790), in Ak. XX, 386 n. This review was written in close collaboration with Kant and is panly based on manuscripts by Kant; however, the passage quoted does not occur in those manuscriprs. 5 Cf. Thompson, op. cit., p. 101. 6 For example Hilary Putnam, 'Mathematics without Foundations', The Journal of Philosophy 64 (1967), 5-22, reprinted in Mathematics, Matter, and Method (Cambridge University Press, 1975, 2d ed. 1980), pp. 43-59. esp. pp. 49, 58-59; also my Mathematics in Philosophy, esp.

pp.21-22, 183-186. Though it is stricter than the notion of logical possibility that does occur in Kant, such a notion of possibility would still have a formal character. 7 Op. cit., pp. 99-100, but Thompson expresses this position with some diffidence. Thompson suggests 'there is constructible' as a reading for the particular ("existential',) quantifier in mathematics, but says that since one must "see (intuit) the consnuctibility, ... There is no need for a

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special symbol by which one represents discursively (asserts that there is) the constructibility one must intuit" (97-100). This seems to me to be a non sequitur. Nonetheless, Thompson's thoughtful discussion raises some difficult issues concerning Kant's. distinction between demonstrations and discursive proofs, which I have not attempted to deal with here. S ct. the comparison of Kant with Frege in 'Objects and Logic', pp. 494-495. A Meinongian view would be of course quite foreign to Kant. 9 'Kant on the Construction of Arithmetical Concepts', Kant-Studien 73 (1982), 17-46.

pp.30-31. 10 Ibid., p. 37. Of course we can now develop first-order arithmetic in a theory of finite sets; the essential ideas for this development were first discovered by Ernst Zennelo in 1908. Such a development has the uncomfortable feature that it singles out more or less arbitrarily a certain~ sequence of sets as "the" natural numbers. A more neutral procedure had been devised earlier by Richard Dedekind in his Was sind und was sollen die Zahlen? (Vieweg, Braunschweig, 1888). Dedekind reads statements about "the" natural numbers as general statements about any "simply infinite system", that is, structure satisfying the Dedekind-Peano axioms. But to develop number theory in this way requires either a second-order theory of finite sets or an axiom of infinity. There is, however, a third possibility which fits Kant's way of talking a little better. This would be to replace talk of numbers by talk of sets modulo cardinal equivalence. Instead of operations on numbers we would have operations on sets: disjoint union for sum, and cartesian product for product. Identity of numbers would be replaced by cardinal equivalence of sets. The prior question, how appropriate it is to talk of sets in the Kantian context, is discussed below. II Ak.XI206. 12 Letter to Rehberg, September 1790, Ak. XI 210. 13 Logik § 21 Note 1 (Ak. IX 102); Metaphysik Volckmann,Ak. xxvm 396. 14 Logik, § 1 Note 2, Ak. IX 91; Wiener Logik, Ak. XXN 909. See Thompson 'Singular Terms and Intuitions', pp. 83-84. 15 Wiener Logik, Ak. XXIV 909. 16 Kanr's Logic (Dissertation, Columbia., 1979), p. 85. 17 It has been disputed whether in the correspondence between the forms of judgment and the categories, Kant intended unity to correspond with singular judgment and totality with universal, as one would expect, or vice versa, as the order in the two tables of the Critique suggests. In my view Michael Frede and Lorenz KrUger have made a convincing case for the former correspondence; see their 'Uber die Zuordnung der Quantitaten des Urteils und der Kategorien der GrOsse bei Kant', Kant-Studien 61 (1970),28-49. 18 §§ 155-164, 'Totale et partiale', reprinted in Ak. XVII 58-61. Some. but not all, of Kant's analysis follows Baumgarten. The close connection between ideas of quantity and of whole and part is shared with Baumgarten; indeed it can be traced back to Aristotle's Categories. The role of a concept in conceptions about quantity (see below) is not in Baumgarten. The Reflections we cite are dated by Adickes between 1780 and the beginning of the 1790's; they are in Vol. XVIII of Ak. and are cited merely by number. Earlier Reflections are briefer and, on the whole, less independent of Baumgarten. (But see Note 28 below.) 19 The relevant sections, all in VoL XXVIII of Ak., are Metaphysik Volckmann (c. 1784/5), pp. 422-428, esp. 422-424; Metaphysik von SchOn (c. 1789190), pp. 504-506; Metaphysik L2 (WS. 1790/1). pp. 560-562; Metaphysik Dohna (1792/3), pp. 636-637; Metaphysik K2 (early 1790's), pp. 714-715. The passage from Metaphysik Lz agrees verbatim with the corresponding

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section of Immanuel Kants Vorlesungen fiber die Metaphysik, ed. by K. H. L. Politz (Etfurr 1821, reprinted Rosswein 1924 and Darmstadt 1964), pp. 31- 32 of the 1924 reprint. These materials are all cited merely by page number. With reference to notes from Kant's lectures, a statement such as "Kant says ... " should be regarded with caution; in my usage it should be regarded as an abbreviation for "Kant is reported to say ... ". 20 I am agreeing with Kemp Smith's translation of Menge at A 140 = B 179, but here he translates it as 'complex'. For my own purpose, a uniform translation is desirable. 21 Metaphysik Volckmann, p. 422. In his German translation of the Inaugural Dissertation, Klaus Reich translates multitudo in § 1 as Menge; see Kant, De mundi sensibiJis atque intelligibilis fomw et principiis (Meiner, Hamburg, 1958), pp. 4-5. 22 Especially Cantor's characterization of a set as "jedes Viele, welches sich als Eines denken Hisst", Gesammelte Abhandlungen, ed. by Ernst Zennelo (Springer, Berlin, 1932), p. 204. I am not pressing any claim of an anticipation of Cantor by Kant; rather, it seems to me that Cantor's explanations are based on older ways of thought and that ideas about whole and part are not entirely absent from his own conception of what a set is. Kant, however, associates Menge with the category of plurality (B Ill). This passage was pointed out to me by Pierre Keller. 23 Metaphysik L 2 , p. 560. 24 B 203, also RefL 5836, 5842. Ct. B 201 n. 25 Allheit, the third category of quantity in Kant's table, is rendered in Latin as totum in the above-quoted passage, but as totalitas in Refl. 5838. A distinction between the two might be made along the lines of that between quantum and quantitas (A 163 = B 204), but Kant does not do so very explicitly. 26 "A compositum, insofar as it is not a part, is a totum" (RefL 5834). 27 Ref!. 5833, 5840. In the fanner matter is said to be compositum, body totum. 28 In one text, Ref!. 4822 (1775/9, Ak, XVII 738), Kant complicates the matter further by saying that in a quantity (Grosse) the whole must be homogeneous with the parts. Here he seems to be thinking of "quantities" in the sense in which it is filled out by a mass term; his example is a quantity of money, and he seems to reject the idea of a quantity of ducats. On this conception, a quantity differs both from a mereological sum and from a set. 29 One would not expect Kant to conceive recognition of the same object at different times on the model according to which an enduring object is a whole that has as parts temporal "stages". Nonetheless, the mathematical representation of time as a line, on which Kant lays great stress, means that persistence through time will have some formal features of extension in space. 30 Metaphysik~p.56l.

31 RefL 5844. In both these passages, Menge could quite appropriately bave been translated 'set'. 32 In one place, however, Kant intimates a distinction between infinity in the sense of nonfiniteness, and unsurpassably large quantity: The former [the concept of the infinite] does not detennine at all, how large something is; however, the concept of maximum does determine quantity. The concept of the infinite shows that my quantum is larger than my power of measuring. Therefore 'God is the infinite being' does not say as much as 'God is the greatest being' (Metaphysik K 2, p. 715). 33 At one point, however, Kant seems to view this as characteristic of quanta in general:

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Every quantum is a compositum whose parts are homogeneous with it. Consequently it is a continuum and does not consist of simple parts (RefL 5847).

Here he goes beyond his usual characterization of a quantum in assuming that the parts are homogeneous not only with each other but with the whole, but the situation is not the special one envisaged in RetL 4822 (see Note 28). He is here apparently thinking of spatio-temporal quanta. 34 Metaphysik Volckmann, p. 423. The issue is complicated by a distinction made in this text between a quantity that is in itself discrete (an sich discretum) and a continuous one that is represented as discrete. It appears that only the latter case will occur in the realm of appearance, but the example of a bushel of corn as a quantum that is discrete because of having parts whose parts are heterogeneous may be intended to illustrate the notion of in itself discrete quantity. 35 Metaphysik L , p. 561. per se is reminiscent of an sich in the corresponding passage of 2 Metaphysik Volckmann (see Note 34), but the characterization of discrete per se is almost opposite. One or the other hearer may have misunderstood Kant. 36 Ak. II 387, trans. by G. B. Kerferd in Kant, Selected Pre-Critical Writings and Correspondence with Beck (Manchester University Press, 1968), p. 47. 37 Ak. II 397; Selected Pre-Critical Writings, p. 62. Reich translates actuatio in concreto as Darstellung im Einzelnen (op. cit., p. 35; see Note 21). 38 Metaphysik Volckmann, p. 423 (in Latin); cf. Metaphysik von SchOn, p. 506; also Metaphysik Dohna, pp. 636-637. 39 'Ka.rlt's Philosophy of Arithmetic', esp. pp. 58-71. But on the treatment there of Kant's conception of mathematical objects, see the Postscript, pp. 73-75, which Section I of the present paper amplifies. Cf.B 16; MetaphysikL 2 , p. 561 (cited above). 41 It is unlikely that is is this concept, rather than the concept of number in general or of a single number such as 5 or 1000, that Kant has in mind when he speaks of representing in an image "in confonnity with a certain concept". For it seems clear from the last sentence of the paragraph that it is the latter concept whose schema is being described; hence it must be the concept of totality or perhaps number. 42 In A 140 = B 179, Kemp Smith translates wenn ichfiinf Punkte hintereinander setze as "if five points be set alongside one another, thus losing the implication of successive "setting". 43 In talking of "symbolic construction" in algebra, Kant does say that algebra "abstracts completely from the properties of the object that is to be thought in tenos of such a concept of magnitude (A 717 = B 745). How far does this "abstraction" extend? Does it make algebm applicable . to objects in general, independently of the forms of intuition? If the role of intuition is only that the signs of a formal calculus are. objects of intuition, and the confonnity of steps to rules is intuitively checkable, then perhaps there is no reason to attribute to the operations any spatiotemporal content or to limit the applicability of algebra to spatia-temporal objects. No such limitation is suggested in Kant's first formulation of these ideas, in Untersuchung iiber die Deutlichkeit der Grundiitze der naturlichen Theologie und der Moral (1764, esp. Ak. II, 291292). With regard to applicability, in the 1788 letter to Schultz discussed below, Kant says quite unequivocally that mathematics is applicable only to sensible things (Ak. X 557). With respect to the content of pure algebra, the matter is less clear; see below. There is some support for the thesis of Alan Shamoon (Kant's Logic, p. 221 n.) that for that domain Kant still held in the Critical period the formalist view expressed in 1764. Shamoon's dissertation contains an interesting discussion of symbolic construction and its relation to ideas of Lambert. 40

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Concerning the Deutlichkeit, my own remaik. ('Kant's Philosophy of Arithmetic', p. 66) that it exhibits a cOImection in Kant's mind between sensibility and the intuitive character of mathematics before he developed the theory of space and time of the Aesthetic was aimed at Jaakko Hintikka's thesis that his own essentially logical analysis of the role of intuition in mathematical proof describes a '''preliminary'' or "earlier" stage of Kant's philosophy of mathematics, at which no connection between intuition and sensibility is made. (See his paper, 'Kant's Transcendental Method and His Theory of Mathematics', in this volume pp. 341-359.) In this cOlll1ection the reader's anention should be called to Mirella Capozzi Cellucci, 'J. Hintikka e il metoda della matematica in Kant', II Pensiero 18 (1973), 232-267. My remark is elabomted on pp. 241-243, but the paper contains a number of further criticisms of Hintikka's conception of a "preliminary" Kantian theory. She is perhaps the only one of Hintikka's critics to engage him on his own grounds, with respect to his use of a Euclidean conception of mathematical proof; see especially Section 7 on ekthesis and logic in Kant. 44 Since Kant's discussion of quantity comprehends the continuous as well as the discrete, knowledge of magnitude involves more than just determination of cardinalities (i.e. counting); it will involve measurement. I have not gone into such issues at all here. Some commentators have read the Axioms of Intuition as concerned with the possibility of measurement of physical phenomena See for example Gordon Brittan, Kant s Theory oj Science (Princeton University Press, 1978). Ch. 4. 45 E.g. Metaphysik Volckmann, p. 424; MetaphysikDohna, p. 637. 46 Hence the centrality to the theory of cardinals of the axiom of choice, which implies that every cardinality can be located somewhere in the sequence of ordinals, and of the continuum problem, which is the question where in the sequence of ordinals the cardinality of the continuum lies. 47 Kemp Smith translates 'consciousness of the synthetic unity of the manifold ... " following Vaihinger, who emended 'Bewussts in des mannigfaltigen Gleichartigen' to 'Bewusstsein der synthetischen Einheit des mannigfaltigen Gleichartigen'. (See Kant, Kritik der reinen Vemwift, ed. by Raymund Schmidt (2d ed., Meiner, Leipzig; 1930), p. 217.) As an interpretation, this seems to me reasonable enough. 48 Ak. X 555. 49 Ak. X 557. 50 Ibid. 51 Ak. XI 205-206 . 52 Cf. 'Kant's Philosophy of Aritlunetic'. With respect to the remarks there (pp. 51ff) about Leibniz's proof in the Nouveaux Essais of '2 + 2 = 4', it can now be observed that similar proofs of '8 + 4 = 12' and '3 X 8 = 24' are to be found in Herder's notes from Kant's lectures on Mathematics, Ak. XXIX 57-58. The lectures would have been in 1762/3, but the notes may be inauthentic or contain later additions by Herder, see Ak. XXIX 658. 53 Ak. XI 209.208. 54 That is, instead ofthe symbol'a'. 55 Ak. XI 209. " Refl. 13 (1790). Ak. XIV 54. 57 Ibid., emphasis mine. By Uh/en Kant evidently means something more general than counting; '.,f2' is called a Zeichen des Uhlens, because the concept it expresses contains a rule for approximating it by mtional numbers. :58 Refl. 6314 (1790). This is one of a group of texts in which Kant returns to the ideas of the Refutation of Idealism. For a discussion of them in that connection, see Paul Guyer, 'Kant's

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Intentions in the Refutation of Idealism', Philosophical Review 92 (1983), 329-383. S9 Cf. 'Kant's Philosophy of Arithmetic', pp. 67-68. 60 Cf. my 'Mathematical Intuition', Proceedings at the Aristotelian Society 80 (1979-80), 145-168; also 'Intuition and the Concept of Number', Seventh International Congress of Logic, Methodology, and Philosophy ojScience, Salzburg 1983, Abstracts, Section 1. pp. 31-33. 61 Cf. 'Kant's Philosophy of Arithmetic', pp. 66-67, and Young's discussion 'of calculation in 'Kant on the Consttuction of Arithmetical Concepts" esp. Section II.

CONSTRUCTION, SCHEMATISM, AND IMAGINATION

Kant maintains that mathematical judgments are synthetic - that we cannot ground them merely through reflection on their constituent concepts. Instead, he argues, we must construct those concepts, i.e., "exhibit a priori the intuition which corresponds to" them, grounding our judgments on what can be made evident only through such construction. (A 713/B 741)1 I first sketch an interpretation of Kant's doctrine, focusing on the construction of arithmetical concepts. I then go on to indicate how an understanding of Kant's view concerning arithmetical construction can shed light on his views concerning imagination. L CONSTRUCTION

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Kant's statement of his view concerning the construction of mathematical concepts is unfortunately opaque. The view itself is also more tangled than his simpler statements would suggest. For the notion of construction turns out to encompass two quite different cases: ostensive construction, on which arithmetical and geometrical judgments are said to rest, and symbolic construction, which is supposed to provide the basis for algebraic judgments. (A 717/B 745, A 734-5/B 762-3) If we focus, however, on what he says concerning ostensive construction, it seems plain what Kant's view is. He holds that arithmetical and geometrical judgments rest, in the end, on the representation of particular things that instantiate our concepts; for he thiuks that only through such representation can we exhibit the content of our concepts, thus making evident the truth of our mathematical judgments. Commentators have been understandably reluctant to take Kant's view at face value, for it is difficult to see how we can ground mathematical judgments, which are supposed to be a priori, and hence universal and necessary (B 3-4), by examining particular collections of strokes or particular geometrical figures. Not surprisingly, alternative interpretations have been developed. Gottfried Martin, drawing on the work of Kant's student Johann

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Schultz, argues that Kant's real if somewhat occluded insight was the recognition that both geometry and arithmetic can be set forth in axiomatic form, and that in both sciences proof depends not only upon definitions but also upon further propositions which we take to be necessary, but which can be denied without self-contradiction. 2 Jaakko Hintikka, developing a suggestion made by E. W. Beth, offers a somewhat different view. 3 He points out that the notion of intnition is essentially tied, according to Kant himself, only to the notion of singularity in representation, not to that of sensibility; for intuition is defined simply as singular representation, or as that which makes singular representation possible. (A 320JB 376-7, A 19JB 33) The connection between intuition and sensibility derives, not from the concept of intuition alone, but from what Kant takes to be an inherent limitation of human - or more generally, of finite, discursive - intelligence, viz., that it can represent particular things only through being sensibly affected. Emphasizing this point, Hintikka argues that what Kant meant, or should have meant, is not that mathematical knowledge depends upon the representation of particular sensible objects, but that it depends upon arguments of a specific kind, in which singular terms play an essential role. What Kant dimly realized, according to Hintikka, is that mathematical knowledge, both in geometry and in arithmetic, depends upon arguments which, when represented in first-order predicate logic, essentially involve the introduction of free variables, i.e., of singular terms which refer to a particular but unspecified member of a certain class of objects, and which thus make possible the drawing of conclnsions about all objects in that class. Both interpretations are intriguing, and both serve to illuminate important aspects of Kant's view. As Charles Parsons has convincingly argued, however, neither is satisfactory as an interpretation of Kant, since neither makes appropriate room for Kant's insistence that mathematical knowledge depends, in the end, on the representation of particulars in sensibility.4 Attempting to make sense of this insistence, at least in the case of arithmetic, Parsons point out that it is natural to think of the numbers as represented to the senses by means of perceptible numeral tokens.' For the numeral systems we use in representing the natural numbers characteristically provide not only names for the numbers, but also models of what we take to be the structure of the numbers themselves. In the system using Arabic numerals and base ten, for example, the numeral '12' serves as a name of the number twelve, but the sequence of numerals from 'I' through '12' also provides a model of what we take to be the structure of the corresponding numbers, since it has an initial element and a successor relation. This point

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holds, moreover, not only for numerals regarded as types, or as abstract mathematical objects, but also for concrete, perceptible tokens of the numerals. If we produce perceptible tokens of the numerals 'I' through' 12' on paper or chalkboard, we have a model of what we take to be the structure of the numbers one through twelve. Since the perceptible tokens can be used merely as representatives of an abstract structure, replaceable by any other instance of that structure, we can use them, as Parsons points out, to verify arithmetical propositions. Parsons makes a related point concetning familiar 'proofs' of elementary arithmetical truths.6 It seems clear that these do not constitute proofs in any sense of the tenn that Kant would recognize, for on Kant's view a proof wonld be a piece of reasoning consisting of judgments and inferences. The familiar 'proofs' are not pieces of reasoning, however, but instead are symbolic structures, structures which may form the subject-matter of reasoning, and which may also be used to support reasoning, but which are not themselves pieces of reasoning. As Parsons suggests, Kant would probably classify such 'proofs' as 'symbolic constructions''? Undoubtedly he would also make much of the fact that they involve constructions which exhibit or are isomorphic to the numbers and their relations, and which thus display or make evident, in a structure that we can perceive, the arithmetical truth being 'proved'. If we represent the numbers by means of a set of numerals defined through an initial element and a successor relation, for example, and if we embed the numeral system in a first-order theory with appropriate 'axioms' and ~rules of inference', we may then 'prove' that seven plus five is twelve. We do so by producing a construction in which the numeral representing the number twelve is replaced, through a series of steps, by the numeral representing seven plus five, and in which the formnla representing the trivial truth that twelve is twelve is transformed into one that represents the non-trivial truth that seven plus five is twelve. The 'proof' works, Kant would no doubt wish to point out, just because the twelfth successor of the initial element in the nwneral set is the fifth successor of its seventh successor, and because a sequence of five translations, licensed by the appropriate 'axioms' and 'rules', therefore resnlts in the replacement of the one numeral by the other. It workS, in other words, just because it exhibits or displays, in a structure that can be made perceptible, the relevant arithmetical truth. Far from refuting Kant's view that arithmetical knowledge rests on the construction of concepts, therefore, the existence of such 'proofs' serves rather to confirm it. 8 Parsons is quite right to insist that on Kant's view, arithmetical knowledge is supposed to rest, in the end, on the representation of sensible particnlars.

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He has succeeded, moreover, in indicating how, by using sensible instances of abstract structures merely as representatives of those struCtures, we may . establish truths about the abstract structures themselves, that are more than 'merely empirical' even though our knowledge of them may rest on concrete perception." Parsons has focused, however, on the possibility of our using symbolic constructions to verify arithmetical truths, while Kant's view, at least in the Critique, is that it is ostensive constructions that provide the ultimate basis for our knowledge of arithmetical truths.!O He does not think of the numbers as non-sensible objects, to be represented symbolically by means of sensible numeral-tokens. He thinks of number, rather, as tied to the concept of quantity, and what he sees as fundamental are therefore sensible constructions that exhibit or display the quantity in question. In the next section I will give some indication as to why he takes this view, and I will try to show that he is right in thinking that ostensive constructions can serve to verify arithmetical truths. Before doing so, however, I want to add a comment on the view that Kant might take of numerals. To the best of my knowledge Kant nowhere discusses either the nature of numeral systems or their role in arithmetical knowledge.!! I suspect that the view he would take, however, is that what is fundamental about such systems - at least those that are adequate for arithmetical purposes - is that they provide procedures by which we can generate ostensive constructions of numerical concepts. The familiar system using Arabic numerals and base ten, for example, provides a procedure by which we can generate an indefinitely long sequence of numeral-tokens. Each numeral has a determinate place within that sequence and can serve as an index of the finite sequence of numerals up to and including itself. What is most important about such sequences of numeral-tokens, Kant would probably argue, is that each such sequence constitutes at the same time a collection of perceptible objects. A sequence of perceptible tokens of the numerals' I ' through' 12' constitutes a collection of twelve perceptible objects, for example. Collections of perceptible numeral-tokens may thus be used, just as may collections of strokes or fingers, to display or make evident corresponding aritlunetical truths. By producing collections of perceptible tokens of 'I' Ihrough '7' and by joining them with perceptible tokens of ' l' thmugh '5', for example, we can make it evident that seven and five are twelve, just as we might do by using two collections of sensible strokes. What is important about sequences of perceptible numeral-tokens, as opposed to collections of other sorts of perceptible things, is simply that they are readily produced, identified, and distinguished, and that each such sequence can readily be indexed by a single one of its member,

viz., the final element. Commanding such a numeral system, we thus have the capacity to generate ostensive constructions by which to make evident elementary arithmetical truths. Because each numeral can serve as "JY'index of the cardinality of the sequence beginning with 'I' and ending with itself,

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moreover, we have the capacity to state arithmetical truths in ways that are economical and readily understood. In writing '7 + 5 = 12', for example, we

are asserting that any collection of the same cardinality as that which results from joining perceptible tokens of '1' through '7' with perceptible tokens of , I' through '5' is of the same cardinality as a collection of tokens of '1' through '12'. This last point is more important than it may first appear. Kant would want to argue, I suspect, that we could not claim to command the concept of number, nor would we be able to articulate or understand arithmetical truths, if we did not have a means for desiguating the quantities that are displayed in various collections of sensible objects. He would probably want to argue, therefore, that command of the concept of number reqnires employment of a numeral system. 12 He would also grant, no doubt, that there is a good deal more to be said about numeral systems than what I have indicated. Ostensive construction, though perhaps fundamental as a basis for knowledge, is also cumbersome, especially with larger numbers. One important feature of the numeral system using Arabic numerals and base ten is obviously that it facilitates calculation, and the connection between calculation and ostensive construction needs to be explored. My interest at the moment, however, is simply to point out that what Kant would see as most important in familiar numeral systems is their use in generating ostensive constructions. It is to the topic of ostensive construction that I now want to tum. I\. SCHEMATA

With these remarks as background, I turn to some of the questions surrounding Kant's claim that arithmetical knowledge rests on the ostensive construction of concepts. As Kant realizes, his claim seems to give rise to inconsistent requirements. On the one hand, to construct a concept ostensively is to represent, in intuition, something that exhibits that concept. In the case of arithmetical concepts, this requires that we represent particular collections of things, and hence that our representation involve sensible content. l3 On the other hand, as Kant says, though what we represent may be particular, it "must in its rep-

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resentation express universal validity for all possible intuitions which fall under the same concept". (A 713/B 741) In explaining how both requirements can be met, Kant refers occasionally to pure, a priori, or non-empirical intuition. Despite what the phrases suggest, careful reading makes it plain that he does not mean to posit a special sort of intuition that miraculously but quite mysteriously enables us to intuit mathematical objects and to ground a priori judgments. His claim is simply that even though we may represent only a particular thing or collection of things, our representation can nonetheless be universaL For we can represent it as an instance of a concept and as nothing but an instance of that concept. Given a concept of the proper sort, moreover, it is possible for us to work Qut, in a particular instance, consequences that will have to hold in all instances of that concept. When we construct an arithmetical concept, for example, Kant's view is that we represent a collection of perceptible particnlars that possesses the number in question, and that we represent that collection as nothing but a collection possessing that number. He also holds that in representing that collection as possessing that number we represent it as conforming to certain 'universal conditions of construction' for the numerical concept in question. (A 714-6/B 742-4) Whatever can be shown to hold of the particnIar collection in consequence of these conditions, he maintains, is thereby shown to

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We may perhaps illuminate Kant's view if we reflect on an example in which we find something analogous to arithmetical construction. To establish how many letters are in the word 'synthetic', we may simply write the word and count its letters. More carefully, we may generate a string of lettertokens, identify it as a (correctly spelled) token of the word, count the number of characters in the string, and thereby determine how many letters are in the word.

Admittedly, the example is humble. Admittedly, too, it is in some respects only an historical accident that the accepted spelling is what it is, and obviously that spelling may change. Still, so long as the accepted spelling remains as it is, any (correctly spelled) token of the word must of necessity contain nine letters. And this is something we can establish by examining a particular, perceptible object: the string of characters printed just above. On these grounds alone the example is interesting. What allows us to use a string of perceptible characters in this way is not difficnlt to see. In the first place, what we determine by counting is not merely the number of characters in a certain ostensively specified string, but

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the number of characters in a (correctly spelled) token of the word. Having determined this, moreover, we know at once how many letters are in the

word, and how many there must be in any other of its (correctly spelled) tokens. For this particular string of characters qualifies as a token of 'synthetic' only because, and only insofar as, it conforms to a rule specifying how that word is to be spelled. 14 We identify the particular string as conforming to this rnIe. We represent it, moreover, as nothing but a string of characters conforming to this rule, ignoring the sizes, colors, etc., of the letter-tokens. Since the rule requires exactly the same sequence of characters

in any token of the

word, and since the same sequence will obviously always contain exactly the same number, we see at once that any other (correctly spelled) token will have to contain nine letters, just as this one does. Besides indicating how we can reach universal conclusions through the

examination of particnlar instances, the example also sheds light on another important point. The rule that determines how a word is to be spelled specifies only what the sequence of letters must be in a token of a word. Since the sequence is unique, the number of letters obviously cannot vary from one token to another. The rule that determines how a word is to be spelled thus determines how many characters there must be in any of its (correctly spelled) tokens. We may be tempted to say that it follows from this how many letters are in the word. It pays, however, to be carefnI. Kant would surely want to insist that the connection in question is not analytic. I think that he wonld be correct, too. For in the first place, even if it does make good sense to say that we have a concept of the word 'synthetic' (not of the property, but of the word itself), and even if certain judgments about the word could be established through analysis of that concept, the facts that we have been' considering are determined, not by the concept of the word, but by something else, viz., by the rule specifying how to identify tokens of the word thus conceived. In the second place, moreover, this rnIe specifies only what the sequence must be in any string of characters that is to qualify as a token of the word: 's', followed by 'y', followed by 'n', etc. To establish how many characters are in this sequence, we need to use the rule to identify a

perceptible token of the word (or something with an appropriately analogous structure) and to enumerate the characters in it. Kant would no doubt argue, accordingly, that the judgment that 'synthetic' contains nine letters is itself synthetic, and indeed that our knowledge of it rests on something very like mathematical construction. Now a word is of course a symbol. A token of that word, however, is not a symbolic representation of the word itself, but a perceptible instance of it.

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When we base a judgment about a word, or about all of its possible tokens, on what we find to be true in just one such token, this is analogous to what we do in grounding a judgment about all collections of n objects on what we find to be true in one such collection. The points that we have made may therefore be extended For while we may not wish to say that a particular collection of seven plus five is a token of the number that it possesses and exhibits, it is nonetheless related to that number in much the same way that a token of a word is related to that word. If we identify a collection of 7 + 5 strokes, we can determine that it numbers 12, and we can see at once that any other such collection must do so likewise. What allows us to do this should now be apparent. When we enumerate the strokes, we determine not merely the number of objects in a certain ostensively specified collection, but the number of objects in a (correctly identified) collection of 7 + 5 . We realize, moreover, that this particular collection qualifies as a collection of 7 + 5 only because it conforms to general rules specifying how such a collection is to be identified Indeed we represent it merely as something that conforms to these rules, ignoring the size, color, composition, etc., of the strokes. We even ignore the fact that the objects in question are strokes, representing them merely as units or 'ones', i.e., as arbitrary instances of an arbitrary concept. We therefore attribute to the collection of strokes only what is required by the general rules that mark it as a collection of 7 + 5. Since it is evident that any collection conforming to these rules will have to have exactly as many members as any other, it is evident that any such collection will have to number 12 just as this one does. It is true, of course, that we may make mistakes. We may judge that the

collection before us is a collection of 7 + 5 when it isn't, or that it numbers 13 when in fact it numbers 12. TIlls does not imply, however, that we cannot establish arithmetical judgments, which are a priori, by examining particular collections of perceptible things. Again the parallels between our two examples are aseful. When we judge that a certain string of perceptible characters is a (correctly spelled) token of 'synthetic', we need to make a number of empirical judgments. We need to judge that the first character in the string is an's', the second a 'y', etc., as well as that no character has been omined, none accidentally included, etc. In enumerating the characters in the string we again need to make various empirical judgments: that all the characters have been enumerated, that none has been counted twice, etc. It is possible for us to be mistaken in any of these judgments. TIlls does not imply, however, that the procedures we have described - the procedures for writing out a word and

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counting its letters - are inadequate to support the judgment that any token of 'synthetic' must have nine letters. For this latter judgment does not rest on the mere fact that we find ourselves saying 'nine' as we reach the last character in the string, but on the judgment that we find ourselves saying 'nine' at tltis point after correctly executing all the relevant procedures. It obviously is not necessary that we should execute the procedures correctly each time we attempt to do so. It is necessary, however, that proper execution of these procedures should yield a unique result. To determine the number of characters in one token of a word is thus to determine the number of characters in any other token. Parallel points can obviously be made concerning the use/of a particular collection of n + m objects to determine how many objects there must be in any such collection. The analogies between the two examples also help to show why Kant holds that arithmetical judgments are synthetic. We cannot give clear cognitive sense to discourse about particular objects, he thinks, uuless we can, at least in principle, represent and identify such objects. Since the representation of particular objects always involves intuition, and since our intuition is always sensible, we cannot give clear cognitive sense to discourse about numbers as objects, Platonists to the contrary notwithstanding. Instead, Kant thinks, we have to construe discourse about number as having to do with the quantity of collections of sensible objects - with the 'numerosity , or 'howmany-ness' Qf such collections. The representation of number therefore requires intuition, and indeed sensible intuition, since for us at least, the representation of particular objects, and hence of collections of such objects, and hence of the 'how-many-ness' of such collections, requires that we be sensibly affected. To represent the number 7 we must, in the end, represent a particular, though arbitrary, collection of 7 perceptible tltings. To represent the number of 7 + 5 we must do likewise. To determine that a collection of 7 + 5 necessarily numbers 12 we must, in the end, enumerate the things in such an arbitrary collection. I5 The reason for this becomes clear once we see the analogy between our two examples. The rules that specify how 'synthetic' is to be spelled are really just procedures for identifying perceptible tokens of that word. To determine how many characters are in the word, we must use these procedures to identify a token and then enumerate its characters. Similarly, the rules that specify how to represent the number 7 + 5 are simply procedures for identifying perceptible collections of 7 + 5. To determine how many things there must be in such collections, we need to use these procedures to identify an arbitrary collection and then enumerate its members. Our knowledge of the arithmetical truth rests, therefore not on the mere concept

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of the sum of 7 and 5, but on the procedures by which we can identify and enumerate collections exhibiting that concept, and indeed on the employment of those procedures in a particular, though arbitrary, case. It is worth noting that Kant's reason for holding that arithmetical judgments are synthetic is extremely general. The point turns simply on the fact that for us, thought is always merely general or discursive, while intuition, which makes the representation of particulars possible. is always sensible. lf we were to suppose, therefore, as Kant seems to do (B 71-2), that there might conceivably be an intelligence in which intuition is sensible, as it is for us, but neither spatial nor temporal, then for such an intelligence arithmetical knowledge would presumably require the construction of concepts, and arithmetical judgments would presumably be synthetic. Yet such an intelligence would presumably identify collections of n objects by means quite different from those that we must employ. For on Kant's view. the procedures that we must finally use all involve the temporally successive act of running through the members of a collection, accomplishing what Kant describes as "the successive addition of unit to (homogeneons) unit". (A 142/B 182) The point can be approached somewhat differently. A moment ago I suggested that we distinguish the concept of a word from the procedures for identifying perceptible tokens of that word. A parallel distinction needs to be drawn between the concept of the number n and the procedures for identifying collections of n perceptible objects. It seems clear that discourse about particular words requires the use of perceptible tokens of those words. For parallel reasons, it seems clear that we cannot make clear sense of discourse about the number n except by employing collections of n sensible objects, or by employing a sensible numeral-token which can function as an index of the number n just because it is the last thing that would be generated in a standard collection of n sensible things, viz., a collection of the first through the nth numeral-tokens. It seems clear. therefore. that we cannot make sense of discourse about numbers without appealing to some procedures by which collections of n objects are to be identified. Kant WOUld, I believe, agree to both points. He would insist, however, that the distinction between the concept and the procedures for identifying instances of the concept still needs to be drawn. The need for the distinction is perhaps more evident in the first case, since the procedures by which we generate and identify tokens of words could obviously be quite different from what they are. The distinction is still needed, however, in the case of numerical concepts. While we may not be able to envision a fundamental procedure for identifying collections of n objects that does not finally involve what Kant describes as the "successive

addition of unit to (homogeneous) unit", and while we therefore cannot give sense to discourse about number without reference to procedUres that are temporally successive, we can see nonetheless that the concept and the procedure are distinct. 16 Kant's view concerning arithmetic involves two distinct theses, then. The first is that because our thought is merely discursive and our intuition always sensible, our arithmetical knowledge rests upon construction of concepts, and our arithmetical judgments are thus synthetic. The second and more specific thesis is that given the form of our intuition, the construction of arithmetical concepts requires that we command procedures for generating or identifying collections of n objects, and these procedures must be temporally successive. As I indicated earlier. Kant refers to these procedures as "uuiversal condi-

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tions of construction" for the corresponding numerical concepts. He also calls

them 'schemata' for these concepts (A 714/B 742, A 142-3/B 182), ascribing them not to the understanding but rather to imagination. It is to this latter point that I now wish to turn. III. IMAGINATION

Kant characterizes imagination as "the faculty of representing in innrition an object that is not itself present". (B 151) One might easily take this to mean that imagination is simply the capacity for mental imaging, for representing in sensible images things not actually present. lf that were what Kant meant, however, then most of what he says about the role of imagination in knowledge would be utterly untenable. For while one might conceivably make a case for the claim that mental imaging plays some causal role in occasioning the development of certain concepts or prompting the making of certain judgments, that is obviously not the claim Kant wishes to make. I suggest, accordingly, that his characterization be read quite differently. It is worth noting that in ordinary speech, when we use 'imagine' and its cognates, we do not necessarily or even characteristically imply that mental imaging must take place. A child at play may be said to imagine that his stick is a gnn. A paranoid may be said merely to imagine that a passing glance is threatening. In neither case, as reflection will show, do we regard it as essen~ tial that episodes of mental imaging should occur. What is important is simply that someone construes, or views, or takes what is sensibly present as something other, or at least as something more, than what it immediately presents itself as being. The child, for instance, treats his stick as a gun. The paranoid construes the innocent glance as threatening.

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These remarks suggest what I take to be Kant's view. He does not hold that imagination is the capacity for the mental imaging of absent objects, as one might first suppose. He holds instead that it is the capacity to construe or interpret the sensibly present as something other, or something more, than what it immediately presents itself as being. It is the capacity, therefore, for representing in inmition something which, strictly speaking, is not present there, at least not fully. These observations on common uses of 'imagine' are useful if they help us to resist the temptation to suppose that Kant must conceive of imagination as the capacity for mental imaging. Kant's view concerning imagination goes well beyond common sense reflection, however. He does not merely hold that when we perceive something, we can also imagine it - i.e., construe or take it - as something other, or something more, than what we perceive it as being. He holds, much more strongly, that to perceive something in the first place -

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to perceive it. that is, as something - involves construing or taking sensible

awareness as the awareness of something, something that might present itself sensibly in manifold other ways, and something that is therefore not present, at least not fully present, in our immediate sensible awareness. This, I suggest, is precisely the point of Kant's well-known assertion that "imagination is a necessary ingredient in perception itself". (A 120n.) He does not mean to assert that perception necessarily involves mental imaging, a claim

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that would be of little philosophical interest even if it were plausible. He means rather to reject the view that perception can be understood as merely a state of passive sensory awareness, as Hwne had thought, and to insist instead that it involves, besides passive awareness, the act of construal or interpretation of that awareness as the awareness of something. This is not the place to attempt to work out the whole of Kant's view concerning imaginationP I do wish to point out, however, that if we accept my suggestion as to what Kant means by 'imagination', we can then build upon the points made in section II both to illuminate Kant's view concerning imaginationand to indicate why he holds that the construction of arithmetical concepts involves the exercise of imagination.

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To conceive the nwnber n, Kant thinks, is to conceive the quantity or 'how-many-ness' of a collection of n things. To represent this quantity, he holds, we must in the end represent a particular, though arbitrary, collection of n perceptible things. The representation of such a collection - as a collection of n - obviously requires sensible presence. It also requires that we command procedures by which we may determine, for this or any collection of sensible things, whether it nwnbers n. Such procedures are general, and

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our representation of them is therefore not merely sensible. Neither, however,

is it merely conceptual. For as we saw in Section II, there is a distinction to be drawn between the concept of the nwnber n and the procedures by which we identify collections of n perceptible objects. The concept is simply the representation of the quantity or 'how-many-ness' of such a collection. The procedures are rule-governed activities that we use to determine whether a

given collection of sensible things exhibits that quantity, activities that require running through the things in the collection sequentially and taking them together as a collection (cf. the use of 'Zusarnmennehmung' at A 99), thus accomplishing what Kant calls "the successive addition of unit to (homogeneous) unit". (A 142/B IS2) Because of the role they play in enabling us to identify sensible things as exhibiting a concept, Kant calls such procedures 'schemata' for the corresponding concepts. The construction of arithmetical concepts is said to involve the exercise of

imagination, not because Kant thinks that such knowledge somehow rests upon mental imaging, but because he thinks that it depends upon our ability to use such general procedures to construe sensible things as constituting collections of definite number. To construe them in this way is to COnstrue them

as something more than what they present themselves as being, since it is to construe them as confonning to certain general rules or procedures. Arithmetical knowledge thus rests upon the exercise of imagination in just the sense that I have suggested that Kant gives to this term in general. To see why Kant maintains that arithmetical knowledge rests on the exercise of 'pure, a priori' (A 141-2/B ISO-I) or 'productive' (B 151-2) imagination, we need to consider somewhat more closely the character of arithmetical concepts and of the procedures by which we identify collections of sensible things as instantiating and exhibiting those concepts. When we construe our sensible awareness as the awareness of, say, a

maple tree, Kant would presumably say that this requires construing it as the awareness of something confonning to certain general procedures, viz., those we would employ to determine that what is present is indeed a maple tree. If

we consider what such procedures might be like, however, it quickly becomes apparent that they will inevitably involve comparison of what is sensibly present now with what has been sensibly present, to myself or to others, on the other occasions. For whether our concept of a maple tree be naive or sophisticated, identification of anything as instantiating that concept will involve a more or less complex web of comparisons between the thing we are concerned with now and manifold other things that have been perceived. To construe our sensible awareness as the awareness of a maple tree is thus to

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construe it as the awateness of something of a determinate kind. But the kind in question is one that can only be specified through reference to other things, previously encountered, which we take to be likewise of that kind. The kind in question, in other words, must inevitably be identified as the same kind as that which was exhibited in other things. To put the point differently, when we identify something as a maple tree, we identify it as reproducing a kind exhibited in other things previously encountered. And this, I suggest, is the important point behind Kant's usage of the term 'reproductive imagination'. His view, once again, is not that imagination is the capacity for mental imaging. 'Reproductive' imagination, in particulat, is not to be understood as the capacity for generating mental images which replicate or reproduce previous sensations. Imagination, as I have suggested, is the capacity to construe sensible awareness as the awareness of something - something which, because of the merely passive character of sensible awareness, cannot be fully present in sensible awareness. 'Reproductive' imagination. in tum, is the capacity to construe our sensible awareness as the awareness of something of a kind whose characterization requires comparison with, and reference to, other things that we have encountered in sensible awareness. Productive imagination, by contrast, is the capacity to constrne or interpret sensible awareness as the awareness of something of a kind whose characterization does not depend upon comparison with other things previously encountered. It is the capacity, accordingly, to constrne what is sensibly present as exhibiting what Kant would classify as pure rather than empirical concepts. The concept of a collection or totality of things is a pure concept, Kant wonld atgue, for it is the concept simply of all the instances of a concept, whatever that concept may be, and hence it is a concept implicit in the mere form of judgment, irrespective of what the content of the judgment may be. The concept of the quantity or 'how-many-ness' of such a collection is likewise pure. To identify a group of perceptible things as a collection of n, of n + m, etc., is thus to identify it as instantiating and exhibiting a kind, but a kind whose characterization does not require comparison with, or reference to, other things previously encountered. (Indeed, the point of importance is precisely that while one may be sensibly awate of the things in a collection, one cannot be sensibly awate of the collection itself. Constrning these things as things in a collection thus involves construing them as instantiating a concept that cannot be explicated, or defined, through reference to sensible awateness.) A concept of this sort is, as Kant likes to put it, one that we contribute to experience, not one that we derive from it. It is a concept, moreover, that Kant holds to be presupposed in the empirical cognition of things,

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and one that Can thus be said to be constitutive or productive of the form of experience, i.e., of all empirical knowledge. The procedures by which we determine that given sensible things instantiate such a concept - the procedures, in the case of arithmetic, by which we determine that a given group of things constitutes a collection of n, of n + m, etc. - ate thus procedures which do not demand the comparison of the things in question with other things previously encountered in sensibility. They are procedures by which we identify sensible things as instantiating concepts that are constitutive or productive of the form of empirical cognition. Since we must command such procedures in order to be able to identify groups of sensible things as collections of n, of n + m, etc., and since we must be able to make such identifications in order to grpund arithmetical judgments, these procedures may be said themselves to be contributed to experience rather than derived from it, and to be constitutive or productive of experience, or empirical cognition. in its fonn. As procedures, distinct from the concepts they enable us to wield, they ate ascribed by Kant to imagination rather than to understanding. As procedures that ate constitutive of empirical cognition, they are ascribed to 'productive' imagination. Insofat as we have shed light on Kant's doctrine concerning the constrnction of arithmetical concepts, we have thus shed light at the same time on his view concerning productive imagination.

University of Kansas NOTES I Quotations are from Kemp Smith's translation of the Critique of Pure Reason, 2nd impression with corrections, Macmillan, London, 1963. 2 Arithmerik und Kombinatorik bei Kant (dissertation done at Freiburg, 1934), Itzehoe, 1938. 3 'Kant on the Mathematical Method', Monist 51 (1967); reprinted in this volume pp. 21-42. 4 'Kant's Philosophy of Arithmetic', in Philosophy, Science. and Method: Essays in Honor of Ernest Nagel, ed. by S. Morgenbesser, P. Suppes, and M. White, St. Martins, New York, 1969; reprinted in this volume pp. 43-79. 5 Ibid., p. 64ff. 6 Ibid., p. 67. 7 Kant introduces the notion of symbolic construction only in his discussion of algebra. Like Parsons, however, I believe that it is legitimate to extend the notion and to describe both the use of numerals in calculation and the use of formulae in logic as involving symbolic construction. For fuller discussion of this point, see my 'Kant on the Construction of Arithmetical Concepts', Kant-Studien 73 (1982). 17-46. 8 As Parsons notes, analogous points can be made for 'proofs' of the son that Leibniz proposed, as well as for 'proofs' of quantificational schemata that are closely related to the arithmetical identities. Ibid., pp. 66-67.

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9 Parsons' views on this matter are developed further in <Mathematical Intuition', Proceedings qjthe Aristotelian Society, Vol. 80 (1979-80), pp. 145-68. 10 Parsons is aware of this point. As he notes, the view he develops and clarifies is more closely related to the one that Kant states in the prize essay of 1764 (Ak. II, 272-301) than it is to the view that Kant states in the Doctrine of Method in the Critique. 11 There is passing reference to our use of a numeral system of base ten at A78{B 104. 12 This may be what Kant has in mind when he says (A78/BI04) that counting involves a synthesis according to concepts "because it is executed according to a common ground of unity, as, for instance, the decad". 13 We may use actual sensible things such as marks on paper. We may, on the other hand, represent things merely 'in our heads'. In either case, sensible content is involved. 14 It might be objected that there really is no general rule specifying how this, or any other word, is to be spelled. What we have, rather, is only a paradigmatic spelling, i.e., a token of the word that we take as correctly spelled, and that we use to judge correctness of spelling in other tokens. The point remains. however, that such a token, taken as paradigm, then serves as the basis of a general rule by which to determine, for any string of characters, whether it is a (correctly spelled) token of that word. 15 Kant would obviously not want to claim that we need to use this procedure each time we ask ourselves what the sum of 7 and 5 is. Nor would he want to deny that we can use other procedures to establish mathematical truths - e.g., the procedures involved in calculation using Arabic numerals and base ten - and that this may, in many cases, be simpler. He would presumably insist, however, that the procedure of ostensive construction is the final standard by which alternative procedures, to say nothing of mere memory, must be judged. For more complete treattnent of this point see 'Kant on the Construction of Arithmetical Concepts'. 16 It is clear that Kant draws a distinction between the concept of quantity or magnitude ('Groesse' ,"quantitatis') and that of number ('Zahl'), for he says that "the pure schema of magnitude (quantitatis), as a concept of the Wlderstanding, is number, a representation which comprises the successive addition of homogeneous units". (AI42/B182) One might conclude from this that since the representation of number depends upon schematization, there is no pure concept of number, or of particular numbers. Such a conclusion would be misleading, however. Kant's view, I suggest, is that we do have pure concepts, not only of quantity or magnitude but of detenninate quantities or magnitUdes, e.g., of the magnitude possessed by a collection of three things. If we refer to such determinate quantities as numbers, then his view, I suggest, is that we do have pure concepts of the numbers. Kant himself does not use 'Zahl' in this way, hOWever. The llse of 'Zahl' is derivative from that of 'zaehlen', the verb for counting or enumerating. Kant's point, I take it, is that we do have a pure concept of determinate quantities (e.g., of three), and that it is only by virtue of our possessing this concept that we are able, in his phrase, to bring to unity the synthesis accomplished through the activity of counting or enumerating. (Cf. A 789/B I 03-4) He also holds, however, that we cannot establish the relationships between three and other quantities merely through reflection on this pure concept; to do that, we need to represent a particular but arbitrary collection of three things, which requires the procedure of enumeration ('zaehlen'). Though the point is admittedly not altogether clear in the first Critique, Kant does state his view in just this way in a letter to Schultz (25 November 1788; AK. X, pp. 528-31). He argues there that "3 and 4, as so many concepts of quantity rGroesse'), can, when put together, yield the concept of a quantity ('Groesse')", but that this is a mere thought, from which we cannot determine what that quantity is. To determine this, we need "the number rZahl') seven", which is "the representation of this concept in an enumeration ('Zusammenzaehlung')". It is only

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through this latter representation - i.e., only through construction of the concept by means of the procedure of enumeration - that we can determine what that detenninate quantity is that we conceive when we conceive the sum of 3 and 4. 17 The ideas sketched here are developed a bit more fully in my 'Kant's View ofImagination', Kant-Srud;en 79 (l988), pp. 140-164.

MICHAEL FRIEDMAN

KANT'S THEORY OF GEOMETRY!

Tractatus 4.0412: For the same reason the idealist's appeal to 'spatial spectacles' is inadequate to explain the seeing of spatial relations, because it carmot explain the multiplicity of these rela-

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Since the important work of early twentieth century philosophers of geometry like Russell and Carnap, Schlick and Reichenbach, Kant's theory of geometry has not looked very attractive. After their work and the work of Riemann, Hilbert, and Einstein from which they drew their inspiration, Kant's conception is liable to seem quaint at best and silly at worst. His picture of geometry as somehow grounded in

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wrong; and there is a consequent tendency to view the Transcendental Aesthetic as an unfortunate embarrassment that one has simply to rush through on the way to the more relevant and enduring insights of the . Analytic. 2 The standard modem complaint against Kant runs as follows. Kant fails to make the crucial distinction between pure and applied geometry. Pure geometry is the study of the formal or logical relations between propositions in a particular axiomatic system, an axiomatic system for Euclidean geometry. say. As such it is indeed a priori and certain (as d priori and certain as logic

is, anyway), but it involves no appeal to spatial intuition or any other kind of experience. Applied geometry, on the other hand, concerns the truth or falsity of such a system of axioms under a particular interpretation in the real world. And, in this connection, it matters little whether

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the physical world - in terms of light rays, stretched strings, or whatever, or in the psychological realm - in terms of ''looks'' or "appearances" or other phenomenological entities. In either case the truth (or approximate truth) of any particular axiom system is neither a priori nor certain but, rather, a matter for empirical investigation, in either physics or psychology. This modern attitude is epitomized in Einstein's famous dictum: "As far as the

laws of geometry refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.'" From this point of view, then, Kant mis-

177 Carl J. Posy (ed.), Kant's Philosophy of Mathematics, 177-2l9. © 1992 Kluwer Academic Publishers. Printed in the Netherlands.

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construes the problem from the very beginning, and, accordingly, his teaching is hopelessly confused. Yet this modern complaint is quite fundamentally unfair to Kant; for, in the first place, Kant's conception of logic is certainly not our modern conception. Our distinction between pure and applied geometry goes hand in hand with our understanding of logic, and this understanding simply did not exist before 1879 when Frege's Begriffsschrift appeared. The importance of relating Kant's understanding of logic to his philosophy of mathematics has been stressed by several recent commentators, notably, by Hintikka and Parsons.' In reference to geometry in particular, however, I think that no one has been as close to the truth as Russell, who habitually blamed all the traditional obscurities surrounding space and geometry - including Kant's views of course - on ignorance of the modern theory of relations and uncritical reliance on Aristotelian subject-predicate logic.' I think Russell is exactly right. but I would like to turn his polemic on its head. Instead of using our modem conception of logic to disparage and dismiss earlier theories of space, we should use it as a tool for interpreting and explaining these theories, for deepening our understanding of the difficult logical problems with which they were struggling. At any rate, this is what I propose to undertake in reference to Kant's theory in what follows.

What is most striking to me about Kant's theory, as it was to Russell, is the claim that geometrical reasoning cannot proceed "analytically according to concepts" - that is, purely logically - but requires a further activity called "construction in pure intuition.~' The claim is expressed most clearly in the Discipline of Pure Reason in its Dogmatic Employment. where Kant contrasts '!Jhilosophical" with mathematical reasoning: Philosophy confines itself to general concepts; mathematics can achieve nothing by concepts alone but hastens at once to intuition, in which it considers the concept in concreto, although still not empirically, but only in an intuition which it presents a priori, that is, which it has constructed, and in which whatever follows from the general conditions of the construction must hold, in general, for the object [Objektel of the concept thus constructed. Suppose a philosopher be given the concept of a triangle and he be left to find out, in his own way, what relation the swn of its angles bears to a right angle. He has nothing but the concept of a figure enclosed by three straight lines. along with the concept of just as many angles. However long he meditates on these concepts, he will never produce anything new. He can analyse and clarify the concept of a straight line or of an angle or of the number three, but he can never arrive at any properties not already contained in these concepts. Now let the geometer take up this

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question. He at once begins by constructing a triangle. Since he knows that the sum of two right angles is exactly equal to the sum of all the adjacent angles which can be constructed from a single point on a straight line, he prolongs one side of the triangle and obtains two adjacent angles which together equal two right angles. He then divides the external angle by drawing a line parallel to the opposite side of the triangle, and observes that he has thus obtained an external adjacent angle which}s equal to ali internal angle - and so on. In this fashion, through a chain of inferences guided throughout by intuition, he arrives at a solution of the problem that is simultaneously fully evident [einleuchtend] and general (B743-745).6

Kant is here outlining the standard Euclidean proof of the proposition that the sum of the angles of a triangle = 180° = two right angles (Elements: Book I, Prop. 32).7 Given a triangle ABC, one prolongs the side BC to D and then draws CE parallel to AB:

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One then notes that a = a' and 13 = W, so a + J3 + y = a' + 13' + y = 180°. Q.E.D. In contending that construction in pure intuition is essential to this proof, Kant is making two claims that strike us as quite outlandish today. First, he is claiming that (an idealized version of) the figure we drew above is necessary to the proof. The lines AB, BD, CE, and so on are indispensable constituents; without them the proof simply could not proceed. So geometrical proofs are themselves spatial objects. Second, it is equally important to Kant that the lines in question are actually drawn or continuously generated, as it were. Proofs are not only spatial objects, they are spatio-temporal objects as well. Thus, in an important passage in the Axioms of Intuition Kant says: I cannot represent to myself a line, however small, without drawing it in thought, that is gradu-

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ally generating [nach und nach zu erzeugen] all its parts from a point. Only in this way can the intuition be obtained.... The mathematics of extension (geometry), together with its axioms, is based upon this successive synthesis of the productive imagination in the generation of figures [Gestalten] (B203-204).

That construction in pure intuition involves not only spatial objects, but also spatio-temporal objects (the motions of points), explains why intuition is able to supply a priori knowledge of (the pure part of) physics: thus our idea of time [Zeitbegriff] explains the possibility of as much a priori cognition as is exhibited in the general doctrine of motion, and which is by no means unfruitful (B49).

In other words, it is the spatia-temporal character of construction in pure intuition that enables Kant to give a philosophical foundation for both Euclidean geometry and Newtonian dynamics. Kant's conception of geometrical proof is of course anathema to us. Spatial figures, however produced, are not essential constituents of proofs, but, at best, aids (and very possibly misleading ones) to the intuitive comprehension of proofs. Whatever the intended interpreration of the axioms or premises of a geometrical proof may be, the proof itself is a purely "formal" or "conceptual" object: ideally, a string of expressions in a given formal language. In particular, then, all that could possibly be missing from a purely "conceptual" or "analytic" derivation of "X's angles sum to 180°" from "X is a triangle" are the axioms of Euclidean geometry. For us, the conjunction of "X is a triangle" with these axioms does of course imply "X's angles sum to 180°" by logic alone; and no spatia-temporal activity of construction in pure intuition is necessary. To be sure, spatial objects may be needed to supply a particular interpretation of our axioms, but this is quite a different matter. Is Kant simply forgetting about the axioms of Euclidean geometry here? This is most implausible, especially since the proof he sketches is Euclid's. No, his clcim must be that even the conjunction of "X is a ttiangle" with these axioms does not imply "X's angles sum to 180°" by logic alone: in other words, that Euclid's axioms do not imply Euclid's theorems by logic alone. Moreover, once we remember that Euclid's axioms are not the axioms used in modem formulations and, most importantly, that Kant's conception of logic is not our modern conception, it is easy to see that the claim in question is perfectly correct. For our logic, unlike Kant's, is po/yodic rather than monadic (syllogistic): and our axioms for Euclidean geometry8 are strikingly different from Euclid's in containing an explicit, and essentially polyadic, theory of order. The general point can be put as follows. A central difference between

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monadic logic and full polyadic logic is that the latter can generate an infinity of objects while the former cannot. More precisely, given any consistent set of monadic formulas involving k primitive predicates, we can find a model containing at most 2'- objects. In polyadic logic, on the other hand, we can easily construct formnlas having only infinite models. Prooftheoretically, therefore, if we carry out deductions from a given theory using only monadic logic, we will be able to prove the existence of at most 2k distinct objects: after a given finite point we will run out of "provably new" individual constants. Hence, monadic logic cannot serve as the basis for any serious mathematical theory, for any theory aiming to describe an infinity of objects (even "potentially"). This abstract and general point can be illustrated by Euclid's proof of the very first Proposition of Book I: that an equilateral triangle can be constructed with any given line segment as base. TIle proof runs as follows. Given line segment AB, construct (by Postulate 3) the circles C j and C2 with AB as radius:

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Let C be a point of intersection of C I and C z and draw lines AC and BC (by Postulate 1). Then, since (by the definition of a circle: Def. 15) AC = AB = BC, ABC is equilateral. Q.E.D. There is a standard modem objection to this proof. Euclid has not proved the existence of point C, he has not shown that circles C I and Cz actually intersect. Perhaps C I and C z somehow "slip through" one another, and there is no point C. Moreover, in modem formulations of Euclidean geometry this "possibility" of non-intersection is explicitly excluded by a continuity axiom, an axiom that (apparently anyway) does not appear in Euclid's list of Postulates and Common Notions. From this point of view, then, not only is

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Euclid's proof "defective," but so is his axiomatization: the existence of point C simply does not follow from Euclid's axioms." Why do we think that the existence of point C does not follow from Euclid's axioms? We might argue as follows, Cover the Euclidean plane with Cartesian coordinates in such a way that the midpoint of segment AB has coordinates (0, 0), point A has coordinates (-t, 0), and point B has coordinates (~, 0), Then the desired point of intersection C has coordinates (0, ";3/2). Now throwaway all points with irrational coordinates: the result is a model in Q2, where Q is the rational numbers. This model appears to satisfy all Euclid's axioms, but, of course, point C does not exist in the model. So our model gives concrete fonn to the "possibility" of non-intersection, a "possibility" that therefore needs to be excluded by a continuity axiom. But perhaps Euclid's formulation does contain such a continuity axiom, if only implicitly. After all, Postnlate 2 states that straight line segments can be produce "continuously" [Xtna ~O
C 1 is contin,uous C2 is continuous :. C exists

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throwaway all points of the plane except the two points A, B. Let the "line" AB be just the pair {A, R}, let the "circle" C 1 be the singleton {B}, and let the

"circle" C 2 be the singleton {A}. Does "line" AB satisfy Postnlate 2? Can it be "produced continuously"? Note again that neither "can be produced" nor "continuous" is logically analyzed: both appear as simple (one-place) predicates. So, from a strictly logical point of view, we can give them both any interpretation we like: let them both mean "has two elements," for example. Then Postulate 2 is obviously satisfied, and so are the other axioms. Hence, Euclid's axiomatization does not even imply the existence of-more than two points. Does this last "counter-example" show that Euclid's axiomatization is hopelessly "defective"? I think not. Rather, it underscores the fact that Euclid's system is not an axiomatic theory in our sense at all. Specifically, the existence of the necessary points is not logically deduced from appropriate existential axioms. Since the set of such points is of course infinite, this procedure could not possibly work in a monadic (syllogistic) context. Instead, Euclid generates the necessary points by a definite process of construction: the procedure of construction with straight-edge and compass. We start with three basic operations: (i) drawing a line segment connecting any two given points (10 avoid complete triviality we assume two distinct points to begin with), (ii) extending a line segment by any given line segment, (iii) drawing a circle with' any given point as center and any given line segment as radius. We are then allowed to iterate operations (i), (ii), and (iii) any finite number of times. Euclid's Postulates 1-3 give the rules for this iterative procedure, and the points in our "model" are just the points that can be so constructed. In particular, then, the infinity of this set of points is guaranteed by the infinite iterability of our process of construction. to More precisely, it is straightforward to show that the points generated by straight-edge and compass constructions (and, therefore, the points required for Euclidean geometry) comprise a Cartesian space (set of pairs) based on the so-called square-root (or "Euclidean") extension Q* of the rationals, where Q* results from closing the rationals under the operation of taking real square-roots. ll In particular, then, the underlying set (Q* X Q*) is ouly a small fragment of the full Cartesian plane JR2, where JR is the real numbers. The former, unlike the latter, is a denumerable'set, and each element is determined by a finite sequence of elementary operations. In this sense, there is no need in Euclidean geometry for anything as strong as a continuity axiom. Compare Euclid's approach to the existence of points - in particular, to the existence of an infinity of points - with that taken by modem axiomatiza-

MICHAEL FRIEDMAN

KANT'S THEORY OF GEOMETRY

tions. The basis of the modem approach, beginning with Pasch in 1882 and culminating in Hilbert's Foundations of Geometry (1899), is to include an explicit theory of order: a theory of the structure and cardinality of the points on a line. Thus, imagine the points on any line to be ordered by a two place relation < of "being-to-the-Ieft-of':

monadic, one can only represent such infinity intuitively:' by fui iterative process of spatial construction. Therefore, since, for Kant, concepts are monadic concepts, our idea of space cannot be a concept:

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(irrefiexivity) (transitivity) (connectedness) (no endpoints) (denseness)

The presence of some such axioms as 1-6 is the chief difference between Hilbert's axiomatization and Euclid's.12 Axioms I - 6 have only infinite models, and, of course, they make an essential use of modem polyadic logic. Note, however, that it is not merely the presence of two-place as opposed to one-place predicates that is crucial here. After all, axioms 1-3 alone certainly have finite models. Rather, the essential new element is the quantifier-dependence exhibited in 4-6: the logical form I/..Cly." This kind of dependence of one quantifier on another cannot arise in monadic logic. where we can always "drive quantifiers in" so that each one-place matrix is governed by a single quantifier. (Thus, for example, l/..Cly(Fx --> Gy) is equivalent to 3xFx --> 3yGy.) Moreover, it is the dependence of one quantifier on another - specifically, of existential quantifiers on universal quantifiers - that enables us to capture the intuitive idea of an iterative process formally: any value x of the universal quantifier generates a value y of the existential quantifier, y can then be substituted for x generating a new value y', and so on. Hence, the existence of an infinity of objects can be deduced explicitly by logic alone. We .can now begin to see what Kant is getting at in his doctrine of construction in pure intuition. For Kant logic is of course syllogistic logic or (a fragment of) what we call monadic logic. 14 So, for Kant, one cannot represent or capture the idea of infinity formally or conceptually: one cannot represent the infinity of points on a line by a formal theory like 1-6 above. If logic is

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Space is represented as an infinite given quantity [Grosse]. Now one must ~ertainly think eveI)l concept as a representation in which an infinite aggregate [Menge] of different possible representations are contained (as their common characteristic (MerkmalD, and it therefore contains these under itself. But no concept, as such, can be so thought as if it were to contain an infinite aggregate of representations in itself. Space is thoug.1it'in precisely this way, however (for all parts of space in ifl-finitum exist simultaneously). Therefore t..'1e original representation of space is an a priori intuition, and not a concept (B40).

At first sight, it is not at all clear what Kant means by the distinction between "containing representations under itself" and "containing representations in itself' here, but our above considerations supply us with a plausible reconstruction. Monadic concepts can, of course, have infinite extensions: an infinite number of objects can happen to fall under any given monadic fonnnla. But monadic concepts (unlike polyadic formnlas) cannot/orce their extensions to be infinite: they do not (and cannot) contain an infinity of objects i.."1 their very idea, as it were. Hence. since the idea of space does have this latter property, it cannot be a (monadic) concept. The notion of infinite divisibility or denseness, for example, cannot be represented by any such formnla as 6: this logical form simply does not exist. Rather, denseness is represented by a definite fact about my intuitive capacities: namely, whenever I can represent (construct) two distinct points a and b on a line, I can represent (construct) a third point c between them. Pure intuition - specifically, the interability of intuitive constructions 15 - provides a uniform method for instantiating the existential quantifiers we would use in formulas like 6; it therefore allows us to capture notions like denseness without actually using quantifier-dependence. Before the invention of polyadic quantification theory there simply is no alternative. Thus, in Euclid' s geometry there is no axiom corresponding to our denseness condition 6. Instead, we are given a uniform method for actually constructing the point bisecting any given finite line segment: it suffices to join C in the figure on page 181 with its "mirror image" below AB - the resulting straight line bisects AB (Prop. 1. 10). This operation, which is itself constructed by iterating the basic operations (i), (ii), and (iii), can then be iterated as many times as we wish, and infinite divisibility is thereby represented. So we do not derive new points between A and B from an existential axiom, we construct a bisection function from our basic operations and obtain the new points as the values of this function: 16 in short, we are given what modern logic calls a Skolem function for the existential quantifier in 6. 17 For Kant,

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this procedure of generating new points by the iterative application of constructive functions takes the place, as it were, of our use of intricate rules of quantification theory such as existential instantiation. Since the methods involved go far beyond the essentially monadic logic available to Kant, he

places are only parts of the same boundless space, related to one another through a certain position. nor can you conceive to yourself a cubic foot unless it be bounded on all sides by the surrounding space.

views the inferences in question as synthetic rather than analytic. I8 Finally, we should note that our modem distinction between pure and applied geometry, between an uninterpreted formal system and an interpretation that makes such a system true, cannot be drawn here, In particular, the only way to represent the theory of linear order 1-6 is to provide, in effect, an interpretation that makes it true. 19 The idea of infinite divisibility or denseness is not capturable by a formula or sentence, but only by an intuitive procedure that is itself dense in the appropriate respect. By the same token, the sense in which geometry is a priori for Kant is also clarified. Thus, the proposition that space is infinitely divisible is a priori, because its truth - the existence of an appropriate "model" - is a condition for its very possibility.2o One simply cannot separate the idea or representation of infinite divisibility from what we would now call a model or realization of that idea; and our notion of pure (or formal) geometry would have no meaning whatever for Kant. (In a monadic context a pure or uninterpreted "geometry" cannot be a geometry at all, for it cannot represent even the idea of an infinity of points.) II

The above considerations make a certain amount of sense out of Kant's theory, but one might very well have doubts about attributing them to Kant. After all, Kant certainly had no knowledge of the distinction between monadic and polyadic logic, nor of quantifier-dependence, Skolem functions, and so on. So using such ideas to explicate his theory may appear wildly anachronistic, and my reading of B40 may appear strained In particular, the . distinction I stressed between "containing representations under itself' and "containing representations in itself' appears capable of a much simpler interpretation.· Kant is merely drawing a contrast between the predication relation and the part-whole relation: the relation of space to spaces is .not one of general concept to its instances but one of (individual) whole to its parts. Therefore, space is a singular representation (intuition), not a general representation (concept). These doubts are reinforced by a glance at § IS.B of the Inaugural Dissertation (1770), where the point is made in exactly this way: The concept of space is a singular representation comprehending all things in itself, not an abstract and common notion containing them under itself. For what you speak of as several

187

Here the distinction between the part-whole relation and the predication relation is quite clear and explicit. Further, there is no confusing - and apparently extraneous - reference to infinity. Yet, from the present point of view, it is precisely this difference between the Critique and the Inaugural Dissertation that is most striking. Kant apparently found it necessary to modify his arguments between 1770 and 1781 (and again between 1781 and 1787, since B40 is itself a revision of the first edition passage at A2S), and we should ask ourselves why. First of all, it is clear that Inaugural Dissertation, § \S.B is, by itself, quite insufficient for Kant's purposes. Kant wants to show that our cognitive grasp of the notion of space is intuitive rather than conceptual or discursive, and the mere contrast between the part-whale relation and the predication relation certainly does not establish this. For, as Kant explicitly recognizes in 1781, there is indeed a "general concept of space (which is common to both a foot and an ell alike)" (A2S), and this general concept of space - which we might represent by Ix is a space' - does bear the predication relation to spaces or parts of space. In other words, there is both the general concept IX is a spacel, which bears the predication relation to spaces, and the singular individual space, which bears the whole-part relation to spaces (and is in turn also related via predication to IX is a space'). The question is: why should the latter have cognitive priority over the former? Wby should our idea or representation of space be identified with the actual individual space rather than the general concept IX is a space'? Moreover, there is also a general concept corresponding to the part-whole relation: the concept Cx is a part ofyl-this would presnmably be an example of the kind of "general concent of relations of things as such" (or simply "concept of relations") to which Kant is alluding at A2S 21 Given this concept and the concept IX is a spacel we can apparently frame a discursive or conceptual definition of the individual space: namely, "that space of which all other spaces are parts." From this point of view, then, it begins to look as if our cognitive grasp of the general concept of space and spatial relations (for example, the part-whole relation) precedes our cognitive graSp of the individual space. Kant rejects this last possibility at A2S = B39: Space is not a discursive or, as one says. general concept of relations of things as such. but a pure intuition. For, first, one can represent to oneself only one space; and when one speaks of several spaces, one means thereby only parts of the same unique space. Nor can these parts precede the

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one all-embracing space, as being, as it were, its constituents (and making its composition possible); on the contrary, they can be thought only in it: Space is essentially one: the manifold in it,. and hence the general concept of spaces as such. depends solely on limitations. It follows that an a priori intuition (that is not empirical) underlies all concepts of space.

One cannot arrive at the individual space by way of the concepts rx is a space' and ex is a part of y'. On the contrary, these concepts are themselves only possible via the intuitive act of "cutting out" parts of space from the singular intuition space. But now the question becomes why should this be so: why should the singular intuition cognitively precede the general concept? In the sentence immediately following the above quotation Kant appeals to our knowledge of geometry: So too are all principles [Grundsa:tze] of geometry - for example. that in a triangle two sides

together are greater than the third - derived: never from general concepts of line and triangle, but only from intuition, and this indeed a priori, with apodeictic certainty.

We find a sintilar appeal in Jnaugural Dissertation, § I5.C: "That there are not given in space more than three dimensions, that between two points there is only one straight line, that from a given point on a plane surface a circle can be described with a given straight liue, etc. - these cannot be concluded from some general notion of space, but can only be seen, as it were, in space in concreto." We also find a reference to "incongruous counterparts" (which are not mentioned in the Critique) and the claim that "here the diversity, namely, the discongruity, can only be noticed by a kind of pure intuition." Finally, Kant appeals to geometrical demonstrations: "Geometry does not demonstrate its own general propositions by thinking an object by means of general concepts as happens with things rational, but by subjecting it to the eyes by means of a singular intuition as happens with things sensitive." In the end, therefore, Kant's claim of cognitive priority for the singular intuition space rests on our knowledge of geometry.22 Our cognitive grasp of the notion of space is manifested, above all, in our geometrical knowledge. Hence, if we can show that this knowledge is intuitive rather than conceptual, we will have shown the cognitive inadequacy of the general concept of space and the priority of the singular intuition. Continuing our line of questions, then, we must ask why, at bottom, is conceptual knowledge inadequate to geometry: why must intuition play an essential role? Surely, the mere assertion that geometrical principles "cannot be concluded from some general notion ... but can only be seen, as it were, in space in concreto" is not expected to convince those, like the Leibnizians and

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Wolffians. who maintain precisely L,e opposite. Now it is at this point that Kant puts forward t,'te "infinity argnment" in the Critique. In the first edition we have § 2.5: Space is represented as an infinite given magILitude. A general concept of space (which is common to both a foot and an ell alike) can detennine nothing in regard to magnitude [Grossel. Were there no limitlessness in the progression of intuition, no concept of relations could, by itself, supply a principle of their infinitude (A25).

Neither the general concept rx is a spaceJ nor any concept of spatial relations (for example, ex is a part of yl) can yield a "pdnciple of infinitude." Such a principle can only be found in the '"limitlessness in the progression" or nnbounded iterability of pure intuition. Hence, since the idea of infinity - in parricular, the idea of infinite divisibility - is all essential part of geometry and therefore of our idea of space, the latter must indeed be a pure intuition and not a concept. The second edition passage at B40 is clearer, for Kant is more explicit that the problem is not with th~ general concept rx is a spaceJ in particular but with all general concepts as such. In the first edition, the problem appears to be merely that rx is a spaceJ says nothing whatever about magnitUde, whether infinite or finite. We might then be tempted to try concepts such as rx is a cubic foot of spaceJ or even rx is a greater space than y'. The passage at B40 (along with the final sentence of A, § 2.5) gets to heart of the matter: without the unbounded iterability of pure intuition, no concept - not even a relational concept like rx is a greater space than y' - can force its extension to be infinite; although, to be ·sure, any concept may have an infinite (possible) extension. The same idea stands out clearly in § 12 of the Prologomena (1783): "That one can require a line to be drawn to infinity (in indeifinitum), or that a series of changes (for example, spaces traversed by motion) shall be infinitely continued, presupposes a representation of space and time that can only depend on intuition, namely, in so far as it in itself is bounded by nothing; for from concepts alone it could never be inferred." Once again, Kant's conception of infinity and infinite divisibility can be clarified by contrasting it with modem formulations. We, of course, can easily represent infinite divisibility by means of concepts like ex is greater than y l - as we did above in the theory of dense linear order. In such a theory the points on a line are taken as primitive, and the line itself is built up from them in just the way Kant says it cannot be: the points relate to the line as "its constituents (and making its composition possible):' Yet what makes this· representation itselC possible is precisely the quantifier-dependence of

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modern polyadic logic: the logical form V.. \13. In the absence of such logical forms - and in accordance with the actual procedure of Euclid's geometry the natural alternative is to represent infinite divisibility by an intuitive constructive procedure for "cutting out" a stualler line segment from any given one: for example, Euclid's construction for bisecting a line segment of Prop. 1.10. Thus, whereas we can represent infinity by ''v'x3y(x is greater than y)', Kant would formulate this proposition by 'x is greater than f B(x)', where fB(x) is the operation of bisection, say. And, in this representation, the idea of infinity is not conveyed by logical features of the relational concept 'x is greater than Y', but by the well-definedness and iterability of the function fB(x): our ability, for al'Y given line segment x, to construct (distinct) fB(x), fB(fB(x», ad infinitum. This, I suggest, is why Kant gives priority to the singular intuition space, from which all parts or spaces must be "cut out" by intuitive construction ("limitation"). Only the unbounded iterability of such constructive procedures makes the idea of infinity, and therefore all "general concepts of space," possible. And, of course, it is this very same constructive ilerability that underlies the proof-procedure of Euclid's geometry.

and eighteenth centuries; that does reqnire genuine continuity - "all" or "most" real numbers; whose modern, "rigorous" formulation requires full polyadic logic - much more intricate fonns of quantifier-dependence than V.. \13; and, finally, whose earlier, "non-rigorous" formulation made an essential appeal (in at least one tradition) to temporal or kinematic ideas _ to the intuitive idea of motion. This branch of mathematics is of course the calculus, or what we now call real analysis. It goes far beyond Euclidean geometry in considering "arbitrary" curves or figures - not merely those constructible with Euclidean tools - and in making extensive use of limit

III

Even if we are on the right track, however, we have still gone only part of the way towards understanding construction in pure intuition. We can bring out what is missing by three related observations. First, as we noted above, the notions of denseness, infinite divisibility, and (even) constructibility with straight-edge and compass do nol amount to full continuity. These notions all involve denumerable sets of points which are but small fragments of the set Ill. of real numbers. Hence, to understand how full continuity comes in we have to go beyond Euclidean geometry. Second, these notions do not (on a modem construal) exploit very much of polyadic logic: just the logical form V.. \13. If t..his is all that were required modern logic would hardly need to have been invented. But we would like to understand why modern logic was invented and, in particular, why it was invented when it was. Third, the procedure of construction with Euc.lidean tools - with straight-edge and compass - does not really exploit the kinematic element that is essential to Kant's conception of pure intuition: no appeal is made to the idea that lines, circles, and so on are generated by the motion of points. So why did Kant think that motion is so important? These three observations are in fact intimately related. For there exists a branch of mathematics which was just being developed in the seventeenth

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operations.

From a modem point of view the basic limit operation underlying the calculus is explained in teITIlS of the Cauchy-Balzano-Weierstrass notion of convergence. Moreover, we also appeal to this notion in explaining the distinction between denseness and genuine continuity, in precisely expressing the idea that there are "gaps" is a merely dense set like the rational nllll1bers. Thus, let s" s" ... be a seqnence of rational numbers that converges to n, say - that approaches n as its limit (for example, let s, = 3.1, s, = 3.14, and in general Sn= the decimal expansion of 1t carried out to n places). This sequence of rationals converges (to "something" as it were), but in the set Q of rational mimbei-s (and even in the expanded set Q* of Euclidean-constructible, numbers) there is no \intit point it converges to. Such limit points are "missing" from a merely dense set like the rationals. A truly continuous set contains "all" such limit pOints. More precisely, using Cauchy's criterion of 1829, we say that a sequence SI' S2" .. converges if 'v'E 3N 'v'm 'v'n [m, n > N ~ ISm - s,1 < Ej,

where £ is a positive rational number and N, m, n are natural numbers. A sequence SI, S2.··· converges to a limit r if "IE 3N "1m [m>N ~ Ism-rl <Ej,

The problem with a merely dense order is that the first can be true even when the second is not, whereas a continuous order satisfies the additional axiom of Cauchy completeness - whenever a sequence converges, it converges to a limit r - which clearly has the logical form:

\l3VV ~ 3r\l3V Note the additional logical complexity of this axiom: in particular, the use of the strong form of quantifier-dependence \13"1. The increase in logical strength that I find so striking here can be best

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brought out if we compare the way points are generated by a completeness axiom with the way they are generated by Euclidean constructions (or, from a modem point of view, by the weaker form of quantifier-dependence V.. '0'3). In the latter case, although the total number of points generated is of course infinite, each particular point is generated by a finite number of iterations: each point is determined by a finite number of previously constructed points. In generating or constructing points by a limit operation, on the other hand, we require an infinite sequence of previously given points: no finite number of iterations will suffice. So limit operations involve a. much stronger and

more problematic use of the notion of infinity than that involved in a shnple process of iterated construction. Let us now return to Kant and the late eighteenth century. We cannot of course represent the ideas of convergence and transition to the limit by complex quantificational fonus like \if 3V. But the idea of continuous motion appears to present us with a natural alternative. Thus, for example, we can easily "construe!" a line of length re by imagining a continuous process that takes one unit of time and is such that at t = -l- a line of length 3.1 is constructed, at t = %a line of length 3.14 is constructed, and in general at t = n/n + 1 a line of length s" is constructed, where s, again equals the decimal expansion of

1t

carried out to n places. Assuming this process in fact has a

tenninal outcome, at t = I we have consttucted a line of length re. In some sense, then, we can thereby "construct" any real number. 23

In this style of representation the notion of convergence or approach to the iimit is expressed by a temporal process: by the idea of one point moving or becoming closer and closer to a second. This intuitive process of becoming

does the work of our logical fonn 113V, as it were. That the limit of a convergent seqnence exists is expressed by the idea that any finite process of temporal generation has a terminal outcome. This idea does the work of our logical form 3113V. In particular, then, what we now call the continuity or completeness of the points on a line is expressed by the idea that any finite motion of a point beginning at a definite point on our line also stops at a definite point on our line,24 What t.~e modern definition of convergence does, in effect, is

replace this intuitive conception based on motion and becoming with a formal, algebraic, or "static" counterpart based on quantifier-dependence and order relations. Now a temporal conception of the limit operation is explicit in the basic lenuna Newton uses to justify the mathematical reasoning of Principia: Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer to each other than by any given difference, become ultimately equal (Book I, § I, Lemma D.25

This lenuna is perhaps best understood as a definition of what Newton means by «quantiiy": namely, an entity generated by a continuous temporal process (it clearly fails for discontinuous "quantities"). Newton's conception of "quantities" as temporally generated is even more

explicit, of course, in his method of fluxions, where all mathematical entities are thought of as fluents or "flowing quantities." For example: I don't here consider Mathematical Quantities as composed of Parts extremely small, but as generated by a continual motion. Lines are described, and by describing are generated, not by any apposition of Parts, but by a continual motion of Points. Surfaces are generated by the motion of Lines, Solids by the motion of Surfaces, Angles by the Rotation of their Legs, Time by a continual flux, and so in the rest. 26

Moreover, Kant appears to be echoing these ideas in an hnportant passage about continuity in the Anticipations of Perception: Space and time are quanta continua. because no part of them can be given without being enclosed between limits (points and instants), and therefore only in such fashion that t.~is part is itself again a space or time. Space consists only of spaces, time consists only of times. Points and iristants are only limits, that is, mere places [Stellen] of their limitation. But places always presuppose the intuitions which they limit or determine; and out of mere places, viewed as constit1-'ents capable of being given prior to space or time, neither space nor time can be composed [zusammengesetzt]. Such quantities [Grossen} may also be calledjfowing Iftiessendl, since the synthesis (of the productive imagination) in their generation [Erzeugung] is a progression in time, whose continuity is most properly designated by the expression of flowing (flowing away) (B211-212).

For Kant, like Newton, spatial quantities are not composed of points, but rather generated by the motion of points. I take Kant's choice of language to be especially significant here, for his "fliessende Grossen" is the standard German equivalent of Newton's "fluents. "27 This expression is used, for example, by the mathematician Abraham Kastner in his influential textbooks on analysis and mathematical physics." Kastner's analysis text attempts to develop the calculus from a "rigorous" standpoint that makes no appeal to infinitely small quantities. In this connection he develops a version of Newton's method of fluxions, and, what is more remarkable for a German author of this period, he argues that Newton's fluxions are in some respects clearer and more perspicuous than Leibniz's differentials. Further, he explicitly applauds Collin Maclaurin's attempt, in his monumental Treatise of Fluxions (1742), to develop the calculus on the basis of a kinematic conception of the limit operation. 29

Without going into detail,'" the most basic ideas of the fluxional calculus are as follows. We start with fluents or "flowing quantities" x, y, conceived as continuous functions of time. We can then fonn the fluxions or time-

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derivatives .t, y, because continuously changing quantities obviously have well-defined instantaneous velocities or rates of change. 31 If we are then given a curve or figure y = f(x) generated by independent motions in rectangular coordinates of the fluents x, y, the derivative (slope of the tangent line) will be dy/dx = y/i<:

particular, since convergence is represented by a continuous, but finite process of temporal generation, the relevant limit point is automatically generated as well. Moreover, although the kinematic interpretation of the calculus certaiuly does not meet modern standards of rigor, it is also not afflicted with the obvious problems about consistency and coherence facing an interpretation based on differentials, infinitesimals, and infinitely small quantities. Indeed, when the kinematic interpretation was explicitly criticized by mathematicians like D' A1embert and I'Huilier in the late eighteenth century, it was not on grounds of coherence and consistency but because it was thought to import a

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"foreign" or "physical" element into pure mathematics. Pure mathematics

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should be independent of and prior to mathematical physics; therefore, it should be developed in complete independence of the idea of motion.34 For Kant, on the other hand, this "mixing" of physical and mathematical ideas is not a defect but a virtue. Since part of the "general doctrine of motion" _ namely, pure kinematics or "phoronomy" - is, in effect, also a branch of pure mathematics, it is possible to hold that litis part of mathematical physics is a priori as well." So an explicit "mixing" of physical and mathematical ideas is essential to the unity of Kant's system.36 But why exactly does the kinematic interpretation fail to meet modern standards of rigar? This is a difficult and fascinating question. For now, however, I shall simply hazard the suggestion that the difference between the iterative infinity involved in Euclidean constructions and the stronger use of

by the "parallelogram of velocities." Finally, we can recover the integral from the derivative via the Fundamental Theorem (which is also understood temporally).32 So all the basic notions of the calculus are explained without ever appealing to differentials or infinitely small quantities. In any case, it is extremely likely that some such understanding of the calculus (the method of fluxions) underlies Kant's insistence on the kinematic character of construction in pure intuition." When he speaks of the "productive synthesis" involved in the ''mathematics of extension," Kant is referring to what we would now call calculus in a Euclidean space; he is not simply thinking of Euclidean geometry proper. And, if this is correct, we can better understand why, given the way concepts such as continuity and passage to the limit are understood, there is no possibility of a distinction between pure and applied - uninterpreted vs. interpreted - mathematics in the modern sense. The only way one can represent continuity, for example, is to provide what we would now call an intuitive interpretation of the continnity Of completeness axiom, an interpretation that necessarily makes that axiom true. In

infinity involved in limit operations plays a central role here. In Euclidean geometry we specify the objects of our investigation - circles, straight lines, and any figures constructible from them - by a well-defined iterative or "inductive" procedure. This specification then underlies our iterative, step-

by-step method of proof: the substitution of a previously constructed objecta given finite straight line segment, say - as argument in a further constructive operation - the construction of a circle based on this line segment as

radius via Postulate 3, for example. By contrast, in the fluxional calculus we have no such specification: no step-by-step procedure (nor any other precisely defined method) for constructing all fluents or, ''jiiessende Grossen" has been given." Sintilarly, our temporal representation of the lintit operation does not proceed by repeated application of previously given functions: each new limit has to be constructed "on the spot," as it were. This, in the end, is perhaps the most frmdamental advantage of the Cauchy-Bolzano-Weierstrass definition of convergence. For our use of the logical form V3V in an appropriate formal system of quantificational logic permits us to reestablish

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iterative met.'lods of proof: we can "handle" Lne points generated via limit operations by rigorous - finitary - deductions. 38

The resulting curve is continuous, but at no point is there a well-defined tangent.4 ' (In eVelY neighborhood of C, for example, both lines AC and FC intersect infinitely many other points of the curve). Thus, no finite segment of

I

Be this as it may, the class of curves generated as fluents or "fliessende

Grossen" proves to be inadequate to the needs of mathematics and mathematical physics. The main problem is that, since continuiry is explained by continuous motion, continuity automatically implies differentiability as well." A curve generated by continuous motion (drawn by the continuous motion of a pencil, as it were) automatically has a tangent or direction of motion at each point. So the class of continuous curves is assimilated to the class of what we now call smooth (differentiable) curves.40 Actually, this is not quite right, for those who employed the method of fluxions of course knew that there are continuous curves that lack tangents at certain points: curves with "cusps" or "comers." However, such curves can be easily comprehended within a kine-

matic understanding of continuity so long as one can think of them as "pieced together" by a finite number of smooth curves so, long as they have a finite number of "isolated" singular points. But what happens if we allow such a process of "piecing together" smooth curves to be itself iterated indefinitely: if we apply limit operations to infinite collections of given smooth or "well-behaved" curves? What we get, of course, includes continuous but nowhere differentiable curves. The most famous examples of such curves, given by Weierstrass in 1872, are constructed via trigonometric series.41 Fortunately, however, we can also give simpler, more intuitive examples. Perhaps the simplest is the Koch curve: we start with a horizontal line segment AB which we divide into three equal parts by points C and D; on the middle segment CD we construct an equilateral triangle CED and erase the open segment CD: we repeat the same construction on each of the segments AC, CE, ED, DB; finally, we continue this process

indefinitely on each remaining segment:

the Koch curve can be drawn by the continuous motion of a pencil: we must

think .of each point as laid down independently, as it were, yet nevertheless in a continuous order. Such continuous but nowhere differentiable curves clearly exceed the scope of the kinematic interpretation: we cannot understand their continuity via the intuitive idea of continuous motion.43 To get a mathematical grip on

this wider class of curves we need a clear distinction between continuity and differentiability; and this, of course, is one of the main achievements of our

modern approach to convergence. We define the continuity of a function fix) at a given point Xo by an expression of the form V3V [Conv(s, xo) & J, Where s is a sequence defined algebraically from fix) and says that s converges tofixo). We define the differentiability affix) at a given pointxo by an expression of the fonn:

V3V [Conv(s', x o)], where s' is a second sequence defined algebraically from fix). By understanding these notions formally rather than intuitively, we can, for the first time, both clearly and precisely distinguish them and clearly and precisely explore their logical relations: differentiability logically implies continuity but not vice versa, for example.44 IV

The present approach to Kant's theory of geometry follows Russell in assuming that construction -in pure intuition is primarily intended to explain mathematical "proof or reasoning, a type of reasoning which is therefore distinct

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from logical or analytic reasoning. Again following Russell, we have sought an explanation for this idea in the difference between the essentially monadic logic available to Kant and the polyadic logic of modem quantification theory. Further, we have tried to link this conception of mathematical reasoning with the very possibility of thinking or representing mathematical concepts and propositions. Thus, for example, "r cannot think a line except by drawing it in thought" (B 154), because only this representation permits me to use the concept of line in mathematical reasoning (such as Euclid's or

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Newton's) where properties like denseness and continuity play an essential role. Yet RuselI's assumption has been vigorously debated. It has been maintained that Kant did not deny, and indeed may have even affirmed, that mathematical inference is logical or analytic; his primary concern, rather, is with the status of the premises or axioms of such inferences. Geometry is synthetic precisely because its underlying axioms are synthetic; the (synthetic) theorems of geometry follow purely logically or analytically. This anti-Russellian view is clearly and forcefully stated by Beck: The real dispute between Kant and his critics is not whether the theorems are analytic in the sense of being strictly [logically J deducible, and not whether they should be called analytic nOw when it is admitted that they are deducible from definitions, but whether there are any primitive propositions which are synthetic and intuitive. Kant is arguing that the axioms cannot be analytic ... because they must establish a connection that can be exhibited in intuition.45

As Beck indicates, this view is attractive because Kant will not be refuted, as Russell thought, by the mere invention of polyadic logic. For even modern formulations of Euclidean geometry like Hilbert's will contain priruitive propositions or axioms, and pure intuition can be called in to secure their truth (to provide a model, as it were).46 Indeed, from this point of view, the discovery of logically consistent systems of non-Euclidean geometry should be seen as a vindication of Kant's conception. The existence of such geometries shows conclusively that Euclid's axioms are not analytic and, therefore, that no analysis of the basic concepts of geometry could possibly explain their truth (as Leibniz apparently thought). Assuming that Euclid's axioms are true, then, there is no alternatiye.but to appeal to a synthetic source: hence pure intuition.47 On the Russell-inspired interpretation developed here, by contrast, there can be no question of non-Euclidean geometries for Kant. Non-Euclidean straight lines, if such were possible, would have to at least possess the order properties - denseness and continuity - common to all lines, straight or curved. And, on the present interpretation, the only way to represent (the order properties of) a line - straight or curved - is by drawing or generating it in the space (and time) of pure intuition. But this space, for Kant, is necessarily Euclidean (on both interpretations). It follows that there is no way to draw, and thus no way to represent, a non-Euclidean straight line; and the very idea of a non-Euclidean geometry is quite impossible." (Another way to see the point is to note that the anti-Russellian interpretation would reinstate precisely the modern distinction between pure and applied geometry argued above to be unavailable to Kant.)

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The anti-Russellian interpretation draws its primary support from B 14: For as it was found that all mathematical inferences proceed in accordance with the principle of contradiction [nach dem Satze des Widerspruchs fortgehen] (which the nature of all apodeictic certainty requires), it was supposed that the fundamental propositions [Grundsatze] could also be recognized from that principle [aus dem Satze des Widerspruchs erkannt wUrdenl. This is erroneous. For a synthetic proposition can indeed be comprehended [eingesehenJ in accordance with [nach] the principle of contradiction, but only if another synthetic proposition is presupposed from which it can be derived [gefolgert}, and never in itself.

Kant seems to be saying that because inference from axioms to theorems was (correctly) seen as analytic, the axioms themselves were (incorrectly) thought to be analytic. But these axioms are really synthetic; for this reason (and only for this reason), so are the theorems. Kant therefore agrees with Russell that the conditional [Axioms ..... Theorems] is a logical or analytic truth;49 his ~ point is simply that the antecedent of the conditional is synthetic. I do not think this reading of the passage is forced on us. First of all, Kant does not actually say that mathematical inference is analytic, nor that the theorems can be analytically derived. The first sentence may mean only that mathematical proofs necessarily involve logical or analytic steps - and, of course, no logical fallacies. 50 Similarly, the last sentence does not explicitly say that the derivation of one synthetic sentence (theorem) from another (axiom) is analytic; the possibility that this derivation is itself synthetic is at least left open. Second, it is assumed that by fundamental propositions [Grundslitze] Kant means axioms, and this is doubtful. Kant's own technical term for axioms is Axiomen (cf. B204-205, B760-762), and at A25 he calls the proposition that two sides of a triangle together exceed the third a fundamental proposition [Grundsatz]. This latter is of course not an axiom in Euclid, but a basic (and therefore fundamental) theorem (Prop. 1. 20). So the error Kant is diagnosing here may not be the (really rather ridiculous) mi~e of transferring analyticity from inference to preruise (axiom), but the more subtle supposition that because logic plays a central role in the proof of basic theorems it is sufficient for securing their truth. A more fundamental problem for the anti-Russellian reading of BI4 is posed by Kant's conception of arithmetic. Kant is supposed to have a moreor-less modem picture of mathematical theories as strict deductive systems. The synthetic character of mathematics depends solely on the synthetic character of the underlying axioms. But this is certainly not Kant's picture of arithmetic. According to Kant, arithmetic differs from geometry precisely in having no axioms, for there are no propositions that are both general and synthetic serving as preruises in arithmetical arguments (B204-206). Thus, our conception of arithmetic as based on the Peano axioms, say, is completely

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foreign to Kant, and one cannot use the model of an axiomatic system to explain why arit.~metic is synthetic: one cannot suppose that arithmetical reasoning proceeds purely logically or analytically from synthetic axioms as premises.:5i Yet arit.lnnetic is the very first example Kant uses (at BI5-16) to illustrate, "-,,d presumably illuminate, the general ideas of B 14. Arithmetical propositions like 7 + 5 = 12 are synt.;'etic, not because they are established by analytic derivation from synthetic axioms (as we would derive them from the Peano axioms, say), but because they are established by the successive addition of unit to unit. This procedure is synthetic, according to Kant, because it is necessarily temporal, involving "t..1-te successive progression from one moment to another" (AI63)." Thus, for example, only the general features of succession and iteration in time can guarantee the existence and uniqueness of the sum of 7 and 5, which, as far as logic and conceptual analysis are concerned, is so far merely possible (non-contradictory). Similarly, only the unboundedness of temporal succession can guarantee the infinity of the number series; and so on. For Kant, then, arit.ljmetical propositions are established by calculation, a procedure that is sharply distinguished from logical argument in being essentially temporal. This is why Kant says that the synthetic character of arithmetical propositions "becomes even more evident if we consider larger nwnbers, for it is then obvious that, however we might turn and twist our concepts, we could never by mere analysis unaided by intuition be able to find the sum" (B 16). The reference to larger numbers makes it clear that intuition is not being called in to secure the truth of basic propositions - such as 2 + 2 = 4, perhaps - by "seuing them before our eyes. "Rather, intuition underlies the step-by-step process of calculation which, in its entirety, may very well not be surveyable "at a glance."" We have now reached the heart of the matter, I think, for it is the idea of a sharp distinction between calculation and logical argument that is perhaps most basic to Kant's conception of the role of intuition in mathematics. Thus, at B762-764 Kant contrasts matllematical and philosophical reasoning. Only mathematical proofs are properly called demonstrations, while philosophy is restricted to logical or conceptual ("acroamatic" or discursive) proofs. The latter "must always consider the universal in abstracto (by means of concepts)," the fonner "can consider the universal in concreto (in the single intuition), and yet still through pure a priori representation whereby all errors are at once made visible [sichtbar]" (B763). That Kant has calculation centrally in mind here is indicated by his refer-

ence to the methods employed in solving algebraic equations (B762), a reference which recalls the even more explicit conception of calculation found in the Enquiry Concerning the Clarity of the Principles of Natural Theology and Ethics (1763). See, for example, the First Reflection, § 2, entitled "Mathematics in its methods of solution [Ausflosungen], proofs, and deductions [Folgerungen] examines the uuiversal under symbols in concreto; philosophy examines the universal through symhols in abstracto":

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I appeal first of all to arithmetic, both the general arithmetic of indeterminate magnitudes [algebra], as well as that of numbers, where the relation of magnitude to unity is determinate. In both symbols are first of all supp~~, instead of the things themselves, together with special notations [Bezeicbnungen] for their increase and decrease, their relations, etc. Afterwards, one proceeds with these signs, according to easy and secure rules, by means of substitution, combination or subtraction, and many kinds of transformations, so that the things symbolized are here completely ignored, until, at the end, the meaning of the symbolic deduction is finally deciphered [entziffertJ.

As the Third Reflection, § I explains, this "symbolic concreteness" of mathematical proof accounts for the difference between philosophical and mathematical certainty. Since philosophical argument is discursive or conceptual, ambignities and equivocations in the meanings of general concepts are always· possible. Mathematics, on the other hand, works with concrete or singular representations that allow us to be assured of the correctness of its substitutions and transformations "with the same confidence with which one is assured of what one sees before one's eyes." As Kant puts it in the Critique, the step-by-step application of the easy and secure rules of calculation "secures all inferences against error by setting each one before our eyes" (B762).54 From the present point of view, the point could perhaps be reconstructed as follows. Mathematical proof, unlike logical proof, operates not only with predicates like 'x is even' and 'x is a triangle', but first and foremost withfunciion-signslike ex + y" and ethe bisector of z.,. In calculation we form functional terms by inserting particular arguments into the function-signs, we set up equalities (and inequalities) between such functional terms, and we substitute one functional term for another in accordance with these equalities. Since hoth the arguments and the values of our function-signs are individuals," the procedure of substitution is to be sharply distinguished from the subsumption of individuals under general concepts characteristic of logical or discursive reasouing. In particular, the essence of the former procedure lies in its iterability: f(a) can be substituted inf(x) to form a distinct functional termfif(a», while it of course makes no sense at all to subsume the predication F(a) under the

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predicate F(x)." Thus, the essentially "extra-logical" form of inference required is that which takes us from one object a satisfying a condition ... a ... to a second object/Cal satisfying another condition _ _ lea) ~ and from there to a third object/(f(a)) satisfying---flj(a)) ---, and so on. Now this conception of the role of calculation and substitution in mathematical proof also applies, mutatis mutandis, to the case of geometry. In Euclidean geometry we start with an initial set of basic constructive functions: the operation fLex, y) taking two points x, y to the line segment between them, the operation fE(x, y) taking line segments x, y to the extended line segment of length x + y, and the operation fc(x, y) taking point x and line segment y to the circle with center x and radius equal to y. We also have a specifically geometrical equality relation (congruence) and, of course, definitions of the basic geometrical fignres (circle, triangle, and so on). Euclidean proof then proceeds somewhat as follows. Given a figure a satisfy-

Nevertheless, there is of course an important difference between the two cases. As already noted above, geometry has axioms whereas arithmetic does not; moreover, geometry uses "ostensive construction (of the objects [Gegenstiinde] themselves)" (B745) in additon to the "symbolic" or "characteristic" (B762) construction conunon to algebra and arithmetic. Thus, Kant's discussion of algebraic construction has a decidedly "formalistic" tone: we "abstract completely from the nature of the object" (B745) and, as the above passage from the Enquiry puts it, "symbols are first of all supposed, instead of the things themselves" and "the things symbolized are here completely ignored." In geometry, on the other hand, such "formalism" is quite inappropriate: geometrical construction operates with "the objects themselves" (lines, circles, and so on). This difference between arithmetical-algebraic construction and geometrical construction is perhaps most responsible for the confusion that has surrounded Kant's theory. For it begins to look as if geometrical inntition has not merely an inferential or calculational role, but also the more substantive role of providing a model, as it were, for one particular axiom system as opposed to others (Euclidean as opposed to non-Euclidean geometry). Intuition does this, presumably, by placing the objects themselves before our eyes, whereby their specific (Euclidean) structure can be somehow discerned. From the present point of view, of course, there can be no question of picking out Euclidean geometry from a wider class of possible geometries. Rather, the difference between geometrical and arithmetical-algebraic construction is understood as follows. Geometry, uulike arithmetic and algebra, operates with an initial set of specifically geometrical functions (the operations f L, fE' and fe) and a specifically geometrical eqnality relation (congruence). To do geometry, therefore, we require not only the general capacity to operate with functional terms via substitution and iteration (composition), we atso need to be "given" certain initial operations: that is, intuition assures us of the existence and uniqueness of the values of these operations for any given arguments. Thus, the axioms of Euclidean geometry tell us, for example, ''that between two points there is only one straight line, that from a given point on a plane surface a circle can be described with a given straight line" (Inaugural Dissertation, § I5.C), and they also link the specifically geometrical notion of equality (congruence) with the intuirive notion of superposition (Prolegomena, § 12).60 Now one might at first suppose that the case of arithmetic is precisely the same. After all, we need inntitive assurance that the successor function, say, is uniquely defined for all arguments. The point, I think, is that the successor

202

ing a condition ... a ... , we construct, by iteration of the basic operations, a new constructive function g yielding an expanded fignre g(a) satisfying a condition - - - g(a) - - - . From this last proposition we are then able to derive a new condition _ _ a _ _ on our original figure a. Whereas the inference from ---g(a)--- to _ _ a __ can be viewed as "essentially monadic," and is therefore analytic or logical for Kant, the inference from ... a .. . to - -g(a)---is not: it proceeds synthetically, by expanding the figure a as far as need be into space the around it, as it were. Since this procedure is grounded in the indefinite iterability of our basic constructive operations, geometry is

synthetic for much the same reasons as is aritlunetic; and, therefore, the case of arithmetic is primary.57 Confirmation is apparently provided by the discussion at B 15-17. For Kant illustrates BI4 at great length with the example of arithmetic and only then touches on geometry, almost as a corollary." Just as little,is any fundamental proposition [Grundsatz] of geometIy analytic. That the straight line between two points is the shortest is a synthetic proposition. For my concept of straight {GeradenJ contains nothing of quantity [Grosse], but only a quality. The concept of shortest is

entirely an addition, and cannot be derived by any analysis of the concept of straight line. The aid of intuition must therefore be brought in, by means of which alone the synthesis is possible (B 16-17).

As the discussion of arithmetic has shown, the general concept of magnitude [Grosse] requires an inntitive synthesis (the successive addition of unit to unit). But geometry requires this concept as well (for example, in connecting the notion of straight line with the notion of shortest line). Therefore, geometry, just as much as arithmetic, is a synthetic discipline."

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Kant is imagining any such process of "visual inspection." It is much more

general form cif succession or iteration common to all functional operations

plausible thai, in precise parallel to his discussion of the angle-sum property at B743-745, he is referring to the Euclidean proof of this proposition (Prop. 1.20): We consider a triangle ABC and prolong BA to point D such that DA is equal to CA:

whatsoever. So it is not necessary to postulate any specific initial functions in

IIII "if,

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'1

definedness of the successor function is guaranteed by the mere form of iteration in general (that is. time). Thus, in explaining why geometry has axioms while arithmetic does not at B205-206, Kant refers to the need in geometry for general "functions of the productive imagination" like our ability to construct a triangle from any three line segments such that, two together exceed the third (this functional operation is of course definable, in Euclidean geometry, from the operations f L , fE' and fc= Prop. l,22). The point, presumably, is that no such specific functional operations need be postulated in arithmetic.6l In any case, the idea that pure intuition plays the more substantive role of providing a model for one particular axiom system as opposed to others - as the anti-Russellian interpretation reqnires - is rather obviously untenable and definitely unKantian. The untenability of such a view has been clearly brought out in an iostructive article by Kitcher." Kitcher supposes that the primary role of pure intuition is to discern the metric and projective properties (the Euclidean structure) of space. We construct geometrical figures like triangles and somehow "see" that they are Euclidean: "[Kant's] picture presents the mind bringing forth its own creations and the naive eye of the mind scanning these creations and detecting their properties with absolute accu-

J

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1'I

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function is not a specific function at all for Kant; rather, it expresses the

arithmetic: whatever initial functions there may be. the existence and well-

I

KANT'S THEORY OF GEOMETRY

racy" (p. 129). It is then easy to show that pure intuition, conceived on this quasi-perceptual model, could not possibly perform such a role. Our capacity for visualizing figures has neither the generality nor the precision to make the reqnired distinctions. Thus, for example, Kant's appeal to the proposition that two sides of a triangle together exceed the third at A25 is considered, and "We now imagine ourselves coming to know [it] in the way Kant suggests. We draw a scalene triangle and see that this triangle has the side-sum property" (p. 125). But this idea quickly founders on Berkeley's generality problem: how are we supposed to conclude that all triangles have the side-sum property and not, say, that all triangles are scalene? (Actually, in this connection, a more relevant dimension of generality is size. In elliptic [positive curvature] space, "small" - relative to the dimensions of the space itself - triangles have the side-sum property while arbitrarily large triangles do not. So one caunot argue from the properties of small, visualizable triangles to the properties of arbitrary triangles.)" It is extremely unlikely, however, that in appealing to intuition at A25

D

B

c

We then draw DC, and it follows that if:ADe. Since the greater angle is subtended by the greater side (Prop. I.l9), DB > BC. But DB = BA + AC; therefore BA + AC > Be. Q.E.D. Intuition is required, then, not to enable us to "read off" the side-sum property from the patticular figure ABC, but to guarantee that we can in fact prolong BA to D by Postulate 2.64(It is precisely this Postulate that fails in elliptic space: straight lines are finite - but unbounded - and caunot be indefinitely prolonged at will). Now, as Heath observes in his commentary on Prop. 1.20: "It was the habit of the Epicureans ... to ridicule this theorem as being evident even to an ass and requiring no proof' (op. cit., p. 287), and one might be tempted to ~uppose that Kant holds a similar view (so that the "visual inspection" . metaphor is appropriate a.fter all). This suggestion is immediately squashed by the important discussion of mathematical method at Bxi-xii, however, Kant applauds Diogenes Laertius for naming "the reputed author of even the least important elements of geometrical demonstrations, even of those which, according to ordinary judgement, require absolutely no proof' and concludes: A new light dawned on the first man (be he Thales or whoever) who demonstrated the isosceles triangle. The true method. be found, was not to inspect what he discerned either in the figure, or in the mere concept of it, and from these, as it were to learn its properties; but to bring forth what he himself had injected in thought [hineindachte] and presented (through construction) a priQri according to concepts.

Iii

I, 206

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Kant's example here, of course, is the discovery (sometimes attributed to Thales) of the Euclidean proof that the angles at the base of an isosceles triangle are equal (prop, L5)," a proof that also proceeds by means of an ingenious expansion of OUl" original triangle into several additional triangles via "auxiliary construction." I do not see how there can be any doubt, therefore, that Kant's "true method" of geometry is precisely Euclid's procedure of construction with straight-edge and compass.66 Once again, we are forced to conclude that the primary role of pure intuition is to undenvrite the constructive procedures used in mathematical proofs. In fact, when Kant himself uses "visual inspection" and "eye of the mind" metaphors, it is almost always in connection with inference and proof. Thus, in the passage from the Enquiry, Third Reflection, § I quoted above, Kant says that we can check the correctness of algebraic substitutions and transformations "with the same confidence with which one is assured of what one sees before one's eyes." At B762 we are told that the procedure of algebra "secures all inferences against error by setting each one before our eyes." The intuition involved here is not a quasi-perceptual faculty by which we ''read off' the properties of triangles from particular figures, but the intuition involved in checking proofs step-by-step to see that each rule has been correctly applied: in short, the intuition involved in "operating a calculus." The only apparent exception of which I am aware is Inaugural Dissertation, § 15.C, where Kant says that some Euclidean axioms (Postulates 1 and 3) are "seen, as it were, in space in concreto." Even here, however, we can construe the primary role of intuition as existential- providing us with well-defined initial functional operations; rather than evidential - yielding substantive knowledge of mathematical truths." Moreover, the fact that Kant does not use such language in the Critique suggests that he himself became sensitive to its possible misuse. Indeed, there is no room in the critical philosophy for the picture underlying the anti-Russellian conception of pure intuition. That conception views Euclidean geometry as a body of truths that selects one structure for space from the much wider class of all possible such structures. Since both Euclidean and non-Euclidean axiom systems are consistent, we need to call pure intuition to provide a model, as it were, for one system rather than another. As it happens, intuition picks out the Euclidean system. 68 The problem is that Kant has no notion of possibility on which both Euclidean and non-Euclidean geometries are possible. His official notion is "that which agrees with the formal conditions of experience (according to intuition and concepts)" (A218). Mere absence of contradiction is quite insufficient to establish a possibility (A220-221); and "To determine [a geometrical

on

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207

figure's] possibility, something more is required, namely, that such a figure be thought under pure [Iauter] conditions on which all objects of experience rest" (A224). Accordingly, Kant complains that "the poverty of our customary arguments by which we throw open a great realm of possibility, of which all that is actual (the objects of experience) is only a small part, is patently obvious" and concludes "this alleged process of adding to the possible I refuse to allow. For that which has still to be added would be impossible" (A23 I). What produces confusion here is the idea that Kant is operating with two notions of possibility; "logical possibility," given by the conditions of thought alone; and "real possibility," given by the conditions of thought plus intuition. One then supposes that the former is a wider genus, containing both Euclidean and non-Euclidean spaces, of which the latter is a species, containing only Euclidean space. But this line of thought employs a notion of logical possibility that is completely foreign to Kant. Kant's conception of logic is not that of modem quantification theory, and he can have no notion like ours of all logically possible structures - all models of consistent first-order (or _ second-order) theories, say. Thus, for example, while there may be no (monadic!) contradiction in the concept of a non-Euclidean figure such as "the concept of a figure which is enclosed within two straight lines" (A22!), this does not mean that there is a possible non-Euclidean strucnrre containing such a figure. For a non-Euclidean structure would have to possess the topological properties (denseness and continuity) common to Euclidean and nonEuclidean spaces, and this, for Kant, is impossible. There is only one way even to think such properties: in the space and time of our (Euclidean) intuition. Considered independently of our sensible intuition, then, the concept of a non-Euclidean figure remains "empty" and lacks both "sense and meaning [Sinn und Bedeutung]" (B 149).69 A closely related point is that pure mathematics is not a body of truths with its own peculiar subject matter for Kant.70 There are no "mathematical objects" to constitute this subject matter, for the sensible and perceptible objects of the empirical world (that is, "appearances") are the only "objects" there are. For this reason, pure mathematics is not properly speaking a body of knowledge (cognition): Through the determination of pure intuition we can acquire a priori cognition of objects (in mathematics), but only with respect to their form, as appearances; whether there can be things that must be intuited in this form, is still left undecided. Therefore, mathematical concepts are not in themselves cognitions, except in so far as one presupposes that there are things that can be presented to us only in accordance with the form of this pure sensible intuition (B 147).

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Hence, only applied mathematics has a subject matter (the sensible and empirical world), and only applied mathematics yields a body of substantive truthsJI Pure mathematics is a mere fonn of representation (on the present interpretation, a fonn of reasoning), whose applicability to the chaotic sensible world must be proved by transcendental deduction. In this sense, pure intuition cannot be said to provide a model for Euclidean geometry at all; rather, it provides the one possibility for a rigorous and rational idea of space. That there is a model or realization of this idea is not established by pure intuition, but by Kant's own transcendental philosophy. In the end, therefore, Euclidean geometry, on Kant's conception, is not to be compared with Hilbert's axiomatization, say, but with Frege's Begriffsschrift.72 It is not a substantive doctrine. but a fOTIIl of rational representation: a form of rational arguroent and inference. Accordingly, its propositions are established, not by quasi-perceptual acquaintance with some particular subject matter, but, as far as possible, by the most rigorous met.'lods of proof ~ by the proof-procedures of Euclid, Book I, for example. There remains a serious question about Euclid's axioms, of course; when pressed, Kant would most likely claim that they represent the most general conditions under which alone a concept of extended magnitude ~ and therefore a rigor0us conception of an external world ~ is possible (see B204). And, of course, we now know that Kant is fundamentally mistaken here. In 1854 Riemann developed the general concept of n-fold extended manifold ~ containing three dimensional Euclidean space as one very special case alongside of more additional possibilities than Kant (or anyone else in the eighteenth century) ever imagined. In 1879 Frege developed a logical framework which ma.lces possible the even more general concept of relational structure - under which is subsumed all models for Hilbert's geometry and even, as we now say, all "logically possible worlds." Yet Kant is surely not to be reproached for failing to anticipate the leading logical and mathematical discoveries of a later age, he is rather to be applauded for the depth and tenacity of his insight into the logical and mathematical practice of his own. The University of ll/inois at Chicago

NOTES 1 Numerous people have given me valuable suggestions and criticisms, which I shall try to acknowledge in the course of the paper. But I am especially indebted to the suggestions and

encowagement of Thomas Ricketts, the suggestions and writings of Charles Parsons and Manley

KANT'S THEORY OF GEOMETRY

209

Thompson, and conversations with William Tait that led to substantial improvements. I am also indebted to the inspiration of Gerd Buchdahl's Metaphysics and the Philosophy of Science (Oxford: Basil Blackwell, 1969), and to helpful comments from Philip Kitcher on an earlier version. Pervious versions of this paper were presented at Columbia University, Rutgers University. the University of Chicago, Indiana University, the University of Wisconsin at Milwaukee, the University of Maryland, and at a Duke University conference on Kant's philosophy of mathematics in the Spring of 1983. 2 One finds this attitude in even as sympathetic and sensitive a commentary as N. Kemp Smith's A Commentary to Kant's 'Critique of Pure Reason' (London: MacMillan, 2nd. ed. 1923), for example, pp. 40-41. 3 A. Einstein, Geometrie and Erfahrung (Berlin: Springer, 1921)., English tranSlation by G. Jeffrey and W. Perrett in Sidelights on Relativity (London: Methuen, 1922). This last appears (somewhat abridged) in H. Feigl and M. Brodbeck. eds., Readings in the Philosophy of Science (New York: Appleton-Century, 1953). 4 For 1. Hintikka see "On Kant's Notion of Intuition (Anschauung)," in T. Penelhum and J. Macintosh. eds., The First Critique (California: Wadsworth, 1969); "Kant on the Mathematical Method," reprinted in this volume pp. 21-42. "Kant's 'new method of thought' and His Theory of Mathematics," Ajatus 27 (1965), pp. 37-43. "Kant Vindicated" and '!(ant and the Tradition of Analysis," in P. Weingartner, ed., Deskription, Analytizitiit, und Existenz (Salzburg: Pustet, 1966). The last two are reprinted in Logic, Language-Games and Information (Oxford: Clarendon, 1973), the preceding one in Knowledge and the Known (Dordrecht: Reidel. 1974) - page references are to these volumes. For C. Parsons see "Kant's Philosophy of Arithmetic,'" reprinted in this volume pp. 43-79; and "Infinity and Kant's Conception of the 'Possibility of Experience' ," The Philosophical Review 73 (1964), pp. 182-197 - both reprinted in Mathematics in Philosophy (Ithaca: Cornell, 1983), page references to that volume. Also relevant are "Frege's Theory of Number," in M. Black, ed., Philosophy in America (London: Allen at;I.d Unwin, 1964 - reprinted in Parsons, op. cit.; "Objects and Logic," The Monist 65 (1982), pp. 491-516; and the superb encyclopedia article "Mathematics, Foundations of," in P. Edwards, ed., The Encyclopedia of Philosophy, 8 vols. (New York: Academic Press, 1977). See, in addition, E. Beth, "Uber Lockes 'Allgemeines Dreieck'," Kant-Studien 48 (1956-1957), pp. 361380, which inspired Hintikka; and M. Thompson. "Singular Terms and Intuitions in Kant's Epistemology," reprinted in this volume pp. 81-107. This last is oriented around the role of intuition in empirical knowledge, but it also contains a very important discussion of mathematics, logic, and the relationship between them. ,~ See especially § 434 of B. RusseU, The Principles of Mathematics (Cambridge: University Press, 1903), entitled "Mathematical reasoning requires no extra-logical element." 6 References to the Critique of Pure Reason given by pagination of the first ("A"'-1781) and/or second ("B"-1787) editions. Here and below translations from Kant's writings generally follow those of N. Kemp Smith, Immanuel Kant's Critique of Pure Reason (New York: St. Martin's, 1929); of P. Carus, Kant's Prolegomena (La Salle: Open Court, 1902); of G. Kerferd and D. Walford, Kant: Selected Pre-Critical Writings (Manchester, England: University Press, 1968) for the Enqu.iry Concerning the Clarity of the Principles of Natural Theology and Ethics and the Inaugural Dissertation; and of A. Zweig, Kant: Philosophical Correspondence, 1759-99 (Chicago: University Press, 1967) for the letter to Johann Schultz of November 25. 1788. 7 T. Heath, The Thirteen Books of Euclid's Elements, 3 vots. (Cambridge: University Press, 2nd. ed.I926).

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8 These received their more-or-less definitive fonnulation' in D. Hilbert, Grundlagen der Geometrie (Leipzig: Teubner. 1899); English translation from the tenth (1968) edition by L. Unger (La Salle: Open Court. 1971). I say "more-of-less" because there remains some confusion in Hilbert about the proper form of a continuity or completeness axiom. 9 A nice introductory account of such "defects" in Euclid is found in H. Eves, A Survey of Geometry, 2 vals. (Boston: Allyn and Bacon, 1963), § 8.1. Heath, op. cit., vol. I. pp. 234-240, provides a very detailed discussion of the "intersection" problem from a modern point of view. As far as I know, the above criticism of Prop.1. I was first made by M. Pasch, Vor/esungen iiber neure Geometrie (Leipzig: Teubner, 1882), § 6. 10 See Eves, op. cit., Chapter IV, for a discussion of the mathematics of Euclidean constructions. I am indebted to William Tait for emphasizing the importance of straight-edge and compass constructions to me, and for helping me to get clearer about their essential properties. 11 See Eves, op. cit., § 9.3. 12 I have left out the continuity axiom (which is of course second-order), so axioms 1-6 will be satisfied in the rationals Q. The resulting Cartesian space Q2 will therefore be insufficient for Euclidean geometry. Nevertheless, as noted above, full continuity is certainly not required, and it suffices to supplement the Cartesian space based on axioms 1-6 with an axiom of intersection for straight lines and circles (this, of course, is where the square-roots come in). See the excellent survey by A. Tarski, "What is Elementary Geometry?" in L. Henkin, P. Suppes and A. Tarski, eds., The Axiomatic Method (Amsterdam: North-Holland, 1959); reprinted in 1. Hintikka, ed., The Philosophy of Mathematics (Oxford: University Press, 1969) - page references to this latter volume. In particular, Tarski gives a set of axioms sufficient to generate a Cartesian space based on the square-root extension Q* (see Theorem 6 governing system E2": the circlelline axiom is A13' on p. 174; like the denseness condition it has the logical form V.. V3). Adding a (firstorder) continuity schema extends our Wlderlying set to a real closed field (see Theorem I governing system q': the continuity schema is AI3 on p. 167; it has the [minimal] logical fonn V.. V333VV). Finally, adding a (second-order) continuity axiom gives us a system essentially equivalent to Hilbert's: the underlying set is precisely R2. 13 Thus, Euclid's Common Notions contain axioms governing an equality or congruence relation and axioms governing the part-whole relation. The point, however, is that such axioms are "essentially monadic" in exhibiting no quantifier-dependence (we could formulate them using universal free variables as in axioms 1-3). Moreover, these Euclidean axioms have finite models: they do not say anything about the cardinality of ,our Wlderlying set of points. (Interestingly enough, Kant explicitly says that these axioms are analytic: cf. B 16-17, AIM.) 14 Kant's actual views on logic involve many subtleties which I here pass over. See, in particular, the interesting discussion in Thompson, op. cit. Thompson argues very convincingly that Kant indeed made one substantial advance in logic by replacing the traditional logic of terms with a "transcendental logic" of objects and concepts: "a logic in which the form of predication is 'Fx' and not'S is P'." (p. 342). Yet I cannot follow Thompson when he says: "The general logic required by Kant's transcendental logic is thus at least first order quantificationallogic plus identity" (p. 334). If we do not limit ourselves to the logical fOTIns of traditional syllogistic logic, Kant's TabJe of Judgements makes no sense. It is more plausible, I think, to equate Kant's conception of logic with, at most, monadic quantification theory plus identity (which, as far as I can see, is all Thompson requires in his fascinating discussion' of Kant, Strawson, and Quine on singular tenns and descriptions: page 96 - especially n. 15). 15 For the centrality of indefinite iterability to Kant's conception of pure intuition, I am indebted, above all, to Parsons, "Kant's Philosophy of Arithmetic," especially § Vll. But see also Parsons,

"Infinity and Kant's Conception of the 'Possibility of Experience'," for doubts about the "psychological" or "empirical" reality of such truly indefinite iterability. Space prevents me from here giving these doubts the extended discussion they deserve. 16 A simpler illustration of these ideas is provided by the theory of successor based on a constant o and a one-place function-sign sex). Instead of saying "Every number has a successor," we lay down the axioms:

o;;/:. sex) sex) = s(y)

-7 X = y.

These axioms have only infinite models, for we have "hidden" the quantifier-dependence in the function-sign sex): we presuppose that the corresponding function is well-defined for all arguments. 17 A Skolem function for y in Vx3yR(x, y) is a functionf(x) such that VxR(x,f(x)); Skolem functions for y, w in Vx3yVz3wB(x,y.z,w) are functionsf(x), g(x.z) such that VxVzB(x,j{x),z,g(x,z)); and so on. See, for example, H. Enderton, A Mathematical Introduction to Logic (New YorJC: Academic Press, 1972), § 4.2 (Here I follow a suggestion by Thomas Ricketts.) 18 These ideas have much in common with Hintikka's reconstruction: see especially "Kant Vindicated," in Logic, Language-Games and Information. As in the present account, Hintikka argues that Kant's analytic/synthetic distinction is drawn within what we now call quantification theory, and Hintikka calls a quantificational argument synthetic when (roughly) ''new individuals are introduced." Thus, synthetic arguments, for Hintikka, will correspond closely to those in which Skolem functions figure essentially. Hintikka also notes the importance, in this connection, of the (often ignored) fact that Kant's logic is syllogistic or monadic (pp. 189-190). (In this respect, Hintikka has indeed made an important advance over Beth. For Beth considers only the trivial procedure of conditional proof followed by universal generalization, and therefore puts forward a conception of the role of "intuition" in proof that_applies equally well to monadic or syllogistic logic: ct. op. cit., §§ 5-7.) Yet Hintikka views the problem of quantificational rules like existential instantiation in rather the wrong light, I think - particularly when he attempts to conceive Kant's "transcendental method" as, in part, ajustification of such rules (see Chapter V of op. cit.). As I understand it, the whole point of pure intuition is to enable us to avoid rules of existential instantiation by actually constructing the desired instances: we do not derive our ''new individuals" from existential premises but construct them from previously given individuals via Skolem functions. 19 Similarly, the theory of successor of note 16 contains an infinite sequence of tenus: the socalled numerals 0, s(O), s(s(O)), and so on, that is itself a model for that theory. Compare Parsons, "Kant's Philosophy of Arithmetic," § VII and Thompson, op. cit., page 101 where this feature of the numerals is connected with Kant's views on "symbolic construction." 20 I am indebted to Philip Kitcher for prompting me to make this last point explicit. The proposition that physical space is infinitely divisible is quite a different matter, however, whose a priori truth requires transcendental deduction: see note 36 and § IV. 11 Thus Kant recogn~es relations: 'x is a part of Y' is not a monadic predicate. Nevertheless, our theory of this relation will still be "essentially monadic" in exhibiting no quantifier-dependence: cf. note 13. 22 But see also the very interesting account in A. Mehrick, Kant's Analogies of Experience (Chicago: University Press, 1973), Chapter I.,A, emphasizing the individuating role of the singular intuition. Thus, for example, two cubic feet of space are not distinguished by the general concept rx is a space1 (nor any other general concept), but only by their "positions" in space

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(pp. 9-14). In this connection, see also Thompson. op. cit. 23 Here, and tlu.-oughout this section especially, I am indebted to clarificatory suggestions from Roberto TorrettL 24 Compare Heath's interpretation of the intuitive content of Dedekind continuity: "[it] may be said to correspond to the intuitive notion which we have that, if in a segment of a straight line two points start from the ends and describe the segment in opposite senses, they meet in a point." op. cit., p. 236. 25 I. Newton, Mathematical Principles of Natural Philosophy (1687), trans. A. Motte (1729), revised F. Cajori (Berkeley: University of California, 1934), p. 29. 26 "Quadrature of Curves" (1692), it, D. Whiteside, ed., The Mathematical Works of Isaac NelIJton, 2 vols. (New York: Johnson, 1964), p. 141 of voL 1. 27 P. Kitcher, "Kant and the Foundations of Mathematics," reprinted in this volume, pp. 109-131, has drawn attention to Kant's use of Newtonian tenninology and the connection between B211-212 and the fluxional calculus: see p. 41. (See also ll. 33.) 28 See Anfangsgriinde der Analysis des Unendlichen (G6rtingen, 1761) and Anfangsgriinde der hOhern Mechanik (GOttingen, 1766). Kant was well acquainted with., and an admirer of Kastner's works. See the reference to the latter in § 14.4 of the Inaugural Dissertation, for example. 29 Following the suggestions of Philip Kitcher, I should also note that Maclaurin's book contains a number of different approaches to the foundations of the calculus. In Maclaurin - and in the Newtonian tradition generally - it is certainly not obv·ious that the kinematic approach is predominant. My point is simply that thls strand of the Newtonian tradition is central to Kant's thinking. 30 See, for example, the very sympathetic account in D. Whiteside, "Patterns of Mathematical Thought in the later Seventeenth Century," Archive for History of Exact Science I (1961), pp. 179-388, especially §§ V, X, XI, making extensive use of the concept of "limit-motions." 31 See the Scholium following Lemma XI of Principia, Book I, for example, where Newton appeals to t.h.e intuitive idea of instantaneous velocity to justify the existence of the required limits: pp. 38-39 of op. cit. 32 See Whiteside, op. cit., pp. 374-376. 33 As far as I know, Kant comes closest to making this explicit in the course of an exchange with August Rehberg in 1790 concerning the nature of irrational numbers and their relation to intuition. In a draft of his reply t-o Rehberg Kant says: "If we did not have concepts of space tr.'len the quantity ..J2 would have no meaning IBedeutung1 for us, for one could then represent every number as an aggregate [Menge] of indivisible units. But if one represents a line as generated through fluxion [durch fluxion], and thus generated in time, in which we represent nothing simple, then we can think l/lO, 1/100. etc., etc. of the given unit" (Vol. 14, p. 53 of the Akademie Edition [Berlin: de Gruyter, 1902-; abbreviated as Ak.] of Kant's Gesammelte Schriften; see also Adickes' note to this passage on pp. 53-54.) The entire exchange (Rehberg's letter: Ak. 12, pp, 375-377; Kant's d>-afts: Ak. 14, pp. 53-59; Kant's reply: Ak. II, pp. 195-199) sheds much light on Kant's view of the relationship between spatiotemporal intuition and arithmetical-algebraic concepts, and calls for a detailed investigation. Some aspects of the exchange are discussed by Parsons, "Arithmetic and the Categories," this volume. pp. 135-15834 See the excellent account in J. Grabiner, The Origins of Cauchy's Rigorous Calculus (Cambridge: rvIIT Press, 1981) and., in particular. the quotation from Bolzano's "Rein analytischer Beweis ... " (1817), § 11: "Ihe concept of Time and even more that of Motion are ... foreign to general mathematics," on p. 53.

35 See §§ X-XV of Kastner's analysis text. entitled "Bewegung gehOrt in die Geometrie." Kastner answers rHuilier's criticism of the idea of motion by drawing a sharp distinction between kinematics ("phoronomy") - where one considers the motion of mere mathematical points independently of t.heir physical properties (such as mass); and dynamics - where one explicitly considers both the physical constitution of such points 8J.,d the forces that produce the motion. The fonner is a branch of pure mathematics and is therefore a priori; the latter is a bra..,ch of physics and is therefore a postedori. In the third edition of his text (1799), Kastner even refers to Kant's Metaphysical Foundations ofNatural Science (1786) for a justification of this distinction. 36 In this connection, see the extremely important footnote at B155: Motion of an object [ObjektJ in space does not belong to a pure science. and consequently not to geometry; for, that something is movable cannot be cognized a priori, but only through experience. But motion, as the describing IBeschreibung] of a space, is as pure act of successive syndlesis of the manifold in an outer intuition in general, and belongs not only to geometry, but even to transcendental philosophy. (That the describing of a space = the motion of a mathematical point is corrfinned by the Observation to Definition 5 in the first chapter ["Phoronomie"] of the Metaphysical FOUJ1.dations of Natural Science.) Thus, Kant does have a distinction between pure and applied geometry although it is certainly not our distinction. In pure geometry we consider figures generated in "empty" space by the motion of mere mathematical points; in applied geometry we consider the actual sensible objects contained "in" this space. That what holds for mere mathematical points in "empty" space holds also for actual sensible objects fou.."1d "in" this space (that pure mathematics can be ,applied) can only be established in transcendental philosophy. See also B206-207. (Joshua Cohen. Ralf Meerbote, and Manley Thompson have all empr.asized the importa..1J.ce of B155 to·me.) 37 We might conceive "fiuents" as smooth (or at least piece-wise smooth) maps from the real numbers ("time") into some smooth manifold (Euclidean three-space, for example). The point, of course, is that precisely these concepts are unavailable to Newton and Kant. 38 This is perhaps part of what Russell had in mind when he praised the "infinitary" power of quantification in 1903: "An infinitely complex concept, though there may be such, can certainly not be manipulated by the human intelligence; but infinite collections ... can be manipulated without introducing any concepts of infinite complexity." Op. cit., § 72. These considerations may also lie behind a temark of Frege from 1891: "the field of mathematical operations that serve for constructing functions has been extended. Besides addition, multiplication, exponentiation, and their converses, the various means of tra.."'1sition to the limit have been introduced - to be sure, without people's being always clearly aware that they were thus adopting something essentially new." "Function and Concept," translated in P. Geach and M. Black. eds .• Translalionsfrom the Philosophical Wlitings ofGottlob Frege (Oxford: Basil Blackwell, 1970), p= 28 - my emphasis. 39 See Whiteside, op. cit., p. 349. 40.Thus, Kant answers a question due to Kastner in the Inaugural Dissertation, § 14.4, by proving that '"the continuous motion· of a point over all the sides of a triangle is impossible." Kant's proof consists in the observation that no tangent or direction of motion exists at any vertex of the triangle. From a modem point of view, then, he "assumes" that a COntinuous map from It (''time'') into space is also a smooth map. 41 See C. Boyer. The History of the Calculus and its Conceptual Development (New York: Dover, 1949), pp. 284-285. (As Boyer points out, the first example of a continuous but nowhere

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differentiable function was given by Balzano in 1834: pp. 269-270). We should also note that such "pathological" functions arise naturally out of Fourier's work in the 1820's on partial differential equations, work that of course is directly inspired by, and has extremely important applications to, problems in mathematical physics. So the difficulty is in no way confined to pure mathematics. This example was published by H. von Koch in papers of 1903 and 1906. See Eves, op. cit., § 13.4. As Eves points out, although the Koch curve is not single-valued, we can easily construct

42

similar single-valued examples. 43 Kitcher, op. cit., stresses the importance of this problem for Kant's conception of pure intuition, a problem that is perhaps even more fundamental than the discovery of non-Euclidean geometries. As Kileher remarks: "The death blow was not struck by Bolyai, Lobachevsky, and Klein but by the men in the tradition which led to Weierstrass's function. continuous everywhere but differentiable nowhere" (p. 123). 44 It is also worth noting that, although the distinction between continuity and differentiability obviously makes an essential (and rather strong) use of polyadic logic, it is not itself a purely logical distinction: the two formulas have the same logical form, they differ only algebraically. However, one finds precisely such a logical distinction in the contrast between pointwise and uniform properties. For example, the distinction between pointwise and unifonn convergence is purely logical: it is a distinction in quantifier order alone. And this distinction, which is at least obscured in the work of Cauchy, is developed with great subtlety and precision by Weierstrass. In Weierstrass's work, we might say, polyadic quantification theory comes fully into its own. It is perhaps no accident, then, that Frege, who was of course intimately acquainted with the foundational (;ontributions of Weierstrass's, invented the first accurate and complete formulation of quantificational logic in 1879. (In this connection, see the description of Frege's "advanced course on Begriffsschrift' in Camap's "Intellectual Autobiography": P. SchiIpp, ed., The Philosophy ofRudoljCarnap (La Salle: Open Court, 1963, p. 6.) 45 L. Beck, "Can Kant's Synthetic Judgements be Made Analytic?" Kant Studien 47 (1955). (pp. 168-181; reprinted in Studies in the Philosophy of Kant (Indianapolis: Babbs-Merrill, 1965) _ page references to this volume: in this case pp. 89-90. (G. Martin takes this point of view to extremes, viewing Kant as a forerunner of "modern axiomatics." See Kant's Metaphysics and Theory of Science, trans. P. Lucas (Manchester: University Press, 1955], for example. Needless to say, such a conception is completely antithetical to the present interpretation.) 46 Thus Hilbert, in his brief Introduction to op. cit., refers to Kant and equates the task of axiomatizing Euclidean geometry with "the logical analysis of our spatial intuition." 47 In the context of contemporary discussion, this view has been articulated most clearly and explicitly by G. Brittan, Kant's Philosophy of Science (Princeton: University Press, 1978). Indeed, after referring to B268: "there is no contradiction in the concept of a figure which is enclosed within two straight lines, since the concepts to two straight lines and of their coming together contain no negation of figure," Brittan says: "It was Kant's appreciation of the fact that non-Euclidean geometries are consistent (possibly something of which his correspondent, the mathematician 1. H. Lambert, made him aware) that, among several different considerations, led him to say that Euclidean geometry is synthetic. The further development of non-Euclidean geometries only confinns his view" (p. 70, n. 4; Brittan follows Martin, op. cit., § 2, in this estimate of the Lambert-Kant connection). I argue in what follows that this view is fundamentally mistaken. Nevertheless, it must be admitted that most of Kant's geometrical examples tum out to be false in non-Euclidean space - and, in fact, like the case of the two-sided plane figure, false in elliptic (positive curvature) space. This is especially intriguing, because one of the more note-

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wonhy statements in Lambert's Theorie der Parallellinien (1766) is the observation that triangles in what we now call elliptic space behave just like ordinary spherical triangles. (See, for example, R. Bonola. Non-Euclidean Geometry [New York: Dover, 1955J, §§ 18-22. We should not forget, however, that Lambert of course proved that elliptic space is impossible; for elliptic space - unlike hyperbolic [Bolyai-Lobachevsky] space - does contradict the remainder of Euclid's axioms: in particular, the assumed infinite extendibility of straight lines.) Further, as Brittan indicates, Kant most likely knew at least something of this important work of Lambert. So it is possible - just possible - that Kant in fact had some inkling of the idea of elliptic space. (Here I am again indebted to William Tait.) In any case, the whole question of Kant's possible insight into non-Euclidean geometry demands a much fuller investigation. (For a start in this direction see R. Torretti, "La Geometria en el Pensamiento de Kant," Annales del Seminario de Metafisica 9 (1974), pp, 9-60.) 48 Compare Beth, op. cit., p. 364: "If one assumes this view of geometrical demonstration [that intuition plays an essential role1, then absolutely nothing follows from the formal possibility of a non-Euclidean geometry, that is, from the formal independence of the Parallel Postulate relative to the remaining axioms of Euclidean geometry. For if we attempt to answer any ge0metrical question on the basis of the remaining axioms, we must (according to Kant) first construct the corresponding figure. This construction will proceed according to the antecedent laws of pure intuition, and therefore the Euclidean answer will come our at the end. The distinction between axioms and theorems will therefore obviously collapse." We should remember. however, that on Beth's own reconstruction this consequence will not hold: the role of "intuition" in inference is nothing but conditional proof followed by Wliversal generalization. See note 18 aQave. 49 But we should remember that the Russell of 1903 still believed thatlogic is synthetic. See op. r, cit., § 434: "Kant never doubted for a moment that the propositions of logic are analytic, whereas he rightly perceived that those of mathematics are synthetic. It has since appeared that logic' is just as synthetic as all other kinds of truth." 50 Compare B84, where the principle of contradiction is said to be a necessary, but insufficient, criterion for all truth, and A15l, where it is asserted that the truth of analytic judgements "can be sufficiently recognized according to the principle of contradiction [nach dem Satze des Widerspruchs hinreichend kannen erkannt werden)." 51 Of course this difference between arithmetic and geometry is explicitly recognized by Beck: he suggests that Kant's discussion of arithmetic is simply inconsistent with the general account of mathematics at B14. See op. cit., p. 89. Compare also Brittan, op. cit.• pp. 50-51. (Martin, on the other hand, goes so far as to attribute an axiomatic conception of arithmetic to Kant: see foc. cit. and especially Arithmetik und Kombinatorik bei Kant [Berlin: de Gruyter, 1975]. Yet Martin relies almost exclusively on the writings of Kant's contemporaries and students - Johann Schultz, in particular - and has no accoWlt of Kant's explicit assertion - repeated in the letter to Schultz of November 25, 1788 - that arithmetic has no axioms. For this reason he is justly criticized by Parsons, op. cit., § m, who rightly points out that what is most striking here is the difference between Kant and Schultz.) 52 Compare B202-203, where a "synthesis of the manifold whereby the representation of a determinate space or time is generated, that is, through the combination of homogeneous units [Zusammensetzung des Gleichartigen]" is said to underly all concepts of magnitude, and A 143: "Number is therefore simply the unity of the synthesis of the manifold of a homogeneous intuition as such, a unity due to my generating time itself in the apprehension of the intuition." See Parsons, op. cit., §§ VI, VII for a rich and penetrating discussion of such passages.

MICHAEL FRIEDMAN

KANT'S THEORY OF GEOMETRY

53 See J. Young, "Kant on the Construction of Arithmetical Concepts," Kant-Studien 73 (1982), pp. 17--46 for a very interesting and helpful discussion of the role calculation in Kant's concep-

together less than two right angles, Postulate 5 tells us not only that we can extend these lines ad infinitum on this side, it also says ,what will happen in the limit: the two lines must meet, and not simply approach one another asymptotically). Second, as argued in § III, geometry for Kant includes the new calculus: the method of fluxions. And this calculus. unlike Euclidean geometry proper, has no basis at all in a finite set of initial constructive functions. So intuition has the even more substantive role of creating each new object "on the spot," as it were. These two complications reflect deep mathematical problems that are only fully solved in the next century through the discovery of non-Euclidean geometries and the independence of the Parallel Postulate, on the one hand, and through the "rigorization" and eventual "arithmetization" of analysis, on the other. In the end., therefore, the relation between arithmetic and geometry remains a source of fundamental, and unresolved, tensions in Kant's philosophy. Yet it is surely remarkable that these tensions arise precisely in connection with the deepest mathematical questions of the time. 61 In this connection, see once again Parsons, op. cit., § VI. Parsons refers to Inaugural Dissertation. § 12. where Kant calls number an "intellectual concept," and to the letter to- Schultz (referred. to in note 58), where Kant explains that "Certainly arithmetic has no axioms [Axiomen], for it properly speaking has no quantum - that is. no object of intuition as magnitude [GrOsse] ~ for its object, but merely quantity [Quantitiit] - that is, a concept of a thing in general through determination of magnitude [Grossenbestimmung]," and "The science of nwnber, notwithstanding the succession that every construction of magnitude [GrOsse] requires. is a pure intellectual synthesis that we represent to ourselves in thought But insofar as magnitudes [Grossen} (quanta) are still to be thereby determined, they must be given to us in such a way that we can apprehend their intuition successively, and thus this apprehension is subject to the conditions of time" (my emp_has~). See also the distinction Parsons draws between "strong intuitability" (geometry) and "weak intuitability" (arithmetic) in "Objects and Logic," § 3. Finally, see Thompson, op. cit., page 97. (Again. these ideas can be profitably compared with Tractatus 6.01-6.031: "Number is the exponent of an operation.") 62 Op. cit.-page references in the present paragraph and the next are to this article. Kitcher himself remains officially neutral on the issue between Beck and Russell. He suggests (§ IV) that pure intuition may playa role in proofs. and even makes some interesting remarks about the use of pure intuition in (Kant's conception of) Newton's fluxional reasoning (p. 122). Nevertheless. Kitcher's setting of the problem only makes sense in the context of an interpretation of the BeckBrittan variety. 63 Closely related considerations are presented by J.Hopkins in his penetrating criticism of Strawson's attempted reconstruction: "Visual Geometry, "The Philosophical Review 82 (1973), pp. 3-34. The basic point is that "in the small," wherein alone "visual inspection" is possible, Euclidean and non-Euclidean geometries are quite indistinguishable. As Hopkins pointedly concludes: "It has always been clear that the observations required to tell between physical geometries could not be made by unaided sight" (p.34). 64 In Kant's technical terminology, Berkeley's generality problem is solved via the distinction between images and schemata - only the latter figure essentially in geometrical proof:

216

tion of arithmetic. 54 These passages, and the closely related passage at B743-746, are illuminatingly discussed by Parsons, loco cit., and especially by Thompson, op. cit., who distinguishes between "diagrammatic" and discursive proofs. Beth, op. cit., was perhaps the first, in the context of contemporary discussion. to emphasize the importance of these passages for understanding Kant's analytic/synthetic distinction. The sharp distinction Kant draws between conceptual ("acroamatic") proof and mathematical proof (demonstration) seems to me to establish part of the Russellian assumption beyond the shadow of a doubt: mathematical reasoning cannot be purely logical for Kant. (What has still to be established is that the inferential use of pure intuition is primary.) 55 See also Hintikka, "Kant on the Mathematical Method," § 6. for an illuminating discussion of the role of function-signs in Kant's conception of algebraic construction. 56 Compare the conception in the Tractatus ofrnathematics as based on "calculation" and "operations" at 6.2-6.241, along with the distinction between "operations" and "[propositional1 functions" at 5.251: "A [propositional] function cannot be its own argument, whereas an operation can take one of its own results as its base." Just as in the Tractatus, however, it is hard to see how such a "calculational" conception can yield more than primitive recursive arithmetic; see Thompson, op. cit., n. 21 on p. 341. An essential difference between Kant and the Tractatus here is that Wittgenstein also applies the notion of "iterative operations" to logic in ..the general form of a proposition" (6), whereas Kant uses this notion to distinguish logic and mathematics. This, of course, is because Wittgenstein is operating in the context of Frege's much stronger logic, where iterative construction of propoSitions via truth-functions and quantifiers plays a central role. 57 Thus I cannot follow Parsons when he draws a sharp distinction between the cases of arithmetic and geometry, and even endorses an interpretation of the geometrical case of the BeckBrittan variety (see op. cit.• §§ II, IV. and p. 58). On the contrary, I think Kant's views can only be understood if we apply the ideas Parsons has developed for the case of arithmetic to the case of geometry also. 58 See also the Enquiry, I, § 2, where the discussion follows the same order, and the letter to Johann Schultz of November 25, 1788, where the priority of arithmetic is stated rather explicitly: "General arithmetic (algebra) is an ampliative [sich erweiternde1 science to such an extent that one cannot name another rational science equal to it in this respect. Indeed, the other parts of pure mathematics [reine Mathesisl await their own growth [Wachstum] largely from the amplification [Erweiterung] of this general theory of magnitude." 59 It is interesting to specUlate on exactly what Kant has in mind in his example of the geodesicity of straight lines. This proposition appears neither as- an axiom nor theorem in Euclid, but it was stated as an assumption by Archimedes (Heath, op. cit., pp. 166-169). Kant does not appear to endorse this idea, and it is perhaps most plausible to suppose that he is referring to the variational methods developed by Euler in 1728 for proving geodesicity. That is, we consider the result of integrating arc-length over all possible (neighboring) curves joining two given points. and we look for that curve which minimizes the integral. Here, of course, the idea of "synthesis" (in the- guise of integration) is especially prominent. - But this is so far just speculation. 60 Serious complications stand in the way of the full realization of this attractive picture. First. of course, Euclid's Postulate 5, the Parallel Postulate, doe~ not have the same status as the other Postulates: it does not simply ''present'' us with an elementary constructive function which can then be iterated (thus, given two straight lines falling on a third with interior angles on one side

217

No image [Bild] could ever be adequate to the general concept of triangle. For it would never attain the generality of the concept which makes it valid for all triangles - whether rightangled, obtuse-angled, etc. - but is always limited to a part of this extension [Sphare]. The schema of a triangle can exist nowhere but in thought and signifies [bedeutet] a rule of synthesis of the imagination in respect to pure figures [Gestalten] in space (A141). (Ralf Meerbote emphasized the importance of this passage to me.) As A165 makes clear, this "rule of synthesis" is nothing but the Euclidean construction of a triangle from any three line seg-

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ment-s such that two together exceed the third of Prop. 1.22. We might represent, it then, by a constructive function fT(x, y, z). which, as Prop. 1.22 shows, is definable from the basic constructive functions fL' f E• and f e. Kant's point is simply that to do geometry we need such (general!) constructive functions (to represent our "existence assumptions"). So we do not establish geometrical propositions by "inspection" of the resulting images, but by rigorous proof from axioms and definitions. (Kitcher is of course perfectly aware of Kant's distinction between images and schemata, but he dismisses it on the grounds that deduction from general schemata would be analytic [op.cit., p. 125]. From the present point of view, of course, such deduction cannot be analytic for Kant. because it employs methods not recognized in syllogistic lor essentially monadic]

Of course Kant is not using our modern distinction between pure and applied mathematics here: see note 36. 72 What I say here actually corresponds more closely to Wittgenstein's conception (in the Tractarus) of Frege's Begriffsschrift than to Frege's own. Frege's own conception is far less ''formalistic.'' In particular, the laws of logic are in a sense scientific laws like those of any other discipline: it is just that they are maximally general laws (containing variables of different levels. but no non-logical constants). See the remarkable series of papers by T. Ricketts - "Objectivity and Objecthood: Frege's Metaphysics of Judgment," in L. Haaparanta and J. Hintikka. eels., Frege Synthesized (Dordrecht: Reidel, 1986); "Generality, Meaning, and Sense in Frege," Pacific Philosophical Quarterly 67 (1986), pp. 172-195; and "Frege, the Tractatus, and the Logocentric Predicament," Nous 19 (1985). pp. 3-15 - which depict both Frege's conception oflogic and the internal pressures pushing that conception towards the Tractatus.

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logic.) The reference to Prop. 1.5 is made explicit in a letter to Christian Schiltz of June 25, 1787, where Kant changes "gleichseitiger" in the printed text to "gleichschenkligter." See Ak. 10,

65

p.466. 66 Hintikka has emphasized the importance of the passage at Bxi - xii (for example, in "Kant's 'new method of thought' and His Theory of Mathematics;' § 2) and the fact that Euclid's proofprocedure provides a model for Kant's notion of construction (especially in "Kant and the Mathematical Method"). Hintikka also rightly emphasizes that Kant's conception of mathematical method is therefore to be sharply distinguished from a naive "visual inspection" view. Yet in his zeal to refute Russell's contention that Kant's view of geometry requires an "extra-logical element," Hintikka overlooks the fact that his own reconstruction of the analytic/synthetic distinction (see note 18) allows us to do justice to both Russell and Kant: Euclidean constructive proofs do indeed require an "extra-logical element" - if logic, as Kant thought,. is syllogistic or monadic .logic (and this, of course, is precisely Russell's point). See "Kant and the Tradition of Analysis," §§ 4-7, especially p. 218, n. 45, where Hintikka is driven to equate Kant's conception of geometrical reasoning with that of Leibniz and Wolff. 67 See B287, where Kant considers Euclid's Postulate 3: such a Postulate "cannot be proved, because the procedure it requires is precisely that through which we first generate [erzeugen) the concept of such a figure." Postulate 3 does not state afact about circles, as it were; rather, it alone makes their rigorous representation first possible. 68 This picture is explicit in Kitcher, op. cit., §§ I, II and Brittan, op. cit., Chapters 1-3 - both of which make heavy use of contemporary "possible worlds" jargon. The same idea is found in Parsons's more circumspect discussion in "Kant's Theory of Arithmetic," pp. 48-49, 58. As Brittan points out (op. cit.• p. 70, n. 4), this picture corresponds precisely to Frege's conception of Euclidean geometry: for example, in Die Grundlagen der Arithmetik (Breslau: Koebner, 1884), pp. 20-21 (translated by J. Austin as The Foundations of Arithmetic [Illinois: Northwestern, 1962], with the same pagination). Yet Frege's conception is surely not Kant's, for it is only possible in tIle context ofFrege's much stronger logic. 69 Kant's conception of possibility can then perhaps be explained as follows. Whereas Kant does distinguish between the conditions of thought alone (the idea of an "object in general") and the conditions of cognition (thought plus intuition), the former does not correspond to our notion of logical possibility but, rather, to the "empty" idea of the "thing-in-itself." Thus, what best approximates our notion of logical possibility is given by the conditions of thought plus pure intuition: _IUlIIle1y, pure mathematics. "Real possibility," then, is given by the conditions of thought plus empiricat intuition: namely, (the pure part of) mathematical physics. SQ "real posibility" most closely corresponds to our notion of "physical possibility." 70 Here I follow Thompson, op. cit .• pp. 99-100. See also Parsons, op. cit., pp. 73-75 ("Postscript").

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KANT'S VIEW OF GEOMETRY: A PARTIAL DEFENSE

Kant's view of geometry has been widely criticised and generally rejected by twentieth-century analytic philosophers. In this paper I shall try to examine what I believe has come to be the most widespread form of their criticism, and I shall suggest that it is inconclusive. Although I have no wish to accept Kant's view of geometry in its entirety, I nevertheless shall try to suggest a partial defense of it. My argument will be that those of its theses which have ~ been most vigorously condemned by analytic philosophers are less indefensible than they have usually supposed. I. OPPOSITION TO KANT'S VIEW OF GEOMETRY

Is the world Euclidean or nonEuclidean in its spatial character? That is, does Euclidean geometry correctly describe the relationships of points, lines, and figures in the world, or does some nonEuclidean geometry do so? This question has seemed important to many twentieth-century analytic philosophers, especially to the logical positivists. The development of their new philosophical position was closely intertwined with their understanding of Einstein's theory of relativity, 1 which they saw as providing a new answer to this question. Schlick,> Reichenbach3 and Carnap,4 with the concurrence of Einstein,5 himself, were originators of what carne to be the new outlook. Russell's views about mathematics6 were an important part of the background to the discussion, of course. Later, Hempel7 and Nagel' offered influential formulations of the new view concerning the status of geometry, and many other writers have endorsed it. The outlook of these thinkers has come to dominate discussion of space and geometry in twentieth-century philosophy of science, and central to it is their thesis that we now know scientifically that the world is nonEuclidean in character. All who accept this thesis find themselves obliged to reject Kant's position concerning the nature of geometry. Indeed, analytic philosophers earlier in this century saw Kant as the chief authority opposing their newly developing outlook regarding mathematics and science. They attacked his doctrine about 221 Carl J. Posy (ed.), Kant's Philosophy of Mathematics, 221-243.

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geometry because it seemed to be a main bastion of the type of theory of knowledge they opposed. Attacks on Kant's philosophy of geometry are less frequent nowadays, presumably because it is now regarded as having been thoroughly exploded; moreover, the weight of his philosophical authority has diminished, so he is no longer seen as an opponent whom it is so necessary to confront. Yet we still find analytic philosophers expounding the view that space is nonEuclidean - and sometimes they have strange and paradoxical things to say as they do so. Let us consider one arresting example of this which occurs in a paper by Hilary Putnam. Then we shall take account of how Kant might have replied.

Putnam thinks that our ability to understand this situation shows how we have liberated ourselves from the faulty, hidebound notion that Euclidean principles possess a priori necessity. Putnam's presentation is distinctly paradoxical, however. According to his verbal account the lines AB and CD that are represented in the diagram are straight lines, and he emphasizes this by calling them "really straight". Yet in his diagram the line segments AB and CD are conspicuously curved in their right-hand portions. They do not look straight to us as we view the diagram, and Putnam intentionally had them drawn so they would not look straight. Something strange is going on. If the lines AB and CD are really straight, why in his diagram does Putnam use curved lines to depict them?

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is arguing against the view that we possess a priori knowledge of important necessary truths (knowledge which Kant would have called synthetic and a priori). Important assertions that are alleged to be necessarily true and knowable a priori always may tnrn out to be false for empirical reasons, Putnam holds. As a major example of this he cites the principles of Euclidean geometry. In the past all these principles were believed to be necessary truths known a priori; but Pntnam maintains that twentieth-century physics has now established that many Euclidean principles are not true. It has been discovered that the space of the world is nonEuclidean, he says, thus overthrowing the old-fashioned view concerning the status of Euclidean principles. Pntnam believes that up-to-date people are learning to accept this conceptnal revolution. To illustrate what he regards as our new freedom of thought concerning the nature of space, Putnam presents a diagram (Figure I), and he explains it by saying: Two straight lines .. _come in from 'left infinity' in such a way that, to the left of EF their distance apart is constant ... while after crossing EF _. . their distance apart diminishes - without it being the case thar they bend ... i.e., they are really straight ... 10

Pntnam realizes that the situation as thus described will seem impossible to those who remain wedded to Euclidean conceptions. He holds, however, that those of us who have understood the theory of relativity will recognize that this situation is possible, and indeed that situations like this do actually occur when the gravitational field is of low intensity to the left of the line EF but grows more intense to the right, especially in the region between points B and D (presumably we are to suppose that there is a concentration of mass there).

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In art, to be sure, curved lines sometimes are used to represent straight ones. This occurs to a limited extent in traditional pictures employing the standard rules of perspective, and in modem art it sometimes occurs in more extreme and idiosyncratic ways. When a drawing is a work of art one cannot

take for granted that in it only straight lines will be used to depict straight lines; the rules of projection may well be more complicated than this. However, in drawing Figure 1, Pntnam was not trying to create a work of art, where part of its interest would stem from the puzzle it presents to the viewer who tries to decipher its mode of projection. Pntnam is supposedly just trying to make clear to us what physical situation it is that he is talking about. In the drawing of two-dimensional geometrical diagrams for purely expository purposes it is a well entrenched convention that straight lines are to be represented by line segments which look straight, and not by line segments which are intentionally made to look curved. Thus in effect Putnam is inviting us to suppose both that lines AB and CD are straight and that they are curved: he tells us that they are straight and he depicts them as curved. He is urging that we should free ourselves from hide-

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bound old preconceptions, and he is treating the idea that straight lioes cannot

(3) that these postulates and theorems are synthetic propositions, not analytic ones.

from misguided prejudices of the past. But is the idea that straight lines cannot be curved one of the prejudices which we should be striving to overcome? Surely not. To be curved is to be not straight. This is no prejudice, but merely a truism; a legitimate truism which modem physics does not overthrow. Of course modem physics may surprise us by discovering that lines which are straight in one sense can be not straight in some other sense; but it

Kant does not explicitly speak about nonEuclidean geometries, of course. In his time they had not yet become known, though they were soon to be developed in nineteenth-century mathematics by Gauss, Boylai, Lobachevsky, and Riemann. Each nonEuclidean geometry, if put into the form of a deductive system, will contain some postulates and theorems contrary to some of the postulates and theorems of Euclidean geometry. For example, while Euclidean geometry contains the theorems "No two straight lines inclose an area" and "The sum of the angles of any triangle equals two right angles," a nonEuclidean geometry will contain the theorems "Some pairs of straight lines do inclose areas" and "Some triangles have angle-sums not equal to two right angles." Suppose Kant had lived longer and had been confronted by nonEuclidean geometries; what would he have said about them? Surely he would have said that any nonEuclidean geometry is an unsound system, in that some of its postulates and theorems will be false when its predicates are interpreted as Kant would have interpreted them. Moreover, Kant would say that they will be necessarily false, and we can know a pliori that they are so. Notice, however, that Kant is not committed to regarding non-

does not tell us that lines which are straight can in that same sense be not

ease in saying this, we are moving into incoherence.

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This passage of Putnam's writing is striking as an instance of how the view that space is nonEuclidean can readily fall into disorder, even when it is being presented by a very able advocate. II Putnam signally fails here to make a clear case for his claim that space is nonEuclidean. This may perhaps lead us to be suspicious of his confident assertion that the Euclidean view of space is wholly' untenable. Was Kant really as wrong about geometry as twentiethcentury analytic philosophers have claimed? Let us consider further what Kant's views about it were, and then we can look more fully at the way they have most often been criticized. 3. BASIC THESES OF KANT'S PHILOSOPHY OF GEOMETRY

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be curved as one of the bad old preconceptions which we are to ontgrow. Surely we all do want to make our thinking flexible, so that we can escape

straight. In consistent presentations of nonEuclidean geometry, just as in Euclidean geometry, it never will be correct to say of straight lines that they are curved. If we carry our flexibility of mind so far that we come to feel at

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In the "Transcendental Aesthetic" of his Critique of Pure Reason Kant develops a philosophical account of space and of geometry. Basic to it is his insistance on the preeminence of Euclidean geometry. According to this part of his doctrine the sensible world (the empirical, physical, phenomenal world) is necessarily spatio-temporal in character, and its spatial structnre is strictly Euclidean. Thus Euclidean geometry is the science which provides the one correct description of the most general spatial aspects of the sensible world. Kant commits himself to the following three philosophical theses about Euclidean geometry; (l) that its postulates and theorems all are true, and hence that any

propositions inconsistent with them are false (2) that each of its postulates and theorems is such that we humans can know a priori that it is true; and

Euclidean systems as inconsistent. This point is sometimes misunderstood

by those who fail to distinguish between necessary falsehood and inconsistency. Kant supposes that we consult pure intuition in order to tell which propositions of geometry are correct; he does not suppose that we can adequately find this out merely by noticing which geometrical propositions (or which sets of them) violate the principles of formal logic. Thus, for him. a set of false geometrical propositions need not be inconsistent in the sense that their logical forms alone prevent them from all being true together. As confinuation that this is his view, let us remember how in his pre-

critical writings on space Kant had explicitly said that it would be false to affirm space to have more than three dimensions, yet that to affirm this need not involve any formal contradiction. 12 There is every reason to think his later critical position agrees with this, so for him the status of the proposition that two straight lioes can inclose an area would be essentially the same as the status he had earlier assigned to the proposition that space has more than three dimensions. From Kant's standpoint the defect in a nonEuclidean system of geometry is that its principles, while not inconsistent by the standards of formal logic, nevertheless include false synthetic propositions, incor-

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rect as descriptions of the nature of space, as we know it a priori through

A system of geometry, as these critics see it, can be studied in two different ways. One way of studying geometry is to make it simply an exercise in logical deduction. Then our concern is merely with whether the theorems of a geometrical system follow from the postulates in strictly valid fashion, solely in virtue of the logical forms of the sentences involved. When we study a system of geometry in this austere spirit, we are doing what has come to be called 'pure' geometry. In contrast to this, a second possible approach involves interpreting the postulates and theorems as statements about the physical world, statements whose truth or falsity is to be established by observation and experiment. When geometrical systems are studied in this way it is said that we are doing 'applied' geometry. The argument against Kant is now posed in the form of a dilemma. It is said that Kant's basic theses concerning geometry must be understood either as pertaining to 'pure' geometry or as pertaining to 'applied' geometry. Whichever alternative is selected, under it Kant's theses will not be jointly tenable. Thus these critics conclude that Kant's view must be rejected. Let us consider the first horn of the dilemma more fully. By 'pure' geometry these critics mean a study which investigates deductive links holding between postulates and theorems solely in virtue of their logical forms. For purposes of such a study we do not care whether the postulates and theorems are true; indeed, we do not care what they mean, or even whether they have been assigned enough meaning so that they are true or false. There are several different ways in which those who have spoken of 'pure' geometry have tried to spell out further how it is to be understood. One way of doing so is to say that we are studying 'pure' geometry whenever we abstract from the meanings and truth-values of the postulates and theorems of a geometrical system and concentrate on those deductive relations between postulates and theorems which arise solely in virtue of their logical forms. When we view geometry in this way there is no need to suppose that the postulates and theorems lack truth values; each can be a proposition about spatial relationships saying something true or false. However, the 'pure' mathematician is utterly unconcerned with what the postulates and theorems mean or with whether they are true, since what is being investigated are their deductive connections only. From this point of view, the study of 'pure' geometry is an enterprise in deductive logic, and it need not involve any substantive know ledge of space. A somewhat different way of spelling out how 'pure' geometry is to be understood is in terms of Russell's notion l6 that each theorem of a system is really a hypothetical proposition which says that if all the postulates hold then a specified consequence will have to hold. Russell's notion was that we

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The three theses stated above by no means exhaust Kant's philosophy of geometry, of course. There are other theses which also are vital to him, but which for our present purposes we may regard as less basic. One is his doctrine of the ideality of space: that things as they are in themselves do not occupy space and have no spatial character, spatiality being an aspect merely of how things appear to us humans (this doctrine is one component of Kant's transcendental idealism 13). Another is his doctrine, mentioned above, that we possess pure intuition of space, and that our knowledge of geometrical relationships results from activities of imaginative mental construction performed in pure intuition. These theses about ideality and pure intuition, though of major importance to Kant's philosophy, are less basic, in that his defense of them rests upon his theses about the preeminence of Euclidean geometry. In Kant's own time his readers must have regarded his doctrines ahollt ideality and pure intuition as the most interesting part of his philosophy of geometry, the part most worthy of their attention. His theses about the preem, inence of Euclidean geometry would have seemed comparatively obvious and uncontroversial. For twentieth-century readers the picture is quite altered. Nowadays few take seriously Kant's doctrines about the ideality of space and the role of pure intuition; these are usually dismissed as unfortunate lapses on his part." Instead, when Kant's view of geometry and space is considered, attention is focused upon his more basic claims about the preeminence of Euclidean geometry. His position is commonly criticized by those who say that there is no type of geometry of which Kant's theses (1), (2) and (3) all hold true. Kant's philosophy of geometry has come to be stigmatized as misguided and outdated, and has remained of interest to most philosophers of science only as a bad old theory the refutation of which should precede the expounding of more modern views. Let us consider the usual twentieth-century criticism of Kant's three theses concerning the preeminence of Euclidean geometry; then we can ask how Kant ntight have responded to it.

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4. THE STANDARD OBJECTION: !PURE GEOMETRY'

Critics of Kant's position have concentrated especially on Kant's three basic theses, and have been determined to show that these cannot all be true together. They have tried to accomplish their refutation of Kant by posing a dilemma. This dilemma has become the most widespread objection to Kant's philosophy of geometry.

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should not call it a theorem of Euclidean geometry that two straight lines cannot inclose an area; instead, we should consider the theorem in this case to be the hypothetical proposition that if all the posrulates of Euclidean geometry are true then no two straight lines inclose an area. To regard a system of geometry as 'pure' in this sense will be to understand all its theorems in this fashion, and to srudy a geometrical system in this spirit will be to establish theorems of this hypothetical type, This will be done by means of formal deductive logic, and the theorems established will be logical truths, Still another way of understanding 'pure' geometry is to regard the system merely a" a scheme of rules for manipulating marks,I7 It will have formation rules, dictating which strings of marks are to count as well-formed formulas of the system, and it will have transformation rules, dictating what steps are pennissible in deriving new formulas from ones already at hand, From this point of view, to srudy 'pure' geometry is to study what derivations can be accomplished within the rules of the game. None of the marks is regarded as having any meaning assigned to it, and the well-formed formulas are not regarded as saying anything true or false, All that matters are the rules of the game and what strings of marks they pennit to be derived. The first part of the argnment against Kant will be that, if we think of him as dealing with 'pure' geometry understood in any,of these three ways, his basic theses will not hold satisfactorily, 'Pure' Eu~iidean geometry will be some kind of exercise in formal derivation, and will not deal with knowledge about the spatial structure of the world. Thus, for example, whether two straight lines can inclose an area is a question which 'pure' mathematics will not even raise, Hence, from the standpoint of 'pure' mathematics, Euclidean geometry can have no preeminence whatever; Euclidean and nonEuclidean systems will be on equal footing, and Kant's claim that Euclidean principles are the true ones will find no support here. To be sure, in the srudy of 'pure' geometry a type of knowledge will be attained: it will be knowledge of relationships of derivability, and it will be a priori, However, Kant would regard that sort of knowledge as analytic, for it is based on formal logic alone. As we have noticed, Kant considers the typical propositions of Euclidean geometry to be synthetic, and not knowable merely on the basis of formal logic. 5. COMPLETING THE OBJECTION: 'APPLIED' GEOMETRY

Let us tum to the second horn of the dilemma, which has to do with what the critics call 'applied' geometry. When we are srudying 'applied' geometry we assign definite meanings to the geometrical terms of a system, and the postu-

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lates and theorems thereby become statements having definite truth values. The predicate "straight" is an especially crucial term; let us suppose we have a system in which "straight" is one of the primitive terms and is interpreted so as to be true of line segments when and only when they are paths which light rays will follow (corrections being made for any variations in the refractive index of the medium). On this interpretation the Euclidean theorem that two straight lines cannot inclose an area will express a proposition which is contingent and empiricaL One cannot know a priori whether two lines which are straight in this sense will be unable to inclose an area; instead, in order to tell whether this theorem is true one mnst observe the physical world, devising empirical hypotheses and testing them by experiment. The story, probably apochryphal, is told of Gauss that, after having thought of nonEuclidean geometry, he asked himself whether Euclidean geometry is true of the world; endeavoring to settle this, he put observers with bright lights on three mountaintops and had them measure the sum of the angles of the closed, three-sided figure created by the rays of light passing from each observer to the next. 18 This would indicate how ready Gauss was to take for granted, as many others dO,I9 that the paths of light rays are straight lines. Gauss's experiment would have been inconclusive, but the idea that experiments of this type should be carried out was shrewd. More recently, in view of the general theory of relativity and the empirical confirmations which have established it, we know that light-ray paths do deviate from what earlier physics would have predicted. The deviations are too slight to be observable in terrestrial measurements, but they become greater, the greater are the distances and the stronger is the gravitational field in the region of space involved. Thus, especially when vast celestial distances are in question, two light-ray paths can inclose a large area, and the sum of the angles of a closed figure whose three sides are paths of light rays can be fur greater than two right angles. If we take it for granted that light-ray paths are straight, then we must regard Euclidean geometry as having been empirically refuted, and we must suppose that the space of the world is a type of nonEuclidean space.

There are other plausible ways of defining straightness that might have been chosen instead, though. One would involve defining a line segment as straight if and only if it is the route along which a measuring rod need be laid down the fewest times in order to Cover the distance from endpoint to endpoint of the segment. Another would involve defining a line segment as straight if and only if it is the path along which a stretched cord will tend to lie, as the tension on it increases without bound. Adopting either of these

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definitions of straightness will also make it an empirical question whether the space of the world is Euclidean. The three ways of defining straightness that have just been mentioned correspond to three types of tests that we use in everyday life for determining whether lines are straight: using light rays (as when the carpenter sights along a board to see whether it is straight), using measnring rods (as when he measures to find the shortest distance between two points, and takes that to be the straight line between them), and using taut cords (as when he snaps a chalk line to establish straightness). Fortunately for the simplicity of our view of nature, it turns out empirically that these three types of procedure march together, yielding congruent answers as to which lines are straight. That is, modern physics tells us that, were we able to use measuring rods or taut cords to layout lines over vast distances, we would observe that pairs of these lines could inclose sizeable areas; and we would observe that the sum of the angles of a closed, three-sided figure constructed from such lines could be considerably greater than two right angles. Thus, measuring rods and taut cords would yield results consistent with the rssults we get using light rays. The position of Kant's critics is that when we assign interpretations to our geometrical terms, we are doing 'applied' geometry. Then our interpreted geometrical postulates and theorems, both Euclidean and nonEuclidean, are going to be contingent, empirical propositions. Moreover, these critics hold that whichever of these interpretations we assigu to the term "straight", some of the postulates and theorems of Euclidean geometry are going to be false; the only geometry all of whose postulates and theorems are true will be some type of nonEuclidean system. Thus the argument of these critics which forms the second horn of the dilemma is that regarding geometry as 'applied' will permit the principles of Euclidean geometry to be meaningful synthetic propositions, but it cannot allow them to be knowable a priori, and it does not in fact allow them all to be true together. Euclidean geometry enjoys no preeminence; instead, the geometry that is true of the world turns out to be a version of Riemannian geometry. Thus the dilemma is completed, and the critics conclnde that whether it is 'pure' or 'applied' geometry that Kant is talking about, in neither case will his set of basic theses be tenable.

The idea is that we see, and visually imagine, according to the principles of Euclidean geometry, even though physics finds that the world has a nonEuclidean structure. Some writers have advocated this view, or attributed it to Kant, or have suggested that Kant's position would be more defensible if he had limited himself to this claim.20 This view presumably has seemed attractive because it retains something for Euclidean geometry to be true of, while conceding that modern physics has found the physical world to be nonEuclidean. Thus we seem to give physics its due, admitting that it alone can settle what the geometry of its

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6. VISIBLE SPACE; AN UNSATISFACTORY DEFENSE OF KANT

Sometimes the view has been put forward that, while Euclidean geometry is not true of the physical world, it is nevertheless true of human visual space.

world is; yet we retain a cut-down version of Kant's insistance on the preeminence of Euclidean geometry by finding a more modest sphere, the sphere of human visual experience, within which Euclidean principles are preminent. The thought would be that each of us has had extensive experience with vision, and is in a position to know with assurance what the structure of our human visual space is. The facts concerning the structure of human visual space presumably are contingent, and our knowledge of them presumably is a posteriori, so Kant's pretensions to a priori knowledge of necessary truths in geometry will have to be set aside. Nevertheless, our extensive experience of seeing enables us to reach conclusions in this area with very high probability; we have done so much seeing that we can say with great assurance that human seeing is Euclidean in its structure. Thus Euclidean geometry retains a modified sort of preeminence. The line of thought which has just been sketched is of interest as an attempt to salvage something from the seeming shipwreck of Kant's philosophy of geometry. However, there are at least two difficulties with this line of thought. First, there is a misrepresentation of Kant's doctrine here. In the Transcendental Aesthetic Kant nowhere limits his discussion to human vision. He does not hold that human awareness of space is peculiarly linked with vision. On the contrary, in considering space, he constantly speaks of 'outer sensibility', meaning thereby all those modes of human perception which yield awareness of objects separate from oneself. Touch and hearing are obviously involved just as much as vision is; this is what Kant thinks, and surely he is right to insist on this. It is implausible to interpret Kant as though he thought that a person blind from birth would not have a Euclidean awareness of space. According to Kant's philosophy, all human beings, whether sighted or blind, represent to themselves the world around them in strictly Euclidean fashion. To alter this part of his teaching would weaken his position rather than improving it. Second, it is a defect of the view under discussion that it seems to be

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opposed to a familiar feature of human visual perception. It is a general ten-. dency in human vision that under most circumstances we learn to see things as we believe them to be. For example, a man commencing to wear new eyeglasses may at first find that the steps of a stair look curved where formerly they looked straight to him. But he believes them to be straight, and after a period of adjustment they come to look straight to him again. Or, to take another example, to a child the railway tracks perhaps look as though they meet in the distance; but as the child matures and learns that tracks do not meet, they cease to look to him as though they did. Usually one fairly quickly learns to see the world as one believes it to be. Therefore, if we now know that the world is nonEuclidean, is it to be expected that our visual experience will continue to be Euclidean? It would seem more likely that our visual experience, especially when we look at the stars, would become adjusted to fit our supposed new knowledge of the nonEuclidean character of physical

terms of the system are assigned no meanings as yet. At this stage, the postulates and theorems of the system lack truth-values and are merely sentential

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terms so as to give truth-values to the postulates and theorems. We must regard Kant and his critics as holding different views about the range of outcomes that can result from giving interpretations here. Kant seems to hold

that there is an interpretation under which the postulates and theorems become a priori truths, and he regards this as the appropriate one; his critics hold to the contrary that under any legitimate interpretation the postulates and theorems become contingent, empirical propositions, some of them false. There will of course be innumerable ways of interpreting the extra-logical terms of the system if all we want to do is give truth-values to its postulates and theorems. For example, we could devise many different interpretations just using the natural numbers as our universe of discourse. Each of the

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Thus I believe we are not doing Kant any service if we try to defend some vestiges of ~s position by turning it into a doctrine about human visual

natural number if and only if the number had some specified numerical property. The postulates and theorems, under such an interpretation, would

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become propositions saying that certain numbers have certain numerical 7. UNSOUNDNESS OF THE DILEMMA

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A better defence of Kant, I believe, will make a more frontal attack on these critics, by calling attention to a gap in the reasoning by which they try to impale Kant on the horns of their dilemma. The critics are trying to insist that when we consider a system of geometry we must either pay no attention to the interpretation of its geometrical terms ('pure' geometry) or we must interpret the terms so that the postulates and theorems become contingent empirical propositions ('applied' geometry). They recognize no third possibility - yet there is a third possibility. When they discuss 'applied' geometry the critics do show that theorems of Euclidean geometry become contingent empirical propositions under some interpretations of their predicates; but this is insufficient. In order for their argument to succeed, they would need to show that under every reasonable interpretation the postulates and theorems become contingent empirical propositions, never necessary propositions. This they do not manage to show, and therefore their dilemma is inconclusive as a refutation of Kant. To put the problem in a general way, suppose we select some good codification of Euclidean geometry and organize it as a formal system. The logical connectives are assigned their usual meanings, but all the extra-logical

properties and that certain numerical relations hold among them. Under interpretations of this sort the postulates and theorems would not become contingent empirical propositions. Instead, they would become propositions of arithmetic, and their truth-values would be knowable a priori. Thus we see that it is perfectly possible to interpret our postulates and theorems so that they become a priori truths. Critics of Kant sometimes have seemed to deny this, and to the extent that they have done so they have spoken misleadingly. Under the type of interpretation just mentioned the postulates and theorems become necessary propositions about numbers~ knowable a priori. However~ any numerical interpretation of geometrical predicates is going to be far removed from what we ordinarily mean by these terms, and the postulates and theorems when understood in this fashion have little relevance to Kant's

philosophy of geometry. Kant is concerned with geometry as a system which speaks about items that are points, lines, and figures in some recognizably normal, familiar sense. His philosophy of geometry does not have to do with systems of propositions which speak only about numbers, or other non-spatial items.

Since our pnrpose is to deal with Kant's philosophy of geometry, we should therefore confine our attention to interpretations which assign to the extra-logical terms of geometry meanings in accord with how we normally

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understand these terms. This will leave us leeway for alternative interprerations only to the extent that the ordinary senses of these geometrical terms are ambiguous. Let us focus again on the term "straight", which has an especially key role in the philosophical opposition between Euclidean and nonEuclidean geometries. There is an important ambiguity about our ordinary use of this term, , because there are two different ways in which our tests for straigbtness can plausibly be understood. Both approaches have real roots in our ordinary understanding of straightness. To see this, consider phenomena of the sort Putnam had in mind. Suppose we layout lines on a vast scale by means of light rays, or even by means of measuring rods or raut cords. If light rays are used we make corrections for any variations in the refractive index of the media through which they pass, so ,that the lines we layout will be the paths which light rays would follow were the medium Wliform. Or, if we are using taut cords or measuring rods, we fuake corrections for any thermal expansion or contraction they may undergo because of variations in temperature in the regions through which they pass; thus we layout our lines as paths along which taut cords would tend to lie or rods would need be laid down the fewest times, were the temperature uniform. However, we make no corrections for gravitational influences. When very long lines are laid out in such fashion, will pairs of these lines be able to inclose areas? And will a closed figure whose three sides are such lines have an angle-sum greater than two right angles? As we saw, the findings of twentieth-centnry physics require us to answer "Yes" to both these questions. But how are these facts to be described in terms of geometry? One way of describing these phenomena is to say that the lines, as thus laid out, are straight and the figures are triangles; this will entail that physical space is nonEuclidean. We shall describe matters this way if we interpret straightness simply in terms of light rays, measuring rods, or raut cords, making corrections only of the type mentioned. A second way of describing the phenomena, however, is to say that gravirational influences bend light rays and shrink measuring rods and raut cords. From this point of view, in order to layout lines that are straigbt we must make corrections for such effects. We shall describe matters in this way if we interpret straightness as requiring whatever corrections are needed in order to preserve Euclidean principles. This approach regards Euclidean principles as built into our conception of straightness, so to speak, and thereby it lets Euclidean geometry stand fast in the face of those phenomena which modem physics tells us to expect.

A critic might perhaps be tempted to accuse this second approach of employing an illegitimately nonempirical conception of straightness. Such a criticism would be misleading, however, for this approach does not treat straightness as a nonempirical or unverifiable characteristic. On the contrary, it leaves it an empirical question which particular lines through the world are straight; operations with ligbt rays, raut cords, or measuring rods will be needed to enable us to detect straightness. 22 If we move from the first to the second approach, those operations retain their centrality to our detecting of straightness. The difference between the two approaches is that while the first countenances corrections for refractive index and for temperatnre ouly, the second adds to these another category of corrections. Kant's modem critics disapprove of doing this, of course. Reichenbach, and others who have followed him, have implied that incorporating further such corrections into our conception of straightness is unacceptable because it makes the over-all theory less simple than it could otherwise be.23 Yet what type of simplicity is in question here: is it merely practical convenience, or is it simplicity of ontological commitments? If the critics were claiming merely mat calculations are easier to carry out when we describe the phenomena in the nonEuclidean style, then their view that the nonEuclidean theory is superior because it is simpler would be unconvincing, for two reasons. For one thing, it does not look as though there need be any significant over-all difference between the two approaches as regards ease of calculation. Perhaps some predictions will be slightly easier to calculate using the nonEuclidean approach, while others will be slightly easier to calculate using the Euclidean approach; yet calculations made from either point of view can readily be transformed into calculations embodying the other point of view. The opposition between nonEuclidean and Euclidean approaches will be rather like the opposition between Fahrenheit and Centigrade, scales of temperature, where for each there are some types of cases in which it is slightly more convenient, yet neither pattern of description can be dismissed as without merit. Furthermore, and more important, convenience of calculation has little bearing on which approach yields the trner description of the world. To demolish Kant's position, the critic needs to show that Euclidean geometry is not trne of the world, and he can hardly esrablish this through appeal to convenience. If the critic's point is to carry weight, surely the type of simplicity he is speaking of must be simplicity of ontological commitments. Here the idea would be that when one theory explains the observed facts in a more parsimonious way than does another competing theory, the former is more proba-

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bly true, and therefore is scientifically more acceptable?' For example, when we seek to explain phenomena of heat, the theory that explains them in terms of the rapid motion of minute parts of hot bodies is to be preferred to the theory that tries to explain these phenomena in terms of the presence in those bodies of quantities of an invisible fluid, phlogiston. The phlogiston theory is less parsimonious, in that it postulates an additional type of substance, while the kinetic theory does not. In order to make his case the critic needs to be able to say that to accept Euclidean geometry plus gravitational force is to adopt an over-all theory that is ontologically less economical, less simple in a sense that makes it less probably true, than what we get if we embrace nonEuclidean geometry without gravitational force. It is a complex and difficult question whether, in the end, this claim is justifiable. 25 I do not propose to take any stand on it here. What I do want to argue is that the claim is not as easy to justify as has been supposed by most critics of Kant's view about the preeminence of Euclidean geometry; and I want to argue that a case can be made, at least up to a point, in favor of the Euclidean approach. In favor of the approach which embraces Euclidean geometry plus gravitational force, it can be said that this approach does not seem to be committed to the existence of any dubious unobserved entity, as the phlogiston theorist was. It would seem that to speak of gravitational force is merely to describe

Yet if one of these approaches is more faithful to what we normally mean by straightness than the other is, this could provide a reason for regarding it as more correct. It is not that ordinary meanings of terms are always prefer-

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how masses move in relation to one another. From this point of view, that

there are gravitational forces is evident from observation, and is not a speculative hypothesis. Thus the opposition between those who retain Euclidean geometry and their opponents who advocate the nonEuclidean account does look like that between advocates of the Fahrenheit scale of temperature and those who favor Centigrade.26 That is, the Euclidean and nonEuclidean approaches seem to be merely two alternative ways of describing the phenomena, not two speculative explanations unequal in their ontological commitments.

I would associate Kant with the Euclidean approach, which insists on interpreting the term "straight" in such as way as to permit the retaining of Euclidean geometry even in the face of surprising observations. For those who adhere to this approach, Euclidean geometry will consist of necessary truths, knowable a priori, as Kant maintained. However, we have not yet found any reason for saying that this is the only correct view of Euclidean geometry; and comparing the opposition between the Euclidean and nonEuclidean approaches to that between Fahrenheit and Centigrade scales of temperature suggests that there will be no reason for regarding one of them as any more correct than the other.

able to innovative technical meanings; the point is rather that we started off

with the question whether space is Euclidean or nonEuclidean, and that question was posed using ordinary language. In responding to it, we should pay attention to what it meant, and try to answer it in its own tenus, rather than evading the question by changing its meaning. Put this way, the issue becomes; does our normal conception of straightness really have built into it the idea that Euclidean principles are to be preserved at all costs, even in the face of surprising phenomena? If it does, we would be justified in preferring the second approach, and hence in concluding that space is Euclidean. Although such an appeal to ordinary language is far removed from Kant's usual way of doing philosophy, someone wishing to defend Kant's view of geometry might wish to make this claim about the normal meaning of "straight", seeking thereby to establish Euclidean geometry as preeminent. I am not so bold as definitely to affirm that preservation of Euclidean principles is a notion built into our nonnal conception of straightness, for the latter seems to me to be too unclear to ground this claim. However, I think it can be argued that the Euclidean approach is at least as faithful as is the nonEuclidean to our normal conception of straightness. In the end, perhaps neither approach is more correct than the other; the two approaches may capture conflicting tendencies both of which are genuinely present in our ordinary conception of straightness, which must therefore be regarded as basically ambiguous in this respect. This line of thought provides a tentative and partial defense of Kant's doctrine that Euclidean geometry is true and that its truth is knowable a priori. We can say on Kant's behalf that there is a not-implausible way of understanding the term "straight" which makes Euclidean geometry into a body of necessary truths, knowable a priori. Even at its best, though, this defense of Kant's theses (1) and (2), is only partial, for it does not establish that the Euclidean approach embodies the only legitimate interpretation of geometry. Still, this partial defense of Kant's position provides it with more support than most of his modem critics have been willing to grant could be found. 8. TWO FURTHER OBJECTIONS

One other move that a critic of Kant might make is to admit that Euclidean geometry can be interpreted so as to be necessary and a priori, but to add the

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reservation that it is then a contingent empirical question whether there are any entities which do satisfy its posmlates.27 The critic is. thinking that, for all we know, Euclidean geometry could be utterly irrelevant to the world, even though it is necessarily true. To explain this position, we might draw an analogy with the principles of genealogy: we k.'10W many necessary, a priori truths about genealogical relationships, for instance that a person's parent's parent is the person's grandparent, and that a person's brother's wife is that person's sister·in-law; yet all these principles would be utterly irrelevant and inapplicable to a world in which there was no sexual reproduction. Similarly, all the principles of Euclidean geometry might be true as hypothetical propositions ("If anything is a triangle then its angle-sum equals two right angles''), yet it seems logically possible that there might be none of the entities it speaks about, such as straight lines and triangles, in which case Euclidean geometry would be irrelevant and inapplicable. In this way the critic argues against Kant that a priori geometrical principles are entirely vapid until they are supplemented by empirical observation of the space of the world. This line of comment is potentially misleading, however. It may seem that Euclidean geometry would be less relevant to the world if the world contained none of the entities of which geometry speaks.28 Yet the proposition that the world contains such geometrical entities as straight lines and triangles is hardly a contingent empirical proposition. No conceivable observations could disconfirm this proposition. Perhaps it is some sort of a bare logical possibility that our observations might be so confused that we could not pick out any lines or figures (though it is scarcely coherent to suppose that 'we' would exist under such conditions). Even this would not establish that there are no such geometrical entities, though; it would show only that we could not identify them. Insofar as we conceive of a physical world at all, we have to regard it as consisting of items disposed at various positions in space, and those positions will have to define points, lines, and figures. 29 One source of misunderstanding here may be confusion about what it is for geometrical entities to exist. When we deal with lines that form the boundaries of an athletic field we mark them out with whitewash; but when we say that the Equator is the line around the Earth equidistant from its Poles, we are not claiming that the existence of this line depends on whether anyone has marked it out - it is sufficient that its locus is specifiable. Similarly, in outer space, when we speak of the straight line which joins the Earth's center of mass with that of the Sun, the existence of this line does not depend on its actually having been marked out, but only on its being specifiable. In general, the existence of points, lines, and figures does not require them to have been

marked out. If there is a physical world at all, it must contain points; and if there are points there must be lines between them, and figures formed by those lines, Thus it is unconvincing to hold in this way that Euclidean geometry can be necessarily true yet may not apply to the space of the world. Still another objection that a critic may raise comes from physics, and relates to the theory of relativity. In the preceding section a defense of Euclidean geometry was suggested that involved treating its principles as necessary a priori truths, though this had to involve supposing that gravitational influence bends the paths of light rays (and shrinks measuring rods and taut cords). If we pursue this line of thinking, it looks as though we shall be assuming that whatever gravitational corrections come into play will depend wholly on the amount of matter in the universe and its distribution. The objection now arises that Einstein's theory of relativity entails a different result, viz., that the nonEuclidean character of paths of light rays depends mlly slightly on the distribution of matter in the universe.3o This seems to indicate that the proposed strategy for defending Euclidean geometry is in conflict with scientific thinking, and therefore is untenable. 31 Complicated issues are raised here, but I think at least a possible line of reply to this objection can be suggested. The defender of Euclidean geometry had better avoid commitment to the idea that it is only in so far as matter has exerted gravitational force on them that light rays deviate from straight-line paths. If that generalization had been true, it would have provided him with a welcome way of explaining of those deviations; but if it is not true, he must seek some further explanation, presumably in terms of an irregularity in the structure of space-time itself. Whatever may be the way in which light rays and other relevant phenomena occur, he is committed to explaining them in some manner consistent with Euclidean geometry. His position has to be that, with ingennity, he can do this.

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9. GEOMETRY AS SYNTHETIC

My claim has been that there is a not implausible way of understanding geometry according to which the principles of Euclidean geometry are a priori and necessary. However, some critics will complain that when its principles are understood in this way Euclidean geometry becomes analytic, rather than synthetic, so that we have not preserved the third of Kant's three basic theses. Unfortunately, the analytic-synthetic distinction, as Kant employs it, is deeply ambiguous. However, there is one defensible sense in which

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Euclidean geometry can be synthetic, under the approach we are considering. 32 One of Kant's conceptions of the synthetic is that synthetic propositions are those deductive proof of which cannot be provided merely on the basis of formal logic and explicit definitions of terms. 33 Suppose that under our Euclidean approach we are unable to give an explicit definition of the term "straight". Perhaps we explain it by reference to operations with measuring rods, but with the open-ended understanding that any needed corrections will be invoked to preserve Euclidean principles; and this yields us no definiens which can be everywhere substituted for the term "straight". With this inexplicit definition of straightness, Euclidean principles cannot be deduced merely from laws of logic plus explicit definitions. Consequently, Euclidean principles will be synthetic in this modest sense. To this extent, Kant's third thesis is defensible. Thus each of Kant's three basic theses about Euclidean geometry can be given more of a defense than the great majority of his twentieth-centnry critics have been willing to suppose. When defended along these lines, Kant's three basic theses will be too modest in content to serve as a good foundation for the further conclusions he wants to draw concerning transcendental idealism and pure intuition. Nevertheless, by understanding Kant in this way we make it possible to offer some defense of his more basic theses, and thus his philosophy of geometry is shown to be less readily refuted than most critics of it have supposed.

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NOTES I Michael Friedman in his Foundations of Space-Time Theories (Princeton, 1983), p. 3, discusses the strong linkage between logical positivism and the theory of relativity. 2 Moritz Schlick, Allgemeine erkenntnislehre (Berlin, 1925), translated as General Theory of - Knowledge (New York, 1974), especially section 29. 3 Hans Reichenbach's view of geometry developed through his close contact with Einstein from 1919 onwards. It was expressed especially in his Relativitatstheorie und erkenntnis apriori (Berlin, 1920), translated as The Theory of Relativity aild A Priori Knowledge (Berkeley, 1965), and in his Philosophie der raum-zeit-Iehre (Berlin, 1928), translated as The Philosophy of Space and Time (New York, 1958). 4 Rudolf Carnap's mature views receive a lively formulation in his Philosophical Foundations of Physics (New York, 1966), Part ID. .s One expression by Einstein of philosophical views about geometry is in his «Reply to Criticisms", especially pp. 676-679, in P. A. Schillip (ed.), Albert Einstein: PhilosopherScientist (Evanston, 1949).

KANT'S VIEW OF GEOMETRY

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6 Russell's best known writings relating to the philosophy of mathematics have little to say specifically about geometry, but advocate an over-all view of mathematics according to which Euclidean geometry could not have the kind of preeminence Kant ascribes to it. See A. N. Whitehead and B. Russell, Principia Mathematica (Cambridge, vol. 1, 1910; vol. 2, 1912; vol. 3, 1913). See also Russell's more popularly written An Introduction to Mathematical Philosophy (London, 1919), especially chapter 18. 7 Carl G. Hempel wrote a pair of widely read articles which came to be regarded as summing up the outlook of logical empiricists concerning the status of mathematics: "Geometry and Empirical Sci:ence" and "On the Nature of Mathematical Truth", both in American Mathematical Monthly, vol. 52 (1945). 8 Ernest Nagel, The Structure of Science (New York, 1961), chapters 8, 9. 9 Hilary Putnam, "The Logic of Quantum Mechanics", in his Philosophical Papers (Cambridge, second edition, 1979), vol. 1, pp. 174-177. 10 Putnam, op. cit., p. 174. II Nor is Putnam alone in allowing his presentation to faIl into disorder in this way. His mentor Reichenbach had become entangled in a similar difficulty in The Theory of Relativity and A Priori Knowledge. There Reichenbach had declared that modern physics shows Euclidean geometry to be wrong (pp. 3-4); presumably he was thinking of astronomical observations which show that the paths of two light rays can inclose an area, and was thinking that we are to identify straight lines with the paths of light rays, yielding the result that two straight lines can inclose an area, contrary to Euclidean geometry. However. elsewhere in his presentation Reichenbach says of such a situation that the rays are deflected and curved as they pass through the gravitational field (p. 11, p. 18). Thus Reichenbach says both that the paths of light rays are straight lines and that these paths are curved. But one and the same line cannot be both straight and curved. 12 Immanuel Kant, Inaugural Dissertation and Early Writings on Space, translated by John Handyside (Chicago, 1929), pp, 5-11. 13 Henry E. Allison in his Kant's Transcedentalldealism (New Haven, 1983) presents one interpretation of transcendental idealism. 14 Kant's theses (I) and (2) would have seemed completely obvious to eighteenth-century readers. Thesis (3) would have seemed more controversial. as Leibniz and Hume had denied it. But the interest of thesis (3) would have lain in what it was thought to imply concerning the nature of mathematical knowledge, and what that in tum would imply concerning metaphysical views such as transcendental idealism. 15 For example, P. F. Strawson in The Bounds o/Sense (London, 1966) treats transcendental idealism as a misguided part of Kant's philosophy. 16 Russell expresses this view in various places, for example in his Principles of Mathematics, 2nd ed. (London, 1937), p. vii. 17 Camap discusses this approach, for example in his Foundations of Logic and Mathematics (Chicago, 1939). This is Volume I, number 3, of the International Encyclopedia of Unified Science. 18 The Encyclopedia Britanica (15th edition, Chicago, 1986), p. 699, describes Gauss as "one of the first to doubt that Euclidean geometry was inherent in nature and thought." 19 See, for example, P. A. M. Dirac, "Development of the physicist's conception of nature", pp. 2-3. in Jagdish Mehra (ed.), The Physicist's Conception o/Nature (Dordrecht, 1973) . 20 'Visual space' is mentioned by Hans Reichenbach in "The Philosophical Significance of the Theory of Relativity", in Albert Einstein: Philosopher-Scientist, edited by Paul A. Schillp

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(Evanston, 1949), p. 299. See also Adolf Griinbaum, "Carnap's Views on the Foundations of Geometry", in The Philosophy of Rudolf Carnap, edited by Paul A. Schillp (La Salle, 1963),

30 For discussion of this point, see Michael Friedman, Foundations of Space-Time Theories (Princeton. 1983). pp. 27-28. 31 I thank Athanasse Raftopoulos and Mauro Dorato for drawing my attention to this objection. 32 I argued to this effect in my Philosophy of Mathematics (Englewood Cliffs, 1%5). ch. 2. More recently Gary Rosenkrantz in the paper cited above bas argued that geometrical principles can be regarded as necessary, a priori, and synthetic. 33 This is the conception of the analytic-synthetic distinction which Gottlob Frege adopts and clarifies in The Foundations of Arithmetic (Oxford, 1953), Section 3. Frege held that geometry is synthetic in this sense.

p.666. 21 Here I do not intend to suggest that it is at all clear what it would mean to see according to Oile type of geometry rather than another. Patrick Heelan in his Space-Perception and the Philosophy of Science (Berkeley, 1983) suggests that many people do visually experience the world as nonEuclidean, and in support of this he refers to the nonstandard type of perspective found in the paintings of Cezanne and others. However, Heelan's account seems to entail, implausibly, that when Cezanne looked at the landscape he saw it according to one scheme of projection, while when he looked at his canvas he saw it according to a different scheme. 22 Of course there will be some line descriptions which pick out lines that must necessarily be straight (e.g., "The straight line between London and Paris", or ''The shortest distance between London and Paris"). However. with a vast range of typical line descriptions (e.g., "The route I lately took from London to Paris") it wiI1 be an empiricaJ question whether the line referred to is straight. 23 Reichenbach. in his The Philosophy of Space and Time. distinguishes between inductive simplicity and descriptive Simplicity. For him. inductive simplicity has to do with the nature of the phenomena being described; where two opposing hypotheses are both consistent with the observed evidence. the inductively simpler hypothesis is more probably true. Descriptive simplicity has to do merely with ease of description; a descriptively simpler hypothesis is more convenient than a descriptively complex alternative, but they describe the same phenomenon and cannot differ in probability. Reichenbach wishes to show that Euclidean geometry is unacceptable, anq to do so he would need to show that some nonEuclidean theory of the world is inductively simpler than any Euclidean theory. His arguments do not succeed in showing that this is so. however. 24 Russell, in Our Knowledge of the External World, lecture IV. stimulated discussion of this idea through his emphasis on Occam's razor as a principle of scientific philosophizing. 25 Michael Friedman in his Foundations of Space-Time Theories (Princeton, 1983) makes an impressive attempt to justify this claim. 26 This is pretty much the view of Henri Poincare. who writes, "What then are we to think of the question: Is Euclidean geometry true? It has no meaning. We might as well ask if the metric system is true ... One geometry cannot be more true than another, it can only be more convenient." Science and Hypothesis (New York, 1952), p. 50 (the original French edition was 1902). 27 Gary Rosenkrantz. in "The Nature of Geometry", American Philosophical Quarterly, vol. 18, no. 2 (April. 1981). argues, as I want to do, that Euclidean geometry can be interpreted so as to be necessary and a priori. However. on p. 108 he says that even if Euclidean geometry is necessarily true. space still may be nonEuclidean; with this I disagree. 28 It seems to be Plato's view that phenomena in the realm of becoming do not have any exact geometrical character, so he could be said to hold that entities of the type discussed in geometry do not exist in the physical world. He does not conclude, however, that geometrical knowledge is unimportant. For Plato, some things in the physical world do come close to having geometrical character; moreover, there is another world containing the geometrical Fonns that have it to perfection. 29 P. F. Strawson in his Individuals (London, 1959). chap. 2, discusses alternative patterns of experience, such as an "auditory world". But it is only when these begin to exhibit something like spatial order that they can be experience of a physical world.

ARTHUR MELNICK

THE GEOMETRY OF A FORM OF INTUITION

~

For Kant, geomeny studies space which is a pure intuition and at the same time a form of empirical intuition. Geometrical statements somehow require or pertain to constructions or productive activity such as drawing. rotating,

•

pointing, bisecting, etc. Space somehow provides the pure manifold which enables thought to pertain to objects. Still more, reality in space is neither finite nor infinite. These are all constraints from Kant's text on what geomeny might be for Kant. The 19th and 20th century have added further constraints on what a justifiable philosophical account of geomeny would have to be. It would have to allow for non-Euclidean geomeny, for the time parameter making a difference to the spatial geomeny (more generally for geomeny being space-time geomeny) and for a connection between geomeny and the distribution of matter. Putting these two classes of constraints together, we get a set of constraints on what a Kantian theory of geomeny that is also in itself plausible would have to be. It is such a theory that I should like to begiu to present in outline fonn. Before beginning we should take cognizance of the fact that for Kant we could verify a priori that geomeny is Euclidean. Further, for Kant the geomeny of space is independent of any time parameter and independent of any material distribution. Thus, satisfying the constraints of a contemporary plausible account of geomeny seems incompatible with its being a Kantian theory. I shall get around this by making a sharp separation of what the subject-matter or topic of geomeny is, from the issue of the verification of geometrical statements. Roughly I shall argue that Kant's theory of the subject-matter of geomeny is compatible with contemporary constraints, and is separable from his theory of geometrical verification. Let us begin then with subject matter of geomeny, viz., Space. Space, for Kant, is the form of empirical intuition. An empirical intuition has sensation as its matter, but is something more than sensation. An empirical intuition involves positing or setting objects outside oneself Kant says. We can take this activity or performance of positing to be pointing or circumscribing, or delineating, or tracing out or otherwise gesturing with one's finger. Instead of

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talking of having a sensation of red, let us talk of using the sound 'R-E-D' as

move thus sweeping out a line with them, that we hold our mental attention

a reactive name; that is, as a reaction to be made only when one is red-wise

outward alongside us as we move. Then we mentally shift or sweep out our attention as we take steps. In this way the focus of attention is at the end of a

affected. In this sense, for example, if a dog has been trained to wag its tail only when presented with what is red, it would have the use of the reactive name 'Red'. To react red, then, is certainly different than first ostending or circumscribing properly before reacting Red. In the latter case one performs spatial-behavior and thereby ostends or indicates or directs attention, thereby producing a singular outer representation. It is spatial behavior or activity, I claim, which is the form of an empirical intuition. Ostending or delineating is productive or a matter of performing. rather than responsive or a matter of

sweep of attention.

Suppose further that this sweep of attention has a sensory upshot so that after moving and circumscribing one can react 'red'. We have then, I suggest, a 'big' or 'non-local' Kantian empirical intuition, the form of which is still productive behavior or activity by foot and finger and whose matter is still passive reaction. We also have a sense in which, as Kant says, the form of intuition precedes our capacity to be affected. One must foot-by-finger

reacting. Note that one may also regard this spatial behavior as directing or drawing out or focusing attention (either one's own or others' attention). Kant would allow that such shifting or focusing of attention can be carried out

perfonn prior to or in order to passively react. The 'scope' of passive reaction

without moving one's finger at all, at least in the case of directing one's own

course of spatializing. An empirical intuition then can extend as far as motive behavior and can include multiple ostensions or resting of attention in the course of step-taking, and can include a multiplicity of sensory reactions. An empirical intuition is not restricted to local or proximate behavior.

attention. He would allow, that is, a purely mental act of shifting or drawing out attention. The important point for us is that this is still an act or a performance, rather than a sensation, and our fundamental thesis is that Space, for Kant, is a matter of spatializing; i.e., it is an activity or mode or productive

behavior, rather than a component of reality that we respond to. Space is a lot bigger than circumscribing or delineating. It is not just that space is locally or in the small a form of intuition. These local spatial acts must be components or in Kant's terms limitations of an ongoing global spatial activity. How then can we direct or shift attention in the large, i.e. beyond tbe reach of our finger gesturing? The answer is by spatializing with our feet; i.e., by moving or taking steps. Suppose that as one moves one holds one's finger outward thus tracing out linearly. At any stage in the step-taking one may cut one's linear progression and delineate or circumscribe with one's finger. The circumscription or ostension then is a limitation of the ongoing linear sweeping. Put in terms of attention. what is happening is that one sweeps out one's attention as one moves and then focuses it by the circum-

scription or delineation. Thus, for example, I can direct your attention by simply ostending or by moving (with or without you) and then ostending. Local proximate finger gesturing then is a special or limiting case of moving and then finger gesturing. Now moving or taking steps (with or without one's finger outsttetched as one moves) is again a piece of behavior or a performance .. It is not merely behavior propadeutic to directing attention but, with the culminating tracing or circwnscription, constitutes behavior that directs attention.

Again, let us suppose that instead of holding our fingers outward as we

(of all possible perception) is constituted by this spatial production in the sense that passively reacting or 'employing reactive names is always in the

Let us turn next to the issue of separability or isolability of this form of intuition. Firstly, one can carry on spatializing behavior independent of sensory upshot or use of reactive names along the way. Secondly, one can direct such behavior in terms that make no reference to how one is affected

along the way. So far we have restricted ourselves to behavior, rather thaJI the regulation of such behavior by thought. The most primitive mode of this regulation would be to command the behavior. When I say that spatializing behavior can be directed independent of empirical reaction, I have in mind such commands as to Take-6-steps-Circumscribe-Squarely or a command such as to Rotate-K-units-Take-L-steps-and-Circumscribe-or-Delineate-withone's-fingers. Here the nature of the instructions makes no mention of how

one is to be affected or to be empirically reactive. An example of a command which does make such mention would be one to Take steps until you can react red. Note that the spatial behavior in every such non-pure command, can be directed also purely in terms of what direction to go off in, how far to go off, what shape one is to delineate with one's fingers, etc. In this regard one ca..Yl direct the full scope of spatializing behavior without any mention of

empirical sensory upshot at all. What we have then is separable or isolable spatial behavior, which can be carried out even if nothing is thereby circumscribed or indicated. Of course, this very same behavior is the form of any positing of reality one does in the course of carrying out the command. To say that the form of empirical intuition is itself given in a pure intuition, I

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ARTHUR MELNICK

THE GEOMETRY OF A FORM OF INTUITION

claim, is to say that the very behavior which underlies and directs our capacity to be affected, can also be carried out fu'1d ordered independent of how one may thereby be affected along the way, Let us now look at a geometer's diagram or construction, and limit ourselves first to a local case of construction with penciL Suppose I draw a triangle. Then I am drawing out your and my own attention with the pencil as an extension of my finger. If I happen to trace or delineate a red patch, then I have thereby circumscribed, indicated, or directed your attention to the red patch. My drawing gestures indeed underly my saying something like 'This red patch'. Of course the geometer constructs in accord with a pure rule. Geometrical constructions are not of the form 'Move the pencil so as to delineate the red portion'. Still, it is the very same behavior, viz., guiding or 'sweeping out' attention, that the geometer performs, as we perfonn in order to ostend red. A geometrical construction then is identical with that behavior which is the form of an empirical intuition, though it can be directed without mention of empirical upshot. If in showing you something, I focus or draw your attention and thus enable you to have a certain sensory upshot, it is likewise true that the geometer is drawing or sweeping out attention by his con-

attention without physical gesturing. I shall keep, however, to the case where one's attention shifts are with physical gestures of the finger and ofthe foot. We have next to ask what about these pure constructions or operations is peculiarly geometrical, or fonns the pa.rticular topic of geometry? I first presume that a geometry is determined by the totality of triangles. Thus Euclidean geometry differs from a.n alternative geometry by the dimensions of its triangles. If, however, we took triangles as the objects of geometry, then different geometries would have different subject-matters. To keep a single subject matter for different geometries, we assume that a tria..,gle is a matter of pairs of different constructions OT operations issuing in coincidence or having coincidence as an upshot. Intuitively, such a pair mllst consist of a straight-line construction, together with a broken line construction. These construction pairs will be invariant from geometry to geometry. The only difference is which pairs of operation ensue in coincidence. We shall make this somewhat more precise in a moment.

248

structions. The geometrical construction is the act of producing, not the image left by the motion of his penciL The geometer points with his pencil, then sweeps out. In drawing an arc between two lines, the geometer rotates his fingers. In drawing a triangle, the geometer meets or coincides with a previous stage of his construction. In drawing a triangle with sides in a definite proportion, the geometer segments his line stopping his sweep to mark cuts in the line. It is the flow of his production that is cut or segmented. It is, we shall argue, operations like pointing, cutting. rotating, sweeping out or flowing, that are, in a sense, the subject matter of geometry. It is not the pictures left as a record of such operations that he studies, but the operations themselves. Further it is these same operations that are involved in a local empirical intuition as its exhibitive or indicative fonn. Let us turn next to carrying out constructions or operations in imagination. Here there is no record left, but one sweeps out in shifting one's focus. Note, one's imaginative or mental shifting is not a construction of or in private space. My attention, for example, traces out a figure in front of me. If asked where I am focusing now, I can point with my finger and you can both focus and point so as to coincide with me. Note further that this purely mental act is ostensive or positing and can be followed by an empirical reaction. I do not wish to either defend or attack Kant on the possibility of sweeping out one's

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For now, let us return to Space as a form of empirical intuition. Suppose we have a language of spatial commands that ensue in empirical reactions and suppose a child and his parent are situated as indicated in Figure I. Suppose further the child sees a toy car and wants his parent to get it or fetch it for him. Behavior in accord with R, will get him in a position to touch the car or delineate the car. Behavior in accord with R2 will get him to the Parent. The determination of whether R 3 is the behavior he should prescribe to the parent is equivalent to the determination of whether the pair consisting of L'te straight-line construction or hehavior R, ends in coincidence with the broken-line behavior R2 + R3. A singular term or representation in this command language is a matter of spatially directing behavior and empirical reaction. Thus, a singular term wonld have the form 'Do R 1 React Car'. Given this notion of a singular term, then to say the subject matter of geometry is the coincidence of pairs of constructions, consisting of one straight line and one broken line, is equivalent to saying that the subject-matter of geome-

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THE GEOMETRY OF A FORM OF INTUITION

try is the equivalence of singular terms or singular representations from different vantage points. Alternatively, we can say that the topic of geometry is

result in or have the upshot of coincidence. Note t.1lat the operations involved,

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the coordination of directing sensory attention across different addressees or

viz., rotating, laying end to end, etc. have their counterparts in our spatial behavior by foot, and we may regard these operations as regularized versions

'speakers' of the sensory-attention directing language. That there should he a geometry of space at all then is equivalent to saying there should be a coordination of singular terms in a language in which a singular term is a matter of directing ostensive foot-by-finger behavior, or a language in which singular terms are directions how to show or exhihit reality. In this way the subject-

of our step-taking. What we seem to end in then is an operational interpreta-

matter of geometry is in terms of the semantical notion of the parity of singu-

verification, and see that for Kant the subject-matter of geometry is construction and that is why, for him, the verification of geometrical statements pertains to constructions, it is not surprising that Kant comes out being a sort of operationalis!. However, I wish to suggest that he is a very special sort of operationalist.

lar representation. Let us return now to a more precise formulation of the pertinent pairs of

constructions or operations. Firstly, what is it to proceed or move straightly? Suppose instead of moving our feet we lay a rod end to end. What is it to lay the rod out straightly? I suggest the following local operational definition of transferring the rod straightly. Rotate the rod so as to produce a circle. Take a smaller rod and measure the circumference of the circle produced and then flip the original rod so that its new position coincides with the point half-way around the circumference of the circle. This is diagrammed as in Figure 2, where the new position of the rod is from B to C!2.

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A straight-line construction then consists of iterations of proceeding straightly according to this local operation. Such a construction forms the first of our pair of constructions which are candidates for coincidence. It should be clear now how to locally define altering the direction of the rod AB, by going to a different position than C!2 on the circle. Then a hent line construction is simply a laying end to end of a rod that proceeds everywhere straightly except at one stage. The subject matter of geometry then is when pairs of operations, one straight-line and one bent, beginning from a common origin

tion of geometry. In one sense, this might have been expected to be the Kantia11 view of geometry, since for Kant determinate spaces are given in construction, and constructions as well as operations are matters of performance. Once we separate the issue of subject-matter from the issue of

Firstly, the operations and ensuing coincidences that interpret geometry are

also the operations or behavior involved in singular representation or directing attention. The semantics of geometry relates to the semantics of singular representation per se, if singular representation is a matter of directing or

governing behavior that gets us affected. These operations then are not idle games or arbitrary semantics for geometry, since the very same operations are literally components of the semantics of empirical representation. Secondly, and equally importantly, the semantics of geometry cannot he carried out in classical fashion, say in terms of quantification over possible operations. To see why not will require a detour into Kant's metaphysics. The fundamental question for Kant is how thought relates to an object. What is it about a thought that makes it bear on or pertain to what is outside or other than itself? Thought does not immediately produce its object, nor in most cases, is the object of thought present to be indicated by thought. How then can I merely by thinking be connected to an object which neither affects me nor is produced by the thought? Part of Kant's answer is that I aln connected by the thought heing productive of that hehavior which gets me into contact with the object, or which makes me liable to he affected by the object. The ouly representative role of thought which enables it to get outside itself is as a guide or rule for the hehavior which enables us to be affected or attain contact. This contrasts with the Leibnizian idea that thought is descriptive and can represent what is outside itself by conceptual individuating conditions. Thus, for Kant, the proper semantics for our language or our thinking is directive or prescriptive or legislative rather than descriptive. This goes for spatial behavior too, which must he regulated by thought not descrihed by it. I do not have time here to go into what such a semantics will look like. It

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suffices to say that a domain generally of a language is not a set of entities, but a procedure or activity, and that the domain in particular of our empirical or factual talk is in part, the spatializing procedure or activity tllat has sensory or affective upshot. When I say then that constructions or operations are the sllbject matter of geometry, ! do not mean that the intended interpretation of geometry consists of " set of consttnctions, acts, or operations. Rather the intended interpretation is a procedure or activity of spatializing. There are no entities, not even act-like entities in Lne semantics, though there is individuation in the sense that the procedure has j,"dividual stages. Geometry, then, literally is to be expressed in the form of rules for behaving. Coincidence then becomes a reaction to one's own productive behavior, rather than a predicate of pairs of constructions. Very roughly, a geometrical statement will have the form 'Do R, Do R2 React Coincide', where the same rule, 'Do R,' say enters into an empirical singnlar tenn s"ch as 'Do R 1 React Red'. Operations are important for Kant because they can be guided by thought, whereas relations or entities can only be described by L':!ought, and descriptive thought is ultimately empty for Kant, no matter how precise or detailed. One thing wrong with objectivist theories of Space, whether relational or absolute, is li,a! on such theories space could only enter descriptively into the representation of objects. Thus, absent objects would have to be represented according to descriptions like '(u) (x is so many meters from this)' or '(u) (x is at place p so many meters from here),. But even purported descriptive reference that includes spatial conditions is ultimately empty alld without determinate connection to any singular reality for Kant. Reality is a reactive upshot of spatial behavior that is guided and directed by thought not a somethiug described, even in spatial terms, by thought. If so, then a consttnctive account of Space is not only an alternative viable account to relational or absolute theories, but the only viable accounL Only if Space is a form of our own behavior, does thought have a systematic and global field of behavior to guide, and so a full scope of reality to represent as the reactive upshot of such behavior. Put simply, there is no determinate empirical representation at all, except if Space is the activity for thought to guide. Thus, geometry too, as the science of space, must be given an operational or procedural interpretation. The whole notion of factual talk collapses with any objectivist theory of Space. This makes it incumbent upon us to make an operationalist theory of geometry consonant with contemporary views4 The first extension of Kant would be to convert geometry from the science of spatial construction t(l the science of spatio-temporal consttnction. This involves understanding how time is a mode of operation or behavior, and

how a langnage expressed in rules can have the past in its scope. The view we have attributed to Kant, that thought can do nothing more than guide behavior, is a kind or pragmatism, and pragmatism seems too limited a view, restricted to~ a future orientation. Much of Kant's Analytic is concerned with how representation of the past is possible, if thought is but a rule or guide for behavior. Talk about the past, for Kant, is not to be reduced to talk about future behavioral consequences. Rather Lltere are specifically past-oriented modes of regulating behavior that are irreducible to future-oriented modes

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such as commanding. For example, one may delay or postpone an empirical

reaction to how one is affected, whiling away the time of delay by tapping or reciting numerals, and then summarize. Summarizing is a past-oriented ana-

logne to the future-oriented comtnanding. Further, thought can regnlate how it is proper or legitimate to be in the course of or in the middle of a procedure, as well as how it is proper to go IL':!ead and begin to proceed. Tne details of how to extend a behavior - legitimating semantics to the past are some-

what involved, but the upshot is that Kant does have a genuine conception of the scope of regulation of behavior covering the past. In terms of this conception one can extend the operational interpretation of geometry to include coincidence of alternative spatiotemporizing constructions or activities. I have worked this out elsewhere up to the case where every two world lines are connected by the ancestral of the relation 'can influence or can _be

influenced by'. The 'mixing' of material distribution with geometty can also be expressed in a purely consttnctive language as a constraint of emph-ical reactions on coincidences of spatiotemporizing constructions. Thus, spacetime still is a matter of construction or activity, but the coincidence reaction to one's own activities are correlated locally with one's empirical reactions to

how one is passively affected. The subject-matter of geometty remains coincidences of constructions, only this time the temporizing component of a construction and the empirical reactions along the way make a difference.

If we focus on the issue of subject-matter then, rather than verification, we can see that the complaint against Kant that progress in mathematics since his time makes geometrical diagrams and intuition generally irrelevant and unecessary misses the mark. Ultimately, to eliminate Kantian inmition is to eliminate constructive behavior or activity as the nature of what space is, which in

tum is not only to remove the topic of geometry, but is also to remove any behavior for thought to regulate. It is thus to remove any real representative worth of empirical thought at all. The progress in mathematics has to be regarded or reinterpreted as progress in providing rules for construction or operation, rather than replacing operations by sheer discursive concepts. So,

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for example, topology has to be regarded as constituting which individual constructions are legitimate, as opposed to geometry which concerns when pairs of such legitimate constructions ensue in coincidence. Having argued, albeit in a sketchy way, for the plausibility and indeed the necessity of a Kantian theory of the subject matter of geometry, we ought to now say something about the other more often discussed issue of the verification of geometry. Why did Kant think geometry could be verified a priori? Let us presume that by carrying ant a local construction we can verify that proximate local triangles are Euclidean. This is a difficult presumption to make because a standard of precision would be required that perhaps cannot be shown or produced in one's operations. However, I am more interested in the issue of why Kant takes the results of a local verification to be transferable to ot.'>er coincidence results. Why, that is, does he think we can anticipate a priori the results of other constructions without having to actually carry them out? Looked at this way, the issue is over homogeneity principles. Although Kant mostly talks of size homogeneity (so that little constructions verify also big results), I shall talk of positional or translational homogeneity. Why should a proximate local construction tell us anything about that same construction as carried out elsewhere? Note, if positional homogeneity obtained, we could verify a priori the complete local structure of Space everywhere and therefore also verify the result of large-scale constructions. I carry out a local construction according to a pair of rules and L':Ie upshot is coincidence. Why think that ca..rrying out that same construction. after an intermediate linear construction (with one's feet say) would also result in coincidence? I tlunk Kant's answer is simply to reiterate that it is the very same construction that is being carried out in both cases, guided completely by the same rules or directions. If Space were something objective, rather than merely our own activity, theu perhaps its features could change from place to place. If space were either a receptacle entity or a system of configurations of empitical items, then the features of this entity or system might very well vary, even the feature of coincidence. Put crudely, maybe the receptacle or else the configuration bends or twists or dips from locale to locale. The point is, as long as we are dealing with something objective, there is no way to verify a priori what its features shall be. On the other hand, if the subject matter of geometry is pure or our own productions then there is nothing objective in the nature of the producing to change the results. The only thing different about the constructions is an intermediate linear construction which separates them. The &1SWer to this consideration is that homogeneity does not follow from purity or constructivity. For example, the

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number series is not homogeneous with respect to the distribution of primes, though there is nothing between two segments of the series, other than an intervening counting or reciting. The answer to Kant is to say that the intervening line construction can ma..ke a difference to coincidence results, that the "unit' of construction may have to include the intervening construction, so that it is literally not the very same construction that is carried out twice. The purity, that is, of geometry seems compatible with positional inhomogeneity, allowing thus, for example, for geometries of variable curvature, and likewise precluding anticipative or a priori verification. Kant is still right, I claim, that verification in geometry pertains to constructions, even if he is wrong about its being a priori. Indeed if the subjectmatter of geometry is coincidence of constructions then of course the verification of geometrical statements 'pertains' to constructions, in the sense that what is being verified is a coincidence of constructions. However. not only isn't the verification of geometry a priori. it is not even pure. The trouble is the incompatibility of spatializing or spatia-temporizing construction. Pairs of constructions cannot be directly carried out together, whereas verification is always ultimately a local matter of having all the required information at once where and when one is. Thus, one needs physical markers or physical siguals to verify coincidence results, but once these are introduced one needs empirical hypotheses about how things move or how forces operate. The very sense of spatial statements includes incompatibility of actually carrying out pairs of constructions, and so the very sense of spatial statements forces a distinction between what is stated and how it can be verified. The purity of the subject-matter of geometry is not only compatible with the empirical nature of its verification, but once homogeneity is given up as an assured ptinciple, the subject matter demands this empirical nature of verification. Another way of stating this point is that Kant's spatial operationalism and the central insight behind it, that thought can do nothing more than guide behavior, demands a wedge between means of verification ana what is thereby verified. His view then that thought obtains real siguificance by legitimating or reguIalating activity is not derived from some verificationist principle, but is an autonomous alternative to verificationism.

University of Illinois.

'1 WILLIAM HARPER

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KANT ON SPACE, EMPIRICAL REALISM AND THE FOUNDATIONS OF GEOMETRY'

I.

SPACE: KANT'S ANSWER TO BERKELEY

1. Kant and Berkeley One of the first reviews (Garve-Feder, 1782) of the Critique of Pure Reason described Kant's system as a form of idealism of a piece with that of Berkeley. Kant (Letter to Garve, August 7, 1783) was not pleased with this comparison. In the Prolegomena (13) he explained that his system, far from agreeing with Berkeley, was the proper antidote to Berkeley's objectionable form of idealism. In an explicit response to the offending review (prolegomena Appendix) Kant claimed that when Berkeley made space a mere empirical representation he reduced all experience to sheer illusion. Kant continued to stress Berkeley's failure to do justice to the special role of space as SOUTce of a priori constraints on experience when he distinguished ·'his view from Berkeley's in the second edition of the Critique (B 69-72, B 274, Note on B xi of Preface). In spite of these protests, quite a number of subsequent writers have offered interpretations of transcendental idealism that would have Kant in basic agreement with Berkeley. Perhaps the most clearly stated example is to be found in Colin Turbayne's classic paper (1955), but any interpretation that construes the manifold in intuitions as sensations or appearances as subjective contents of experience will make Kant's position true to the spirit of Berkeley's point of view. I shall use Turbayne as an example; but, if the interpretation I propose is correct then the way Kant uses space to support his empirical realism makes his position quite different from Berkeley's or from any kind of phenomenalism or any empiricism based on subjective experiences. Turbayne claims that Kant's main argument against transcendental realism was anticipated by Berkeley's main argument against materialism. He breaks the argument down into six steps which I paraphrase roughly as follows: The transcendental realist supposes that external objects of perception have an existence by themselves independently of what we can perceive. (2) What we can be immediately aware of is only the contents of our own representations. (1)

257 Carl J. Posy (ed.), Kant's Philosophy of Mathematics. 257-291.

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(3) Therefore, it is impossible to understa"ld how we could arrive at knowledge of external

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objects and we are led to skepticism about therr existence. (4) We are alSO led to skeptical idealism - the doctrine that we can only know the contents of our own representations. (5) Skepticism about external objects can be avoided by giving up transcendental realism and adopting transcendental idealism - the doctrine that external objects are appearances and so are contents of representations. (6) This supports empirical realism - the doctPJle that we have immediate perception of external objects.

Something like this kind of argument against 't:Fal1scendental realism does seem to be an important part of Kant's Copernican revolution in philosophy.' Turbayne uses quotations from Berkeley and Kant to illustrate u'leir agreement at each step. The last two steps represent the official position which combines transcendental idealism with empirical realism.

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Space is one issue on which Kant and Berkeley clearly differ. Kant held that we have knowledge a priori about space while Berkeley held that all spatial concepts are merely empirical. According to Berkeley (Theory of Vision 153-154) even the three-dimensionality of space is something that must be inferred from experience by associating visual with tactual sensations. As we said above. Kant regarded Lhis rejection of a priori constraints on space as a fatal flaw in Berkeley's account of the difference between truth and error. According to Turbayne (p. 236), Berkeley based the distinction between truth and error on the coherence of our ideas with one another in experience and Kant is committed to the same kind of account. He suggests (pp. 243-244) that Kant's appeal to their differences over space was no more than an attempt to keep his readers from realizing that this basic position was essentially the same as that of u':!e infamous Berkeley.3

Fifth Step Kant: (Transcendental Idealism). External bodies are mere appearances, and are therefore nothing but a species of my ideas, the objects of which are something only through these ideas. Apart from them they are nothing (A 370, Cf. A 491, Prolegomena 13).

Berkeley: As to what is said of the absolute existence of lL."1.thinking things without any relation to t.qeir being perceived, that seems petfectly unintelligible. Their esse is percipi, nor is it possible they shouid have any existence, out of the minds or thinking things which perceive them. (Prin.3).

Sixth Step

Kant: (Empirical Realism). I leave things as we obtain them by the sense their reality (Proleg, 13). In order to arrive at the reality of outer objects, I have just as little need to resort to inference as I have in regard to the reality of the object of my inner sense ... For in both cases alike the objects are nothing but ideas, the immediate perception of which is at the same time a sufficient proof of their reality. (A 371) ... An empirical realist allows to matter, as appearance, a reality which does not permit of being inferred, but is immediately perceived. (A 37 N).

Berkeley: [am of the vulgar cast, simple enougIl to believe my senses and leave things as I find them (Hylas ill). I migh.t as well doubt of my own being, as of the being of those things I actually see and feel ... Those immediate objects of perception, which according to you, are only appearances of things, I take to be the real things themselves. ~ If by material substance is meant only sensible body, that which is seen and felt ... then I am more cert:aill. of matter's existence than you, or any other philosopher, pretends to be (Hylas III).

As these quotations show, Berkeley cerntinly did not describe his position as one which reduces all experience to iIlusion.2 He regarded his idealist account of bodies as the proper defense of common sense empirical realism against skepticism. This is exactly the virtue Kant claimed for his own transcendental idealism.

2. Sellars' objections to phenomenalism On Berkeley's version of the coherence account Macbeth's dagger is illusory because he cannot grasp it or cut with it - the sense data involved in his experience do not fit into the sort of coherent pattern with other sense data that constitutes seeing a real dagger. Unifonnities among our sense data let us coordinate sight with touch and make a host of specific correlations among our subjective experiences. \Vhen these uniformities break down we find that our judgments have been in error. When Macbeth sees the dagger apparition hover Ln the air, he has some grounds for judging that it is not a real daggerreal daggers don't appear to hover in the air without visible snpport. Upon attempting to grasp it he would have more evidence - real daggers resist when grasped. On this view such breakdowns among the unifonnities that constitute experience of real daggers are what make Macbeth's dagger count as illusory. In order to accurately reconstruct the common sense distinction between truth and error we must be able to account for cases where one person's experience is ground for judging correctly that another's judgment is in error, even if the other doesn't realize his errOr. When I am situated so as to see that you are standing in front of an empty facade, as you claim to be in front of a house, I can correctly judge that you are in error even if you don't think so. The uniformities that ground an adequate coherentist account must apply impersonally to the experiences of all of us. In addition to applying impersonally to different observers the unifonnities

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that account for the difference betwen truth and error must apply to possible as well as actual experieuces. Macbeth·s dagger is illusory even if neither he nor anyone alse ever actually tries to grasp it or cut with it. It is illusory because one would not be able to succeed if he were to make the attempt. Even if no one is situated to observe them a real house has other sides and

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insides. Were one to make the appropriate observations he would have the appropriate experiences. Any idealist who leaves everyday empirical things as they are must believe and have good reason to believe many counterfacmals of this sort. In his discussion of phenomenalism Wilfrid Sellars (1963. pp. 60-106) argues that such counterfacmals cannot be analysed into uniformities among acmal sense contents. He points out that in order to specify the appropriate antecedents for the counterfactuals in _question one needs to refer to external objects. One striking example is the need for antecedents such as looking from different perspectives. The counterfacmal arrangements of bodies in space that would re-position the observer with respect to h'1e object would, themselves, have to be formulated in terms of counterfacmal as well as acmal sense contents. Thus the very conditions that would be used to define these

possible sense contents would have to be based on other conditionals of the same sort. According to Sellars (1963, p. 80), a phenomenalist might reply by claiming that there are independent general laws about sense data that do not need to be formulated by reference to external bodies and which can be supported by induction based on actual sense data alone. Sellars' answer is that what the phenomenalist needs are generalizations which would apply impersonally, but the best that the phenomenalist can get are uniformities that are valid only for his own particular experience.4 For Kant the fundamental a priori constraint that space is three dimen-

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sional together with the a priori constraints on shape and perspective that can be established by geometrical constructions provide richer material for generating an empirical realism. I shall argue that the species of representation Kant uses to account for external bodies is the kind of objective perception exemplified by observations of those perceptible feamres presented by a three-dimensional object at a specific location and orientation with respect

to the observer. Kant's a priori constraints build in the assumption that such an object has another side even if only one side is being observed. They also require that the object has a determinate shape that is systematically related to an indefinitely large array of perspectives from which it could be observed.

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These spatial assumptions provide exactly what is needed to get around the objections we have considered. The antecedents of the required conterfacmal conditionals can be cashed out as specific alternative arrangements of the object and h':le observer's body in space. Grasping at the dagger is bringing one's hand into the appropriate location and orientation as one squeezes. Similarly, the antecedents relevant for observing other parts of the house are generated by specifications of locations and orientations for the body of an observer relative to the house. The various uniformities on shape and perspective that support the specific content of these counterfactuals are impersonal in just the way required. That

a quarter-shaped object will present a circular aspect to an observer who looks at it from a perspective orthogonal to and centered on its head's side and present an elliptical aspect to one who looks at it from an appropriately different angle is not something idiosyncratic to any particular observer. Such laws are part of what is to count as nonnal observation of shaped objects in

space.' To the extent that Berkeley's position is vulnerable to these objections to phenomenalism, while Kant's a priori constraints on space get around them, it is plausible to argue that Kant can be taken at his word when the claims that Berkeley reduced experience to illusion when he made space a merely empirical representation. 6

3. Refutation of idealism In addition to the objecrions we have considered Sellars (1963, pp. 83-84) also argues that a phenomenalist is committed to the external world of bodies in space and time when he refers to perceivers and their personal identities. This is one of Kant's own arguments. It is a major theme in the transcendental deducrion and the refutation of idealism. Several other writers, including notably Peter Strawson (1966), have argued that Kant is correct on this point because the path traced out by a person's body as he moves about over time through an enduring world of external bodies in space is all that provides for his ability to collect his various subjective episodes into an experience belonging to a single person. Margaret Wilson (1972, pp. 597-606) has suggested that the foregoing objecrion only shows that one must use external body concepts to describe the subjective contents of experience and does not show that judgments about external objects have to be known to be true. She argues that even if this undermines a position which attempts to reduce all experience to sensory

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contents alone it need not be decisive against Descartes' more modest skeptical position. According to Wilson (ibid., p. 603) Descartes' only essential contentions are (1) Our most confident ordinary employment of physical object concepts is in a significant sense compatible with the non-existence of physical objects, and (2) Judgments which purport only to describe our experience, without claiming the actual existence of entities other than ourselves, are not similarly challengeable.

She uses the following example of Descartes' demon hypothesis in action to support the claim that these contentions are plausible: Consider, the Cartesian may say, the case of a man approaching an oasis across the desert. First he perceives only the tops of the palm trees. After a while he perceives the trunks. Although his perceptions of the trunks occur after his perceptions of the leafy tops, he will naturally take both to be perceptions of one set of stable objects, not of temporally successive sets of objects. As he gets nearer, he sees a bird in one of the trees. He sees the bird stretch open its beak, then close it, then fly off. Then he hears a shrill note. While he perceives the bird-flight before the bird-cry, he takes it to have occurred afterwards. In other words, he implicitly makes all the usual distinctions between subjective and objective time order, in complete conformity with the examples of the Second Analogy. Now let us suppose (1) that the oasis was a mirage; or (2) that the man was not awake; Or (3) that he was in the clu~ches of a deceitful demon or super-scientist, who was in some manner providing him with a fantastic series of perceptual experiences. Certainly, the Cartesian will continue, there is a sense in which this man not deceived about the character of his own perceptual experiences. Yet he certainly was deceived in taking them directly to represent an outer reality. Now, how can we ever be sure that our 'outer experience' is not deceptive in precisely this manner, etc .. .. ?

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The science fiction version of the demon hypothesis is especially compelling today. How do I know that I'm not just a brain in a vat. Perhaps my present experiences, and indeed my whole life's experiences, are nothing but responses by my brain to artificial inputs provided by ingenious super scientists. This kind of hypothesis seems to obviously a coherent possibility in principle, even if it cannot be achieved yet by today's scientists, that it has revived the demon hypothesis as an epistemological puzzle of concern to philosophers. 7 The two contentions Wilson acribes to the Cartesian correspond to the first two steps in the paralogism argument that made Kant look like Berkeley. 8 They have the effect that our judgments about the snbjective contents of our experiences are immediate, but that the existence or non-existence of external objects is independent of our judgments about them. The demon hypothesis challenges our ability to arrive at knowledge of external objects (step 3) and invites the skeptical conclusion that all my knowledge is limited to the subjective contents of my own experience.

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On Berkeley's version of it, the idealist move in step 5 gives up the Cartesian contention (I), but keeps contention (2). According to Berkeley external bodies are accounted for by visual and tactual ideas that contain nothing beyond what is immediately given to the senses. On his view the esse of these ideas is percipi so that my judgments about what I perceive immediately are incortigible9 Kant's skeptical idealist (step 4 in the paralogism argument) conld say that he gives up (1) and that he interprets each perception claim about external bodies as asserting no more than the subjective content of that particular perception. He could then say, as Berkeley does, that in his view, our perceptions of external objects are immediate; but, such a view would certainly not provide for an empirical realism. 10 The Berkeley that Turbayne shows us would interpret my judgments about external objects as asserting appropriate uniformities among the subjective conients of my experience." On the assumption that (contrary to what I have argued above) these uniformities can be made available within Berkeley's framework, this more realistic kind of subjective idealism does defuse the skeptical argument. On this idealistic assumption the demon hypothesis is incoherent because the truth of my claim that external bodies exist comes down to the sarne thing as having my subjective experiences satisfy the appropriate uniformities. On the interpretation I shall propose. there is an important difference between the ways Kant and Berkeley give up contention (I). According to transcendental idealism external bodies are accounted for by appearances and appearances can be immediately perceived by us. The difference is that the appearance I perceive now is correctly construed as an object the existence of which is independent of my perception of it. On this interpretation, appearances are objective rather than merely subjective contents of perception and my judgnients about them are not incortigible. Even though appearances are empirically real so that they are independent of anyone's actual perception of them they are not transcendentally real because they are not independent of what conld be perceived by observers like us. Kant still has available an idealistic answer to the demon hypothesis. On the interpretation I shall defend I can assume that my judgment abont an appearance I perceive is false only by assuming that it fails to cohere with a host of other claims about outer appearances which I assume to be true. There is, on this view, no way to coherently assume that all my judgments abont outer objects are false. By giving this objective account of appearances Kant has broken the connection between immediate perception and incortigibility. He holds both that my perception of the appearance presented to my senses now is immediate

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experiences are construed as supporting nothing beyond incorrigible claims about subjective contents then they cannot support any such objective time order. On the assumption that all my outer experience is hallucination I have given up any grounds on which I could know that the temporal order my experiences seem to have is the one they actually do have. The demon hypothesis is incoherent because on it there is no way to prevent my experience from collapsing into a solipsism of the present moment. 14

and that it includes an objective judgment that can in principle be mistaken. On this view the fact that my present judgment is corrigible does not mean that it is in any way doubtful. My perceptions of the outer appearances presented to me are not mediated by any more direct perceptions of tIle subjective contents of my experience. Even though they are corrigible they are as

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immediate and certain as any perceptions I can have. Kant's transcendental idealism gives up contention (2), as well as contention (I). On his view my judgments about the subjective contents of my own experience are no more imIllediate than my judgments about the outer appearances I am presented with. According to the refutation of idealism (B275-279), my having knowledge of the determinate temporal sequence of my subjective experiences depends upon my having determinate knowledge about outer things. If this is correct, then the need, pointed out by Sellars and Strawson, to appeal to my body's path through an objective world in order to know how my subjective experiences fit together in time requires knowing the truth of some judgments about outer things.I2 Though Kant and the skeptical idealist agree in treating my judgments about external objects and my judgments about my subjective experience as equally immediate they do so in opposite ways. Where the skeptical idealist would treat claims about external objects as incorrigible claims about subjective experience, Kant would treat my judgments about my subjective experiences as no less corrigible than my judgments about external objects. Recently Hilary Putnam (1981, pp. 1-20) has argued that the demon hypothesis '1 am a brain in a vat' is self refuting because if it were true I would be unable to use the word 'vat' to refer to actual vats in the world. Putnam (p. 62) points out affinities between his views, Kant's position on sensations, and Wittgenstein's private language argument. Other writers including Sellars (1963, Chapters 3 and 5; 1968, Chapters I and II) and Jonathan Bennett (1966, pp. 202-209) have also given interesting arguments in support of Kant's position that bring out affinities with Wittgenstein. 13 1 think that Kant's argument can be profitably interpreted along the lines these writers suggest. For Kant, just as for Putnam, the demon hypothesis about my own case gives up what is needed in order for me to make some objective reference it

requires me to make. Unlike Putnam, Kant focuses on the objective reference required to have knowledge of the temporal order of my own experiences. If my past tense judgments are to connect together in an appropriate way then I must be able to use 'now' to demonstratively refer to a location in an objec-

tive time order that defines a past for these judgments to refer to. If my

II.

KANT'S EMPIRICAL REALISM

1. Appearance: The undetermined object of an empirical intuition

The following taxinomy can help to explicate the species of representation 15 that count as intuitions:

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The genus is representation [Vorstellung] in general 'repreasentatio). Under it stands the representation with consciousness (perceptio). A perception which relates only to the subject as a modification afits state is sensation [Empfindung] (sensario),. an objective perception is cognition [Erkenntnis] (cognitio). This is either intuition [Anschauung] or concept [Begriff] (intuitus vel conceptus). The fonner relates immediately to the object and is single, the latter relates to it mediately by means of a feature [Merkmal] which several things may have in common. The concept is either an empirical or a pure concept; and the pure concept, so far as it has its origin only in the understanding (not in the pure image of sensibility), is called a notion (Notio). A concept formed from notions and transcending the bounds of experience [Erfahrung1 is the idea [Idee] or concept of reason [Vemunftbegriff]. (A 320/B 377).

This passage distinguishes intuitions from sensations on the one hand and

concepts on the other. I shall deal with the distinction between intuitions and concepts before dealing with the distinction between intuitions and sensations. An intuition is single and relates to its object immediately while a concept

relates to its object mediately by means of a feature which several things can have in common. An intuition is single in that it is a singular representation -

one that can have only one particular object - while a concept can be satisfied by many distinct instances. In this respect, the distinction corresponds roughly to that between an individual referring expression and a predicate expression in symbolic logic. Hintikka ,has argued (1969) that this sort of logical distinction between particular ideas and general concepts captures the essence of Kant's use of inmition. Other writers (Parsons, 1964; Sellars, 1968; Howell, 1973) have argued that, on Kant's view, a demonstrative element is essential to any intuition. The emphasis, in this passage, on the immediacy with which an inmition is in relation to its object may support

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t.'1ese writers. Our problem is to get clear about the sort of empirical intuitions t.~at Kant uses to account for external bodies. For these intuitions at least, a demonstrative reference to a specified actual instance is essential. Consider the empirical intuition I have as I observe three coins arranged on the desk before me. Presumably, Lhe object of this intuition is a complex of several individual things. It may be, as Kant sometimes suggests, that any such inmition of a complex is a complex of simpler intuitions; but, even if this were so, a complex intuition would still be an intuition. My intuition can still be singular in Lhat it unambiguously designates this one instauce of the arrangement of coins. The importaut singularity of intuitions, at least of empirical intuitions of the sort that concern us here, is to refer demonstratively to a single instauce. It does not matter whether the specified instauce turns out to be simple or complex. In the example under consideration, the object of my intuition is whatever is actually present now at the location I specify when I refer to the coins before me on my desk. Kant distinguishes intuitions from sensations in the following manner: an intuition is a cognition or objective perception while a sensation only relates to the subject as a modification of its state. Since Kant would regard Berkeley's ideas as mere sensations, this distinction between intuitions and sensations is vital to the difference between his transcendental idealism and Berkeley's subjective idealism. Adequate treatment of this difference will require explicating the role of sensation in empirical intuitions. This explication will benefit from a consideration of additional passages in which Kant distinguishes between empirical and pure intuitions. Among these passages the following paragraph deserves to be quoted in full because this will help set the stage for the explication to follow: Our knowledge [Erkenntnis] springs from two fundamental sources of the mind; the first is the receiving of representations (the receptivity for impressions), the second is the power to know Ierkennen] an object through these representations (spontaneity for concepts); through the first an object is given to us, through the second it is thought in relation to that representation (which is a mere detennination of the mind). Intuition and concepts, therefore, constitute the elements of all O!lr knowledge [Erkenntnis], so that neither concepts without an intuition in some way corresponding to them, nor intuition without concepts, can yield knowledge [Erkenntnis]. Both may be either pure or empirical. They are empirical when they contain sensation (which presupposes the actual presence of the object), and when there is no admixture of sensation with the representation they are pure. Sensation may be called the material of sensible knowledge. Pure intuition, therefore, contains only the fonn under which something is intuited, and pure concept only the form of the thought of an object in general. Only pure intuitions or concepts are possible a priori, empirical ones must be a posteriori (A 50/B 74-A 51/B 75).

According to this passage, an intuition is empirical when it contains sensa-

EMPIRICAL REALISM & THE FOUNDATIONS OF GEOMETRY 267

tion; moreover, sensation presupposes the actual presence of the object, and it is through receptivity for impressions that objects are given to us. The passage also implies that all my information which is not contributed by my own mental apparatus must come from the input of sensations to that apparatus. That there is a tight correspondence between such characteristics as the shape and hardness of a physical object, e.g. a rubber ball, and the kind of experience I have when I look at and handle it, is one of the faruiliar facts of iife. When I look at the ball, what I see depends on the ball, the perceptual circumstances, and my psychoiogical circumstances; but, given the specification of these contingencies it is independent of my decision. 16 I can decide to look or not to look but, if I look, what I see is not all up to me. According to Kant, sensations are essential to this independence. In this system they link up my mental machinery with the world: The effect [Wirkung] of an object on the faculty of representation, so far as we are affected by it, is sensation. That intuition which is in relation to the object through sensation is called empirical. (A 19-20/B 34)

Notice that Kant does not say that sensation is perception of what it corresponds to. On the contrary, in all three passages quoted above, he carefully restricts sensation to a modification of the state of the subject only, to that representation which is a mere determination of the mind, and to the effect of an object on the facnlty of represenation so far as we are affected by it. The sort of perception which is in relation to an object, through sensation, is empirical intuition - not sensation itself. How does an empirical intuition contain sensation? Consider the following proposal: Just as an instauce of a sign desigu can function as a token for a sentence in so far as it is subjected to rules that govern correct usage for that sentence, so also may an array of sensations function as the token for an empirical intuition.17 The connection between any token and the represenation it is used to token is provided by the rules that govern correct tokening of the type of representation in question. An empirical intuition is just a sensation episode that is subjected to rules appropriate for tokening that specific type of intuition. According to this proposal, seeing that there are coins on my desk is distinguished from merely having sensations of a certain kind in that seeing is subject to rules that govern judgments about objects of experience while mere sensations are not. This makes the distinction between an empirical intuition and mere sensation analogous to the distinction between asserting that there

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are coins over there and merely mouthing the words. On the other hand. also according to this proposal, seeing that there are coins over there is distinguished from merely asserting the corresponding judgment in that seeing required being in perceptnal circumstances appropriate to produce the corresponding sensations while asserting does not require any such immediate relation to the object of the judgment. The last passage quoted, where sensations and intuitions were characterized' continues with the following sentence, which characterizes appearance as the undetennined object of an empirical intuition:

other things, that coins be rigid enough to resist when touched and that they have appropriate boundaries on the sides not being observed. The additional content provided by the empirical concept of a coin connects the appearance presented to me now wit"!:l other appearances that are not now presented to me but would be presented to an appropriately located observer. Were! to pick up and exaruine one of the coins I would be presented with an appearance that included tactual as well as visual inforarntion. All of the directly presented features, the shape, texture and resistance presented to my fingers as well as the shape presented to my sight are located in one space relative to the location and orientation of my body.19 As I construe II1em here, Kant's appearances are just those objective properties of actual things in space that follow geometrically from those perceptible features that would be presented directly to the senses of an appropriately situated human observer. In many passages Kant tells us that imagination is the process by which sensations are worked up into empirical intuitions. According to this picture we can think of an outer appearance as that set of sensible features which I intuit in an object simply through the taking up of sensations into the imagination according to the general rules for having an outer intuition at all. The -appearance is the content of a minimally conceptualized intuition. The object of such an intuition is characterized as this something - qua having the perceptible features generated by the imagination under the guidance of only my present sensations and the pure concept of a spatial object. Even when the ascription of content to the object of my outer intuition is limited in this way, the general spatial rules require some definite connections between the appearance actually presented to me and further appearances that would be presented to appropriately located observers. For example, (as we remarked above, Note 5), the shape presented to me is systematically related to what would appear to other perspectives. This comntitment to further appearances makes the ascription of even these most directly perceptible features both objective and corrigible. I take this to be Kant's main point when he insists that empirical intuitions are objective perceptions and not mere sensations.

The undetennind object of an empirical intuition is called appearance (A 20/B 34)

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In so far as it is an object of an outer intuition an appearance must be subject to the general rules that characterize the pure concept of an object in space. These rules are generated by the role of space as the pure form of all outer intuition and require that the object have a location with respect to the observer in three-dimensional space as well as satisfy all the constraints on shape and perspective that can be established by geometrical constructions. The appearances Kant uses to account for external bodies are objects of outer intuitions; therefore, whatever may be undetermined about them, they must at least satisfy these general rules. I propose that the object of an outer empirical intuition is undetermined in so far as it is subjected to none but these general rules for objects in space. Consider again the empirical intuition I have as I observe the coins on my desk. I leave the object undetermined when I limit my judgment about it to just those perceptible features actually presented to me now, together with whatever these features imply according to the general rules governing objects in space. The appearance is simply something - qua presenting the aspect of a specific triangular array of three dime-shaped objects viewed from my relative location and perspective. The basic idea here is that the features which generate the content of an appearance are just those perceptible features that are actnally exposed to the appropriate senses of the observer. 18 The shapes on occluded sides of the coins are not part of the content of this appearance. The very sarne appearance could have been presented by rods embedded in the desk with exposed ends shaped like tops and edges of dimes; it could also have been presented by an appropriately focused hologram. An hallucination would not count as observing the Sarne appearance even though I might tuistake one for such an observation. When I judge that what I see are coins, I subject the object of my intuition to rules that require more of it than just this appearance. I require, among

2. Transcendental idealism and empirical truth Kant's most developed exposition of the way his transcendental idealism supports an empirical realism is to be found in the long paragraph (A 190-191; B235-236) which opened his first edition version of the second analogy and was retained unchanged as the third paragraph in his second edition version. I

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shall attempt to show how my explication of Kant's basic conception of appearance as the nndetermined object of an empitical intuition illuminates what I take to be the two centtal ideas Kant inttoduces in this celebrated passage.20 One of these is a transcendental sense of 'appearance' according to which even such a complex solid object as a house counts as an appearance. The other is an account of empitical truth that is objective and yet avoids the demon argument which plagues the transcendental realist conception of truth as correspondence with things as they are in themselves. Kant tell us that a house is not a thing in itself, but an appearance. He explicates this by glossing 'appearance' as "a representation the transcendental object of which is unknown". He then asks what we are to nnderstand by the connection of the manifold in the appearance itself, when an appearance is neverthelsss not anything in itself. Now, as soon as I unfold the transcendental meaning of my concepts of an object, I realize that the house is not a thing in itself but only an appearance, that is, a representation, the trimscendental object of which is unknown; therefore, what am I to understand by the question: how the manifold may be connected in the appearance itself (which is yet nothing in itself)?

When I judge that what I see before me is a house, I ascribe more to the object of my experience than just those directly perceptible features that are now afforded to my senses. These additional ascriptions go far beyond what the general concept of an object in space requires in order that the shape from other perspectives cohere geometrically with what I observe. Accordingly the house before me is not as undetermined an object of empitical intuition as the perspective-bound appearances I have been explicating. Lewis Beck (1978, pp. 143, 146) is surely correct that Kant uses a thicker notion of appearance when he applies it to such complex objects as houses. 2l He calls this Kant's transcendental sense of 'appearance' and identifies the more perspective relative notion I have been explicating with what he takes to be Kant's contrasting empirical sense of 'appearance'.22 Kant's rhetorical question, at the end of this passage, can be understood as asking how the manifold of a transcendental appearance can be independent even though it contains nothing beyong contents of representations. Kant's somewhat enigmatic answer is given in the next passage, which immediately follows his question in the text. That which lies in the successive apprehension is here viewed as representation, while the appearance which is given to me, notwithstanding that it is nothing but the swn of these representations, is viewed as their object; and my concept, which I derive from the representations of apprehension,. has to agree with it.

EMPIRICAL REALISM & THE FOUNDATIONS OF GEOMETRY 271

That which lies in the successive apprehension is presumably a manifold of empitical appearances. These appearances have a two-sided character. On the one hand each is the content of an empirical intnition and therefore can be viewed as a representation. On the other hand, as actual features of objects in space, they are connected with one another in a manner that is independent of anyone's apprehension of them. The empirical intnitions in my apprehension make demonstrative reference to a spatio-temporal vicinity. The independence of the object of my experience is provided by all the additional empitical appearances to be found in that viciuity. In this way a transcendental appearance tlIat contains nothing beyond contents of representations can, nevertheless, be viewed as the independent object that my concept has to agree with. My concept is derived from the representations of apprehension, in that the perceptible features actnally presented to me lead me to judge that what is before me is a house, rather than (say) a ship, tree, or an empty stage prop. If my judgment is correct, this concept has to agree with whatever turns out to be tl,e actual object of my experience. This demonstrative reference, rigidly denoting whatever is at a spatio-tem. pora! v.icinity, is the most important contribution of Kant' a priori requirement that the object of an outer empirical intuition have a determine location relative to the body of the observer in three-dimensional space. The specific geometrical constraints on the relation of three-dimensional shape to perspective also play an important role. They provide a framework which allows the identification of the house as an independent empitical object which underlies all the appearances in its manifold. This object is whatever affords the mereological sum of all the three-dimensional shaped surfaces revealed in these various empitical appearances. It is the empirical substance of which the various perceptible feamres revealed in these appearances are determinations. ..On this account when I look at a quarter from a perspective 45° from perpendicular to its head side, I directly see its non-occluded snrface as sometiling shaped like an appropriate part of a three-dimensional disk located and oriented in the way specified. There is none of that difficult business of seeing it as elliptical but judging it to be ronnd which plagued G. E. Moore. I think that allowing for direct perception of oriented shaped surfaces in three-dimensional space is fundamental to any Kantian account of how observers from different perspectives can see over-lapping parts of the sarne empitical substance. The following acconnt of empitical truth completes Kant's explication of how to construe appearance as the fonnal-being referred to by the representations in my apprehension:

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One soon sees that though truth is agreement of the cognition [Erkenntnis] wiL1. the object, only the fannal conditions of empirical truth can be in question here, and appearance in contrast with the representations of apprehension can be represented as an object distinct from them only if it [the appearance] stands under a rule, which distinguishes it [the apprehension of this appearance] from every other apprehension and makes necessary some one particular kind if connection of

Ill. PHAENOMENAo APPEARANCES THOUGHT ACCORDING TO THE UNITY OF THE CATEGORIES

the manifold. That in the appearance which contains the condition of this necessary rule of apprehension is the object (A 191/B 236).23

So far as we have explicated it, Kant's empirical realism supports the common sense realm of candlesticks, ships and houses against skeptical reductions to subjective contents of experience; but, our explication has been lintited to observables in a sense close to that advocated by va." Fraassen (1980) in his anti-realist constructive empiricism. It would be disappointing for some of those who see Kant as providing a foundation for scientific methodology to find that his empirical realism does not support existence claims about the non-observables posrulated by modem science. In a passage at (A 249) Kant tells IlS that:

Consider my observation of my own house as I stand before it. Let A be the proposition that what is before me now is that particular house. If it is empirically true that A obtains here-now, then the object of my experience must agree with my cognition - i. e., with my judgment that A is the case here now - and must also be representable as something independent of the representations in my apprehension of it. Therefore, whatever is before me over there must satisfy the condition of a rule that distinguishes it from any possible object of experience that fails to be an instance of A. This rule distinguishes my apprehension, qua an apprehension of an instance of A, from any ot.;'er

apprehension - i. e., from any apprehension of anything that fails to lle an instance of A. In this example the representations of my apprehension are my perceptions of the perceptible features actually presented to me. These would include the shape relative position and orientation of the facing surfaces, etc. The object

of my experience is whatever is present at the appropriate, spatio-temporal vicinity of the location to which I now refer demonstratively. There are many

!

more perceptible features there to be observed than the ones now presented to me. It is this demonstrative reference to an inexhaustibly rich source of addi-

tional perceptible features that gives the object of my experience its independence from the representations in my apprehension of it.

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I believe that this account of empirical truth is the heart of Ka."t's Copernican revolution in epistemology. In place of the transcendent notion of truth as correspondence with the way things really are in themselves, he gives us empirical truth as correspondence with what can count for us as the actual objects of our experience. This transcendental idealism avoids the Cartesian argument for skepticism at least as well as Berkeley's subjective idealism. The demon hypothesis cannot be empirically true because it assumes away the demonstrative reference to an independent object of experience required to provide the empirical content that could make it true. The

advantage over Berkeley is that it provides for the independence and objectivity required by our common sense empirical distinctions between truth and error.

I. The principle of extensive magnitudes

Appearances. so fur as they are t.1tought according to the unity of the categories. are called

Phaenomeoa

The various categorical principles Kant argues for impose additional constraints on the basic idea of apprearance as the undetermined object of an empirical intuition. I think that these constraints transform the account of empirical truth by adding comntitments that go beyond observables. I shall illustrate this point by considering some consequences of the axioms of intuition.

According to the Axioms of Intuition all appearances are extensive magnitudes. When Kant tells us that these magnitudes are determinate he is requiring that, for example, at any instant in time ratios of lengths along any specified dimensions of an object in space determine specific real numbers. It

is important to note t.lJ.at Kant takes this commitment to determinate extensive magnitudes as a constitutive condition on appearances. When explicating outer-appearance as the undetermined object of an empirical intuition I cbumed that the content of such apprearances is lintited to features to objects in space that are directly accessible to the senses of observers like us together with whatever these features imply according to the rules constituting the general concept of an object in space. We have seen that the qualitative geometrical constraints on perspective require that judgments about shapes directly presented to an observer at one perspective carry systematic comntitments to further aspects that would be presented to observers at other appropriately oriented perspectives. Now we see that the general concept of an object in space also carries commitment to determinate extensive magnitudes. Even when I limit my judgment about the object of my empirical intuition to

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the outer appearance presented to me ! must include in the content of my

judgment that, at any given instance, e. g. now, and relative to any appropriate specification of a standard, the length along each specifiable dUnension of the object has a detenninate value. Consider a spatial example where lengths are being compared. Let A 1••• An be some finite sequence of propositions such that each Ai asserts that the distance (relative to a specific meter stick) between the centers of mass of two quarters on my desk now falls in the i th of n adjoining tiny length segments. We are to make these segments small enough so that each Ai is below the threshold of human sensory detection, but large enough so that the disjunction of all the A;s is something we can observe to be true. This holds if we limit our observations to whatever comparisons of length we can establish with unaided sight and touch as we lay the meIer stick across the coins, and il continues to hold even if we allow what we observe to be enhanced by the best measuring instruments science can provide. The principle of extensive

magnitude makes commitments that go beyond the resolving powers of human observation even if these powers are extended by instruments. An appearance includes the specification that each spatial dUnension in it be a detenninate extensive magnitude. But, even given the specification of an appropriate standard and time, does it also include specification of what the exact value of each of these magnitudes is? On the account I have been proposing the answer to this question is no, because these exact values are

not implied by observable features even under the most lavish construal and application of mathematical rules constituting the general concept of an object in space. This has the effect that the disjunction A 1 V ..• v An will be empirically true even though none of its disjuncts is. Similar examples will show that an existential statement can be empirically true even when each of

its instances is empirically indeterminate. Indeed, it will be empirically true that each magnitude has some determinate value, even though for each magnitude it will be empirically indetenninate exactly what this value is. Another option would be to include the specification of the exact value of each magnitude as part of the empirical content itself. This would remove any empirical truth value gaps generated by the commitment to determinate extensive magnitudes, but it would lead to a problem pointed out by Charles Parsons (1964). On this option Kant is faced with a dilemma. Either he would have to claim that humans have the ability to, in principle, make infinitely fine discriminations of extensive magnitudes! or he would have to claim that what counts as empirical content is not determined by the discriminations that humans, even in principle, could make. Neither hom of this dilemma is very

EMPIRICAL REALISM & THE FOUNDATIONS OF GEOMETRY 275

attractive. The first hom would seem to commit him to someLhing patently false. Certainly our best instruments now fall far short of the precision required, and, though we can expect improvements that allow c1oserapproximations. nothing in the way such improvements have been made in the past

makes it plausible to suggest that such approximations could even in principle culminate in exact values. The second horn of the dilemma, on the other hand, would seem to fiy in the face of the whole idea of Kant's Copemican revolution in epistemology. If empirical content is inaccessible to OUf senses, even augmented by the best instruments we could in principle, devise, then

how could we know empirical truth from error? On the option I propose this dilemma is avoided in a way that seems in keeping with Kant's Copernican revolution and with his specific account of the a priori as something we impose on nature. The principle of extensive magnitudes shows that the basic account of empirical truth carries commitment to a more detenninistic ideal in which the value of each magnitude is exactly seuled. Any coherent way of filling out the specifications required by this ideal that is left open by what is settled by the empirical content will act as an admissible valuation in a supervaluation semantics appropriate to the account of empirical truth. 24 Whatever holds according to every way of filling out the ideal by arbitrarily assigning these values in some coherent way will count as empirically true. Therefore, the principle of extensive magnitudes contributes considerable strength to the account of empirical truth, even if we allow that the exact values of these magnitudes are empirically indeterminate. This way that empirical truth carries commitment to a more determinate mathematical ideal turns it from a fairly restrictive observationalism to a possible foundation for scientific realism, without violating the spirit of Kant's Copernican revolution in Epistemology.

2. The principle of the first analogy Though I shall not argue the point here, I think that Kant's arguments for the principles of the analogies can best be understood as an attempt to show that extension of the empirical content that can be appealed to in the account of empirical truth beyond what is presented to the observer here-now to additional observables at other times and places reqnires commitment to enduring substances and causal laws. If, as seems to be the case, the ouly candidate for the enduring substances that underly many observable changes are nonobservables, then the first analogy, as well as the axioms of intuition, will carry commitment to the empirical reality of some non-observable entities.

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Kant tells us (B 233) that the principle of the First Analogy can be expressed by the requirement thai all change is alteration - a succession of opposite determinations of a substance which abides. This principle requires that any change which we might be inclined to describe as the destruction of an empirical thing must be a succession of opposite states of some underlying subslance which persists through the change. Consider one of those familiar white styrofoam coffee cups. Now destroy it by smashing it to pieces. If both the before and the after states have to be determinations of the same substance then that substance cannol be (he cup which was destroyed. What is available to both persist through and be releVal1! enough to ground this change? One obvious candidate is the mereologicai SU01 of all the little styrofoam particles. If the after state is these particles all jumbled about in disarray while the before slate was these same particles assembled togeLlter into the cup, the change can be a proper alteration. Now burn the little pieces of styrofoan•. What is available to count as a substance which underlies this change? Presumably, some postulated collection of non-observable entities - something like molecules or atoms. 25 Thus, the principle of the First Analogy, together with such familiar happenings as destructions by burning which break something up into parts smaller than we can observe, seems to carry commitment to just the sort of non-observable theoretical entities dear to the hefu-t of a scientific realist.

3. Indeterminacies We have seen how the principle of extensive magnitudes makes commitments

that generate empirical truth value gaps. The same would hold for commitments generated by the First Analogy. Even if some version of the kineticmolecular theory of gasses turned out to be empirically true of some specified volume of gas there would not be any specific assignment -of positions and momenta to the individual particles (at any given time) that would be singled out as the unique empirically true one. Only the existential proposition that there was some such distribution of momenta would be empirically true. There would be a very large range of possible assignments of these maguitudes that would be equally compatible with the apprearances - roughly all ones which afford average kinetic energy values within the observable tolerances. This possibility of truth value gaps is a feature wbich my account of empirical truth shares with Carl Posy's (1983, 1984) inmitionistic rendering of Kant's transcendental idealism. I think any account of Kant's position which

EMPIR!CAL REALISM & THE FOUNDATIONS OF GEOMETRY 277

takes seriously his relativization of empirical trut.h to possible objects of experience will have to allow for such indetenninacies. Posy's intuitionisti-

cally motivated aCCOU!1t is one way to do this. My account of empirical truth with its supervaluation way of dealing with indetenninacies is another. Posy

(1983) has done an admirable job showing how his proposal can illuminate Kant's difficult discussion in the First Antinomy. I think the supervaluation approach can offer a comparably illuminating analysis of this difficult passage, but the details will have to wait for another occasion. IV.

GEOMETRY

1. Kant's commitment to a priori constraints on space

We have noted (Section I) that, according to Kant, Berkeley reduced all experience to sheer illusion when he made space a merely empirical representa-

tion. This suggests that Kant's commitment to the claim h'tat geometry provides knowledge a priori of constraints on objects of outer sense is deeper than his desire 10 provide a philosophy of mathematics. It suggests that he thought these a priori spatial constraints are what prevent his apprearances from collapsing into. merely subjective contents of experience - that they.are what separate his empirical realism from the objectionable form of idealism he attributed to Berkeley. My account of Kant's empirical realism appeals to the constraint that any object of an outer intuition must bave a determinate location and orientation relative to the body of the observer, and to the numerous specific constraints

on shape and perspective that can be revealed in geometrical constructions. These spatial constraints provide a framework within which appearances can be construed as objective features of things our senses can carry immediate

information about. If this account is correct then Kant's defence of empirical realism is based on these constraints on space. So, Kant may have been justified if he thought that his defence of empirical realism would be threatened unless he could appeal to such knowledge a priori about space.26 The sort of constraints on shape and perspective that my account of Kant's empirical realism needs can be illustrated by Shimon Ullman's (1979) various structure from motion theorems. Ullman's basic theorem concerns

orthographic projections. 27 Given three distinct orthographic projections of four non-coplaner points in a rigid configuration the structure and motion compatible with these views is uniquely detennined (up to a reflection about the image plane).

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Ullman (pp. 134-6) discusses a neat experiment in which visual information processing suppor.able by this theorem can be observed. Moving points are projected onto a screen with motions compatible with ortbographic projections of points on the surfaces of two rotating transparent coaxial cylinders. As you look you cannot help but see them as points on the rotating rigid three dirnensional cylinders. This suggests that we visualize as though we operated with a wired-in program which first looks for some possible rigid body in relative motion interpretation of the sensory input. We can see UIlman's

theorem as providing constraints on what can count as a rigid body in relative motion interpretation. mlman's shape from motion theorem tells us that a rigid configuration that afforded this ortbographic projection

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EMPIRICAL REALISM & THE FOUNDATIONS OF GEOMETRY 279

straints that apply to anything that can be an outer object of experience for us. On this view geometrical constructions provide us with a way of making the pure form of our outer intuition transparent to ourselves. 2. Kant's account of geometrical constructions Perhaps the most salient example of geometrical construction in Kant's writing (A 716-17; B 744--5) is the one used in Euclid's proof of Proposition 32 (in the Elements) - that the sum of the interior angles of a plane triangle equals a straight angle (180°). Euclid's proof of this proposition appeals to proposition 29 about various equal angles made when a straight line falls on two parallel lines. The following diagram is a construction which shows that all the marked angles must be equal.

to one perspective, and this ortbographic projection

to a perspective corresponding to a 45° rotation to the right could not afford this ortbographic projection

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I think that, according to Kant, anyone who properly understands this diagram cannot help but be compelled to see that all the marked angles must be equal - and that this would hold for any straight line falling on two parallei lines. The heart of Euclid's proof of proposition 32 is the following construction.

D

to a perspective corresponding to the opposite 45° rotation. I shall call this my salient illustration. According to Kant such constraints on shape and perspective are built into the structure of space which is the pure form of outer intuition. He holds that geometrical constructions offer us a priori knowledge of these structural con-

B

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From proposition 29, or by inspection of the present diagram, we see that angle ABC equals angle DCE and angle BAC equals angle ACD. From the way we constructed auxiliary lines CE and CD we can now see immediately that ACB plus ACD plus DCE equals BCE, since these three angles together just are the straight angle BCE. I think this proof shows the iotuitive force that geometrical constructions provide. It is very hard to reason through this diagram without feeliog compelled to accept Euclid's general proposition that the sum of the ioterior angles of a plane triangle equals a straight angle (180'). Kant attempts to justify this compUlsion by his account of geometrical constructions. The following passage gives Kant's explanation of the role of the triangle diagram as a constructive definition of the concept of a planetriangle.

schema of the triangle can exist nowhere but in thought. It is a rule of synthesis of the imagination, in respect to pure figures in space_ (A 141/B 180)

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To construct a concept means to exhibit a priori the intuition which corresponds to the concept_ For the construction of a concept we therefore need a non-empirical intuition. The latter must, as intuition, be a single ebject, and yet none the less, as the construction of a concept (a universal representation), it must in its representation express universal validity for all possible intuitions which fall under the same concept_ Thus I construct a triangle by representing the object which corresponds to this concept either by imagination alone, in pure intuition, or in accordance therewith also on paper, in empirical intuition - in both cases completely a priori, without having porrowed the pattern from any experience_ The single figure which we draw is empirical, and yet it serves to express the concept, without impairing its universality.

The single empirical figure I draw functions as the pure intuition which underwrites a real definition of the geometrical concept of a plane triangle. As a real definition it displays sure marks by which to identify any figure that is to count as a plane triangle and it also provides an actual instance which shows that this concept is not empty. (parsons, '1969; Beck, 1956)28 Kant goes on to tell us more about how it is that this siogle empirical figure can serve to express a pure geometrical concept without impairing the generality of that concept.

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For in this empirical intuition we consider only the act whereby we construct the concept, and abstract from the many determinations (for instance, the magnitude of the sides and of the angles), which are quite indifferent, as not altering the concept 'triangle' _(A 714/B 742)

These remarks can be usefully amplified by the followiog passage from the schematism. No image could ever by adequate to the concept of a triangle in generaL It would never attain that universality of the concept which renders it valid of all triangles, whether right-angled, obtuse-angled, or acute-angled; it would always be limited to a part only of this sphere_ The

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The diagram can express the pure iotuition of the geometrical concept of a plane triangle in so far as my reasoning about it appeals only to the schema of this concept. The schema is a rule for the synthesis of imagination reqnired to construct any ostensive representation of a plane triangle. The following passage contrasts what Kant calls the ostensive character of this sort of geometrical construction with the symbolic constructions to be found in algebra. thus in algebra by means of symbolic construction. just as in geometry by means of an ostensive construction (the geometrical construction of the objects themselves), we succeed in arriving at results which discursive knowledge could never have reached by means of mere concepts. (A 717/B 745)

It also illustrates Kant's commitment to the claim that Euclid's proposition 32 does not follow analytically from the mere concept of a plane triangle. The result depends essentially on the additional content provided by the construction. The schema for this pure concept is embedded in and shows us constraints on our framework for ostensively recogniziog figures in space. Thus, this pure intuition reveals a general constraint on space as the pure form of outer sense. For Kant, as I understand him, what I see immediately when I recognize a plane figure as a triangle is gnided by the very sarne rules I would use to construct an image of a triangle 10 imagination or to draw my own diagram of a triangle on paper. These are also the sarne rules I would follow to trace out a plane triangle with my finger or with the path of my whole body as I walked out a triangular pattern on a football field. The plane on which I conctruct or recognize the triangle must be oriented relative to my body in three-dimensional space. Even 10 the imagination, I think, Kant would claim, the plane on whiCh a plane geometry construction is carried out is imagined as oriented in a three-dimensional space relative to a point of view. 29 I think for Kant these general rules for recognizing or constructing any plane triangle support the auxiliary construction of lioes CD and CE in the same plane. They also underwrite the intuitive reasoniog whereby I am compelled to recognize that the sum of the ioterior angles equals the straight angle BCE. Sioce this construction and intuitive reasoning is supportable from the schema for the general concept of a plane triangle, the conclusion I reach for the figure under consideration must hold for any fignre I could ostensively recognize to be a plane triangle.

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3. What about the non-Euclidean geometry of modern physics?

I used Euclid's constructive proof that the sum of the interior angles of a plane Triangle equals 180° to explicate Kant's account of geomeTrical constructions as an endorsement for the intuitive compulsion a proof like this provides. The fact that Kant cites this as a paradigm example of geomeTrical construction gives some snpport to those who claim that he was comntitted to the a priori application of Euclidean geometry to the space in which we can apprehend outer objects of experience. It is now well known that the sum of the interior angles of a plane Triangle is a key mark discriminating between Euclid's geometry and the various non-Euclidean geomeTries of constant curvature. GeomeTries of positive curvature make the sum of the interior angles greater than 180°. This is clearly illustrated in Poincare's model of plane Riemannian geometry in which the plane is identified with the outer surface of a Euclidean sphere. GeomeTries of negative curvature all make the sum less than 180°. These include the classic hyperbolic geomeTries of Lobachevsky and Bolyai. In a geometry with variable curvature the sum of the interior angles of a plane Triangle marks the local curvature of the plane on which the Triangle is constructed. If the sum is 180 0 then the space is locally Euclidean. Kant has some good company if he was committed to Euclidean geometry _ even among mathematicians who were aware of non-Euclidean geometries. I think it was an appreciation of just the sort of intuitive compulsion Kant's theory of constructions attempts to explicate that led Frege (1959) to claim that only Euclidean geometry fits our intuition and led Poincare (1898) to suggest that Euclidean geometry ought to be retained even at considerable cost in additional complexity to physical theory. Nevertheless, I think that today most of us, children of the relativistic age as we are, would regard it as hopelessly Quixotic to continue to claim that Euclidean geometry is the correct geometry of the physical objects we meet in space-time. The weight of evidence is too solidly lined up behind modern physical theory. Does this not show, therefore, that the very foundation of Kant's empirical realism has been overturned by modern physics? Strawson's (1966) attempt to save something of Kant's account of spaceby making it apply to a merely visual geometry - will not do. This attempt and others like it (e.g. Walker's, 1978) which remove the clash with physical theory by giving up comntitment to objective constraints on physical things, will not preserve the fundamental role of space as a framework within which appearances can generate an empirical realism. 3o I think Melnick (this

EMPIRICAL REALISM & THE FOUNDATIONS OF GEOMETRY 283

volume) is absolutely correct in his identification of the subject matter of Kant's account of geometry with the structure of our framework for meeting things outside us. Kant's space is the physical space we move our bodies around in. His straight lines correspond to rigid rods that can be rotated inside their boundaries and to paths of light rays. I think one can keep this physical interpretation of the subject matter of geometry and keep enough of Kant's account of geomeTrical verification to give the constraints his empirical realism needs without flying in the face of modern pbysics. The key idea has been put forward by James Hopkins (1973) in an interesting paper attacking Strawson. It is this: What we can establish by geometrical constructions is limited by our perceptual capacities.

Hopkins (p. 24, 25) points out that we could not take in any diagram that accurately represented the relative sizes and distances between two stars. Either the dots representing the stars would be too small to see 01: the distances would be so great we could not survey the diagram. I think what Hopkins is pointing out is correct and important. The limitations on what we can use diagrams to represent are quite significant. Even two parallel lines one centimeter apart - each say O. 5 mm thick and 150 meters long could not be taken in by us. If we got far enough back to take in the end points· we would be so far back that we would not be able to resolve the separate lines. This shows that Euclid's parallel's postulate could not be established by any geomeTrical construction we could carry out. If Kant had claimed to be able to establish this postulate by constructions he would have violated his own basic injunction about extending concepts ouly valid for objects of experience beyond the limits of what we can experience. Any specification of what happens as parallel lines are extended indefinitely would correspond to an ideal of pure reason not to a principle constitutive of possible objects of experience. If we use the supervaluation method of representing commitment to the possibility of some such idealization then Kant's account of geomeTrical construction would comntit him only to the envelope corresponding to a whole family of geomeTries each of which captured local constraints on threedimensional shape and perspective up to tolerances provided by our perceptual capacities.

When you or I carry out our construction for the sum of the interior angles of a plane Triangle the inmitive compulsion our result carries is not misleading, so long as we recognize that what we establish only holds up to tolerances provided by our perceptual capacities. Similarly, we really can constructively establish Ullman's various shape from motion theorems up to

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such tolerances. Even though the constraints we get this way are vague and have vague limits on their vagueness. they have considerable bite. Lots of hypotheses are clearly beyond the tolerances allowed. For example, there could not be any rigid object, about as wide as my head, that would generate, at about arms length, the three orthographic projections specified in my salient illustration (Section IV, I) of what Ullman's theorem rules out. Such constraints capture a good deal of what Kant wanted from his account of geometry as a source of a priori knowledge. They depend on pervasive and accessible features of the actual capacities our sensory systems have and of the environment in which they have evolved to operate. These are very broad and deep features of what Wittgenstein called our form of life. They are not something we could change by adopting new social conventions. Nor could we find out tomorrow, on the basis of some new theory, that we have been wrong about these things all along. This does not imply that these constraints could never possibly change, only that to change mem we would have to undergo rather gross changes in our bodies or their local environments. If the behaviour of light rays and measurably rigid rods were to change so dramatically that they become unreliable as indicators of shortest paths between macroscopic local locations, then perhaps the constraints would change. I believe Kant would say that such a change would be impos. sible.31 Rather than just follow him in this. I want to point out the difference between having such physical changes actually begin to happen (which would be rather noticeable) and changing our theories about what has been happening all along. This difference points out an important sense in which geometrical constructions give us theory independent constraints on observation. It suggests to me that a Kantian alternative to some of the excesses of the last twenty years is to rush away from the idea of an observation theory distinction.

NOTES

I •

* The earliest ancestor of this paper was a talk I gave at the Canadian Society for the History and Philosophy of Science in 1974. In 1978 I commented on Colin Turbayne at the Rochester Conference honouring Lewis Beck's retirement. This led me to develop the argument in Section I. The first written draft was in May 1982 and its first public presentation was at a conference at the University of Western Ontario in Spring of 1982. A version was delivered as a lecture in my graduate seminar as Visiting Professor at Princeton in spring of 1983. Versions were also presented at a Duke Conference and at a Columbia University Philosophy Department colloquium. I am grateful for the insightful questions and comments received from many of the people who heard one or another of these presentations.

EMPIRICAL REALISM & THE FOUNDATIONS OF GEOMETRY 285 Special thanks are due to Robert Butts, Dan Garber, Ralf Meerbote, Calvin Nonnore. George Pappas, Margaret Wilson and Bas van Fraassen. Section n benefited from acute stylistic criticism generously provided by Paul Kirchner. I Most (but not all) of the passages Turbayne uses to support Kant's commitment to this argument come from the fourth Paralogism (A 367-380) which was dropped from the second edition of the Critique. The new Refutation of Idealism (B 275-279) in the second edition uses a different argument (see Section 3). One can also, perhaps, quibble over some of the steps and the way they are arranged to bring out the affinity to Berkeley. Nevertheless, I think it is fair to say that Kant remained committed to something like this argument. RaJf Meerbote (correspondence with me) disputes this. I think he is correct unless special care is taken with step 2. See Note 8 below for a suggested interpretation of step 2 under which the argument is compatible with the refutation of idealism. It is under this interpretation (which differs from Turbayne's Berkleyean construal of (2) that I hold it plausible be assume that some such argument is important to Kant's Copernican revolution in philosophy. 2 Several writers, e. g. N. K. Smith (Commentary, p. 156) have taken Kant's accusations of illusiONism as evidence that he misunderstood Berkeley's position. What George Miller (1973, pp. 316-322) has called the traditional view of the relation between Kant and Berkeley would explain these apparent misunderstandings on the hypothesis that Kant only knew Berkeley's work through distorted second hand sources. Turbayne (pp. 225-227), Miller (op. cit.) and Henry Allison (1973. pp. 43-45) have made it plausible to assume that Kant had far more access to Berkeley's work than the traditional view would allow. In particular they point out that a Gennan translation of Berkeley's dialogues was readily accessible to Kant. The hypothesis that Kant actually read the dialogues allows one to entertain the view that Kant's reference to the 'good Berkeley' in his B 70 passage we cannot blame the good Berkeley for degrading bodies to mere illusion: which Turbayne finds evidence of animus and Allison of condescension is really only irony obtained by applying to Berkeley the very same rhetorical device he applies to Hylas (the defender of common sense realism) in the dialogues-

Phil: "Have patience, good Hylas, and tell me once more whether there is anything immediately perceived by the senses expect sensible qualities. I know you asserted there was not; but I would now be infonned whether you still persist in the same opinion. " 3 Turbayne, Margaret Wilson (1971), Goerge Miller (1973) and Henry Allison (1973) all point out that in Kant's day Berkeley's position was regarded very unsympathetically. 4 George Pappas brought to my attention James Cornman's (1973) defence of the idea there can be laws COlUlecting sense data. I believe this defence will not work if sense data are to be construed as incorrigible subjective contents of experience (see Note 14). 5 It is not swpising that Sellars, the author of the objections to phenomenalism I have been considering, should take such rules governing what he called point-of-viewish aspect of perception as the key to an interpretation of Kant's transcendental idealism. In 'Kant's Transcendental Idealism' 1975 and in 'The Role of Imagination in Kant's Theory of Experience' 1978, Sellars proposes what I take to be just the sort of account I shall defend here. Indeed, this paper can be well constnied as an attempt to make some of the details of this kind of account more explicit and to document more extensively its textual support in Kant's writing. 6 Margaret Wilson (1971), George Miller (1973) and Henry Allison (1973) all argue impressively against Turbayne that it was reasonable for Kant to draw the conclusion that Berkeley's

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qeatment of space renders his position unable to support an empirical realism. I agree with most of what these writers have to say and offer these additional arguments in support of the view that Kant's conclusion about Berkeley's position is true as well as having been reasonable for him to

draw. 7 John Pollock (1974) and Hilary Putnam (1981) are but two of many recent examples. 8 It is well known that Kant explicitly addressed this paralogism against Descartes' position (A 368), so it is not too surprising that the assumptions of the demon hypothesis should correspond to the assumptions leading to the skeptical nadir of the transition from transcendental realism to transcendental idealism. Some care must be taken with step 2. For one thing, it will tum out that for Kant, unlike Berkeley, immediate awareness need not mean unchallengeable. For another, Kant will distinguish between subjective and objective contents of representations. A Berkelean construal of step 2 on which the content in question is subjective and inunediacy implies not open to challenge is what corresponds to Wilson's contention 2. On this construal, I shall argue, Kant's transcendental idealism gives up contention (2) as well as contention (1). If step two is interpreted so as to include objective content and immediate awareness so as to allow for corrigibility then Kant does not give up step 2. Under this interpretation (in which step (2) is not the same as Wilson's contention (2» Kant's Fourth Paralogism argument, in which step (2) is retained, is quite compatible with his refutation of idealism, which I shall argue rejects Wilson's contention (2). 9 Berkeley's subjective idealism is a salient example in a long tradition of phenomenalistic empiricism that is characterized by the attempt to ground all acceptable knowledge claims is sensations or some kind of incorrigible data base in experience. This tradition, which includes Hwne, Mill, Russell at some stages, the Camap of the Aujbau, and C. I. Lewis, is still alive today in the work of R. M. Chisholm and John Pollock. All of these writers including the most sophisticated agree with Berkeley in holding to contention (2) and most of them give up contention(1) in some way or other. In recent years, perhaps to a great extent due to the influence of Wittgenstein, this idea of a secure data base incorrigible claims about subjective contents of experience has lost power. As this has happened more and more Kant scholars have opted for objective rather than subjective readings of transcendental idealism. If my interpretation is correct then this has been a good trend, for Kant' position always was distinctively different from Berkeley's in that his appearances should never have been construed as incorrigible data. 10 Turbayne (pp. 232-3) considers this skeptical idealism (step 4) to be the first stage of the solution to the skeptical problem. I think Kant considered this position as no better than what Turbayne calls the deepest skepticism of step 3. 11 Note that, on Turbayne's version of it, Berkeley's position would also have to allow thatjudgments about external objects could be mistaken. Even if my judgment about the subjective content I now have were incorrigible my judgment that uniformities appropriate to the claim that there is a real dagger there obtain is subject to error. 12 Of the two interpretations of Kant's refutation of idealism argument suggested by WIlson (1972, pp. 604-605) this is the one that she grants would make trouble for her. It is also the one that best coheres with what Kant says the argument proves (B 275). 13 Barry Stroud (1968) also makes interesting comparisons between Kant's refutation of idealism argwnent and Wittgenstein's private language argumenL Unlike Sellars and Bennett, however, he did not actually propose an argument for Kant's conclusion. 14 This collapse into the solipsism of the present moment provides an additional compelling argument against the hypothesis that there are laws about sense data construed as incorrigible

EMPIRICAL REALISM & THE FOUNDATIONS OF GEOMETRY 287 subjective reports that could be learned empirically from observed regularities in one's subjective experience (See Note 4. ) I explore connections between the foregoing interpretation of the Refutation of Idealism and the Second Analogy Passages on the distinction between subjective and objective succession in Harper (1984). That paper also uses the nice passage from Wilson as a paradigm of the Demon argument and the Cartesian asswnptions it requires. I first saw this kind of interpretation of Kant's Refutation of Idealism, where the key argument is the failure of subjective idealism to support objective truth conditions about past subjective experiences, in Jonathan Bennett (1966). Paul Guyer (1983) has recently provided an extensive discussion of the origin and interpretation of Kant's Refutation of Idealism which also makes the core of the argument depend on these considerations. 15 It is worth remarking that Kant uses 'representation', and each sub-heading on this list, ambiguously as a token term to refer to particular mental episodes (representings) and also as a type term to specify a kind of representation qua - what it represents and how it represents it. The important use is the type use. I shall attempt to restrict my use of 'representation' to it and reserve representing for the token use. When, for example. I speak of my empirical intuition of this coin I shall be speaking of a kind of representing which can be speCified as a representing of this coin under certain perceptual circwnstances. According to this usage the same intuition could have tokened by another person or have failed to be tokened at all should it have been that someone else Or no one at all had satisfied the relevant perceptual circumstances. 16 Van Fraassen (Lecture at University of Western Ontario, fall tenn 1981) had recently suggested that one's inability to believe there is no ball in his hand when he is confronted with it does nouhow that belief is not voluntary any more than the fact that one cannot steal when he is in his own bathtub surrounded by only his own possessions shows that stealing is not voluniary. According to van Fraassen circumstances can sometimes constrain belief. Presumably, one of the most important kinds of constraint is provided by perceptual circumstances. It is to our relative inability to ovenide these circwnstances that I point when I speak of the independence of percep_ tion.

17 Sellars (e.g. 1963a) has emphasized the idea that entities of various kinds could play in thought a role suitably analogous to the role placed by a sentence, e. g. This is a quarter-shaped object before me, i'1 English. My proposal here is designed to be in the spirit of his general use of dot quotes and his own accounts of Kant's intuitions (e. g. 1968, Chapt. L 1975,1978). 18 This account of appearances is very close to Sellar's account of what we see of an object (e.g. 1981, Sections 15-24). It is also very close to Gibson's account of 'affordances' for human per_ ception (Gibson, 1966, 1979). I hope to explore some of the connections with Gibson's work in a later paper. Indeed, I expect that some of Kant's use of geomeny to ground his acCOunt of appearances as undetermined objects of empirical intuition can help answer some of the objections (e.g. in J. A. Fodor and Z. W. Pylyshyn, M. I. T. Occasional paper # 12) that have been raised against Gibson. 19 One of the salient differences between colors and such geometrical properties as shape is that the same geometrical property can often be presented to touch as well as to sighL This legitimate and important integration of sense realms, noted as early as Aristotle, provided motivation for the infamous distinction between primary and secondary qualities. According to Berkeley (Theory of Vision, 121-146) vision is only presented with colour, and colour is never presented to touch. so that these two sense realms are entirely distinct. He insists

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that the shapes presented to touch and the shapes presented to sight are entirely different species related milch as the combinations of writer's letters are related to the sounds of speech. In 'The Perception of Shape' (1983a) David Sanford has offered an excellent exploration of and defence for the claim that the shapes we see are the same as the shapes we touch. This is the best discussion of the issuef know of. 20 In 'Kant's Empirical Realism and the Distinction Between Subjective and Objective Succession' (Harper, 1984) I presented a line by line interpretation of this paragraph. This section of the present paper is, mostly, a summary of the main points in that interpretation; however, it does contain some additional remarks I hadn't thought to make before. 21 Beck's paper contains an admirable brief gloss of Kant's entire third paragraph in B. A good deal of my longer interpretation was cast in the fonn of commentary and expansion on Beck's gloss. 22 Beck distinguishes three distinct versions of Kant's empirical sense of 'appearance'. See (Harper, 1984) for my exposition of their relation to my explication of appearance as undetermined object of an empirical intuition. 23 This gloss of the two occurences of 'it' in the translation was suggested to me in correspondence by Lewis Beck. It is a salient part of the reading he provides in Beck (1978),

pp. 144-146).

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24 See van Fraassen (1966) for the basic account of supervaluation. See Thomason (1973) and Hans Kamp (1981) for applications of supervaluations to problems of indeterminacies generated by vagueness. 25 The Metaphysical FoWldations of Natural SCience suggests that Kant himself would opt for a plenum account based on postulated centers of inverse square attractive forces and inverse cube repulsive forces. 26 I have only argued that Kant's own defence of empirical realism is based on his appeal to a priori spatial constraints. I have not attempted the ambitious task of showing that an empirical realism that does not presuppose such constraints is impossible. Nevertheless, I hope the arguments of this section will make it implausible to suppose that one can recover an empirical realism without presupposing some such constraints. More importantly, I hope to have shown that the sort of constraints Kant's defence requires are not SO very implausible to presuppose. 27 Orthographic projection is not an accurate representation of the visual infonnation afforded to an ob$erver surveying a relatively large object before her. Ullman explores several other projection schemes. One is a perspective projection scheme according to which three views of five elements are mostly sufficient to uniquely specify the configuration and relative motion. This scheme is better at discriminating reflections in the image plane, but not so efficient at ruling out other alternative configurations and motions. The most sophisticated model of a projection scheme for human vision he considers is a polar-parallel projection scheme. Points corresponding to local texture patterns on a large surface are treated as approximately parallel projections to give detailed information about the local surface shape, while the larger structure is pinned down by polar-projection of these various local textured areas. This scheme apparently provides a fair approximation to the strengths and weaknesses of actual human visual discrimination. 28 Beck (1956) and Parsons (1969) have made a convincing case that Kant interprets such constructions as real definition of a geometrical concept. Real definitions are not analytic nor are they to be merely conventional stipulations. I have tried to make it plausible that geometrical constructions can play such an exalted role. 29 David Sanford (1983b) has an interesting discussion of the commitment to orientation relative to a point of view of a visual field. [think Kant would agree with this and would extend the point to imagination as well as perception.

I

EMPIRICAL REALISM & THE FOUNDATIONS OF GEOMETRY 289 30 Reid's Geometry of Visibles has been reswrected as one of a number of proposals for construing visual geometry as non-Euclidean (Angell, 1974; Daniels 1972). Strawson's attempt to save Euclidean geometry by dividing viSUal geometry from physical interpretations plays right into the hands of these advocates for non-Euclidean geometries of visual experience. 31 I do think that it is rational now to proceed as though such changes could not happen. Indeed I think that we all really do proceed this way. We cannot help but make them into conceptual commitments for us. I also think that conceptual commitments can be rationally changed (Harper,

1978).

REFERENCES Allison, H.: 1973, 'Kant's Critique of Berkeley', Journal of the History

of Philosophy 11,

43-63. Angell, R. B.: 1974, 'The Geometry ofVisibles'.Nous 8 87-117. Beck. L w.: 1956, 'Kant's Theory of Definition', Philosophical Review 65, 179-19l. Beck, L. W.: (ed.): 1972, Proceedings of the Third International Kant Congress, Reidel, Dordrecht. Beck, L. W.: 1978, Essays on Kant and Hume, Yale University Press, New Haven. Bennett, 1.: 1966, Kanfs Analytic, Cambridge University Press, Cambridge. Berkeley, G.: 1954, Three Dialogues Between Hylas and Philonous ed. by C. M. Turbayne, Bobbs Merri1l, Indianapolis. Berkeley, G.: 1965, A Treatise Concerning the Principles of Human Knowledge, ed. by C. M. Turbayne, Bobbs Merrill, Indianapolis. Berkeley, G.: 1969, An Essay Towards a New Theory of Vision. ed. and introducted by A. D. Lindsay, New York. Cornman, 1. W.: 1973, 'Theoretical Phenomenalism', NOlls 7, 120-138. Daniels, N.: 1972, 'Thomas Reid's Discovery of a Non-Euclidean Geometry', Philosophy of Science 39, 219-234. Euclid: 1956, Elements, Trans. by T. C. Heath, Dover, New York. Fodor, 1. A. and Pylyshyn, Z. W.: 'How Direct is ViSUal Perception?', MIT Occasional Paper # 28. Frege, G.: 1959, The Foundations ofArithmetic, trans. by 1. Austin, Blackwell, Oxford. Gibson, 1. 1.: 1966, The Senses Considered as Perceptual Systems, Houghton Mifflin, Boston. Gibson, 1. 1.: 1979.An Ecological Approach to Visual Perception, Houghton Mifflin, Boston. Guyer, P.: 1983, 'Kant's Intentions in the Refutation of Idealism" Philosophical Review 92,

329-383. Harper, W.: 1978, 'Conceptual Change, Incommensurability and Special Relatively Kinematies', Acta Philosophica Fennica 30, 429-460. Harper, W.: 1981, 'Kant's Empirical Realism and the Second Analogy of Experience', Synthese

47,465-480. Harper, W.: 1984. 'Kant's Empirical Realism and the Distinction between Subjective and Objective Succession', in Harper and Meerbote (eds.). Harper, W. and R. Meerbote (eels.): 1984, Kant on Causality. Freedom and Objectivity. Minneapolis, University of Minnesota Press. Hintikka, 1.: 1965, 'Kant's "New Method of Thought" and His Theory of Mathematics', Ajatus

27.37-97. Hintikka, J.: 1967, 'Kant on the Mathematical Method', The Monist 51, 352-375; reprinted in this volume, pp. 21--42.

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Hintikka. J.: 1969, '00 Kant's Notion of Intuition (Anscbauung)', in Penelhum and Macintosh (eds. ). Hintikka, r.: 1982. 'Kant's Theory of Mathematicals Revisited', in Mohanty and Shehan (eds.). Hopkins, J.: 1973. 'Visual Geometry'. The Philosophical Review, pp. 3-34. Howell, R.: 1973. 'Intuition, Synthesis, and Individuation in the Critique of Pure Reason', Nous 7,207-232. Kamp, H.: 1981, ''The Paradox of the Heap', in U. Monnich (ed.), ASpects of Philosophical

Logic, Dordrecht, pp. 225-277. Kant, I.: 1950. Prolegomena to Any Future Metaphysics, in Mahaffy-Carus-Beck trans., Bobbs Merrill. Indianapolis. Kant, I.: 1963, Critique of Pure Reason, transl. by N. K. Smith. St. Martin's Press, New York. Kant, I.: 1967, Letter to Garve, August 7,1783 in A. Zweig (eel., trans.), Kant: Philosophical Correspondence, University of Chicago Press, Chicago. Kant, I.: 1968, Kants Werke (Academic-Textausgabe), Walter de Gruyter & Co., Berlin. Kant, 1.: 1970, Metaphysical Foundations of Natural Science, trans. by I. Ellington, Bobbs Merrill, Indianapolis. Melnick, A.: 1984, 'The Geometry of a Form of Intuition', Topoi, v. 3; reprinted in this volume

1

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pp.145-155. Miller, G.: 1973, 'Kant and Berkeley', Kant Studien 64, 315-335. Mohanty, I. N. and Shehan, R. W. (eds.): 1982, Essays on Kanfs Critique of Pure Reason. University of Oklahoma Press, Norman, Oklahoma. Parsons, C.: 1964-, 'Infinity and Kant's Conception of the "Possibility of Experience"', Philosophical Review 73,183-97. Parsons, C.: 1969, 'Kant's Philosophy of Arithmetic', in Morgenbesser et al. (eds.), Philosophy. Science, and Methodology: Essays in Honor of Ernest Nagel, St. Martin's Press, New York; reprinted in this volume pp. 43-79. Penelhum and Macintosh (eds.): 1969. The First Critique, Wadsworth, Belmont, California. Poincare, H.: 1898, 'On the Foundations of Geometry', The Monist 9, 1-43. Pollock, J.: 1974, Knowledge and Justification, Princeton University Press, Princeton. Posy, C.: 1983, 'Dancing to the Antinomy: A Proposal for Transcendental Idealism', American Philosophical Quarterly 20, 81-94. Posy, C.: 1984, 'Kant's Mathematical Realism', The Monist 67 (1984); reprinted in this volume pp.293-313. Putnam, H.: 1981, Reason, Truth and History, Cambridge University Press, Cambridge. Sanford, D.: 1983a, 'The Perception of Shape', in C. Genet and S. Shoemaker (eds.), Knowledge and Mind: Philosophical Essays. Oxford, pp. 130-158. Sanford. D.: 1983b, 'Impartial Perception', Philosophy 58, 392-395. Sellars, W.: 1963, Science, Perception and Reality, Humanities Press, New York. Sellars. W.: 1963a, 'Abstract Entities', Review of Metaphysics 16, 627-671. Sellars, W.: 1968, Science and Metaphysics, Humanities Press, New York. Sellars, W.: 1975, 'Kant's Transcendental Idealism', in Laberge et al. (eds.), Ottawa Congress on Kant in the Anglo-American and Continental Traditions, Editions d'Universite d'Ottawa, Onawa. Sellars, W.: 1978, 'The Role of Imagination in Kant's Theory of Experience'. in Johnstone (eds.), Categories: a Colloquium, Penn. State Press, pp. 231-245. Sellars, W.: 1981, 'Foundations for a Metaphysics of Pure Process', The Monist 64, 1-88. Smith, N. K: 1929, A Commentary to Kant's Critique of Pure Reason, Macmillan, London. Strawson, P.: 1966. The Bounds of Sense, Methuen, London.

EMPIRICAL REALISM & THE FOUNDATiONS OF GEOMETRY 291 Stroud, B.: 1968, 'Transcendental Arguments', Journal ofPhiiosophy6S, 241-256. Thomason, R.: 1973, 'Supervaluations, The Bald Man and The Lottery', Unpublished Mimeo, University of Pittsburgh. Torretti, R.: 1978, The Philosophy o/Geometry from Riemann to Poincare, Reidel, Dordrecht. Turbayne, c.: 1955, 'Kant's Refutation of Dogmatic Idealism', The Philosophical Quarterly 20, 225-244. Ullman, S.: 1979, The Interpretation of Visual Motion, MIT Press, Cambridge (Mass.). van Fniassen, B.: 1966, 'Singular Terms, Truth Value Gaps, and Free Logic', Journal of Philosophy 63, 481-495. van Fraassen. B.: 1980, The Scientific Image, Oxford University Press, Oxford. Walker, R. C. S.: 1978, Kant, Routledge and Kegan Paul, London. Wllson, M.: 1971, 'Kant and the Dogmatic Idealism of Berkeley', Journal of the History of Philosophy 9, 459-476. Wllson, M.: 1972, 'On Kant and the Refutation of Subjectivism'. in L. W. Beck (ed.), (1972).

CARL J. POSY

KANT'S MATHEMATICAL REALISM

Though my title speaks of Kant's mathematical realism, I want in this essay to explore Kant's relation to a famous mathematical anti-realist. Specifically, I want to discuss Kant's influence on L. E. J. Brouwer, the 2Oth-cenrury Dutch mathematician who built a contemporary philosophy of mathematics on, constructivist themes which were quite explicitly Kantian. I Brouwer's

theory (called intuitionism) is perhaps most notable for its belief that constructivism (whatever that means) requires us to abandon the traditional (classical) logic of mathematical reasoning in favor of a different canon of reasoning, called intuitionistic logic. Brouwer thought that classical logic is intrinsically bound up with a nonconstructive (or "realistic") view of mathematics~ This means that, according to Brouwer, when we do mathematics we must give up bivalence (the principle that a given sentence either is true or is

detenninately false), we must no longer use such familiar logical laws as excluded middle, and we must sometimes forebear from the classic method

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of reductio ad absurdum. All of these are intuitionistically invalid classical principles. Now scholars generally agree that Brouwer did borrow Kantian themes. But there is very little consensus about which details he borrowed. And there is certainly little agreement about how this same set of doctrines could foster such radically different programs as intuitionism and Hilbert's formalism. (Hilbert too was an avowed Kantian.) So I have set myself the task of trying to give a precise - though perhaps anachronistic - account of those Kantian themes Brouwer might be developing in his critique of classical logic. I will, indeed, consider two rival accounts of Brouwer's debt to Kant. This task is especially interesting to me, because I am fond of using intuitionistic techniques to interpret and vindicate some Kantian arguments - not so much arguments about mathematics but mainly ones about empirical science. These are arguments in which Kant defends his famous transcendental idealism, TI, (the view that empirical objects are mere appearances) from its rival 'transcendental realism, TR, (the view that these same objects are things in themselves). I think of this project as exploring Kant's debt to 293 Carl J. Posy (ed.J, Kant's Philosophy of Mathematics, 293-313.

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Brouwer. Any philosopher owes a debt (even posthumously) to the progenitor of techniques that can vindicate some of his reasoning. But the point is that I can't just go about attributing intuitionistic logic to Kant simply because it salvages this or that argument. To make this project plausible, I'll have to find in TI a philosophical ground to support these logical moves. This ground can be the theme Brouwer borrowed from Kant. As I said, I'll suggesttwo rival interpretations of Tl to do that job: the first, a fairly standard ontological one, unfortunately fails, both as a reading of Kant and of Brouwer. The second, which applies some currently popular notions from the philosophy of language, does work for Kant (and perhaps for Brouwer). But my main thesis now is that this second interpretation has a most remarkable consequence. When these linguistic notions are clarified and applied to Kant, they show him to be an intuitionist for empirical science, but a realist, or at least an advocate of classical logic, for mathematics! So my remarks will fall under four main topics:

beginning is ruled out, because the event of creation ex nihilo could have no observable cause. [A427/B455].) The idealist escapes this dilemma. For, according to Kant, TI's world need be neither finite nor infinite. (A504/B532). In fact, it is neither finite nor infinite. (A520/B548). Passages containing this attack on TR have themselves suffered much critical attack. Jonathan Bennett says that Kant's discnssion of measurability simply distorts the properties of discretely ordered series.3 Bertrand Russell tells us that Kant is unable to express clearly the infinity of a class (in this case the class of past durations), and that Kant's basic move is undermined by the discovery of infinite numbers.4 There's a logical question as well: Why is Tl free from the disjunction (finite vs. infinite past) while TR is not? Lastly, there is a simple question of consistency: How can Kant square his claim that TI's world is neither finite nor infinite with his subsequent claim just a page later that the idealist's world does indeed extend infinitely into the past? (A521/B549). (Perhaps Kant is distinguishing, in the later passage, between an actual and potential infinity. But if that is so why isn't the realist entitled to the same distinction? Realism about physical objects is perfectly compatible with denying actual infinity.) [2] Here is where I think some contemporary insights and techniques will help. Observe first that Bennett is right in one respect: Kant is indeed Concerned with a discrete linearly ordered series, a sequence of equal temporal durations (say, days) stemming backwards from some fixed moment. Indeed we can say (without anachronism) that the elements are ordered by a relation satisfying the axioms of a linear order. Now suppose we introduce a very simple formal first order language, Lr (T for temporality), whose variables range over these equal durations. And suppose Lr has a single, nonlogical, binary symbol, 'B', which denotes that discrete ordering of durations. B (x, y) means that interval x began at a moment distinguishably prior to the beginning of y. In this language we can symbolize the claim about a finite beginning by

I. Display Kant's debt to Brouwer. II. Describe the standard interpretation, and why it fails. (I will do these first two very briefiy.J2 III. Describe the linguistic reading, and clarify some of its oft rnis~d details. IV. Apply this reading to Kant, and use it to support the snrprising conclusions I mentioned. I will conclude with some general remarks about how all of this interacts with some other themes in the Critique of Pure Reason. and with a discussion of the sort of anachronism that mayor may not be lurking in all this. 1. KANT'S DEBT TO BROUWER,

[I] Let me tum to my first task and qnickly describe an instance where current intuitionism may help us interpret a controversial Kantian defense of TI. I have in mind Kant's attempt, in the famous "Antinomy" chapter, to produce an internal contradiction in the realist's empirical world view - in particular in his view about the age of the universe. According to Kant, the transcendental realist is committed to the view that the world either had a finite beginning in time or extends infinitely into the past. Kant then proves that neither alternative is possible, and thus reduces TR ad absurdum. (An infinite past is impossible, he says, because it is impossible to measure such a long temporal stretch. [A 426/B454]. And a finite

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(I) 3xVy"B(x,y).

The claim about an infinite past becomes (2) V:i3Y#xB(y, x).

Let me step out of my announced order for a minute to say that I think this simple translation into 4, which seems so natural today (languages like LT compose our modern lingua franca), this simple move is probably the most

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anachronistic move I will make. I'll discuss that a bit later. But I'll also point out that it's no less foreign to Brouwer (perhaps it is even more so) than it is to Kant. But if, for the moment, we do allow this barbatism, and we do assume that (I) and (2) are fair representations of Kant's alternatives, then the first group of objections melts away (by force, to be sure). There are no ordinal irregulatities hidden by (I) and (2); each is perfectly consistent with the linear order of time. (2) is precisely the expression of infinity that Russell ultimately advocates, and the existence of infinite numbers is (sttictly speaking) irrelevant to this pair offonnulas. As for the second problem, the logical one, that dissolves as well, if we associate TR with classical logic and TI with the logic that has been fonnulated for intuitionism. For the result of disjoining (I) and (2) is a classical logical truth, but not an intuitionistic one. As for the final problem - the one about the consistency of Kant's denial that the world is infinite with his later claim that the world does reach infinitely into the past -let me explain that latter remark. Let's assume for the moment that TI does link physical existence with experience. Still, according to Kant, we can never be satisfied with partial scientific results. We never rest easy with just those facts and objects which we have experienced. This is dictated by what he calls the "faculty of reason", the faculty that governs theoryfonnation and our general inferential activity. In particular this part of our personalities demands that we must always consider how any series of cosmological discoveries can be extended to discoveries documenting ever more remote past durations.

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always to push for further extensions of any sequence of discoveries, combined with the belief that such extensions will be found but the refusal to act as if they have been found. 5 There is a simple Kripke model which shows the compatibility of (3) with the axioms of linear order. The nodes of the model are artanged as a single infinite chain {a,} ,. The domain of a, is the set {I, 2, 3, ... n. And we interpret B(j, k) as j > k. (Numerical increase represents increasing remoteness into the past.) Now for-

mulas (2) and (3) are not inmitionalistically equivalent. (Indeed, (I) and (2) are both false in t1tis same Kripke model.) That explains Kant's dual attitudes toward infinity. The world is not infinite in the sense of fonnula (2), but is in the sense of (3). Classically, (2) and (3) are logically indistingnishable: so the realist is indeed barred from this resolution of the antinomy. That solves all the problems I raised above. [3] As it turns out t1tis same device is quite useful in interpreting several other passages in the Critique where the regulative effect of the faculty of reason is invoked. Thus, for instance, the principle of causality which is defended in the second analogy, needs something like this to reconcile it with the arguments of the first and third antinomies. 6 And the model structures that logicians use to study intnitionism provide handy devices for depicting the notions of "objectivity" and "unity" that figure prominently in Kant's Transcendental Deduction.7 There are other applications as well.

So there are exegetical advantages to assuming that the transcendental realism/idealism debate includes this modern debate about the logic of empir-

If (as the idealist requires) this feature of the mind has a role in forming

ical discourse. But as I said at the start, even if we ignore the inherent

our scientific picture of the empirical world, then that science must satisfy our

drive for "completeness" of the series. But, at the same time, pure reason

anachronism, still all of this becomes philosophically interesting only if we understand TI in a way which ma.kes it reasonable to link it with intnitionism.

alone cannot provide us with the observations and experiences we haven't yet

Let me tofu to that question.

had. (Observations and experiences require the faculty of "intuition.") Kant encapsulates all this in the slogan that the faculty ofreason posits the infinity of the world as a regulative ideal. Butnaming it isn't explaining it. Once again intuitionism comes to the rescue. We can symbolize the effects

of all of these seemingly diverse pulls in fonnula (3): (3) Vx - - 3yox B(y, x)

The intuitionistic double negation here means that given any duration, x, we cannot help but eventually discover a distinct y which is prior to x. This

gives a sympathetic reading to Kant's notion of a regulative idea: the drive

2. ONTOLOGICAL IDEALISM

[4] There is a more or less standard interpretation of TI which at first sight might do the job. This is the view which makes TI into a literal physical constructivism - called phenomeualism - and according to which material objects are entirely human constructions.

This interpretation is standard because it works within the widely accepted correspondence theory of truth and preserves the familiar truth tabular meanings of the logical connectives.

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And indeed it does fit many Kantian texts. Physical objects are often described as constructed ("synthesized") from the material of intuition. 8 Mathematical objects are thought of as similarly constructed, though out of different, more rarified material.9 This reading is consistent with the general assumption that Kant meant intuitions (both empirical and mathematical intuitions) to apply directly to their objects in a de re relation. 10 And since it views mathematical objects as reified versions of the principles governing the construction of empirical objects, it does a good job of explaining Kant's views on applied mathematics. I should add that this picture of the scientist and mathematician somehow "constructing" their objects seems to fit a number of Brouwerian passages as well." [5] Nevertheless this picture of constructivism won't work for Brouwer or for Kant. Though it does invalidate bivalence,12 still it fails for Brouwer because it does not generate the full intuitionistic logic. It will not for instance validate - - (Pt V - Pt) and other tautologies that figure prominently in many Brouwerian arguments. So this ontological constructivism - even if it is a Kantian theme - can't be one that Brouwer borrowed in order to ground his logical beliefs. 13 But in fact it fails as a reading of Kant too, and for three reasons. First, this interpretation makes it impossible to understand the realist's reasoning in the antinomy. If physical events and objects are (for the realist) mind - independent things in themselves, why should their measurability or immeasurability in any way affect the truth of judgments about their size? Secondly, if objects are constructed out of the material of sensation, what hope is there for the existence of objects too faint, too far, or too small to be perceived? Yet Kant clearly held that things of this sort (distant stars, miniscrile particles and the like) most definitely are physical objects, and do exist. l ' Finally, this reading blurs any appreciable difference between Kant and Berkeley. The two will share the sarne theory of truth and the sarne phenomenalist conception of empirical objects. This is bad on its own (phenomenalism for material objects is an abhorent doctrine); but it's even worse, because Kant explicitly separates himself from Berkeley on several occasions. l5 So, all in all, we have to look for some other philosophical ground generating intuitionistic logic - both in mathematics, and (for Kant) in empirical science. Michael Dummett, building on suggestions stemming from Heyting (and really aimed at mathematics) has provided a philosophical argument which (if sound) will invalidate bivalence and will generate an intuitionistic logic. It is

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the argument from the so called assertability theory of meaning. 3. ASSERTABILISM REFINED

[6] The main thrust of this argument is really qnite general, and can easily apply to empirical as well as to mathematical language. It goes like this: Given some fragment of language (say Lro or the language of number theory), the main premise of the argument claims that we must replace the standard referential notion of truth for sentences of that fragment with a notion of "sufficient evidence" (warranted assertability). Instead of speaking about what must be the case in the world (or in some model) for a given senteriCe to be true, we speak about what sort of evidence will suffice to warrant the speech act of asserting the sentence. Indeed we must replace the classical notion of a model (or possible state of affairs) with the notion of an evidential state: a collection not of objects, but of bits of evidence sufficient to support or refute some set of sentences. Whatever logical role standard truth in a model used to play in the language this notion of assertability at an evidential situation now plays. In particular it is assertability that is now to be preserved by valid inferences, and it is assertability that figures into the meanings of the logical particles. These facts will then fix the corresponding logic for that language. And that will be intuitionistic logic, because, as it tums out, the orily statements guaranteed to be assertable at every evidential state are precisely the intuitionistic tautologies. Or so the argument goes. Now this argument - or at least its assertabilist premise - has gained a certain notoriety; mainly, I think because of Durnmett's spirited defence of thit main premise by use of arguments about the tearnability of language, and the possibility of communication. But I also think that some confusions have snuck in under all this hullabaloo over the replacement of truth by proof. For one thing. because sufficient evidence is now the criterion of semantic success (i.e., of truth), some people have slipped into the habit of saying that propositions (understood assertabilistically) are about units of evidence. (That the objects of mathematics are proofs, and the objects of empirical science are bits of empirical evidence). This is a confusion. Assertabilism per se says nothing about the objects of discourse. Indeed, the assumption that a theory of truth involves a theory of objects is a correspondentist assumption to start with. You can be a perfectly good assertabilist and simultaneously hold that objects of one sort or another are totally mind-independent.

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Secondly, all this jwnping and shouting about the assertabilist premise has obscured the fact that assertabilism alone will not suffice to generate an intu-

(A4) (A -7' B) is assertable in a situation iff, if A is assertable in the situation B is. (AS) VxA is assertable in a situation iff Aa is thus assertable for every a spoken about by the given language. (A6) 3xA is assertable in a situation iff for some such a, Aa is thus assertable.

itionistic logic. In order to go from assertabilism to intuitionism we have to

settle four additional matters. I call these additional considerations (i) epistemic myopia, (ii) assertability conditions, (iii) constructive optimism, and (iv) the long vs, the short semantic view. Let me briefly explain them in order, [7] First, these evidential states that replace models must be limited in a certain way. That is, we must conceive of these states in such a way that it is always possible that at least some proposition or other being evaluated at a given epistemic situation is undecided at that point by the ''relevant evidence."

- This is certainly true of our ordinary notion of empirical evidence (at some point in time), Thus I may learn tomorrow that there is oil under all that worthless clay in my backyard. But right now, I don't have enough evidence to confirm that claim; so I can't assert it. And that is a fact about my current epistemic state. Now of course for rigor we have to be more general, and speak of epistemology in the abstract, or of the general theory of epistemic states. But if there's to be any hope of scotching bivalence, we had better see to it that our epistemology does impose some sllch limits on its notion of evidential state. I call this the condition of "epistemic myopia." For it rules out the infamous "God's eye" view of knowledge. It admits a notion of gradual evidential growth, and says that sometimes that growth takes you beyond the evidence you happen to see within a given evidential state. (Formally, you might think of the nodes in a Kripke or Beth model as a progression of evidential states.) [8] The second thing we have to do is agree upon some general conditions which (once we set ground-rules for asserting atomic claims) will tell us about the compound propositions assertable in a given evidential state. As it turns out, that's less trivial than you might think. In particular it's not clear that we can simply substitute "epistemic situation" for "model" and plug in "assertability" for "truth" in the standard truth conditions derived from Tarski. Table I shows what this would look like. Table I (AI) (A & B) is assertable in an epistemlc situation if and only if A and B are both tl;tus assertable. (A2) -A is assertable in a situation iff A is "negatively assertable." [Ordinarily this is interpreted as A ~~, -1 some known falsebood.] (A3) (A V B) is assertable in a situation iff either A is thus assertable or B is.

This approach falters though because it is unenforceable in practice. Clause (AS) for instance is impractical when we are talking about very large collec-

tions of objects. And (A6) - which is a version of the requirement for "constructive existence proofs" - simply clashes with common practice and with almost all learned opinion. No one - not even a constructivist - always sits ~_d waits to actually prove an instance before asserting an existence claim

3xPx. Quite the same holds for disjunction in (A3). The problem is that these clauses aren't merely myopic about assertability - they're just plain blind. Only things that are concretely touched or bwnped into in an evidential state COlmt as evidence at that state.

Now mathematical assertabilists often solve this problem by importing what they call an "effective procedure." Four our purposes we can think of this as a definable operation which takes pieces of evidence into pieces of

evidence and which guarantees to produce an output for any given input (within a finite time). If you admit effective procedures (or better the recognition of effective procedures) as relevant to assertability then you get the assertability conditions I've sketched in Table II. Table II (A3), (A V B) is assertable at a situation iff that situation includes (recognition of) an effective procedure which results in evidence confirming A or evidence confirming B. (A4), (A ---+ B) is assertable in a situation iff that situation contains (recognition of) an effective procedure converting evidence for A into evidence for B. (AS)' (Ax) A is assertable in a situation iff the situation contains (recognition of) an effective procedure taking evidence that a exists into evidence for Aa. (A6), (3x) A is thus assertable iff the situation contains (recognition of) an effective procedure which will produce evidence for Aa for some a.

This is still a constructivism. It still requires that existentials be wituessed and that the assertability of disjunctions entail the assertability of one of the disjuncts. But it's a mild constructivism - it allows me to be satisfied with a guarantee of eventual success instead of the "blind" requirements of the stricter view.

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On this milder view I'm entitled to assert right now that some large number [say lOw + I] is either prime or composite - because I know a method to figure that out; and I know that once I start I'm guaranteed success. On the strict view I have no such rights. So. even if I adopt a basic assertabilist view of truth, still I will have to choose between the strict constructivism of Table I, or the milder version of Table 2. That's the second matter to be settled. [9] My third point is that this difference - between ntild and strict constructivism - is a difference that makes a difference for our concern with intuitionistic logic. For suppose I do espouse a basic assertabilism and equate truth with knowledge, and suppose I do admit that knowledge is limited at any given point. But now suppose I adopt a mild constructivism, and I couple all of this with a sort of brazen optimism. Suppose I claim that every proposition (of the language in question) will (or at least can) be constructively decided. (I.e., - suppose I am prepared to assert that given a meaningful proposition, and given enough time, I will be able either to prove it constructively or refute in constructively.) This isn't so outlandish a belief as one might think. It's precisely Hilbert's view about large parts ofmathematics. 6 But this Hilbertian optimism, which can be expressed in formula 4, combines with my ntild constructivism to validate excluded middle, and in fact the full classical logic.

Peircian assertabilism which defines truth as eventual assertability (Le., assertability in some eventually realised epistentic situation). The Peircian picture is one in which mankind progresses from epistemic situation to situation, learning (and asserting) new things, and adding them to an ever growing stock of recoguized truths. To be true is to have been put in this stockpile of assertable truths. Thongh tied to assertability, truth is now a tenseless notion. (Thus if I strike oil tomorrow it is just true outright that there is oil under my yard.) Now Dummett disparages this picture. Essentially it loosens truth from the grip of immediate assertability, and he thinks that's a step towards realism. But he's wrong. Indeed, his own linguistic support of basic assertabilism automatically supports this tenseless variety as welL 17 [To be sure, this Peircian long view faces some problems about corrigibility and convergence. But as things tum out these ills plague the local version as well.] The fact is that this Peircian version is a legitimate assertabilism. Recoguizing this fact is the final connection I mentioned. For when we combine this view with the Hilbert style constructive optintism, then the result is not intuitionism at all. It is a bivalent language - no matter which style of constructivism we adopt! All told, with these three latter issues (style of constructivism, optimism, long vs. short semantic view) to be settled, I've distinguished eight different assertabilist positions. Some will indeed invalidate bivalence, some will yield intuitionistic logic for any language to which they are applied. But some will not. I've tabulated the possibilities on Table Ill. So clearly, the move from assertabilism to intuitionism is far from guaranteed.

(4) Va3p [Pr(p, a) v Pr(p, lXl] (where a ranges over the propositions of our language, p over pieces of evidence (proofs) and Pr is a proof predicate.) The reason: Well think about any formula of the form (A v -A) at some epistentic situation. I ntight not be in a position to assert either disjunct but I am assuming that I can assert formula (4) with A for a, and this itself provides the _guarantee I need in order to assert the disjunction. So - unless the assertabilist can refute this optintism or constrain it - he must either ,adopt the stingy strict constructivism of Table r or tread the primrose path right back to classical logic. [10] Finally, as for bivalence: Notice that up till now I have been speaking as if our assertabilist premise simply equates truth with assertability-at-anepistentic situation. (So those propositions which are actually assertable right now, are the only ones which are actually true, and those not currently assertable are either false or without truth value.) Certainly that's the way Dunnnett looks at things. But by putting things this way I am automatically excluding the prominent

TableID Strict Constructivism: Short Semantic View (i) Non-optimistic

(1) non-bivalent

(2) no classical logic (ii) Optimistic

(b) Long View (i) Non-optimistic

(I) non-bivalent (2) no classical logic

(1) non-bivalent

(2) no classical logic

(ii) Optimistic

Mild Constructivism: (a) Short View

(I) bivalent (2) No classicallogic. 18

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(1) non-bivalent (2) no cIassicallogic (1) non-bivalent

(2) classical logic (b) Long View: (i) Non-optimistic

(1) non-bivalent

(2) no c1assica11ogic (li) Optimistic

(1) Bivalent

(2) classical logic

4. KANT THE ASSERTABILIST

[II] Now did Brouwer use an assertabilism to motivate his rejection of classicallogic? Two remarks are in order: (a) Brouwer would not accept Dummett's linguistic arguments supporting the general assertabilist premise for mathematics. These rest on considerations that are foreign to Brouwer's professed outloOk. 19 (b) On the other hand there are other places (eg. his proof of the bar theorem and in some posthumously found notes) where he endorsed the assertabilist definition of some logical particles. 20 And clearly he did identify Hilbert's constructive optimism with classical logic and bivalence. 21 So the evidence is mixed. But if a general assertability reading does rehabilitate Brouwer (perhaps by force) then it can do the same for Kant. There are powerful reasons to accept this reading. If Tl can be some version of assertabilism which does yield an intuitionistic logic, that would account for the idealist's moves in the antinomy. Since assertability is ontologically neutral, there would be no problem with phenomenalism. Moreover we could begin to make sense of the realist's arguments in the antinomy. He would be a constructive optimist - the sort who demands constructive verification for his judgment but who confidently assumes that he can always get it. That will by why he adoprs classical logic, but also connects measurability to truth. If we do read Kant this way, then we will have to understand intuitions (empirical or mathematical) as units of evidence and not as the building blocks of objects. Indeed, intuitions would be the special sort of evidence demanded by synthetic atomic (or at least singular) judgments. [Pt is

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assertable in an evidential state, just in case that state contains an intuition of t which is also a P -experience.] So we would have to speak of intuitability de dicto and to think of Kant's elaborate apparatus of faculties as a theory about different varieties of evidence. 22 But, once again, even granting that Kant's transcendental idealism is an assertabilism the main thing we have to do is find some internal cues to place it at a position on Table III which generates an intuitionistic logic. I think there are some textual hints which do this job. Thus, for instance on the question· of constructivism, Kant comes down clearly on the mild side: The postulate bearing on the knowledge of things as actual, does not indeed demand immediate perception (therefore sensation of which we are conscious) of the object whose existence is to be mown. What we do, however, require is the connection of the object with some actual perception in accordance with the analogies of experience ... (A255/B272)

The "actuality" Kant is talking about here is simply the existence of empirical objects. He is admitting that even if I don't strike oil, still I tuight have enough geological knowledge to read signs which will allow me to assert right now that there's oil under my yard. Much the same holds for unobservable empirical objects (distant stars. small particles, etc.) about which we have enough scientific evidence to assert their existence. I think on the question of the long (peircian) vs. short (Dummettian) semantic view, Kant is equally clear: To call an appearance a real thing prior to our perceiving it, either means that in the advance of experience we must meet with such a perception, or it means nothing at all ... (A493/B521)

Here he is admitting nonverified truths, but ouly on the ground that they are eventually verified. He says similar things a bit later about historical truths as well. This is a Peircian position. Now, consulting Table III, we see that if our Kantian idealist is indeed a mildly constructive Peircian, then all our logical questions come down to the issue of constructive optimism. If he is an optimist (and believes that all empirical questions can ultimately be answered), then it's classical logic and bivalence for him. If not, then not. Here too Kant's view (qua transcendental idealist) is loud and clearly announced (at least for empirical languages): In the explanation of natural appearances much must remain uncertain. and many questions insoluble, because what we know of nature is by no means sufficient. in all cases, to account for what has to be explained... (A477 /B505)

There's no optimism here. So for Lr it is intuitionistic logic and irs atrendant complications. Thus, if we accept the assertabilist reading of Kant, and if

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Dummett and Heyting are right about Brouwer, then we do find in Kant a philosophical theme regarding empirical discourse, whose logical consequences >1-TO developed by the inmitionists for mathematical discourse. That, I think, is the better theory of what Brouwer borrowed from Kant. But there is one important difference. On this interpretation the application of Kant's general views to the case of mathematics will not Jead him to be a

These are not isolated passages. They represent a general Kantian theme: Proper rational sciences - disciplines concerned ultimately with the nature of our representations - are legitimately optimistic. The issue comes down in the end to a matter of control. In empirical science the material out of which we fashion our evidence is sensory. And that means we must depend on passively receiving it. We may try to search for the needed intuitions and put ourselves in position to receive them. But there is no guarantee that such a search will succeed or will even terminate. In the rational sciences the material is within us. So. as Kant says, "it is impossible to plead unavoidable ignorance." (A477/B505). We must take care to note that it is only the internal (spontaneous) origin of .its evidence that justifies, for Kant, this sanguine attitude towards the rational sciences. Forgetting that is precisely the realist's error in the antinomy: He wants to combine the optimism of pure reason with a constructivism based on empirical inmitions. That is the combination which Kant drives ab absurdum. But retwning to our case of mathematics we can now see that when we combine Kant's generally far-sighted (peircian) assertabilism, and his overall mild constructivism, with the optimism generated by mathematical intuition, we have the urunistakable recipe for a bivalent language and a full classical logic! I would conclude that while he might be an intuitionist for empirical science, Kant would simultaneously advocate classical logic for mathematical science. [12] That's the conclusion I announced at the outset. But if you look at it for a moment - it may seem to you that I've defeated my own purpose. Think all the way back to my solution of the antinomy about the age of the universe. I said that Kant advocates the "regulative" infinity of the universe in the sense of formula (3). This means in our current terminology that at an evidential state - at any point in which we have documented some finite part of the past - we are enjoined to search further for even more remote past durations. Moreover, we are guaranteed that this search will be fruitful. We will be able to document that past. (That is, the Kripke model will have further nodes.) This guarantee is provided by "transcendenral philosophy" - it follows from the conditions of the possibility of experience. So it is legitimate. But then, we ought to combine that guarantee with Kant's mild constructivism to warrant the claim right now that these remote (not yet documented) durations do exist. I know this reasoning is quite general so I find myself in the following simation: I can imagine myself given a duration x (at some evidential state) - and

proto ~ intuitionist in that field. II

Two facts are clear about Kant' views on the "relevant evidence" for mathematical statements. (a) He does say that we construct our mathematical inmitions, and (b) mathematical activity is, for him, purely intellecmal. (It belongs to the faculty of reason.) There's considerable controversy about how to interpret Kant's claims that the mathematician "constructs his objects in pure intuition". All I need say now is that this is not the "strict constructivism" described before. (Kant's not saying that I have to have constructed an object with the property P before I can assert 3xPx.) I think what we have here is simply a combination of his general view that synthetic atomic claims must rest on intuitions, with his special theory that mathematical (pure) intuitions are wholly the product of the mind. Whether or not pure intuitions are the sort of things we can experience, and whether or not this experience has some sensory element, one thing is certain. Pure intuitions are not passive or "receptive." They are, as Kant sometimes says, "spontaneous" mental representations. They stem from the mind alone. That above all distinguishes them from the empirical intuitions

needed for ordinary science. Other disciplines which share this penchant for internally, or spontaneously generated bits of evidence include ethics and the so called "transcendental philosophy," which studies the conditions of possible experience. Now according to Kant, these disciplines enjoy a certain epistemic advantage. Since their subject matter is internal to 'the mind, it is, he says, more readily known, . .. every question arising within their domain should be completely answerable in terms of what is known, inasmuch as the answer must issue from the same sources from which the question proceeds. (A476/B504) 1~1l

II",·

You couldn't ask for a clearer statement of the optimistic attitude fostered by these disciplines. But if you do, Kant himself provides it: It is not so extraordinary as at first seems the case that a science should be in a position to demand and expect none but assured answers to all the questions within its domain, although up to the present they have perhaps not been found. In addition to transcendental philosophy there are two pure rational sciences ... , pure mathematics and pure ethics. (A480!B508)

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I ask about the existence of a prior state (i.e., I ask about 3y BIy, x)). But then recalling the reasoning just completed - I confidently assert 3y BIy, x). Is this not a general procedure? So can I not say Vx3y BIy, x)? Precisely the outlawed infinity of formula (2). I've brought back the the dirty bath water with the baby! The answer in brief is that this bit of reasoning will not suffice to justify the existential claims in question. Look back at the passage at (A255/B272) I quoted from the Postulates in order to support my initial claim that Kant is a mild constructivist in general. Notice that it is causal information (an empirical causal law) that counts as a guarantee of existence here. My point: That sort of evidence can count, but evidence from pure reason can't count. What we have here is what Kant (in , the "Amphiboly") calls the "Transcendental Location of Concepts" (A268/B234), the association of concepts with specific faculties. I would interpret this as a classification of concepts according to the faculty which may be invoked to provide evidence for jndgments in which that concept occurs. The notion that I have symbolized by "B" is an empirical one. It requires empirical evidence, empirical intuitions, provided by the faculty of sensibility. To be sure the faculty of reason, does govern some concepts and judgments which have empirical import. It does for instance structure the arena for empirical searches 24 (In the jargon I employed above we can say that it does build the model structures which we use to depict our empirical evidential situations.) Thus is guides our empirical searches for actual objects. But that alone doesn't allow it to get inside and provide the intuitions or connections which ground that "actually." This, I think is the heart of Kant's famous "crit-

pretations can have no solid ground to stand on. This accusation, in the present context, really has three prongs: It is addressed against

- ical" turn. 5. ANACHRONISM

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[13] If we do read Kant in this general assertabilist way, and if Dummett is right about the origins of Brouwer's intuitionism, then there certainly is a profound debt there. (Indeed, we would also have a theory about the origins of some of Hilbert's doctrines as well.) To be sure many details have to be filled in and questions answered before this becomes a full fledged theory of Kant. But let me close by addressing (briefly) just one of them. The question of anachronism, that I promised at the start. The point of this accusation is that Brouwer was writing in a scientific climate so foreign to Kant that, as history, my proposed Brouwerian inter-

(al My use of formal langnages (like L T ) to express empirical and philosophical claims for Kant. (b) My use in particular of a full first order language with its quantifiers and multiple quantifiers; and (c) The general assertabilist premise as a reading of Kant's notion of truth. [l4] I'll speak to these in order. Regarding the first, clearly there is no issue here of LT as a formal syntax. That device is as benign as using English to 'describe Kant's views - so long as we don't falsely ascribe any implicit meanings or attitudes hidden in the formalism. The real question is the autonomy of logic as a topic neutral science. It is ultimately to emphasize that notion that formal languages came into their own. Two comments are in order. First, this question of autonomy was itself a matter of heated dispute between Brouwer and Hilbert themselves. Hilbert was quite taken with the autonomy of logic. Brouwer denied the autonomy of logic, and as a consequence he disparaged the widespread formalization of mathematical theories. But this Brouwerian view has not deterred his students; and the formal study of intuitionism is the order of the day, among intuitionists and nonintuitionists alike. And so at the very least we might say that whatever success these methods have in expressing Brouwer's ideas they can have for Kant as well. But in fact (and this is my second comment), on this dispute about the autonomy of logic Kant stands closer to Hilbert. Kant was perfectly at home with the idea of a theory of judgments which is topic neutral. The unschematized categories and his whole notion of general logic are two levels of abstraction which fit into this general way of talking. Kant wasn't as clear on this as Russell and Hilbert, but there is no great distortion in using their devices to illuminate his thinking. [15] Tuming to the second accusation: Really the force behind this accusation is Russell. And really the force behind Russell's objection is his conception of infinity. Russell was very much concerned with the question of how our finite powers of thought and imagination can properly grasp the mathematical infinite. He came to believe that the V3 quantifier combination does this job, and captures the notion of infinity in a way that Kant (who had no quantifiers) could never achieve.

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Russell, I think, was quite right that his work does crystallize a notion of infinity which is thoroughly unKantian, a notion based ultimately on the idea of an arbitrary (perhaps even undescribable) function, This is essentially a set theoretic notion, and it should be contrasted with the idea which underlies Kant's conception of infinity. That is the idea of a continued iteration of some simple process, like describing or dividing a line segment, or like generating a sequence of events and their predecessors. But Russell was wrong in his implication that this is an illegitimate (or contradictory) notion of infinity, It is in fact Brouwer's notion of infinity, (It was captured at first in his claims about the "primary intuition of the continuum," and ultimately refined in his theory of infinitely proceeding sequences.)25 Brouwer and Kant were concerned with precisely the same question as Russell, how can a finite intelligence deal meaningfully with the infinite. And they came up with the idea of a sequential process underlying both temporal (and for Kant at least) spacial infinity.26 Moreover, Brouwer and some of his followers have shown that many parts of mathematics can be built on this more dynamic conception of infinity. And Russell was also wrong in thinking that his static infinite has sale claim on the \13 quantifier sign design. To whatever extent we succeed in formalizing intuitionistic mathematics using the machinery of quantified logic to that extent we have expressed the Kantian' infinity in a first order language. If this is a reconstruction (or classical interpretation), then again if it works for Brouwer, it can work for Kant as welL Another way to put this: Brouwer differed with Hilbert not only on the autonomy of logic and language, but also on the notion of infinity. Hilbert did hold a more Russellian notion of infinity. Kant, we saw, looks more like a proto-Hilbertian on the first question. But on the second question, the dynamic vs. the static conception of infinity, Kant seems to be a forerunner of Brouwer. This alone, however, will not lead him to chuck classical logic for mathematics; because, we have seen, of his Hilbert-like optimism. [16] Finally, on the question of assertabilism for Kant, let me say first of all that it is no more anachronistic to attribute an assertability theory of truth to Kant than a referential theory. Both the correspondence and the assertabilist notions of truth that we have today are expressed in terms unknown to Kant. Insofar as our project is to recreate Kant's actual state of mind, we might want to say that he was struggling (sometimes successfully, sometimes inexpertly) to express a view which we today have more clearly stated. Kant clearly thought that TI had "logical" upshots. And the best way to generate these sorts of logical consequences is by means of what we today would call a theory of truth. There is no anachronism in assuming that he had an inkling

of connections and consequences which we today can establish with formal rigor. But I myself prefer a less temporally chauvinistic attitude. The fact is that we are still concerned with many of the same questions that engaged Kant, in particular the questions surrounding infinity. But we are still far from the goal of full understanding. We don't yet have a satisfactory idealistic theory of truth, or even a fully satisfactory understanding of quantification. And so it seems to me that we can't help but profit from testing our ideas on Kant's problems.

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NOTES I See for instance Brouwer's remarks in Chapter 11 of his dissertation. Over de grondslagen der wiskunde. (Translated on pages 11-101 in L E. J. Brouwer, Collected Works v.I. North Holland,

Amsterdam, 1975. Hereafter I will make page references to this anthology when citing Brouwer's works. I will abbreviate it CW.) See also Brouwer's suggestion in "Intuitionism and Formalism" (1913, CW page 123) that his own position accepts Kant's views on the a priority of time, and rejects Kant's notions about space. 2 I have discussed these matters a bit more extensively in "Dancing to the Antinomy", "Transcendental Idealism and Causality," and "The Language of Appearances and Things in Themselves". "Dancing to the Antinomy" American Philosophical Quarterly, vol. 20, (1983), pages 81-94. 3 J. Bennett, Kant's Dialectic, Cambridge University Press, London, 1974, section 40. 4 B. Russell, Principles of Mathematics, Norton. See especially section 435. 5 It is tempting to confuse this notion of regulativity with Kant's notion of continuation in indeftnitum (A51O-11/B538-9). I fell into this mistake in "Dancing to the Antinomy" as did Bennet in Kant's Dialectic section 46. But the in infinitum/in indefinitum distinction concerns only the initial conditions on a given regressive series. Thus for instance the series of divisions discussed in the second antinomy is given regulatively, but nevertheless is continuable in

infinitum. 6 See my 'l'ranscendental Idealism and Causality" in Kant on Causality, Freedom and Objectivity, edited by R. Meerbote and W. Harper. University of Minnesota Press, Minneapolis, 1984. I have discussed some of these notions in ''Transcendental Idealism and Causality". 8 This is especially prominent in the A-Deduction, A 103 ff. 7

To know anything in space (for instance a line) I must draw it. and thus synthetically bring into being a detenninate combination of the given manifold.... Actually this is a theme that goes back in Kant's writing at least as far as the Prize Essay of 1763: A cone may signify elsewhere what it will: in mathematics it originates from the arbitrary representation of a right angled triangle rotated on one of its sides. The explanation obviously originates here, and in all other cases through synthesis. (AK II, 296)

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\0 See-C. Parsons "Mathematical Intuition", Proceedings of the Aristotelian Society, vol. 80 (1979-80), pages 145-68.

22 I have treated this in more detail in ''The Language of Appearances and Things in Themselves," Synthese, vol. 47, (19S1), pages 313-352. 23 "For the natural appearances are objects which are given to us independently of our concepts, and the key to them lies not in us and our pure thinking, but outside us; and therefore in many cases, since the key is not to be found, an assured solution is not to be expected."

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... there are iterative complexes of sensations whose elements are permutable in point of time. Some of them are completely estranged from the subject. They are called things. For instance individuals, i.e., human bodies, the home body of the subject included. are things, ... Mathematics wmes into being when the two-ity created by a move of time is divested of all quality by the subject, and when the remaining empty form of the common sub-stratum of all

two-ities, as basic intuition of mathematics, is left to an unlimited unfolding creating new mathematical entities .... ("Consciousness, Philosophy and Mathematics" 1948, CW, p. 480.) See also Brouwer's remarks about the construction of mathematical objects at the end of Chapter I in his dissertation. 12 That's because some constructions may leave some properties of the constructed objects eternally undecided. 13 In general _ under the minimal deviation from the truth tabular meanings - if A is truthvalueless so too are -A and thus -A. Now take A as (Pt V -Pt) with Pt undecided. 14

Thus from the perception of the attracted iron filings we know of the existence of a magnetic matter pervading all bodies, although the constitution of our organs cuts us off from all immediate perception of this medium ... The grossn~ss of our senses does not in any way decide the form of possible experience in general. (A226/B273) (See also A52-2/B550). 15 See Bxl(n), B70-71, B274 and the "Appendix" to the Prolegemena. 16 See D. Hilbert, "Mathematische Probleme",Arch, d. Math. u. Phys. (3), 1901; and "UberDas Unendliche". Math., Ann., (96), 1926. 17 Consider for instance Dwnmett's argument from language learning (i.e., from the premise that assertability conditions are inevitably the novice's first contact with meaningful complete sentences; see Truth and Other Enigmas, Harvard, University Press, Cambridge MA:, 1978, pages 217ff.) TIris argument certainly encompasses the Peircian assertabilism. For the novice can't escape the preponderance of deferred justifications in our everyday discourse. Quite similarly, Dwnmett's Wittgensteinian argument (as found in Truth and Other Enigmas pages 216-27 and Elements of Intuitionism, Oxford, 1977, pages 360-89) can be adapted to the Peircian case as well. The Peu'cian, like the Dwnmettian, ties "understanding" a sentence to a grasp of its assertability conditions, and thus does link understanding with actual behavior. 18 It might seem strange that the optimist should reject classical logic, and even stranger that he should do this while advocating the long semantic view. But you must recall that logic per se is linked with asserting behavior rather than with beliefs about the future. Under the strict constructivist view we cannot use these guarantees to assert as yet unwitnessed existentials or undecided

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disjunctions. 19 See for instance the criticism of Hilbert towards the end of Chapter ill in Brouwer's dissertation, and the similar criticism almost fifty years later in "Historical Background, Principles and Methods of Intuitionism" (1952), CW pages 50S-15. 20 See Ober Definitionsbereiche von Funktionen" (1927). CW pages 390-405, and the editor's note (4) (CW page 603) to "The non-equivalence of the constructive and the negative order relation on the continuum" (1949) CW pages 495-96. 21 See for instance "Intuitionistiche Betrachtungen iiber den Formalismus" (192S) CW pages

409-14.

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(A480-1/8508-9). 24 This is how I Wlderstand the Appendix to the "Dialectic", entitled ''The RegUlative

Employment of the Ideas of Pure Reason." See in particular A547-8TB674-6. 2S Compare the dissertation description of the "basic intuition" of continuity with the presentation of the continuum as a "spread" in subsequent writings, e.g., Begrundung der Mengenlehere unabhangig yom logischen San yom ausgeschlossenan Dritten, CW 150-221. 26 I w!luld distinguish here between notions like this iterative conception of infinity and notions which depend intrinsically on multiple quantifier shifts (e.g., the arithmetical hierarchy). The latter, I think, are wholly language bound, and do not correspond to any specific Kantian ideas. 27 Since the first appearance of this paper I have published four additional pieces which are relevant to the topics I have covered here. "Autonomy, Omniscience and the Ethical hnagination" (in Y. Yovel, ed. Kant's Practical Philosophy Reconsidered, Kluwer, 1989) explores the parallels in Kant's practical philosophy to the view that I have here called his "mathematical realism." "Where Have All the Objects GoneT' (The Southern Journal of Philosophy (1986) XXV, Supplement) touches on Kant's views about the nature of mathematical objects. This ontological theme as well as some phenomenological and semantic issues are taken up in "Mathematical as a Transcendental Science." (in D. FlZillesdal et. al. eds., Phenomenology and the Formal Sciences, University Presses of America, 1990). That paper also expands on the remarks in note (5) above concerning the "Second Antinomy." Finally, "Kant and ConceptuaJ Semantics" (Topoi, 10, 1991) qualifies my attribution to Kant of a modem assertabilism and considers his anti-realism in the context of his Leibnizian background.

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A great deal of excellent work on Kant's philosophy of mathematics has been done in the recent past, much of it facilitated by a deep knowledge of the traditional criticisms made of his philosophy and by contemporary developments in logic and set theory. In particular, I have in mind papers by Michael Friedman, Jaakko Hintikka, Charles Parsons, Carl Posy, Manley Thompson, and J. Michael Young.' Any new discussion of the topics they address should be able to presuppose some acquaintance with them. Their net effect has been to make Kant's position, in its general outlines, philosophically credible as well as historically interesting, perhaps especially as concerns arithmetic. Moreover, with the exception of Posy, whose commentary on Kant takes place against a background provided by Brouwer and the Dutch Inmitionists, there is a surprising degree of consensus among them. There remain important differences, of course, and a great deal of disagreement concerning the details. But in this paper I will be more interested in the similarities, and will in any case spend an inadequate amount of time on the details. My approach will be to isolate one or two of the main problems that Kant's philosophy of mathematics is designed to solve, relate them to a central theme, that there are deep difficulties with the "inferenrial" interpretation shared by almost all of the commentators mentioned, and draw more attention than is usual to his various remarks about algebra These remarks serve to clarify the concept of "consttuctibility" on which his position turns, and to underline some of the difficulties to which its analysis leads. They also serve to undennine the twin tendencies among most of the above-mentioned commentators to ascribe "symbolic consttuctions" to arithmetic, identifying arithmetic with algebra in the process, and to logic (variously understood), assimilating logic to mathematics. Both of these tendencies, as I hope to show, are mistaken. In my view, Kant is working his way to a rather modern and abstract conception of algebra as consisting of sets on which certain iterable operations are defined, a fact that demonstrates his originality and insight while it explains his obscurity.2 315 Carll. Posy (ed.), Kant's Philosophy o/Mathematics, 315-339. rr"l laO? KIr""'Ar-

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There are several themes running through all of Kant's work. A main one is that a merely conceptual determination of the objects of experience is never adequate; what he calls "intuitions" (Anschauungen) are also required. That the objects of our experience are determinable in a variety of ways is a fact. That they must be determined in certain of these ways follows from the argument of the Transcendental Deduction: application of the concept of the self as subject of experience necessitates, among other things, that the objects of that experience are spatial, enduring, and causally connected. Since such determination of objects is both "intuitive" and necessary, it is, in Kant's own vocabulary, synthetic a priori. What Kant adds by way of explanation is that this is possible only if the determination involves both a passive content, contributed "by the world," and an active form, contributed "by us." Thus the first half of the Critique of Pure Reason. In the second half, Kant shows why and in what ways the adequate determination of other types of so-called "transcendent" objects is not possible, and hence that we cannot have anything amounting to knowledge of them. Now as I understand Kant's strategy in this connection, it is to begin with what might appear to be the most problematic case for his thesis, the case of mathematics. For the objects of mathemarics, in his view, are completely determined; we have knowledge about them, indeed a priori knowledge of the most secure kind. What he needs to show is that even in the case.of mathematics a merely conceptual determination is not complete. Even mathematics requires (sensible) intuitions for an adequate determination of its objects, hence even mathematics is synthetic (as well as a priorz). Once convinced of this, we should more readily accept the general thesis that there is no (adequate) determination without (sensible) intuitions. Let me try to spel! this theme out in more detail, adding qualifications where important. First, Kant often talks about the determination of objects. But he also, and apparently interchangeably, talks about the verification and falsification of propositions. Thus, to determine an object (or type of object) a with respect to a property P is simply to verify or falsify the proposition "a is P." We can always replace talk about objects and properties with talk about the trnthvalues of propositions. This is iruportant for the following reason. It is unclear whether Kant has a view concerning the role and status of "mathematical objects" per se, still less whether they exist in any sense of the word. 3 It is arguably the case that he thinks mathematical propositions are true of concrete and not merely abstract objects, if they are trne at aiL But he does think

mathematical propositions are trne and false and hence in this minimai sense we can talk about "mathematical objects" and their determination. To talk in this guarded way, it should be clear, is not to quantify over such "objects:" With this qualification well in mind, I will continue to talk, as does Kant, about "mathematical objects" (numbers, plane figures, et. al.) and their determination; among other things, it allows for their convenient comparison with "empirical objects" and ''transcendent objects." Second, Kant claims that every well-formed mathematical proposition is decidable; in this sense, all "mathematical objects" can be completely determined. Mathematics, he says, may ... demand and expect none but assured answers to all the questions within its domain, although up to the present they have perhaps not been found (CPR, A480JB508).

So far as I can see, Kant gives no argument for this claim and we, chastened by the Godel results, know that it is false. Yet it seems to figure as an important premise in his argument that methematical determination requires intuitions. I am not at all sure why he thought that it was trne (over and above the fact that everyone up to the present century thought so). Perhaps he believed that it follows from the doctrine of the spontaneity of ''pure intuitions" that we can always actively geuerate the ''pure'' intuitions required· for the further determination of mathematical objects, and do not have to wait passively for the "empirical" intuitions which might otherwise be unavail5 able, but this use of the doctrine, with a qualification to be noted in the third paragraph hence, would seem to have no support other than the claim that every well-formed mathematical proposition is decidable. Of course, Kant

invokes the doctrine of "pure intuition" for a number of reasons; but none of the other reasons, however good, guarantees the ready availability of the intuitions demanded. We use the metaphor of "maker's knowledge"6 in transcendental philosophy and pure ethics as well as in pure mathematics,7 to explain what has already been assumed, that every proposition within these particular domains is decidable. The situation with regard to mathematics in this respect contrasts sharply with the situation in the natoral sciences. In the explanation of natural appearances much must remain uncertain, and many questions

insoluble, because what we know of nature is by no means sufficient, in all cases, to account for wh~lt has to be explained (CPR, A477/B505).

The, surrounding context implies that what is at stake here is the receptivity of the "empitical" intuitions on the basis of which empirical objects are determined. Just as the spontaneity of "pure" intuitions guarantees the decidability

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of mathematical propositions, so the receptivity of "empirical" intuitions, and the corollary that we must wait for "the world" to provide them, precludes our knowing in advance when and if they will occur. 8 At the same time, the complete determination of empirical objects, the objects of experience, is not required by the argument of the Transcendental Deduction, although it

remains an ideal of reason. But Kant also suggests a rather different line of argument for the contrast between mathematics and the natural sciences as concerns their respective decidability.9 Our experience of an object is never completely determinate

because to know a thing completely, we must know every possible [predicate], and must detennine it thereby, either affirmatively or negatively_ The complete determination is thus a concept. which, in its totality. can never be exhibited in concreto (CPR, A573/B610).

An empirical object can never be completely determined. The question is, why not? It is an aspect of Kant's general thesis that we can know in abstracto of any object that it has one or the other of two contradictory predicates or properties, but at least in a great number of cases we can kuow which of these two it has in fact only when the object is given in concreto, in intuition. Only then can we decide, on the basis of an "empirical" intuition, whether it is P or not-Po Now according to Kant, an object given in concreto is always given as a spatial-temporal particular. The further determination of it is thus always with respect to the forms of space and time. But then an object cannot be detennined with respect to every possible predicate, because at least some predicates (in Thompson's example, having an immortal soul) do not apply to spatial-temporal particulars. A completely determinate object would be possible only if some of our intuitions were "intellectual," but again according to Kant, such intuitions are, at least for us humans, impossible. iO This line of argument suggests a way in which Kant's claim concetuing the decidability of mathematical propositions might be supported. If we held that the incompleteness of empirical objects resulted solely from the fact that some among their possible properties could not be had by spatial-temporal particulars, then mathematical objects might be held to be completely detenninate in the sense that all of their possible properties were, appropriately, spatial temporal.ll This suggestion receives some small additional support from the fact that in the passage quoted earlier at A480/B508, Kant is careful to say that pure mathematics can answer all questions within its domain; within its domain, possible properties do not outrun those which in principle it is possible to ascribe on the basis of spatial-temporal intuitions, "pure" or otherwise.

Having clarified, if not also secured, its first premise, we can now outline Kant's master argument for the claim that the determination of mathematical objects (the decidability of mathematical propositions) requires intuitions. 1. Mathematical objects are completely detenninable (the decidability thesis).12 2. Concepts do not completely detennine mathematical objects (the conceptual under-detennination thesis).l3 3. Either concepts or intuitions determine objects (the framework thesis). 4. Therefore, intuitions are required to complete the detennination of mathematical objects. Of course, Kant does not quite leave it at this. He makes two important additions to the argument, both having to do with the elaboration of its third premise. First, there is a subsidiary argument that goes roughly as follows: (i) All intuitions are either sensible or intellectual (spatial-temporal arnot). (ii) But we humans (i.e., beings endowed with our perceptual abilities and conceptual capacities) are not capable of intellectual intuitions. (iii) Therefore, all our intuitions are sensible. (iv) Therefore, the intuitions required to complete the determination of mathematical objects are sensible. This argument, set out in the Transcendental Aesthetic, has nothing specifically to do with mathematics, although it compels Kant to provide an elaboration of its corollary. Thus, second, although it is one of Kant's framework principles that all representations are either concepts or intuitions, and hence that there is no other way in which objects can be determined, he also thinks that one must provide an explanation of just how sensible intuitions, in the cases of arithmetic and geometry, are required, of just how they serve to complete the determination of matllematical objects. We might call this the "intuitivity thesis." It is that sensible intuitions are required, in ways that can be spelled oul in detail, to determine mathematical objects. 'JYpically it is supported by adducing paradigmatic examples. There are two separate parts to Kant's case. On the one hand, the conceptual under-determination thesis mnst be established. On the other hand, the intuitivity thesis, and the kinds of explanations it involves, must be established. Any account of Kant's position which fails to do both is itself seriously incomplete, although it might be

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claimed (as certain commentators do implicitly) that either suffices to estab.lish the conclnsion of the master atgument, and the synthetic chatacter of

denied. IS Systems of non-Euclidean geometry ate logically consistent. But if the truth of the axioms of Euclidean geometry cannot be determined conceptually, then, on the assumptions that they ate true and that the concept/intuition distinction is exhaustive, an appeal to intuition must be made to detennine their truth. Intuition alone serves to pick out one atnong the various self-consistent geometries. There ate two further variations on this line of interpretation that should be mentioned. On one, it is pointed out that the axioms or first principles of a given mathematical theory typically under-detennine its objects. Consider the axioms for an elementary Euclidean geometry.19 These axioms chatacterize sets of objects, the various set-theoretical structures that satisfy them. But the axioms do not succeed in completely characterizing these sets; atnong other things, the sets that satisfy the axioms most closely resembling those that Kant had in mind differ in catdinality. But again, if the axioms do not succeed in completely characterizing the mathematical objects to which they apply, and all mathematical objects can be characterized completely, then appeal must be made to intuition. This variation has a straightforwatd application to arithmetic as well. We can take the Peano Axioms as characterizing the basic elements of arithmetic, numbers. That they under-detennine them, in the appropriate sense, follows from the well-known fact that very differeni set-theoretical structures satisfy them. On the other variation on this line of interpretation, it is pointed out that on Euclid's fonnulation of them, the axioms for elementary geometry ate incomplete; certain theorems cannot be proved on their basis. In the case of arithmetic, an even stronger result can be demonstrated: elementary number theory is essentially incomplete in the sense that not all true mathematical propositions can be proved using a fiuite set of axioms (and finitary methods ofproof).2o There is much to be said for this line of interpretation, on either of its variations, patticulatly from a contemporary point of view. There is a cleat sense in which Euclid's axioms for geometry don't, and any set of axioms for arithmetic can't, completely determine the objects within their domains. On the assumption of the decidability and framework theses, appeal to intuition must be made. But there is also a great deal to be said against this line of interpretation.21 In the first place, Kant does not have in mind anything like the modem picture of mathematical theories as deductive systems, theorems derived from well-defined sets of axioms. Although he does refer to the axioms (more

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It is not possible to survey all of the attempts to establish one or the other of the conceptual under-detennination and intuitivity theses. I will look at two already well established in the literature, more to clarify the theses and the sense of "detennination" at stake than anything else, and the "calculational" variation on one of them. We should then be in a position to exatnine Kant's various rematks on algebra and to detennine their bearing. We can begin by effecting a preliminary classification of attempts to establish the two theses on the assumption that, for Kant, mathematical proposi15 tions can be decided on the basis of something like a deductive atgument. Eventually this assumption will be given up. Thus some commentators focus attention on the premises of such arguments - the axioms, basic propositions, or principles of arithmetic or geometry.16 Other commentators focus instead on the logic invoked to derive their conciusionsP On one line of interpretation, the axioms, basic propositions or principles do not serve to detennine completely the objects which they chatacterize. On the other line of interpretation, inferences from the~e as premises. using the monadic or Aristotelian quantification theory available to Kant, ate not capable of deciding all mathematical propositions within their domain, although using general quantification theory or other specifically mathematical fonns of reasoning would. On both, further detennination or decision requires intuitions (in senses of that tenn that have not yet been specified). The first line of interpretation is suggested most directly by the following well-known passage: For as it was found that all mat.i.ematical inferences proceed in accordance with the principle of contradiction (which the nature of all apodeictic certainty requires), it was supposed that the fundamental propositions [Grundsatzel could also be recognized from that principle. This is erroneouS. For a synthetic proposition can indeed be comprehended in accordance with the principle of contradiction, but only if another synthetic proposition is presupposed from which it can be derived, and never in itself (CPR, B14).

The appatent implication is that those (Leibniz and his followers) who correctly saw that the inference from axioms to theorems is logically valid, mistakeuly held that the axioms themselves ate conceptual truths. That they ate not conceptual truths follows from the fact that they can be consistently

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precisely, the postulates) of Euclidean geometry, he expressly denies that arithmetic has axioms,22 and suggests rather (as well shall see shortly) that arithmetical propositions are established on the basis of calculation, not argumentation. Nor does he make a sharp distinction between the syntactically characterized axioms and the set-theoretical structures that satisfy them on which this line of interpretation depends. This is to say that for all of its independent philosophical plausibility, this line of interpretation is seriously anachronistic. In the second place, and much more serious, the appeal to intuition on this line of interpretation is completely vacuous. It is part of Kant's master argument that an explanation be provided of precisely how intuitions further determine the basic objects of mathematics, numbers for example, and plane figures. But there is no such explanation forthcoming on this line. To precisely what is appeal made in picking out, e.g., elementaty Euclidean geometry from the set of logically consistent geometries? There is nothing in our visual experience, still less in the character of drawn figures, that would allow us to discriminate between them. The difficulty is even more marked in the case of arithmetic: how does "intuition" (whatever it ntight be) serve to decide for or against particular set-theoretical constructions of the natural numbers? Thus, while a good, albeit seriously anachronistic argument can be made on this first line of interpretation for the conceptual under-determination thesis, it does not provide us with the materials for understanding, let alone the reasons for accepting, the intuitivity thesis. It remains, indeed, something of a mystery on this line of interpretation how the further requisite deterntination of mathematical objects is to be carried out. The other interpretation focusses not on the axioms and first principles, but on the inferential procedures used. There are, in fact, a number of passages in which Kant emphasizes the character of the reasoning used in mathematics and connects it directly to the synthetic a priori status of mathematical propositions. For example, in the Critique of Pure Reason at B744-745:

line parallel to the opposite side of the triangle. and observes that he has thus obtained an external adjacent angle which is equal to an internal angle - and so 00. In this fashion, through a chain of inferences guided throughous by intuition. he arrives at a solution of the problem that is simultaneously fully evident and general (my italics).

Suppose a philosopher be given the concept of a triangle and he be left to find out, in his own way, what relation the sum of its angles bears to a right angle. He has nothing but the concept of a figure enclosed by three straight lines, along with the concept of just as many angles. However long he meditates on these concepts, he will never produce anything new. He can analyse and clarify the concept of a straight line or of an angle or of the number three, but he can never arrive at any properties not already contained in these concepts. Now let the geometer take up the question. He at once begins by constructing a triangle. Since he knows that the sum of two right angles is exactly equal to the sum of all the adjacent angles which can be constructed from a single point on a straight line, he prolongs one side of the triangle and obtains two adjacent angles which together equal two right angles. He then divides the external angle by drawing a

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Here it seems explicit that it is the inference, and not the axioms, that requires intuition. 23 Now what is it about mathematical reasoning that is intuitive? This line of interpretation proceeds somewhat as follows. 24 Kant's conception of "logic" was thoroughly Aristotelian, which is to say approximately eqnivalent to monadic quantitification theory. Now the striking thing about this theory is that monadic formulas always have finite realizations Or models, a fact which is closely linked with their decidability. Polyadic formulas, on the other hand, often have only denumerable models, as, for example, when existential are dependent on universal quantifiers, as in (x)(EyJ FxY" But mathematical reasoning involves, in a variety of ways, the introduction of an unlintited (or infinite) number of new individuals, in guaranteeing the closure of the basic arithmetical operations, for example, or in proofs which require the density (or continuity) of the Euclidean straight line. Io its essentially "finite" character, monadic quantification theory does not harbor the resources to prove all of the mathematical theorems that polyadic quantification theory does. Identifying conceptual determination with monadic provability, Kant saw that many theorems reqnired, in a sense to be indicated momentarily, an appeal to intuition. TIris is to put a complex view too simply. But the main point should be clear: monadic quantificatioo theory, together with the various axioms and basic propositions of geometry and arithmetic, does not serve to determine completely the objects at stake, and this in large part because the determining procedures needed are not invariably finitaty. Polyadic quantification theory, on tltis line of interpretation, thus belongs to mathematics. What is particularly appealing about this line is that, unlike the first, it ties the conceptual under-determination thesis to the intuitivity thesis in a very plausible way. The existence of the requisite number of points cannot be demonstrated from the traditional axioms of Euclidean geometry if we have no more than monadic quantification theory at our disposal; monadic formulas cannot ''force'' models that have enough objects in them. We must tum, instead, to their "construction,"25 which in this case involves the continued bifurcation of a line segment originally given in concreto, in intuition, as a spatial o~ect. 26 Similarly, the determination of particular numbers, as the result of employing the basic arithmetical operations, is a calculational

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procedure which presupposes the continued iterability of these operations. This calculational procedure, in so far as it takes time, is in a fundamentally Kantian sense "intuitive." Monadic methods do not fully determine mathematical objects; their complete determination requires an appeal to spatial objects and temporal activities, i.e., to what is apprehended under the forms of intuition. There are, I believe, a great variety of things wrong with this interpretation, despite its considerable merits.27 But we should confine our attention to those that deal most directly with the theme of detennination.28

procedures involved in the second line of interpretation. This emphasis, quite apart from the monadic/polyadic distinction to which Friedman ties it (for reasons to be spelled out in the next paragraph) is worthy of closer examination, particularly since it brings out certain crucial features of mathematical reasoning and sets the stage for an exaruination of Kant's remarks on algebra which, in my view, are incompatible with it. There are three corollaries to the "calculational" interpretation of Kant's position. One, arithmetic is primary, and a better case can be made for its "intuitive" character than for geometry.32 Two, calculational procedures are in a fundamental sense "symbolic" and thus there is, from Kant's point of view, no reason to distinguish very sharply between arithmetic and algebra.33 Three, since the procedures of quantification theory, monadic or polyadic, are in generally this same sense "symbolic," there is, again from Kant's point of view, no reason to distinguish very sharply between "logic" and mathematics, despite his intentions to the contrary.34 All three of these corollaries are, in my view, mistaken, in large part because they misconstrue the role of algebra and the allied concept of a "symbolic construction" in Kant's thought. Such at least is one burden of this paper. To the extent that the calculational view implies them, it must be rejected, and with it the most promising way in which the second line of interpretation of Kant's position has been put. What exactly is the calculational view? Emphases vary among the commentators listed, but this is perhaps the core of the view.35 Mathematics is, Kant says, the science of quantity or magnitude. Maguitudes are determined, by way of a construction, through the application of calculational procedures. The case of arithmetic is paradigm. To determine some sum, for example, is to calculate it, to successively iterate the operation of addition, a unit at a time, until the sum is reached. The procedure, successfully performed, both guarantees the "existence" of the required sum and verifies that it is correct. The possibility that a given magnitude can be constructed is thus no more than the possibility that a given operation can be repeated, and that the rules governing it be followed. There are two "intuitive" aspects to this sort of activity. One is that it takes time; on this interpretation, the metaphor of "construction" is taken rather literally, the determination of maguitude is sometlting one does. Whether the result is to be taken as a "making" or a "finding," it involves a progressive enumeration that is temporal, hence intuitive. The other aspect is that traditional monadic logic does not have the conceptual resources to represent it. 36 To represent a progression. in fact, requires formulating a set of axioms for arithmetic or using set-theoretical concepts to define the natural numbers. Friedman emphasizes that mathematics requires, notably

In the first place, there is no reason in Kant's text (over and above the presumed correctuess of this line of interpretation) to think that he had some insight into the fact that any consistent monadic formula has a fiuite model; this result was first demonstrated, after all, in the 20th century. Nor is there any suggestion in the more mathematically sophisticated Leibniz, so far as I can see, that the use of "transfinite" methods goes hand in hand with the adoption of a polyadic notation; in common with a long tradition, he thought (incorrectly) that polyadic formulas were reducible to monadic ones. Nor, from a contemporary point of view, does Kant's presumed assimilation of polyadic quantification theory to mathematics do him much credit, for he assumes (as we have seen) that mathematics is generally decidable, whereas we know that polyadic quantification theory is not. This is obviously not intended as a criticism of Kant; the point is simply that if he had the kind of insight with which he is credited into the model-theoretic character of Aristotelian logic, then there is perhaps some reason to think that he would have understood that polyadic or general quantification theory is undecidable (since there is no effective way to generate counter-examples in models that contain at least a denumerable number of objects).29 In the second place, the distinction between monadic and polyadic quantification theory does not square very well with the general characterization Kant gives of "logic." There is, of course, no doubt that by "logic" he intended the traditional Aristotelian, essentially monadic, theory. But his characterization of logic as "empty," "the mere form of knowledge," etc.,30 includes (universally free) polyadic quantification theory plus identity as well. The latter theory is in the same precise sense "empty" in that its theorems hold in or of all models, including the model that contains no objects. 31 Although we can, trivially, prove a great many more propositions using a polyadic notation, and in this sense polyadic procedures, the objects that realize formulas in this notation are not more precisely "determined." These criticisms aside, I want to follow up the emphasis on calculational

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in the detennination of irrational magnitudes, the unlimited iteration of particular operations and the indefinite extendibility of the number series on which it depends. Now construction on the calculational view is "symbolic" in this sense.37 Numbers are represented by numerals, sensible tokens which are in this sense -themselves "intuitive." The numerals function as names or "symbolic constructions" of numbers, but they also, and more importantly, model the structure of the number series.

mentary, rejecting the majority tradition, also tends to suggest that a better case can be made for the synthetic a priori status of arithmetical than of geometrical propositions.

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In the system using Arabic nwnerals and base ten, for example, the nwneral '12' serves as a name of the number twelve, but the sequence of numerals from' 1' through' 12' also provides a model of what we take to be the structure of the corresponding numbers, since it has an initial element and a successor relation.... If we produce sensible tokens of the numerals' I' through '12' on paper or chalkboard, we have a model of what we take to be the structure of the nwnbers one through twelve. Since the perceptible tokens can be used merely as representations of an abstract structure, replaceable by any other instance of that structure, we can use them to verify them, ... , to verify arithmetical propositions.38

In fact, of course, we operate by and large with numerals when we construct magnitudes, particularly when the magnitudes are so large that we run out of fingers and toes, relying on familiar properties of the base ten system of representing nwnbers. This calculation with numerals, which are no more than symbolic constructions of numbers, does not simply abridge the otherwise laborious process of adding, subtracting, etc., numbers unit by unit; it affords a verification of its result because, once again, there is a fundamental isomorphism between the numerals and the structure of the number series. Arithmetic thus conceived consists not so much of a body of general truths or axioms as of calculational techniques, methods for finding magnitudes or solving equations, although these tecltniques rely on the general properties of both numbers and numerals already mentioned. Arithmetic is therefore prior to geometry in at least two senses. In the first place, it is more general. It has to do with the calculation of any magnitudes and does not have to do, in particular, with spatial (or, for that matter, temporal) objects. This generality derives from the fact that arithmetical constructions are purely symbolic, whereas geometrical constructions are invariably ostensive. In the second place, the iteration of what are basically arithmetic operations on which calculation depends is presupposed by geometry as well, for example in gnaranteeing both the indefinite extendibility and bifurcation of given line segments. Geometry, one might say, is applied arithmetic, although as already indicated, on the present line of interpretation no very sharp line can be drawn between pure and applied mathematics. Recent com-

4

As mentioned, I think that Kant's views on algebra tbrow a great deal of light on his philosophy of mathematics and his use of the concept of constructibility. I also think that his views on algebra are incompatible with certain aspects of the "calculational" interpretation of his position, although in other respects they support and elaborate it. I begin with a much-commented passage at A716/B744 of the Critique of Pure Reason: But mathematics does not only construct magnitudes (quanta) as in geometry; it also constructs magnitude as such (quantitas), as in algebra. In this it abstracts completely from the properties of the object that is to be thought in tenns of such a concept of magnitude. It then chooses a certain notation for all constructions of magnitude as such (numbers). that is, for addition, subtraction, extraction of roots, etc, Once it has adopted a notation for the general concept of magnitudes so far as their different relations are com:erned, it exhibits in intuition, in accordance with certain universal rules, all the various operations through which the magnitudes are produced and modified. When, for instance, one magnitude is to ~ divided -by another, their symbols are placed together in accordance with the sign for division, and similarly in other processes; and thus in algebra, by means of a symbolic construction, just as in geometry by means of an ostensive construction (the geometrical construction of the objects themselves), we succeed in arriving at results which discursive knowledge could never have reached by means of mere concepts (my italics).

This passage is problematic in several respects. But it is a key to understanding Kant's position. As is invariably the case with Kant, it turns on distinctions, here between magnitudes (quanta) and magnitude as such (quantitas) and between ostensive and symbolic constructions. The distinctions are clearly and quite closely related. The paradigm of an ostensive construction, at least in this passage, is provided by geometry, where the postulates and definitions afford the "synthetic" means to carry out the construction of particular figures which as such are "ostended" and the eventual objects of (visual) perception. But it is important to note that for Kant arithmetic as well is "ostensive."39 As much is indicated by the passages at B 15-16 and elsewhere that indicate how numbers can be "ostended" by fingers, strokes on a page, and so on, all of them spatial representations. In both cases, arithmetical as well as geometrical, the ostension is taken to provide us with a kind of evidence on the basis of which the

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corresponding claim (that 7 + 5 = 12, that the sum of the interior angles of a triangle is equal to two right angles) may be verified. The ostension, we might say, controls the claim.40 Now it is also true that Kant distinguishes between arithmetic and general arithmetic and couples the latter with algebra.41 The evident parallel between the two types of arithmetic is that they involve the same combinatorial operations. 42 The difference between them, as Kant puts it in #2 of the First Reflection of the Enquiry Concerning the Principles of Natural Theology and Ethics is that general arithmetic has to do with indeterminate magnitudes and arithmetic proper with numbers ("where the ratio of the magnitude to unity is determinate"). Michael Friedman suggests that "the arithmetic of numbers is concerned only with rational magnitudes, whereas general arithmetic or algebra is also concerned with irrational or incommensurable magnitudes."43 This is fine as far as it goes, but it does not go far enough. First, the reason why arithmetic proper is concerned only with rational magnitudes is that only rational magnitudes can, in the appropriate arithmetical sense, be "ostended." There are no determinate magnitudes, no homogeneous combinations of ostensive representations of numbers, corresponding to the irrationals. Second, general arithmetic has to do not simply with the "construction" of numbers, whether rational or not, but with "magnitude" per se, including the definition of complex curves. Kant has a much more general concept of the "indeterminate" than Friedman allows. For him, algebra is a logic of types, not of individuals, and applies as much to geometry as to arithmetic.44 Third, all "determinate" mathematical objects are in certain respects qualitative, "... for instance, the difference between lines and surfaces, as spaces of different quality, and with the continuity of extension as one of its qualities ..." (CPR, A715/B 743). But algebra considers these objects in their merely quantitative guise, as magnitudes as such, quantitas rather than quanta.45 Moreover, the qualitative/quantitative contrast which is here used to characterize algebra would seem to have little to do with the rational{rrrational contrast that Friedman wants to take as primary. But the most important and immediate point is simply this: arithmetic proper has an obviously "intuitive" foundation, ostensive constructions on the basis of which sample arithmetical claims may be verified, algebra or general arithmetic does not. 46 The subject matter, in the sense already introduced, "controls" the adequacy of its descriptions. In the case of algebra, on the other hand, the "subject matter" does not exist unless and until various combinatorial operations have been performed, i.e., the quantities involved are reached by way of an "analytical" characterization.47 Given that they cannot in general be "ostended,"

what other sorts of controls might there be on their admission? Or, in the absence of appropriate ostensions, do we simply dismiss them as "empty" and "unreal"?48 This is a problem for Kant, which the tendency to conflate arithmetic and algebra as part of the calculational interpretation simply sidesteps. There is another closely connected problem: given that algebra is, from one point of view, a system of rules (and not a set of determinate objects) licensing various combinatorial operations, what in tum licenses or legitiruizes the rules? These same points can be reinforced from a different direction and in an alternate vocabulary. Ostensive constructions can be identified with the use of "synthetic" methods, symbolic constructions can be identified with the use of "analytic" methods, as these terms were commonly understood in the geometrical tradition.49 The question then is this: does every analytic ("symbolic") construction have to be backed, as Descartes, for example, insisted, by a synthetic ("ostensive") proof, and, if not, what guarantees the adequacy of analytic constructions? Kant responds directly to the first of these questions in the controversy with Eberhard. The situation is straightforward with respect to the tradition of Greek geometry, he indicates, for on that tradition every proof given is synthetic and hence "constructive." Thus, one of Kant's favorite examples. . Apollonius first constructs the concept of a cone, i.e., he exhibits it a priori in intuition (this is the first operation by means of which the geometer presents in advance the objective reality of the concept). He cuts it according to a certain rule, e.g., parallel with a side of the triangle which cuts the base of the cone at right angles by its summit, and establishes a priori in intuition the attributes of the curved line produced by this cut on the surface of the cone. Thus, he extracts a concept of the relations in which its ordinates stand to the parameter, which concept, in this case, the parabola, is thereby given a priori in intuition. Consequently the objective reality of this concept, Le., the possibility of the thing with these properties, can be proved in no other way than by providing the corresponding intuition. 5o

But Kant is also very much aware that since the 16th century and the work of Vieta and Descartes, and particularly in the work of Newton, analytic methods had been generally adopted.51 So with this fact in mind, he continues: One could ... address to the modern geometers a reproach of the following nature: not that they derive the properties of a curved line without first being assured of the possibility of its object (for they are fully conscious of this together with the pure, merely schematic cOnStruction, and ~ they also bring in mechanical construction afterwards if it is necessary), but that they arbitrarily think for themselves such a line (e.g., the parabola through the fonnula ax = y2), and do not, according to the example of the ancient geometers, first bring it forth in a conic section. This would be more in accord with the elegance of geometry, an elegance in the name of which we are

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often advised not to completely forsake the synthetic method of the ancients for the analytic

have a special status. The point here, rather, seems to be that intuition is needed to verify the correct application of these rules, that these rules are used to construct maguitudes, and that therefore algebra, like geometry and arithmetic, is synthetic. 56 From this point of view, I think the case of algebra, much more clearly than the other two, demonstrates the fundamental correctness of the line of interpretation that emphasizes the "intuitive" character of the reasoning involved in mathematics, although it would be a tuistake, we have seen, to isolate its temporal character. Algebraic objects are conceptually under-determined in a much more obvious way than numbers or plane figures. It is the reasoning about them that is guided throughout by intuition,

method which is so rich in inventions. 52

With some Wlcertainty, I read this passage as follows. Contemporary geometers, despite their nse of analytic methods, are always in a position to demonstrate the objective reality (real possibility) of the curves with which they work. But this does not require that every analytic construction be backed by a synthetic proof. That is, it would be more "elegant" to bring it forth in a classically geometrical construction, and we should not "completely forsake" the synthetic method of the ancients, but these are not required by the possibility of 1:.1e Curve analytically characterized. Undoubtedly Kant was aware of the fact that many of the curves thus characterized could not be brought forth by the conic sections or any of the other constructive methods available to the Greeks, or to Descartes for that matter. How, then, is the "possibility" of such curves to be established? The passage originally quoted at A7l6/B744 of the first Critique is, I think, even more explicit that geometrical diagrams do not have to be supplied for algebraic results in its sharp separation between ostensive and symbolic constructions. For the requirement that every analytic demonstration be backed by a traditional synthetic proof just is the requirement that all constructions be ostensive. At the same time, Kant wants .to provide some sort of founda~ tion for these algebraic results. He does it by introducing the notion of a symbolic construction that is in some sense "intuitive" and therefore "synthetic," in the process going well beyond what Descartes, and a fortiori the Greeks, had meant by "construction,'" "intuit," and "synthetic."S3 What we "intuit" are

the operations performed on certain otherwise unspecified "objects" by way of their symbolic representation. It is just because the results of the combinatorial operations in algebra cannot invariably be ostended that the control is placed not on the exhibition of objects but on the carrying out of rulegoverned operations. This leaves questions concerning the combinatorial operations themselves,

or the rules which license them, still open. That is to say, the symbols are manipulated according to certain "universal rules" and, I take it, the application of these rules is guided, or "controlled," by our intuition of the symbols (in the paradigm case, marks on paper) manipulated.54 But it is never made clear what the status of these "rules" - for addition, subtraction, multiplication, division, and the extraction of roots - is. He denies that arithmetic or

algebra have axioms in severa! places,55 nor would it help to solve the question of status if the universal rules were "axioms" for Kant for he never really says why the Wldoubted axioms (read "postulates") of Euclidean geometry

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the reasoning being in some sense constitutive. For the rest, whatever else

might be said about them, the rules and combinatorial operations they define are, at least for Kant, clearly a priori, although how intuitions, pure or otherwise, might guarantee their status is, so far as I can see, never indicated. But all the master argument requires is that their application be "intuitive" in some general sense.

A great deal more needs to be said, of course, about the notion of a "symbolic construction." For the moment, however, I want to underline two of its very genera! implications. The first is that whereas in geometry and arithmetic the objects (numbers and figures) ostended "control" the resulting claims about them, in algebra the claims, in the form of the symbolic results of the various combinatorial operations we perform, "control" the objects. An object is adntissible on this latter tack if it is constructed in the right sort of way. We have no independent access to the object, no ostension to which we can appeal: from one point of view this is just the difference, Kant rightly saw, between a variable and a constant. The other, related implication is that Kant is working his way towards the concept of a relational structure. What a symbolic equation "pictures," so to speak, is a relation between magnitudes

(as such, and otherwise unspecified). The "objects" are simply whatever satisfies the relational structure. Thus "algebra abstracts completely from the properties of the object that is to be thought in terms of such a concept of magnitude." But to say this is to say that the problem is no longer the "determination" of particular mathematical objects, but the determination of general relational structures. Algebraic objects remain relatively indeterminate. 57 Relational structures, in turn, are "determined" by being "bnilt up" in certain ways, that is, by being "constructed." The construction makes use of symbols, sensible tokens, and presupposes the iteration of the basic operations; to this extent, it is "intuitive." But it is maiuly guided by the combinatorial rules of construction. In moving to a more modem conception of

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algebra, Kant strains his doctrines of "determination through iutuition" to the limit.58

great deal of controversy iu the 17th and 18th centuries concerniug algebra and its foundation. 59 The general tendency was to say that its procedures, barely systematized at the time, were merely "heuristic;" they could be justified ouly on the basis of the fact that they "worked." On my suggestion, Kant provides something like this answer, although he prepares the way to it much more carefully by showing why a reduction of it to either arithmetic or geometry would be iuappropriate. Here as elsewhere his position is thoroughly "anti-reductionist." Second, Kant appears to waver on a central issue,

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In fact, I think that iu moviug to a more modem conception of algebra, Kant was forced to go beyond the doctrine of "determiuation through intuition" as it is illustrated and understood in the cases of geometry and arithmetic. Although this is pure speculation on my part, I suggest that reflection on the character of algebraic methods, and iu particular on the "indeterminate" character of algebraic variables was one source of his central philosophical problem, how to secure the "objectivity" of at least some of our judgments about experience.

The specifically algebraic problem, as I reconstructed it, was to "determine" the relational structures to which application of the combinatorial rules gives rise, in the absence of ostensive coustructions to which appeal might otherwise be made. Kant's solution, certaiuly nowhere explicit iu the text, is that these relational structures are "determined" by way of a direct application to the objects of our experience, that is, the objects that have been fully conceptualized in accord with the requiremeU(s of the argnment of tJie Transcendental Deduction. Algebra and algebraic results cannot be justified piecemeal. Rather, they are to be justified in so far as they are generally true of the objects of experience. Or, to put it much too simply, what guarantees the objective reality (real possibility) is not ostensive construction; what guarantees it is its wholesale application, the fact that it gives empirically verifiable results, that it "work." Algebra, iu this respect uulike arithmetic and geometry, is legitituized iu practice. Kant generalized this presumed reflection very roughly as follows. Our scientific claims about the world constitute a relational structure or series of

such structures. The "objects" postulated or presupposed by the truth of these claims are iu this sense under-determined. Their future determiuation, which as we saw earlier is completable but never complete, is by way of a rather global fitting of this structure to our experience (itself made possible by the application of the Categories). Algebra gives us the outline of a form of the world which is validated by its empirical applications. Here are two small pieces of evidence, not for the generalization, which is admittedly vague and intended as no more than suggestive, but for the claim about algebra. that demonstrating the objective reality of its "objects" requires the machinery of the transcendental philosophy. First, there was a

whether the mathematician can or cannot demonstrate the "real possibility"

of the objects with which he or she deals. In the Critique of Pure Reason, A223-4/B271, Kant writes: It does, indeed. seems as if the possibility of a triangle could be known from its'concept in and by itself ... , for we can, as a matter of fact, give it an object completely a priori, that is, we can construct it. But since this is only the fann of an object, it would remain a mere product of the imagination, and the possibility of its object would still be doubtful. To detennine its .possibility, something more is required, namely, that such a figure be thought under no conditions save those

upon which all objects of experience rest.

But iu a passage already quoted from his polemic against Eberhard, iu the course of which he announces that he wants to clarify his earlier discussion of "construction of concepts" in the Critique, K;mt writes: Consequently, the objective reality of this concept [of a parabola} Le., the possibility of a thing with these properties, can be proven in no other way than by providing the corresponding intuition ... ,60

a view very much reiuforced iu his letter to Reiuhold of May 19, 1789. The second passage suggests that the mathematician can demonstrate real possibility, by way of an ostensive construction of his or her concepts, while the first passage appears to deny it. I am not sure how these passages, and others like them, are to be reconciled. But it might be the case that in the former Kant has mathematics as generalized to iuclude algebra in mind. For algebra, unlike arithmetic and geometry, gives us the mere form of an object, quantity without quality. Quality, iu the form of experience, is hence needed to secure "reality." In the latter passage, Kant has geometry or arithmetic proper iu mind, for in these cases ostensive construction already includes the sort of possibility of perceptual experience that assures the "real possibility" of their objects. There is, however, no direct textual evidence for this reconciliation, and the passage at A223-4/B271 specifically mentions an ostensively constructible triangle. Fiually, despite the similar abstractness of the two discipliues, and the

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same sort of emphasis on combinatorial and transformational rules, algebra should be distinguished from "logic." For algebra has the kind of empirically verifiable applications to the objects of experience that logic, at least as Kant conceives it, can never have. 6J

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I See Friedman, "Kant's Theory of Geometry," reprinted in this volume, pp. 177-219, and "Kant on Concepts and Intuitions in the Mathematical Sciences," Synthese 84 (1990), pp. 213-257; Hintikka, the various essays on Kant reprinted in Logic. Language-Games, and Information (Clarendon Press, 1973) and Knowledge and the Known (Reidel, 1974), "On Kant's Notion of Intuition," in Penelhum and Macintosh. eds.• The First Critique (Wadsworth. 1969), "Kant's Theory of Mathematics Revisited," in Mohanty and Shehan, eds." Essays on Kant's Critique of Pure Reason (University of Oklahoma Press, 1982), and "Kant's Transcendental Method and His Theory of Mathematics." in this volume pp. 341-359; Parsons, "Kant's Philosophy of Arithmetic," reprinted with a Postscript in this volume pp. 43-79 and "Arithmetic and the Categories," in this volume pp. 135-158; Posy, "Kant's Mathematical Realism.," in this volume pp. 293-313 and "Mathematics as a Transcendental Science," in F~llesdal et. al. eds. Phenomenology and the Formal Sdences (University Presses of America. 1990); Thompson, "Singular Terms and Intuitions in Kant's Epistemology," this volume, pp. 81-107; Young, "Kant on the Construction of Arithmetical Concepts," Kant-Studien 73 (1982), pp. 17 -46, and "Construction, Schematism, and Imagination," in this volume, pp.159-175. 2 For more on this historical development, and Kant's place within it, see my "Algebra. Constructibility, and the Indetenninate," in Brittan, ed., Causality, Method, and Modality: essays in honor of jules Vuillemin (Kluwer, 1991). pp. 99-123. 3 In a crucial, albeit very difficult passage from the Methodology in the first Critique, Kant says: " ... in mathematical problems there is no question of ... existence at all, but only of the properties of the objects in themselves, [that is to say], solely in so far as these properties are connected with the concepts of objects" (A719JB747). 4 In "Singular Terms and Intuitions in Kant's Epistemology," Thompson argues persuasively that we cannot quantify over them. 5 See Posy, "Kant's Mathematical Realism," p. 307 6 See Hintikka, "Practical vs. Theoretical Reason - An Ambiguous Legacy," reprinted in Knowledge and the Known. 7 See the first Critique, A480!B508. 8 See Posy, "Brittanic and Kantian Objects," in den Ouden, ed., New Essays on Kant (Peter Lang, 1987), pp. 29-46. Posy argues that the contrast between the definability of mathematical concepts and the Wldefinabi1ity of empirical concepts, on which Kant insists (CPR, A728!B756ff.), is itself a corollary of the spontaneity/receptivity contrast. 9 See Thompson, "Singular Terms and Intuitions in Kant's Epistemology," pp. 86ff., whose accoWlt I follow closely. 10 Objects which cannot be given in concreto, under the forms of space and time, might be

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called ''transcendent'' objects. Examples are, of course, God and the souL Although a minimal conceptual detennination of these objects is possible, by way of certain analytic propositions (e.g., "God is omnipotent"), they cannot be further detennined by either "pure" or "empirical" intuitions. An object which cannot in principle be further determined is Dot properly an object of knowledge; the real possibility of its concept cannot be demonstrated. There is a kind of contin~ uum here: mathematical objects are completely determined, empirical objects are adequately determined, transcendent objects are inadequately determined, where "adequately" means "so as to meet the requirements of the Transcendental Deduction." 11 I realize that the consequent of this conditional needs a great deal more clarification and support, but its import should be apparent. 12 Notice that this premise is. independent of the requirements of the arguments of the Transcendental Deduction that an object of experience be adequately determined It is primarily this fact, I think, that allows mathematics to be discussed'first, in the Aesthetic, before the considerations central to the Analytic are advanced. 13 In the passage immediately preceding that quoted in footnote 3 at CPR, A719!B747, Kant says: "There is indeed a transcendental synthesis framed from concepts alone, a synthesis with which the philosopher is alone competent to deal; but it relates only to a thing in general, as defining the conditions Wlder which the perception of it can belong to possible experience." I take this to mean that a complete conceptual determination (one sense of "synthesis") of a "thing in general" is possible, but presumably every other object, things in particular, are conceptually under-determined. Of course, Kant intends 2. to illustrate and support the more general thesis, and not the other way around. 14 In Kant's Theory of Science I implied, mistakenly, that the conceptual Wlder-determinatiQn thesis entails the intuitivity thesis. Perhaps the converse holds, however, a fact that would rationalize the otherwise excessive attention paid to the explanations Kant offers in the case of such examples as " + 5 = 12" and "the sum of the interior angles of a triangle is equal to two right angles." Puzzling over these examples, without first having some grip on a general strategy, is futile and worse. 15 For present purposes it is Wlimportant whether he had anything like the modern concept of a mathematical proof in mind or how he understands his problematical distinction between demonstration and discursive proofs (CPR, A735!B763ff.). 16 They include, in more recent times: LoW. Beck, "Can Kant's Synthetic Judgme-nts Be Made Analytic?" Kant-Studien, 47 (1955), pp. 166-81; Gordon Brittan, Kant's Theory of Science (Princeton University Press, 1978), chapters 2 and 3; Ernst Cassirer, "Kant und die modeme Mathematik," Kant-Studien, 12 (1970), pp. 1-40; Gottfried Martin, Kant's Metaphysics and Theory of Science, trans. Lucas (Manchester University Press, 1955). 17 They include: E. W. Beth, "Uber Locke's 'Allgemeines Dreieck'," Kanr-Studien, 48 (195657), pp. 361-80; Bertrand Russell, The Principles of Mathematics (Cambridge University Press, 1904), especially #434; and the papers by Friedman, Hintikka, Parsons (who endorses this view for arithmetic), and YOWlg mentioned in the first footnote. 18 See the CPR, B268: "there is no contradiction in the concept of a figure which is enclosed within two straight lines, since the concepts of two straight lines and of their coming together contains no negation of a figure." .19 See Alfred Tarski's paper, "What is Elementary Geometry?" in Henkin, Suppes, and Tarski, ·eds., The Axiomatic Method (North-Holland, 1959). Depending on [he precise formulation of "elementary geometry" chosen, of course, different metamathematical results can be proved. 20 On this variation, the Godel results, far from showing that Kant is wrong about the

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decidability of arithmetic, show that Kant is right (as reconstructed) in denying that truth in mathematics can be identified with provability in a fonnal system; their truth must rest on extraconceptual or ''perceptual'' considerations. GOdeI himself seems to urge something like this possibility in "What is Cantor's Continuum Problem," reprinted in Benaceraff and Putnam, ed5., Philosophy o/Mathematics: Selected Readings (Prentice-Hall, 1964). 21 All of it in the two articles by Friedman cited above. Despite my many reservations about his positive account of the issues involved, he has devastated attributing this sort of conceptual under-deterrnmation thesis to Kant. 22 First Critique, Al 64JB204. 23 Both Friedman, "Kant on Concepts and Intuitions in the Mathematical Sciences," and Hintikka, "Kant's Theory of Mathematics Revisited," try in interesting ways to reconcile this passage with that quoted earlier at B 14. Perhaps it is enough to regard these passages as supporting the conceptual under-detetmination and intuitivity theses respectively (the passage at Bl4 is clearly directed against Leibniz and the claim that mathematical objects can be conceptually determined, whereas the passage at B744-745 contrasts the "intuitive" methods of the mathematician with the "conceptual' methods of the philosopher) and that more in the way of reconciliation is not needed. 24 Interestingly, all of the recent work on Kant's philosophy of mathematics mentioned in the first paragraph of this paper makes contact at one point or another with this line (parsons seems to think that the first line of interpretation is more plausible in connection with geometry and the second in the case of arithmetic). All tend to identify construction, at least in arithmetic. with calculation. and all tend to assimilate the idea of calculation to "symbolic construction." I here follow Friedman in "Kant's Theory of Geometry." Hintikka begins in the same place. the disti~­ tion between monadic and polyadic quantification, but moves in a,rather different direction. 25 "To construct a concept means to exhibit a prior{ the intuition which corresponds to the concept," CPR, A713!B741. This characterization is notoriously problematic. It would seem to follow from it that concepts for which no intuition can be exhibited a priori are not constructible. But without any further qualification, the concepts (or, equivalently in the case of geometry, the curves) proscribed by this criterion would seem to be few in number. e.g., curves everywhere continuous, but nowhere differentiable. Everything turns, of course, on what it means to "exhibit a priori" an intuition corresponding to a concept. It turns out, on the present line of interpretation, that in the paradigm case this involves not so much drawing a line as calculating a number. 26 Euclid's actual procedure, given by Friedman. is somewhat more complicated. but the essential point is the same: the "existence" of an infinite number of points is guaranteed by the iterability of particular constructive procedures. 27 For one thing, Kant's text abounds with arguments which are expressly analytic and patently polyadic, e.g., at CPR, A766/B794. 28 The most detailed criticism of Friedman's use of the concept of infinity, and of his reading of the passage at CPR, B40 on which a case for that use is based, was made by Manley Thompson in comments at the American Philosophical Association Central Division Meeting in the spring of 1987. Hintikka·s position has been subjected to extensive criticism, perhaps in the most damaging way by Parsons, "Mathematical Intuition," Proceedings of the Aristotelian Society, N.S. 80 (1979-80), pp. 142-168,

"constructive" although it is not without many difficulties. See. for example, Parsons, "Infinity and Kant's Conception of the 'Possibility of Experience'," reprinted in Mathematics in Philosophy. 30 See the first Critique, A52/B76, A151/B19Off., and the opening paragraphs of his lectures on logic. 31 See R.K. Meyer and K. Lambert, "Universally Free Logic and Standard Quantification Theory," Journal a/Symbolic Logic, 33 (l968), pp. 8-26. 32 See Friedman, "Kant's Theory of Geometry," p. 202: "the case of arithmetic is primary." 33 See Thompson, "Singular Terms and Intuitions in Kant's Epistemology." p. 98: "A numeral is thus a symbolic construction of a number, and in arithmetic by further symbolic constructions from numerals (by calculation) we discover further numerical properties of numbers, just as in . geometry by further ostensive constructions from figures we discover further geometrical properties of figures." 34 See Parsons, "Kant's Philosophy of Arithmetic," p. 67: "The considerations about the role of symbolic operations in mathematics apply equally to logic and therefore undetmine Kant's apparent wish to distinguish them on this basis.," and Thompson, "Singular Terms and Intuitions in Kant's Epistemology," p. 106, n. 23: "Since general logic so conceived [as first-order quantification theory + identity] contains symbolic constructions and demonstrations, it would seems at least in this sense to be something Kant would have to regard as a branch of mathematics." 35 Following Friedman, "Kant on Concepts and Intuitions in the Mathematical Sciences," who in turn draws on Parsons. Thompson. and Young. 36 See Parsons, "Arithmetic and the Categories." p. 149: "finite iteration is an abstract coun~r­ part of the notion of successive repetition. But to describe it was quite beyond the logical and mathematical resources of Kant and his contemporaries; the task was first accomplished in the 1880's by Frege and Dedekind." 37 Following Young, "Construction, Schematism, and Imagination," although Young is careful to point out, as noted below, that for Kant the constructions on which arithmetical reasoning rests are in fact ostensive. 38 Ibid.• p. 124. 39 Friedman, "Kant on Concepts and Intuitions in the Malhematical Sciences," argues for including both arithmetic and algebra under the heading of symbolic construction. In so doing he seems to me to miss the point of Kant's distinctions. Young, "Kant on the Constructions of Arithmetical Concepts," thinks that it would be a proper extension of Kant's view to regard calculations as ~ymbolic constructions, for reasons already advanced. but he notes correctly that in the basic case arithmetical constructions are ostensive. Indeed, he finds it more plausible to regard our knowledge of arithmetical truths as resting on ostensive constructions than geomenical truths. for a collection of n things, in his view, can represent the number n in a way that allows us to use the collection to gain knowledge about the number, whereas the parallel claim about the geometrical constructions has little support. 40 We have to distinguish between the process of calculation and its product. In arithmetic, only the latter is ostended. Only because of the underlying isomorphism with the ostensive construction is the correctness of the symbolic construction with numerals assured. In this sense the numerals "represent" and do not simply "symbolize" as do algebraic variables. 4,' As in a letter to Johann Schultz of November 25, 1788. The letter is included in Amuif Zweig, trans. and ed. Kant: Philosophical Correspondence, 1759-1799 (University of Chicago Press. 1967).

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29 There is something misleading, in any case. about attributing the view that mathematics requires models containing at least a denumerable number of elements to Kant, for it i.s clearly his view that all numbers are finite. Numbers are determinate quantities and "a determinate yet infinite quantity is self-contradictory" (CPR, A521/B555). Kant's view of infinity is thoroughly

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42 It is possible that Kant identified arithmetic proper with the four basic combinatorial opera_ tions and distinguished it from general arithmetic or a1gebra in part on the basis of the fact that the latter includes the extraction of roots as well. As Michael Young has pointed out to me. the expectation of closure under the four basic operations would lead to the introduction of negative numbers, but not to the introduction of irrational numbers. Moreover, there is a sense in which negative numbers can be "constrUcted," as the difference between pairs of positive (and hence ostendibie) numbers, while L"Tational numbers cannot be "constructed" in the same way as ratios of pairs of positive numbers. This consideration would support Friedman's emphasis on the rationa1{llTIltional number distinction as the key to the difference between arithmetic and algebra. but, first, it is not in my view fundamental (the operation on variables is) and, second, I have not been able to determine whether Kant thinks extraction of roots does not belong to arithmetic proper. What could be a principled reason for its exclusion, particularly if one took arithmetical constructions as symbolic rather than ostensive? 43 "Kant on Concepts and Intuitions in the Mathematical Sciences." 44 Kant had a copy of Descartes' Geometry in his library and was presumably familiar with Descartes' views on algebra and analytic (as contrasted with "synthetic") geometry. An adequate description of the algebraic background of Kant's thought would also have to include Leibniz, whose work on algebra is extensive, and Euler, whose Anleitung zur Algebra (published in 1770) is widely regarded as the best algebra text of the century. Newton's views on algebra will be discussed shortly. 45 See the letter to K.L Reinhold of May 19, 1789: "The mathematician cannot make the least claim in regard to any object whatsoever without exhibiting it in intuition (or, if we are dealing merely with quantities without qualities. as in algebra, exhibiting the quantitative relationships thought under the chosen symbols) ... "In The Kant-Eberhard Controversy, trans. Allison (Johns Hopkins Press, 1973), p. 167. The intuition to which the arithmetician appeals is not, at least in the basic case, "symbolic." although it is. of course, "representative." 46 It is simply a fact, according to Kant, that these constructions show the "reality" of whole and rational numbers and their various rule-governed combinations. Perhaps it should be made explicit that it is only insofar as magnitudes also have qualities that they can be given an ostensive construction, only insofar as they have qualities that their "reality" be shown. Mere quantities can only be given a symbolic construction. 47 Note in the passage at CPR, A7161B744, that a symbolic construction "exhibits in intuition ... all the various operations through which the magnitudes are produced and modified" (my italics), whereas an ostensive construction exhibits an objecT corresponding to a concept. Note also as against those who emphasize the time-taking, hence "intuitive" character of arithmetical calculation that Kant here talks about representing the operations involved in algebraic manipulation and not carrying them out. 48 The mathematical tradition until- well into the 18th century (and in many cases beyond) rejected negative roots of quadratic equations as "false" and "unreal" simply because they could not be represented, although they could, of course, be symbolized. 49 In this tradition, algebra is identified with "analysis." Thus Fran~ois Viera (1540-1603), the "father" of modem algebra, entitled his work Introduction to the Analytical Art. 50 The Kant-Eberhard Controversy, p. 110. In an important gloss on this passage in his letter to Reinhold of May 19, 1789, Kant reinforces the point: if Eberhard understood the example, he would see "that the definition which Apollonious gives, e.g., of a parabola, is itself the exhibition of a concept in intuition, namely the intersection of a cone under certain conditions, and in establishing the objective reality of the concept, that the definition here, as always in geometry, is at

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the same time the construction of the concept." (Ibid., p. 168). He adds that the actual drawing of the parabola, given the parameter, follows as a practical corollary and has nothing to do with the theoretical point at issue. 51 The case of Newton is typically complex and illustrates the lack of clarity concerning the relationship between geometry and algebra, synthetic and analytic methods, and the foundation for each, which in fact extends well into the 19th century. On the one hand, he says in his Universal Arithmetic (1707) with apparent approval that "the Modems advancing yet much further [than the plane, solid, and linear loci of the Greeks] have received into Geometry all lines that can be expressed by Equations," and we know that in his own work he made rather free-wheeling use of analytic methods. Yet on the other hand, in a letter to David Gregory (1661-1708), Newton remarks that "Algebra is the analysis of bunglers in arithmetic," and he continued to take the Greek mathematical tradition as his standard of rigor. Perhaps for reasons of the "elegance" which Kant mentions in the Eberhard controversy (see below), Newton composed the Principia according to the synthetic-geometrical method, apparently no more than assuming that the synthetic proofs he offered were adequate to reach the results that he had in fact arrived at analytically. See Morris Kline, Mathematics: The Loss of Certainty (Oxford University Press, 1980), chapterV. 52 The Kant-Eberhard Controversy, p. 111. 53 According to Kline, Mathematics: The Loss of Certainty, p. 125. Newton tried to ground algebra by arguing that "the letters in algebraic expressions stand for numbers and no one can doubt the certainty of arithmetic." What this means, apparently, is that algebraic results are simply generalized versions of arithmetical truths and in some sense reducible to them. It is noteworthy that Kant does not avail himself of this strategy, nor could he, on my interpretation, sin~ arithmetic proper affords ostensive constructions on the basis of which its truths are immediately verifiable. 54 See the Critique of Pure Reason, A7341B762. 55 Notably in the Axioms of Intuition, A1631B204. 56 More than anyone else, Gottfried Martin in Arithmetic and Combinatories: Kant and His Contemporaries, trans. Wubnig (Southern Illinois University Press, 1985) has stressed the role played by algebra in Kant's philosophy of mathematics and the ways in which it is "creative" and hence, in another sense of the word than we have used so far, "synthetic." Martin goes astray in insisting that Kant was the first to formulate certain "axioms" of arithmetic (such as the rule of commutation), but he was right to draw attention to the status of the rules used in combinatorial operations. 57 In which case Kant's "decidability" thesis needs to be reconstrued. - 58 Descartes held that algebraic "objects" were admissible just in so far as they could be given a geometrical (spatial) construction. Newton held that algebraic "objects" were admissible just in so far as they could be given an arithmetical (numerical) construction. In distinguishing between arithmetic and geometry, on the one hand, and algebra, on the other, Kant would seem to reject both of these moves. But when the possibility of such ostensive constructions (even at second hand) is rejected, intuition as so far understood comes to playa reduced role. 59 See William S. and Martha Kneale, The Development of Logic (Oxford University Press, 1962). p. 308. 60 The Kant-Eberhard Controversy, p. 110. . 61 I have learned a great deal from all ofthe authors listed in the first paragraph. Michael Young , has made very helpful comments on an earlier draft of section 4 of the paper. James Allard raised the right sort of questions throughout, though I wasn't able to answer all of them.

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JAAKKO HINTIKKA

KANT'S TRANSCENDENTAL METHOD AND HIS THEORY OF MATHEMATICS

I. THE AIM OF THIS PAPER

·This paper has a dual aim. On the one hand, it is a part of a larger attempt to understand the nature of Kant's ideas of transcendental method and transcendental knowledge and their implications, for instance, the question as to what the objects of transcendental knowledge are. On the other hand, I am outlining once again what I take to be the true argumentative structure of Kant's doctrines of the mathematical method, space, time, and the forms of inoer and outer sense. The link between the two is that on my interpretation Kant's theory of mathematics offers an excellent example of the applications of his transcendental method. Moreover, after having recently defended my construal of Kant's views on mathematical reasoning and their foundation on historical and textnal gronnds, it may be in order· to try to vindicate it in another way, to wit, by relating it to the overall natnre of Kant's philosophy, including his idea of transcendental knowledge. I suspect that this may be a better way of convincing my colleagnes than nitty-gritty analyses of Kantian texts. At the same time, this approach offers me a chance of indicating some of the consequences of my results concerning Kant's theory of mathematics for the rest of his philosophy. It tnms out that the observations we can make in pursuing this line of thought have also interesting consequences for our contemporary thought in the philosophy of logic and mathematics. 2. KANT'S CONCEPT OF THE TRANSCENDENTAL

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A starting-point is offered by Kant's crucial concept of the transcendental. How does Kant use the term? Kant introduced the term "transcendental" in a slightly different way in the first and in the second editions of the Critique of

Pure Reason: A version:

I'call transcendental all knowledge which is concerned, not so much with objects. as with

B version: I call transcendental all knowledge which is concerned. not so much with objects, as

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our concepts a priori about objects in general. (A 11-12.)

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with mode of knowledge of objects in general, in so far as this [knowledge] is to be possible a priori. (B 25.)

These definitions or characterizations seem to identify transcendental knowledge simply and solely with the kind of knowledge we have in general epistemology. In order to appreciate the point of Kant's characterizations we must look up his further explanations concerning transcendental argumentation. The obvious text to consult is his definition of the most important type of _transcendental argument, viz. transcendental deduction: The explanation of the manner in which concepts can thus relate a priori to objects I entitle their transcendental deduction. (A 85 = B 117.)

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- The "manner in which concepts can ... relate to objects" includes for Kant ~eir role in our knowledge-acquiring activities ("the ... strivings of OUf faculty of knowledge". as Kant puts it in A 86 = B 118). Indeed. Kant clearly

indicates that a transcendental deduction is an" explanation and a justification - of certain kinds of a priori knowledge. (See A 87 = B liD.) Now how is it that we can according to Kant achieve knowledge a priori? What are the "strivings of our faculty of knowledge" Kant has in mind? An answer is obtained from B xviii. There Kant says that he is "adopting as our new method of thought. .. that we can know a priori of things only what we ourselves put into them". In the same spirit, Kant says in B xiii that "reason has insight only into that which it produces after a plan of its own". Hence a transcendental deduction deals, according to Kant, essentially with the activities

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through which we "put things into objects" and so "produce" the objects of our knowledge. It is presumably in order to make room for this specific emphasis that Kant changes the wording of his characterization of the concept of transcendental knowledge in the second edition of the first Critique. All told, Kant's characterization of transcendental knowledge thus has to be taken in combination with his explanations of his "Copernican revolution", i.e., with his "new method of thought" which according to him is based on that insight that reason "must adopt as its guide ... that which it has itself put into nature" (B xiv). In other words, in transcendental knowledge we are dealing not so much with our know ledge as a static system or with our own

concepts thougbt of as inert tools, as with our mode of knowledgeacquisition. Not only does Kant acknowledge that we have to do something to reach the knowledge we in fact have; be goes to the idealistic extreme and intimates that our synthetic knowledge a priori is in the last analysis about

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what we have ourselves brought about. The "mode of knowledge" Kant speaks of in the B version has to be thought of as an activity on the knower's part, not as a natural phenomenon, for it is only through an activity on our own part (acting qua rational human beings) that we can create or (in Kant's own word) "produce" a priori knowledge or "put it into objects". The poignancy of the term 'transcendental" as used in Kant - or in any true Transzendentalphilosophie - is therefore due largely to its being used to mark the role of human knowledge-seeking activities in our total strucrure of knowledge and their contribution to what this knowledge - at least its a priori " component - is about. To use the term "transcendental" of a srudy of the general features of our own conceprual system or of "descriptive metaphysics" is thus deeply misleading as long as the role of the acrual human activities in creating and maintaining our conceptual system is not recog-

nized. This point has been obscured by several different factors. For one thing, an interesting fallacy is lurking in the wings at this point. Frequently, indeed usually, the kind of emphasis I am placing on the knowledge-seeking activities of the human mind is set aside by philosophers who claim that to take Kant to emphasize human activities means turning Kant's theory into something that has merely psychological or merely anthropological interest. For these activities may seem to be conditioned mainly by the contingent nature of the individual human agents involved in them. This objection is nevertheless fallacious, for it overlooks the possibility that the principles governing the operations of the human mind - and of the entire human being - are not merely psychologically or anthropologically conditioned but can be guided by objective rules, to the extent that these rules can become transcendental conditions of experience. Indeed, this overlooked possibility is inter alia shown by the analogy Kant draws between himself and Copernicus. For how can objects confonn to our concepts unless we do something to them? And the whole point of Kant's analogy is of course to emphasize the significance to our "motions", i.e., our doings, as creating the transcendental conditions of

experience. (See B xvi - xvii.) Hence the fallacy I am diagnosing is deeply anti-Kantian. This interpretational fallacy has a neat lingnistic counterpart in twentiethcentury philosophy. Often it is assumed that all srudy of the use of language, called pragmatics, is merely a part of psychology or sociology (or perhaps anthropology). This is fallacious for it overlooks completely the possibility that the use of langnage is governed by laws that can be srudied objectively in abstraction from the idiosyncracies of particular language users in the same

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way as syntax is studied in abstraction of the graphological and orthographic peculiarities of language users. This could be called the fallacy of pragmatics as an empirical science, and the other fallacy we have encountered is its transcendental counterpart. It nevertheless seems to me that Kant himself fell victim to this fallacy to some extent, in that he typically speaks of the subjective conditions of our knowledge rather than of those of our knowledge-seeking activities. This does not yet make a decisive difference, however, and it is partly due merely to his "Aristotelian mistake" to be discussed later in this paper. Here we are also already catching a glimpse of the larger problems I mentioned in the beginning of this paper. One reason why philosophers have not paid more attention to such idealistic pronouncements of Kant's as I have quoted lies in their peculiar ambiguity. Taken literally, Kant seems to be saying that what our synthetic knowledge a priori is really about are things of our own making and doing, what we ourselves put in to objects. Yet in some of the passages from which my quotes came Kant is emphasizing the contribution of our activities to our knowledge of objective realities. (In B xiii - xiv he is speaking of the knowledge we have in experimental physics.) This poses a perplexing problem. What is the relevant knowledge about, anyway? How can our activities and their products - which is what Kant seems to be talking about - contribute to our knowledge of physical reality? These questions are part and parcel of the dialectic hidden in Kant's concept of transcendental knowledge which I am trying to get at in this paper. The same perplexing problem meets us in Kant's explanations of the force of his notion of the transcendental. The explanations I quoted above from A 11-12 = B 25 identify transcendental knowledge with knowledge which is about ("is concerned with") our knowledge rather about its objects. But elsewhere Kant equates transcendental knowledge with a special kind of knowledge about objects, viz. "a priori knowledge in so far as it relates to objects and can be applied to them",' based on pure concepts and pure intuitions. In this use, transcendental knowledge is contrasted by Kant to empirical knowledge. How can Kant thus identify knowledge about our mode of knowledge with a certain kind of knowledge about objects? Our own active role in the genesis of our knowledge explains part of the problem, for the knowledge a priori, which is called transcendental, applies to objects according to Kant because we have made it apply to them through our knowledge-seeking activities. But another part of the puzzle remains. How can one and the same knowledge be about objects and also about our mode of knowledge? This is one of the

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larger problems which I ultimately want to get at and which I am trying to illustrate in this paper. 3. KANT'S TRANSCENDENTAL THEORY OF MATHEMATICS, SPACE. AND TIME

In order to find material for such illustrations, we can use the insights already reached. The partial insights we have reached are important both historically and systematically. Both these aspects are illustrated by an analysis of the first major use Kant made of the basic idea of his Transzendentalphilosophie. It is his theory of the mathematical method, space, time, and senseperception. I am tempted to say, it is his "Transcendental Aesthetic". Unfortunately, the true argumentative structure of this part of Kant's work is hidden in the first edition of the Critique of Pure Reason even more thoroughly than the force of his term "transcendental" was hidden in. his halfhearted explanation of the term quoted above. And when Kant clarified the structure of his line of thought in the Prolegomena (see especially sections 7-11, whose counterpart in the first Critique is A 46-49 = B 64-66), he unfortunately left the real basis of his argument hanging on a footnote reference to his discussion of the mathematical method at the end of the Critique of Pure Reason (A 713 ff = B 741 ff). This basis has been misunderstood almost universally. Since I have given full analyses of it elsewhere, I may perhaps be relatively brief here in serting Kant's record straight. 2 Contrary to what is often thought, the datum Kant is trying to account for in the Transcendental Aesthetic' is not that we can obtain synthetic a priori knowledge in mathematics by means of a special source of knowledge called "intuition". According to this fallacious view, this source of knowledge operates like mathematical imagination, and it is for the purpose of appealing to it that we use constructions (figures) in geometry and comparable aids to the intuition in other parts of mathematics. This view is grundfalsch, totally wrong. It misconstrues the force of the term "intuition" (Anschauung) in Kant, and it neglects Kant's own explicit injunctions against all appeals to geometrical imagination. "If he [sc. a geometer] is to know anything with a priori certainty, he must not ascribe to the figure anything save what necessarily follows from what he himself set into it in accordance with his concept [of the figure]" (B xii). Can one rule out more explicitly all appeals to intuition in geometrical proofs? This presupposes that what Kant meant by intuitions are simply those Vorstellungen which represent their objects as particulars, without the help of

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general concepts. I have atgued this point repeatedly in the past (see Note 2 above). This has provoked some discussion. Now there is a supremely conclusive test available to us to decide whether my interpretation of Kantian intuitions is correct. This test is to see whether my interpretation makes sense of Kant's problem of the possibility of synthetic knowledge a priori in mathematics. Kant's real problem is to explain why we can obtain synthetic knowledge a priori in mathematics, not by means of "intuition", but by means of what twentieth-century logicians would call instantiations, that is, by means of considering "arbitrarily chosen" representatives of general concepts in mathematical arguments. For such representatives of patticulats (individuals) '-- are precisely what I have shown Kantian intuitions to be. Their introduction is what Kant defines constructing to be. (See Kant's definition of this term in A 713 = B 741.) The problem about the use of such instantiation methods is that in them we introduce a representative of a patliculat entity a priori, without there being any such entity present or otherwise given to us. More fully expressed, Kant's problem is not how instantiations ate used in logical or mathematical arguments. He is accepting the traditional concept of a mathematical (for us: logical) argumentation in which appeals to geometrical "intuition" in our sense play no role. Kant's problem is: how can such argumentation, prominently including apparent anticipations of absent patticulars in instantations, yield knowledge which is applicable to all experience a priori. All this is amply shown by Kant's own words. In the Prolegomena, Section 8, Kant writes about the use of intuitions (patlicular representations) a priori. Le., so as to anticipate their objects:

even when they are used in the absence of their objects? It is here that Kant's transcendental method comes into play. Since "reason has insight ouly into that which it produces after a plan of its own", the explanation of the universal applicability of knowledge obtained by using instantiation methods, i.e., by anticipating certain properties and relations of patticulars, can only lie in the fact that we have ourselves put those properties and relations into objects in the processes through which we come to know individuals (patticulars). Then the knowledge gained in this way must rellect the structure of those processes, and is applicable to objects only in so far as they are potential targets of such processes. Now what are these processes? How is it that we do come to know particulars? Following a long philosophical tradition. Kant answers: by senseperception: "Objects are given to us by means of sensibility, and it alone yields us intuitions ... [IJn no other way can an object be given to us" (A 19 = B 33). This answer goes all the way back to Aristotle, according to whom "it is sense perception alone which can grasp patliculars". (Analytica Posteriora A 18,81 b6.) Given this assumption, Kant concludes that the properties and relations with which mathematics deals are put into objects by us in sense-perception. Hence (Kant says) these properties and relations are due to the structure (form) of our faculty of sense-perception. These forms Kant identifies with space and time. Thus our mathematical knowledge rellects the form of the processes by means of which we allegedly come to know patticulars, and is applicable to objects only qua objects of sense-perception. Kant expresses the conclusion just indicated as follows (Prolegomena, Section 9):

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But with this step the difficulty seems rather to grow than to decrease; For now the question runs: How is it possible to intuit anything a priori? Intuition is a representation, such as would depend directly on the presence of the object. Hence it seems impossible to intuit anything a priori originally, because the intuition would then have to take place without any object being present, either preVIously or now, to which it could refer ....

1

Thus the usual interpretation - or at least one variety of it - tums Kant's problem neatly upside down. Kant is not trying to explain how intuitions can yield knowledge in virtue of their especially immediate relation to their objects. He is explaining why certain intuitions (viz. the instantiating terms used in logic and mathematics) can yield synthetic a priori knowledge even when their objects are absent. This passage is not about our representations (intuitions) of physical objects, either, as some scholars have clai.med. It is plainly Kant's own canonical and completely general statement of his transcendental problem of the possibility of mathematics. But how can intuitions yield knowledge a priori

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If our intuition had to be of such a nature that it represented things as they are in themselves, no intuition a priori could ever take place and intuition would be empirical every time.

This is Kant's transcendental assumption, which implies that the properties and relations mathematics deals with do not belong to things as they are in themselves, but are put into them by ourselves. Accordingly, he concludes: There is thus only one way in which it is possible for my intuition to precede the reality of the object and take place as knowledge a priori, namely, if it contains nothing else than the form of sensibility which in me precedes all real impressions through which I am affected by objects. (Kant's emphasis).

Here we can see both Kant's Aristotelian premise ("the form of sensibility ... precedes all real impressions ... by objects") and his main conelusion (an intuition a priori can yield knowledge only "if it contains nothing

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else than the form of sensibility"). A little later (Section II) Kant notes how this restricts the applicability of mathematics:

our knowledge acquisition which he elsewhere highlights, not the least in his use of the concept of the transcendental, as we saw earlier. Given this one fallacious step, Kant can claim that mathematics is based on the structure of our faculty of sense-perception and reflects this structure. The main remaining step he took is to identify this structure with space and time. This further step is not discussed here. In spite of the fact that Kant takes this one fallacious step, it is bad interpretation and bad philosophy for the commentators to make Kant's mistake for him. The interpretation I criticized above is fanlty not the least because it disregards the distinctively transcendental element in Kant's thinking in this department. On this interpretation, Kant's theory of space and time as the respective forms of our outer and inner sense differs little from any old naturalistic explanation of any old naturalistic phenomenon. This interpretation, however you propose to develop it, is bound to overlook the transcendental element in Kant's arguments about space and time in the "Transcendental Aesthetic", and also to overlook the precise force of such Kantian terms as "intuition" (Anschauung) there. In contrast to it, on my interpretation his argnments suddenly make perfectly good sense. For instance, the interpretation of an intuition as a Vorstellung of a particular is shown to be correct by the way Kant argues in A 24-25 ; B 39:

Pure mathematics, as synthetic knowledge a priori, is only possible because it bears on none other than mere objects of senses ....

4. KANT"S ARISTOTELIAN MISTAKE EXPOSED

This Kantian argument depends essentially, not only on his general transcendental position, but also on the assumption that the particular objects to which . mathematics applies are always given to us by sense-perception. But is Kant right in thus assuming that the process by means of which we become aware of the existence of individuals is sense-perception? In spite of its plausibility and wide currency, I believe that Kant's assumption is deeply wrong. Indeed, I have suggested that it is Kant's basic fallacy in his first Critique. Kant is asking the wrong question here. He is in effect asking: Is perception involved in all our processes of coming to know particulars? It is not unreasonable

(although not unchallengeably obvious) to opt for an affirmative answer to this question. But this is not the appropriate question here. In Kant's own terms, if there is something else involved in our cognition of particulars than perception, we could have smuggled the requisite things into the patticnlar objects of our knowledge in that other component. What Kant oUght to have asked therefore is this: Is perception all that is involved in our coming to know particular objects, especially their existence? Is sense-perception the way of cognizing individual existence in general? And as an answer to these questions, Kant's doctrine is hopelessly wrong. The most general description of the ways in which we do reach the information which we actually have about individuals (especially their existence) is not passive perception, but active seeking and finding. Only rarely can we relax and sit back and wait passively until the right particnlars show up in our sense-perception. Typically, we have to get up and actually look for the particular objects of our knowledge. Indeed, we can speak of seeking and finding even in areas where sense-perception is not involved at all, e.g., in dealing with numbers and other abstract entities. Langnage-games of seeking and finding (as we might call them) are thus much better candidates for the role of the general activities by means of which we come to know particulars than perception. Hence Kant's Aristotelian view concerning intuitions and perception is wrong. We can now also see that Kant's mistaken view was not only myopic, but un-

transcendental and hence un-Kantian. He overlooks here the active element in

Space is not a discursive or, as we say, general concept of relations of things in general,. but a pure intuition. For ... we can represent to ourselves only one space. (Emphasis added.)

Here the uniqueness (particnlarity) of space is used as a reason for the intuitivity of its representation, thus bringing out the force of the term "intuition" in his argument. In drawing conclusions from his 'Transcendental Exposition of the Concept of Space' Kant writes (A 26; B 42): Space does not represent any property of things in themselves. nor does it represent them in their relation to one another. That is to say, space does not represent any determination that attaches to the objects themselves, and which remains even when abstraction has been made of all the subjective conditions of intuition. For no determination ... can be intuited prior to the existence of the things to which they belong (emphasis added), and none, therefore, can be intuited a priori. ... Space is . .. the subjective condition of sensibility, under which alone ourer intuition is possible for us.

This passage should be compared with the quote from the Prolegomena, Section 8, above. Both make the same point, even though the Prolegomena passage is considerably more explicit and clearer. But quite unmistakably Kant is saying in the Critique passage, too, that (,'1e use of intuitions is impos-

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sible if it deals with "detenninations" of things in themselves, that is to say, independent of the "subjective conditions" of our knowledge of particular objects. Intuitions used a priori (that is to say, used so as to precede their objects) can yield knowledge only if they pertain to the subjective conditions of knowledge, that is, to our knowledge-seeking activities and their products. These Kant tacitly identifies with sense-perception ("subjective conditions of sensibility"). Essentially the same things can be said of Kant's conclusions concerning space and time in the Transcendental Aesthetic, viz. in A 25 = B 42 and A 32-33 = 48-50. Hence the line of thought I have ascribed to Kant is not resrricted to the Prolegomena but is clearly present also in the first Critique.

games" of seeking and finding and sees in our logical knowledge merely the structure of these games writ large. Such an approach to logic is represented by the "game-theoretical semantics" which I have developed in the last few years 3 The upshot of our hindsight-motivated correction to Kant is thus -literally - a transcendental deduction of game-theoretical semantics. Gametheoretical semanticists are hence the true Kantians among contemporary theorists of logic, we can conclude. This should alone suffice to show that transcendental arguments are not dead (i.e., without topical interest). The interest and value of this born-again Kantian argument is undoubtedly seen best from the specific conclusions to which it leads us. Several of them have a hauntingly familiar transcendental quality, at least for a true Kantian. Kant rries to analyze the nature of perceptual and apperceptual processes and of the application of concepts to perceptual raw material in order to gain further insights into the limits of the legitimate use of the categories of understanding, which he relates to the different logical fonns of judgment. Somewhat analogously, a deeper examination of the preconditions of the "language-games" of seeking and finding leads us to interesting insights concerning the presuppositions and limitations of different kinds of logics, including a better grasp of the limitations of classical logic and the classical concept of model in logic.4 These further applications of Kantian ideas are closely connected with his fundamental transcendental viewpoint. According to this view, our system of logical truths is detennined by the structure of our knowledge-seeking activities, which in my model are the language-games of seeking and finding. Any change in the rules for these Hgames" or in their preconditions will be reflected by the structure of our logical system. For instance, if the strategies which are available to the players of my semantical games are resrricted to computable (recursive) ones (and the overall games divided in a certain sense

Thus a closer exartlination of Kant's actual argumentation confirms my

interpretation of the structure of his line of thought. 5. A TRANSCENDENTAL DEDUCTION OF GAME-THEORETICAL SEMANTICS

In spite of Kant's fallacious idea that sense-perception is the way of gaining knowledge about particulars, his theory is extremely interesting, and offers rich material for further discussion. One reason why I have expounded Kant;s line of thought as fully as I have is that, in spite of this one slip, the rest of his argument seems to me sound. Moreover, the one mistake Kant makes can

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easily be corrected, as we saw, by substituting the activities of seeking and finding for passive sense-perception in Kant's line of thought. The right conclusion from the reconstructed Kantian argument is therefore implicit in what I have already said. Obviously, it will be that the knowledge obtained by means of instantiation procedures deals "really" with our rule-governed activities of seeking and finding, and reflects the structure of these activities. Admittedly, a tenninological adjusttnent is needed here. The knowledge thus obtained was identified by Kant with mathematical knowledge, but it is easily seen that what it really amounts to is the kind of knowledge we obtain in logic, especially (but not exclusively) in first-order logic (quantification theory). After all, instantiation rules are the cornerstone of logic, especially the ground-floor logic just mentioned. Moreover, this logic is precisely what goes into the kinds of mathematical arguments Kant actually used as his paradigm cases. (They were mostly arguments used in elementary geometry developed in the traditional manner a la Euclid.) All told, the corrected Kantian line of thought thus becomes an argument for an approach to logic both to fonnal logic and to Sprachlogik - which focuses on the "Ianguage-

into subgames), we obtain certain nonclassical interpretations of logic which

include prominently GOde!'s famous functional interpretation of first-order logic and arithmetic.5 Again, our usual logic applies to objects only in so far as they are potential objects of our activities of seeking and finding. This is analogous to Kant's claim that mathematics applies to objects only qua objects of senseperception. Such preconditions to our use of logic and mathematics are usually taken to imply that the objects in question must not change while the games in question are played on them, in other words, that the world does not change while we investigate it, more precisely, that the world does not change between the moves of the semantical games. However, it is clear that

"-,

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not all changes in the objects make them ineligible to serve as objects of seeking and finding. Hence, we can broaden onr concepts of logic and of model in logic so as to accommodate certain changes in the world while we play langnage-games of seeking and finding in it. The resnlt is the extension of the classical concept of model which Rantala has called urn model. 6 Normal (classical) models then tnrn out to be a special case of urn models, viz. invariant urn models (models that do not change in tandem with onr steps of seeking and finding). This widening of the classical concept of model is promising, and has already given rise to interesting applications.7 As we can see, it is transcendental in spirit, for it deals with the preconditions of onr "mode of knowledge" of the world in the sense of the preconditions of onr language-games of seeking and finding - with "the subjective conditions" of our knowledge, as Kant might have put it. This reconstructed modem counterpart to Kant's theory may seem to belong to a sphere of ideas altogether different from Kant's. The logical procedure of instantiation does not seem to have any conceprua! relation to geometry, even though it has sometimes been related to sense-perception or sensory imagination. Games of seeking and finding played on discrete individuals seem to be a far cry from the continuous Euclidean geometry Kant was talking abont. I believe, however, that by inquiring into the constitution of those discrete physical objects whose availability is a presupposition of my games of seeking and finding, we are forced back to emphasizing the need of geometry frameworks as a prerequisite of the individuation of the objects of seeking and finding. This line of thought has been explored elsewhere. 8 I cannot discuss it adequately here. It offers a very interesting and promising second line of defense to latter-day Kantians. I might perhaps put the same point as follows: logic reflects those aspects of the structure of onr activities of seeking and finding. These activities are not independent of the structure of the medium in which the relevant search is conducted. In the long run, this structure has to be brought in, and then we are led back to ideas that resemble Kant's theories more closely than the logic of seeking and finding does.

great deal else in his philosophy. In order to see this, it is advisable to return to the concept of the transcendental. There is a facet of this concept in Kant's Critique of Pure Reason which we have not yet touched. It is the facet of this concept that ties it to the title of Kant's book. Kant was not only interested in describing and studying our knowledge-seeking processes and their input into the total edifice of our knowledge. His primary aim was a critical one. He wanted to stake the legitimate bouudaries of those processes. This is what makes his book into a Critique of pure (i.e., a priori) reason. This purpose is sigualled by many Kantian concepts and terms. One of these concepts is the bastard half-brother of the concept of the transcendental, to wit, the concept of the transcendent. Kant's main explanation of the difference between the two is given in A 295-296 = B 352-353:

6. KANT'S CONCEPT OF TRANSCENDENCE

".

Turning Kant's 'Transcendental Aesthetic' into a "transcendental deduction of game-theoretical semantics", by correcting his Aristotelian mistake, thus leads to several interesting observations. In another sense, however, Kant's mistake is much harder to correct, viz. harder to correct without affecting a

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We shall entitle the principles whose application is confined entirely within the limits of possible experience, immanent; and those ... which profess to pass beyond these limits, transcendent . ... Thus transcendental and transcendent are not one and the same. [For instance} the principles of pure understanding ... allow only of empirical [employment} and not . .. employment extending beyond the limits of experience. A principle, on the other hand, which takes away these limits, or even commands us actually to transgress them, is called tran-

scendent.

The picture that emerges here is seen more clearly by recalling Kant's earlier explanation of the notion of transcendental (discussed above). As was pointed out, transcendental knowledge can be understood as dealing with, not just our general conceptual system. not even with the activities that serve to constitute this system, but more specifically with the legitimate limits of these activities. Now it seems that the contrast between the tenns "transcendental" and "transcendent" is that the former pertains to the boundaries of the use of understanding, which Kant identifies with the limits of possible experience, whereas the latter refers to transgressions and even denials of these boundaries. This seems indeed to be the basic idea in Kant. However, the actual situation turns out to be more complicated than this. Kant himself uses the term "transcendental" also of some types of steps beyond the limits of experience. (These must not amount to denials of the boundaries of understanding, however, nor instigations to transgress the boundaries.) This is vividly illustrated by the very passage I quoted from A 295-296 = B 352-353. Its middle part reads in its entirety as follows: [For instance] the principles of pure understanding, which we have set out above, allow only of empirical and not of transcendental employment, that is, employment extending beyond the limits of experience. A principle, on the other hand . ...

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Is this a confusion or perhaps even a slip on Kant's part? Admittedly, there is a royal confusion between the two terms in parts of later literature. Moreover, Kant was not entirely blameless, either. However, I believe (but cannot argue here) that there is a deeper reason for Kant's hesitation about the proper use of the terms "transcendental" and "transcendent" and about their relation. Indeed, this uncertainty about the proper relation of the transcendental and the transcendent turns out to be a symptom of the vety "paradox of transcendental knowledge" which is the problem which I would ultimately like to solve. Another notion Kant uses to highlight his critical theme is that of thing in . itself (Ding an sich). It is an inescapable shadow of the basic idea of Kant's transcendental philosophy. As soon as we consider our knowledge of objects as being obtained by means of certain knowledge-seeking activities, we ipso facto make it impossible for ourselves to consider them at the same time "in themselves", that is, to consider them independently of those knowledge-

the categories in the apperceptive assimilation of sense-experiences to the texture of our knowledge. All this has to be re-evaluated if Kant's philosophy is to be purged of his crucial ntistake. Needless to say, I cannot attempt this monumental task here. Attempts have been made repeatedly both by Kant and by his followers to do better justice to those fundamental ideas of his which are independent of the mistake. For instance, one can start from empirical objects in Kant and consider things in themselves as the outer limit of what these phenomenal

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seeking activities. The outer limits where the legitimate uses of our cognitive

activities must stop will the mark the limit beyond which things are transcendent, an sich. The concept of things in themselves, like the concept of the transcendent, is thus in effect part and parcel of Kant's project of marking the boundaries of our knowledge-acquiring activities. c

7. KANT'S MISTAKE AND THE TRANSCENDENCE OF THINGS IN THEMSELVES

One sense in which it is very difficult to correct Kant's mistake (his identification of the way we gain knowledge of the existence of particulars in general with sense-perception) is that its consequences are virtually impossi-

ble to disentangle from the rest of Kant's philosophy. Here I can only call the reader's attention to some of the most obvious consequences. It was precisely

Kant's Aristotelian mistake that turned that inevitable dark side of his tran-

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objects could become if we considered them in themselves, i.e., abstracted

from them all traces of our knowledge-seeking activities, But as long as one assigns to sense-perception the role it in fact plays according to Kant in the acquisition of our knowledge, it is hard to see that one cannot help ending up with the noumenal objects allover again. However, this theme cannot be

developed fully here, even though it is extremely important for our understanding and evaluation of Kant's philosophy, In particular, I don't want to deny that there were forces in Kant's own thinking which pushed him towards a position much closer to mine than his official one, It is not hard to find specific manifestations of such forces. Within Kant's

own system, there is for instance a reluctant and indirect adntission of the conceptual fact that the logic of our knowledge of particulars (intuitions) is the logic of existence and universality, that is, twentieth-century philosophers' "first-order logic". For him, there are certain principles of understand-

ing which correspond to the categories of quantity, which in tum correspond to the different quantities of a judgment: universality, particularity and singnlarity. These features of propositions Gudgments) are precisely what firstorder logic deals with. Now - prima facie quite surprisingly - the principles of understanding which correspond to these categories are according to Kant the axioms of intuition. (See A 161-166 = B 200-207,) However, in the light of what we have found, this identification is really not vety surprising. It is merely Kant's oblique recognition of the fact that the logic of existence and

scendental position, things in themselves, from epistemological restraints on

universality is the "logic" of the axioms of intuition, i.e., of the axioms for

our knowledge into metaphysical reifications, For it was the assumption concerning the role of perception in our knowleage-seeking that implied that the epistemological unreachability by our knowledge-seeking activities, which things in themselves by definition enjoy, amounted to transcendence with respect to sense-perception, that is, noumenal existence. In other words, the self-same Aristotelian assumption is the ultimate reason why Kant identified the limits of the legitimate use of the categories of understanding to possible sense-experience, and forced him to seek the ground of the applicability of

particular representations. Moreover, it is hard to avoid the impression that, for the purpose of the first axiom of intuition, the connection between intuitions and sense-perception

postulated by Kant is otiose, The axiom is formulated by Kant as saying that "all intuitions are extensive magnitudes". Unpacked, it says that all particulars are subject to geometrical and kinematic conditions. For this purported conclusion, it is certainly easier to argue in tenns of our concepts of existence

and universality than in terms of the idea of intuitions as always being given

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to us in sense-perception. It is certainly much more difficult to think of forms of existence of particulars which are not spatial and temporal than to imagine possible forms of perception of which the same is true. This point is especially persuasive if some form of game-theoretical semantics is adopted. For how can one possibly look for a search for particulars except in space and time? Even though we can perhaps in the last analysis make sense of nonspatial and nontemporal seekings and fiudings, the paradigm case can nevertheless be maintained to be the spatio-temporal one. Whatever further arguments one may want to put forward here, these observations strongly suggest that the viewpoint from which we are looking at his theory of space, time, mathematics, and particular existence is one toward which Kant himself was pushed to some extent by his own conceptual framework. Admittedly, Kant tries to give a deeper foundation to the alleged connection between particulars and sensibility by means of his examination of the threefold synthesis through which objects of experience are according to him constituted. But this attempt is a belated one. The plausibility of Kant's account of synthesis is no greater than that of Kant's initial Aristotelian mistake. It may be prima facie more plausible to think of the particular objects of our experience as being constituted in perception and apperception than as constituted to be potential objects of seeking and finding. On reflection, it is seen that this plausibility is merely another form of the same Aristotelian fallacy. The true account of individuation and identification will center crucially on the re-identification of objects in space and time, i.e., on those very properties of theirs which make them potential objects of seeking and finding (cf. Sec. 5 above). In other ways, too, can it be seen that objects are in our actual conceptual system individuated primarily to be reidentifiable objects of search and recognition.

tation of, sometimes to criticize, the philosopher in question but the method is the same in both cases. Usnally, the constructive results are inconclusive. An interpretational problem which can be solved by assembling a collection of quotations is not worth raising. By the same token, it is a cheap shot to point out that a proposed interpretation does not innnediately square with what for us is the literal meaning of the words of the philosopher in question. What is missing from discussions which tum on this kind of argumentation are two things. First, they miss the dynamics of a philosopher's thought: his problems, his tacit conceptual assumptions, his favorite modes of argnmentation, and his different (and frequently mutally incompatible) attempts to solve his problems. Second, what is often missing is the realization that a historical philosopher's words and other conceptual philosopher's words and assumptions are different from ours. As a consequence, the argnmentative structure of a philosopher's thought is easily misunderstood. There are many examples of these phenomena in those parts of recent Kantian studies which are relevant to my own interpretations. To take examples from the best literature, Jill Buroker9 discusses Kant's remarks on the problem of incongruent counterparts without any deeper awareness of the force of Kant's notion of intuition nor of the relation of this part of his thought to the overall argumentative structure of Kant's philosophy of mathematics. Likewise, Charles Parsons, in his paper on Kant's philosophy of arithmetic,1O never really raises the questions of what Kant's own central problems were and how various pronouncements are related to his overall line of thought. Yet this makes a crucial difference to Parsons' own argnments. For instance, in criticizing my interpretation of Kant's notion of intuition as amounting to a singular Vorstellung (and nothing more, as far as the force of the term is concerned), Parsons correctly notes that Kant seems to consider on a large number of occasions the immediate perception-like relation of an intuition to its object as one of its important characteristics. However, he never asks where these occasions occur. Now it easily turns out that each and every one of them occurs in the systematic order of things after Kant has established to his own satisfaction that all intuitions, not just empirical ones, have an essential relation to sense-perception. (Empirical intuitions are assnmed to be given by perception and a priori ones have been argned by Kant to be based on the form of our sense-perception.) Hence these passages have no relevance whatsoever to the question of the force of the term "intuition" in Kaut. ll

8. KANT'S TRANSCENDENTAL VANTAGE POINT AS GUIDE OF INTERPRETATION

There is a solid methodological reason why it is important to examine Kant's philosophy of mathematics in the light of his basic ttanscendental viewpoint. This importance is perhaps best seen by asking: How is an interpretation of a historical figure like Kant to be judged? Almost always, what one finds in books and papers in the history of philosophy are attempts to reconcile an interpretation with the letter of all the different relevant texts of some famous philosophers. This is sometimes done for the purpose of proving an interpre-

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There perhaps are no crucial experiments in science. There are, however, crucial tests in the history of philosophy. No matter what you can, for instance, say of Kant's use of the term "intuition" (Anschauung) elsewhere, an interpretation which does not enable us to make sense of the very argument by means of which Kant arrives at his theory of space and time is not only wrong; it is a nonstarter. My focus on Kant's transcendental method is useful in that it shows what the crucial arguments in Kant are that can serve as "crucial tests" of this kind. What we have to do is to therefore look at the arguments whicb Kant employs to establish the link between a priori inmitions and sensibility. This is precisely what I did above in Section 3. And then it turns out that Parsons·' view is not only awkward, it is diametrically opposed to what Kant is assuming in his argument. (A priori inmitions are not characterized by an especially immediate relation to their objects; they are precisely inmitions used in the absence of their objects.) Thus even a minimal attention to the structure of Kant's reasoning immediately settle an interpretational issue. An emphasis on Kant's transcendental method is useful precisely because it forces us to attend to the structure of Kant's overall argumentation. It was, for instance, my attention to Kant's transcendental method which led me to inquire into the knowledge-seeking activities which according to Kant give us our knowledge of particnlar objects. And this question of course, was what led me to recognize Kant's important Aristotelian mistake. In general, attention to Kant's method leads us to try to understand the argumentative struc-

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ture of Kant's theory of mathematics. The other major flaw in the recent literature, viz. insufficient attention to the iroport of Kant's key concepts, is best cured by close attention to the liter-

ature which formed the background of Kant's theories. I believe that from that literature one can find much more evidence for my interpretations than has been spelled out yet. Unfortunately, this is not the occasion to look for such evidence.

Note: This paper is closely related to the first half of my contribution to the 1981 Cambridge Conference on Transcendental Argumentation, which has appeared in German translation under the title "Das Paradoxon transzendentaler Erkenntnis" in the proceeding volume of the Conference, entitled 8edingungen de,. Moglichkeit, Klett-Cotta Verlag, Smttgart, 1984, ed. by W. Vossenkuhl and E. Schaper. The work on the changes from and additions to that earlier version has been supported by NSF Grant # BNS 8119033. Boston University

359

I See Carl Christian Erhard Schmid, Worterbuch zum leichteren Gehrauch del' Kantischen Schriften, vierte Ausgabe, 1798, reprinted by Wissenschaftliehe Bucbgese1lschaft, Darmstadt, 1980, p. 525. 2 See the essays on Kant reprinted in my books, Logic Language-Games. and Information, Clarendon Press, Oxford, 1973. and Knowledge and the Known, D. Reidel, Dordrecht, 1974, as well as 'Kant's Theory of Mathematics Revisited', in J. N. Mohanty and R. W. Shehan (eds.), Essays on Kant's Critique of Pure Reason, U. of Oklahoma Press, Norman, Oklahoma. 1982, pp. 201-215 (reprinted from Philosophical Topics 12, NO.2 (1982». 3 See Esa Saarinen (ed.), Game-Theoretical Semantics, D. Reidel. Dordrecht. 1979; Jaakko Hintikka. 'The Game-Theoretical Semantics: Insights and Prospects'. Notre Dame Journal of Formal Logic 23 (1982), 219-241; Jaakko Hintikka, The Game of Language, D. Reidel,

Dordrecht, 1983. See here Jaakko Hintikka, Transcendental Arguments Revived', in A. Mercier and M. Svilar (eds.), Philosophers on Their Own Works. Vol. 9. Peter Lang, Bern, 1982, pp. 115-166. 5 See Kurt GOdel, Ober eine bisher noch niehl beniitzte Erweiterung des finiten Standpunktes'. in Logica: Studia Paul Bernays dedic-ata (no editor given), Editions du Griffon. Neuchatel. 1959, translated as 'On a Hitherto Unexploited Extension of the Finitary Standpoint'. Journal of Philosophical Logic 9 (1980), 133-142, and cf. Jaakko Hintikka, 'Game-Theoretical Semantics: Insights and Prospects' (Note 3 above). Further reference to the literature is given in both these places. 6 See Veikko Rantala, 'Urn Models', in Saarinen (ed.) (Note 3 above). 7 See Veikko Rantala, Aspects of Definability (Acta Philosophica Fennica. Vol. 29, Nos. 2-3, North-Holland, Amsterdam. 1977); Jaakko Hintikka, 'impossible Possible Worlds Vmdicated', in Saarinen (ed.) (Note 3 above). 8 On the subject of individuation and identification, see Jaakko Hintikka and Merrill B. Hintikka, 'Towards a General Theory of Individuation and Identification', in W. Leinfellner, E. Kraemer, and J. Schank (eds.). Language and Ontology: Proceedings of the 1981 International Wittgenstein Symposium. Holder-Pichler-Tempsky, Vienna. 1982, pp. 137-150. 9 Jill Vance Buroker. Space and Incongruence, D. Reidel. Dordrecht. 1981. 10 Charles Parsons, 'Kant's Philosophy of Arithmetic', in this volume pages 43-79; orginally in Sidney Morgeobesser et al. (eds.), Philosophy, Science, and Method: Essays in Honor of Ernest Nagel, S1. Martin's Press, N. Y., 1969, pp. 588-594. 11 Some critics of my earlier work have thought that I interpret any representative which for conceptual reasons stands for only one entity as an intuition. No, of course not. An intuition according to Kant represents its object qua particular, i.e., without the help of general concepts. Hence, e.g., the Vorstellung that goes together with a definite description is not an intuition for Kant, even though it can stand for only one object.

4

f""

CONTRIBUTORS

II

.L

STEPHAN BARKER Department of Philosophy The Johns Hopkins University Baltimore, MD 21218 USA

ARTHUR MELNICK Department of Philosophy University of Illinois, Urbana·Champaign Urbana, IL 61801 USA

GORDON G. BRITTAN. JR. Department of History and Philosophy Montana State University Bozeman. MT 59717 USA

CHARLES PARSONS Department of Philosophy Harvard University Cambridge, MA 02138 USA

MICHAEL FRIEDMAN Department of Philosophy University of Illinois at Chicago Chicago, IL 60680 USA

CARLJ.POSY Department of Philosophy Duke University Durham. NC 27708 USA

WILLIAM L. HARPER Deparnnent of Philosophy University of Western Ontario London, Ontario Canada

MANLEY THOMPSON Department of Philosophy University of Chicago Chicago, IL 60637 USA

JAAKKO HINTIKKA Department of Philosophy Boston University

J. MICHAEL YOUNG Department of Philosophy University of Kansas

Boston. MA 02215

Lawrence, KS 66045

USA

USA

PHILIP KITCHER Department of Philosophy University of California. San Diego La Jolla, CA 92093 USA

361

-'

INDEX

Bennett, J. 16, 17,264,286,287,289,295, 3ll Berkeley, G. 8, 86, 123-124,204,217, 257-263,266,272,277,285,286,287, 289,298 Beth, B.W 4, 41, 42, 45, 54-56, 57, 75, 76, 160,209,211,215,216,300,335 Bivalence 14,293,298,300.302-303,304, 305,307 Black,~. 77,130,209,213 Bolyai, J. 123,214,215,225,282 Bou.mo,B. 191, 195,212,214

Abstract ideas, Theory of 123 Adickes. E. 212 Alexander the Commentator 35-36. 42

Algebra 14--15, 26-27, 31-32, 39, 56, 57, 61,65,97, ll8-l20, 121-122, 139, 150-152,156,173,201,203,206,216, 281,315,320,325-326,328, 330-334,337,338,339 Allard, J, 339 Allison,H. 16, 17,241,285,289,338 AngeU, RB. 289 ApoUonius 329, 338

Bonola, R. 215 BOy'er, C. 214

A~ces268-272,273,277,288,313

Apperception 351, 356 Archimedes 216 Argll1ld,l22 Aristotle 34-36, 41, 42, 47,154,287,347 Arithmetic 3-4, 10, 15,26-28,31-32, 43-79,98,106,117-120,121,130, 135-136, 144--146, 149-152, 153, 159-162, 163, 173, 199-204,215,216, 217,233,315,320,321-322,323, 325-326,327-328,330,333,336,337, 338,339,351,357 Axioms for 12, 50-56, 61, 63, 120~ 160, 199-200,203,204,215,217,321-322, 325,339 Assertibilism (see also Truth) 13-14, 299-303,304-310,312,313 Austin. J. 289

Brittan, G. 14-15,71-72,78-79,157,214, 215,216,217,218,334,335 Brodbeck,~.209

Brouwer, L.E.J. vii, 1, 16. 17,69,74.

293-294,296,298,304,306,308-309, 310, 3ll, 312, 315 Buchdahl, G. 209 Buroker, J. 357, 359 Butts, R. 285

Cajori, F. 212 Calculation 163,200-202,216,235,322, 323-326,336,337,338 Calculus see Real analysis Cantor, G. 142, 149, 155,336 Camap, R. 5,177,214,221,240,241,242, 286 Carus, P. 209, 290 Cassirer, E. 335 Caregories 9, 10,63,74,96, 102-103, 135-136, 140-147, 149, 151-152, 153, 154,155,162,174,273,309,332,351, 354

Barker, S. 5-6, 12 Banmgarten, A. 141, 154 Beck, J.S. 78 Beck,L.W. 198,214,216,217,270,280, 284,288,289,290,291,335 Benacerraf. P. 129.336

363

T

I 364

INDEX

of Quantity 9-10, 135, 140-145, 147, 149,151-152,153,155,162,174,355 Cauchy, A. 131, 191, 195,212,214 Cauchy completeness 191 Celluci, M.e. 157 Cezanne, P. 242 Chisholm, R.M. 286 Cohen, J. 213

Concepts (see also Categories) 44-45. 47. 49,51,63-64,60-70,74,75,81-103, 104,112,123-124,140,143,152, 164, 171-173,174,185-189,201,217,265, 266,271,280-281,308,319,336 Singular 3,45-46, 70, 89, 91,103,140 General 3, 6, 21, 23-29, 33, 34, 75, 81, 83-84,186-189,201,217

1, l

II

['I. I I

,I.

I !

;1

I ,,

, I ,I j

!

Unity of 88-89 Consistency 225, 295-296, 321

Construction 4, 7,10,21-24,27-32,34-38, 40,41,55-62,65-68,73-74,76,77, 97-101,106,114-115,118-122,124, \31,135-138,139-140,145-146,148, 151-152, 156, 159-165, 169, 170-171, 173-175,178-180,183-185,190-192, 194-197,202,203,205-206,210,211, 215-218,226,248-255,298,306,312, 315,322-323,325-333,336, 338-339,345,346 Arithmetical 98, 148, 164, 170-171, 173, 203,326,338,339 Geometric 21-22, 29-32, 36, 61, 97-98, 100,114-1\5,121-122,124,140,152, 159,162-163,183,185,190,192,202, 203,217,248-254,260,268,277, 278-284,288,326,330,339 Ostensive 161-163, 174,203,281,326,

327-330,332-333,337,338,339 Symbolic 65-67, 97-98,100-101,106, 118-1\9, 139, 151, 156, 159, 161-162, 173,203,211,281,315,325-327, 329-331,336,337,338 Constructivism 293, 298, 301-303, 304.

305,307,308,312 Contintuity 182-183, 190-194, 196, 197, 198,210,214.313 Dedekind continuity 212 Convergence 191-192, 195, 197,214

365

INDEX

Cornman, J. 285, 289 Counterfactuals 8, 260-261 Couterat, L. vii D' Alernben, J. 195 Daniels, N. 288, 289

Decidability 317-319, 320, 321, 323-324, 335-336,339 Dedekind, R. 149, 154, 182, 337 Dedekind-Peano axioms 154 den Ouden, 334 Descartes, R. (see also Skepticism, Cartesian) 30-31. 44, 45. 262, 286,

329,330,338,339 Differentiability 196-197.214 Diogenes Laertius 205 Dirac, P.AM. 241

Dorato, M. 243 Durnmett, M. 298-299, 302, 303, 304, 305, 306,308,312

Eberhard.J.G.1, 16, 137, 153, 329, 333, 338,339 Ecthesis 28-30, 33, 34-35, 42, 157 Edwards, P. 209 Effective Procedures 301 Einarson, B. 42 Einstein, A. 177,209,221.239.240 Ellington, J. 290 Enderton, H. 211

Euclid 22,28-31,34,36,49,55,76, 180-184,190,197,198,199,206,210, 215,216,218,279,281,282,283,289, 321,336,350 Euler,L.130-131,216,338 Eves. H. 210, 214 Evidence 299-301, 304-308, 327 Evidential states 299-301. 305. 307. 308

Feder, J.G.II. 257

Frede,M 154 Frege,G.I, 16, 17,50,58,60,62, 112, 147, 149,150,178,208,213,214,216,218, 219,243,282, 289, 337 Friedman, M. 6, 7,12,240,242, 243, 315, 325,328,334,335,336,337,338 Garber, D. 285 Garve, C. 257, 290 Gauss, K.F. 122,225,229,241 Geach, P. 130,213 Genet, C. 290 Georneny 3, 5, 8, 11-12,22, 28-32, 38, 46, 48-50,52,54-55,57-62, 71, 86, 97, 113-116,117,118,120-126,130,139, 150,159-160,177-208,210,213, 214-215,216-217,218,22\-240, 241,242,243,245,249-255,277-284, 287,289,320,321,323,325,326,327, 329-330,333,336,337,338-339,352 Abstract 5, II Analytic 30-32, 338 Applied 5, 177-178, 186, 198,213, 227-230, 232 Euclidean 5, 22, 28-32, 49, 57, 61, 114, 130,177,179-183,185,190,198,202, 203,205-206,208,210,214,216-217, 221-226,228,229,230-232, 234, 235-240,241,242,245,249,279-281, 282,321-322,323,350,352 Non-Euclidean 12,49,58,60,198,203, 206,214-215,217,221-224, 225-226,228-229,230,234, 235-237,245,282-284,289,321 Pure 177-178, 186, 198,213,227-228, 232 Gerllardt, C.J. 76 Gibson, JJ. 287, 289 GOdeI, K. 317, 335-336, 351, 359 Goodstein, R.L. 106

Feigl, H. 209 Fluxional Calculus 193-197,212.217 Focus 246-248

Grabiner, J. 212

Fodor, J.A. 287, 289

Griinbaum, A. 242 Guyer. P. 157,287,289

Forms. Plato's theory of 242 Fourier. J. 214

Frnenke1, AA 77

Greenberg,M. 16, 17 Gregory, D. 339

Handyside,J.241

Harper, W. II, 16, 287, 288, 289, 311 1I~ T.41, 205, 210, 212, 216, 289 lIeelan,P.242 lIempel, C.G. 5, 221, 241 lIenkin, L. 210, 335 lIenrich, D. 153 lIerder, C. 157 lIeyting, A. 298, 306 Hilbert, D. vii, 74,130,177,184,198,208, 210,214,293,302, 303, 304, 308, 309, 310,312 Hintikka. KJ. vii, viii, 3, 4. 15-16,41, 45-46,54-55,57,67,69-74,75,76, 77,78,81-84,95,98,101,103,105, 106,130,157,160,178,209,210,211, 216,218,219,265,289,315,334,335, 336,359 lIintikka, M.B. 359 lIopkins, J. 217, 283, 290 lIowell, R. 70-72, 78, 79, 265, 290 Hume, D. 2. 170.241,286 Idealism Berl<eleyan 86, 257-258, 263, 266, 272, 277,286 "Refutation of' 152, 157,261-265,285.

286,287 Transcendental 2, 5, 7, 8,13-14,146, 226,240,241,257-258,263-264,266, 269,272.276,285,286,293-297, 304-305,311 I'Huilier 195, 213

imagination 10,23,95,106,159,169-173, 204,217,248,269,280,288,309,345, 352 PToductivel06.172-173,204 Reproductive 172 Incongruent counterparts 357 Infima species 89, 92-93 Interpretation 356-358

Intuition 2, 3-5, 9, 10, 14, 16,21,23-28, 33-35,37,44-47,49-51,54-57, 60-74,81-107,111-118,120-123, 125-129,130,131,137-139,145-146, 149-152,156,157,160,164,167-169, 170,177-178,180,185-189,198,200, 202,203-208,209,211,212,214,215,

-.-----

366

INDEX

216,217,218,225-226,253,265-270, 273,280-281,287,296,298,304-308, 316-327,329-332,335,336,338,339, 345-350,355,357-358,359 Empirical 85-87, 101, 103,218,245, 247-249,266-271,273,280,298,308, 317-318,335,357 Fonus of 2, 46, 48, 50, 62-64, 68, 90, 92, 99,104,105,113-115,117-118,145, 149,151-152,156,169,245-249,268, 278-279,324 Intellectual 47, 72, 86, 90, 101, 131,318, 319 Pure 24, 46, 55-56, 60, 65, 68-69, 83, 85-87,94,96,101,104, lll, 114-116, 117-118,121-123,125-129,130,137, 138,139,164,178,185,187-189,198, 204,206-208,211,214,215,216,217, -218,225-226,240,245,247,266,279, 280-281,306, 317,335 Intuitionism 1. 13-14.69,77,101, 276-277,293-294,296-300, 302-307,308-310,315 loogh, J.J. 75 Jeffrey, G_ 209 Johnstone, P. 290 Judgements 3-4, 7-8,50-51, 67, 70, 81-84,88-89,92,99,101-102, 103-105,109-111,125,140-141,154, 159,165-169,261-264,267-268,271, 273-274,286,308,309,351,355 Analytic 50-51, 67, 88, 110 Empirical 81-83, 99,101,166,267-268 Singular 70,83-84,89,92, 103, 104, 140-141,154 Synthetic 3, 4, 7, 8, 88, 109-111, 125,

159,165,167-169,305 Kamp, H. 288, 290 Kastner, A. 193.212,213 Katz, J. 153 Keller, P. 155 Kerferd, G_ 130, 156,209 Kirchner. P. 285 Kitcher, Patricia 129 Kitcher, Philip viii, 11, 16,204,209,212.

INDEX

214,217,218 Klein F. 123,214 Kline, M_ 339 Kneale, M_ 339 Kneale, W_ 339 Knowledge 3, 15, 3~, 44-45, 56, 64, 68-69, 73,75,85-87,92-95,99,101-102, 11O-111, 114, 116, 118-120, 122-129, 130,131,137-139,146,150,160-163, 168-169,171,173,180,188,207,209, 222,226,228-230,231,233,237,242, 262,264,266,300,305,316,335, 342-350,352-355 A priori 3,15,37,56,73,86,99,106, 110-111,116,128,130,138,180,222, 226,228-230,231,233,237,259,277, 278,284,316,342-348 Empirical 92, 95,173,209,344

197-198,199,200,207,210,211,214, 215,216,218,219,225,227-228, 232-233,240,265,293-294, 295-300,302-307,309-311,312, 315,320,323-325,328,334,337,346, 350-352,355 Aristotelian 7, 34-35, 42, 47,102,178, 320,323-324 Autonomy of 309

Infinitary 213, 309-310, 313 Monadic 7, 54,160,180---181,183-186, 197,210,211,218,323-325,336 Platonist view of 69 Po1yadic 180-182, 185, 190-191, 197, 198,214,323-325,336 Transcendental 96, 102-103, 210 Lucas, P. 214, 335 Lucasiewicz, J. 42

Immediate 44, 114, 120,264

Mathematical 3, 5, 64, 68-69, 75, 112, 116,118-120,122,123-129,130,131, 137-139,160-163,168-169,171,228, 241,337,347,350 Transcendental IS, 341-344, 353 Knowledge-seeking activity 343-344, 350, 353-355,358 Koch curve 196-197, 214

Kraemer. E. 359 Kripke model13, 297, 300, 307 KrUger, L 154 Laberge, P. 290 Lambert, J.H. 214-215 Lambert, K. 337 Leibniz, G. 49, 51-53, 66, 76,116,157,

193,198,218,241,251,313,320,324, 336,338 Leinfellner, W. 359 Lewis. c.I. 286 Lindsay, AD. 289 I-obachevsky,N_ 123,214,215,225,282 r-ocke, J_ 44,123-124 Logic (see also Quantification theory) 13-14,22,34-35,39-40,42,47-50, 51,54,56,58-61,64-65,67,71,84, 92, %,101-103,104,106,135-136, 140-141, 178, 180-186, 190-191, 195,

Maas, J_G_E_ 42 Macintosh,J_H_ 75,103,130,209,290,334 Maclawin, C. 193,212 Mahaffy, J. 290

Mahoney,M.129 Martin, G. vii, 15, 52~54. 159,214,215, 335,339 Mathematical objects 10, 12, 14,58, 63-64, 73-74,101,106,135-136,138-140, 146,151,156,161,164,207,298,299, 316-319,321-323,328,331,333,336, 338,348,351 Mathematical proofs 3,10,14,15,27-28, 31-33,36,55-57,59,61,66-67,100, 120-121,137-139,153,157,160,161, 179,197,199,200-202,206,216, 217,218,299,321,335 Meerbote, R. 213, 217, 285, 289, 311

Mehra,1.241 Meinong, A. 139

Melnick, A. 6-7, 16, 17,211,282,290

Method Analytic 30-31,329-330,339 Mathematical 21-42, 66, 205-206, 218 Synthetic 30---31, 329-330, 339 Tr.uo«ndenuil 15-16,211, 341,347-358 Meyer, R.K. 337 Mill, J_S_ 286

367

Miller, G. 285, 290 Model Theory, 324 Models Of geometry 122, 182-183, 186, 198, 203,204 Urn 352 Mohanty, J_N. 290, 334, 359 MOrullCh, U. 290 Moore, G.B. 271 Morgenbesser, S_ 103, 173,209,290,359 Motte, A. 212 Nagel, E. 5, 221, 241

Necessity 109-112

Nevnon, 1.22, 122,130, 131, 192-193,198, 212,213,217,329,338,339 Norrnore, C. 285

Notions 86 Number 3, 9-10, 63, 68, 83, 97-101,177, 122, 135-136, 139-141, 142, 144-152, 154,156,160,162-163,166-168, 170-171,174,215-216,217,233,321, 326,328,336,337 Concept of 3, 9-10, 63, 68, 83, 97,101, 117,135-136,140,142,145,146-152, 156,162-163,167-168,170-171,174, 217 Existence of 53,98-101. 139 Nwneral system 10,64-65,98, 101, 160-163,174,211,326,337 Objective validity 5, 7, 8, 9,11-12

Occam's razor 242 Operationalism 248-255 Order, theory of 180. 184. 186. 189

Pappas, G_ 285 Parsons. C. vii, viii. 3, 4, 9. 12, 16, 17,71, 81,101,103,104,106,107,126-127, 160-162,173,174,178,208,209,211, 212, 215-216, 217, 218, 265, 274, 280, 288,290,312,315,334,335,336,337, 357,358,359 Pasch, 11_ 184,210 Paton, HJ. 41 Patzig. G. 42

Peano, G. 154, 199, 200, 321

f<1' 368

INDEX

Peano Axioms 199.200,321

Peirce, C.S. 106,303,305,307,312 Penelhum, T. 75, 103, 130,209,290,334 Perrett, w. 209 Phenomenalism 8, 257. 260-261. 285, 286. 297,298,304 Plato 86, 242 Poincare, H. 5,130,242, 282, 290 Politz. K.HL. 155 Pollock, J. 286, 290 Possibility 48, 58, 68, 74, 78-79, 106, 127-128,137-138,153,186,206-207, 218,333 Logical 48, 58, 88, 106, 127-128, 153, 207,218,238 Mathematical 68, 137-138 Physical 68. 127.218

Real 74, 78-79,127,207,218,330, 332-333,335 Possible worlds 109-110, 114,208,218

Posy,C.153,276-277,29O,315,334 Poznanski, RU. 77 Putnam, H. 106, 153,222-224,234,241, 264,286,290,336 Pylyshyn, Z.W. 287, 289 Quanta 142-145, 147-149, 153, 155-156, 193,217,327-328 Quantification theory 9, 35, 40, 47, 56, 61, 92,96,99-103, 106, 182, 185-186, 195,207,210,211,213,214,310-311, 320,323-325,336,337,350 Quine, W.V.O. 27, 76, 105,210 Rabin, M.O. 77 Raftopoulos, A. 243 Rantala, V. 352, 359 Real analysis 181 Realism Empirical 8, II, 257-258, 260, 263, 265-272,273,277,282-283,286,288 Scientific 275-276

Transcendental 7, 13,257-258,263,270, 286,293,294-298,304,307 Reasoning (see also Logic) Geometric 3 Mathematical 38-39, 322-323. 331

Rehberg, A. W. 140, 150, 151, 154, 212 Reich, K. 155, 156 Reichenbach, H. 5,125,131,177,221,235, 240,241,242 Reid, T. 289 Reinhold, KL. 333, 338 Relational structures 331-332 Relativity, theory of 5,221.222. 229. 239,

240 Ricketts, T. 208, 211, 219 Riemann. G. 177, 208, 225, 282 Robinson. A. 77 Rosenkrantz, G. 242, 243 Ross, W.D. 41 Russell, B. vn, 1,7,16,17,41,78,177,178, 197-199,209,213,215,217,218,221, 227,241,242,286,295,296,309-310, 311,335 Saarinen, E. 359 Sanford, D. 288, 290 Schank,J.359 Schaper, E. 358 Schemata 10, 63, 124, 135, 146, 147-148, 163-169,171,217,218,281 Schilpp, P. 214, 240, 241, 242 Schtick,M.177,221,24O Schmid, C.C.E. 359 Schmidt, R 157

Schulrz, J. 52-54, 56, 63, 76, 98,101,137, 138-139,149,151,159-160,209,215, 216,337 Schlitz, C. 218

Seeking and finding (activity of) 348, 351-352,356 SellaIS, W. 260, 261, 264, 265, 285, 286, 287,290

Senmntics299, 305, 350-352 game-theoretical 350-352, 356 of geometry 251-252 Sensations 265-269, 312 Sense perception (see also Sensibility) 15, 23,35-40,85,170,258,260,262-263, 288,312,345,347-350,351,352, 354-356,357 Sensibility (see also Sense perception) 46-48,56,58,60,62-63,65-66,68,

369

INDEX

70-71,73,82,85,87,98,104,150, 160, 308, 358 Sets (see also Set theory) 63, 64. 142-143. 149,154,155,315,321 Set Theory 9, 58-61, 64, 136, 154,310, 315,321-322,325 Shamnon, A. 78, 140, 156 Shehan, R.w. 290, 334, 359 Shoemaker. S. 290 Simplicity (of theories ) 235-236. 242 Skepticism (Cartesian) 258, 262-263. 264, 272,273,286 Skolern function 185, 186.211 Smith, N.K. 16, 17,40,75,77,103,105, 153,155,156,157,173,209,285,290 Solipsism 265, 286-287 Space 3, 5-8,11,16,24,46,49,58,62,64, 68-69,74,77,86-87,92,94,96-97, 101,104,113-118,125-126,128,130, 131,142,144,147,150,152,178, 185-190,193-194,198,204-208,211, 212,213,215,221-222,224-227, 229-232,234,237-239,242, 245-246,249,251-254,257,259261,268,271,273-274,277-279, 281-283,286,311,345,347,349,356 Absolutist theory of 252 Apriorityof5-6. 16, 113. 186 Critical theory of 68, 113-116, 128. 185-190 Empirical reality of 8 Empty space 213 Ideality of 226 Intuitivity of 3,6-7,226 Physical space 211, 232, 234, 238-239, 283 Relational theory of 116, 252 Spatializing activity 6, 246-253 Visual space 230-232, 242, 282 Staal, J.P. 75 Strawson, P.F. 105, 129,210,217,241,242, 261,264,282,283,290 Stroud, B. 286, 290 Suppes, P. 103, 173, 209, 210, 335 Synthesis Figurative 149 Intellectual 105-106, 149-151,217

Transcendental 99, 105,335 Taiminen, L. 219 Tai~ W. 209, 215 Tarski, A. 210, 300, 335 Thales 205, 206 Theophrastus 42 Things-in+thernselves 16,49,218.270,293, 298,347,350,354-355 Thomason, R. 288, 290. Thompson,~.vli,4, 12, 14,77,78,79.138, 153-154,209,210,211,213,216,217, 218,315,318,334,336,337 Time 4, 11,50,58,60,62-64,67-69,87, 90,92,94,96-98,101,105,117,142, 144, 146-148, 150-152, 155, 180, 193, 200,213,245,252-253,261,264-265, 296,311,312,324,325,345,347,356 Torretti, R. 212, 215. 291 Transcendent objects 334-335 Truth 49, 58, 215, 242, 259, 272, 299, 302-303,304,309,310-311,336 and Existence 235-238 and Meaning 227-229, 232-233 Coherentist theory of 259-260. 263 Correspondence theory of 270-272, 297. 299,310 Empirical truth 269-277 Truth value gaps 274-276 Turbayne, C. 257, 259, 263, 284, 285, 286, 289,291 Ullman, S. 277-278, 283-284, 288, 291 Unger, L. 210

Vaihinger, H. 41, 157 van der Wal, G.A. 75 va." Fraassen, B. 11,273,285,287,288, 291 Verification 139, 245, 251, 253, 254-255, 283,304,305,316,326,328,332,334, 339 Vieta, E 329, 338 Visual inspection 115-116, 129, 204-206, 217,218,280 von Koch. H. 214 Vossenkuhl, W. 358

=~-

370 Walford, DE. 130,209 Walker. R.C.S. 282, 291, 359 Wallies, M. 42

Wang, Hao 77 Wechsell22 Wedberg, A. 42 Weierstrass, K. 191. 195, 196.214 Weierstrass' function 123.214

~

INDEX WIlson, M. 129,261-262, 285, 286, 287, 291 Wittgenstein, L. 216, 219, 264,284,286, 312 Wolff, R.P. 131 Wubnig. C. 339

Weingartner, P. 41, 42, 76, 209

Young,I.M. 10, 139, 158,216,315,335, 337,338,339

White. M. 103, 173. 209 Whitehead, A.N. 241 Whiteside, D. 212, 213

Zweig, A. 209, 290, 337

Zennelo, E. 154

SYNTHESE LffiRARY Studies in Epistemology, Logic, Methodology, and Philosophy of Science 1. J, M, Boch&lski, A Precis of Mathematical Logic, Translated from French and German by 0, Bird. 1959 ISBN 90-277-0073-7 2. P, Guiraud, Problemes et methodes de la statistique linguistique, 1959 ISBN 90-277-0025-7 3, H. Freudentltal (ed.), The Concept and the Role 0/ the Model in Mathematics and Natural and Social Sciences. 1961 ISBN 90-277-0017-6 4. E. W. Beth, Formal Methods. An Introduction to Symbolic Logic and 10 the Study of Effective Operations in Arithmetic and Logic. 1%2 ISBN 90-277-0069-9 5. B. H. Kazemier and D. Vuysje (eds.J, Logic and Longuage. Studies dedicated to Professor Rudolf Carnap on the Occasion of His 70th Birthday. 1%2 ISBN 90-277-0019-2 6. M. W. Wartofsky (ed.), Proceedings of the Boston Colloquium/or the Philosophy of Science, 1961-1962. [Boston Studies in the Philosophy of Science, Vol. I] 1963 ISBN 90-277-0021-4 7. A. A. Zinov'ev, Philosophical Problems of Many-valued Logic. A revised edition, ediled and translated (from RussianJ by G, Kiing and D.D. Corney. 1963 ISBN 90-277-0091-5 8. G. Gurvitch, The Spectrum 0/ Social Time. Translated from French and edited by M. Korenbaum and P. Bosserman. 1964 ISBN 90-277-0006-0 9. P. Lorenzen, Formal Logic. Translated from German by FJ. Crosson. 1965 ISBN 9O-277-008O-X 10. R. S. Cohen and M. W. Wartofsky (eds.J, Proceedings of the Boston Colloquium/or the Philosophy 0/ Science, 1962-1964. In Honor of Philipp Frank. [Boston Studies in the Philosophy of Science, Vol. II] 1%5 ISBN 90-277-9004-0 11. E, W. Beth, Mathematical Thought. An Inttoduction 10 the Philosophy of MatheISBN 90-277-0070-2 matics. 1965 12. E. W. Beth and J. Piaget, Mathematical Episterrwlogy and Psychology. Translated from French by W. Mays. 1966 ISBN 90-277-0071-0 13. G. Kiing, Ontology and the Logistic Analysis of Language. An Enquiry into the Contemporary Views on Universals. Revised ed., translated from German, 1%7 ISBN 90-277-0028-1 14, R. S. Cohen and M. W. Wartofsky (eds.J, Proceedings of the Boston Colloquium/or the Philosophy of Sciences, 1964-1966. In Memory of Norwood Russell Hanson. [Boston Studies in the Philosophy of Science, Vol. III] 1967 ISBN 90-277-0013-3 15. C. D. Broad, Induction, PrObability, and Causation. Selected Papers. 1968 ISBN 90-277-0012-5 16. G. Patzig, Aristotle's Theory of the Syllogism. A Logical-philosophical Study of Book A of the Prior Analytics. Translated from German by J. Barnes. 1968 ISBN 90-277-0030-3 17. N. Rescher, Topics in Philosophical Logic. 1968 ISBN 90-277-0084-2 18. R. S. Cohen and M. W. Wartofsky (eds.J, Proceedings of the Boston Colloquium/or the Philosophy of Science, 1966-1968, Part l. [Boston Studies in the Philosophy of Science, Vol. IV] 1969 ISBN 90-277-0014-1 19. R. S. Cohen and M. W. Wartofsky (eds.J, Proceedings o/the Boston Colloquium/or the Philosophy 0/ Science, 1966-1968, Part II. [Boston Studies in the Philosophy of Science, Vol. V] 1969 ISBN 90-277-0015-X