Brouwer meets Husserl: On the Phenomenology of Choice Sequences

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BROUWER MEETS HUSSERL

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

Editor-in-Chief:

VINCENT F. HENDRICKS, Roskilde University, Roskilde, Denmark JOHN SYMONS, University of Texas at El Paso, U.S.A.

Honorary Editor: JAAKKO HINTIKKA, Boston University, U.S.A.

Editors: DIRK VAN DALEN, University of Utrecht, The Netherlands THEO A.F. KUIPERS, University of Groningen, The Netherlands TEDDY SEIDENFELD, Carnegie Mellon University, U.S.A. PATRICK SUPPES, Stanford University, California, U.S.A. JAN WOLEN´SKI, Jagiellonian University, Kraków, Poland

VOLUME 335

BROUWER MEETS HUSSERL ON THE PHENOMENOLOGY OF CHOICE SEQUENCES

by

Mark van Atten CNRS Paris, France

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 ISBN-13 ISBN-10 ISBN-13

1-4020-5086-0 (HB) 978-1-4020-5086-2 (HB) 1-4020-5087-9 (e-book) 978-1-4020-5087-9 (e-book)

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com

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All Rights Reserved © 2007 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands.

in memoriam Gian-Carlo Rota 1932–1999

Duo sunt nimirum labyrinthi humanae mentis, unus circa compositionem continui, alter circa naturam libertatis, qui ex eodem inﬁniti fonte oriuntur. Leibniz, ‘De libertate’

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 1

An Informal Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Method, and an Assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 5 5 7 7

3

The Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 9

4

The Original Positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Incompatibility of Husserl’s and Brouwer’s Positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Two Sources of Mutual Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Similarity of Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Initial Plausibility of Both Positions . . . . . . . . . . . . . . . . . 4.3 Resolving the Conﬂict: The Options, and a Proposal . . . . . . . . . 4.3.1 Deny That Some Mathematical Objects are Intratemporal, Dynamic and Unbounded . . . . . . . . . . . . . 4.3.2 Deny That Mathematical Objects are Omnitemporal . . . 4.3.3 Deny That Mathematical Objects are Within the Temporal Realm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Deny That Mathematics is About Objects . . . . . . . . . . . . 4.3.5 A Proposal: The Heterogeneous Universe . . . . . . . . . . . . .

11

vii

11 17 18 27 37 37 39 40 40 51

viii

5

Contents

The Phenomenological Incorrectness of the Original Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Phenomenological Standard for a Correct Argument in Ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Husserl’s Weak Revisionism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Husserl’s Implied Strong Revisionism . . . . . . . . . . . . . . . . . . . . . . 5.4 The Incompleteness of Husserl’s Argument . . . . . . . . . . . . . . . . . . 5.4.1 From Atemporality to Omnitemporality . . . . . . . . . . . . . . 5.4.2 Possible Inﬂuence of Husserl’s Informants . . . . . . . . . . . . . 5.5 The Irreﬂexivity of Brouwer’s Philosophy . . . . . . . . . . . . . . . . . . .

53 55 59 67 67 72 74

6

The Constitution of Choice Sequences . . . . . . . . . . . . . . . . . . . . . 6.1 A Motivation for Choice Sequences . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Choice Sequences as Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Choice Sequences as Mathematical Objects . . . . . . . . . . . . . . . . . 6.3.1 The Temporality of Choice Sequences . . . . . . . . . . . . . . . . 6.3.2 The Formal Character of Choice Sequences . . . . . . . . . . . 6.3.3 The Subject-dependency of Choice Sequences . . . . . . . . .

85 85 89 95 96 97 98

7

Application: An Argument for Weak Continuity . . . . . . . . . . . 103 7.1 The Weak Continuity Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.2 An Argument That Does Not Work . . . . . . . . . . . . . . . . . . . . . . . . 105 7.3 A Phenomenological Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

8

Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

53

Appendix: Intuitionistic Remarks on Husserl’s Analysis of Finite Number in the Philosophy of Arithmetic . . . . . . . . . 113 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Name and Citation Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Preface

This is an analysis, using Husserl’s methods, of Brouwer’s main contribution to the ontology of mathematics. The discussion is essentially self-contained, but, depending on one’s background and purposes, one may wish to consult further literature. An introduction, from an equally phenomenological point of view, to Brouwer’s intuitionism as a philosophical foundation of mathematics is [3].1 There are many introductions to phenomenology. I mention Husserl’s own [128] and [130], the latter of which Gödel considered a ‘momentous lecture’;2 the wide-ranging, historiographical [203]; and the more problem-oriented [198], [201] and [248]. A short intellectual and psychological biography of Husserl is [236]; on Brouwer’s life there is now the two-volume biography by Dirk van Dalen [60, 63]. There are also entries on Brouwer and on Husserl in the Stanford Encyclopedia of Philosophy on the internet [2, 15]. Paris, April 2006

MvA

ix

Acknowledgements

This is a record of my gratitude to the following people for discussion, comments, criticism, and advice when I was writing and rewriting this book. Most of all, I am indebted to Dirk van Dalen, Charles Parsons, and Richard Tieszen. In addition, I thank Richard Cobb-Stevens, Fabio D’Agati, Stephen Donatelli, Igor Douven, Michael Dummett, Ruurik Holm, Leon Horsten, Piet Hut, Hidé Ishiguro, Juliette Kennedy, Georg Kreisel, Menno Lievers, Dieter Lohmar, Per Martin-Löf and his students, Sebastian Luft, Carl Posy, Robin Rollinger, Stanley Rosen, the late Gian-Carlo Rota, the late Karl Schuhmann, Rochus Sowa, Göran Sundholm, Harrie de Swart, Robert Tragesser, Anne Troelstra, Wim Veldman, Albert Visser, Thomas Vongehr, Freek Wiedijk, Olav Wiegand, Palle Yourgrau, and a number of anonymous referees. For discussion of versions of the appendix, I thank Carlo Ierna, Robin Rollinger, and Richard Tieszen. In its various stages, I have worked on the manuscript at the Departments of Philosophy at Utrecht University, Harvard University, and the Catholic University of Louvain; at the Mittag-Leﬄer Institute in Djursholm; at the Department of Mathematics at the University of Helsinki; and at the Institut d’Histoire et de Philosophie des Sciences et des Techniques (CNRS/Paris I/ENS) in Paris. I am grateful to these institutes for hosting me, and to their faculty, staﬀ and students for their kindness, help, and tea. The Department of Philosophy at Utrecht University together with the Netherlands Organisation for Scientiﬁc Research (NWO) supported my one year’s stay at Harvard’s Philosophy Department. NWO also funded my long visit to the Mittag-Leﬄer Institute, and the Fund for Scientiﬁc ResearchFlanders (FWO) made it possible to spend a term in Helsinki. This ﬁnancial support was much appreciated. Material from the manuscript was presented at ‘Logic, Methodology and Philosophy of Science’, Cracow, 1999, at the ‘Husserl Arbeitstagung’, Cologne, 1999, at ‘Logique et Phénoménologie’, Paris, 2000, at ‘History of Logic’, Helsinki, 2000, at the ‘Roskilde Summer School on the Philosophy of Mathematics’, 2000, at ‘Existence in Mathematics’, Roskilde, 2000, at ‘Foundations xi

xii

Acknowledgements

of the Formal Sciences II’, Bonn, 2000, and at seminars in Utrecht, 1999, Louvain, 2000, and Dublin, 2001. The ﬁrst version of the appendix was written for, and presented at, the annual meeting of the Husserl Circle at Fordham University, New York, 2003. Subsequent versions were presented at the ‘Conference on the Philosophy of Mathematics’, Tokyo, 2003, at the ‘LERIUconference on Formal Concepts’, Geneva, 2003, and at seminars in Stockholm, 2003, Helsinki, 2003, and Paris, 2004. I thank the organisers for providing me with these opportunities, and the audiences for their questions, remarks, and criticisms. Rudolf Bernet kindly provided me with a copy of the recent English translation [131] of Husserl’s Philosophie der Arithmetik [107]. Several sections have appeared elsewhere before, in part or whole, with changes to adapt them for separate publication. I thank the respective publishers and editors for their kind permission to make use of the following material here: Parts of sections 4.3.1 and 5.4.2 were included in ‘Phenomenology’s Reception of Brouwer’s Choice Sequences’, in Oskar Becker und die Philosophie c der Mathematik, Wilhelm Fink Verlag: München, 2005, pp. 101–117. 2005 V. Peckhaus. Section 4.3.4 appeared as part of a larger paper with Dirk van Dalen and Richard Tieszen, ‘Brouwer and Weyl: the Phenomenology and Mathematics of the Intuitive Continuum’, Philosophia Mathematica 10, no. 3 (2002):203c 226. 2002 R.J. Thomas. Sections 5.1-5.3 appeared as ‘Why Husserl Should Have Been a Strong Revic sionist in Mathematics’, Husserl Studies 18, no. 1 (2002):1-18. 2002 Kluwer Academic Publishers. Section 5.5 appeared as ‘The Irreﬂexivity of Brouwer’s Philosophy’, Axc iomathes 13, no. 1 (2002):65-77. 2002 Kluwer Academic Publishers. An earlier version of section 7.3 was transformed into part of a paper with Dirk van Dalen, ‘Arguments for the Continuity Principle’, The Bulletin of Symbolic Logic 8, no. 3 (2002): 329-347. The present section 7.3 has in turn been rewritten from that. Scattered parts of the main text found their way into a talk titled ‘Brouwer, as Never Read by Husserl’, published in Synthèse 137, no. 1 (2003):3–19, an issue that contains the proceedings of the conference ‘History of Logic’, c Helsinki, 2000. 2002 Kluwer Academic Publishers.

Acknowledgements

xiii

The appendix appeared, under the same title, in the Graduate Faculty Philosc ophy Journal, 25(2), 2004, pp. 205–225. 2004 Graduate Faculty Philosophy Journal and Mark van Atten. For permission to quote from material in the Brouwer Archive (Utrecht) and the Husserl Archive (Louvain), I am much obliged to their respective directors, Dirk van Dalen and Rudolf Bernet.

1 An Informal Introduction

1. What is the aim of this book? The aim is to use phenomenology to justify Brouwer’s choice sequences as mathematical objects. 2. First of all, what is a choice sequence? Imagine that you have a collection of mathematical objects at your disposal, let’s say the natural numbers. Pick out one of them, and note the result. Put it back into the collection, and choose again. You may choose a diﬀerent one, or the same. Note the result, and put it back. For example, perhaps you chose 12, 3 Making further choices, you may arrive at 12, 3, 81, 12, 221 and you can continue from there. A choice sequence is what you get if you think of the sequence you are making as potentially inﬁnite. The two sequences given above are initial segments of the choice sequence. Initial segments are always ﬁnite. We cannot make an actually inﬁnite number of choices, but we can always extend an initial segment by making a further choice. This potential inﬁnity of the choice sequence we indicate by three dots: 12, 3, 81, 12, 221, . . . 3. How are they used? There is an age-old problem in mathematics how to analyse the straight line (‘the continuum’). Traditional mathematics thinks of the straight line as a large number of isolated points lying next to each other, like grains of sand. As Aristotle already pointed out, the problem is that this isolation breaks the line’s continuity. A line is continuous through and through; a continuum is not made up from grains of sand but rather from strings of 1

2

1 An Informal Introduction

melted cheese. The mathematician L.E.J. Brouwer was the ﬁrst to show how to rectify the situation mathematically: his choice sequences provide a means to give a mathematical form to the strings of cheese. 4. You say you want to give a phenomenological justifation of choice sequences as mathematical objects. Is there a need, then, for a justiﬁcation? Yes. Most mathematicians refuse to accept Brouwer’s choice sequences as mathematical objects: these sequences depend on the individual’s choices and they grow in time, but none of the objects that traditional mathematics talks about are like that. They are too strange. A small group of mathematicians however has accepted choice sequences, and they continue to develop Brouwer’s ideas. 5. So mathematics is not a uniﬁed science and mathematicians actually disagree among each other as to what objects they are talking about? Yes, this is the situation (and choice sequences are not even the only disputed objects). The reason that this is not generally known is perhaps that the mathematical objects that non-mathematicians use in daily life (such as ﬁnite numbers, fractions, real numbers generated by an algorithm, and geometrical shapes) are not among the bones of contention. But speciﬁc views on the nature of mathematical objects may introduce (or, alternatively, rule out) speciﬁc constraints on what mathematical objects can exist. Hence, with diﬀerent philosophical views may come diﬀerent kinds of mathematics. Therefore, even someone whose primary interest is doing mathematics rather than philosophising about it will at some point have to engage in some philosophy, or at least to acknowledge that there is a philosophical question to be answered. 6. Didn’t Brouwer have a justiﬁcation of his own? If so, why not use that one? Brouwer indeed had a justiﬁcation of his own, but it was based on a background philosophy that is defective in such a way that it cannot be used to justify the introduction of choice sequences. 7. If Brouwer’s justiﬁcation doesn’t work, why turn speciﬁcally to Husserl for another one? It seems natural to me to look to Husserl for an alternative, most of all because I believe that phenomenology in general grants the power to understand, see into, see insightfully, and so think and justify cogently. But there are also speciﬁc circumstances that bring Brouwer and Husserl close together. First, Husserl’s general philosophy is very similar to Brouwer’s, without suﬀering from the defect I referred to above. Second, Husserl was very interested in phenomena that Brouwer also studied, such as time, which is itself an example of a continuum (think of the idiom ‘a timeline’).

1 An Informal Introduction

3

Third, in Husserl one ﬁnds analyses of the philosophical aspects of the notions of object and sequence as such, which will be helpful. 8. But if the philosophies of Brouwer and Husserl were that close, then why didn’t Husserl himself come up with choice sequences? Apart from the fact that one does not always come up with a good idea when one has all its ingredients in hand, from various things that Husserl said about mathematics it follows that he would not have accepted choice sequences when asked, let alone have been led to consider their possibility himself. For example, he claims that mathematical objects exist eternally and never change; choice sequences, on the other hand, come into being (at the moment one begins to make choices) and change over time (as with each new choice they grow longer). Husserl’s claims are characteristic of the tradition that he, orginally a mathematician, was trained in, and which is still dominant today. But a closer look at the motives that led Husserl (and others) to make these claims will show that these motives can also be honoured by somewhat weaker claims which, in contrast, do not rule out choice sequences. A crucial part of the argument will consist in showing that the fundamental tenets of Husserl’s phenomenology did not force him to make the strong claims that he in fact made. 9. Let me see if I get this right. You base a defense of Brouwer’s mathematical innovation on a philosophy, Husserl’s, that Brouwer himself did not embrace, while Husserl, in turn, had a conception of mathematics that would not embrace Brouwer’s idea? Yes. Brouwer’s idea was good, but his background philosophy is not capable of justifying it; Husserl’s background philosophy can justify it, but this has always been obscured by various of Husserl’s speciﬁc claims about mathematics. These speciﬁc claims however can be shown to be unwarranted by his own standards. So I defend Brouwer’s idea by Husserl’s means, even though Husserl himself would have said this cannot be done. In other words, I exploit the possibility that there could be a diﬀerence between Husserl’s utterances on a certain subject and what his philosophy actually implies about it. In the case of mathematics, I argue, there is indeed such a diﬀerence, which, moreover, opens up suﬃcient space for choice sequences to ﬁnd a place in Husserl’s philosophy. 10. Aha. Now, does the phenomenological analysis of choice sequences you give provide us with suﬃcient insight so that we not just accept them as mathematical objects, but also see how to go ahead with developing out of them a better mathematical theory of the continuum? Yes. For technical reasons, it would not be possible to develop much actual mathematics from choice sequences unless a certain crucial principle (called ‘the continuity principle’, but in a sense which is diﬀerent from the continuity of a line) holds for them. Without this principle, choice

4

1 An Informal Introduction

sequences would in a mathematical sense just be curiosities. Brouwer freely used the principle, but for its validity only plausibility arguments were advanced; however, it will turn out that the phenomenological analysis of our processes of thinking when we create choice sequences provides a justiﬁcation of the principle in question. 11. The motto of this book is taken from Leibniz. Would he have accepted choice sequences? No. According to Leibniz, the objects of pure mathematics exist in God’s mind, God exists outside of time, and mathematics in no way depends on God’s will. In such a setting, growing objects that depend on choices would not have been recognisable as mathematical objects. There is no evidence that Brouwer knew the passage in Leibniz from which the motto is taken, or similar ones; but, as it happens, Brouwer’s mathematical model of the one notion that Leibniz mentions, the continuum, depends precisely on the other one, freedom.

2 Introduction

2.1 The Aim The aim is to use phenomenology to justify Brouwer’s choice sequences3 as mathematical objects.4

2.2 The Thesis One correct, phenomenological argument on the issue whether mathematical objects can be dynamic (e.g., choice sequences) is not Husserl’s (negative) argument, but a reconstruction of Brouwer’s (positive) one.

2.3 Motivation The thesis involves a meeting of the thoughts of Brouwer (1881-1966) and Husserl (1859-1938); as their careers overlapped for some thirty years, this naturally suggests the question whether they ever met in person. They did, in April 1928, when Husserl came to the Netherlands to deliver his ‘Amsterdamer Vorträge’ [103]. On the 30th of that month, Brouwer wrote to a German friend, ‘Here, at the moment, Husserl is darting around, which strongly draws me in’ [63, p. 567, trl. Dirk van Dalen].5 That the appreciation was mutual is clear from Husserl’s report to Heidegger of May 5: Among the most interesting things in Amsterdam were the long conversations with Brouwer, who made a quite distinguished impression on me, that of a wholly original, radically sincere, genuine, entirely modern man. [128, IV:p. 156, trl. mine]6 However, nothing is known about the content of these conversations, nor, for that matter, about possible further exchanges between them.7 They have

5

6

2 Introduction

never discussed each other’s work,8 yet there has always been a close (conceptual and factual) link between Husserl’s phenomenology and Brouwer’s intuitionism (or constructivism in general). In one sense, this is not surprising [141, p. 99]: in both strands of thought the main principle is that all genuine knowledge refers back, directly or indirectly, to intuitions: experiences in which objects are given as themselves. Cases in point are Becker [11], Heyting [88] and Weyl [238, 239] who have applied phenomenology to argue in favour of (parts of) intuitionism. Heyting’s well-known interpretation of the logical constants [85, 87], for instance, uses the phenomenological concepts of intention and fulﬁlment to analyse intuitionistic ideas about meaning. However, another aspect of intuitionism is completely at odds with Husserl’s philosophy of mathematics, and this aspect concerns the nature of the mathematical universe. According to Husserl, the mathematical universe is static: its objects are ﬁnished (or complete) and mathematical truths and objects are omnitemporal (‘allzeitlich’, e.g., in Experience and Judgement [124, section 64]). Brouwer, in contrast, regards the universe as a construction of the mathematician. Hence it is not omnitemporal and, moreover, it is dynamic in the sense that some objects, namely, choice sequences, are open-ended and are developed in time. He showed how, if choice sequences are accepted as genuine mathematical objects, one can develop a rich and constructive theory of the continuum.9 Brouwer’s argument is based on a background philosophy that, as I attempt to show in chapter 5, is in fact incapable of justifying anything. I look to phenomenology for an alternative foundation of parts of intuitionism. But Husserl emphatically denies what Brouwer aﬃrms, i.e., the possibility of dynamic mathematical objects. Who is right? A glance at the reasons Brouwer and Husserl give for their respective positions leads to the following observation. Brouwer appeals to acts of construction and free choice, against the background of his mystical theory of mind. These ideas can be found in Brouwer’s writings from the early Life, Art and Mysticism [22] to the mature and elaborate ‘Consciousness, philosophy and mathematics’ [39]. Husserl, on the other hand, often states without further argument, and never even mentioning choice sequences, that mathematical objects are static. He considers it simply part of the meaning of mathematical statements that mathematical objects have this property. Examples from respectively the early and later Husserl can be found in his Logical Investigations [113, p. 134] and the already mentioned Experience and Judgement [124, section 64]. But this is precisely what Brouwer contests, and he does give arguments for doing so. Moreover, the appeal in these arguments to certain acts makes Brouwer seem, in this matter, the real phenomenologist of the two. This suggests that, even if we shift the background from Brouwer’s own speciﬁc philosophy to phenomenology, an argument can be found for a dynamic universe by reconstructing Brouwer’s argument (where ‘to reconstruct’ means following

2.5 The Literature

7

an argument closely, changing it where necessary, trying to preserve as much of the conclusion as possible). Note that, logically, there are three possible conclusions: 1. All mathematical objects are omnitemporal. (Husserl) 2. No mathematical objects are omnitemporal. (Brouwer) 3. Some mathematical objects are omnitemporal, some are not. My argument here concerns the third of these. In particular, I argue that choice sequences are an example of dynamic, and hence not omnitemporal, objects in mathematics. Whether there are other dynamic mathematical objects I will leave an open question. As an argument against Husserl’s thesis, one counterexample suﬃces. This is why the thesis speaks of one rather than the correct argument. That choice sequences might be justiﬁed on phenomenological grounds is yet just an idea. To put it in a metaphor that Husserl liked to use: if the suggestion is of any real value, we should be able to get small change for its large banknotes. The aim is to see if that can be done. If it can, we will have a way to do justice to some of Brouwer’s ideas without compromising a phenomenological point of view.

2.4 Method, and an Assumption The framework I will adopt is that of Husserl’s transcendental phenomenology, as outlined in his Formal and Transcendental Logic [112] and the Cartesian Meditations [126]. It suggests that ontological questions in a priori sciences are to be settled by attempting a constitution analysis; this determines my method. I will assume without further argument that this framework is, by and large, correct.10

2.5 The Literature In the early literature (Weyl [238, 239], Becker [10, 11], Kaufmann [139]) choice sequences are discussed extensively. But none of these discussions relates them to Husserl’s ideas about temporal aspects of mathematical objects. (It is true that Becker [11] investigates temporal aspects of mathematics in depth, but he does so within the context of Heideggers’s Being and Time, which is rather diﬀerent from Husserl’s framework, which I adopt here.) On the one hand this should not be surprising, Husserl published very little (and only late) on these matters in the period in which Weyl, Becker and Kaufmann wrote.11 On the other, he was in close contact with these authors, so one would have expected the matter to come up in private communications. (See note 8.)

8

2 Introduction

The question of dynamic objects has not received due attention in recent phenomenological literature: Tieszen [214] and Lohmar [152] discuss aspects of intuitionism, but not this question.12 Lohmar [153] mentions that the eternity (i.e., omnitemporality or atemporality) of mathematical objects would pose a problem for intuitionism, but he in no way questions this property himself. Bachelard [9] makes only a few passing remarks on intuitionism, and objects to its reformist pretentions. Schmit [189] conﬁnes his discussion of constructivism to its use within classical mathematics. Rosado Haddock [183], in spite of his book’s title (Edmund Husserl’s Philosophy of Mathematics in the Light of Modern Logic and Foundational Research), hardly mentions intuitionism at all. The exception is Tragesser’s discussion of choice sequences [218, ch. 4]; he makes the connection with the idea of diﬀerent ontological regions, each with their own appropiate logic. Even though the conﬂict between Husserl’s and Brouwer’s views is not brought out, his exposition sets the stage to do so.

3 The Argument

3.1 Presentation The argument for the thesis runs as follows: 1. According to Husserl, mathematical objects are static. (Premise) 2. According to Brouwer, there is at least one kind of mathematical object, the choice sequence, that is not static. It is dynamic. (Premise) 3. Husserl’s and Brouwer’s conclusions are contradictory. (From 1 and 2) 4. Transcendental phenomenology provides the full ontology for the a priori sciences. (Assumption) 5. In transcendental phenomenology, ontological questions in the a priori sciences are decidable. (Elucidation of 4) 6. Measured by phenomenological standards, Husserl’s argument is not correct. (Premise) 7. Measured by phenomenological standards, Brouwer’s argument is not correct. (Premise) 8. There must be a third, phenomenological argument for either Husserl’s or Brouwer’s conclusion. (From 3, 5, 6, and 7) 9. Brouwer’s argument can be reconstructed in phenomenology. (Premise) 10. One correct, phenomenological argument on the issue whether mathematical objects can be dynamic is not Husserl’s (negative) argument, but a reconstruction of Brouwer’s (positive) one. (From 8 and 9)

3.2 Comments The intended meanings of various terms in this argument are speciﬁed as we go along. The assumption 4 and its elucidation 5 are explained in chapter 5. For ‘reconstruction’, I refer to the introduction, 2.3; for ‘(transcendental) phenomenology’, to the introduction, 2.4, and chapter 4.

9

10

3 The Argument

The main work is done in chapters 4–6, in which I argue for the truth of the premises in my argument. Each of these chapters is of a diﬀerent nature. Chapter 4 (concerning steps 1 and 2 of the argument) is expository. Chapter 5 (steps 6 and 7) is critical and presents two argumenta ad hominem. The term is not meant pejoratively, but expresses that the original arguments of Husserl and Brouwer are attacked in their own terms. The Brouwer case has the form: ‘According to his own principles, he cannot argue for his possibly true conclusion’. The Husserl case has the form: ‘According to his own principles, his conclusion is not drawn correctly’. In terms of Johnstone’s analysis of such arguments [135], I argue that in Brouwer’s case, the mismatch between argument and conclusion is implied by the content of his (general) position (‘charge of self-disqualiﬁcation’ [135, p. 91]). Husserl’s, on the other hand, is of a more accidental kind: it can be remedied without giving up his (general) position (‘charge of dogmatism’ [135, p. 86]). Chapter 6 (step 9) is constructive. Its positive result is that choice sequences are, from a phenomenological point of view, acceptable mathematical objects. Chapter 7 presents an application of my analysis to one of the key questions about choice sequences once they are admitted. It is a phenomenological justiﬁcation of the continuity principle.

4 The Original Positions

The purpose of this chapter is to bring out a conﬂict between Brouwer’s and Husserl’s philosophies of mathematics. I will begin by locating a particular point of disagreement, and pin down what it consists in. Disagreement by itself does not warrant speaking of a conﬂict. A further condition should be fulﬁlled, namely, the presence of mutual pressure. Two sources of such pressure are identiﬁed. Finally, we have to consider the ways in which the conﬂict could be resolved, and ﬁnd one that is consistent with our phenomenological approach.

4.1 The Incompatibility of Husserl’s and Brouwer’s Positions One of the characteristics of mathematical objects, according to Husserl, is that they are at any point in time always exactly the same. On his view, these objects are related to time in the sense that they ﬁgure in our temporal thought processes, but they are not individuated in time. One cannot tell one from another by referring to a moment of coming into being, and they do not acquire identifying properties over time. Husserl calls such objects ‘irreal’ (e.g., ‘mathematical and other irreal objects’ [124, p. 312]), and explains that notion thus: Real objectivities are joined together in the unity of an objective time and have their horizon of connection; to the consciousness we have of them there belong, accordingly, horizon-intentions which refer to this unity. On the other hand, a plurality of irreal objectivities, e.g., a number of propositions belonging to the unity of a theory, does not have for consciousness such horizon-intentions referring to a temporal connection. The irreality of the proposition as the idea of a synthetic unity of becoming is the idea of something which can appear in individual acts in any temporal position, occurring in each as necessarily temporal and temporally becoming, but which is the same ‘at all 11

12

4 The Original Positions

times’. It is referred to all times; or correlatively, to whatever time it may be referred, it is always absolutely the same; it sustains no temporal diﬀerentiation, and, what is equivalent to this, no extension, no expansion in time, and this in the proper sense. [110, p. 259]13 Now compare this to Brouwer’s construal of one particular kind of mathematical object, the mathematical point or, equivalently, real number. Brouwer considers rational intervals of the form a a+2 λν = ν−1 , ν−1 2 2 (where ν and a are natural numbers) and then proceeds with the following deﬁnition: We . . . consider an indeﬁnitely proceedable sequence of nested λintervals λν1 , λν2 , λν3 , . . . which have the property that every λνi+1 lies strictly inside its predecessor λνi (i = 1, 2, . . .). Then [according to the deﬁnition of λ-intervals] the length of the interval λνi+1 at most equals half the length of λνi , and therefore the lengths of the intervals converge to 0 . . . We call such an indeﬁnitely proceedable sequence of nested λintervals a point P or a real number P . We must stress that for us the sequence (1) λ ν1 , λ ν2 , λ ν3 , . . . itself is the point P ; not for example ‘the limiting point to which according to classical conception the λ-intervals converge and which could according to this conception be deﬁned as the unique accumulation point of midpoints of these intervals’. Every one of the λ-intervals (1) is therefore part of the point P . [47, p. 69, original emphasis, trl. mine]14 In a footnote to this deﬁnition, Brouwer adds: An indeﬁnitely proceedable sequence is in general no fundamental [i.e., lawlike] sequence, as during its generation free choice of its elements is not excluded. [47, p. 69n. 1, trl. mine]15 (Brouwer’s ﬁrst published formal deﬁnition of ‘choice sequence’ is encapsulated within his deﬁnition of ‘spread’ (which he called ‘Menge’, set); see p. 49.) He compares his notion to the classical one: For us, a point and hence also the points of a set, are always unﬁnished and often permanently indeterminate objects. This in contrast to the classical conception, in which a point is determinate as well as ﬁnished. [47, p. 71, trl. mine]16

4.1 The Incompatibility of Husserl’sand Brouwer’s Positions

13

A natural question here is why Brouwer thinks of points in this untraditional way. I will come to that later (section 4.2.2); but let me ﬁrst spell out in what way the two quoted positions are incompatible with each other. Brouwer notes that his conception diﬀers from the classical conception in two respects, which he describes using the two pairs of opposites determinate-indeterminate (‘bestimmt-unbestimmt’) and ﬁnished-unﬁnished (‘fertigwerdendes’). An object is ﬁnished if it does not develop through time, and unﬁnished exactly if it is not ﬁnished. (The reference to time is implicit in ‘ﬁnished’; Brouwer also mentioned time explicitly, for example in the quotation on p. 14.) By ‘determinate’, Brouwer understood ‘given by a law’. This is probably not the way a classical mathematician would understand it: he would say that ‘the reals are ﬁxed by the magic of the principle of the excluded middle [PEM], not by honest calculation’[60, p. 393]; also [40, p. 142]. What is it that makes a particular mathematical object determinate? We have to try to ﬁnd a criterion that is neutral with respect to intuitionist and classical mathematics. For if we place ourselves on the classical view, its universal adherence to PEM trivially ensures that all objects are determinate, without telling us why some of the intuitionists’ objects are not. Likewise, we cannot exclude that some objects are indeterminate intuitionistically, but determinate classically. The criterion should be neutral with respect to the question whether PEM is valid universally. A ﬁrst try at such a neutral criterion would be, ‘An object is determinate exactly if for all properties PEM holds with respect to that object, and indeterminate exactly if it is not determinate’. However, this will not do, for if a universe contains both determinate and indeterminate objects, among a determinate object’s properties will be its standing in particular relations with indeterminate objects. But then there will be relations between determinate and indeterminate objects that depend on indeterminate properties of the latter. This way a determinate object ‘inherits’ an indeterminate property from another, indeterminate object in the universe. Because of this inheritance, not all of the determinate object’s properties obey PEM.17 A second try would be to take a hint from the objection to the ﬁrst, and exploit the distinction between relations that are constitutive of an object’s identity and those that are not. The idea is that the determinate object would be the same whether there are indeterminate objects present or not. Such inherited relations as just mentioned would then be coincidental and not part of the identity of the determinate object. Take, for example, the undoubtedly determinate number 2, and an indeterminate object such as the lawless sequence α that I have just begun. Assume further that the ﬁrst member of α has already been chosen, and that it happens to be 2. Then the relation ‘R(x, y): x is the ﬁrst member of choice sequence y’ holds for x = 2 and y = α. But whereas R(2, α) is constitutive of the identity of α (if its ﬁrst member were not 2, it would be a diﬀerent

14

4 The Original Positions

sequence), this is not the case for 2 itself: if α had been diﬀerent, or had not existed at all, 2 would still be 2. The idea is now that, given that 2 is a determinate object, this must be so solely in virtue of the properties that constitute its identity. We arrive at a reﬁnement of the ﬁrst proposal by deﬁning ‘Determinate objects are those for which PEM holds for all relations that constitute its identity’. But this still will not do. In intuitionistic logic, A(x)∨¬A(x) is true only for predicates A that are decidable; classical logic does not have this restriction. As a consequence, there may be objects that are determinate classically, but not intuitionistically. Hence the second try at deﬁning ‘determinate’ does not result in a generally applicable criterion. However, this second objection also shows that there is a condition for being indeterminate according to both classical and intuitionistic logic: an object is indeterminate if there is an undecidable relation that is constitutive for that object’s identity and for which excluded middle does not hold. Brouwer admitted such objects: In intuitionist mathematics a mathematical entity is not necessarily predeterminate, and may, in its state of free growth, at some time acquire a property which it did not possess before. [43, p. 114] Observe that a property such as ‘The number n occurs in the choice sequence x’ is constitutive of the identity of x, but is generally undecidable and does not satisfy PEM. Brouwer did acknowledge the validity of PEM for predications that we can decide by a ﬁnite construction [39], even when it concerns predications of choices that are yet to be made. Suppose we have chosen no more than the ﬁrst 10 values of a sequence α, that there is no 2 among them, and that there is no restriction on future choices. Then Brouwer would say, on the one hand, that it is true that ‘Either the 20th element is odd, or it is even’; on the other, that it is not true that ‘Either a 2 does occur in α, or it does not’. In the latter case, because the choices are free and the sequence can be proceeded beyond every ﬁnite bound, no ﬁnite construction can decide the question. In section 6.3.3, I discuss the question whether choice sequences are really chosen freely. None of the four combinations of determinate/indeterminate and ﬁnished/unﬁnished is conceptually impossible. ‘Mathematical objects are all ﬁnished and not indeterminate’ is the case of classical mathematics. For a position to be incompatible with Brouwer’s, it suﬃces to disagree on just one of these aspects. The classical conception disagrees on both; let us see how matters stand with Husserl. Husserl says, in eﬀect, that mathematical objects are ﬁnished objects (by ‘ﬁnished’ I do not wish to imply that they were once begun, just that there is nothing to be completed about them). They are not subject to diﬀerentiation or growth as time passes. During his philosophical development, Husserl changed his understanding of that stability.18 At ﬁrst these objects were characterised as atemporal, existing wholly outside of time; later this became

4.1 The Incompatibility of Husserl’sand Brouwer’s Positions

15

omnitemporal (‘allzeitlich’, e.g., Experience and Judgement [124, section 64c]), existing at every moment in time. (The quotation on p. 11 is from the later period; Husserl’s reason for this change of mind is explained in section 5.4.) However, both interpretations express the invariance of these objects relative to the passage of time; they are, to use a term that I will deﬁne more precisely in a moment, static. Now static is precisely what Brouwer’s points generally are not. I say ‘generally not’ because, although on a strict reading each Brouwerian point is something unﬁnished or becoming (‘etwas werdendes’) and hence not static, such a narrow interpretation hides the fact that in a whole class of cases the incompatibility seems wholly inconsequential. If a point is given by a law, one may individuate the point by referring to that law. As a law is a ﬁnite object, it can be speciﬁed once and for all, and hence a Husserlian could regard lawlike points as static, too. Therefore ‘static’ and ‘ﬁnished’ are not synonymous.19 The disagreement with Brouwer in this case would not be a substantial one. However, non-lawlike sequences (in the extreme case, lawless sequences) are a diﬀerent matter altogether. When free choices are involved, there is no other way to individuate a sequence other than speciﬁying the moment it was begun.20 Such a sequence begins somewhere in time, and grows in a way that cannot be fully predicted beforehand (by seeing what a law speciﬁes; partial predictions may be read oﬀ from restrictions, if posed). By its very nature, a non-lawlike sequence is an unﬁnished object; its additional indeterminateness justiﬁes the richer qualiﬁcations ‘open-ended’ or ‘dynamic’. The non-lawlike sequences in no way ﬁt into Husserl’s picture. It is not that Husserl demands that all mathematical objects are lawlike (i.e., given by a construction method or algorithm); he does not. True, Husserl repeatedly says that mathematics is a matter of ‘constructions’, but Lohmar [152, p. 195] noticed that Husserl most of the time uses that term as equivalent to ‘deduction’. For example, Husserl compares phenomenology to mathematical disciplines and asks about the former: Must we look for ‘fundamental formations’ here, and from them derive by construction, i.e., deductively by a consequential application of the axioms, all the other essential formations in the province and their essential determinations? [120, p. 165, emphasis mine]21 And: Mathesis universalis . . . is, for apriori reasons, a realm of universal construction . . . In it occur, as the highest level, the deductive systemforms and no others. [106, p. 103]22 This is also the sense in which Hilbert used ‘construction’. For example, in 1922 Hilbert claimed that his formalism was the true constructivism: In my opinion, only the path taken here in pursuit of axiomatics will do full justice to the constructive tendencies, to the extent that they are natural. [157, p. 200]23

16

4 The Original Positions

Husserl’s use of the term may very well have been the result of Hilbert’s inﬂuence.24 On this construal, existence of a point amounts to derivability of a classical existential sentence (i.e., without the demand for a witness, or a method to exhibit one). It is rather that Husserl conceives of the points not given by a law as nevertheless ﬁnished, static entities, as everywhere in mathematics. Therefore, if we agree to classify Brouwer’s lawlike points as static objects as well (for the reason given above), the following expresses at least partly the incompatibility of their positions: 1. Mathematical objects are static. (Both early and later Husserl) 2. Some mathematical objects are dynamic. (Brouwer) That mathematical objects are static was expressed by Husserl in terms of their temporality: initially by saying that they are atemporal (see note 18), then that they are omnitemporal; this change is discussed further in section 5.4.25 As I follow the philosophy of the later Husserl, ‘Husserl’s position’ will refer to the position that mathematical objects are omnitemporal. We then need to describe Brouwer’s position in commensurable terms; I propose ‘intratemporal’, and use the following related deﬁnitions, recapitulating some of the above: static–dynamic: An object is static exactly if at no moment are parts added to it, or removed from it. It is dynamic exactly if at some moment are parts added to it, or removed from it. temporal–atemporal: An object is temporal exactly if it exists in time, and atemporal exactly if it does not exist in time. intratemporal–omnitemporal: A temporal object is omnitemporal exactly if it is static and exists at every moment. A temporal object is intratemporal exactly if it is not omnitemporal. Thus, a temporal object is intratemporal if it is dynamic or does not exist at every moment, or both. An atemporal object is necessarily static; conversely, a dynamic object is necessarily temporal. An intratemporal object may be bounded or unbounded intratemporal. A bounded intratemporal object is a temporal object the existence of which has a beginning and an ending; an unbounded intratemporal object is an object that is intratemporal but not bounded. At least part of the incompatibility of Brouwer’s and Husserl’s views can now be reformulated, in my own terms, more explicitly: 1. All mathematical objects are omnitemporal. (Husserl) 2. Some mathematical objects are intratemporal, dynamic and unbounded. (Brouwer) Is there more to the incompatibility? In other words, does the other potential source of incompatibility, the issue of determinateness (p. 13), contribute

4.2 Two Sources of Mutual Pressure

17

to it? A classical mathematician, clinging to PEM as a universal law, no doubt would reject non-lawlike sequences because of their indeterminateness. But Husserl’s logic is not necessarily classical. He recognises that to regard PEM as a law in many cases is an idealisation, and as such open to critique (and, perhaps, criticism) [112, section 77]. The idealisation expressed in this law is that every judgement can be given adequately, i.e., with full evidence, which is either positive or negative evidence [112, p. 200-201].26 But the apropriateness of that idealisation also depends on the nature of the objects that these judgements are about. In other words, a region of objects may be such that it does not allow for PEM. The theme that diﬀerent ontological regions may require diﬀerent logics has been much elaborated by Robert Tragesser [218, ch. 4]; in particular, he points out that choice sequences, because of their open-endedness, form a region for which PEM is not appropiate. (The question why Husserl did not come up with that example is adressed towards the end of section 5.3.) Given the considerations in the preceding paragraph there is no reason to believe that the indeterminateness of non-lawlike sequences by itself would make Husserl object to them. But he could argue against such sequences (as mathematical objects), from his point of view, by focusing on their temporal properties directly. (Even constructivists other than intuitionists need not accept the idea of dynamic objects. Markov for example accepts only sequences constructed according to a recursive algorithm; such sequences are, as ongoing constructions, intratemporal and unbounded, but not dynamic.) So the incompatibility between Brouwer’s and Husserl’s positions should be understood as only involving the aspect of time. Having thus clariﬁed the incompatibility, why can’t we just leave it at that? Why couldn’t a Husserlian simply dismiss the disagreement by saying something of the form27 ‘I believe p, and Brouwer believes not-p. Brouwer is, of course, out of his mind’ ? In other words, why is what we have here not merely an incompatibility of views but a genuine conﬂict that demands resolution?

4.2 Two Sources of Mutual Pressure The pressure comes from two facts. The ﬁrst is that the respective conceptions of knowledge of Brouwer and Husserl are, at root, similar enough to raise the question how their claims on the temporal character of mathematical objects can ﬂatly contradict each other (p. 16). The similarity of these epistemologies makes one suspect that an argument based on one of them may have a close analogue based on the other. In eﬀect, this allows for transfer of arguments. But then the arguments in favour of the one claim can function as arguments against the other. Let me explain. Suppose we have two very similar philosophical systems, p1 and p2. In p1, let a1 be an argument to conclusion c1. Similarly, let a2

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4 The Original Positions

be an argument to conclusion c2 in p2. Assume further that c1 and c2 are incompatible conclusions. Now, consider a1. It is constructed from certain elements in p1. As p1 and p2 are very similar, it is at least plausible that in p2 we can ﬁnd analogues to those elements of p1. But if there are, we could try to construct, in p2, an argument a1* analogous to that for c1 in p1. The conclusion of a1* would be the analogue of c1; like c1, it is probably incompatible with c2. Thus, the analogy might allow us to construct, within p2, a counterargument to c2. And, by parity of reasoning, in p1 there might be an argument a2* against c1. This is just a plausibility consideration. Detailed analysis may show that this construction cannot be carried out after all, but the pressure consists exactly in putting such analysis on the agenda. In the present case, the two questions motivated by the similarity of systems are: 1. Can Brouwer’s argument be reconstructed within Husserl’s phenomenology? 2. Can Husserl’s argument be reconstructed within Brouwer’s philosophy? In chapter 5 it will turn out that the important question is the ﬁrst one. The second fact that generates pressure between Brouwer’s and Husserl’s positions is that, by an appeal to mathematical experience (or practice), both have a high prima facie plausibility. This fact has force in the discussion between Brouwer and Husserl only because in their approaches to knowledge actual, cumulated experience is the starting point, in a way to be explained. (Note that neither Husserl nor Brouwer began by asking about the relation between mathematical objects and time, but arrived at this question in an eﬀort to account for mathematical knowledge.) These two facts will now be discussed in some more detail. 4.2.1 Similarity of Methods Both Husserl (after 1906/07) and Brouwer were struck by the fact that our experiences (in the widest sense of the word) are before anything else subjective. There is a ‘mineness’ to my experience that precludes it from being part of the objective world at the same time. Subjectivity makes objectivity understandable, not the other way around. Accordingly, Brouwer and Husserl hold that any philosophical account of our experience ultimately refers back to essential properties of subjectivity. That qualiﬁes both philosophies as transcendental [175].28 Husserl writes: [Transcendental philosophy] is the motif of inquiring back into the ultimate source of all the formations of knowledge, the motif of the knower’s reﬂecting upon himself and his knowing life in which all the scientiﬁc structures that are valid for him occur purposefully, are stored up as acquisitions, and have become and continue to become freely available . . . This source bears the title I-myself, with all of

4.2 Two Sources of Mutual Pressure

19

my actual and possible knowing life and, ultimately, my concrete life in general. The whole transcendental set of problems circles around the relation of this, my ‘I’ – the ‘ego’ – to what it is at ﬁrst taken for granted to be – my soul – and, again, around the relation of this ego and my conscious life to the world of my which I am conscious and whose true being I know through my own cognitive structures. [108, 97–98]29 This subjectivity (or ‘letzte Quelle’) cannot be understood as being part of the objective world. Husserl had three arguments for this transcendental turn: a Cartesian, a psychological, and an ontological one. In the course of time he came to reject the ﬁrst two, which I will mention here to bring out the contrast with the third, clarifying its sense. The ontological way is the one that Husserl travels in the work that serves as one of my methodological refererence points, the Formal and Transcendental Logic [112]. For discussion of the intertwined and sometimes confused development of these three ways, and Husserl’s reasons for ﬁnally abandoning the ﬁrst two, I refer to Kern [141, section 18]. The following characterisations are adapted from Kern. The Cartesian way 1. Philosophy is to be an absolute science, developing itself from an absolutely certain beginning. 2. This beginning cannot be found in any knowledge of a transcendent world, as the existence of such a world may be doubted and is therefore not absolutely evident. 3. But the beginning can be found in the cogito, the content of which cannot be doubted and is absolutely certain. 4. The world is intentionally contained in this cogito, as cogitatum. Its existence is not certain but its presence as cogitatum qua cogitatum is. The cogito cannot be identiﬁed with anything in the world, because the world, and hence everything in it, is subject to a doubt that the cogito and its content are not. The psychological way 1. Some sciences study corporealities: chairs, human beings, works of art are all considered as ‘things’, abstracting from all psychological aspects; this suggests a complementary science that studies the psychological, abstracting from all corporeal aspects. 2. Just abstracting from the human body does not do away with intentional reference to wordly objects as such. Instead of the meant object we should consider the object meant as meant. What is of interest in pure psychology is the perceiving as such, not whether the perceived object is real. 3. But one-by-one reduction will not do. Every intentional act has a horizon that implies further worldly objects, and the world as such. To arrive at

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4 The Original Positions

the purely psychological, every implication of something non-psychological should be dropped. A universal reduction is necessary. What is reduced to is not wordly, it is the transcendental. The ontological way 1. Every positive (or objective) ontology leaves the relation with subjectivity unthematised. In this sense such an ontology remains one-sided. 2. Thematising this relation requires a radical reﬂection which focuses on how the objective is intentionally implied in the subjective. 3. This subjectivity (or transcendental ego) should not be construed as somehow objective or as part of a positive ontology; doing so would be to unthematise the relation again. Brouwer also sees that subjectivity is to be explained in its own, nonworldly terms; for example, he says that ‘things’ (physical objects, including ‘the home body of the subject’) are ‘completely estranged from the subject’ [39, p. 1235] (more on this in section 5.5). Characteristically, what he oﬀers is a statement of his position rather than an argument for it. Although perhaps regrettable from a dialectical point of view, this trait is actually in accordance with the content of some of those statements. Brouwer holds that everyone can only try to ﬁnd the evidence for himself, and language should not be trusted as an instrument to share insights: In default of a plurality of mind, there is no exchange of thought either. Thoughts are inseparably bound up with the subject . . . By so-called exchange of thought with another being the subject only touches the outer wall of an automaton. This can hardly be called mutual understanding. [39, p. 1240] One can see Brouwer’s view at work in his (draft of a) letter to Weyl from 1920, concerning Weyl’s ‘Foundational crisis’ [238] (which paper will be discussed in section 4.3.4): The fact that we disagree on some minor points can only have a stimulating eﬀect on the reader. Of course, you are fully entitled to formulating these diﬀerences of opinion . . . These are not matters of discussion; they can only be a matter of decision through individual concentration. [157, p. 102]30 Even so, it seems that Brouwer has an argument for his turn to subjectivity and that this is similar to Husserl’s ontological argument: as long as you describe objects, the (irreducible) subject remains out of sight. For example, speaking of languages in which the subject and other people are treated as objects in a causal world, Brouwer says that the part assigned to the subject individual in this language is analogous to those assigned to object individuals, whereas the subject itself

4.2 Two Sources of Mutual Pressure

21

is ignored in it. In this way civilised languages . . . suggest a sameness for such totally diﬀerent phenomena as acts of the subject and acts of object individuals are. [39, p. 1239] For Brouwer as well as for Husserl, the relation between the subjective and the objective is not one between two separate relata. The subjective in some sense contains the objective. In Husserl, this sense is that of intentional implication; in Brouwer, experiencing a world of objects is a particular phase of consciousness (more on this in section 5.5). Brouwer speaks of the phases consciousness has to pass through in its transition from its deepest home to the exterior world in which we cooperate and seek mutual understanding. [39, p. 1235] According to Husserl and Brouwer, then, the description of consciousness as such cannot make any presuppositions about an existing world or about objects in an existing world. The phenomenological idiom is that their theories are not mundane. The transcendental ego is not part of a world. Here is one diﬀerence between both of them and Kant [141, p. 239]. For Kant, the transcendental ego is not accessible to experience, hence not empirical, but it is still part of the noumenal world (‘das eigentliche Selbst, so wie es an sich existiert, oder das transzendentale Subjekt’ [137, B520]); both Husserl and Brouwer deny that there is a noumenal world. Husserl and Brouwer describe the transcendental ego eidetically, i.e., in terms of its essential properties. In Brouwer in particular this takes the form of a description of an idealised subject31 (the ideal mathematician), which he came to call the ‘creating subject’.32 Describing essential properties and describing an idealised subject here amount to the same, as the idealisation involved is that of abstracting from empirical limitations, and essential properties are those that govern any instance, empirically possible or not. It will be clear by now that I do not suspect Brouwer of psychologism, pace Dummett [68, p. 609] and Parsons [169, pp. 213–214]. The subject that Brouwer describes, I hold, is a transcendental subject, not a psychological subject. That Brouwer meant it as such, is corroborated by a letter he wrote to van Dantzig, dated August 24, 1949, now in the Brouwer Archive in Utrecht. In this letter, he distances himself from ‘those, who cannot recognise the “creating subject”, because they look at intuitionism from the psychologistic point of view’, and speaks of his ‘conviction, that psychologistic interpretations of intuitionistic mathematics, however interesting, will never be adequate’.33 In summary, Brouwer and Husserl share both a transcendental and an eidetic turn. There may be substantial diﬀerences between their ways from here. At ﬁrst sight, Husserl and Brouwer seem to have diﬀerent ideas about the transcendental ego and intersubjectivity.34 However, it is not clear to me to what extent these diﬀerences are real. To go into these matters here would distract from my present aims, but they would be a central topic in a phenomenological critique of intuitionism. It is certain that Brouwer and

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Husserl disagree about what genetic functions (or structures) there are. I will say something about this at the end of this subsection. What matters now is the basis that they share. I will explain how their standing on common ground contributes to making the incompatibility into a conﬂict. (Diﬀerences between Brouwer and Kant are discussed by Parsons [169] and Posy [175]; between Husserl and Kant, by Kern [141] and Mohanty [163].) They also share the Kantian idea that concepts need complementary intuitions to arrive at knowledge.35 (Parsons [169, p. 211] remarks that even more closely related to Kant’s conception of intuition than Brouwer’s is Hilbert’s. See also note 267.) In intuition, objects are actually given as themselves (as opposed to being merely thought, or merely intended). Such knowledge may be called ‘authentic’ (‘originär’). (Husserl and Brouwer accepted intuition of abstract objects, which Kant did only insofar as it concerns the forms of intuitions of physical objects (i.e., space and time). See note 216.) Transcendental philosophy is aware that knowledge is never immediately given but is an accomplishment of consciousness. Intuitions, to be obtained, generally require a series of mental acts. This series of acts has a speciﬁc structure that depends on the kind of object to be intuited. The contents of our stream of experiences do not follow one another randomly but are systematically related. There are structures that govern the ﬂow of consciousness. Even what is normally called ‘a ﬂash of insight’ is a systematic whole or synthesis of acts. To bring out and describe those syntheses is the task of genetic analysis. The way philosophy bears on epistemic claims is that it tries to make explicit the structure of the series. It may turn out that some alleged knowledge cannot thus be explicated; after all, we do make mistakes. These are in their own way performances of consciousness. At a certain stage of the genesis, a step was taken that does not fall under the essence of truth-giving acts. By disclosing this, philosophy performs a critical function, it shows that the knowledge claim is unjustiﬁed. How this works out in Husserl’s framework, I will discuss later on: the critical function, in the context of mathematics, in section 5.1; Husserl’s later, deepened understanding of genesis, in section 5.4; its application to choice sequences, in chapter 6. The three examples of genetic accounts that I want to give now, I take from Brouwer, who is far less explicit about it than Husserl. The ﬁrst can be found in Brouwer’s thesis from 1907. It concerns the relations between mathematics, language, and metamathematics. It is the one place where Brouwer explicitly calls his account ‘genetic’ [23, pp. 173–175]. What he does is to start from purely mathematical constructions and to describe how, through repeated acts of reﬂection and subsequent description, logic (as a linguistic rendering of structures in mathematical thought) and metamathematics arise. At the ﬁrst level there is the pure mathematical constructing; these constructions are described linguistically on the second level. We can go on and study those linguistic descriptions mathematically in turn, bringing out general structures in them.36 Going on in this way, Brouwer distinguishes mathematical systems of the ﬁrst, second, third order, and

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23

naturally this can be continued (although he doubts whether that yields anything essentially new, p. 175). Let me bring out two important aspects of this example. First, by showing how logic and metamathematics arise through (certain) acts of reﬂection and description, Brouwer justiﬁes them; this is because of the constructivist nature of intuitionism. Secondly, having analysed this genesis into several phases, he uses the genetic account as a basis for criticism of other philosophies of mathematics. He considers the various conceptions (in his thesis, those of Hilbert, Peano, and Russell) answerable to his genetic schema, indicating its transcendental nature: every factual development can be judged in its light. For example, at step 6 he writes: 6. The mathematical study of this language. This step is essential in Hilbert’s work, in contrast to that of Peano and Russell. [44, p. 95]37 He then takes Hilbert to task for concluding from a linguistic system to the primary intuition that it accompanies, going against the genetic order [p. 176n. 1]. The general point is that if you can give a genetic account such that it appeals only to certain vindicated kinds of acts, that counts as a justiﬁcation; whereas if that is not possible, that counts as a ground for repudiation. That is not how Hilbert would have looked on the matter, but Brouwer and Husserl repeatedly speak of the possibility that science misunderstands itself. That is not to imply that the scientiﬁc results are wrong, but it means that scientists may give completely mistaken interpretations of their scientiﬁc accomplishments. (Such cases illustrate the distinction between ‘natural’ and ‘philosophical’ attitudes.) A second example of a genetic account in Brouwer, also in his dissertation, is his criticism of Cantor’s second number class, i.e., all countable ordinals as a totality. Brouwer rejects this totality because, he claims, a correct genetic account cannot be given: here [Cantor] mentions something which cannot be thought of, i.e., which cannot be mathematically constructed; for a totality constructed by means of ‘and so on’ can only be thought of, if ‘and so on’ refers to an ordertype ω of equal objects, and this ‘and so on’ does not refer to an ordertype ω, nor to equal objects. Here Cantor loses contact with the ﬁrm ground of mathematics. [44, p. 81, original emphasis]38 A third case where the combination of genetic analysis and criticism appears in Brouwer’s writings is ‘Historical background, principles and methods of intuitionism’ [40], which begins: The historical development of the mental mechanism of mathematical thought is naturally closely connected with the modiﬁcations which, in the course of history, have come about in the prevailing philosophical ideas ﬁrstly concerning the origin of mathematical certainty, secondly

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concerning the delimitation of the object of mathematical science. [40, p. 139] By ‘mental mechanism of mathematical thought’ Brouwer means the actual practice of mathematics, as becomes clear in the next sentence, which begins by stating that ‘the mental mechanism of mathematical thought during so many centuries has undergone so little fundamental change’. The various modes of thinking that mathematicians have accepted in their practice are, according to Brouwer, directly related to their ideas as to why mathematics is certain and what mathematics is about. He then goes on to show this connection in the case of various positions which he identiﬁes as classical mathematics, old formalism, pre-intuitionism, and new formalism. He concludes that none of them can give a satisfactory account of the intuitive continuum, and then introduces intuitionism as intervening at this point, by two acts: The ﬁrst act of intuitionism completely separates mathematics from mathematical language, in particular from the phenomena of language which are described by theoretical logic, and recognises that intuitionist mathematics is an essentially languageless activity of the mind having its origin in the perception of a move of time, i.e., the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory. If the two-ity thus born is divested of all quality, there remains the empty form of the common substratum of all two-ities. It is this common substratum, this empty form, which is the basic intuition of mathematics. ... The second act of intuitionism . . . recognises the possibility of generating new mathematical entities: ﬁrstly in the form of inﬁnitely proceeding sequences p1 , p2 , . . ., whose terms are chosen more or less freely from mathematical entities previously acquired; . . . secondly in the form of mathematical species, i.e., properties supposable for mathematical entities previously acquired. [40, pp. 140–141, 142] From a phenomenological point of view, these acts are given in terms of eidetic structures of consciousness. In eﬀect, Brouwer criticises the other positions mentioned for making claims about mathematics that are not in accordance with these structures (or disregard them). This third example brings us to the matter of history. In any transcendental account, the explanandum is experience. Brouwer and Husserl look upon the practice of mathematics not as the employment of fruitful techniques but as the collecting of experiences of certain objects (intellectual experience). But whose experience? There is the experience of the individual and that of the mathematical community, the latter in the synchronic and diachronic sense – there is a tradition, and it will be continued. The ‘who’ of mathematical experience (in its broadest sense) is ‘anyone’ (e.g., [126, section 49],

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[100, Beilage III]). The individual mathematician ﬁnds himself in a body of experience that is there already; his further development is a process of give and take between himself and the tradition, to which the individual in turn contributes. This ‘anyone’, precisely because of its universality, can only be described by describing the transcendental subject. It pertains to the structures of consciousness not of any particular individual at any particular point in time, but of everyone as a matter of essence. That is why Husserl and Brouwer, in giving their philosophical account of mathematics, take a turn to history, not to ask factual questions but to inquire into the essential structure of any genesis of the knowledge we actually have (or claim to have). A well-known application is Husserl’s account of the origins of geometry [100, Beilage III]. The investigation into the origins is not meant as historical research in the common sense of the word. ‘Origins’ here refers to the properties of the transcendental ego that have to be in place if it is to conceive of any geometry; these determine how the development of geometry is possible, and set the precise boundaries for the space within which any factual development can unfold itself. The history as it actually happened is only one of the possibilities. Against the background of a genetic account of knowledge, every claim to knowledge raises the issue of what in experience motivates that claim. What earlier performances of consciousness are the conditions of possibility of the claim? Even mistakes are commonly explained by pointing out factors that invited (or motivated) them. Knowledge (putative knowledge) is justiﬁed (accounted for) by referring it back to genetically earlier performances of consciousness. If there is to be a real conﬂict between the two statements on the temporal character of mathematical objects on p. 16, there should be strong motivations for each of them coming from diﬀerent corners of mathematical practice. Before searching for such support, we should consider the following line of thought. Husserl and Brouwer diﬀer on the question what kinds of genetic functions there are (constitution and construction (or generation), respectively). Isn’t it possible that the conﬂict arises because Brouwer might allow particular structures that Husserl can’t?39 That the root of the conﬂict does not lie here should be clear from the fact that these two concepts seem to be related by subsumption: everything that Brouwer understands by ‘construction’ and ‘generation’ seems to fall under Husserl’s concept of constitution. Again, the problem of the exact determination of their relation would have to ﬁnd its place in a phenomenological critique of intuitionism. Perhaps one could say that constitution is construction and generation plus ideas of reason.40 But it is in any case clear that whatever the precise meanings of ‘construction’ (or ‘generation’) and ‘constitution’ are, both should be able to deal with our factual ability simply to start choosing a sequence of numbers. In the conﬂict at hand, the real question then is not so much whether there is a genetic account that justiﬁes choice sequences as objects per se (there should

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be – see section 5.5), but whether they are to be considered mathematical objects, as Brouwer claims (see section 6.3). For example, one alternative position would be to accept choice sequences as objects, but not as objects of pure mathematics. This seems to have been Gödel’s view, as described in Kreisel’s Memoir: He never questioned the possibility of a part of mathematics which is intended to be about our own ‘constructions’ or choices . . . But he did not regard that part as at all useful for mathematics itself, let alone the whole of legitimate mathematics. [147, p. 202] Gödel’s Platonism excludes that pure mathematics is about our own ‘constructions’ or choices, but not that there are parts of mathematics that can be applied to describe these. The theory of choice sequences would then be a matter of applied mathematics, just as, say, mechanics; and properties of choice sequences would not lead to conﬂicts with the nature of purely mathematical objects, for choice sequences would inhabit a diﬀerent ontological region. In a draft letter to Bernays from July 1969, Gödel reports that it now seems to me, after more careful consideration, that choice sequences are something concretely evident and therefore are ﬁnitary in Hilbert’s sense, even if Hilbert himself was perhaps of another opinion. [77, p. 269] In the letter he actually sent at the end of that month, he says that choice sequences seem to him ‘to be quite concrete, but not to extend ﬁnitism in an essential way’ [77, p. 271]. The formulation he chooses in a manuscript from 1972 (a revision of his Dialectica paper from 1958) is slightly weaker; commenting on Gentzen’s (necessarily, non-ﬁnistic) consistency proof of arithmetic from 1936, he says that ‘a closer approximation to Hilbert’s ﬁnitism can be achieved by using the concept of free choice sequences’ [75, p. 272, footnote c]. Note that to accept (any non-lawlike variety of) choice sequences as objects of ﬁnitary mathematics would imply that ﬁnitary mathematics is not a proper part of classical mathematics. In so far as ﬁnitary mathematics is conceived of as a (schematic) theory of our handling of intuitive objects, this may indeed be the case [4, p. 425]. That would go together well with Gödel’s view, as reported in a letter from Shen Yuting to Hao Wang of April 3, 1974, that ‘classical mathematics does not “include” constructive mathematics’;41 even though there Gödel will rather have been concerned with the more general fact that constructive mathematics by deﬁnition involves explicit reference to properties and activities of a mathematician, whereas classical mathematics does not. (For Gödel’s use of the continuity principle for choice sequences as a heuristic, see the digression on p. 103; a more elaborate discussion of Gödel’s views on, and work related to, intuitionism, is found in [8].)

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Wittgenstein’s view on choice sequences is similar to Gödel’s view as described by Kreisel. Reacting to Weyl, Wittgenstein wrote: A freely developing sequence is in the ﬁrst place something empirical. It is nothing but the numbers that I write down on paper. If Weyl believes that it is a mathematical structure because I can derive a freely developing sequence from another by means of a general law, e.g., m1 , m2 , m3 , . . . m1 , m1 + m2 , m1 + m3 , . . . then the following is to be said against it: No, this shows only that I can add numbers, but not that a freely developing sequence is an admissible mathematical concept. [246, p. 83] Wittgenstein’s particular criticism of Weyl here is obviously cogent; but it does not suﬃce for showing that choice sequences are not mathematical objects. 4.2.2 Initial Plausibility of Both Positions Brouwer and Husserl point to diﬀerent aspects of mathematical practice that speak in favour of their respective positions. In fact, when Husserl and Brouwer became aware of these aspects, they changed their respective earlier philosophies of mathematics substantially. Husserl moved from psychologism to essentialism; Brouwer changed his theory of the continuum. Husserl Consider the point of view of the ‘working mathematician’ (as described, for example, in the essays gathered by Tymoczko [228]): 1. There are mathematical objects out there, and the task is to describe them and theorise about them. In general, there are diﬀerent ways of describing the same objects. You make conjectures about these objects, which turn out to be right or wrong; but if you can prove a mathematical theorem, it can never become a falsehood later on (there may have been hidden assumptions in our ﬁrst proof, but we consider the ideal case where all assumptions are explicit). 2. Furthermore, the truths of mathematics are, in contrast to those arrived at in the empirical sciences, not contingent. Problems that are still open nevertheless have a ﬁxed answer, that may or may not be found. 3. This peculiar nature of mathematical truths goes hand in hand with the idea that the domain of objects must also be of a diﬀerent nature from that of the empirical sciences. Mathematical objects are not in the world, but in a realm of their own. This view implies a rejection of psychologism and other forms of reductionism. It rejects all attempts to say that mathematics is ‘really’ about, or dependent

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in its content on, something else. When pressed by reductionistic philosophers, mathematicians may change their account, but back at their desks they comfortably assume their usual attitude again (‘the typical working mathematician is a Platonist on weekdays and a formalist on Sundays’ [83, p. 11]). And with good reason, it seems: isn’t it true that mathematical theorems are established once and for all? Isn’t it true that mathematical truth is independent of geography and culture, and can in principle be shared by everyone, living anytime, anywhere?42 Isn’t mathematics independent of what happens to be going on in space and time? One should think that any philosophy of mathematics should to a large extent do justice to this experience. A glance at the genesis of Husserl’s thought shows that the need to account for this aspect of mathematics was always one of the driving forces in its development.43 Husserl was originally trained as a mathematician in Berlin, one of the leading departments of the time. There he studied with Kummer, Kronecker and Weierstraß; in particular the latter made a very strong impression on him: It was my great teacher Weierstraß who, in my student years, by his lectures on the theory of functions raised in me the interest in a radical foundation of mathematics. I came to appreciate his efforts to transform analysis, which was such a mixture of rational thought and irrational instinct and feeling, into a rational theory. [191, p. 7, trl. mine]44 Husserl clearly shares the point of view of the working mathematician, and philosophical accounts of mathematical knowledge should respect it: If we likewise, untroubled over all controversy, for example, between Platonism and Aristotelianism, speak of numbers of ‘the’ number series, of propositions, of pure classes and types of formal as well as material givens (as, for example, the mathematician as arithmetician or geometrician does), we are not yet epistemologists. We follow the evidence which such ‘ideas’ give us . . . But all the same it is a primary and entirely necessary step for the posing of intelligent epistemological questions to talk in this way and not to allow oneself right away to be talked out of ideas as givens. [114, p. 39]45 How this demand played an important role at two junctures in Husserl’s development – the abandonment of Brentano’s conception of intentionality in favour of his own, and the move from descriptive psychology to transcendental phenomenology – I will now indicate. Scientism (or naturalism), widespread in the nineteenth century (and later), is the view that everything is to be explained by the natural sciences. Applied to logic and mathematics, this scientism often took the form of psychologism. In reality, this position stated, the truths of logic and mathematics are grounded in the empirical science of psychology.

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But here a distinction must be introduced. On the one hand there is genetic psychologism, which wants to account for the validity of laws of logic and mathematics by giving causal explanations, referring to human beings as part of the objective world. On the other hand there is descriptive psychologism, the position that any account of logic and mathematics can only be built upon a purely psychological description of the relevant phenomena. (That precludes bringing in causal considerations into the account later.) (Seebohm [193] gives a detailed taxonomy of these and other varieties of psychologism.) Where genetic psychologism comes up with psycho-physiological explanations, descriptive psychologism rejects those and conﬁnes itself to a pure description of the lived experience of human beings in the world. Brentano stated as its task: To give clarity about what inner experience immediately shows, hence not a genesis of facts, but ﬁrst and foremost a description of the subject-matter. This part is not psycho-physiological, but purely psychological. We must in advance know how the facts look: and this is shown by an inner glance into the psychical. [181, p. 24]46 Husserl’s ﬁrst book, the Philosophy of Arithmetic from 1891 [107], was a psychologistic account of mathematics; in the next, the Prolegomena to the Logical Investigations (1900), he denounces psychologism, suggesting that he no longer agrees with his earlier view in the Philosophy of Arithmetic. But in fact, the psychologism of the Philosophy of Arithmetic is a descriptive one, whereas what he rejects in the Prolegomena is genetic psychologism. As Husserl explicitly states [113, p. 262], the Prolegomena themselves are still written from the point of view descriptive psychology. At the time, he was in the middle of his project of applying descriptive psychology to problems in the foundations of logic and mathematics. He had started that undertaking as a ‘wholeheartedly orthodox Brentanist’ [181, p. 44]; and while a shift away from that position began shortly after the publication of the Philosophy of Arithmetic, he surely remained within the Brentanist school. This shift had to do with a discussion with another student of Brentano’s, Twardowski, which led Husserl to develop an alternative to (the early) Brentano’s conception of intentionality [181, ch. 5]. Brentano held that intentional objects are immanent objects, that is to say, they are contained (‘inexistent’) in the particular intentional act: Every psychical phenomenon is characterized by what the Scholastics of the Middle Ages called the intentional (or sometimes the mental) inexistence of an object, and what we should like to call, although not quite unambiguously, the reference to a content, the directedness toward an object (which in this context is not to be understood as something real) or the immanent-objectquality (immanente Gegenständlichkeit). Each contains something as its object. [203, p. 36]47

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A particular instance of this thesis was defended extensively in Twardowski’s Habilitationsschrift ‘Zur Lehre vom Inhalt und Gegenstand der Vorstellungen’ (1894), where the meaning of a name is identiﬁed with the presentation which is manifested in the particular act of uttering that name. Husserl came to see that, on the contrary, meanings are not parts of acts as Twardowski suggests. In a review (1896) of Twardowski’s book, Husserl gives three objections [116, pp. 349–350]: 1. A name may have a generally accepted meaning, while the manifested presentations may vary from one person (or occasion) to the other; the use of the name ‘tree’ may arouse an image of an evergreen in me but a linden in you. 2. Presentations in distinct individual acts are at best similar, whereas meanings remain identical. Presentations that accompany the same name can vary greatly, but it makes no sense to speak of such variations in the case of meanings. 3. The objection that goes to the heart of the matter and lies at the root of the other two is that contents are parts of individual acts but meanings are not: The content is, as such, an individual psychical datum, a being which is here-and-now. But the signiﬁcation is nothing individual, nothing real, never at any time a psychical datum. For it is identically the same ‘in’ an unlimited manifold of individual acts separated in reality. That there is and can be something like minding what is identically the same ‘in’ distinct acts – this is the primal fact, certiﬁed through Evidence, underlying all knowledge and ﬁrst giving it any sense whatsoever. – The content resides in the representation as a real constituent, but the signiﬁcation does so only functionally. It would be absurd to conceive of it as a real fragment or as a part of the representation. [127, p. 389]48 Intentionality, Husserl concludes, should be conceived of in such a way that an act can intend a meaning without containing it. Meanings are not immanent objects as Brentano’s notion of intentionality has it. Acts and their contained parts are particulars, while meanings are universals. Husserl applied the same reasoning to the objects of logic and mathematics, as we will see in a moment when discussing the Prolegomena. This change in Husserl’s understanding of intentionality is why he, while working on the Prolegomena, can write to Natorp on January 21, 1897, that this work is directed against Brentanism, even when Husserl’s general outlook at the time could still be classiﬁed as that of a descriptive psychology: I am working on a longer paper, which is directed against the subjectively psychologizing logic of our times (against the standpoint, that is, that I used to support when I was a student of Brentano). [128, V:p. 43, trl. mine]49

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And that Husserl here has his new idea about intentionality in mind transpires from the next sentence: In the course of this, there will no doubt be ample opportunity to deal with the diﬀerence in question [i.e. the disagreement with Twardowski on intentionality, which Natorp had referred to in a previous letter]. [128, V:p. 43, trl. mine]50 Husserl comes to reject genetic psychologism in the Prolegomena by exposing several of its implausible consequences and attacking several of its presuppositions. The heart of the book is formed by sections 44–48, where it is pointed out that genetic psychologism (in logic) does not accord with the meaning of logical statements. Logical statements simply are not about psychological events, nor are such events implied by them. Genetic psychologism is a relativism because it makes logical truth dependent upon events in the world, hence on empirical matters. But, Husserl points out, the content of logic has nothing to do with contingent facts. The truths of logic and mathematics are independent of the changing facts of the world. They do not deal with what is in spacetime, not with the ‘real’ as Husserl calls it. From Lotze he borrows the term ‘ideal’ [116, p. 156] to describe the other realm, that which logic is concerned with.51 Mathematics and logic are ideal sciences, ﬁnding out the ideal truths about ideal objects. What is real is the individual with all its constituents: it is something here and now. For us temporality is a suﬃcient mark of reality. Real being and temporal being may not be identical notions, but they coincide in extension . . . Should we wish, however, to keep all metaphysics out, we may simply deﬁne ‘reality’ in terms of temporality. For the only point of importance is to oppose it to the timeless ‘being’ of the ideal. [111, pp. 351–352]52 Husserl here says that ideal objects, and this includes the mathematical ones, are outside of time altogether. From 1917 on, he would say they are inside of time, but at every moment the same. In section 5.4, I explain his reasoning behind this, and why his argument is in some sense incomplete. Shortly after publishing the Logical Investigations, Husserl realised that the framework of descriptive psychology is still not able to ground logic and mathematics. It is true that descriptive psychology describes experiences purely psychologically, not psycho-physiologically (hence Husserl’s later praise for Brentano [181, p. 49]). But while this is a step forward compared to genetic psychologism, it still describes rexperiences as experiences of a being in a world. It does not describe the world directly, but the world and its a priori structures are implied by the descriptions; for in descriptive psychology, consciousness is viewed as consciousness of a being that is localised in objective space and objective time. Hence material essences are presupposed in these descriptions, but the evidence for material essences is always weaker

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than that for formal essences; and it is impossible to ground the evidence for the completely formal in the weaker material kind. So although the descriptive psychology of the Prolegomena is able to show that logical and mathematical objects are not mental objects, it is not able to ground logic and mathematics. In a review from 1903 of Elsenhans’ article ‘The relation of logic to psychology’ [72], Husserl declares But it is absolutely imperative for any self-consistent epistemology, or philosophy in general, that one eﬀect with ﬁnality the separation in principle between purely immanent phenomenology or critique of knowledge, which holds itself free from all suppositions extending beyond the content of the given, and empirical psychology, which, even where it merely describes, makes such suppositions; and that one accordingly does not confuse, as is commonly done, questions about ‘origin’ in the critique of knowledge with those about psychological origin. [127, p. 252]53 This insight contributed to his rejection of naturalism in philosophy – note that to ‘self-consistent epistemology’ he adds ‘or philosophy in general’. This rejection in turn opened the way to the phenomenological reduction, ﬁrst hinted at in 1904 [141, p. 182] and described around 1905/1906. With this, Husserl took his transcendental turn. In a diary note of September 25, 1906, Husserl looks back on his work since his Habilitationsschrift of 1887, and notes that the central question has been that of how the subjective and the objective are related to each other: And while laboring over projects concerning the logic of mathematical thought, and of the mathematical calculus in particular, I was tormented by those incredibly strange realms: the world of the purely logical and the world of actual consciousness – or, as I would say now, that of the phenomenological and also the psychological. I had no idea of how to unite them; and yet they had to interrelate and form an intrinsic unity. [127, pp. 490–491]54 This retrospective note by Husserl leads to the conclusion of this historical outline. It is not just that Husserl’s transcendental philosophy is concerned with the task of providing an account of the objectivity of mathematics. It is that, by and large, his perceived need to do justice to this objectivity was responsible for the main turning point in Husserl’s philosophy, that from descriptive psychology to transcendental phenomenology. The development of his thought toward that later system was partly guided (and always checked against) this part of our mathematical experience. Brouwer Brouwer is the one who showed how to incorporate into mathematics a point that was already made by Aristotle and before: a set of discrete points cannot

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not represent a geometrical or intuitive continuum. Aristotle writes in the Physics: For whoever divides the continuous into two halves thereby confers a double function upon the point of division, for he makes it both a beginning and an end . . . and by the very act of division both the line and the movement cease to be continuous. [1, book VIII, ch. 8] and Weyl, speaking of the set-theoretical continuum, emphasised: Do not forget that in the ‘continuum’ of the real numbers, the individual elements are in fact precisely as isolated from one another as, say, the whole numbers. [240, p. 122]55 The intuitive continuum cannot be conceived of as a set of distinct atoms. Note that the cardinality of such a set does not inﬂuence the argument; rather, it is its discreteness that poses the problem.56 Brouwer, too, from the very beginning insisted that continuity and discreteness are irreducible, complementary notions: So where in that ur-intuition continuous and discrete occur as inseparable complements, both having equal rights and both equally clear, it is impossible to insulate oneself from one of them as an original entity, and then to construct it from the other one which has been set apart by itself; for it is already impossible to consider that other one by itself. [44, p. 17]57 However, the continuum as a whole was given to us by intuition; a construction for it, an action which would create from mathematical intuition ‘all’ its points as individuals, is inconceivable and impossible. [44, p. 45]58 Posy [176, p. 312] points to a passage in Kant that is very similar to these two just quoted: The property of magnitudes by which no part of them is the smallest possible, that is, by which no part is simple, is called their continuity. Space and time are quanta continua, because no part of them can be given save as enclosed between limits (points or instants), and therefore only in such fashion that this part is itself again a space or a time. Space therefore consists solely of spaces, time solely of times. Points and instants are only limits, that is, mere positions which limit space and time. But positions always presuppose the intuitions which they limit or are intended to limit; and out of mere positions, viewed as constituents capable of being given prior to space or time, neither space nor time can be constructed. [138, B211]59 The historical interest of this parallel derives from the fact that Brouwer studied Kant closely when writing his dissertation. However, neither in the

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ﬁnished work, nor in the drafts now in the Brouwer Archive in Utrecht, do we ﬁnd Kant’s passage mentioned. In his later writings, Brouwer is not as explicit about this primitive intuition of the continuum anymore. It has been suggested that Brouwer abandoned this primitive intuition. For example, Placek writes: With the introduction of the intuitionistic concept of set (called species) and by generalizing the notion of numerical sequence (dubbed spreads), Brouwer introduced the species of real numbers by showing that it is equivalent to the species of spreads of rational numbers. Consequently, the continuum disappeared from his account of intuition, as the succession of elements turned out to be prior to continuity. [174, p. 28, original emphasis] and Troelstra and van Dalen: Around 1917 Brouwer must have realised that from an intuitionistic point of view, there was no objection against considering sequences obtained by successive choices . . . This move enabled Brouwer to reinstate the ‘arithmetical’ account of the continuum, but now in an intuitionistic context; and this seems indeed to provide a much more satisfactory intuitive grasp of the continuum than to postulate the continuum as a primitive intuition. [226, p. 642]60 But Brouwer could only have given up the primitive intuition of the continuum if he would have been prepared to reject the complementarity (in intuition) of continuity and discreteness. That he indeed was not prepared to do so is clear from a remark in his lecture in Vienna on March 14, 1928. Having introduced the notions of choice sequence, spread (then, in German, still called ‘Menge’), as well as the representation of the unit continuum as a ﬁnitary spread (fan), he says: After reﬂection on the mathematical ur-intuition of two-ity – the basis of the whole of intuitionism – the introduction of the spread construction on which the ﬁnished overdenumerable power of the continuum is based, does not require further discussion . . . In the light of intuitionism the original interpretation of the continuum of Kant and Schopenhauer as pure a priori intuition, mentioned at the outset, can in essence be upheld. [157, pp. 57–58, trl. modiﬁed]61 To emphasise the point, in the margin of a copy of the published version of this lecture he wrote: In the continuum lecture, add at the end of [section] I that the continuum is all the same immediately given in the ur-intuition, just as in Kant and Schopenhauer. [Brouwer Archive, trl. mine]62

4.2 Two Sources of Mutual Pressure

35

The situation was correctly described by Heyting: From 1918 on Brouwer no longer mentions the continuum as a primitive notion. He can do without it because the spread deﬁned above represents it completely, as far as its mathematical properties go. [90, p. 84, emphasis mine] From the start, Brouwer did see the need for a mathematical analysis of the intuited continuum, which would otherwise not be mathematically tractable in any way. When Brouwer began his work, there were two types of mathematical analyses of the continuum. Analyses of the ﬁrst kind are those by Cantor and Dedekind, which depend on actually inﬁnite, non-denumerable sets. The other type is constructivistic and can be found in the work of the semi-intuitionists (as they were labeled afterwards), e.g., Baire, Borel, Lebesgue, and Poincaré – Brouwer later referred to them as ‘old-intuitionists’ [36] or even ‘pre-intuitionists’ [40]. The theory of the Cantorian type theory was rejected by Brouwer in his dissertation, because its non-denumerable sets are not acceptable to a constructivist. (On p. 23, I discussed Brouwer’s argument here as an example of genetic analysis.) Instead, he set out to do, independently, what the semiintuitionists had done before (Brouwer probably ﬁrst learned about them at the 1908 conference in Rome [60, pp. 203–204], where Borel stated his views on the continuum in a lecture titled ‘On the principles of set theory’ [19]). The idea is that if you cannot construct all points of the continuum, work with just those that you can construct and don’t bother about the rest. Brouwer called the scale of constructable points, that he superimposed on the intuitive continuum, the ‘measurable’ continuum; similar constructions of the semiintuitionists were the ‘reduced’ and the ‘practical’ continua. One knows for sure that the measurable continuum is poorer than the intuitive one, for the constructions are denumerable (denumerably unﬁnished, in Brouwer’s more speciﬁc terms), while Cantor showed that the points on the continuum are not [23, pp. 10, 62]. Brouwer could also have objected to both the Cantorian and the semiintuitionist analyses (with whose position he still more or less identiﬁed in his inaugural lecture of 1912 [25, 26]) by appealing to his general argument of the inseparability of the discrete and the continuous. But all by itself that objection would at that point only have had a negative eﬀect, as he did not have a truly alternative analysis yet. As far as he could see then, this failure to do justice to non-discreteness would be the case in any mathematical analysis of the continuum. Instead, it would be better to settle for a view like that of the semi-intuitionists and obtain some mathematics a constructivist could work with. But in 1917 [28, pp. 440–441], Brouwer writes that, for reasons internal to intuitionism, he is no longer satisﬁed with his theory of the measurable continuum that he had given in his thesis. He discovered that it depended on two suppositions whose intuitionist acceptability had yet to be shown (the

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suppositions of the individualised constructability of a point set and of the internal dissectability of every individualised point set). At ﬁrst he accepted these on the pragmatic ground that they were indispensable to get a viable theory. He could have gone on to clear up those two suppositions, or ﬁnd alternatives for them. But in fact he came up with a theory that superseded all those of the semi-intuitionist variety. Brouwer found that he could actually dispense with the questionable principles when he realised that his philosophy allowed for unﬁnished, non-lawlike objects. These were the choice sequences.63 Moreover, unlike its semi-intuitionist predecessors this new theory was able to capture the full continuum of our intuition (this claim will be substantiated phenomenologically in chapter 6). From then on, Brouwer did not want anyone to doubt whose contribution this was; for example, in a letter of January 28, 1927, he complains to Fraenkel: ‘And so I would like to beg you urgently not to continue the expropriation which the German mathematical review literature has practised on me, by making me share what is my exclusive personal intellectual property with Poincaré, Kronecker and Weyl’ [61, pp. 290–291].64 In his second Vienna lecture (1928) [36] and in the overview article of 1952 [40], he would also criticise the semi-intuitionist continua by an argument meant to show that they have measure zero; however, as we now know, that argument is not entirely conclusive [221, p. 481], [226, p. 640]. And the argument from measure seems not to have been necessary anymore, for the argument against discreteness had received a positive twist with the introduction of choice sequences. The primary motivation for the introduction of choice sequences was to come up with a mathematical theory that can handle the irreducible opposites discrete-continuous.65 Brouwer realised that the intuitive continuum should not, as in the Cantorian tradition, be analysed using the element-set relation, but that of part-whole. (This motivation will be treated in more detail in chapter 6.) He introduces unﬁnished, non-lawlike objects, which can by their nature not be considered as discrete elements forming a set. That is not to say that they cannot be individuated, one can always pick out a particular sequence by referring to the point in time it was begun; it is just that extensional identity of two individuated sequences is not generally decidable. Suppose we have individuated two sequences, of which the initial segments are identical, the ﬁrst-order restrictions on each are compatible, and at least one of them is non-lawlike. Then the open-ended character of the non-lawlike sequence entails that, despite the identical initial segments, still nothing can be said as to whether the whole sequences are extensionally identical. Brouwer proposed the ﬁrst mathematical theory that captured a part of our experience that has always been with us, the non-discreteness of the intuitive continuum; as Valéry observed, ‘The most novel novelties give birth to very old consequences’ [229, p. 51, trl. Robin Rollinger]66

4.3 Resolving the Conﬂict: The Options, and a Proposal

37

4.3 Resolving the Conﬂict: The Options, and a Proposal Let me put what I have found so far in a systematic form. I began by isolating two propositions that express respectively Brouwer’s and Husserl’s position on the temporal aspect of mathematical objects (p. 16). Then it became clear that much could be said for the plausibility of each. Part of their appeal lies also in what they are directed against: Husserl against psychologism and reductionism, Brouwer against atomistic conceptions of the continuum. Such considerations make both positions seemingly acceptable. However, this is where things become problematic: they are not consistent with one another. That both positions were arrived at in genetic analysis only increases the mutual pressure. If we add, for a reason that will become clear in a moment, two theses presupposed by both positions, we are dealing with this group of propositions: P1. P2. P3. P4.

Mathematics is about objects. Mathematical objects are within the temporal realm. Mathematical objects are omnitemporal. Some mathematical objects are intratemporal, dynamic and unbounded.

Taken together these propositions, although as I explained plausible in themselves, are not consistent; they form an aporetic cluster [179]. It is clear that to restore consistency, a subset of these propositions will have to be rejected; if not wholesale, then at least partially. P1 and P2 are presupposed by both P3 and P4; hence giving up P1 or P2 are among the possibilities to resolve the conﬂict. That is why they are listed explicitly. I will indicate, in reverse order, how rejection of any of P1–P4 may resolve the conﬂict. This (negative) step also presents a need for the further (positive) one of providing an alternative view that can deal with the prima facie intuitions and plausibility considerations mentioned in the previous section, either by answering to them or by arguing why they should be discarded after all. But we will see that not every way to resolve this aporetic cluster will be equally acceptable within the phenomenological framework adopted here. 4.3.1 Deny That Some Mathematical Objects are Intratemporal, Dynamic and Unbounded One could deny that some mathematical objects are intratemporal, dynamic, and unbounded. That simply is to give up Brouwer’s idea completely and to endorse Husserl’s position. One could even do this and still accept choice sequences as objects, by denying that they are objects of pure mathematics, and treating them as empirical instead (Gödel’s position quoted on p. 26). In any case, what could the motive for denying P4 be? Doing so involves abandoning, or not acknowledging, a strong intuition about the continuum. But continuity and discreteness are equiprimordial: one cannot be reduced to the other (as explained in the part on Brouwer in section 4.2.2).

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What if a mathematician would recognise the distinction between the mathematical (or arithmetic) continuum and the intuitive (or geometrical) continuum, but argue that mathematics has no need for an analysis of the latter? Then there would be no need for choice sequences. For example, G. H. Hardy wrote: The aggregate of all numbers, rational and irrational, is called the arithmetic continuum. It is convenient to suppose that the straight line . . . is composed of points corresponding to all of the numbers of the arithmetic continuum, and of no others . . . This supposition is merely a hypothesis adopted (i) because it suﬃces for the purposes of our geometry and (ii) because it provides us with a convenient illustration of analytical process. [79, p. 24] The answer is that to settle for an admittedly incomplete, yet convenient analysis is, philosophically, simply ducking the issue. A speciﬁc case of denying P4 are the arguments formulated by Felix Kaufmann [139]. The general objections to denying P4 will work here, but the speciﬁcity of Kaufmann’s arguments allows for additional ones. First, Kaufmann uses an epistemological argument to reach an ontological conclusion: For Brouwer’s view that mathematical facts change with mathematical knowledge implies that there is something cognizable that is created only by being cognized, which runs counter to the nature of cognition. For . . . all cognition presupposes an object that must be thought of as existing independently of its being cognized. [140, pp. 56–57]67 What is correct in this passage is that the object of knowledge is never constituted, let alone created, in the act of knowing itself. The act of knowing only thematises. In this sense one can say that the object of knowledge is always transcendent to the act of knowing. In the case of choice sequences, the acts of choosing the next number and coming to know what the next number is are hard to separate. But even then, we ﬁrst have to thematise that choice sequence as an object that can be extended; the choice sequence is transcendent to the particular act in which we choose the next number. But this foundation relation between act and object of knowledge requires only that the object be given as identical, which does not mean that its properties can never change. Change of properties depends on an underlying substrate that remains identical. So for Kaufmann’s argument to work, one would have to show in addition that a choice sequence does not really have such an underlying substrate, in other words, that ‘the identity of a choice sequence’ is just a manner of speaking. Kaufmann adds a second, weaker (because less general) argument against choice sequences in mathematics: their dependence on time would conﬂict with the essential properties of mathematical objects.

4.3 Resolving the Conﬂict: The Options, and a Proposal

39

Brouwer himself reaches a determination of real number very similar to that given here, as soon as we leave aside the time interpretations that accompany his account. These must absolutely be eliminated, if we are to gain an accurate grasp of the speciﬁc meaning of mathematical aspects. We must be absolutely clear that an ‘irrational number’ is not at all something that ‘becomes’. [140, p. 111n. 29]68 But we should reject that part of Brouwer’s doctrine which is essentially based on the introduction of the concept of time into mathematics. [140, p. 58]69 In chapter 6 I will try to show how, contrary to Kaufmann’s contention, choice sequences can be disclosed as mathematical objects. 4.3.2 Deny That Mathematical Objects are Omnitemporal There are two forms that a denial of P3 may take. P3a. No mathematical object is omnitemporal. This would be unacceptable for the same reasons as denying P1 would. It would imply discarding all evidence there is for mathematical objects having this property (e.g., the number 2). P3b. At least some mathematical objects are not omnitemporal. To deny P3 in this way implies the rejection of an implicit assumption Brouwer and Husserl shared, the assumption that the ontological region of mathematical objects is temporally homogeneous. This means that Brouwer as well as Husserl holds that all mathematical objects are alike with respect to the temporal: Mathematical objects are temporally homogeneous, because they are all intratemporal (Brouwer) Mathematical objects are temporally homogeneous, because they are all omnitemporal (Husserl) Denying this shared assumption of temporal homogeneity means to allow that in the mathematical universe dynamic and static objects can co-exist. An alternative interpretation of P3b would be: ‘There is no essence of mathematical objects. They do not form a universe together in any systematic sense. Instead, there is a family resemblance among them. But then perhaps they need no longer be regarded as temporally homogeneous.’ This interpretation would need to be supplied with substantial evidence before it has any force, but one can say right away that neither Brouwer nor Husserl would adopt it. It is not open to Brouwer, because for him all mathematical objects are created in the free unfolding of the empty two-ity. The constructions are wellfounded, giving objects a genetic uniformity, which justiﬁes speaking about them as forming a universe in a pregnant sense.

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Husserl would not take the alternative line either, on account of his form of essentialism. Mathematics deals with the hightest empty essence ‘anything whatever’ and all the specialisations thereof (section 5.1). 4.3.3 Deny That Mathematical Objects are Within the Temporal Realm This alternative amounts to a wholesale rejection of Brouwer’s view. It excludes dynamic objects, which are by deﬁnition within the temporal realm, whereas Brouwer’s position derives its accordance with our basic intuition of the continuum from its essential use of such objects. But denying P2 is also to reject the later Husserl’s shift from atemporality of ideal objects to omnitemporality. I explain why he did so in section 5.4; but, anticipating that discussion, it should be said that this alternative is incompatible with our methodological choice for Husserl’s transcendental phenomenology. 4.3.4 Deny That Mathematics is About Objects There are a trivial and a non-trivial way to do this. 1a. The trivial one is this: ‘Mathematics surely seems to be about objects, but we are misled. It actually deals with logic (or marks on paper, or language games, etc.).’ There would be a diﬀerence between what we seem to do and what we actually do; correspondingly there would be mathematics in its experienced form and in its true form. If true mathematics is not about objects of any kind, then, trivially, no conﬂict over the temporal aspects of objects could arise. This is a clear-cut reductionism. Husserl is a thorough anti-reductionist, urging to take things as they come: No conceivable theory can make us err with respect to the principle of all principles: that every originary presentive intuition is a legitimizing source of cognition, that everything originarily (so to speak, in its ‘personal’ actuality) oﬀered to us in ‘intuition’ is to be accepted simply as what is presented as being, but also only within the limits in which it is presented there. [120, p. 44]70 and The evidence of irreal objects, objects that are ideal in the broadest sense, is, in its eﬀect, quite analogous to the evidence of ordinary so-called internal and external experience, which alone on no other grounds than prejudice is commonly thought capable of eﬀecting an original Objectivation . . . In just the same fashion, we say, there belongs to the sense of an irreal object the possibility of its identiﬁcation on the basis of its own manners of being itself seized upon and had. Actually the eﬀect of this ‘identiﬁcation’ is like that of an ‘experience’. [106, pp. 155–156]71

4.3 Resolving the Conﬂict: The Options, and a Proposal

41

On this basis Husserl ﬁrmly holds that mathematics is about objects: that accords best with our experience and ﬁts into phenomenological epistemology. Brouwer would also reject 1a right away, and for very similar reasons. According to him, mathematical judgements are introspective reports on languageless mental constructions and buildings. He actually deﬁnes mathematics as the subject that deals with all that can be generated from the ‘Urintuition’ of two-ity (section 5.5). On his view, it makes no sense to deny that mathematics is about objects. 1b. The other way to reject this common assumption P1 is to hold that only part of mathematics deals with objects, and the rest only seems to. Mathematics is then divided in two parts. To make such a construal workable, one needs a criterion to decide which part a mathematical statement belongs to, and a procedure to translate statements about illusory objects into statements about the genuine objects. (Correspondingly, a distinction is induced between illusory statements and genuine statements.) Then the conﬂict between Brouwer and Husserl would be resolved if the dynamic objects happen to fall in the illusory class and the genuine ones are all static. Translations One might think that the existence of translations from statements that quantify over choice sequences into equivalent ones that do not, points to this second alternative, 1b. For simplicity, I will discuss the case of lawless sequences, but the arguments are general. Troelstra [220, ch. 3], developing earlier work by Kreisel [146], presents a formal system LS describing lawless sequences, together with a mapping τ into a subsystem without variables for lawless sequences IDB1 , such that 1. τ (A) ≡ A for A a formula of IDB1 2. LS A ↔ τ (A) 3. (LS A) ⇔ (IDB1 τ (A)) Aren’t such translations a means to explain lawless sequences away? The phenomenological answer would be ‘no’, for three reasons. First, there is an argument that could be used from any philosophical point of view. The translation yields equivalences, statements of the form A ↔ τ (A) where A quantiﬁes over lawless sequences and τ (A) does not. But as A is a component of this statement as a whole, the lawless sequences are still involved. Being part of a genuine statement, A cannot be regarded as illusory. For this reason, Kreisel [146, pp. 225–226] suggests that such translations should not be understood (as they usually are) as ‘elimination theorems’, but as giving ‘a complete analysis of lawless sequences’. Should one maintain that the translation does eliminate quantiﬁcation over lawless sequences by deﬁning it contextually, then there are still the second and third objections.

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Secondly, to speak of ‘elimination’ falsely suggests that the intuitive continuum can be regarded as a discrete point set after all. Even if statements quantifying over lawless sequences are equivalent to statements that do not, the fact remains that we should think of the intuitive continuum as analysable into, among others, such sequences and not into elements of a set.72 (This is treated in somewhat more detail in chapter 6, where I argue that only the notion of lawless sequence fully reﬂects both the inexhaustibility and nondiscreteness of the intuitive continuum, and that all other types of choice sequence are modiﬁcations of this full notion.) Thirdly, these translations occur in the context of axiomatisations of choice sequences. These systems should not be confused with the sequences themselves. Axiomatisations are a way to present mathematical content, but they are not identical with it [187]. Lawless sequences have been axiomatised in different, not always equivalent ways (e.g., Kreisel [146], Myhill [166], Troelstra [219]); nevertheless something remains the same, namely, these axiomatisations are all about lawless sequences. (As the biologist Jean Rostand noticed, ‘Theories pass. The frog remains.’) Clearly it is not the case that we were ﬁrst in the dark about this and then learned it from ﬁnding mappings between the formalisms. These sequences (or any other mathematical object) cannot be identiﬁed with any particular axiomatisation (let alone formalisation). Moreover, doing so would force one to accept the implausible view that Brouwer’s theorising before the introduction of axiomatic theories was, in fact, about nothing. One may expect that Brouwer would have argued against the translations option along the lines of the second and third objection, because he views mathematics as essentially languageless (ﬁrst act of intuitionism, quoted on p. 24; this is also suggested by Vesley [231, p. 326]). What is more, for him it is clear from the outset that choice sequences are objects, he holds that they are freely generated in the mind (second act of intuitionism, p. 24). Weyl Hermann Weyl’s work of the early 20s [238, 239] may be interpreted as an earlier, diﬀerent attempt at 1b.73 Weyl, along with Oskar Becker, was one of the ﬁrst to interpret intuitionistic logic and mathematics from the standpoint of Husserl’s phenomenology. As Weyl’s literary style is excellent, his work must have warmed more people to the study of intuitionism than Brouwer’s own publications. Indeed, as Skolem remarked in 1929, The works of Brouwer are hard to read, and because of that, intuitionism has become known rather through some works of Weyl. [196, p. 217, trl. mine]74 But Brouwer was not altogether happy with Weyl’s contributions. In the Brouwer Archive, there is a manuscript (likely from the late 20s) in which Brouwer writes

4.3 Resolving the Conﬂict: The Options, and a Proposal

43

As creative Neo-Intuitionist besides Brouwer, only his student Heyting may qualify, who has reconstructed the geometric axiomatics on the basis of the new intuitionism. Weyl however can, to date, hardly be mentioned in this context: true, he has published, as a ﬁrst, halfunderstanding follower of Brouwer, some attempts at popularisation, but the ingredients of his own contained in them are all wrong and mislead the public. [trl. mine]75 Indeed, the position Weyl arrived at diﬀered in some important respects from Brouwer’s.76 It is not always clear to what extent these diﬀerences arose because of Weyl’s own phenomenological research, or because of elements that he simply took from Husserl’s writings. It will turn out that Weyl’s position is in fact not entirely coherent. Hence his work allows diﬀerent interpretations, depending on how incoherencies are resolved. My reading of Weyl will be such that it yields a potential counterargument to choice sequences as individual mathematical objects. Otherwise, Weyl’s work would not jeopardise the position I defend. Weyl seems to recognise only two concepts of sequences, the lawless and the lawlike. The term ‘lawless’ was introduced in print only later, by Kreisel [146], following a suggestion by Gödel). As Kreisel explained in a letter to Heyting in 1963, Gödel says they should be called ‘absolutely lawless’ and not ‘absolutely free’ because freedom implies that one may impose a law. The meaning is to be this: I decide in advance never to impose a restriction . I think he is right.77 Indeed, ‘lawless’ has become the standard term. Weyl [157] speaks of ‘the freely developing choice sequence’ (p. 100), which is ‘unrestricted by any law in the freedom of its development’ (p. 94); as he nowhere speaks of (allowing or forbidding) higher-order laws, this latter qualiﬁcation indicates that by laws he means ﬁrst-order restrictions, and hence ‘freely developing sequence’ does not mean ‘sequence developing in any arbitrary way’ but ‘lawless’. The lawless sequences, according to him, are not individual objects, whereas the lawlike are: ‘a single particular (and up to inﬁnity determined) sequence can only be deﬁned by a law’; and ‘the individual real number is represented by a law’ [157, p. 94]. (As we saw, Brouwer thinks of a point as ‘always something becoming and often remaining undetermined’, so for Brouwer nonlawlike sequences are individual objects just as well.) However, Weyl also holds that there is a use in mathematics for the concept of lawless sequence, in that it allows one to conceptualise the intuitive continuum in the right way. ‘It is one of the fundamental insights of Brouwer that number sequences, developing through free acts of choice, are possible objects of mathematical concept formation’ [157, p. 94]. This seems to be a way to do justice to both Husserl’s and Brouwer’s intuitions while avoiding actual conﬂict, and therefore it must be treated at some length. Brouwer received a copy of Weyl’s manuscript for the ‘Foundational

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crisis’ paper, and commented on it. We will quote from Brouwer’s reaction later on (see [57] for the historical details). Weyl claims that the range of the existential quantiﬁer over sequences is only the lawlike sequences: ‘The expression “There is” commits us to Being and law, while “every” releases us into Becoming and Freedom.’ [157, p. 96]78 Weyl’s explanation seems to be that it is essential to pure mathematics that its objects can be coded in natural numbers: Every application of mathematics must set out from certain objects that are to be subjected to mathematical treatment, and that can be distinguished from one another by means of a number character. The characters are natural numbers. The connection to pure mathematics and its constructions is achieved by the symbolic method, which replaces these objects by their characters. The point geometry on the straight line is, in this way, based on the system of the abovementioned dual intervals, which we are able to identify by means of two whole number characters. [157, pp. 100–101]79 When reading this quotation one has to be aware that for Weyl ‘application of mathematics’ does not always mean what it normally would, and does not necessarily contrast with what is usually understood as pure mathematics. In Weyl’s usage, the point geometry on a line already is an application of mathematics. The coded basic objects are rational segments. This example also illustrates that the substitution of numbers for objects in the process of symbolisation cannot be a mere labeling of those objects. Some information needs to be preserved, depending on what we want to use the mathematics for. When that is done, Weyl continues, we are back in the realm of ‘pure mathematics and its constructions’. From this we conclude that according to Weyl, all individual objects of pure mathematics can be coded into natural numbers. Laws, being ﬁnitely speciﬁable, can be coded in natural numbers and therefore Weyl, by the criterion above, accepts lawlike sequences as genuine (individual) objects of mathematics. Lawless sequences cannot be thus coded, hence are not to be regarded as individual objects. This gives Weyl’s game away: although he recognises a role for lawless sequences in conceptualising the continuum, in the end mathematics is only about numerically codiﬁable objects. Weyl says that from conceptual truths about lawless sequences, one arrives at genuine mathematical statements by substituting lawlike sequences for lawless sequence variables: The proper judgments that can be gained from these universal judgments come into being . . . in the case of the freely developing choice sequence, by substituting for it a law φ that determines an individual number sequence in inﬁnitum. [157, p. 100]80

4.3 Resolving the Conﬂict: The Options, and a Proposal

45

The concept of lawless sequence is meaningful, but there are no individual mathematical objects falling under it. They are beyond the reach of Weyl’s methods of construction. Only objects that can be coded into natural numbers can be constructed in that strong sense. Weyl, however, has to accept lawless sequences as objects of some, albeit not mathematical, sort. Perhaps they could be accepted as empirical objects, as Gödel seems to suggest [147, p. 202]. In Husserl’s terms, Weyl and Gödel could think of such sequences as ‘real’, whereas the concepts or essences of which they are instances are ‘ideal’. An incoherence in Weyl’s conception of choice sequences seems to be that he on the one hand only talks about lawlike and lawless sequences, but on the other hand also gives the following example of forming a sequence n by operating on a lawless sequence m [238, p. 50]:81 n(h) = m(1) + m(2) + . . . + m(h) The trouble here is that n, because of its dependence on m, is neither lawlike nor lawless. It isn’t lawlike, because it functionally depends on a lawless sequence; it isn’t lawless, because it depends on another sequence at all. (So lawless sequences are not closed under even the most innocent operations, except for the identity operation.) Brouwer was aware of this need for types of choice sequence in between lawless and lawlike, we turn to that below. Leaving this matter aside for the moment, let us review the argument that Weyl develops to defend the thesis that the concept of lawless sequence has mathematical applicability: 1. The concept of lawless sequence is an object for mathematical concept (Begriﬀ) formation. (premise, [157, p. 94]82 ) 2. Lawless sequences may be so chosen as to follow a lawlike sequence. (premise, implicit) 3. All individual (i.e., lawlike) sequences can be embedded in the continuum of lawless sequences. (from 1 and 2; see [157, p. 100]83 ) 4. It may be part of the meaning of ‘lawless sequence’ that it does or does not have a given property E. (premise, [157, p. 96]84 ) 5. Hence there will be general judgement directions for lawlike sequences. (from 3 and 4; see [157, p. 100]85 ) Premise 2 is implicit in Weyl’s following passage: Then it can happen that it is part of the essence of a developing sequence, that is, of a sequence where each individual choice step is completely free to possess the property not-E. This is not the place to discuss how such insights into the essence of a developing sequence are to be gained. Yet only this sort of insight provides a justiﬁcation for the fact that, when given a law φ by someone, we can, without examination, reply: The sequence determined in inﬁnitum by this law does not possess the property E. [157, p. 96]86

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4 The Original Positions

This premise 2 is already problematic. One can see the idea behind it – can’t a lawless sequence be chosen such that it happens to coincide extensionally with a lawlike sequence? But the answer is ‘no’. One cannot specify a lawless sequence α by saying that it follows some lawlike sequence a; for that would contradict the lawless character of α. As non-lawlike sequences are necessarily unﬁnished objects, one cannot say that a law and a sequence of free choices may simply be alternative ways to describe the same completed inﬁnite sequence. (One might be able to establish that an already chosen initial segment of a lawless sequence coincides with an initial segment of a lawlike sequence.) Weyl goes on to argue that there is no corresponding general theory of functions. He recognises three kinds of functions (all constructive, i.e., laws): functiones discretae: N → N, from numbers to numbers (i.e., lawlike sequences) functiones mixtae: (N → N) → N, assigning numbers to choice sequences, or N → (N → N), from numbers to lawlike functions functiones continuae: (N → N) → (N → N), from choice sequences to choice sequences The argument is based on the ‘fact’ (that is, if one accepts premise 2) that functiones discretae, and only those, can be embedded into a conceptually acceptable continuum. That embedding is based on the concept of lawless sequence; but there is, according to Weyl, nothing analogous to that in the case of functiones mixtae and continuae. There is no continuum then that the mixtae and continuae can be embedded in, hence, there is no interpretation of general statements about them, and there is no general theory of functions. This point Weyl stresses again and again: But let me emphasize again that individual, determined functions of this sort occur from case to case in the theorems of mathematics, yet one never makes general theorems about them. The general formulation of these concepts is therefore only required if one gives an account of the meaning and the method of mathematics; for mathematics itself, and for the content of its theorems, it does not come into consideration at all . . . For these theorems, as far as they are self-suﬃcient and not purely individual judgments, are general statements about numbers or choice sequences, but not about ‘functions’. [157, p. 106]87 Sets of functions and sets of sets, however, shall be wholly banished from our minds. There is therefore no place in our analysis for a general set theory, as little as there is room for general statements about functions . . . As far as I understand, I no longer completely concur with Brouwer in the radical conclusions drawn here. After all, he immediately begins with a general theory of functions (the name ‘set’ is used by him to refer to what I call here functio continua), he looks at properties of functions, properties of properties, etc., and applies the

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identity principle to them. (I am unable to ﬁnd a sense for many of his statements). [157, p. 109]88 Arithmetic and analysis merely contain general statements about numbers and freely developing sequences; there is no general theory of functions or sets of independent content! [157, p. 110]89 Now we turn to premise 5. In modern language, Weyl’s semantics of ∀αA(α) is that A(a) holds for every lawlike a. The universal quantiﬁer ranges over lawless sequences, but statements about individual sequences must be about lawlike ones. Let us repeat this quotation: The proper judgments that can be gained from these universal judgments come into being . . . in the case of the freely developing choice sequence, by substituting for it a law φ that determines an individual number sequence in inﬁnitum. [157, p. 100]90 This semantics for universal quantiﬁcation is the clearest sign that Weyl does not accept non-lawlike sequences as individuals. However, given Weyl’s stress on the concept of lawless sequence, and given his mentioning of constructing new sequences through operations on other sequences, one may conjecture that he had in mind something like the following. A plausible attempt to obtain a satisfying universe of choice sequences is to start by accepting the lawless sequences and then get further sequences by applying lawlike continuous operations to them. One might hope to get from this all the sequences one needs, without opening the gates to many diﬀerent primitive types of sequence. But Troelstra [220, ch. 4] has shown that this attempt must fail, in so far as it allows counterexamples to basic principles such as weak continuity (see above). So Weyl’s eﬀort to avail himself of the use of the concept of lawless sequence without allowing them into his mathematical ontology suﬀers from incoherence (implicitly admitting sequences in between lawless and lawlike), a faulty premise (after all, lawless sequences cannot be stipulated to follow lawlike sequences), and two unwelcome consequences. The ﬁrst is that it requires an unnatural (asymmetrical) interpretation of the quantiﬁers, according to which the universal quantiﬁer ranges over lawless, but the existential over lawlike sequences. The second is that it forbids a general theory of functions. (We will see how Brouwer reacts to especially this second consequence.) Moreover, Weyl’s approach is subject to two phenomenological reservations. First, that it should be an essential property of mathematical objects that they can be coded into natural numbers might be true, but in phenomenology one needs a constitution analysis to justify such a claim. Unless that is done, it is only a presupposition. (I am attempting here to show that this presupposition is actually false, by bringing out evidence that non-lawlike sequences indeed are mathematical objects (and hence that Husserl’s ‘omnitemporality’ condition is a false presupposition too).)

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Secondly, there is Husserl’s anti-reductionism, applied to choice sequences: the discussion should be carried on in terms of the (counter)evidence we can get for these alleged objects. This application of anti-reductionism, an essential part of Husserl’s transcendental idealism, to choice sequences can be connected to Troelstra’s distinction of three approaches to the study of choice sequences [222, pp. 201–203]: Analytic: one starts with (a conceptual analysis of) the idea of an individual choice sequence of a certain type (say τ ) and attempts to derive from the way such a choice sequence is supposed to be given to us (e.g., from the type of data available at any given moment of its generation) the principles which should hold for choice sequences of type τ . Holistic: one accepts (in one way or another) the universe of all intuitionistic (choice) sequences (or alternatively, ‘quantiﬁcation over choice sequences’) as a single primitive intuition. Figure of speech: this approach might be presented as an extreme variant of the holistic approach and regards all talk about choice sequences as a ﬁgure of speech, which expresses that ‘quantifying over all sequences’ means somehow more than just ‘quantifying over all determinate sequences’. Troelstra states a preference for the analytic approach and proposes to take recourse to the other two only when it has demonstrably failed. He has this preference because the analytic approach enables one to study the role of intensional aspects in intuitionism. There are many diﬀerent types of choice sequences, depending on the information we have about them, and this leads to the question what the eﬀect is of varying types of intensional information. The analytic approach is the one that is sensitive to this variety. Husserl’s principled anti-reductionism may serve as a theoretical underpinning of this preference.91 But Weyl does not conduct all of his investigation according to Husserl’s methodology, for although there is in Weyl’s work a phenomenological analysis of the continuum, there is no corresponding analysis of choice sequence as objects. Husserl, to whom Weyl sent an oﬀprint of his ‘Foundational crisis’ paper, did not react explicitly to Weyl’s theory. Brouwer did. It is important to see how Brouwer’s view on choice sequences diﬀers from Weyl’s: for Brouwer they are individual objects, individuated by their moment of beginning, hence not omnitemporal. They cannot be given to us (extensionally) via a linguistic representation, but that does not bother him as he holds that mathematics is an activity independent from language to begin with. For Weyl, on the other hand, the impossibility of a ﬁnite numerical coding (and such coding is equivalent to the existence of a linguistic representation) rules out their status as mathematical objects. Brouwer writes, in an undated draft of a letter to Weyl: ‘In your restrictions of the object of mathematics you are in fact much more radical than I am’ [57, p. 148].92

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In that same draft, Brouwer comments: It seems to me that the whole purpose of your paper is endangered by the end of the second paragraph of page 34.93 After you have roused the sleeper, he will say to himself: ‘So the author admits that the real mathematical theorems are not aﬀected by his considerations? Then he should no longer disturb me!’ and turns away and sleeps on. Thereby you do our cause injustice, for with the existence theorem of the accumulation point of an inﬁnite point set, many a classical existence theorem of a minimal function, as well as the existence theorems for geodesics without the second diﬀerentiability condition, loses its justiﬁcation! [57, p. 149]94 Brouwer does accept non-lawlike sequences as mathematical individuals. They are free constructions of the subject (the ‘second act of intuitionism’, mentioned above). So when Weyl writes The concept of a sequence alternates, according to the logical connection in which it occurs, between ‘law’ and ‘choice’, that is, between ‘Being’ and ‘Becoming’ [157, p. 109]95 Brouwer comments in the margin for me ‘emerging sequence’ is neither one; one considers the sequences from the standpoint of a helpless spectator, who does not know at all in how far the completion has been free. [57, p. 160]96 In other words, whether a sequence should be considered as an individual object does not depend on our knowledge of its being lawlike or not. Brouwer agrees with Weyl that the concept of law is extensionally indeﬁnite, but he does not accept Weyl’s further premise that sequences can only be individuated by a law; they are always individuated by their moment of beginning. This amounts to allowing dynamic objects as individuals. Hence it makes sense to build a theory on basis of these sequences. They can be elements of a spread (Brouwer’s term is ‘Menge’): A spread is a law on the basis of which, if again and again an arbitrary complex of digits [a natural number] of the sequence ζ [the natural number sequence] is chosen, each of these choices either generates a deﬁnite symbol, or nothing, or brings about the inhibition of the process together with the deﬁnitive annihilation of its result; for every n, after every uninhibited sequence of n − 1 choices, at least one complex of digits can be speciﬁed that, if chosen as n-th complex of digits, does not bring about the inhibition of the process. Every sequence of symbols generated from the spread in this manner (which therefore is generally not representable in ﬁnished form) is called an element of the spread. We shall also speak of the common mode of formation of the elements of a spread M as, for short, the spread M . [29, pp. 2–3, original emphasis, trl. Dirk van Dalen and me]97

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In 1947, Brouwer added the following clariﬁcation to this deﬁnition: Because mathematics is independent of language, the word ‘symbol’ (‘Zeichen’) and in particular the words ‘complex of digits’ (‘Ziffernkomplex’) must be understood in this deﬁnition in the sense of ‘mental symbols’, consisting in previously obtained mathematical concepts. [38, original emphasis] In the next sentence after the above deﬁnition of a spread, Brouwer refers to these elements as ‘choice sequences’: ‘Wenn verschiedene Wahlfolgen . . .’. Later, Brouwer directed his attention from choice sequences as elements of a spread to the way individual choice sequences are given to us [221]. Brouwer notes in his draft letter to Weyl that spreads can accomodate the three kinds of functions that Weyl describes. For the details I refer to [57]. The second case of the functiones mixtae requires a law that assigns numbers to choice sequences. Here Weyl appeals to a continuity principle (for lawless sequences, now known as the ‘principle of open data’): A law that, from a developing number sequence, generates a number n that is dependent on the outcome of the choices is necessarily such that the number n is ﬁxed as soon as a certain ﬁnished segment of the choice sequence is present, and n remains the same however the choice sequence may further develop. [157, p. 95]98 . And later he adds that for every mode of construction of functiones mixtae (‘Erzeugungsgesetz’), whatever it may be, the following ‘essential insight’ holds: According to this law, there will always be a moment in the developing sequence where the sequence, regardless of how it develops, creates a number. This feature is all that is essential to the f.[unctiones] m.[ixtae]. [157, p. 105]99 Brouwer, in contrast to Weyl, realised that one needs other non-lawlike sequences than just the lawless, and formulated a continuity principle on the universal spread, of which all sequences, lawless, lawlike and all in between, are elements: A law that assigns to each element g of C [the universal spread] an element h of A [the natural numbers], must have determined the element h completely after a certain initial segment α of the sequence of numbers of g has become known. But then to every element of C that has α as an initial segment, the same element h of A will be assigned. [29, p. 13, trl. Dirk van Dalen and me]100 Brouwer neither then nor later gives a justiﬁcation that this principle indeed holds for the universe of all choice sequences (I attempt this in chapter

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7) but he never questioned it, and it shows that Brouwer was aware that choice sequences come in a wide variety. We have to conclude that Weyl’s method does not succeed in both saving the conceptual advantages of non-lawlike sequences and escaping ontological commitment to them. However, as van Dalen remarks, it would be rather unfair to ask for today’s insights in Weyl’s pioneering paper. The surprising thing is not that Weyl’s conception of choice sequence was awkward, but that Brouwer hit on the right notion right away. 4.3.5 A Proposal: The Heterogeneous Universe It seems that neither Husserl nor Brouwer can deny any of the other three shared assumptions P1, P2 and P4 without getting into conﬂict with their basic philosophical views. I want to suggest that in the case of P3, the situation is diﬀerent.101 To Brouwer, it is essential to admit intratemporal mathematical objects; but it is not essential to Husserl’s philosophical system to hold, even though he in fact did, that there cannot be such objects. Here we have an ontological presupposition of Husserl’s (pace Lohmar [153, p. 79]). Now I can formulate my proposal. In this chapter, I have argued that the conﬂict between the positions of Husserl and Brouwer is best represented by the following aporetic cluster: P1. P2. P3. P4.

Mathematics is about objects. Mathematical objects are within the temporal realm. Mathematical objects are omnitemporal. Some mathematical objects are intratemporal, dynamic and unbounded.

What I want to suggest, and work out in what follows, is that this cluster can be made consistent by giving up P3 and replacing it by the weaker P3*: P3*. Some mathematical objects are omnitemporal, some are not. That is, I suggest that the mathematical universe is not temporally homogeneous. Further, I claim that this new cluster can be ﬁtted into Husserl’s general philosophy, which leads to a revision of his philosophy of mathematics. These claims I will argue for in chapters 6 (by showing how objects as described in P4 are constituted (in the strict sense)) and 5 (by showing that Husserl’s transcendental phenomenology warrants this kind of revisionism in mathematics). The idea behind my approach is analogous to a suggestion by Gödel, which was put in print by Kreisel. The suggestion is that ‘what characterises the diﬀerence between e.g., the idealist and the realist view is what aspects of (crude) experience (in this case, mathematical experience) are regarded as signiﬁcant and suitable for study’ [144, p. 190].102 On this view, the diﬀerent traditional schools in the foundations of mathematics each are an exaggeration of a particular aspect of mathematical experience.103 In his Memoir of Gödel, Kreisel mentions that he held that view:

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In his publications Gödel used traditional terminology, for example, about conﬂicting views of ‘realist’ or ‘idealist’ philosophies. In conversation, at least with me, he was ready to treat them more like diﬀerent branches of the subject, the former concentrating on the things considered, the latter on the processes of acquiring knowledge about these objects or processes . . . Naturally, for a given question, a ‘conﬂict’ remains: Which branch studies the aspects relevant to solving that question? [147, p. 209, original emphasis]104 In section 4.2.2, we saw what diﬀerent parts of experience Brouwer and Husserl select for study, and precise formulation of the conﬂict leads to the idea of these parts being compossible, by reconsidering the temporal properties of the mathematical universe. Nonetheless, as a matter of fact Husserl does argue for P3, not P3*, and Brouwer does claim something stronger than P4, to wit, ‘All mathematical objects are intratemporal, and some of them are dynamic and unbounded’. That is incompatible with P3* as well. So neither Husserl’s nor Brouwer’s original position squares with my proposal. Therefore, to defend this proposal, I should be able to explain why their arguments are not correct from the phenomenological point of view.

5 The Phenomenological Incorrectness of the Original Arguments

5.1 The Phenomenological Standard for a Correct Argument in Ontology The purpose of this section is threefold. First, to provide a standard to evaluate Husserl’s and Brouwer’s original positions by. Secondly, to let this standard be the methodological clue to the reconstruction of choice sequences in chapter 6. Thirdly, to justify the revisionism implied in that reconstruction; which is all the more urgent since, as I will argue below, the kind of revisionism needed is not apparent in those of Husserl’s writings on which I rely. As the ﬁrst and second will have to be addressed in serving the third, this section will take the form of an argument for revisionism. ‘Revisionism’ is understood here, following Crispin Wright, as the term that applies to ‘any philosophical standpoint which reserves the potential right to sanction or modify pure mathematical practice’ [247, p. 117]. I want to make a distinction between weak and strong revisionism. The point of reference is the actual practice of mathematics. Weak revisionism then potentially sanctions a subset of this practice, while strong revisionism potentially not only limits but extends it, in diﬀerent directions. In strong revisionism, certain combinations of limitation and extension may lead to a mathematics that is no longer compatible with the unrevised one. May lead, not necessarily leads: it is all a matter of reserving rights; whether there is occasion to exercise them is a further question. To illustrate these categories, let me give examples of non-revisionism, weak revisionism, and strong revisionism. Non-revisionism can be found in Wittgenstein’s Philosophical Investigations, where philosophy can neither change nor ground mathematics: Philosophy may in no way interfere with the actual use of language; it can in the end only describe it. For it cannot give it any foundation either. It leaves everything as it is.

53

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It also leaves mathematics as it is, and no mathematical discovery can advance it. A ‘leading problem of mathematical logic’ is for us a problem of mathematics like any other. [244, p. 49e]105 Whether Wittgenstein’s claim of non-revisionism is consistent with the rest of his philosophy is a matter of discussion, and a large part of Wright’s study [247] from which I took the deﬁnition of revisionism given above is devoted to this question. But the quotation certainly serves as an illustration of an attitude; the same goes for the examples to follow. (Other examples of non-revisionism are naturalism and Hao Wang’s ‘factualism’ [232]; see [170, p. 110].) An example of weak revisionism is Hilbert’s Program. In his lecture ‘The new grounding of mathematics’ from 1922, he said: The goal of ﬁnding a secure foundation for mathematics is also my own. I should like to regain for mathematics the old reputation for incontestable truth, which it appears to have lost as a result of the paradoxes of set theory; but I believe that this can be done while fully preserving its accomplishments. The method that I follow is none other than the axiomatic. [157, p. 200]106 This statement clearly shows the two deﬁning characteristics of weak revisionism: the eﬀort to provide mathematics with a foundation and the wish to preserve all of mathematics as actually practised. Brouwer’s intuitionism is an example of strong revisionism. It both limits and extends actual practice (in a way that is incompatible with classical mathematics, e.g. in intuitionistic analysis all full functions are continuous). In the overview article of 1952, ‘Historical background, principles and methods of intuitionism’, Brouwer described the eﬀect of introducing intuitionism on mathematics: In this situation intuitionism intervened with two acts, of which the ﬁrst seems necessarily to lead to destructive and sterilizing consequences; then however, the second yields ample possibilities for recovery and new developments. [40, p. 140] More speciﬁcally, then, the claim that I want to defend now is that, even though his explicit claims are only for what I would call weak revisionism, from his own general views Husserl should have derived a strong revisionism.107 My argument is this: 1. Husserl presents a weak revisionism. (Premise) 2. Husserl’s weak revisionism implies a strong revisionism. (Premise) 3. So Husserl’s position in fact warrants a strong revisionism. (1,2)

5.2 Husserl’s Weak Revisionism

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5.2 Husserl’s Weak Revisionism Premise 1 has to be supported by textual evidence, and therefore I will quote amply (the argument for premise 2 is in section 5.3). We will see that in most of these quotes Husserl discusses not just mathematics but a priori sciences in general; however, for the purposes of the present discussion I will read them only as statements about mathematics. Such textual evidence is already found in Husserl’s Prolegomena of 1900 [113]. Section 71 of that work is entitled, signiﬁcantly, ‘Division of labour. The work of the mathematicians and that of the philosophers’. In that section, Husserl describes how he views the task of philosophy with respect to mathematics: It does not seek to meddle in the work of the specialist, but to achieve insight in regard to the sense and essence of his achievements as regards method and manner . . . Philosophical research so supplements the scientiﬁc achievements of the natural scientist and of the mathematician, as for the ﬁrst time to perfect pure, genuine, theoretical knowledge. The ars inventiva of the special investigator and the philosopher’s critique of knowledge, are mutually complementary scientiﬁc activities, through which complete theoretical insight, comprehending all relations of essence, ﬁrst comes into being. [111, I:p. 245]108 The revisionism described here is a weak one, as it is not concerned with the content of mathematics but with its epistemological methodology. Philosophy should provide the foundation and insight for mathematics. As we will see in a moment, it is a view that Husserl expressed all through the development and reconceptions of phenomenology in later years. After the Prolegomena, he wrote on the philosophy of science most extensively in the posthumously published Ideas III [99] (which its editor, Marly Biemel, named ‘Phenomenology and the foundations of the sciences’); and later detailed discussions can be found in the diﬀerent versions of the Britannica article [103] and Formal and Transcendental Logic. In Ideas III, again the stress is on methodology, as in the Prolegomena; but also a new and essential idea is introduced, namely, that of transcendental phenomenology as providing the universal ontology. By this Husserl means that transcendental phenomenology forms a (particular) unity with the ontologies of the particular sciences: Everything that the sciences of the onta, the rational and empirical sciences, oﬀer us (in the enlarged sense they can all be called ‘ontologies’, insofar as it becomes apparent that they are concerned with unities of the ‘constitution’), ‘resolves itself into something phenomenological’ . . . What must be accomplished here . . . , that which we have in view here under the title ‘regression to the constitutive

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absolute consciousness’, presupposes a transcendental phenomenology in a deﬁnite sense. [117, pp. 66–67]109 (A brief hint at this idea already appeared in Husserl’s reﬂections immediately after his lecture series The Idea of Phenomenology of 1907 [97, pp. 13–14].) To elaborate this point, I want to refer to the Britannica article of 1927 and Formal and Transcendental Logic of 1929.110 The interest of the Britannica article for my discussion derives from two facts: ﬁrst, in versions three and four of it Husserl gives a direct argument for his conception of transcendental phenomenology as providing the universal ontology; secondly, it is here that he for the ﬁrst time connects the paradoxes in the foundations of mathematics with this discussion of ontology. The ﬁrst and second did not contain the closing discussion of ontology; the fourth, which is the one published in the Britannica, is mainly the result of cutting down the third because of space limitations.111 Husserl’s Amsterdam Lectures were a revision of that fourth version. A part on ontology however, which according to Husserl’s notes was meant to be included [103, p. 349n. 1], was never written. It is a pity that he did not get round to that, as it is very probable that Brouwer was in the audience (see the draft letter of organiser Pos to Husserl [128, IV:p. 441]). The argument in the Britannica article runs, in Husserl’s own phrases, as follows: 1.1. ‘Transcendental phenomenology is the science of all conceivable transcendental phenomena’ (Premise) 1.2. ‘All entities get their ontological sense from intentional constitution’ (Premise) 1.3. ‘This phenomenology is eo ipso the absolute, universal science of all entities’ (1,2) A few lines further on, Husserl restates this conclusion in other words: 1.3* ‘We could even bring up the traditional expression and broaden it by saying: Transcendental phenomenology is the true and genuinely universal ontology.’ [129, p. 150]112 Premise 1.1 means that transcendental phenomenology is universal in that it deals with all conceivable constitutive possibilities of the transcendental ego. Premise 1.2 is a version of the fundamental principle of transcendental idealism, as Husserl developed it from 1907 onwards [97]. It singles out a special group among all transcendental phenomena, namely those experiences in which the intentional object is given as itself, as existing. In Cartesian Meditations, Husserl calls the corresponding group of constitution processes cases of strict (‘prägnant’) constitution [126, section 23]. There are correspondences between the essence of an object and the strict constitution of such an object. That is, the way in which an object is given as an identity through various acts is characterised by a rule that is speciﬁc for that kind of object

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[115, p. 330], [112, section 98]. Strict constitution is constitution according to this rule. Finally, existence is the objective correlate of strict constitution [126, section 26]. By ‘ontology’, in 1.3*, Husserl means the systematic unity of all a priori sciences [103, p. 519]. This unity derives from the fact that all a priori concepts, like everything else, acquire their sense in the one process of constitution. Husserl has in mind here a science of possible being, not just of actual being as a more traditonal conception of ontology alluded to in 1.3* would have it, and he traces the idea of such a science back to Leibniz: Leibniz already had the fundamental insight that in every genuine theoretical knowledge and science the knowledge of possibilities must precede the knowledge of actualities. Accordingly, for every kind of real and ideal sphere of being he required the appurtenant a priori sciences as such of pure possibilities (for example, even a pure grammar, a pure doctrine of law, and so forth). [129, p. 150]113 The second point of interest of the Britannica article is its treatment of the paradoxes in the foundations of mathematics. Husserl interprets them as symptoms of the lack of philosophical foundation: The fundamental principle of the method of mathematics is being shown to be inadequate, and the much admired evidence of mathematics is being shown to need critique and methodological reform . . . The conﬂict over the ‘paradoxes’, that is, over the legitimate or illusory evidence of the basic concepts of set theory, arithmetic, geometry and the pure theory of time and so on . . . has revealed that, as regards their whole methodological character, these sciences still cannot be accepted as sciences in the full and genuine sense: as sciences thoroughly transparent in their method and thus ready and able to completely justify each methodical step. [129, p. 151, trl. modiﬁed]114 A weak revisionism as described here is called for as the cure to develop mathematics into a true science. Unless that task has been completed, mathematics is, properly speaking, not a science but a useful technique: Everywhere we observe, as in the setting up of other epistemological problems, the repeatedly cited error of accepting the sciences as something that already exists as though inquiry into foundations signiﬁed only an ex post facto clariﬁcation or, at most, an improvement that would not essentially alter these sciences themselves. The truth is that sciences that have paradoxes, that operate with fundamental concepts not produced by the work of originary clariﬁcation and criticism, are not sciences at all but, with all their ingenious performances, mere theoretical techniques. [106, p. 181]115 So the task of philosophy with regard to the sciences is to transform them from techniques to genuine (i.e., insightful) knowledge, by clariﬁcation (‘Klärung’) of their concepts:

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The point is to lead the sciences back to their origin, which demands insight and rigorous validity, and to transform them into systems of cognition based on insight by work that clariﬁes, makes distinct, and grounds ultimately, and to trace the concepts and statements back to conceptual essences, themselves apprehensible in Intuition, and the objective data themselves, to which they give appropriate expression insofar as they are actually true. [117, p. 83]116 With this clariﬁcation comes the possibility of rejecting supposed objects. This is made manifest in the introduction of Formal and Transcendental Logic: Original sense-investigation signiﬁes a combination of determining more precisely the vague indeterminate predelineation, distinguishing the prejudices that derive from associational overlappings, and cancelling those prejudices that conﬂict with the clear sense-fulﬁlment – in a word, then: critical discrimination between the genuine and the spurious. [106, p. 10]117 This ‘critical discrimination between the genuine and the spurious’ is what makes ontological questions decidable. Hence Husserl can claim, in the fourth version of the Britannica article, that in a priori sciences that have a phenomenological foundation, no foundational crises could occur. The clariﬁcation that transcendental phenomenology supplies guarantees that there will be no paradoxes [103, p. 297].118 Given this function of clariﬁcation, evidently Husserl’s published philosophy of mathematics can have as domain only current practice, concepts that are pre-given; one cannot clarify something that isn’t already there: The goal of clariﬁcation can also be understood in the sense already explained as that of producing anew, as it were, the concept already given, nourishing it from the primal source of conceptual validity, i.e., intuition, and giving it within the intuition the partial concepts that belong to its originary essence. [117, p. 88]119 (for a similar passage [112, p. 188]). And in the introduction to Formal and Transcendental Logic, he says of the method of clariﬁcation: The explication begins with the theoretical formations that, in a survey, are furnished us by historical experience in other words: with what makes up the traditional Objective content of formal logic and puts them back into the living intention of logicians, from which they originated as sense-formations. [106, p. 10]120 Thus, clariﬁcation is a retrogressive inquiry back to sense-conferring living intention, a genetic analysis. In summary of the argument for premise 1 of the main argument, what Husserl presents is a form of revisionism because of the demand for a philosophical foundation of mathematics, without which it could be a technique

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but not a true science; and it is a weak revisionism because its method, clariﬁcation, can only act on the concepts of current mathematical practice. There is the possibility of rejecting parts of existing practice – the cancellation (‘Durchstreichung’) of intentions; but there is no evidence in his writings for strong revisionism, the introduction of new concepts on philosophical grounds, and indeed Husserl wants to hold to a division of labour between mathematics and philosophy.

5.3 Husserl’s Implied Strong Revisionism However, that Husserl’s weak revisionism does imply a strong revisionism (premise 2 of the main argument) is what I will now argue for. The basis is a certain diﬀerence between constitution in the formal and non-formal a priori sciences. I ﬁrst present the argument as a whole, and then explain terms and comment on the steps. 2.1 Formal objects have no hyletic-material binding. (Premise) 2.2 Strict (‘prägnant’) constitution is constitution with evidence and in accordance with laws of essence for that object. (Premise) 2.3 Strict constitution of formal objects demands no speciﬁc hyletic-material data. (2.1,2.2) 2.4 Existence is the objective correlate of strict constitution. (Premise) 2.5 For formal objects, transcendental possibility implies existence. (2.3,2.4) 2.6 Clariﬁcation aims at strict constitution of objects that are already ﬁguring in actual practice. (Premise) 2.7 In purely formal sciences, the capacity for clariﬁcation is exactly the capacity for strict constitution. (2.5,2.6) 2.8 The objects ﬁguring in actual practice need not exhaust the totality of objects that are possible according to essence. (Premise) 2.9 So Husserl’s weak revisionism implies a strong revisionism. (2.7,2.8) As I am here trying to make a point about the nature of Husserl’s weak revisionism, it will not come as a surprise that three of the ﬁve premises already made their appearance in the foregoing exposition of that revisionism. 2.2, (a version of) 2.4 (the principle of transcendental idealism), and 2.6 we already saw in the previous section. The meaning of the part 2.1–2.5 can be clariﬁed by contrasting the position of Husserl to (one aspect of) that of Nicolai Hartmann in Possibility and Actuality [80].121 I have chosen Hartmann, because his general position on ideal objects largely coincides with Husserl’s, but Husserl wants to add an extra condition, perhaps following Kant [137, B638]. Hartmann [80, Kap. 41] recognises two diﬀerent senses in which an ideal object can be ‘possible’, in modern language: Logical possibility: (Hartmann: ‘logische Möglichkeit’) An object is logically possible exactly if the corresponding existence statement has a model.

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An example considered by Hartmann is a square circle [80, Kap. 41e]. If we take ‘square’ and ‘circle’ as primitive terms, then a square circle is logically possible because ‘there is a square circle’ has the form ∃x(P x ∧ Qx), which has a model. This may not be the intended model, but that is no longer a matter of just logic. There is no contradiction in the logical form.122 An example of an object that is not logically possible is something that is round and not round: ∃x(P x∧¬P x) has no models, it is a formal contradiction. Conceptual possibility: (Hartmann: ‘Wesensmöglichkeit’) An object is conceptually possible exactly exactly if it is logically possible and moreover there is no material contradiction involved. A square circle is not conceptually possible: although it fulﬁls the condition of logical possibility, it is materially contradictory. A square cannot be a circle at the same time. Hartmann writes: There is much that is possible logically but not conceptually. Logic has to take ‘impossible objects’ into account, that is, objects that are not possible conceptually (a square circle); depending on what pregiven properties are recognised (let us say, of a circle), such objects are possible or not. The ‘impossibility’ of such objects is precisely an impossibility of being (here, of a geometrical kind), not a logical one. [80, p. 323, trl. mine]123 Hartmann’s claim is that for ideal entities, conceptual possibility and conceptual existence (‘Wesenswirklichkeit’) are co-extensive: [Actuality and non-actuality] have, as concerns ideal being, no role of their own to play in addition to the possibility of being and nonbeing. The former are posited with the latter, as a matter of course, and do not signify anything ‘more’ than being possible and not being possible. [80, p. 318, trl. mine]124 Husserl, using slightly diﬀerent terminology, recognises this distinction between logical and conceptual possibility. In the fourth of the Logical Investigations, which is devoted to the theory of pure grammar, he speaks of ‘formal possibility’ and ‘material’ (or ‘synthetic’) possibility [121, section 14]; but his transcendental turn of a few years later induces a third sense of possibility: Transcendental possibility (my term): An object is transcendentally possible exactly if it is conceptually possible and moreover can be strictly constituted (that is, ideally, with full evidence). For example, the number 3 and the set of all even natural numbers. Conceptually possible, but transcendentally impossible entities are, it would seem, non-well founded sets. From the point of view of transcendental idealism, they could therefore not be accepted as existing objects. Hartmann, on the other hand, would ascribe mathematical existence to them, as long as they are not internally inconsistent.

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The diﬀerences between Hartmann and Husserl arise because Hartmann considers ontology and epistemology as completely independent of each other, also in the case of ideal objects: For cognition cannot at all be implied by the existence of its object. Ideal being in itself is as indiﬀerent to cognition of the ideal as real being is to cognition of the real. [80, p. 464, trl. mine]125 Hence in ontology of ideal objects, he recognises only what he calls internal constraints, i.e., formal and material consistency. Husserl, on the other hand, thinks there is a further constraint, the external one implied by transcendental idealism: an object can be said to exist if and only if it can ideally be brought to consciousness [115, section 142] – a principle clearly relating ontology and epistemology. This is external because the constraint arises not from the alleged entities themselves but from properties of consciousness. It is external in exactly the same sense that strong revisionism invokes considerations external to mathematics, namely, philosophical ones. Steps 2.1-2.4 of the argument describe this notion of transcendental possibility. An object cannot be strictly constituted if it is transcendentally impossible to do so. It is against this background that we reach an even stronger version of 2.5: for formal objects, transcendental possibility and existence are co-extensive.126 (In the argument 2.1–2.9, I need only one direction of the resulting biconditional.) This claim can be found almost literally in Experience and Judgement (provided ‘Möglichkeit’ is interpreted as transcendental possibility): All existential judgments of mathematics, as a priori existential judgments, are in truth judgments of existence about possibilities. [110, p. 371]127 How is this claim defended? Husserl views the objects of pure mathematics as formal objects. Correspondingly, the ontological region of pure mathematics is made up of the empty form ‘anything whatever’, and all specialisations thereof: This gives rise to the idea of an all-embracing science, a formal mathematics in the fully comprehensive sense, whose all-inclusive province is rigidly delimited as the sphere of the highest form-concept, any object whatever (or the sphere of anything-whatever, conceived with the emptiest universality), with all the derivative formations generable (and therefore conceivable) a priori in this ﬁeld – formations that always go on yielding new formations as products generated in a constructing that is always reiterable. Besides set and cardinal number (ﬁnite and inﬁnite), combination (in the mathematical sense of the word), relational complex, series, connexion, and whole and part, are such derivatives. [106, pp. 77–78]128

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To say that formal objects have no hyletic-material binding is to say that the hyletic-material is not part of the essence of a formal object, or put diﬀerently, that the sensory or the material is not, however indirectly, part of the meaning of the formal. [112, p. 33] The non-formal (‘das Sachhaltige’), on the other hand, does have hyleticmaterial content, and this gives rise to a diﬀerence in strict constitution: Every Apriori with a material content . . . demands a return to intuition of individual examples . . . The evidence of laws pertaining to the analytic Apriori needs no such intuitions of determinate individuals. It needs only some examples or other of categorialia; even categorialia having indeterminately universal cores will do (as when propositions about numbers serve as examples). They may indeed point back intentionally to something individual; but they need not be further examined nor explicated in this respect. [106, p. 213]129 These ‘unbestimmt allgemeine Kerne’ justify the inference from 2.3 and 2.4 to 2.5: purely categorial objects may be constituted on the basis of any hyletic-material content whatsoever, all that matters is form.130 This means, ﬁrst, that mathematical objects are not arbitrary, but are in their constitution constrained by the laws of categorial formation [122, section 62]; and, secondly, that one may use spontaneous imagination to start the constitution process. This is all similar to Brouwer, who says that ‘mathematics comes into being when the two-ity created by a move of time is divested of all quality by the subject’, thus leaving ‘the empty form of the common substratum of all two-ities’ [39, p. 1237]. Husserl and Brouwer agree that (in Husserl’s terms) the formal is founded on the hyletic: the constitution of any formal object has to start with some hyletic material, but precisely this material is then abstracted from. Hence, the particular feature of mathematical essences is that they govern a priori possibilities, possibilities that are independent of particular sense data: All mathematical propositions of existence have this modiﬁed sense: ‘There are’ triangles, squares, polygons of any increasing number of sides; ‘there are’ regular polyhedrons of ﬁfty-six lateral surfaces but not of any number of such surfaces. The true sense is not simply a ‘there is’ but rather: it is possible a priori that there is. [110, p. 370]131 Negative existential propositions have the function of separating out the invalid concepts, the expressions corresponding to no essence. [117, p. 71]132 As a consequence, to reject a particular mathematical practice it is not necessary (though surely suﬃcient) that it give rise to contradictions. It may also be that fulﬁlment of the intentions that make up this practice simply is beyond the power of the laws that govern categorial formation. The point, in both cases, is to show that a meaning can never be fulﬁlled. (In the second case, we can still have consistency.)

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Now consider the following objection to 2.5: ‘Aren’t there many concepts that one would take to be mathematical and that allow for conceptual possibility (i.e., according to essence), but of which we don’t know whether there are objects corresponding to them or not?’133 To answer this question, we begin by noting that ‘concept’ is ambiguous between ‘meaning’ and ‘essence’. That we can form a meaning (‘Bedeutung’, or, generalised also to non-linguistic acts, ‘Noema’) does not imply that there is an object falling under it. The origins and formation of a meaning can be studied largely independently of the question whether it refers. If no object corresponding to it exists, there is also no essence of such an object: In the case of logical signiﬁcations, we see now that what is thought as such (logical signiﬁcation in the noematic sense) can be ‘countersensical’, that it – which, after all, ‘exists’ within the category of being ‘logical signiﬁcation’, and more generally, ‘noema’ – has its actual being, as for example, the thought signiﬁcation ‘round rectangle’ . . . The essence of what is signiﬁed is also something other than the signiﬁcation. There is no essence ‘round rectangle’; but in order to be able to judge this, it is, presupposed that ‘round rectangle’ is a signiﬁcation existing in this unitariness . . . To posit signiﬁcations and to posit objects are two diﬀerent things. [117, pp. 73, 76]134 Husserl distinguishes valid meanings from invalid meanings. The valid meanings are those for which the possibility of fulﬁlment is not a priori excluded, or, in other words, for which fulﬁlment is ideally possible: The word-signiﬁcations can be valid as logical essences only if according to ideal possibility the ‘logical thinking’ actualizing them in itself is adaptable to a ‘corresponding intuition’, if there is as corresponding noema a corresponding essence that is graspable through intuition and that ﬁnds its true ‘expression’ through the logical concept. [117, p. 23]135 Here is how Husserl applies this criterion to geometry: A geometric judgment is valid only if the idea, the essence, is space and spatial formation, or, speaking in terms of extension, if a spatial formation is possible. [117, p. 70]136 Further on in that example he says that an existence proof consists in showing that there truly is in space according to its essence a geometric essence corresponding to this shape-concept (a freely formed logical signiﬁcation). So long as the proof of existence is not carried out, no geometric judgment as to nature (e.g., about the nature of regular bodies of ten surfaces) can be made. Every valid geometric judgment posits eidetic particulars (which is equivalent to a positing of corresponding essences as objects) that altogether make up the province of ontology circumscribed by the validly posited regional idea. [117, pp. 70–71]137

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Having thus justiﬁed 2.5, let us now relate conclusion 2.9 to mathematical practice. Let A be the class of objects that ﬁgure in actual practice at a particular moment. From a revisionist point of view (weak or strong), some of them are admissible, others may not be. Let C be the class of objects that will pass the revisionist criteria. Then A − C is the class of objects rejected on revisionist grounds. Strong revisionism allows for the possibility that C − A is inhabited, i.e., that one can indicate admissible mathematical objects that are found not by mathematical but by philosophical considerations. For Husserl, making such considerations requires that we leave the natural attitude that the working mathematician is in. The most interesting cases of strong revisionism are those that are not compatible with existing practice. As we saw in the discussion of the Britannica argument in section 5.2, Husserl frames his weak revisionism in terms of strict constitution. In Formal and Transcendental Logic, he describes the activity of his revisionism as a kind of creation (of our awareness of the object, not of that object itself): But this signiﬁes that here such criticism is creative constitution of the objectivities intended to each in the unity of a harmonious givenness of that objectivity itself, and creation of their respective essences and eidetic concepts. [106, p. 180]138 Stated in these terms, 2.7 would read that the capacity to ‘re-create’ formal objects that already ﬁgure in practice, is also the capacity for ‘creating’ formal objects that do not. Hence 2.9 says that the same methods that enable Husserl to exercise his weak revisionism in fact make a strong revisionism possible, namely, by shifting attention from A – the objects in current practice – to C – the objects that are acceptable from a philosophical point of view, whether they ﬁgure in current practice or not. I surmise that much of Brouwer’s development can be seen as such a shift. Brouwer’s introduction of choice sequences, when he realised that his theory of the subject allowed him to do so, is an example. An objection to this argument for strong revisionism in Husserl might be found in the fact that Husserl himself sees the role of phenomenology in mathematics diﬀerently. He recognises that there are diﬀerent conceptions of mathematics, but he assumes that they all have a common core that is responsible for their being mathematics. The task of phenomenology then would be to study this core meaning. This view can be found in Formal and Transcendental Logic [112, pp. 12, 14]. An even more interesting place is in an as yet unpublished manuscript from 1932 [95], because it is one of the few occasions in his writings where Husserl mentions Brouwer [190]: Should one, in one’s judgement of mathematics, the total sense of which depends entirely on these [foundational] concepts, follow a Hilbert, a Brouwer, or whom else? Can we be so sure, although exactly that is communis opinio today, that classical mathematics and likewise physics was not better advised? But we will do no better there. It was

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never ﬁnished but itself becoming, and so the problem repeats itself, the impossibility of a deﬁnite choice that determines the norm for us. Meanwhile it quickly becomes apparent that actually it is not that important to make such a choice by deciding in favour of some camp or some leading researcher. Everyone who has studied mathematics knows the general phenomenon called mathematics – mathematics as this exact science, which is becoming at any time and through all time, in which it was and still is, the one in becoming that has become at every present, from present to present all the same unitary in its continuing development, in spite of all the discrepancies that always exist between the researchers and between the conceptual, the theoretical formations they have produced. [95, 20b–21a, trl. mine]139 Husserl argues that phenomenological investigation of the foundations of mathematics should not follow the lead of any particular foundational program. His reasoning here seems to be that, as the science ever grows, there can be no ﬁnal and univocal selection of concepts. Phenomenology should study the core meaning of mathematics instead; evidently he views Formal and Transcendental Logic as such a study [95, 24a]. However, although a convincing case could be made that the core meaning of mathematics lies in its formality, a limitation of phenomenological attention to this core does not square with phenomenology as giving the universal ontology as described in section 5.2. In that conception, to which I hold here, any foundational concept proposed in any program is subject to critical investigation. A suspicion that there perhaps never will be a deﬁnitive set of all concepts is no reason to stop judging the philosophical merits of any speciﬁc concept. Moreover, from this quotation it transpires that Husserl thinks of all the alternative foundational programs as positive sciences, concerned as they are with ‘theoretical formations’. But that makes him overlook the possibility that there might already be a program that is not a positive science, but roots in considerations about the knowing subject. It seems Schmit does the same when he describes Husserl’s investigations as investigations that go beyond the foundation implied by intuitionism which is internal to science, and aim at a clariﬁcation of subjectivity as such. [189, p. 119n. 262, trl. mine]140 One might argue that Brouwer’s intuitionism, unlike Husserl and Schmit suggest, is a program that begins with the subject. (A source that makes this clear could have been known to Husserl, i.e., Brouwer’s ﬁrst Vienna lecture of 1928 [35]; see p.128.) Related to this is another objection to my interpretation of phenomenology as a strong revisionism, an objection that comes from Bachelard [9]. Consider her following paragraph, where she discusses Husserl’s silence on intuitionism (I split it in 3 parts to comment upon):

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(1) Interpreting Husserl’s silence on this point, we think that he remains an orthodox mathematician who is obliged to consider intuitionist mathematics as a mathematics on the fringe of classical mathematics. (2) One can rightly object that in wanting to ‘ground’ mathematics through a fundamental reform, intuitionism only succeeded in setting up a new mathematics alongside classical mathematics. The intuitionist conceptions, while giving themselves over to a sort of intentional inquiry, concluded in such an amputation of traditional mathematics that they cannot claim to institute a grounding sense-investigation for mathematics. (3) Husserl, on the contrary, is unwilling to sacriﬁce any part of classical mathematics. [9, p. 123; original emphasis] Concerning (1) and (3), I think that Bachelard is right in ascribing this attitude to Husserl. That would explain how, in Formal and Transcendental Logic, while claiming, as we saw, to investigate the core of any mathematics, he can arrive at conclusions that contradict intuitionism, such as the omnitemporality of all mathematical objects [112, p. 164]; the somewhat unlikely way out would be to deny that intuitionist mathematics exists. But I also argued above that this attitude runs counter to his own general views on ontology.141 As for (2), there is a diﬀerence between pure meaning-investigation (noematic analysis142 ) on the one hand and analysis of essences (eidetic analysis) on the other. Bachelard in eﬀect criticises intuitionism for not limiting itself to purely noematic analyses. And indeed it does not; but to do so has never been its aim in the ﬁrst place.143 The project of intuitionism is not to analyse the meaning of concepts in the various conceptions of mathematics. It rather strives to ﬁnd out, by philosophical means, what is possible in mathematics and what is not. In phenomenological terms: intuitionism attempts eidetic analysis.144 And that makes all the diﬀerence. Eidetic analysis is critical, it separates ‘Wesen’ (essence) from ‘Unwesen’ (non-essence) [99, p. 87]. Noematic analysis is not critical; it conﬁnes itself to a pure investigation of meaning (keep in mind note 142), and does not ask the further question whether an intention of that meaning can (ideally) be fulﬁlled; but, according to the principle of transcendental idealism, it is the answer to that question that is decisive in ontological matters.145 For example, noematic analysis will not tell us that round squares cannot exist; that judgement requires eidetic investigation (for this and the previous paragraph, [99, section 16]). Also note how (2) stresses only the negative aspect of intuitionism while neglecting the new possibilities openened up by it. One could say that the ﬁrst act of intuitionism is an act of weak revisionism, and the second act one of strong revisionism. In summary, according to the phenomenological standard for a correct argument in ontology, such an argument should attempt a strict constitution

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analysis within a reﬂexive, transcendental framework. I have argued that this standard opens the way for Husserl to strong revisionism in mathematics. In the next two sections, I argue that Husserl’s and Brouwer’s original arguments fall short of this standard.

5.4 The Incompleteness of Husserl’s Argument 5.4.1 From Atemporality to Omnitemporality In section 4.2.2, I discussed Husserl’s motives for holding that mathematical objects are invariant with respect to time. In the Logical Investigations, he held that they are because such objects exist outside of time altogether. They are atemporal (‘unzeitlich’) [121, p. 129]. Conceptually, there is a second possibility for expressing invariance with respect to time, the possibility of omnitemporality: following the deﬁnition in section 4.1, a temporal object is omnitemporal exactly if it is static and exists at every moment. For mathematical practice, whether its objects are omnitemporal rather than atemporal, or vice versa, does not make any diﬀerence. Both alternatives imply that mathematical objects never come into being, never change and that, correspondingly, mathematical truths are true for all times. From a phenomenological point of view however, the question points to a deep problem of constitution. (This asymmetry between mathematics and phenomenology expresses that the former remains in the natural attitude, while the latter performs the transcendental reduction.) Husserl may not have considered the alternative of omnitemporality at the time of writing the Logical Investigations, but it is certain that he knew of it only a few years later. The notion was discussed, and advocated, by Anton Marty, in his Investigations on the Founding of General Grammar of 1908 [158]. Husserl owned a copy and annotated it in the margins; it can be found in Husserl’s library at the Archives in Louvain. From these notes it becomes clear that Husserl then rejected omnitemporality.146 Marty favoured omnitemporality over atemporality: In the Logical Investigations Husserl ascribes a timeless existence to that which he distinguishes as ideal from the real, whereas the real is allegedly temporal. I cannot regard a distinction of this kind as justiﬁed. Everything that we conceive of as being, it seems to me, we conceive of as being in time. Which already follows from the fact that we conceive of it either as persistent and perseverant, or as changing. The timeless in the strict sense is neither the one nor the other. [181, p. 231, trl. continuation mine]147 Next to this Husserl wrote a shorthand note: ‘Of course, the concept which I thereby have, after all, is not Marty’s concept of the real ?!’ [181, p. 232]148 It is true that Marty has a somewhat diﬀerent conception of ‘real’ and ‘ideal’ (for

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my discussion it is not relevant what Marty’s conception exactly was). But still Husserl’s remark is puzzling, for Marty begins his passage by adopting, for the sake of argument, Husserl’s notions from the Logical Investigations. After all, Marty starts from ‘that which he [Husserl; emphasis mine] distinguishes as ideal from the real’. Marty acknowledges that sometimes when we present an object, we leave the temporal aspect indeterminate, as when we do not have any particular moment or interval in mind; and that sometimes we present an object as existing at all times. These circumstances, Marty holds, are still not suﬃcient to conclude to atemporality: But neither when the temporal determination remains indeﬁnite, nor when we explicitly lift any restriction on the temporal determination, does it seem appropriate to me to speak of a timeless existence. Something existing that neither has become nor will pass, is not thereby timeless, but rather in every time. [158, p. 328, trl. mine]149 Marty does not deny that there are ‘eternal truths’ such as ‘a triangular rectangle is impossible’. But this should be understood as ‘true in the present, past, and future’, not as ‘atemporal truth’ [158, p. 329]. In another note in the margin, Husserl writes ‘Timelessness of the ideal disputed’ [181, p. 232n. 2].150 But Marty does not reject just the idea that the ideal is atemporal, but the very idea of atemporal existence. However, Rollinger may have gone too far when he says that Husserl, in his not entirely accurate note, ‘somewhat trivialises the dispute’ [181, p. 232n. 2]. For it seems rather doubtful that one could, on the one hand, accept atemporal existence, and, on the other hand, deny that ideal objects are atemporal. They are the prime candidates for atemporal existence. However, in 1917/1918, Husserl suddenly embraces omnitemporality, not atemporality, for ideal objects. In a manuscript from that time that would later be used for Experience and Judgement, he characterises an ideal object thus: It is referred to all times; or correlatively, to whatever time it may be referred, it is always absolutely the same. [110, p. 259]151 and Such an irreality has the temporal being of supertemporality, of omnitemporality, which, nevertheless, is a mode of temporality. [110, p. 261]152 (‘Irreality’ is a term Husserl uses for idealities when he wants to stress that they are not individuated in time (are not ‘real’).) Husserl now adopts Marty’s idea that he had rejected before; surprisingly, he never refers in his treatment of this subject to Marty’s book of 1908.153 This is all the more surprising because the reasoning behind it will turn out to be analogous to Marty’s.

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This move is of particular relevance for the present discussion: admitting that ideal objects are not outside of time is a necessary (although not suﬃcient) condition for admitting choice sequences. What were Husserl’s reasons? In fact, the thesis of omnitemporality is a by-product of a much more general development in Husserl’s thought, the expansion of the static phenomenology of Ideas I of 1913 into the genetic phenomenology that began to blossom in the early 1920s. This is a complicated and intricate transformation, but I will be brief about it (the transformation is treated in detail by, for example, Sokolowski [199]). Static phenomenology takes an ontological region as starting point, and then proceeds to clarify the actual and possible ways in which these stable objects come to givenness and manifest themselves in immanent consciousness. It takes the ﬂow of time into consideration, but only to the extent that it follows unities through time. It traces the correlations between constituting consciousness and constituted object, and is therefore directed at what is already ‘objective’, i.e., what is already an intentional unity. Static phenomenology is the phenomenology of the ready-mades. Genetic phenomenology investigates not so much the correlations between the ﬂow of consciousness and the objects given in it, as how these correlative systems come into being. It is, in this sense, an investigation into origins (see p. 25). It clariﬁes where the sense of identity, where these ontological regions ‘come from’. Genetic phenomenology is a genealogy [124, p. 1]. The strict constitution of an object, that is, constitution of an object as self-given (as deﬁned on p. 56), is statically analysed by Husserl as the thematisation of an invariant in the ﬂow of experience. Such an act of thematisation is called ‘objectifying’ [124, section 13] or ‘doxic’ [115, e.g., section 114]. A true or genuine object is something that we can always identify again, perhaps in changing modes of givenness (e.g., perception changing into memory). As Husserl says in Experience and Judgement, It is precisely this identity, as the correlate of an identiﬁcation to be carried out in an open, boundless, and free repetition, which constitutes the pregnant concept of an object. [110, p. 62]154 Note that the time-constituting ﬂow itself is not an object in this sense. The ﬂow of time is not a substrate which is altered or not altered in the course of its perception [104, pp. 74–75]. Genetic analysis made Husserl realise that identies are not only disclosed as invariants through time, as just described, but that the temporal ﬂow is also a condition for their being identities: The ‘object’ of consciousness, the object as having identity ‘with itself’ during the ﬂowing subjective process, does not come into the process from outside; on the contrary, it is included as a sense in the subjective process itself – and thus as an ‘intentional eﬀect’ produced by the synthesis of consciousness. [109, p. 42]155

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That is, if an object is given to us as identical, that is not because that object announces itself as such. Rather it is a performance of consciousness, a result of synthesis, that establishes the sense ‘identical object’. And the form of identity syntheses, Husserl says, is that of internal time consciousness:156 The operations of the synthesis in internal time-consciousness . . . as belonging to the lowest level, necessarily link all others. Timeconsciousness is the original seat of the constitution of the unity of identity in general. [110, p. 73]157 By this, Husserl means that performing an identity synthesis presupposes a temporal structure; it is only because there is an internal time ﬂow that the experiences of the subject present themselves as temporally ordered, as having a beginning and an ending in time, as occurring simultaneously or one after the other; as candidates for synthesis, so to speak. This is the way in which time is a condition for identity. Up till about 1917, Husserl thought that ideal objects were atemporal. Given the above, this cannot be right. On the one hand, if an object is supposed to be essentially outside of time altogether, its identity cannot depend on temporal structures. On the other hand, Husserl discovered that time not only discloses but also conditions identity. Husserl concluded that atemporal objects are really omnitemporal. Only this accords with the fact that the constitution of any genuine object must be constitution in time (note the analogy with Marty’s ‘Everything that we conceive of as being, . . . we conceive of as being in time’, p. 151). Now a further distinction must be introduced. All objects are constituted in the ﬂow of consciousness. That is, they are given to us during a certain span of internal time, which Husserl calls the ‘givenness-time’ [110, p. 255].158 Higher-order acts build up another temporal form of objects, which Husserl calls the ‘essential time’ [110, p. 255].159 of these objects. Diﬀerent objects may have diﬀerent essential times. For example, physical objects typically are given to us as existing during a certain ﬁnite interval of objective time. In the case of immanent objects, such as sensations, givenness-time and essential time coincide: a sensation endures as long as it is given (p. 306). Husserl claims that mathematical objects have as essential time omnitemporality (pp. 309– 314). In a discussion of the question how a categorial object is given to us (section 64c of Experience and Judgement), Husserl therefore says that ‘a temporal form belongs to it as the noematic mode of its mode of givenness’ [110, p. 258].160 But although atemporality is rejected, its essential motivation, according to Husserl, remains in force: a mathematical object ‘properly speaking . . . has no duration as a determination belonging to its essence’.161 In other words: the noemata of acts in which mathematical objects are given have, like all noemata, a temporal binding (because the noema is ‘reell’, i.e., a part of the act, which has its particular place in the ﬂow of time).162 But their noematic essence (‘noematisches Wesen’ [115, section 94]) has not: the

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temporal determination in the noema (the speciﬁc binding to time), is not part of what makes these objects the mathematical objects they are. At any time they are (or would be, or would have been) given as identically the same. Therefore they are omnitemporal. (This is exactly the same move Marty made earlier: existence that is not bound to any point or interval in time is not timeless, but in all times.) Given the criteria for a phenomenological argument in ontology arrived at in section 5.1, we can certainly accept Husserl’s argument in so far as it says that mathematical objects are temporal, as opposed to atemporal. But his actual claim of omnitemporality is stronger. Omnitemporality would forbid choice sequences, at any rate the non-lawlike, whose unﬁnished character requires a temporal determination for their individuation. I now want to argue that Husserl’s own methodology suggests a way to criticise this stronger claim. Husserl frames the claim of omnitemporality as a matter of essence, and he is explicit about the method to disclose such essences. Essences are determined in eidetic variations, so Husserl must have looked at series of mathematical objects and brought out in eidetic variation that this temporal noematic element does not enter into the representation of those objects [124, p. 309]. That is, they are objects for which change (development) in time is not part of their noematic essence, because they are ﬁnished objects; which causes the contingency of the binding. Non-essential temporality motivates omnitemporality (as opposed to atemporality; because every object necessarily is constituted in the ﬂow of time). If an object is atemporal, it is by implication static. That opens the way to one reason for coming to reject atemporality, i.e., if one ﬁnds that certain mathematical objects, supposedly all atemporal, are not static. This was not the road Husserl took. His motivation to reconsider atemporality was not the question whether all mathematical objects are static, but whether objects can be constituted as atemporal. In other words, it was a problem in the theory of constitution that prompted Husserl to revise his ideas on the temporal characteristics of mathematical objects, not any problem in mathematics itself. To bring in temporal aspects of mathematical objects is a necessary (yet, on its own, not suﬃcient) condition for accepting choice sequences as mathematical objects. Why didn’t Husserl also take the further step? What I want to suggest is that Husserl’s eidetic variations were one-sided, caused by an ontological prejudice that mathematical objects must be ﬁnished objects. Husserl neither then nor later considered choice sequences. His eidetic variations are done (or taken for granted) only on ﬁnished objects, e.g., in Experience and Judgement [124, section 64c]; see Bachelard’s comments, quoted at the end of section 5.1. His variations only included classical objects, and therefore the essence arrived at is not necessarily that of the temporal characteristics of mathematical objects in general, but of ﬁnished mathematical objects.

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Alternatively, he could have tried to show that choice sequences are not really mathematical objects and are for that reason excluded from eidetic variations on mathematical objects. Such an argument is not to be found in Husserl either. The case that choice sequences indeed are mathematical objects, I try to make in chapter 6. This methodological blind spot of Husserl is not surprising around 1917. After all, Brouwer did not develop choice sequences in print until the years 1918-1923, in papers that did not loudly announce the introduction of this breakthrough. Some of these papers were hard to ﬁnd even for professional mathematicians, and when found, properly understood by only a few [60, pp. 312–314, 327]. Husserl, who had obtained his doctorate in mathematics in 1883 (p. 28), had, in his intense concentration on philosophy, eﬀectively become a non-mathematician over the years (he was aware of this, as I show in the next subsection). So one can be sure that he did not see the relevant work of Brouwer then. But some of Husserl’s students and correspondents did (see, for example, the references given by Weyl in 1921 [238, p. 39n. 1] and by Becker in 1923 [10, p. 404n. 5] (probably not coincidentally the same that Weyl gives)). 5.4.2 Possible Inﬂuence of Husserl’s Informants It seems Husserl did come to know about choice sequences no later than early 1922. In 1921, Weyl’s ‘Foundational crisis’ [238], originally meant for Husserl’s Yearbook,163 discussed them; Weyl sent Husserl a reprint, with dedication,164 and in January 1922, Becker handed in his Habilitationsschrift to Husserl (published in the Yearbook the next year [10]). Becker discusses both Weyl and Brouwer. On April 9, 1922, Husserl wrote to Weyl: How strongly interested my circle here in Freiburg is in your works, is shown by the Habilitationsschrift of Dr. Becker, which has now been ﬁnished and presented to the Faculty. I have studied it thoroughly, and have reviewed it with highest praise. [128, VII:p. 293, trl. mine]165 So Husserl cannot have missed the choice sequences. But he was well aware of his lack of (updated) mathematical knowledge, and of his dependence on informants. I quote some passages from his correspondence that reﬂect this awareness. When thanking Weyl for sending him Space, Time, Matter (June 5, 1920), Husserl said: Really study the work I cannot do yet . . . , but I am looking forward to the holidays and will at once have an excellent student of mathematics give reports on it to me and discuss the ideas with him. [128, VII:p. 289, trl. mine]166 According to the editors of the Briefwechsel, the excellent student is Oskar Becker.

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In a letter to H.J. Pos (December 26, 1927), concerning Husserl’s upcoming visit to Amsterdam, the latter confesses: Mr. Brouwer, whom I look forward to meeting, I will certainly disappoint. For at present the philosophical-mathematical is somewhat distant from my mind and I would not like to speak about that from behind the lectern. It would take me to much time to make myself familiar with it again – much though I have worked on it in the past. [128, IV:p. 442, trl. mine]167 And in letters to Felix Kaufmann of January 8 and June 27, 1931, one reads: Since the work168 goes into mathematics, I ask you to have it carefully read through with regard to that by a professional mathematician. The smallest mistake or even just diﬀerence from the usual style of presentation will lead to a big fuss from the mathematical readers. Indirectly, that will ruin a positive general reception . . . As Mr. Becker has been taking care of the editorial work for me for years now, I cannot but present this work to him as well, and he too is a professional mathematician. [128, IV:p. 180, original emphasis, trl. mine]169 and Don’t forget about expert veriﬁcation of your mathematical treatise by a mathematician. It is not easy when you have trouble with mathematicians. [128, IV:p. 181]170 (Karl Schuhmann told me that Husserl never had a mathematician check his own Formal and Transcendental Logic of 1929.) These three, Becker, Kaufmann and Weyl, must have been Husserl’s best informants on contemporary mathematics, and intuitionism in particular; especially Weyl. But they would not have made Husserl reconsider his claim of omnitemporality, for each was, in one way or another, unsympathetic to the idea of choice sequences as genuine objects (in a Husserlian framework). The case of Weyl I discussed in section 4.3.4: he thought that we needed the concept of non-lawlike sequence, but should not accept any mathematical object as falling under that concept. We should recognise only lawlike sequences as genuine objects: only they are ‘being’ (‘seiend’), non-lawlike sequences are ‘becoming’ (‘werdend’). Moreover, lawlike sequences can be conceived of as omnitemporal objects. Weyl’s article must have, or would have, made Husserl think that non-lawlike sequences, which are intratemporal, are a façon de parler and not really necessary to admit into one’s ontology. Becker treated choice sequences twice, in his Habilitationsschrift of 1923, and in the treatise Mathematical Existence, which appeared in Husserl’s Yearbook in 1927. Only in 1923 is he concerned with choice sequences as a tool to analyse the intuitive continuum. In 1927, he looks for solutions to the mathematical continuum problem, which is a set-theoretical problem, and therefore

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has, as Becker observes, not much to do with choice sequences ([11, p. 606]; also [10, p. 417]). In both writings, Becker closely follows Weyl’s exposition of 1921 [238]. Moreover, he seems to identify the positions of Brouwer and Weyl. For example, in 1923 he speaks at the same time of the ‘Brouwersche Theorie des Kontinuums’ and the ‘Brouwer-Weylsche Theorie’ [10, pp. 414–415] (he repeats this in Mathematical Existence [11, p. 604]). (Husserl, after supervising Becker’s habilitation, adopted the latter term from Becker, an example is Husserl’s letter to Weyl of April 9, 1922 [128, VII:p. 294].171 ) More importantly, when explaining Brouwer’s theory in 1927, Becker often repeats Weyl: e.g., It is simply not even the same concept of sequence that can be used now in arbitrary generality and then in uniquely determined lawlikeness. [11, p. 603, trl. mine]172 In fact, Brouwer objected when Weyl expressed this idea, and was quick to point out (to Weyl) that there were further substantial diﬀerences (explained in section 5.3). One can assume that from his readings of Weyl and Becker, Husserl got the same picture of choice sequences, i.e., Weyl’s.173 This picture was not completely in accord with Brouwer’s own conception, and in particular discouraged the view of non-lawlike choice sequences as genuine objects. Kaufmann’s case I discussed in section 4.3.1. He argued that, on epistemological grounds, choice sequences cannot be mathematical objects at all. In conclusion of this section, we can say that Husserl did not investigate choice sequences on his own initiative; perhaps he had grown too far apart from the frontiers of foundational research in mathematics. He never included them in his eidetic variations on the temporal characteristics of mathematical objects, but neither did he oﬀer a separate argument for not doing so. Weyl, Becker, and, later, Kaufmann, brought choice sequences to Husserl’s attention, and by 1922 he must have known of them. However, none of these three advocated the conception of (non-lawlike) choice sequences as genuine mathematical objects; perhaps therefore Husserl did not feel the need for an analysis of these objects of his own. Measured by the standard for phenomenological arguments in ontology set in section 5.1, Husserl’s argument for omnitemporality simply falls short, by not considering all relevant candidates in his constitution analyses.

5.5 The Irreﬂexivity of Brouwer’s Philosophy Brouwer tried to go ‘from philosophy to mathematics’ and grounded his intuitionistic mathematics in a more general philosophy.174 This background philosophy can be characterised as a transcendental one. That is, it purports to explain how a non-mundane subject builds up its world in consciousness.

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It is a radical transcendental philosophy in that this ‘world’ does not contain just physical objects but everything, including abstract objects and the mundane subject (the subject as part of the world). From the empirical point of view, such a non-mundane subject is an idealised one. Like fellow transcendentalists Kant and Husserl, Brouwer sought to account for mathematics by referring to structural features of acts of this idealised subject [175]. Because of its solipsistic tendencies (but see [174]) and its mystical characteristics, this part of Brouwer’s thought is generally dismissed. In fact, more sense can be made of it than is commonly assumed (see van Dalen’s overview mentioned in note 174); however, the claim I want to defend here is that, even if we grant Brouwer these features, his background philosophy could still not function as a basis for intuitionist mathematics. This philosophy, taken at face value, is not able to reﬂect on itself. It cannot thematise itself and, a fortiori, it can neither account for itself, nor be self-critical. Therefore it cannot do as the grounding for mathematics that Brouwer wanted it to be. For what is its grounding force, if it cannot be distinguished from mere conjectures that would certainly strike us as less solid than mathematical knowledge?175 There are diﬀerent ways of interpreting my claim. For example, one may be of the opinion that Brouwer’s own philosophy is actually irrelevant to the viability of intuitionism in mathematics. Then what follows can be regarded as providing one precise argument in support of that opinion. Moreover, it is an argument that does not depend on an evaluation of solipsism or mysticism. Or one may rather see it as further encouraging attempts at close reconstruction of Brouwer’s ideas within more acceptable frameworks, for example along the lines of the phenomenological approaches of Weyl, Becker, and Heyting (e.g. [238], [11], [87]). This is the interpretation I will use. In any case, it is simply a historical point: Brouwer’s philosophy was, as a matter of principle, not able to deliver what he wanted from it in mathematics. Related to this is that, even if one denounces intuitionism or prefers an independent, perhaps less ‘obscure’ argument for it such as Dummett’s, that does not make precise evaluation of Brouwer’s own views unnecessary. Every philosophy deserves to be judged on its own merits. Before embarking on the argument, let me address the following objection to even undertaking it.176 ‘Brouwer thinks that mathematics is justiﬁed to the extent that it conforms with intuition and with certain properties of intuition that establish limitations on what we can know. What more could be needed? Why is reﬂexivity needed?’ This objection already grants Brouwer precisely that which is under discussion. How did Brouwer come to know these properties? Was he justiﬁed in ascribing these properties to intuition? Speciﬁcally, could the justiﬁcation that he himself came up with actually do its job? I will argue that the answer is no, because his background philosophy is not reﬂexive, as explained above. The reﬂexivity does not come in only after the mathematics has been

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vindicated along the lines mentioned in the objection, it bears on that vindication itself. The force of my argument would not be clear if all philosophies could be accused of this kind of irreﬂexivity; but clearly that is not the case. For example, nothing in naturalism stands in the way of naturalist talk about naturalism, and nothing in phenomenology hinders phenomenologists in reﬂecting on and reﬁning phenomenology (in fact, reﬂexivity is one of its basic features [115, sections 77–79]). The following argument I set forth against the background of Brouwer’s last and most detailed philosophical exposition, ‘Consciousness, philosophy, and mathematics’ [39]. The precise meaning of the terms used will be made clear as we go along. 1. Brouwer’s philosophy asserts that there are three (or four) kinds of consciousness: stillness, sensational, mathematical, and (perhaps) wisdom. (Premise) 2. But correctly making this assertion requires a kind of consciousness that is itself neither stillness, nor sensational, nor mathematical, nor wisdom. (Premise) 3. So Brouwer’s philosophy cannot reﬂect on itself. (1,2) To argue for the truth of premise 1, I will bring it out as a distinction that Brouwer stops just short of introducing explicitly, while clearly making it. I start from the ﬁrst paragraph: First of all an account should be rendered of the phases consciousness has to pass through in its transition from its deepest home to the exterior world in which we cooperate and seek mutual understanding. [39, p. 1235] Consciousness in its deepest home is the Urstate of consciousness. In this state, consciousness ‘seems to oscillate slowly, will-lessly, and reversibly between stillness and sensation’ [39, p. 1235]. There is no sense of time in it. From this original state an ‘exodus of consciousness’ [39, p. 1238] can start. This exodus happens in three phases, which Brouwer names the ‘naive’, the ‘isolated causal’, and the ‘social’ [39, p. 1238]. In the naive phase the subject comes to distinguish between the present sensation and those of the past. During the following, isolated causal phase, out of sensations thus organised in time grows consciousness of things (among them individuals), and causal relations. The things make up the ‘exterior world of the subject’ [39, p. 1236]. In the social phase, the end of the exodus, we become conscious of a world as shared with other individuals, with whom we can cooperate in our dealings with nature and things. So Brouwer recognises four states of consciousness, the deepest home and the three transitory phases. To all appearances, this division is meant as a proper one, i.e., exhaustive and consisting of mutually exclusive terms. Each

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transitory phase presupposes the previous ones and therefore we can speak of the distance of a given phase from the deepest home. There is a second distinction relating to consciousness, one that Brouwer clearly uses but does not deﬁne explicitly. States of consciousness can be classiﬁed not only according to distance from the deepest home, as above, but also according to the basic structure of their content. The categories in this classiﬁcation I call kinds of consciousness. As Brouwer clearly takes his fourfold division of states of consciousness to be exhaustive, a likewise exhaustive list of kinds of consciousness can be obtained by considering, for each state, which kinds it allows for. I think that there are three, and maybe four, which I call stillness, sensational, mathematical,177 and wisdom. (Except for ‘sensational’, for which he gives no alternative, these are terms used by Brouwer.) In the deepest home, as we saw, there are two kinds of consciousness, stillness and the sensational. Note that in Brouwer’s usage of the word, ‘sensations’ need not be sensory; for example, he speaks of ‘sensations of vocation and of inspiration’ [39, p. 1236]. The naive phase is opened by the phenomenon of a move of time. ‘By a move of time a present sensation gives way to another present sensation in such a way that consciousness retains the former one as past sensation’ [39, p. 1235]. Furthermore, by distinguishing between present and past sensation consciousness places itself at a distance from both and from stillness. Consciousness thus distanced Brouwer calls ‘mind’. It is here that the subjectobject distinction is introduced: mind, in the function of a subject, experiences both the past and present sensation as object. This ‘two-ity phenomenon’ can be iterated when yet another sensation becomes the present one, and so on, thus leading to ‘a world of sensations of motley plurality’. Some sensations turn out to be inextricably bound to the subject; these are the ‘egoic’ sensations. All egoic sensations of an individual together are called the soul of this human being [39, p. 1235]. The opposite of ‘egoic’ is ‘estranged’. We will see in a moment that things in the exterior world are associated with such estranged sensations. Mathematics seems to have its origins in the naive phase as well, as it comes 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 substratum of all two-ities, the basic intuition of mathematics, is left to an unlimited unfolding. [39, p. 1237] In the isolated causal phase, Brouwer distinguishes types of entities that we can be conscious of, i.e., causal sequences, things (in particular, individuals), and objects in general. These are all identiﬁed with certain iterative complexes of sensations (sequences of sensations that occur repeatedly), diﬀering in their additional properties. Iterative complexes come in two varieties, those whose elements can be permuted in time and those whose elements cannot (this may be a matter of gradation).

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The ﬁrst variety Brouwer does not name as a whole. Some of them, those that are completely estranged from the subject, he calls ‘things’; one may suggest the use of ‘object complexes’ for the whole variety. The second variety of iterative complexes are the causal sequences. Their deﬁning characteristic is that these are sequences such that ‘if one of its elements occurs, all following elements are expected to occur likewise, in the right order of succession’ [39, p. 1235]. As an illustration of the diﬀerence between causal sequences and object complexes: you cannot switch the cutting of an onion and your subsequent crying without breaking the relation between the two, but you can look at the diﬀerent sides of the onion in any temporal order you want. The social phase begins when causal attention detects acts of the other indivduals. This opens the way to organised cooperation of a group of individuals. A prominent form of this is scientiﬁc thinking, ‘which in an economical and eﬃcient way catalogues extensive groups of cooperative causal sequences’ [39, p. 1237]. However, the kinds of consciousness involved here are still just the sensational and the mathematical, and no new ones are found. It is not easy to determine what Brouwer means by ‘wisdom’, or even whether it is a kind of consciousness or not. The sharpest characterisation is not in ‘Consciousness, philosophy, and mathematics’, but in ‘The unreliability of the logical principles’ [24] of forty years earlier, where it is said that ‘wisdom abolishes the discernment between the subject and something diﬀerent’ [44, p. 108].178 But it seems the same is meant in the later text, as there it is stated several times that wisdom is not found in causal thinking, for which that separation is a necessary condition. The following are two examples of such statements. Searching for wisdom, we may ﬁnd it in knowing that causal thinking and acting is non-beautiful and hard to justify, and that in the long run it brings disappointment. [39, p. 1240; original emphasis] If the delusion of causality could be thrown oﬀ, nature, gradually resuming her right, would be (except for her bondage to destiny) generous and forgiving to a mankind decausalised and subsiding to more modest and harmonious proportions. [39, p. 1242] It is not clear to me to what state of consciousness wisdom belongs. On the one hand, it seems that this is the deepest home. For example, at the end of a passage where he describes how ‘there may be wisdom in a patient tending towards reversible liberation from participation in cooperative trade and from intercourse presupposing plurality of mind’, he writes that ‘perhaps at the end of the journey the deepest home vaguely beckons’ [39, p. 1242]. Then wisdom would be a third kind of consciousness in the deepest home, for I take it that it is neither a sensation nor stillness. (Also, ‘in wisdom . . . the perception of time is no longer admitted’ [44, p. 108].179 ) This leaves only the deepest home, because the other states require the move of time to come into being.

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On the other hand, Brouwer often speaks of wisdom as consisting of thoughts that can be expressed, for example when he cites passages from the Bhagavad-Gita [39, p. 1241–2] (see also the quotation below). And the place for thoughts that can be expressed can hardly be the deepest home. To resolve this unclarity, I suggest that the word ‘wisdom’ as Brouwer uses it is ambiguous. It may mean ‘pure wisdom’, which somehow resides in the state of the deepest home, but which is of yet another kind than stillness or sensation; and it may mean ‘practical wisdom’, which is any knowledge in another state that points to, or tends to, the way back to the deepest home. But as we will see in the discussion of premise 2, for the present argument it will suﬃce to know that one of the deﬁning characteristics of wisdom, whatever that may be in the ﬁnal analysis, is its dissociation from causality. In sum, we have found that Brouwer recognises three (or four) kinds of consciousness: stillness, sensational, mathematical and (perhaps) wisdom. This establishes premise 1. For the truth of premise 2, I consider each of these four kinds and ask whether the assertion could have its origin in consciousness of that kind. First, that the assertion cannot have its origin in stillness follows immediately from the meaning of the latter term. In stillness, there is no activity at all. Secondly, the assertion in question clearly is not purely mathematical; it does not deal with purely mathematical entities, i.e., entities created starting from the empty two-ity. Thirdly, the assertion cannot be one of pure wisdom either. Introducing a threefold (or fourfold) distinction requires a mathematical operation. But in wisdom there is no place for a mathematical understanding, and one cannot coherently make an assertion while at the same time denying something that is presupposed by that assertion. Mathematics depends on the subject-object distinction, while in wisdom precisely that distinction is rejected. That is why Brouwer can write: ‘in wisdom, which abolishes the discernment between the subject and something diﬀerent, there is no mathematical understanding’ [44, p. 108].180 One may suggest that the assertion, although not part of pure wisdom for the reason just given, could be an approximation to it. Then the assertion would perhaps keep most of its value (i.e., as practical wisdom). In a sense, Brouwer allows for such approximations, namely, statements of (applied) mathematics that express wisdom in a one-sided or distorted way. This is implied by a characterisation of mysticism that he gives:181 Perhaps the greatest merit of mysticism is its use of language independent of mathematical systems of human collusion, independent also of the direct animal emotions of fear and desire. If it expresses itself in such a way that these two kinds of representations cannot be detected, then the contemplative thoughts – whose mathematical restriction appears as the only live element in the mathematical system – may

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perhaps again come through without obscurity, since there is no mathematical system that distorts them. [206, p. 398, trl. modiﬁed]182 But it is hard to see how this possibility could be of any help here. If, as we just concluded, introducing distinctions runs counter to the fundamental nature of wisdom, how then could adding a distinction still lead to an approximation of wisdom? Fourthly, it cannot be a case of causal consciousness. The reason is that causal attention only organises sensations and does not add anything qualitatively diﬀerent to them; a fortiori, consciousness itself is inaccessible to causal attention. To see the validity of this claim, consider the following argument (the numbering indicates that it deals with premise 2 of the argument above): 2.1 2.2 2.3 2.4 2.5 2.6

Sensations are individuated in time. (Premise) Consciousness is prior to the move of time. (Premise) Consciousness cannot be individuated in time. (2.2) Hence consciousness is not a sensation. (2.1,2.3) Consciousness is neither a sensation nor a complex of sensations. (2.4) Consciousness is inaccessible to causal attention. (2.5)

In the whole argument, ‘consciousness’ refers to consciousness as such, and not to any of its particular kinds or stages. By premise 2.2 I mean that consciousness can exist without the move of time, but not vice versa. The priority is ontological. Step 2.3 is a speciﬁcation of premise 2.2. For speciﬁc varieties that arise after the introduction of the move of time, the step need not be valid; but here it is, as the argument is one about consciousness as such. The inference from 2.1 and 2.3 to 2.4 is licensed by the general principle that two objects that cannot be individuated by the same principle cannot be of the same kind. From step 2.4 we get to 2.5 by another general principle, namely that if objects of kind K are individuated in time, then so are complexes of objects of kind K. In this case, complexes of sensations would be the result of activity of the subject. Finally, we get 2.6 from 2.5 by the deﬁnition of causal attention as the ‘freewill-phenomenon’ that ‘performs identiﬁcations of diﬀerent sensations and of diﬀerent complexes of sensations’ [39, p. 1235]. The textual evidence for premises 2.1 and 2.2 is not hard to ﬁnd in ‘Consciousness, philosophy, and mathematics’. Let me cite its second paragraph in full (the ﬁrst one was quoted on p. 76). I split it in two: (1) Consciousness in its deepest home seems to oscillate slowly, will-lessly, and reversibly between stillness and sensation. And it seems that only the status of sensation allows the initial phenomenon of the said transition. This initial phenomenon is a move of time. (2) By a move of time a present sensation gives way to another present sensation in such a way that consciousness retains the former one as a past sensation, and moreover, through this distinction

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between present and past, recedes from both and from stillness, and becomes mind. [39, p. 1235, original emphasis] According to the second part, it is by a move of time that one present sensation makes place for another. The former is retained in the present one as ‘past’. A further move of time iterates the process. The structural aspect of this process relevant here can be represented in a simple diagram: 1 2 (1) 3 (2 (1)) .. . When sensation 2 occurs, sensation 1 is retained as past; then a move of time gives way to sensation 3 as present, retaining 2 as past, which retained 1 as past. And so on. The structure of time is a nested one, so with each sensation a diﬀerent moment in time is associated. Conversely, this association serves to individuate sensations; this yields premise 2.1. Regarding premise 2.2, the ﬁrst part of the quoted paragraph says that consciousness can exist without having sensations, but that the status of sensation is a necessary condition for a move of time to occur.183 Hence a move of time depends for its existence on consciousness, but not vice versa. Thus consciousness is (ontologically) prior to the move of time. (For a remark on the word ‘seems’, see p. 83.) This conclusion is corroborated by Brouwer’s statement further on that one can always ‘easily [realise] temporary reﬂuence to the deepest home leaving aside naivety, through the free-will-phenomenon of detachment-concentration’ [39, p. 1238]; recall from p. 77 that by ‘naivety’ Brouwer means the phase of consciousness where the move of time occurs. An objection here may be that the word ‘temporary’ suggests that permanent reﬂuence is not possible and that this indicates that somehow consciousness cannot exist entirely independent from time. However, I think that ‘temporary’ was put in to justify the ‘easily’, and not to suggest that permanent reﬂuence is impossible in principle. Such an addition does not appear in the earlier formulation of this idea in the ﬁrst of the two 1928 Vienna lectures: Everyone can have the inner experience, that he can at will dream himself to be without time awareness and without the separation of the I and the world of perceptions, or bring about this latter separation by his own eﬀort.184 Likewise, one of the rejected passages of his dissertation reads: ‘We can go even further and say that the creation of time as a matrix of moments is a free act of ourselves’ [206, p. 401, modiﬁed].185 Both quotations state that time is a free creation of consciousness, which implies that (a certain deep level of) consciousness can exist independently from time.186

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This concludes the subordinate argument, begun on p. 80, for premise 2. It is able to stand by itself but is corroborated by the cited earlier passages in Brouwer’s writings. Moreover, it completes the main argument. Finally, I have to address two objections. The ﬁrst is that Brouwer sometimes is self-critical. Let me discuss two examples. One case concerns the Cantor-Bendixon theorem (every closed, wellconstructed point set can be divided into a perfect and a denumerable point set). In a book review of 1914, Brouwer considered this theorem evidently true and not even in need of a proof [27, p. 79]. Only a few years later he rejects this view and now the same theorem is said to be false [30, p. 953n. 9]. Then the objection might be that this reversal of view would not be possible according to my argument. Now I do not mean to rebut this objection by saying that such selfcriticism, when it occurs, is not part of Brouwer’s oﬃcial philosophy (in the sense that the oﬃcial philosophy would not allow for it and that in such cases Brouwer does not practise what he preaches). It is rather that Brouwer’s philosophy, as mentioned in the introduction, describes an ideal(ised) mathematician (for further discussion, see the Appendix). Various contingent factors (such as limitations of memory and attention) could inﬂuence the acts of an actual subject. It is such factors that Brouwer held responsible for his change of view. He commented that the said review was written at a stage where he had found the general principles of intuitionism, but was not yet completely clear about their consequences [30, p. 953n. 9]. Another example of Brouwer’s self-criticism brings out the same point. In ‘Historical background, principles and methods of intuitionism’, he discusses choice sequences and restrictions on the freedom of choosing numbers. Then he adds a footnote: In former publications I have sometimes admitted restrictions of freedom which regard also further restrictions of freedom. However this admission is not justiﬁed by close introspection. [40, p. 142] The root of the error, Brouwer says, is that the original act of introspection was not careful enough; but such a lack of care is a problem that only the empirical, not the idealised, subject can have. And sometimes, Brouwer himself is hindered by slipping into classical thought. In 1952, he criticised one of his former publications in this way: In §1 of my essay ‘Begründung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossenen Dritten, Zweiter Teil: Theorie der Punktmengen’ which appeared in 1919, I tried among other problems to circumscribe the fragments of the Bolzano-Weierstrass theorem and of Cantor’s main theorem which can be preserved in intuitionistic mathematics. Reading over these developments to-day, one ﬁnds that they are obsolete and in need of radical recasting. We must acquiesce

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to the fact that intuitionistic ideas penetrate mathematics only slowly and that remnants of unclear classical ways of thinking are only gradually removed. [41, p. 516] What matters here is not so much whether the details of Brouwer’s diagnoses are correct. The point is that they aim at shortcomings of the empirical subject, not of the idealised subject of the transcendental philosophy. What Brouwer ﬁnds fault with are not the basic properties of the idealised subject as stated in his philosophy, but with his own thinking about them. The second objection is that Brouwer in ‘Consciousness, philosophy, and mathematics’ nevertheless speaks about ‘the joyful miracle of the selfrevelation of consciousness’ [39, p. 1238], which implies the presence of a self-reﬂexive structure in consciousness. Now this may be meant as a metaphor, but that would take the sting out of the objection. So for the sake of argument, let me assume that it is to be taken literally. Then my reply would be that, in the way Brouwer understands it, this is not quite self-revelation. He says it is ‘apparent in egoic elements of the object found in forms and forces of nature’ [39, p. 1238]. But egoic elements are sensations, and part of the subordinate argument just concluded is that consciousness itself is neither a sensation nor a complex of them. It follows that what can become apparent in egoic elements is not consciousness itself but, at best, consciousness as sensationally given. (But it is hard to see how this consequence could be correctly asserted within Brouwer’s philosophy: doing so would require the access to consciousness that this philosophy, I argued, denies.) One might take the tentativeness in the ﬁrst part of the quotation on p. 80, expressed by ‘seems’ (twice), as an indication that Brouwer may have had some notion of this (note that this tentativeness is absent in the description of other phases of consciousness in ‘Consciousness, philosophy, and mathematics’). But that would imply a radical scepticism about consciousness itself, thus certainly making it impossible to ‘render an account of the phases of consciousness’, as Brouwer claims to do [39, p. 1235]. It is clear that, in order to fulﬁll its transcendental task, a reﬂexive structure should be added to Brouwer’s framework. Doing so is in any case warranted phenomenologically (in a sense not limited to Husserl). It would certainly require revisions of other parts of that framework to make it ﬁt. Whether the result would be close to Kant, close to Husserl, or yet something else, depends on how one resolves further issues in Brouwer interpretation that are beyond the scope of my present concerns; I will leave the question open for now. In conclusion, Brouwer’s philosophy does not meet the phenomenological standard for a correct argument in ontology (section 5.1). It has no place for self-reﬂexivity, and therefore has no grounding force. If we are to admit choice sequences into our ontology (to arrive at a conceptually more satisfying version of analysis than the classical), a new argument is called for. The next chapter aims to provide this.

6 The Constitution of Choice Sequences

In section 5.1, it was argued that with regard to pure mathematics, phenomenology is capable of ontological judgements. Generally, complete justiﬁcation for asserting the existence of a supposed object consists in giving a strict constitution analysis; what is speciﬁc to the case of pure mathematics is that the laws governing strict constitution of its objects are precisely the laws of categorial formation. In a slogan, for formal objects, transcendental possibility and existence are equivalent. What has to be shown, then, is that choice sequences can be strictly constituted as formal, or purely categorial, objects. (In Tragesser’s felicitous turn of phrase, ‘something is recognizable as being a mathematical object if it can be recognised that it can be completely thought through mathematically’ [217, p. 293].) This will be attempted in two steps. The ﬁrst is to show that choice sequences can be constituted as objects at all, the second, that this is constitution of purely categorial objects.

6.1 A Motivation for Choice Sequences Brouwer proposes to analyse the intuitive continuum mathematically by employing choice sequences. A choice sequence is a potentially inﬁnite sequence of which the subject chooses the elements, one after the other. In making its choices, the subject enjoys complete freedom; this includes the freedom to impose restrictions on its own choices. In the limiting case, such restrictions amount to making the choices according to a law. As described in section 4.1, Brouwer deﬁnes a point of the continuum (or real number) P as a choice sequence of nested rational intervals λνi : P = λ ν1 , λ ν2 , λ ν3 , . . . This approach is easily generalised to n-dimensional Cartesian spaces. Such a sequence is unﬁnished and, except when lawlike, indeterminate to a varying extent. 85

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What motivates deﬁning points in this way? Here we have an instance of that harmless circularity that characterises all understanding, known as the hermeneutic circle. Mathematical theorems depend on deﬁnitions, but these deﬁnitions do not come out of the blue: they are motivated by what kind of theorems one wants to prove using them. That is, we use choice sequences to analyse the continuum, but how we deﬁne choice sequences in turn depends on a prior understanding of the structure of the intuitive continuum. Notice that the intuitive continuum itself is not, in Husserl’s sense, an object of pure mathematics. It is not a wholly formal object, but contains sensuous (‘sinnliche’) elements. It is not purely categorial. On the other hand, the choice sequences that Brouwer introduced to analyse the continuum mathematically, are objects of pure mathematics (or rather, are meant to be such objects, for to show that they are, is precisely the task at hand). The reason is that the numbers in a choice sequence specify (boundaries of) intervals of the continuum, but are not those intervals themselves. The problem at hand is how choice sequences are constituted, not how the intuitive continuum is. The intuitive continuum itself is constituted in the lifeworld, our everyday world of pre-theoretical experience. I will take that constitution as a given; for details, see in particular sections 9a and appendices II and III of [100], as well as [10] and [209]. Here is the main idea. According to Husserl, the intuitive continuum is an ideal, a limit (‘Limes’), obtained by eidetic variation from strokes and lines in our everyday world, in particular from our experience of straightening them out and dividing them. From these experiences the ideals arise of the limit of straightening out, and of the limit of dividing lines (in the sense of ‘inﬁnitely repeatable’). As he elaborates in the Crisis, In place of real praxis – that of action or that of considering empirical possibilities having to do with actual and really [i.e., physically] possible empirical bodies – we now have an ideal praxis of ‘pure thinking’ which remains exclusively within the realm of pure limit-shapes . . . But in this mathematical praxis we attain what is denied us in empirical praxis: ‘exactness’; for there is the possibility of determining the ideal shapes in absolute identity. [108, pp. 26–27]187 In Husserl’s account, then, the constitution of the intuitive continuum shows a dependency on acts in our everyday world. (Such a dependency does not imply a form of psychologism, it is just that such an ideal object is given to us through such acts.) Brouwer and Weyl stressed the inexhaustibility and, in particular, the nondiscreteness of the (intuitive) continuum (as discussed in sections 4.2.2 and 4.3.4).188 The phenomenological description can be carried one step further: what is the condition of possibility for those two aspects? In other words, what is it about the intuitive continuum that lets us see those aspects? It is the homogeneity [10, p. 426] of the continuum: it is a whole of which the parts are fused with it, in the strong sense that they are also continuous;

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the same holds for parts of parts, and so on. Husserl calls wholes of this kind ‘extensive wholes’ (‘extensive Ganzen’) and characterises them by saying that Here the parts of the parts are parts of the whole in exactly the same sense as the original parts were. [111, p. 470]189 This homogeneity accounts, ﬁrst, for the non-discreteness of the continuum, i.e., the fact that it has no ultimate parts; for any supposed ultimate part would be homogeneous with the whole continuum, hence could be further divided after all. This stands in contrast to, for example, an inﬁnite point set. The latter may be divided into subsets, but there are ultimate parts, i.e., the (singletons corresponding to the) elements of the set. Surely, an inﬁnite point set may be dense, but it makes no sense to say that its ultimate parts, the elements, are dense; but every part of the continuum is itself continuous, therefore it has no ultimate parts. This is why it cannot be analysed as a set of discrete elements. Secondly, from this homogeneity follows the inexhaustibility of the continuum; if the continuum does not consist of ultimate parts or atoms, there is, a fortiori, no exhaustive enumeration of the parts of the continuum. (There is no implication in the other direction: an uncountable point set consists of atoms, its elements, but is not exhaustible either.) This particular part-whole relationship lets the continuum be given to us with a horizon ‘and so on’, i.e., we see that after every determination of parts, however many, there will be still further ones (compare section 51b of Experience and Judgement [124], which I will use again below).190 The deﬁnition of choice sequences is motivated by this inexhaustibility and non-discreteness of the (intuitive) continuum.191 The collection of laws is denumerable; choice sequences may follow a law, but need not. The possibility of non-lawlike sequences represents the inexhaustibility of the continuum. Also, the identiﬁcation of points with unﬁnished sequences of nested intervals expresses the non-discreteness of the continuum, by making the question of extensional identity undecidable, not just for epistemological but for ontological reasons. Two sequences may have identical initial segments, but in general we cannot tell yet whether they will not diverge later on, because they are unﬁnished objects (p. 36).192 The claim that choice sequences do truly capture the intuitive continuum can now be supported by pointing out that their deﬁnition represents the two essential properties we discern in the intuitive continuum. Here the earlier constructivist analyses of the continuum (section 4.2.2) fell short. The analysis so far suggests that, genetically, wholly lawless sequences are primordial: the homogeneity of the continuum is only fully reﬂected in the choice of the next nested interval if that choice may be made wholly freely. Other types of sequence are arrived at by modifying the concept of lawless sequence, through the speciﬁcation of various restrictions. The general concept ‘choice sequence’ is constituted in eidetic variation on sequences of these diﬀerent types.

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What motivates the constitution of diﬀerent types? The free choice of elements allows free choice of restrictions on choices of elements. Operations on various types may lead to yet further types (there is an example on p. 45). The advantage of restrictions is that they allow more deﬁnite assertions about the sequences falling under them [222, p. 225]. Such restrictions are freely chosen, because the homogeneity of the continuum does not force them upon us. This is a feature of all extensive wholes, of which the continuum is an example: In an extended whole there is no division which is intrinsically primary, and no deﬁnitely delimited group of divisions forming the ﬁrst grade in division; from a given division there is also no progress determined by the thing’s nature to a new division or grade in division. We could begin with each division without violating an intrinsic prerogative. [111, pp. 470–471]193 So restrictions are self-imposed. As Brouwer noted on a reprint (now in the Brouwer Archive in Utrecht) of the ﬁrst ‘Foundation’ paper, The arbitrariness of this ‘restriction condition’ associated with a ﬁnite choice sequence lends this choice sequence, and hence also its extensions, a new arbitrariness. [32, original emphasis]194 Moreover, as the intervals chosen at stages following the introduction of a restriction retain the character of a continuum, the continuum does not force us to hold on to that restriction in our further choices. (For the same reason, restrictions can be freely and meaningfully chosen at any stage in the development of a sequence: at any stage we are facing an interval which is continuous, so there is always something for which posing restrictions makes sense.) Let me introduce a distinction between two kinds of restrictions, provisional ones and deﬁnitive ones. The deﬁnitive restrictions are those that we stipulate to keep to in the whole further development of the sequence. Provisional ones are those that we use for the time being, but may lift later on. This distinction will be the basis for an argument for the continuity principle in chapter 7. For now, I will leave it at pointing out where it comes from.195 Historically, the wish to reﬂect the inexhaustibility and non-discreteness of the intuitive continuum mathematically was the original motivation for introducing choice sequences. But these objects, once obtained, can be separated from their historical origin, and indeed other uses have been found. They are also at work in completeness proofs for intuitionistic predicate logic developed by Beth, de Swart, and Veldman (see, e.g., ch. 5 of [71] for a discussion); choice sequences together with Kripke’s schema make it possible to deﬁne the (intuitionistic) powerset of N as a spread [53];196 and Gödel offered an interpretation of Bernays’ proof of the well-foundedness of 0 that uses choice sequences [77, pp. 268–271] (see also p. 26). Accordingly, below, in dealing with the question why choice sequences are mathematical objects at all, what motivates these objects will play no important role.197

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6.2 Choice Sequences as Objects A choice sequence is begun at a particular moment in time, and then grows as we choose further numbers. This process is generally open-ended and may be continued forever. There is a clear, implicit understanding in this way of speech that a choice sequence is an object of some sort. In the following I want to make this understanding explicit, to disclose in virtue of what such a series of choices forms a unity, that is, in virtue of what it constitutes an object. The following discussion builds further on section 5.4. We have to see whether they really are objects in the strict sense to begin with. Recall from section 5.4 that this means that we have to see whether descriptive analysis exhibits an activity that founds a subsequent act in which a suitable invariant is thematised. The activity that founds the self-givenness of choice sequences is that of choosing. In this activity choice sequences are pre-constituted (‘vorkonstituiert’ [124, p. 64]), meaning that after this activity one only needs to carry out an objectifying act to constitute the object. Such an objectifying act thematises the invariant that established itself in passive synthesis.198 Similarly, natural numbers are arrived at in objectifying acts after the activity of counting. Husserl proposes the general picture in Experience and Judgement (one of his examples is that of a set, section 61. Other examples that can be interpreted in this schema of activity followed by objectifying act, are given by Mac Lane [156, section I.11]). That choice sequences change over time not only does not rule out they are invariant over time in some respect, such an invariance is presupposed by it. Only some substrate that remains identical can change, or remain the same, over time. (In section 5.4 it was remarked that for this reason, the ﬂow of time is not itself an object in the strict sense. That ﬂow cannot change, rather it is a condition for change.) In the case of choice sequences, what is this substrate, this invariant? I will look at two alternatives before introducing what I think is the right one. First, the invariant is not any concept, however speciﬁc, that a sequence would fall under. A concept and an object falling under that concept are two diﬀerent things. In the case of choice sequences this becomes clear when we consider that a sequence consists of linearly ordered parts, while a concept governing it does not. Also, the temporal characteristics of a concept and a sequence to which that concept applies are not always the same. A non-lawlike sequence unfolds in time, a concept does not. Consider, for example, the concept expressed by ‘The lawless sequence begun by me exactly one year from now at 6.58 pm’. The meaning of this expression, and hence the concept, is invariant over all time, past, now, or future. It does not change; it is omnitemporal. Now consider that meaning not just by itself, but as part of an act in which I intend that choice sequence (that meaning is the act’s ‘noematic nucleus’199 [115, sections 90–91]). Because of the nature of a lawless sequence, that intention could not get any fulﬁlment before I start making choices at the moment

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in time that the noematic nucleus refers to. So although there is an invariant meaning in my experience, there can not be an invariant which fulﬁls intentions with that meaning as noematic nucleus before the moment I start choosing. In other words, the object isn’t there until I begin choosing (although the meaning I would employ in describing that very same object is); the object is intratemporal, the meaning omnitemporal. Here the case of lawlike sequences is a little diﬀerent (as is also observed by Becker [11, p. 449]). In a higher order act we may abstract from the unfolding of a lawlike sequence and regard it as non-essential for its identity: at any moment the unfolding would have proceeded in exactly the same way, so we may instead thematise that way of unfolding, which is just another way of saying that we thematise its law. The unfolding itself then is not representative for the lawlike sequence. This is why Husserlians can conceive of lawlike sequences as static objects, even when Brouwer insists that all sequences are unﬁnished and do develop in time (section 6.1). But here still the ﬁrst diﬀerence mentioned applies: a law and the sequence speciﬁed by it are not identical. The information one might want to extract from a lawlike choice sequence can already be found in the law itself. The law may go proxy for the sequence. A second alternative for the invariant we are looking for would be the initial segments. Indeed, once values are chosen for an initial segment they cannot be changed later on. But these segments cannot be the invariant that are the evidence for choice sequences as genuine objects. Diﬀerent choice sequences may have the same initial segment. What makes them diﬀerent is their intensional properties, and such a diﬀerence need not consist in more than having been begun at diﬀerent times. (Think of two lawless sequences, one started at t0 , the other at t1 , and with the same course of values so far.) In the unfolding of the as yet open horizon, they may come to diverge. The reason why this second alternative does not work provides the clue to what seems to be the right invariant. The ‘and so on’ that indicates continued choices is recognised by Husserl as a categorial form (Urteilsform):200 There appears here the new form of determination: ‘and so on,’ a basic form in the sphere of judgment. The ‘and so on’ enters into the forms of judgment or it does not, depending on how far the thematic interest in S [the object of which our experience is to be explicated] extends; therefore, it produces diﬀerences in the forms of judgments themselves. [110, p. 218]201 In considering initial segments as the identity-constituting invariant, precisely this horizon ‘and so on’ is not thematised, is left out of our interest. In doing so, we miss the fact that choice sequences are unﬁnished objects. I will now work out the suggestion that what remains invariant is the character of the sequence as a developing sequence, a development that started at a particular point in time. To bring this invariant to light, ﬁrst of all notice that in forming a choice sequence, an identical process is continued:

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Every mental process with several members, progressing in an orderly manner, carries with it such an open horizon; it is not one next unique member which is prescribed but the continuance of the process itself, which thus always has the intentional character of an open process. [110, pp. 217–218]202 The relation between the ‘progress in an orderly manner’ and the openendedness of the process is also noticed by Brouwer, in a footnote in his dissertation: Where one says: ‘and so on’, one means the indeﬁnite repetition of the same thing or operation, even though that thing or operation may be deﬁned in a complex way. [23, p. 143, original emphasis]203 The same idea later also appeared in Wittgenstein’s Tractatus, 5.2523: The concept of successive applications of an operation is equivalent to the concept ’and so on’. [245]204 To apply this idea to choice sequences, one has to consider the more or less free choice of a number an operation too. In 1907, Brouwer obviously did not, but his coming to accept choice sequences as intuitionistic objects by and large had to consist in accepting this as an operation. This is conﬁrmed in a lecture from November 1951, where Brouwer says that the extension of mathematics with choice sequences is justiﬁed not by utility considerations, but by the fact that it is ‘an immediate consequence of the selfunfolding’ [45, p. 93]. In Husserl’s terms, the selfunfolding (ﬁrst act of intuitionism) introduces the required categorial form. The process of choosing extends identically the same, unﬁnished object. The object grows but it is the same object that grows. Some substrate must remain identical if the idea of change (in this case, growth) of an object is to make sense. At this point the distinction between the process and the choice sequence constituted in it must be kept in mind. There is the identity of the process, and the identity of the sequence. These are not the same. The process may have all kinds of characteristics that the sequence does not share. For example, the process consists of acts, the sequence does not. A conspicuous consequence is that the time span between successive acts of choice does not show up in the sequence (it plays no representing role). Also, the process may involve revising intensional properties (e.g., abandoning a provisional restriction; I come back to this in chapter 7) and such revisions along the way need not leave a trace in the sequence itself. That is, if a sequence is given to us at time tk in the form ‘initial segment and some intensional properties’, that by itself may not be enough to tell whether at a time tj (j < k) the sequence had not only a smaller initial segment but also diﬀerent intensional properties; on the other hand, we could tell by looking at the whole process so far.

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One could thematise the process and then consider that as an object, but that object is not the choice sequence, for the reasons just mentioned. But there is a close relation between the process and the sequence: the identity of the sequence is founded on the identity of the process.205 That is, there is no identity of the sequence without the identity of the process. We see the identity of the sequence through the identity of the process. That is why both are individuated by the same moment of beginning: the sequence only comes into being at the moment the process of growth (by choice) starts. (For omnitemporal objects, the moment of constitution does not individuate the object; at any other moment it could, in principle, be constituted as exactly identical. Not surprisingly, Husserl works out the relation between moment of constitution and temporality of the object in the case of abstract objects only for omnitemporal objects, e.g., in appendix XIII to the lectures on time consciousness [104].) An act of abstraction leads from process to sequence. Some aspects of the process are abstracted from in all cases, such as the temporal intervals between successive choices, mentioned above. What other aspects can be abstracted from depends on the kind of sequence. Two temporally diﬀerent processes may constitute the same lawlike sequence; in such cases the temporal properties may be abstracted from without loss of the sense of identity of the sequences. In open-ended sequences, on the other hand, the horizon ‘and so on’ plays a representing role too. Its particular moment of occurrence cannot be abstracted from, because that might lead to identiﬁcation of diﬀerent non-lawlike sequences that share their initial segments so far. Putting it all together, a choice sequence is constituted as an individual object in the following stages: 1. Keeping in retention, or recollecting, the process as it has developed till now. A newly chosen number is meant to extend the process of successive choices that we have obtained so far. The ﬁrst step, then, is to locate that process in the past. In case we have chosen the previous number only a moment ago, this locating does not involve active recollecting, as the process then is still held in retention and has not sunk back in the past yet. 2. Re-presentation of the process. The process has to be thematised not as having been extendable once, but as being extendable now. The temporal horizon that the process had at the moment we halted its development has to be made actual, or present, again. If we remember the process as openended after the most recent choice, it has to be given as open-ended again now. This asks for the actualisation of the protentive (or anticipatory) intentions associated to the process. Such intentions may be ‘empty’ in the sense of not prescribing any speciﬁc element(s); this is diﬀerent for lawlike sequences. 3. Choice of the next element. If this next element satisﬁes all restrictions that are in eﬀect at this moment, the choice fulﬁls the protentive intention

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directed at an immediately following new element. This fulﬁlment, which is an act of synthesis, glues the present act of choice to the process as we had it so far. Other elements in the temporal horizon of the process are carried over into the extension, such as ‘and so on’, or, ‘and so on, subject to restriction R’. This is also the stage where restrictions can be lifted, revised, or added. 4. Sinking back into retention of the sequence, and return to stage 1. Due to the ﬂow of time, the just fulﬁlled intention does not remain present but sinks back into retention and possibly further into the past. Founded on these four acts, and occurring between going back from 4 to 1, there is 5. Apprehension of the identity of the process through its categorial form of ongoing process (mentioned in Husserl’s quotation on p. 90); this identity functions as foundation for the identity of the choice sequence; constitution of the choice sequence by (partial) abstraction from the underlying process. The way in which a choice sequence is an object has much in common with the way another, more familiar type of object is: a melody. An ongoing melody is experienced as an identity even though it may not have been completed yet. In fact, in his analyses of temporal objects, Husserl often used examples from music: the constitution of an identical tone through the diﬀerent phases of its duration, of a melody through its development, of a whole concert out of its movements, of a piece of music as an object in culture. Of special interest for our case is Husserl’s reaction (around 1905) to Meinong’s explanation (1899) of how a melody is constituted as an object. (This discussion is documented in supplementary texts 29–34 of the Husserliana volume on internal time consciousness [104], and its editor’s footnotes.) Phenomenologically, a melody is more than just a succession of separate tones. It forms a unity of these tones. The question Husserl disagreed about with Meinong was: On what moment is a succession of tones that is heard in a succession of perceptions, heard as a melody? Meinong’s answer was that this moment occurs only after all tones of the melody have been heard. In one simultaneous looking back at all tones, their unity as a melody is synthesised. He arrives at this conclusion because he conceives of time as a chain of separate points that have no extension. This leaves at any moment only room for what is perceived ‘now’, i.e., in that very moment itself; there is no perception of a succession, because that requires retention of notes as past, a current tone, and perhaps an anticipation of tones to come. For the same reason, the only possibility left to perceive a whole melody is that all its tones are given simultaneously. But in our experience, perception of a melody does not happen the way Meinong says it does. We perceive a melody already as such as it is still going on, when we may not even know when it will end. According to Meinong, we would ﬁrst have to

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wait till the succession of tones has stopped, and only then could interpret this succession as a melody. This is not only at odds with how we do experience melodies, but also leaves the further question how in a simultaneous grasping of all tones, an progression is constituted [123, p. xxvii]. Husserl’s criticism of Meinong can be phrased in terms of a distinction that the latter introduced himself. Objects are either temporally distributed, or temporally indistributed (‘zeitlich (in)distribuiert’). Temporally distributed objects are those that necessarily occupy an interval (i.e., more than one moment) in time and hence cannot be given in a single moment. A melody clearly is a distributed object; a colour is indistributed. Against Meinong’s position, Husserl stresses that the perception of an object that is distributed in time does not occur in one single moment but is itself distributed in time. This is possible because moments are not (as Meinong assumes) unextended points, but a ‘now’ always has a halo around it of retentions, keeping the connection with the immediate past, and protentions, looking forward to the immediate future. In the ﬂow of time these halos overlap. A melody is given to us in continuous syntheses of the halos in which the successive tones are perceived; generally, distributed objects are given in distributed perceptions. Perception of a distributed object is itself distributed. This explains how we can come to perceive a melody as such immediately after its beginning. The syntheses set in right away, and to grasp the unity of the melody we only have to thematise them. A direct analogy between melodies and choice sequences can now be drawn, as unﬁnished objects are distributed objects, while ﬁnished objects are indistributed. As described above, choice sequences are constituted in a process of successive (though not necessarily continuous) syntheses. As so often, also in this analogy the lawlike sequences take a special place. The fact that a law speciﬁes each successive number guarantees the unity of the corresponding sequence. For this aspect a term is ﬁtting that was used in a discussion of Meinong’s distinction by L.W. Stern: The expression ‘momentary whole of consciousness’ is not so much meant to refer to those contents that indeed last for only a moment (the existence of which is most uncertain), as to, quite generally, those that, apart from their possible duration, are at every moment complete; this means that all elements that belong together or that are needed to produce the apprehension are given isochronously, so that the extension in time does not constitute an integrating factor. [205, p. 326n. 1, trl. mine]206 Similarly, we can say that lawlike sequences are given ‘isochronously’, because the law itself can go proxy for the sequence it determines. All information one needs for the unfolding of the sequence is already contained in the law. But for non-lawlike sequences, their unfolding in time is an integrating factor. Only in the actual successive choices are its elements speciﬁed.

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Even if we would leave open whether choice sequences are mathematical or empirical objects, I suggest that they are objects at all for the same reasons that melodies are. Non-lawlike sequences are improvisations, lawlike sequences are note-for-note instructions (not: actually inﬁnite scores).207 The indeterminateness of non-lawlike choice sequences is something that they have in common with ﬁctional objects (on most accounts of ﬁctional objects). For example, just as there is no fact of the matter whether Hamlet sports shoes of size 9 or not, there is no fact of the matter whether the number 2 does or does not occur in a sequence α in which we so far have made the choices 1, 3 and 5, and on which we have as yet imposed no restrictions. But the diﬀerence is obvious. The choice sequence is something we literally bring about in the process of its construction; so if at a given moment there is no fact of the matter whether 2 does or does not occur in α, that is not because α would be ﬁctional, but because α, in terms of its individual elements as well as the restrictions imposed, has not yet been suﬃciently developed for there to be such a fact. Whenever there is no fact of the matter whether a given choice sequence has a given property, the possibility of there coming to be a fact of the matter is, as a matter of essence, always open. Crucially, this is not the case for ﬁctional objects, of which the absence of this possibility is one of the deﬁning features. That in the case of a choice sequence the possibility of our bringing about a fact of the matter is always open, is because a choice sequence comes into being by just appropriately thinking about it, whereas that will not suﬃce to transform ﬁction into fact. This is an application of what was argued for in chapter 5: for purely mathematical objects, transcendental possibility and existence are equivalent.

6.3 Choice Sequences as Mathematical Objects Now that we have these objects, why should we accept them as mathematical objects? From the traditional (also Husserl’s own) point of view, there are two obvious concerns. The one is that choice sequences are unﬁnished and hence time-dependent objects. Unlike the number 2 or the sinus function, choice sequences come into being at a particular moment in time, and grow from there on. Doesn’t such a dependence on time rule out that choice sequences are purely mathematical objects? The other doubt derives from the subjectdependency of choice sequences. As long as we think of mathematics as the activity of the one creating subject, who is the ideal mathematician, there is no question. But as soon as we drop this idealisation and consider real mathematicians, we have to ask whether the acceptance of choice sequences squares with the intersubjectivity of mathematical truth. These two doubts do not threaten the coherence of choice sequence as a mathematical concept. I will argue this by pointing to a counterexample to the claim that the concept is incoherent for the two proposed reasons. Having seen

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the counterexample, we want to ﬁnd out what makes such a counterexample possible. Consider the lawless sequences, in fact the strongest case of timedependent, subject-dependent objects (as elaborated in chapter 4 and below, lawlike sequences, at the other end of the spectrum, need not be considered essentially dependent on time or subject). For this class we have the KreiselTroelstra translation (discussed in section 4.3.4), which shows that sentences quantifying over lawless sequences are equivalent to other sentences that do not, and whose mathematical nature goes unquestioned; the translation depends only on the notions of natural number, constructive function, and inductively deﬁned neighbourhood function [220, pp. 33–34]. But then the sentences that are translated must also be mathematical. Translations suﬃce to show that the concept of choice sequence, and speciﬁcally, the concept of lawless sequence, is mathematically coherent (cf. [71, p. 222]). Incidentally, one of Husserl’s arguments against psychologism (and one might have felt that lawless sequences are psychological entities if anything is) was that mathematical statements do not imply psychological ones [113, section 23]; now we see that undoubtedly mathematical statements do imply statements that quantify over lawless sequences, for the equivalences the translation yields work both ways. By itself, this argument for the coherence of the concept does not compel one to admit that there exist objects falling under that concept; one may still hold that quantiﬁcation over lawless sequences is deﬁned contextually. In the presence of the constitution analysis of the previous section, however, that would make no sense, for then we know that lawless sequences, like those of other types, are indeed given to us as genuine objects. Further phenomenological description will now reﬁne this point and clarify the mathematical nature of choice sequences. Thereby some of the conditions of possibility of such translations as mentioned are disclosed. 1. It is not their reference to time as such that distinguishes choice sequences from other mathematical objects, but the particular way in which they refer to time. 2. This particular reference to time does not threaten their status as formal objects. 3. Their subject-dependency (part of which is the freedom of generation) poses no problems for mathematics. 6.3.1 The Temporality of Choice Sequences According to the later Husserl, all mathematical objects refer to time (section 5.4). He drew the further conclusion that this reference takes the form of omnitemporality. The way he arrived at this conclusion I already criticised there – he did not consider choice sequences at all; and as just explained, lawless sequences, for which intratemporality is essential, can be strictly constituted

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and are mathematically coherent. More generally, intratemporality is essential for all non-lawlike sequences. It is implied by the openness of their horizon: because the horizon is open, they are essentially unﬁnished, and if two would have the same initial segment (and compatible intensional properties), that does not imply they are identical choice sequences; in each case we can still proceed diﬀerently. The sequences may still come to diverge, and the only way to distinguish two such sequences now is by reference to the moment they began. Hence the diﬀerence with lawlike sequences, where one knows that the same law, followed at any other point in time, would yield the same elements in the sequence. Husserl contrasts the ideal objects of mathematics, which he says are omnitemporal, to the real or individual objects, which are individuated in time.208 But choice sequences, instead of being necessarily inﬁnitely temporal in two directions (and thereby omnitemporal), would be so in just one direction (the future). In that case still what is proven once, is proven forever. But, as we saw in chapter 4 it was this feature of mathematics that led Husserl to say that mathematical objects are static and omnitemporal. (Proofs at later stages of the object’s development may be sharpenings of earlier ones.) It follows that inﬁnite temporality only in the direction of the future preserves the original motivation for the thesis of omnitemporality. For ﬁnished objects, omnitemporality is evident. The sense of being a ﬁnished object, i.e., one that does not change over time, contains the sense that at every moment it will be constituted as extensionally the same (we may have knowledge of this extension to diﬀerent degrees). Otherwise, if at t1 the object is given as diﬀering from itself at t0 , it must have changed in between, which its ﬁnished character forbids. The diﬀerence between being omnitemporal and being inﬁnitely intratemporal in the direction of the future shows up, not in the further development of mathematics, but in the evaluation of counterfactuals. Counterfactuals in the past tense about omnitemporal objects can be true (‘Had the question of the irrationality of π been conclusively investigated a hundred years before Lambert did so in 1761, people would have arrived at the same answer.’), but not those about open-ended objects (think of ‘Three years ago I would have generated exactly the same lawless sequence as I am doing now’ – sameness of open-ended objects implies sameness of temporality). 6.3.2 The Formal Character of Choice Sequences Now the further question comes up whether objects that essentially refer to time can still be formal objects. In other words, is an individual choice sequence also a purely categorial object? Doesn’t the necessary reference to time make a choice sequence a mixed object (and hence not purely mathematical)? No. Mixed categorial objects are not pure in the sense that they depend on speciﬁc sensuous elements or sense data (see section 5.1). These data are immanent objects. But whereas ‘red’ is represented by sense data

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‘red’, there are no ‘time sensations’ on basis of which internal time would be constituted. The reason is that such time sensations would have to be constituted as unities in immanent consciousness, but the constitution of any identical object presupposes the absolute ﬂow of time (section 5.4).209 The reference to time does not make choice sequences from otherwise pure into mixed categorial objects. It follows that if, with Husserl, we deﬁne mathematics as dealing with the purely formal (p. 61), then this does not yet imply a decision on temporal aspects, precisely because temporal aspects are not sensuous. This argument for the formal character of choice sequences in spite of their dependence on time would not go through for the earlier Husserl, whose position I am not following, who still held that time consciousness was a matter of apprehending time sensations.210 In that conception, formal objects cannot depend on time, for the presence of time-sensations rules out complete formality. That implication was not what led Husserl to reject time sensations, it was the considerations of the absolute ﬂow that made him do so. Why does Husserl say, in Experience and Judgement [124, sections 64–5] that irreal (or ideal) objects are ultimately distinguished from real ones because the former are omnitemporal, and not ‘because the former are nonsensuous’ ? Because not all irreal objects are free of sensuous elements, only the formal-mathematical ones. Geometrical objects, for example, have sensuous components,211 but are omnitemporal all the same. 6.3.3 The Subject-dependency of Choice Sequences A sequence that is open-ended (non-lawlike) is bound to the particular subject generating it, in the following way. Imagine I am choosing such a sequence, and pause after the ﬁrst 53 numbers. Then you ask me what they are, and what restrictions I have posed on the further development of the sequence. Armed with this information you tell me, ‘The 54th number in your sequence is 2; I have just chosen it to be’. Must I accept this? No, the initial segment that I have generated is given to me with an open horizon (possibly limited by restrictions; suppose for the sake of argument that there are two or more alternatives left for the next choice, so that there is room for me to diﬀer from you). Moreover, the openness of the horizon refers to my freedom to choose; this was one of the motivations to deﬁne choice sequences the way they are (section 5.1). I am not forced to accept your choice for the 54th number in my sequence, although I may do so. What really happened is that you began your own choice sequence by copying the initial segment of mine; we are both free to continue from there in diﬀerent ways. The point of the example is that it is essential to a non-lawlike sequence to have a bearer or owner, and such a sequence cannot be handed over from one subject to the other.212 But this does not lead to unshareable truths. A nonlawlike sequence develops in stages, and at any stage the only information the subject has consists of an initial segment and, possibly, a number of intensional

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properties such as self-imposed restrictions on future choices. Initial segments and sets of restrictions are ﬁnite, therefore all information can be shared. All the subject knows about a particular sequence is intersubjectively accessible. And, I will claim in a moment, it is equally important that the categorial form ‘choice sequence’ can be shared between diﬀerent subjects.213 In chapter 7, we will see the continuity principle for choice sequences, which says that, assuming the domain is the universe of all choice sequences, predications and functions of choice sequences depend only on a (ﬁnite) initial segment. One may ask: ‘What do we need choice sequences for, instead of just all ﬁnite sequences? One never has to construct an individual choice sequence (in a mathematical proof).’ This objection presupposes what it explicitly denies. The use of ﬁnite sequences for the purpose of saying something about the continuum presupposes the existence of choice sequences, as we must see those ﬁnite sequences as initial segments of choice sequences. (Hence, that the categorial form ‘choice sequence’ is shareable is just as necessary as the communicability of ﬁnite segments.) For we can be sure that the intuitive continuum is not made up of parts coded by ﬁnite sequences.214 Here we seem to have a case (pace Kreisel215 ) where thinking of mathematics in terms of objects yields more than thinking of it in terms of objectivity (understood as idealised intersubjectivity), or more accurately, where objectivity presupposes the object. Another aspect of the subjective nature of choice sequences, besides the fact that they cannot be shared, is that they are generated freely. One may wonder, ﬁrst, whether this is really possible, and, secondly, if so, whether this does not make everything arbitrary. Can choice sequences indeed be generated freely? Consider the following objection: ‘Non-lawlike sequences do not exist. True, to us they seem to involve free choice, but, metaphysically, they are really lawlike as well’. In analogy to Kantian terminology,216 ‘Our choice sequences are phenomenally free, but noumenally lawlike. Therefore, non-lawlike sequences do not really exist’. As a consequence, choice sequences would not be able to capture the whole continuum. This objection presupposes ‘things in themselves’ (noumenal objects) in the realist or objectivistic interpretation of that term. According to that reading, noumenal objects are the genuine objects themselves, understood as being independent from any acts of knowledge. They not only transcend consciousness, but are unreachable for it; they can only be conceived of as the unknowable causes of its experiences.217 Husserl rejects this dichotomy between phenomenon and noumenon, for objects of any kind, and replaces it by a universal correlation of being and consciousness. This correlation is the essence of his transcendental idealism. As he phrases it in Ideas I, Of essential necessity (in the Apriori of the unconditioned eidetic universality) to every ‘truly existing’ object there corresponds the idea

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of a possible consciousness in which the object itself is seized upon originarily and therefore in a perfectly adequate way. Conversely, if this possibility is guaranteed, then eo ipso the object truly exists. [120, p. 341]218 Because of this correlation, within a Husserlian framework sequences that for all we can know are lawless but are really lawlike, are ruled out. If such a sequence is really lawlike then, as an ideal possibility, we can come to know it as such. (And as it cannot be that the whole intuitive continuum can be analysed by just lawlike sequences, even if it were the case that some seemingly lawless sequences are really lawlike, they cannot all be lawlike.) Let me address the second objection to freely generated sequences now, the objection that this introduces arbitrariness into mathematics. Gödel (as related by Wang) observed that Brouwer does not mean arbitrary creations. Rather he means creation according to certain principles. The central and appropiate concept for Brouwer is construction rather than creation. We construct something out of something given. [234, p. 248] Indeed, choice sequences are formed according to the laws for forming categorial objects (section 6.1). This contrast between arbitrary creation and construction also answers the following objection to intuitionism, formulated by Lohmar: Other doubts in turn are directed at the view that the objects of mathematics are generated in the mathematician’s acts. As a counterreaction to Platonism, this view is certainly understandable; and it clearly points to the contribution of actions to the constitution of mathematical objects. However, it conceals the fact that cognition and itself-givenness of mathematical connections are founded on something which, in the activity that leads up to them, occurs passively. In mathematics, too, all we can do is to bring ourselves to the point where cognition either takes place or not. [152, p. 248, trl. mine]219 The construction of non-lawlike sequences is free, but not arbitrary. The choices will have to satisfy certain constraints, depending on their application; in the case of analysis, there is the general constraint of determining nested intervals. This constraint is not our free creation, but is motivated by the nature of the intuitive continuum. If there are additional ones, these cannot be completely arbitrary either, as they have to respect the general constraint, and they have to be compatible with one another. Finally, once these constraints are in place, whether a proposed number actually satisﬁes them is not up to us. This is the ultimate, passive element in the formation of choice sequences. In the above, I have brought out how choice sequences are constituted as mathematical objects. In the ongoing process of choosing numbers in a

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sequence, an invariant establishes itself, which, when thematised, functions as the foundation for the identity of the choice sequence as mathematical object. This is the conclusion of a descriptive analysis. However, it is not the end of constitution analysis as such. For, given that intentions directed at categorial objects can be fulﬁlled, there is the further question of the constitution of the fulﬁlling evidence. This is a general, still open problem in the theory of categorial intuition. I will not go into it, as we do not need an answer to this further question now that we have, descriptively, disclosed choice sequences as mathematical objects (Lohmar oﬀers some suggestions on this matter [152, pp. 60, 69]). Choice sequences are genuine objects, preserve the original motivation for omnitemporality, are formal, and do not lead to truths that are not intersubjectively shareable. They are self-given in a process of free choice. The next chapter shows an application of this phenomenological analysis.

7 Application: An Argument for Weak Continuity

In chapter 5, I explained Husserl’s principle that transcendental phenomenology provides the ontology for the a priori sciences. What the basic objects, and the axioms governing them, in an a priori science are, is to be disclosed in phenomenological analysis. Now I want to show this idea in action: the constitution analysis of choice sequences, undertaken in the previous chapter, yields a justiﬁcation of one of the basic principles for choice sequences.

7.1 The Weak Continuity Principle Consider two-place predicates A(α, x), where α ranges over choice sequences and x over natural numbers. Extensionality is deﬁned in the usual way: Extα,x A(α, x) =df α = β ∧ x = y ∧ A(α, x) → A(β, y) The weak continuity principle for numbers in the theory of choice sequences reads ¯ =α WC-N ∀α∃x(A(α, x) ∧ Extα,x A(α, x)) ⇒ ∀α∃m∃x∀β[βm ¯ m → A(β, x)] where α and β range over choice sequences, m and x over natural numbers, and αm ¯ stands for the initial segment of α of length m. The principle says that, if you have a construction that assigns a number to every choice sequence, all the information it needs to do so is an initial segment of that sequence. More information (intensional properties) may be given, but is not required. (Digression: In conversation with Hao Wang, Gödel claimed, ‘In 1942 I already had the independence of the axiom of choice [in ﬁnite type theory]. Some passage in Brouwer’s work, I don’t remember which, was the initial stimulus.’ [235, p. 86]. In his notes on the proof from 1942 (Arbeitsheft 14, item 030032 in the Gödel Nachlaß, Firestone Library, Princeton), one sees how Gödel used the continuity principle for choice sequences to deﬁne a notion of ‘intuitionistic truth’ for propositions about inﬁnite sequences (e.g., p. 14 103

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(clause 1), p. 15 (clause 6), p. 16 (in ‘Df. ‘W”)). By that year, Gödel may have seen the statement of the principle in [29, p. 13], [31, p. 189] or [33, p. 63]. For more on this, see the section ‘Inﬂuences of intuitionism on Gödel’s work’ in [8].) A special case of WC-N is the one where the predicate A is required to be graph-extensional, notation GExtα,x A(α, x). It means that A refers to the choice sequence only through its graph, i.e., only to its values. Correspondingly, GWC-N is WC-N for graph-extensional A: ¯ = α ¯m → GWC-N ∀α∃x(A(α, x) ∧ GExtα,x A(α, x)) ⇒ ∀α∃m∃x∀β[βm A(β, x)] Graph-extensionality is not the same as the usual notion of extensionality. The latter applies to a predicate A(α, x) if, whenever it is given that two sequences α and β coincide extensionally, they bear the relation A to the same number(s) x. In that case, it is not excluded that x depends not just on the graph of α and β but on intensional properties. One can prove the following theorem: Assume the creating subject generates choice sequences as individual objects, and can therefore enumerate the sequences generated so far. Then WC-N does not hold generally for extensional predicates. The proof proceeds as follows. Assume we have an operation F (not necessarily a function) that enumerates the choice sequences. Then ∀α∃n(α = F (n)) Put G(α, n) ≡ α = F (n) to get ∀α∃nG(α, n) and apply WC-N. This yields ¯ =α ∀α∃m∃n∀β[βm ¯ m → G(β, n)] But this means that the same n will be paired to diﬀerent choice sequences, which conﬂicts with the notion of ‘enumeration’. The conclusion is that in the presence of an enumeration, WC-N does not hold generally. Note that the predicate G is not graph-extensional.220 It is not clear how to formalise the notion of graph-extensionality, or whether this is even possible. A trick found by Beeson [12] (which is the basis for the proof mentioned in note 220), shows that a predicate that is extensional in the usual sense can be based on an intensional property. By deﬁnition, that is not the case for a predicate that is graph-extensional. From this one might conjecture that graph-extensionality, like intensional equality, has to be taken as primitive, and I do so here. For the intuitionistic reconstruction of analysis, it suﬃces to have GWC-N instead of WC-N. The following two theorems show that intuitionistic analysis is not just an amputation of classical mathematics, but contains new results that are classically not acceptable. Veldman [230] has shown that from WC-N (in fact, GWC-N) one can derive

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the continuity theorem: a real function whose domain of deﬁnition is the closed segment [0, 1] is continuous on [0, 1].221 the unsplittability of the continuum The continuum cannot be split into two non-trivial subsets: if R = A ∪ B and A ∩ B = ∅, then A = R or B = R. Brouwer did not explicitly state the continuity theorem, instead he proved the stronger uniform continuity theorem: a real function whose domain of deﬁnition is the closed segment [0, 1] is uniformly continuous on [0, 1]. Brouwer used the bar theorem222 for this and seems to have believed that the continuity theorem can only be obtained as a corollary from the uniform continuity theorem [230, p. 4]. Veldman’s proof shows that the continuity theorem does not require the bar theorem, and can be proven directly from WC-N (GWC-N). This reﬂects that continuity is a local phenomenon, whereas uniform continuity is not. Likewise, Brouwer in his proof of the unsplittability of the continuum appealed to the fan theorem, where the simpler WC-N (GWC-N) suﬃces, for unsplittability is a direct consequence of the continuity theorem:223 suppose R = A ∪ B and A ∩ B = ∅, then f deﬁned by 0 if x ∈ A f (x) = 1 if x ∈ B is total and therefore, by the continuity theorem, continuous. But then f must be constant, so either R = A or R = B. An instance of unsplittability is that it is false that every real number is either rational or irrational. For if it were, we would obtain a non-trivial splitting of the continuum by assigning 0 to rational, and 1 to irrational reals.

7.2 An Argument That Does Not Work Given these consequences, the question arises, what justiﬁes WC-N? One might think along the following lines. The logic of open-ended objects requires that, to prove an existential quantiﬁcation over them, we construct a witness (in the limiting case, by unrestricted choice), or a method that is guaranteed to produce one. But, whereas choice sequences proceed inﬁnitely, constructions are always ﬁnite. By the time the construction is completely carried out, no more than a ﬁnite number of choices in the sequence have been made. So if there is a construction for a witness, it must be based on no more than an initial segment α(0), α(1), . . . , α(m − 1) of the sequence α; hence any other sequence β with coinciding initial segment should yield the same witness. But we have to be careful here, as this idea does not really justify WC-N.224 That it does not, is shown by the following example (told to me by van Dalen):

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Imagine two players, I and II.225 I starts to generate a choice sequence α, and the task of II is to tell what the 100th number in α is. (Thus we have deﬁned A(α, x) ≡ α(100) = x.) Now I begins by choosing, say, 8, 4, and then 3. The ﬁrst three values are not enough information for II to complete his task, so he asks for more. Instead of choosing the 4th value, I says, ‘The sequence will from now remain constant’. II immediately knows that the 100th value of α is 3. On the other hand, we cannot conclude that any β with initial segment 8, 4, 3 will have 3 as its 100th value. The intensional property that becomes known with I’s disclosure that the sequence will be constant ever after the third choice, spoils the simple continuity argument. The moral is that a full justiﬁcation of WC-N should take into account that, besides an initial segment, we may know all kinds of intensional properties of a sequence. It not always clear exactly what inﬂuence the presence of such information has on operations that should be applicable to all choice sequences; and as Troelstra (e.g. [220, p. 151], [222]) has noted, justiﬁcations of WC-N so far have not gone beyond plausibility considerations.

7.3 A Phenomenological Argument Here I want to give a derivation, by a phenomenological argument, for GWC-N. The argument is meant as an exercise in ‘informal rigour’: The ‘old-fashioned’ idea is that one obtains rules and deﬁnitions by analyzing intuitive notions . . . Informal rigour wants (i) to make this analysis as precise as possible . . . and (ii) to extend this analysis, in particular not to leave undecided questions which can be decided by full use of evident properties of these intuitive notions. [145, pp. 138–139] (The term ‘informal rigour’ was coined by Kreisel; Gödel then encouraged him to develop and advocate the notion, while warning him that mathematicians would not be enamoured of the idea.226 ) My strategy is to take the analysis of the intuitive notion of choice sequences, obtained in chapter 6, and extend it by making explicit the application of Husserl’s principle of the noetic-noematic correlation [115, section 93]. Roughly, this principle states that the structure of the way an object is given to us (the noema) is parallel to the structure of the acts in which that object is intended (the noesis). In the case of choice sequences, this leads to the question: in what ways can the freedom the subject enjoys in the process of generation be reﬂected in the intensional properties of the sequences themselves? That gave the idea of provisional restrictions, and it also gives rise to the already familiar concepts of lawlike and lawless sequences. What else does it yield? Let us look at a particular description by Brouwer of choice sequences as freely generated objecs (from a note related to his manuscripts for the 1927 Berlin lectures):

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The freedom to proceed with the choice sequence can after every choice be arbitrarily restricted (possibly in dependence on events in the world of the mathematical thought of the choosing person, imposed on the choosing person) (resulting e.g., in complete determination, or determination by a spread law). The arbitrary nature of this restriction, permitted at each new choice as long as the possibility to proceed is retained, is an essential element of the free becoming of the element of the spread, as is the possibility to link to every choice a restriction of the freedom to make further restrictions of freedom etc. [221, p. 473]227 (Compare also the quotation on p. 88.) In section 6.1, this freedom that Brouwer talks about was brought out phenomenologically. A sequence acquires intensional properties through the subject’s free decision to conduct the generation of the sequence in a certain way. Restrictions can be of diﬀerent orders, e.g., ‘I will pose no ﬁrst-order restrictions’ is a second-order restriction that captures the lawless sequences. Restrictions should be decidable, otherwise the subject will not be able to conduct its choices in accordance with them. So one can look at the process of generating a choice sequence as choosing, at each stage i, not only an element ni , but also a (ﬁnite) number ki of restrictions of diﬀerent orders, Ri1 . . . Riki . Now I take up again the distinction introduced in section 6.1, the division of restrictions in two kinds, which I called provisional and deﬁnitive (p. 88). The subject can make any revision in its ideas about how to go on generating a choice sequence as long as 1. this revision does not go against any deﬁnitive intensional properties of the sequence, and 2. this revision admits the existing initial segment, and 3. after the revision, it is still possible to extend any admissible initial segment.228 So the constraint on revisions is that under the new conception the sequence obtained so far must be allowed as initial segment. (One could say that otherwise the subject shifts to a diﬀerent choice sequence.) The decision at a particular stage may also be not to change the intensional properties and to go on as before. Thus we have deﬁnitive intensional properties, which have the form ‘from now on, restriction Rik holds, and it will not be revised anymore’ (with or without a speciﬁed initial segment); provisional intensional properties, which have the form ‘for an unspeciﬁed number of stages, restriction Rik holds’. As long as an intensional property of a particular sequence is not deﬁnitive, that property can be changed at some (or any) stage. (Because it is up to the

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subject’s own choice to make an intensional property either provisional or deﬁnitive, what kind it is dealing with is decidable.) If several restrictions are in eﬀect, their conjunction counts as provisional exactly if one or more of its members is. Note that, for an individual choice sequence to be given to us, it is not required to have any deﬁnitive property. Unless the subject explicitly lifts a provisional restriction, it remains in eﬀect at subsequent stages. This requirement of explicitness is a consequence of the creating subject’s full responsibility for the objects in intuitionistic mathematics. In this circumstance lies the answer to the following question:229 ‘If I know that you imposed a provisional restriction at some earlier stage, what more do I know that I should not have known if you had imposed no restriction?’ The extra information that a provisional restriction gives you consists in the following. If I tell you that I will not begin by lifting the provisional restriction, then you do know something about my next choice of a term, namely, that it has to respect this restriction. Had I imposed no restrictions, then you would not have known this. In the literature there is a particular type of choice sequence that can now be interpreted as an implicit application of the distinction between provisional and deﬁnitive restrictions. This is the hesitant sequence [226, p. 208]: A hesitant sequence (say β) is a process of generating values β0, β1, β2, . . . ∈ N such that at any stage we either decide that henceforth we are going to conform to a law in determining future values, or, if we have not already decided to conform to a law at an earlier stage, we freely choose a new value of β . . . The decision whether or not to conform to a law may stay open indeﬁnitely. Thus, a hesitant sequence from the start is a provisionally lawless sequence (which the subject at any stage may decide to turn into a deﬁnitively lawlike sequence). In fact, Brouwer already employed the device of provisional restrictions, without drawing further attention to it, in his argument from 1927 that every full function on the reals is negatively continous.230 The particular fragment of the argument that interests us here is that where, departing from a given choice sequence 1 , the following directions for generating a choice sequence 2 are given: By means of an unlimited sequence of choices of λ-intervals, [we construct] a point 2 of the unit continuum in such a way that we temporarily choose, for every natural number n that we have already considered, the ﬁrst n intervals identical with the ﬁrst n intervals of 1 but reserve the right to determine, at any time after the ﬁrst, second, . . . , (m − 1)th, and mth intervals have been chosen, the choice of all further intervals (that is, of the (m + 1)th, (m + 2)th, and so on) in such a way that [either 2 coincides with 1 , or they are apart]. [81, p. 459]231

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The provisional condition on 2 is that it should follow 1 ; an explicit decision to diverge from 1 is required (and suﬃcient) to lift it. The purpose that provisional conditions serve in Brouwer’s argument as a whole is to guarantee the possibility of generating discontinuities; this should be suﬃcient to suggest the argument for the continuity principle (generalised to include conditions of n-th order) that I will present next. As Brouwer never gave an explicit argument for that principle, I do not know to what extent the argument that he nevertheless must have had in mind resembles the one that follows; perhaps the two are very similar. From the recognition of the two types of restrictions, deﬁnitive and provisional ones, an argument for GWC-N can be derived. The assignment of a number to a choice sequence requires a construction. At the time of construction, only an initial segment of the sequence is known, as well as some intensional properties. Of these properties, the restrictions may be provisional and therefore open to revision. In this setting, the intensional properties that a sequence has at any particular stage do not necessarily characterise that sequence also at all later stages. After all, these properties might be changed along the way. That in turn means that a construction that depends on these intensional properties may yield diﬀerent results at diﬀerent stages. However, it is part of the notion of proof that once a proof has been given, it remains valid forever. In particular, once we have a proof that a relation A(α, x) holds between a choice sequence and a number, this relation holds forever. The trouble with a construction that depends on provisional properties is that it cannot guarantee this lasting validity: should the provisional properties change, the outcome of the construction might as well, and the proof no longer goes through. Therefore it would contradict the notion of proof to count such a construction as a proof of A(α, x). Let me illustrate this by looking at the counterexample to the simple argument for WC-N (and GWC-N) on p. 106 again. Suppose we have, of a choice sequence α, the initial segment 8, 4, 3, and the deﬁnitive restriction ‘From now on, α is constant’. From this information, we can immediately conclude that α(100) will be 3. Now suppose we have, of a choice sequence β, the initial segment 8, 4, 3, and the provisional restriction ‘From now on, β is constant’. In this case we are not entitled to conclude that β(100) will be 3, as between our choices of β(3) and β(100), we might decide to lift the provisional restriction; and if we do so, we can consistently choose β(100) diﬀerent from 3. To conclude the argument for GWC-N: given GExtα,x A(α, x), any intensional property that might be useful in constructing an x such that A(α, x), should already be extractable from the ﬁrst-order restrictions, as only these have a direct bearing on the graph of α. But, as we just saw, if we allow provisionality, restrictions cannot be depended on when constructing such an x. Hence the construction can only depend on the other information that we have. This other information is just the initial segment; which is exactly what GWC-N asserts.232

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A condition for this argument to work is that the universe of choice sequences under consideration contains, for each element α, the ‘provisionalised’ version α as well. To say that α is the ‘provisionalised’ version of α is to say that they have the same initial segment and are subject to the same restrictions, but in the case of α all of these restrictions are provisional. In the universe of all choice sequences, this requirement is met by deﬁnition. The argument for continuity from provisional restrictions presented here has now found application also in accounting, in the theory of equity valuation [204]. Finally, I want to mention a suggestion of Troelstra’s. It is possible to model choice sequences with provisional conditions by projections of lawless sequences, as in the following example. Think of a choice sequence as a sequence of triples (n0 , R0 , t0 ), (n1 , R1 , t1 ), (n2 , R2 , t2 ), . . . Here, ni is the number chosen at stage i, Ri is a (possibly provisional) restriction, and ti is a sequence of integers all ≤ i in absolute value. If +j appears in ti , this means that restriction Rj is made deﬁnitive; −j means that restriction Rj is lifted. Within this setting, we may consider the sequence of triples as lawless; what is projected from it is a choice sequence with variable secondorder restrictions. No mathematical theory of this projection model has been developed yet. (For the modelling of choice sequences with only deﬁnitive ﬁrst-order restrictions by projections, see [225] and [94].)

8 Concluding Remarks

One correct, phenomenological argument on the issue whether mathematical objects can be dynamic is not Husserl’s (negative) argument, but a reconstruction of Brouwer’s (positive) one. This I have argued by an attempt to justify, phenomenologically, the existence of one particular kind of dynamic object, the choice sequence. The phenomenological analysis was then applied to yield, in an attempt at informal rigour, a justiﬁcation of the weak continuity principle for numbers. Four areas for further research suggest themselves. 1. The argument entails that the mathematical universe is temporally heterogeneous: diﬀerent mathematical objects may bear diﬀerent relations to time. It would be interesting to see how this relates to more general ideas found in (formal) ontology and mereology; also, to see if yet other parts of mathematical experience are connected to this heterogeneity. 2. Parts of the argument may contribute to a full phenomenological critique of Brouwer’s original intuitionism. Such a critique should go beyond just choice sequences and, following Richard Tieszen’s suggestion [214, p. 179], investigate the intuitionistic notion of construction, questions of idealism, psychologism, and solipsism in mathematics, and the theory of meaning as implied by intuitionism; [174] and [3] are steps in that direction. 3. The justiﬁcation of the continuity principle in chapter 7 may be a starting point for investigations of related principles. 4. One may investigate exactly why choice sequences would, or would not, ﬁt into speciﬁc alternative views on the ontology of mathematics.

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Appendix: Intuitionistic Remarks on Husserl’s Analysis of Finite Number in the Philosophy of Arithmetic

The following is an attempt at some remarks on Husserl’s Philosophy of Arithmetic from an intuitionistic, by which I mean a Brouwerian, point of view. More speciﬁcally, I will conﬁne myself to some intuitionistic musings on Husserl’s analysis of the concept of ﬁnite number, which he developed in his Habilitationsschrift of 1887, On the Concept of Number, and presented again in 1891 in the ﬁrst four chapters of the Philosophy of Arithmetic. It will turn out that the intuitionistic account of number is very diﬀerent from Husserl’s. They strongly disagree already about the number 2. On the other hand, at root the projects of Husserl and Brouwer are much alike. As it is this fact that lends their disagreement much of its signiﬁcance and makes it pressing, I will begin there. In the Philosophy of Arithmetic, which he dedicated to his teacher Brentano, Husserl is interested in developing a descriptive psychology of mathematics. This implies that he is concerned with the way humans actually carry out mathematical activities; not in the sense of experimental psychology, but rather as a theory of correct performance, to borrow Chomsky’s term.233 The aim is to describe our mathematical representations, the components out of which they are built, and the diﬀerent ways in which these components can be combined [21, p. 1]. In particular, Husserl hopes to trace the origins of the concepts we use in mathematics, such as the concept of number. As Dallas Willard once explained it, ‘to give the origin of a concept is, for Husserl and his fellow workers in this early period, to describe the essential course of experiences through which one comes to possess a concept’ [241, p. 106]. The value of exhibiting the origins of a concept is that in doing so, that concept itself becomes clearer to us. Carl Stumpf, with whom Husserl studied in Halle in 1886 and 1887 while writing his Habilitationsschrift, was particularly clear about this. In 1873 he wrote, The question ‘Whence arises a representation?’ is of course (though this is not always done) to be clearly distinguished from the other question, ‘What is its knowledge content, once we have it?’ However,

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these two questions are methodologically related, insofar as the question about the origin of a representation leads us to the separate parts of which it is composed, and therefore yields a more precise grasp of its content. [119, p. 87]234 Husserl’s aim to give an account of mathematical knowledge is also Brouwer’s; and in working towards its achievement, Brouwer in eﬀect also resorts to descriptive psychology, even though he never mentions that term or the name of Brentano. But the function of such a descriptive psychology for Brouwer is not just that of an investigation into origins. It is also a propaedeutic to ontological investigations. Before indicating in what sense Brouwer’s project, too, rests on descriptive psychology, it should be said that the ﬁndings of descriptive psychology do not, by themselves, have any implications as to what mathematical objects exist and what the foundation relations between them are. This was not altogether clear to Husserl when writing the Philosophy of Arithmetic, but it is exactly the critical point he would later make about that work in the introduction to both editions of the Prolegomena to the Logical Investigations. Rather, any ontological investigation of mathematics presupposes an analysis in the sense of descriptive psychology. We ﬁrst need to analyse the content of our mathematical concepts, partly by analysing how we arrived at them; only then can we ask whether anything in mathematical reality – however one construes that – corresponds to them. Incoherent concepts or merely consistent concepts are just as much objects of study for descriptive psychology. Indeed, in Brouwer’s work one can ﬁnd analyses how people have arrived at a certain concept or idea that according to him corresponds to no mathematical reality, for example in his discussion of the case of the principle of the excluded middle [35, p. 158].235 Having investigated how a particular concept has been arrived at, which is a piece of descriptive psychology, Brouwer then evaluates that against a certain standard of correctness. It is there that factors come in that fall outside a project of pure description. What is the standard of correctness to which Brouwer holds? For Brouwer, the situation is the following: according to him, we construct the numbers, and all other mathematical objects, ourselves. ‘Mathematics is a mental construction,’ Brouwer said [38]. The material out of which the mind creates mathematics is the pure intuition of time.236 Intuitionistic mathematics studies what can be given to us in intuition, be it immediately or after carrying out a construction method. Descriptive psychology is involved in the following way. If the analysis into the origins of a certain concept shows that it can be formed in accordance with certain principles of creation, this means, according to Brouwer, that the concept is mathematically real. If, on the other hand, it cannot be formed that way, this means that the concept is not mathematically real. So we see that descriptive psychology comes to contain ontology if one adds the premise that the objects in question are created in the mind. For if an object is created in the mind, then the acts in which that

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object is given to us are at the same time the acts in which that object comes into being. In that setting, questions of ontology are equivalent to questions of psychological origins. Such are my reasons for holding that, in their initial phases, the projects of Husserl and Brouwer coincide. In particular, both require ﬁrst of all a descriptive analysis of our handling of authentic representations (‘eigentliche Vorstellungen’). Human psychology being what it is, these are our representations of small and simple objects, such as sets of 2 or 3 elements. ‘One can hardly count beyond three in the authentic sense,’ Husserl famously asserted in the ﬁfth thesis appended to his Habilitationsschrift of 1887.237 The projects of Husserl and Brouwer come to diﬀer only after this requirement of descriptive analysis has been met. Brouwer’s project is an extension of Husserl’s, in the following sense. Brouwer’s interest is in the concept of number. In particular, he is interested in its intrinsic properties. The fact that actual humans cannot have authentic representations of large numbers is due, he says, to limiting properties of our memory and capacity for attention. These properties are not implied by the concept of number as such. In that sense, these limiting properties of our memory and so on are, relative to the concept of number, contingent. If we would have better memories, we would have a greater number of authentic representations than we have now, but the concept of number would not change. In fact, diﬀerent actual people diﬀer in their ability to have authentic representations of numbers, yet for all of them the concept of number is the same. The way to abstract from such diﬀerences is to introduce various idealisations, but there are diﬀerent ways of going about this. Brouwer chose to theorise about an ideal mathematician, whose mind is like ours, but perfect.238 The idealisations Brouwer makes are therefore far from arbitrary. He accepts just those idealisations that satisfy two conditions. On the one hand, they should take away factors that hinder the free employment of the capacities that we actually have. (What these capacities are is to be determined by inner perception and this is therefore an ‘empirical’ question in Brentano’s sense; see footnote 233.) On the other hand, they should not alter the nature of those capacities themselves. (This is parallel to Brentano’s anti-inﬂationism [182, p. 258].) Examples of such hindering factors are lack of time, space, attention, or motivation. In particular, Brouwer notices, it is only because of such factors that we need to take recourse to language in mathematics, in the form of symbolic systems and formalisms. As the ideal mathematician could always work with authentic representations and would never forget them, for the ideal mathematician there would be no need for language: Premising that rational reﬂection leads to the conclusion that the exactness of mathematics, in the sense that mistakes and misunderstanding are excluded, cannot be secured by linguistic means, the question

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arises whether it can be secured by other means. The answer is that the languageless constructions which arise from the self-unfolding of the basic intuition, are exact and true, by virtue of their very presence in the memory, but that the human faculty of memory which must survey these constructions, is by its nature limited and liable to error, even when it seeks the support of linguistic signs. Thus for a human mind equipped with an unlimited memory, pure mathematics, practised in solitude and without using linguistic signs, would be exact, but the exactness would be lost in mathematical communication between human beings with unlimited memory, because they would still be thrown upon language as their means of understanding. [44, p. 443, original emphasis]239 Mathematics is, as Brouwer put it, ‘an essentially languageless activity of the mind’ [40, p. 141], and it is mathematics in this sense that is the object of study in intuitionism. Brouwer only idealises from what is already present in our experience: extrinsic, negative elements are taken away, but no intrinsic, positive ones are added. This ensures that the fruits of intuitionistic reﬂection, although no longer descriptive of our actual practice, will nevertheless pertain to the essence of our mathematical capacities. Troelstra has called intuitionism a schematic description of human mathematical activity;240 again in Chomsky’s terms, one could also say that intuitionism is a theory of competence rather than performance.241 In particular, intuitionism goes beyond a theory of correct performance, for correct performance is still aﬀected by the limitations that are irrelevant to competence.242 In order to be clear about what Brouwer’s introduction of the notion of an ideal mathematician is supposed to accomplish, one may draw the obvious analogy with Turing’s analysis of a human computer, who mechanically computes numbers (nowadays commonly referred to as a computor). The computer that Turing describes is, he argues, able to compute ‘all numbers which would naturally be regarded as computable’ [227, p. 135]. Yet, this computer is not someone we can actually turn to when we run out of resources while trying to carry out a certain computation. It would be a category mistake to inquire, upon reading Turing’s paper, for his computer’s telephone number. For it is rather a conceptual device to enable us to study computability in its essential aspects. At the same time, Turing makes it clear that his analysis is meant to be of epistemological interest: ‘The real question at issue is “What are the possible processes which can be carried out in computing a number?” ’ [227, p. 135]. This epistemological interest controls the idealisations that Turing is willing to make from the capacities of an actual human computer. For example, potentially inﬁnite time and a potentially inﬁnite piece of scrap paper are allowed for, but a ﬁnite bound is set on the number of symbols observable on the paper by the computer at any one moment, as otherwise they cannot be taken in ‘at a glance’. The concept of computation that Turing

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has in mind is such that possible limits on time and paper available do not enter into an answer to the question ‘What is a computation?’, whereas a ﬁnite bound on the information intake from the paper at each step of the computation does.243 Similarly, Brouwer’s ideal mathematician is not someone whom we can ask for help when we do not see the solution to a certain mathematical problem. Rather, it is a conceptual device to enable us to study constructability in its essential aspects. The concept of construction that Brouwer has in mind is such that possible limits on time and memory available do not enter into an answer to the question ‘What is a construction?’, unlike, for example, the limitation that the subject cannot complete inﬁnite tasks. Brouwer’s rationale for making this choice is that, according to him, limitations such as the ﬁrst two do not reﬂect an essential property of the mind but the third one does. The idealisations Brouwer makes are controlled by what he takes to be the essential properties of mind.244 The conception of mental capacity Brouwer adopts may be called, following Charles Parsons, ‘transcendental’ [171, p. 341].245 To say that intuitionism is a theory of competence rather than of correct performance, is just another way of making this point. It is because of this trait of intuitionism that Brouwer could reply to David van Dantzig, who had taken him to task for not having considered a number of limitations on psychologically real subjects, that ‘psychologistic interpretations of intuitionistic mathematics, however interesting, can never be adequate’.246 On the same grounds, it can be argued that ﬁnitism (strict or not) and intuitionism are no rival theories, as a theory of competence is not concerned with the issues of feasibility that interest the ﬁnitist.247 At the end of chapter XI of the Philosophy of Arithmetic, in the section on inﬁnite sets [107, pp. 218–221], Husserl comes close to describing Brouwer’s ideal mathematician, who is the subjective correlate of a theory of competence. Husserl there makes the same point as Brouwer did. Speaking of what Husserl calls the ‘so to speak, contingent limits’ to our knowledge, he explains: An extension of the capacity for representation which would put it in a position to grasp, in the collective manner, groups of a hundred, a thousand, or a million elements in genuine engagement is quite conceivable. And hence our intention which underlies the symbolic representation of such large groups gives no occasion for logical scruples. That intention tends toward the actual representation of collections which, if not within our scope, yet fall within that of an idealised capacity of human knowledge. [131, p. 231]248 If we recall Husserl’s assertion, quoted above, that ‘one can hardly count beyond three in the authentic sense’, a question arises as to the exact nature of the idealisation(s) Husserl envisages in the present passage. Does he mean that, we can idealise the capacity to count in the authentic sense from 3 to any speciﬁc n, or does he mean that we can idealise that capacity to one bounded by no speciﬁc n? In the ﬁrst case, a subject idealised in this particular way

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would again have to rely on symbolic systems when dealing with numbers larger than n, whereas in the second case such a need would never arise. Be that as it may, clearly Husserl does not go on to thematise that second possibility, and, moreover, that would not have been in his speciﬁc interest. He does not want to idealise that far, for to do so would take him too far away from descriptive psychology of actual humans, for whom there are always various limitations of the type mentioned, and who therefore always will have to resort to symbol systems at some point. This choice on Husserl’s part also means that his theory in the Philosophy of Arithmetic, therefore is a theory of (correct) human performance rather than one of competence. Characteristic for this choice is his comment on Brix earlier on in that work: From the possibility of continuing the successive positing of units arbitrarily far, Brix draws the highly precarious conclusion . . . that in such a way arbitrarily large numbers can be formed. But the mere succession of the repeated positings does not yet guarantee any synthesis, without which the collective unity of the number is inconceivable. It is precisely upon the inability actually to carry out such a synthesis, as we shall later discuss, that every attempt to form an authentic representation of the higher groups and numbers factually runs aground. Through the simplest of experiments Brix could have convinced himself that even nineteen positings of units are not clearly distinguishable from twenty, unless by the indirect means of symbolizations that serve as surrogates for syntheses actually carried out. [131, p. 31n. 2]249 Regarding the particular case of limitations of our memory, it is worth noting that in the lectures on internal time consciousness from 1905, Husserl says that, although our actual retentions fade, a consciousness in which they never vanish is ‘idealiter’ possible [104, footnote 1 to section 11]. This possibility corresponds to the wider notion of an ideal subject or mathematician, that is, one with potentially inﬁnite memory, that Brouwer is concerned with. From this diﬀerence in character between Brouwer’s and Husserl’s theories (competence versus performance), it follows that their projects as such are not conﬂicting, as is sometimes suggested [178, p. 112n. 18]. Nevertheless, in their concrete analyses it is possible to ﬁnd a conﬂict. If, as I have been arguing, Brouwer as well as Husserl are engaged in descriptive psychology, then the following disagreement between them cannot be merely apparent. In their speciﬁc investigations into the origin of ﬁnite number, Husserl and Brouwer arrive at opposite answers to the question which concept of number is basic. Husserl thinks that the psychologically basic concept of number is that of cardinal number (‘Anzahl’). Brouwer thinks that the psychologically basic concept of number is that of ordinal number (‘Ordnungszahl’). Husserl’s account of number in the Philosophy of Arithmetic begins with a claim that ordinal numbers are founded on cardinal numbers. In the introduction, Husserl writes:

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As whole numbers [Anzahlen] refer to sets, so ordinal numbers refer to series. But series are ordered sets . . . If the common signiﬁcations of these names are attended to, then it remains correct that the concept of ordinal number includes, and thus presupposes, that of the whole number [Anzahl], as the manner in which the terminology has developed rightly indicates. [131, p. 12]250 The relation between sets and sequences given here is familiar from Bolzano and Cantor, and Husserl may well have taken it from them.251 Husserl concludes that it remains for him to ﬁnd the origin of cardinal numbers: In every case, therefore, an analysis of the concept of whole number [Anzahl] is an important prerequisite for a philosophy of arithmetic. And it is its ﬁrst prerequisite, unless it should turn out that logical priority belongs to the ordinal concepts, as is maintained from another quarter. The possibility of an analysis of the whole number concept that is done in complete disregard of the ordinal will supply the best proof of the inadmissibility of that view. [131, p. 14]252 About Husserl’s analysis of cardinal number that then follows I do not have much to say, as what I will be concerned with is just the claim from which Husserl starts and which I just quoted: the claim that ordinal numbers are founded on cardinal numbers. But the result of Husserl’s analysis is that the concept of cardinal number is based on the two concepts of ‘mere something’ and conjunction. As Dallas Willard once put it, ‘a number is a property of similar groups which are seen or conceived of as mere “somethings” joined by a mere “and”.’253 After all, when counting the number of elements in a set we are neither interested in qualitative aspects of these elements nor in particular relations these elements may bear to one another. To see the various things that are counted together is, Husserl says, to see them in a ‘collective combination’.254 Brouwer conceives of numbers as mental constructions. What is the building material out of which they are constructed? Immediately given to us is what Brouwer called the ‘two-oneness’ or, and this is Brouwer’s contribution to the language of Shakespeare and the King James, the ‘two-ity’: Intuitionistic mathematics is an essentially languageless activity of the mind having its origin in the perception of a move of time, i.e., of the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory. [40, p. 139] The two-ity is the experience of a present and a past moment as two moments held together in the mind. In Husserl’s terms: in the present, the immediate past is kept in retentional awareness; in that particular, intentional sense, the immediate past is contained in the present. What is in between the immediate past and the present is the continuous ﬂow of time. This is not to suggest that the present and the immediate past are discrete points and, as such, are no part of the temporal continuum. Rather, they are small, non-overlapping parts

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of that continuum. (In the note passages in Husserl’s what Brouwer says. I will example, sections 6 and 18 Brouwer continues:

following, there will be a number of occasions to writings that may serve to elaborate or amplify not make a systematic attempt at this. See, for of Husserl’s Consciousness of Internal Time.)

If the two-ity thus born is divested of all quality, there remains the empty form of the common substratum of all two-ities. It is this common substratum, this empty form, which is the basic intuition of mathematics. [40, p. 139] Brouwer speciﬁed that the two parts are connected by a continuum, and, as these all have their origin in the basic and immediate phenomenon of a move of time, he insisted that the discrete and the continuous are mutually dependent and mutually irreducible notions [23, p. 8]. Moreover, the two parts of the two-ity are not independent from each other, as the abstract parts of timeconsciousness of which they are the form are not. In the two-ity, one part is embedded in the other. This embedding determines an order relation. The empty two-ity deﬁnes the ordinal number 2. By a further act of abstraction, one obtains, from the empty two-ity, the number 1. As Brouwer explains in his dissertation, The ﬁrst act of construction has two discrete things thought together (also according to Cantor . . .); F. Meyer . . . says that one thing is suﬃcient, because the circumstance that I think of it can be added as a second thing; this is false, for exactly this adding (i.e., setting it while the former is retained) presupposes the intuition of two-ity; only afterwards this simplest mathematical system is projected on the ﬁrst thing and the ego which thinks the thing. [44, p. 97, original emphasis] Thus, genetically speaking, the ordinal number 2 is more basic than the ordinal number 1. Accordingly, Brouwer begins a (much later) sketch of ‘the inner experience’ as follows: ‘twoity; twoity stored and preserved aseptically by memory; twoity giving rise to the conception of invariable unity’ [45, p. 90]. The experience of the two-ity gives rise to further natural numbers as follows. As time ﬂows on, the moment which is ‘now’ does not permanently keep that quality. It sinks back into the past and thus becomes part of the past. This way, the later part of the two-ity falls apart into a new two-ity, containing the old one as one of the two parts. The form of this whole now corresponds to 3. Proceeding this way, one obtains the ordered sequence of the natural numbers. Brouwer says that mathematics ‘comes into being by self-unfolding of the basic intuition’ [38].255 The general notion of ordinal number in turn is obtained by reﬂecting on this self-unfolding.256 In reﬂection, one recognises the common mode of generation of the individual numbers.257 This mode of formation depends on the empty two-ity as the starting point together with the iterability of the formation of two-ities. In turn, the empty two-ity and its iterability are

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abstracted from the inner structure of the experience of the move of time. The structure Brouwer describes here is in fact much the same as that in the analyses of time that Husserl embarked on shortly after the Philosophy of Arithmetic. Husserl’s famous diagrams come to mind.258 Brouwer developed his views on time independently from Husserl, but he did have at least a passing acquaintance with work of Fechner and Lotze.259 The two-ity even leads beyond the natural numbers. The very insight that the process of embedding two-ities can be iterated gives rise to the inﬁnite ordinal ω (the ﬁrst number after 1,2,3,. . .). Through this insight, the intuitionist has a constructive concept of inﬁnity without presupposing that one can construct each natural number in practice. This inﬁnity has to be thought of as potential and not actual: on the one hand, the subject can start a potentially inﬁnite task, but on the other hand, it can not complete it.260 From ω one gets further countable ordinals, for example ω + 1 by taking ω and 1 together in a two-ity. Cardinals, for Brouwer, have their origin in the ordinals. As in classical mathematics, he says that two sets are of the same cardinality if one can establish a 1-1 mapping between their elements.261 Unlike classical mathematics, however, Brouwer deﬁnes a set in terms of ordinals, roughly as a mapping from ordinals to whatever objects the elements of the set are chosen from. This is not so strange as it may seem: after all, the basis of all intuitionistic mathematics is the sequence given by the self-unfolding of the two-ity, so whatever else we are going to construct has to obtained from that. Such a mapping is the means to run through the elements of a set, and thus to constitute the set: in the case of ﬁnite sets, this takes the form of a simple enumeration of the elements, in the case of denumerably inﬁnite sets, of an inductive deﬁnition. Brouwer also showed how, by generalising his deﬁnition of a set and by admitting so-called choice sequences (of ordinals), one can obtain sets of a more complicated type that are necessary to model the continuum (for the exact deﬁnition of this type of set or ‘spread’, see p. 49). I will not go into the technical details; what matters here is the foundation relations: what is based on what? Intuitionistically, cardinals are founded on sets, and sets are founded on ordinals. Hence, by transitivity, cardinals are founded on ordinals. This is quite the opposite of Husserl’s view. Now, if, as I have argued, Husserl and Brouwer both engage in descriptive psychology, where does this contradiction come from? Husserl holds that order is not part of the concept of cardinal number [107, p. 32]: this may be accepted as a conceptual truth, and Brouwer does so. But, as we saw, Husserl also holds that sequences are ordered sets and that therefore the concept of cardinal is more basic than that of ordinal. Now, one should certainly accept that a sequence may be deﬁned by specifying an ordering of a set. But this fact does not, by itself, imply that sequences are ordered sets, or that the concept of sequence presupposes the concept of set. To reach that conclusion, one would need an additional argument to the eﬀect that the deﬁnition of a sequence as an ordered set is the only one

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possible. In particular, in a psychological analysis of number, it would have to be shown that the only way to obtain (a representation of) a sequence of ‘mere somethings’ is by ordering a previously obtained set of ‘mere somethings’.262 And Husserl never establishes this. Ironically, in his text ‘On the theory of the totality’ of 1891, Husserl does make the symmetrical point that the fact that the numbers can be put in a sequence does not, in and of itself, imply that the numbers are founded on the concept of sequence [107, p. 400]. So to the extent that Husserl’s analysis is correct, it leaves open the possibility that the concept of sequence is not dependent on that of set. It moreover leaves open the possibility that, on the contrary, the concept of set is dependent on that of sequence. This is the root of the disagreement between Husserl and Brouwer. For if the concept of set would be dependent on that of sequence, the concept of cardinal would then be dependent on that of ordinal. According to the more detailed descriptive analysis that Brouwer makes, this possibility in fact obtains. What Brouwer’s analysis shows is that there is a sequence that we can become aware of as such without ordering a previously unordered set. No active ordering on our part is needed to arrive at that sequence; only an act of thematisation. I am referring to the two-ity and its unfolding. On Brouwer’s account, in a reﬂexive act, the structure of internal time awareness is thematised. This yields ﬁrst of all the two-ity as an object; this operation can be iterated. Then various abstractions are made. For example, the time between the experiences of successive elements is left out of account; and in the constitution of a pure (ordinal) number, all content of those experiences is abstracted from as well. What remains is the pure structure of the nesting. In the process of abstraction, the speciﬁcally temporal aspects are done away with; and because of the way cardinals are founded on ordinals, these aspects do not form part of the concept of cardinal number either. In this sense, Brouwer agrees with Husserl that time as such is no part of the concept of cardinal number. But an abstraction (in the sense of a dependent moment) of time awareness – its nested structure – is part of the concept of ordinal number, for it provides their order. And hence, to the extent that the notion of cardinality refers to ordering because invariance under ordering is part of that notion, the nested structure is part of the concept of cardinal number as well. In the circumstance that time awareness provides the order of the ordinal numbers lies the diﬀerence with a phenomenological theory of number that, like the intuitionistic theory, has ordinals as the basic concept, the theory developed by Gurwitsch in lectures at the New School in 1973.263 Questions about that theory have been raised by McKenna: Gurwitsch was quite explicit, especially in conversations that I had with him, that the thematizing consciousness was not the disclosing of an organization already somehow contained in the gardener’s procedure [of examining the roses in his garden one after the other], but

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rather was the bringing about of a new one on the basis of an examination of that procedure . . . It is one whose parts have a very speciﬁc relationship to one another, a relationship which was brought about by the thematizing consciousness. The diﬃculty is in seeing how that organization is brought about.264 On the intuitionistic account the parts already are related to one another, as that relation derives from an intrinsic feature of the awareness of time. Another phenomenological account, that given by Tieszen [214, ch. 5] here (and elsewhere) agrees with the intuitionistic one; it is based on the structure of time awareness, and gives ordinals genetic precedence over cardinals, the reason being that ‘to be conscious of a cardinal number n it is necessary to abstract from the ordering of units in a construction’ [214, p. 105]. Another reason is also suggested: ‘for larger constructions we cannot determine the number of units in one act but must succesively run through the units to determine number’ [214, p. 105]. Indeed, it is a well known phenomenon that people (as well as various kinds of animals) are able to recognise small quantities at a glance, in one act, without counting; this is referred to as ‘subitisation’. However, it seems to be rather a phenomenon of pattern perception and memory (as the ability to subitise disappears very quickly for more than a handful of objects). As such, the ability to subitise is no indication whatsoever of a grasp of the concept of number. The latter must involve a grasp of the structure that the number series exhibits.265 When someone subitises a certain quantity of objects and then conceptualises this by saying, for example, ‘there are four items here’, this conceptualisation must all the same depend on counting, either by quickly running through the objects or, in the case of a recognisable pattern (e.g., the dots on a face of a die), on having run through objects forming the same pattern on a prior occasion. It is for this reason that subitisation will play no role in the intuitionistic account of number.266 On the basis of the foregoing considerations, I should like to formulate the following two theses: 1. Brouwer gave an account of ordinal numbers which is descriptively more accurate than Husserl’s in Philosophy of Arithmetic. 2. The psychologically basic concept underlying that of number is not that of set, but that of sequence (and hence, ordinals take genetic priority over cardinals). But perhaps someone will raise the following objection to the second thesis. The comparison between Husserl and Brouwer may show that Brouwer’s account is an argument against Husserl’s. But it does not show that there could not be yet another and perhaps even more basic way to arrive at the concepts of ordinal and cardinal number. Why isn’t the unfolding of the twoity, just as contingent, relative to the concept of number, as, for example, the imperfections of our memory?

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Brouwer’s answer follows from his paper from 1948, titled ‘Consciousness, philosophy, and mathematics’ [39]. From what he says there it follows that the genesis of the two-ity, which depends on the ﬂow of time, is part of the genesis of intentionality itself (Brouwer does not use the term ‘intentionality’). Before having a closer look at Brouwer’s paper, let me explain the relevance of this consequence. In eﬀect, Brouwer presents us with two options. The ﬁrst is to forget about the notion of intentionality altogether, or at least to deny its relevance to a philosophy of mathematics. As mathematical knowledge is a performance of consciousness, and the characteristic feature of consciousness is intentionality, this option is less than inviting.267 The other option is to accept the relevance of intentionality. But then, Brouwer argues, one also has to accept the relevance of time-awareness and of the two-ity, for it is through them that intentionality arises. And once one has the two-ity, one obtains, in reﬂection on it, intuitionistic mathematics. Here is the relevant passage of Brouwer’s.268 First of all an account should be rendered of the phases consciousness has to pass through in its transition from its deepest home to the exterior world in which we cooperate and seek mutual understanding ... Consciousness in its deepest home seems to oscillate slowly, willlessly, and reversibly between stillness and sensation. And it seems that only the status of sensation allows the initial phenomenon of the said transition. This initial phenomenon is a move of time. By a move of time a present sensation gives way to another present sensation in such a way that consciousness retains the former one as a past sensation, and moreover, through this distinction between present and past, recedes from both and from stillness, and becomes mind. As mind it takes the function of a subject experiencing the present as well as the past sensation as object. And by reiteration of this twoity phenomenon, the object can extend to a world of sensations of motley plurality [44, p. 480]. This is in fact very similar to what Husserl would come to say about the relation between time and intentionality in what has become known as the C-manuscripts in the 1930’s. James Mensch explains Husserl’s idea that the awareness of time is the condition of possibility of the subject-object relation as follows: Object in German is Gegenstand. It is that which stands against the subject. This ‘againstness’ indicates a certain nonidentity, a certain distance, between the subject and its object. In transcending temporal positions, the now of my actively constituting subject constantly opens up a temporal distance between itself and the constituted positions . . . In this, it also opens up the original distance between itself

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and its objects which are positioned in deﬁnite stretches of successive time. All objectiﬁcation involves such positioning in successive time. [160, p. 62, original emphasis] To constitute this distance involves holding two diﬀerent things together – the subject and the object – , and hence the two-ity. Note that when Brouwer, in a quotation I gave earlier, explains that the notion of twoity is simpler than that of unity, he says that unity arises ‘only afterwards [when] this simplest mathematical system is projected on the ﬁrst thing and the ego which thinks the thing.’ As a consequence, no intentional act could be prior to the genesis of the two-ity. This is why Brouwer calls the move of time the ‘Ur-phenomenon’. But then the concept of sequence or self-unfolding two-ity cannot be contingent relative to any other property of intentional life. However, the following question arises. In his work on time consciousness, which Husserl began in 1893, only two years after publishing the Philosophy of Arithmetic, Husserl analyses the structure of internal time awareness as a tripartite one: retention, primordial impression, and protention. In Brouwer’s analysis, only the ﬁrst two elements have played a role. What about the third? Why does Brouwer arrive at a two-ity and not at a three-ity [45, p. 90] as the basic intuition of mathematics? Retention and primordial impression have a property in common that protention does not have, and which is relevant here. In retention and in the primordial impression, something is presented ; they are perceptions. A protention on the other hand is just an anticipation, albeit not necessarily one that is as active as the usual sense of the word suggests. It is a reference to the immediate future, just as a retention is a reference to the immediate past. Unlike the primordial impression and a retention, however, a protention does not present something that is or has been experienced. More speciﬁc to the context of mathematics, a protention does neither present something immediately given nor (the result of) a construction. Therefore, it cannot be part of the basis of mathematical constructions. However, there is a way in which protention does play a role in Brouwer’s analysis: it is constitutive of the awareness that the two-ity can be iterated.269 That is, protention is constitutive of our awareness that time not only has moved but will move on. This is part of the meaning of Brouwer’s term that we saw above, the ‘self-unfolding of the two-ity’. It is also at the basis of the constitution of the notion of choice sequence. Indeed, Brouwer came to say that the ‘extension [of mathematics by choice sequences] is an immediate consequence of the self-unfolding’ [45, p. 93n]. But it is precisely because of this essential involvement of protention that proceeding sequences are never ﬁnished, are never completely constructed. In the foregoing, I have given two arguments, a negative one and a positive one. The negative one is that Husserl’s analysis of the notion of number contains a non sequitur: from the fact that a sequence can be deﬁned by ordering a set, it does not follow that a sequence, genetically speaking, is an ordered set. The positive one is an argument to the eﬀect that the intuitionistic analysis

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is an improvement over Husserl’s in the Philosophy of Arithmetic. Brouwer’s analysis shows the primordiality of the concept of sequence and, in particular, its priority over the concept of set. It does so by bringing out the structure of the two-ity and its self-unfolding. From this structure, Brouwer obtains the sequence of ordinals as well as the notion of sequence itself. Whether one ﬁnds Brouwer’s account in itself convincing is a separate question. Brouwer’s analysis involves the structure of time awareness. Husserl overlooks that structure in his considerations about a possible role of time in the Philosophy of Arithmetic. Yet, when Husserl was writing the Philosophy of Arithmetic and its predecessor On the Concept of Number, he knew very well, from his study of Brentano, Stumpf, and others, that time awareness has an inner structure. In particular, he knew about retention (then still called ‘primary memory’ (‘primäre Erinnerung’)), the feature of time consciousness that deﬁnes Brouwer’s two-ity. I ﬁnd it hard to guess why Husserl did not bring this notion into play when considering the relation between time awareness and number.270 Dallas Willard has drawn attention to the fact that Husserl’s analysis of number never changed much from the one he had given in the Philosophy of Arithmetic [241, p. 108]. In various of Husserl’s works after that, one will ﬁnd remarks that are easily related to Brouwer’s account of the notions of number and sequence, e.g., Husserl’s analysis of the sequence as a categorial form in Experience and Judgement [124, section 51b]. In fact, all the elements of Brouwer’s account can be found throughout Husserl’s work – yet not that account itself.

Notes

1

Errata to On Brouwer [3]:

1. ﬁrst page of Preface, line 7: for ‘paraphrase’ read ‘adaptation’ 2. p. 5 line 7: the reference ‘[45, p. 136]’ is not correct. The page number is, but the work in question is D. van Dalen (2001), L.E.J. Brouwer en de Grondslagen van de Wiskunde, Utrecht: Epsilon. 3. p. 15 line 6: for ‘dinner’ read ‘lunch and tea’ 4. p. 15 line 8: for ‘the evening’ read ‘it’ 5. p. 16 line -10: for ‘proof object’ read ‘objectiﬁed proof’ 6. p. 26 line -3: for ‘if for some k ≤ n’ read ‘for the smallest k ≤ n such that’ 7. p. 28 line -11: for ‘A ∨ B ’ read ‘¬(¬A ∧ ¬B )’ 8. p. 48 line 24 omit ‘thin’ 9. p. 86 delete footnote 18 (it is correct only in so far as Heyting is concerned) 2

Item 050120.1 in the Gödel Nachlaß, Firestone Library, Princeton. On Gödel’s interest in Husserl’s phenomenology, see [7]. 3

Dutch: ‘keuzerijen’, German: ‘Wahlfolgen’, French: ‘suites de choix’.

4

Here I ﬁnd myself in agreement with Dummett, who writes:

Intuitionists are engaged in the wholesale reconstruction of mathematics, not to accord with empirical discoveries, nor to obtain more 127

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fruitful applications, but solely on the basis of philosophical views concerning what mathematical statements are about and what they mean. Individuals may be converted to the intuitionistic viewpoint, without wishing thereafter to scrutinize more closely the philosophical arguments for or against it, just as they may be converted to a religious faith without wishing to become theologians: but intuitionism will never succeed in its struggle against rival, and more widely accepted, forms of mathematics unless it can win the philosophical battle. If it ever loses that battle, the practice of intuitionistic mathematics and the metamathematical study of intuitionistic systems will alike become a waste of time. [66, p. ix] The view I am arguing against is eloquently presented by Bishop: Brouwer became involved in metaphysical speculation by his desire to improve the theory of the continuum . . . In Brouwer’s case there seems to have been a nagging suspicion that unless he personally intervened to prevent it the continuum would turn out to be discrete. He therefore introduced the method of free-choice sequences for constructing the continuum . . . This makes mathematics so bizarre it becomes unpalatable to mathematicians, and foredooms the whole of Brouwer’s program. [16, p. 6]. 5

Brouwer to Eva Wernicke. ‘Hier tummelt sich jetzt Husserl herum, wobei ich stark mit herangezogen werde.’ [63, p. 567n. 18]. 6

‘Mit das Interessanteste waren in Amst[erdam] die langen Gespräche mit Brouwer, der einen ganz bedeutenden Eindruck auf mich machte, den eines völlig originellen, radikal aufrichtigen, echten, ganz modernen Menschen.’ 7

Husserl owned an oﬀprint of Brouwer’s paper from 1928, ‘Intuitionistic reﬂections on formalism’ [34] – incidentally, the only of Brouwer’s papers that Husserl seems to have had a copy of. That paper had been communicated by Brouwer at a meeting of the Dutch Academy of Sciences of December 17, 1927, and published in the proceedings of the Academy in 1928. Although I do not know the date on which these proceedings came out, it is not unlikely that Brouwer had given an oﬀprint of the paper to Husserl on the latter’s visit to Amsterdam, which took place at the end of April 1928. It also appeared in the Proceedings of the Prussian Academy of Sciences in Berlin dated May 5, 1928 [62, p. 28]. Unfortunately, the oﬀprint Husserl owned has been missing from the Husserl Archive for years now (which means that we cannot tell whether that oﬀprint came from Amsterdam or from Berlin, nor, worse, whether Husserl had made notes on it); I only know that Husserl had it because it is listed in an old and abandoned catalogue of his personal library, and I am grateful to Thomas Vongehr for calling my attention to this fact. In

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a meeting prior to Vongehr’s discovery, Byung-Hak Ha had already emphasised to me the relevance of that paper to Husserl’s concerns in Formal and Transcendental Logic, which was written and published in the year after the visit to Amsterdam. It is not clear to me what to make of the fact that in that work, Husserl does not refer to Brouwer’s paper. 8 As for Husserl, this needs some qualiﬁcation. It is clear from his correspondence and unpublished manuscripts that he was aware of Brouwer’s project. Also, Weyl, (Felix) Kaufmann and especially Becker will have told him about intuitionism, and he read their work. But it seems he never discussed the subject in any depth. By way of an explanation, Husserl himself wrote that his own investigations were leading into other areas than mathematics, and that it would be too much of an eﬀort to catch up with recent developments (see the quotation from his letter to Pos, on p. 73). Husserl’s contact with mathematics at the time after the introduction of choice sequences is treated in section 5.4. 9

As mentioned in the text, Heyting formalised intuitionistic logic and gave a meaning explanation in Husserl’s terms. But although at that time Heyting also gave a formalisation of choice sequences [86], he did not supply a corresponding meaning explanation for them. Göran Sundholm has been stressing this point since 1983 [211, pp. 163–164], [212, p. xiv], but it is still not the commonplace it deserves to be. 10

Mohanty [162] and Seebohm [192] discuss the possibility of transcendental phenomenology. Tito’s defense of transcendental phenomenology [215, chs. 4 and 5] is excellent. 11

The idea of omnitemporality of mathematical objects is already in the Bernau research manuscripts from 1918. However, the ﬁrst published hint of this idea appeared only in the Formal and Transcendental Logic of 1929 [112, pp. 164, 167], and the ﬁrst detailed treatment in the posthumous Experience and Judgement from 1939 [124, section 64]. Lohmar [154] has traced the connections between this treatment and the 1918 manuscripts. 12

In a review of Tieszen’s book, van Dalen [56] pointed to the possibility of a universe that contains, in my terms, static as well as dynamic objects. 13

‘Die realen Gegenständlichkeiten schließen sich in der Einheit einer objektiven Zeit zusammen und haben ihre Zusammenhangshorizonte; zu ihrem Bewußtsein gehören demgemäß Horizontintentionen, die auf diese Einheit verweisen. Hingegen eine Mehrheit irrealer Gegenständlichkeiten, z.B. mehrere Sätze, die zur Einheit einer Theorie gehören, sind nicht bewußt mit solchen Horizontintentionen, die auf zeitlichen Zusammenhang verweisen. Die Irrealität des Satzes als Idee einer synthetischen Werdenseinheit ist Idee von Etwas, das an jeder Zeitstelle auftreten kann, an jeder notwendig zeitlich und zeitlich

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werdend auftritt, und doch ‘allzeit’ dasselbe ist. Es ist auf alle Zeiten bezogen, oder auf welche auch immer bezogen, immer absolut dasselbe; es erfährt keine zeitliche Diﬀerenzierung und, was damit äquivalent ist, keine Ausdehnung, Ausbreitung in der Zeit.’ [124, p. 311]. 14

‘Wir . . . betrachten eine unbegrenzt fortsetzbare Reihe von ineinander geschachtelten λ-Intervallen λν1 , λν2 , λν3 , . . . von der Eigenschaft, dass jedes λνi+1 im engeren Sinne im vorhergehenden λνi (i = 1, 2, . . .) gelegen ist. Nach [der Deﬁnition der λ-Intervalle] ist dann die Länge des Intervalles λνi+1 höchstens gleich der Hälfte der Länge von λνi und daher konvergiert die Länge dieser Intervalle gegen Null . . . Ein derartige unbegrenzte Folge ineinander geschachtelter λ-Intervalle nennen wir einen Punkt P oder eine reelle Zahl P . Wir betonen, dass bei uns die Folge λ ν1 , λ ν2 , λ ν3 , . . .

(1)

selbst der Punkt P ist, nicht etwa “der Grenzpunkt, auf welchen sich nach der klassischen Auﬀassung die λ-Intervalle zusammenziehen und der nach dieser Auﬀassung etwa als einziger Häuﬀungspunkt der Intervallmittelpunkte deﬁniert werden könnte”. Jedes der Intervalle (1) gehört dann zum Punkte P .’ 15

‘Eine unbegrenzt fortsetzbare Reihe ist im Allgemeinen keine Fundamentalreihe . . . da während ihrer Entstehung freie Wahl der Elemente nicht ausgeschlossen ist.’ 16

‘Bei uns sind ein Punkt und daher auch die Punkte einer Menge immer etwas Werdendes und manchmal etwas dauerhaft Unbestimmtes, im Gegensatz zur klassischen Auﬀassung, wo der Punkt sowohl als bestimmt, wie als fertig gilt.’ 17

This was observed by Dirk van Dalen.

18

Before the work that led to the Prolegomena in 1901, Husserl did not thematise this stability and its relation to time at all, because he did not recognise ‘ideal objects’. Then still following Brentano, he regarded every object as either physical or psychical. But it is precisely this Brentanism that led Husserl to the impasse for which the doctrine of ideal objects was his way out; more on this in section 4.2.2. The contrast between an early and a later Husserl, on the issue of temporality of mathematical objects, is therefore that between the Husserl of the Prolegomena and the one from 1917 onwards. 19

In chapter 6 this will turn out to be an example of Meinong’s distinction between ‘zeitlich distribuierte’ and ‘zeitlich indistribuierte Gegenstände’. 20 I assume that the subject is the ideal mathematician, who begins only one sequence at a time. It may seem that the subject could generate n sequences in parallel by using codings of n-tuples into single numbers, but that is not really the case: when decoding the n-tuples, the elements of the

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diﬀerent sequences come out sequentially, hence they are temporally ordered. A description how to generate a particular sequence is not identical to that sequence itself (section 5.2). One may, however, think of the coded n-tuples as a lawless sequence, and the n sequences as projections from that sequence. But note that these n sequences then can’t be lawless themselves, because of their dependency on the sequence of coded tuples; the alternative is to consider that sequence as a mere process, and consider the n sequences as lawless choice sequences [221, p. 215]. 21 ‘Haben wir auch hier “Grundgebilde” zu suchen und aus ihnen alle anderen Wesensgestaltungen des Gebietes und ihre Wesensbestimmungen konstruierend, d.i. deduktiv unter konsequenter Anwendung der Axiome, herzuleiten?’ [115, p. 153, emphasis mine]. 22

‘Die mathesis universalis . . . ist aus apriorischen Gründen ein Reich universaler Konstruktion . . . Darin treten als höchste Stufe die deduktiven und keine anderen Systemformen auf.’ [112, pp. 107–108]. Further examples are in Ideas I [115, p. 127] and Formal and Transcendental Logic [112, p. 102]. In manuscript K I 26/58a (before 1910), Husserl seems to use ‘Konstruktion’ in the modern sense (although there is room for an alternative interpretation): ‘Konstruktion ist eine Erzeugung eines Objektes oder Objektgebildes durch Grundoperationen.’ (This passage was brought to my attention by Karl Schuhmann.). 23

‘Erst der hier in Verfolgung der Axiomatik eingeschlagene Weg wird, wie ich glaube, den konstruktiven Tendenzen, soweit sie natürlich sind, völlig gerecht.’ [92, p. 160]. 24

Hesseling [84, ch. 4] discusses the widely diverging uses of the term ‘construction’ in the foundational debate in the 1920’s. In the passages quoted, surely neither Husserl nor Hilbert is thinking of, e.g., Gödel-Herbrand equations for recursive functions, or of Post systems; in such cases one sometimes speaks of ‘constructions’ as well. In any case, that use of ‘construction’ is equally far from Brouwer’s. 25

Atemporality and omnitemporality are two diﬀerent conceptions of eternity. The following I obtained from Sorabji’s historical discussion [202, ch. 8]. What Plato says about eternity is somewhat ambiguous between the two conceptions. Among the ﬁrst to be explicit about the diﬀerence are Plutarchus and Philo Judaeus; later, but better known, are passages in Plotinus’ Enneads (3.7.2, 3.7.6) and Boethius’ De Trinitate (4.67–4.77). The context of these occurrences is not mathematics but the question how the One (or God) diﬀers from the Universe. 26

This dependence of an ‘objective law’ on idealisations of subjective capacities is part of Husserl’s transcendental idealism, where all objects are

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constituted objects, all truths constituted truths. Speaking of the basic concepts and the (basic and derived) laws of logic, he writes: To every operational law of the theory of forms there corresponds a priori a subjective law concerning the constitutive subjectivity, a formal law relating to every conceivable judger and his subjective possibilities of forming new judgments out of old ones. [106, p. 182] (‘Jedem operativen Gesetz der Formenlehre entspricht apriori eine subjektive Gesetzmäßigkeit in Hinsicht auf die konstituierende Subjektivität, eine formale Gesetzmäßigkeit, bezogen auf jeden erdenklichen Urteilenden und seine subjektiven Möglichkeiten, aus Urteilen neue Urteile zu bilden.’ [112, p. 190]). 27

Paraphrasing Johnstone [135, p. 37]. The example is not without instantiations. 28

This is not Brouwer’s terminology but, in accordance with my methodological alliance (Introduction, 2.4), Husserl’s. One of the points of this section is that much of Brouwer’s position can very well be described in Husserl’s terms. 29

‘[Transzendentalphilosophie] ist das Motiv des Rückfragens nach der letzten Quelle aller Erkenntnisbildungen, des Sichbesinnens des Erkennenden auf sich selbst und sein erkennendes Leben, in welchem alle ihm geltenden wissenschaftlichen Gebilde zwecktätig geschehen, als Erwerbe aufbewahrt und frei verfügbar geworden sind und werden . . . Diese Quelle hat den Titel Ich-selbst mit meinen gesamten wirklichen und vermöglichen Erkenntnisleben, schließlich meinem konkreten Leben überhaupt. Die ganze transzendentale Problematik kreist um das Verhältnis dieses meines Ich – des “ego” – zu dem, was zunächst selbstverständlich dafür gesetzt wird: meiner Seele, und dann wieder um das Verhältnis dieses Ich und meines Bewußtseinslebens zur Welt, deren ich bewußt bin, und deren wahres Sein ich in meinen eigenen Erkenntnisgebilden erkenne.’ [100, pp. 100–101]. 30

‘Daß wir beide über einige Nebensachen verschieden urteilen, wird auf die Leser nur anregend wirken. Allerdings haben Sie mit Ihrer Formulierung dieser Meinungsverschiedenheiten vollkommen recht; . . . es läszt sich aber darüber nicht diskutieren, diese Sachen sind nur durch individuelle Konzentration zu entscheiden.’ [57, p. 167]. 31 Parsons [169, pp. 213, 223n. 13] discerns traces of naturalism in Brouwer’s philosophy. The passage Parsons refers to is from the ﬁrst Vienna lecture: ‘Mathematical Attention as an act of the will serves the instinct for self-preservation of individual man; it comes into being in two phases: time awareness and causal attention’ [157, p. 45]. (‘Die mathematische Betrachtung kommt als Willensakt im Dienste des Selbsterhaltungstriebes des

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einzelnen Menschen in zwei Phasen zustande, die der zeitlichen Einstellung und die der kausalen Einstellung.’ [35, p. 153]) This motivation by the problem of survival makes Parsons doubt whether Brouwer is ‘wedded to any notion of the a priori’ [p. 213], and, given human limitations, whether ‘the iterability of intuitive construction as Brouwer conceives it is compatible with its being a capability of the self’ [p. 223n. 13]. Let me say right away that I do not know how to square my interpretation of Brouwer with this particular statement, but the later Brouwer is concerned with accounting for mathematical certainty (e.g. [40, p. 139]), and does not mention survival or other naturalistic notions as motive for mathematics anymore (e.g. [39]). This may suggest a change of Brouwer’s mind. Another possible answer, phrased in Husserl’s terms, would be to point out that the motives to perform the transcendental reduction have to be present already in the natural attitude. Luft [155] oﬀers an extensive discussion and defense of this suggestion. It should be noted, however, that Luft’s discussion also suggests that according to Husserl the problem of survival would not be able to function as the required motive. Note that Brouwer also discusses naturalistic motives in the ‘signiﬁc dialogues’ from 1922, published in 1937 [44, pp. 450–451]. 32

In the literature widely known as ‘the creative subject’, but the participle yields a better translation of Brouwer’s ‘scheppende’. Hiding it in a footnote, Roberts [180, p. 248n. 27] aptly remarks ‘The creative subject is cognate, probably, with Husserl’s notions of internal time consciousness.’ In chapter 6, I will analyse what Brouwer would call ‘the generation of a choice sequence by the creating subject’ as the constitution, founded on internal time consciousness, of a choice sequence by the transcendental ego. 33 ‘hen . . . die het intuïtionistische “scheppende subject” niet kunnen erkennen, doordat zij ten aanzien van de wiskunde . . . een psychologistisch standpunt innemen’; ‘Mijn geloof, dat psychologistische beelden der intuïtionistische wiskunde, hoe belangwekkend ook, nimmer adequaat zullen zijn.’ For the full text and a discussion, see [3, ch. 6]. Brouwer’s letter shows that, pace Niekus, ‘creating subject’ cannot be taken to denote ‘any mathematician, which could be ourselves’ [167, p. 6]. Moreover, Niekus claims that a proper understanding of Brouwer’s creating subject requires keeping the notion ‘having a proof of φ now’ distinct from ‘having a proof of φ in n steps from now’; but Brouwer explicitly makes the idealisation of reducing the latter to the former:

The case that [p] has neither been proved to be true nor to be absurd, but that we know a ﬁnite algorithm leading to the statement either that [p] is true, or that [p] is absurd, obviously is reducible to the ﬁrst and second cases [i.e., to having proved p or to having disproved p]. [45, p. 92].

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This reduction is made also when the n steps involve making free choices, as is clear from [42, pp. 12–13]. Speaking of a species K of choice sequences (here, ‘arrows’) barred by a species of nodes C(K), Brouwer there remarks: The deﬁnition of a crude bar means that for every arrow α of K the order n(α) of the postulated node of intersection with C(K) must be computable, however complicated this calculation may be. The algorithm in question may indicate the calculation of a maximal order n1 at which will appear a ﬁnite method of calculation of a further maximal order n2 at which will appear a ﬁnite method of calculation of a further maximal order n3 at which will appear a ﬁnite method of calculation of a further maximal order n4 at which the postulated node of intersection must have been passed. The fact that the methods of calculation of the various orders themselves come to appear at various orders of the choice sequence indicates the possibility that these methods of calculation depend on the choices made in between the orders in question. For the reduction that Brouwer proposes, the relevant aspect is not the algorithmic character as such but rather the ﬁniteness; would Brouwer really have said that a ﬁnite algorithm does permit said reduction but the instruction to make a speciﬁed ﬁnite number of choices does not? That is very unlikely, as what matters here is only that one is dealing with a construction that can be completed; whether the process of completion proceeds by algorithm or by choice does not make a relevant diﬀerence for that. 34

A non-phenomenological recent study of intersubjectivity and intuitionism is that by Placek [174]. 35

As Kant has it:

Intuition and concepts constitute, therefore, the elements of all our knowledge, so that neither concepts without an intuition in some way corresponding to them, nor intuition without concepts, can yield knowledge. [138, B74] (‘Anschauung und Begriﬀe machen also die Elemente aller unsrer Erkenntnis aus, so daß weder Begriﬀe, ohne ihnen auf einige Art korrespondierende Anschauung, noch Anschauung ohne Begriﬀe, ein Erkenntnis abgeben können.’ [137, B74]) and Therefore all concepts, and with them all principles, even such as are possible a priori, relate to empirical intuitions, that is, to the data for a possible experience. Apart from this relation they have no objective validity, and in respect of their representations are a mere play

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of imagination or of understanding. [138, B298] (‘Also beziehen sich alle Begriﬀe und mit ihnen alle Grundsätze, so sehr sie auch a priori möglich sein mögen, dennoch auf empirische Anschauungen, d.i. auf Data zur möglichen Erfahrung. Ohne dieses haben sie gar keine objektive Gültigkeit, sondern sind ein bloßes Spiel, es sei der Einbildungskraft, oder des Verstandes, respective mit ihren Vorstellungen.’ [137, B298]). 36

Husserl sketches a very similar picture in section 42 (especially subsection a) of Formal and Transcendental Logic [112]. 37

‘6. Het wiskundig zien van die taal; dezen stap uitdrukkelijk te doen, is iets essentieels bij Hilbert in onderscheid van Peano en Russell.’ [23, p. 174] The reference to Hilbert is the 1904 Heidelberg lecture [91]. 38

‘[Cantor] spreekt hier van iets, wat zich niet laat denken, d.w.z. zich niet wiskundig laat opbouwen; immers een geheel, geconstrueerd met behulp van “en zoo voort” laat zich alleen denken, als dat “en zoo voort” op een ordetype ω van gelijke dingen slaat; maar het “en zoo voort” hier slaat niet op een ordetype ω, en ook niet op gelijke dingen. Cantor verliest dus hier den wiskundigen bodem.’ [23, pp. 145–146, original emphasis]. 39

One has to keep in mind that there may be a diﬀerence between what Husserl actually says, and what his position really commits him too. 40 As suggested by Husserl’s Ideas I [115], sections 142–144; these sections are also appealed to in Lohmar’s criticism [152, p. 195] of Becker’s approach [10, pp. 387ﬀ., 414]. Lohmar writes:

It becomes clear that Becker’s essay is determined to appropriate Husserl as a philosopher of the intuitionism of Brouwer and Weyl . . . Incidentally, Becker is well aware that, by basing his reading on intuitionistic concepts, he is changing the meaning of Husserl’s intentions. [trl. mine] (‘Es wird deutlich, daß Beckers Schrift darauf angelegt ist, Husserl als Philosoph des Brouwer-Weylschen Intuitionismus zu vereinnahmen . . . Becker ist sich seiner Umdeutung der Husserlschen Intentionen durch die Lesart gemäß der intuitionistischen Begriﬀe im übrigen durchaus bewußt, vgl. p. 413 [10].’) This quotation is also interesting in itself, because to speak of one ‘BrouwerWeylsche Intuitionismus’ is problematic precisely when it comes to choice sequences, as I argue in section 5.3. 41

Item 013133 in the Gödel Nachlaß, Firestone Library, Princeton. Compare [147, p. 202].

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42 The ways in which these truths are presented, or are presented best, may be culturally dependent. This belongs to the ﬁeld of ethno-mathematics. 43

An early and detailed study of the inﬂuence of Husserl’s mathematical training on his phenomenology was written by Picker [173]. 44

‘Mein großer Lehrer Weierstraß war es, der in mir in meinen Studienjahren durch seine Vorlesungen über Funktionstheorie das Interesse für eine radikale Begründung der Mathematik weckte. Ich gewann ein Verständnis für seine Bemühungen, die Analysis, die sosehr ein Gemisch rationalen Denkens und eines irrationalen Instinkts und Takts war, in eine rationale Theorie zu verwandeln.’ 45

‘Wenn wir ebenso, um allen Streit etwa zwischen Platonismus und Aristotelismus unbekümmert, von Zahlen “der” Anzahlenreihe, von Sätzen, von reinen Gattungen und Arten formaler wie materialer Gegebenheiten reden, wie es z.B. der Mathematiker als Arithmetiker oder Geometer tut, so sind wir noch keine Erkenntnistheoretiker; wir folgen der Evidenz, die uns solche “Ideen” gibt; . . . Aber freilich ist es ein erster und durchaus notwendiger Schritt für die Stellung vernünftiger erkenntnistheoretischer Fragen, so zu reden und Ideen als Gegebenheiten nicht gleich wegreden zu lassen.’ [96, p. 131]. 46

‘Klarheit darüber zu geben, was die innere Erfahrung unmittelbar zeigt; also nicht eine Genesis der Tatsachen, sondern zunächst erst Beschreibung des Gebietes. Dieser Teil ist nicht psychophysiologisch, sondern rein psychologisch. Vorweg müssen wir wissen, wie die Tatsachen aussehen: und dies zeigt ein innerer Blick ins Psychische.’ [181, p. 24]. From unpublished lecture notes by Brentano. 47

‘Jedes psychische Phänomen ist durch das charakterisiert, was die Scholastiker des Mittelalters die intentionale (auch wohl mentale) Inexistenz eines Gegenstandes genannt haben, und was wir, obwohl mit nicht ganz unzweideutigen Ausdrücken, die Beziehung auf einen Inhalt, die Richtung auf ein Objekt (worunter hier nicht eine Realität zu verstehen ist), oder die immanente Gegenständlichkeit nennen würden. Jedes enthält etwas als Objekt in sich.’ [20, Buch 2, Kap. I, section 5]. Oskar Kraus, whose edition of Brentano is used here, adds a footnote: ‘Br. gebraucht hier “Inhalt” gleichbedeutend mit “Objekt”. Später bevorzugt er den Terminus “Objekt” ’. 48

‘Der Inhalt als solcher ist ein individuelles, psychisches Datum, ein jetzt und hier Seiendes. Die Bedeutung aber ist nichts Individuelles, nichts Reales, nie und nimmer ein psychisches Datum. Denn sie ist identisch dieselbe “in” einer unbegrenzten Mannigfaltigkeit individuell und real getrennter Akte. Daß es so etwas wie Meinen identisch desselben “in” verschiedenen Akten geben kann und gibt – das ist die aller Erkenntnis zugrundeliegende, ihr überhaupt

Notes

137

erst Sinn verleihende und durch Evidenz bezeugte Urtatsache. – Der Inhalt wohnt der Vorstellung real ein, die Bedeutung nur funktionell; es wäre absurd, sie als reales Stück oder als Teil der Vorstellung zu fassen.’ [116, p. 350]. In contrast to Brentano in the quotation in note 47, by ‘Inhalt’ Husserl does not mean ‘object’ but ‘content’, i.e., all that is noticeable in a particular act. 49 ‘Ich arbeite an einer größeren Schrift, welche gegen die subjektivistischpyschologisirende Logik unserer Zeit gerichtet ist (also gegen den Standpunkt, den ich als Brentanos Schüler früher selbst vertreten habe).’ 50

‘Dabei wird es wohl nicht an Gelegenheit fehlen, den fragl[ichen] Unterschied mit zu behandeln’. 51

Husserl later acknowledged his great indebtedness to Lotze in his own overcoming of psychologism. In 1913, he wrote: For the fully conscious and radical turn and for the accompanying ‘Platonism’, I must credit the study of Lotze’s logic. Little as Lotze himself had gone beyond [pointing out] absurd inconsistencies and beyond psychologism, still his brilliant interpretation of Plato’s doctrine of Ideas gave me my ﬁrst big insight and was a determining factor in all further studies. Lotze spoke already of truths in themselves, and so the idea suggested itself to transfer all of the mathematical and a major part of the traditionally logical [world] into the realm of the ideal. With regard to the logic which before I had interpreted psychologistically and which had perplexed me as a mathematical logician, I, thanks to a fortunate circumstance, no longer needed extensive and detailed deliberations as to its separation from the psychological [sphere]. [114, p. 36]. (‘Die voll bewusste und radikale Umwendung und den mit ihr gegebenen “Platonismus” verdanke ich dem Studium der Logik Lotzes. So wenig Lotze selbst über widerspruchsvolle Inkonsequenzen und über den Psychologismus herausgekommen war, so steckte seine geniale Interpretation der platonischen Ideenlehre mir ein erstes grosses Licht auf und bestimmte alle weiteren Studien. Schon Lotze sprach von Wahrheiten an sich und so lag der Gedanke nahe, alles Mathematische und ein Hauptstück des traditionell Logischen in das Reich der Idealität zu versetzen. Ich brauchte hinsichtlich der Logik, die ich früher psychologistisch gedeutet hatte, und die mich als mathematischen Logiker in Verlegenheit gesetzt hatte, dank einem glücklichen Umstande nicht mehr lange und ins einzelne gehende Überlegungen hinsichtlich der Scheidung von Psychologischem.’ [96, p. 129]). 52

‘Real ist das Individuum mit all seinen Bestandstücken; es ist ein Hier und Jetzt. Als charakteristisches Merkmal der Realität genügt uns die Zeitlichkeit.

138

Notes

Reales Sein und zeitliches Sein sind zwar nicht identische, aber umfangsgleiche Begriﬀe . . . Soll aber Metaphysisches ganz ausgeschlossen bleiben, so deﬁniere man Realität geradezu durch Zeitlichkeit. Denn worauf es hier allein ankommt, das ist der Gegensatz zum unzeitlichen “Sein” des Idealen.’ [121, p. 129]. 53

‘Geradezu entscheidend für eine widerspruchsfreie Erkenntnistheorie und Philosophie überhaupt ist es aber, daß man endlich die prinzipielle Sonderung vollzieht zwischen der rein immanenten Phänomenologie und Kritik der Erkenntnis, die sich von allen über den Inhalt des Gegebenen hinausgehenden Suppositionen freihält, und der empirischen Psychologie, welchem auch da, wo sie bloß beschreibt, solche Suppositionen macht, und daß man demgemäß nicht, wie es gewöhnlich geschieht, die Fragen des erkenntniskritischen Ursprungs und die des psychologischen Ursprungs vermengt.’ [116, pp. 207–208]. 54

‘Und während ich mich mit den Entwürfen zur Logik des mathematischen Denkens und insbesondere des mathematischen Kalküls abmühte, peinigten mich die unbegreiﬂich fremden Welten: die Welt des rein logischen und die Welt des Aktbewußtseins, wie ich heute sagen würde, des Phänomenologischen und auch Psychologischen. Ich wußte sie nicht in eins zu setzen, und doch mußten sie zueinander Beziehung haben und eine innere Einheit bilden.’ [101, p. 294]. 55

‘Man vergesse nicht, dass im “Kontinuum” der reellen Zahlen die einzelnen Elemente genau so isoliert gegeneinander stehen wie etwa die ganzen Zahlen.’ [237, ch. 2, section 6, n. 2]. 56

The history of the debate about the atomistic conception is treated by e.g., Weyl [238, 239], Laugwitz [151], and Bell [14]. 57

‘Waar dus in die oer-intuïtie continu en discreet als onafscheidelijke complementen optreden, beide gelijkgerechtigd en even duidelijk, is het uitgesloten, zich van een van beide als oorspronkelijke entiteit vrij te houden, en dat dan uit het op zichzelf gestelde andere op te bouwen; immers het is al onmogelijk, dat andere op zichzelf te stellen.’ [23, p. 8]. 58

‘Het continuum als geheel was ons echter intuïtief gegeven; een opbouw er van, een handeling die “alle” punten er van geïndividualiseerd door de mathematische intuïtie zou scheppen, is ondenkbaar en onmogelijk.’ [23, p. 62]. 59

‘Die Eigenschaft der Größen, nach welcher an ihnen kein Teil der kleinstmögliche (kein Teil einfach) ist, heißt die Kontinuität derselben. Raum und Zeit sind quanta continua, weil kein Teil derselben gegeben werden kann, ohne ihn zwischen Grenzen (Punkten und Augenblicken) einzuschließen, mithin nur so, daß dieser Teil selbst wiederum ein Raum, oder eine Zeit ist. Der Raum besteht also nur aus Räumen, die Zeit aus Zeiten. Punkte und Augenblicke sind nur Grenzen, d.i. bloße Stellen ihrer Einschränkung; Stellen aber setzen jederzeit jene Anschauungen, die sie beschränken oder bestimmen

Notes

139

sollen, voraus, und aus bloßen Stellen, als aus Bestandteilen, die noch vor dem Raume oder der Zeit gegeben werden könnten, kann weder Raum noch Zeit zusammengesetzt werden.’ 60

Van Dalen no longer holds this view, see [63, p. 565].

61

‘Die Einführung der Mengenkonstruktion, auf welcher also die fertige überabzählbare Vielfachkeit des Kontinuums beruht, bedarf nach stattgefundener Besinnung auf die mathematische Urintuition der Zweiheit, welche dem gesamten Intuitionismus zugrundliegt, keiner weiteren Besinnung . . . Die anfangs erwähnte Betrachtung des Kontinuums als reine Anschauung a priori nach Kant und Schopenhauer [behauptet] sich im Lichte des Intuitionismus im wesentlichen.’ [36, p. 6]. 62

‘In continuum voordracht, aan het slot van I toevoegen, dat het continuum dus toch weer uit de oerintuïtie onmiddelijk gegeven is, juist als bij Kant en Schopenhauer.’ 63

Brouwer seems to have accepted choice sequences as objects in intuitionistic mathematics in 1914 [27, p. 79]. But he ﬁrst realised how to exploit them while giving a course on set theory in 1916–1917, for the essential principle of continuity is scribbled in the margin of those lecture notes [60, pp. 240–241]. Troelstra [221] describes Brouwer’s development of choice sequences; Jervell [134], the introduction of choice sequences in relation to the debate on the axiom of choice. 64

‘Und da möchte ich Sie nun dringend bitten, dass Sie nicht die Expropriation kontinuieren, die die deutsche referierende mathematische Literatur an mir verübt hat, indem sie mich dasjenige, was mein ausschliessliches persönliches geistiges Eigentum ist, mit Poincaré, Kronecker und Weyl teilen lässt.’ [58, p. 30] 65

An observation stressed by Robert Tragesser in a contribution to the mailing list FOM (Foundations of Mathematics) of March 4, 1998 (archive at http://www.cs.nyu.edu/mailman/listinfo/fom): Here I present a reasonable alternative to the Byzantine (and I think wrong) Dummett/Tennant ‘rescue’ of Brouwer. It rests on the argument that the foundational problems dropping out of the ﬁnite/inﬁnite were not as central to Brouwer as those dropping out of discrete (or set)/continua. Regarding Dummett, see also note 144 on p. 151. 66

‘Les nouveautés même nouvelles enfantaient des conséquences très anciennes.’

140

Notes

67 ‘Denn Brouwers Auﬀassung, daß sich die mathematischen Tatsachen mit der mathematischen Erkenntnis wandeln, schließt ein, daß es ein zu Erkennendes gebe, welches erst durch die darauf bezogene Erkenntnis geschaﬀen wurde, was dem Wesen der Erkenntnis zuwiderläuft. Denn jede Erkenntnis setzt . . . einen unabhängig von seinem Erkanntwerden als bestehend zu denkenden Gegenstand voraus.’ [139, p. 65]. 68 ‘Brouwer selbst gelangt zu einer Bestimmung der reellen Zahl, mit der die hier gegebene starke Ähnlichkeit aufweist, sobald man von der bei ihm mitauftretenden zeitlichen Interpretation absieht. Deren Ausschaltung aber ist unbedingt erforderlich, wenn man den speziﬁschen Sinn mathematischer Gegenstände präzis erfassen will. Die “Irrationalzahl” ist – darüber muß man sich völlig klar sein – keineswegs ein “Werdendes”.’ [139, p. 128n. 2]. 69

‘Derjenige Teil der Brouwerschen Lehre aber, der wesentlich auf der Einführung des Zeitbegriﬀes in die Mathematik basiert, ist abzulehnen.’ [139, p. 68]. 70

‘Am Prinzip aller Prinzipien: daß jede originär gebende Anschauung eine Rechtsquelle der Erkenntnis sei, daß alles, was sich uns in der “intuition” originär, (sozusagen in seiner leibhaften Wirklichkeit) darbietet, einfach hinzunehmen sei, als was es sich gibt, aber auch nur in den Schranken, in denen es sich da gibt, kann uns keine erdenkliche Theorie irre machen.’ [115, p. 51]. 71

‘Die Evidenz irrealer, im weitesten Sinne idealer Gegenstände ist in ihrer Leistung völlig analog derjenigen der gewöhnlichen, sogenannten inneren und äußeren Erfahrung, der man allein – ohne einen anderen Grund als den eines Vorurteils – die Leistung einer ursprünglichen Objektivierung zutraut . . . Ebenso, sagen wir, gehört zum Sinn eines irrealen Gegenstandes die ihm zugehörige Identiﬁzierbarkeit auf Grund der ihm eigenen Weisen der Selbsterfassung und Selbsthabe. Der Leistung nach ist sie also wirklich so etwas wie eine “Erfahrung”.’ [112, pp. 163–164]. 72

Related to this objection is the point made by Dummett (in a discussion of translating a diﬀerent system for choice sequences, CS, into IDB1 ) [70, p. 222] that ‘[No one] would seek to construct, e.g., a theory of real numbers in terms of real number generators taken as choice sequences if these appeared only in the thorough disguise provided by the translation into IDB1 ’. 73

The following discussion of Weyl’s eﬀort is much indebted to van Dalen’s ‘Hermann Weyl’s intuitionistic mathematics’ [57]. 74

‘Die Arbeiten Brouwers sind schwer lesbar, und der Intuitionismus ist deshalb mehr durch einige Arbeiten Weyls bekannt geworden.’ 75

‘Als schaﬀender Neo-Intuitionist kommt neben Brouwer ausschliesslich dessen Schüler Heyting in Frage, der die geometrische Axiomatik auf der

Notes

141

Grundlage des neunen Intuitionismus neu gestaltet hat. Weyl aber kann bis heute in diesem Zusammenhange kaum erwähnt werden: zwar hat er als erster, halbverstehender Nachfolger Brouwers einige Popularisierungsversuche publiziert, aber die in derselben erhaltenen eigenen Zutaten sind sämtlich falsch und führen das Publikum irre.’ 76 Grattan-Guinness [78] mentions that, in opposition to Hilbert, Brouwer and Weyl each had alternative versions of the continuum, and then adds: ‘Perhaps sensing weakness in the camp, [Fraenkel] [73] noted diﬀerences between the two in a long piece for the Jahrbuch [über die Fortschritte der Mathematik]’. But note that Fraenkel does not bring out any diﬀerence not already mentioned by Weyl himself. 77

Kreisel to Heyting, Princeton, February 4, 1963. Heyting Nachlaß, Rijksarchief Noord-Holland, Haarlem, item Bkre 630204. 78

‘Das “es gibt” verhaftet uns dem Sein und dem Gesetz’ [238, p. 52].

79

‘Jede Anwendung der Mathematik muß ausgehen von gewissen, der mathematischen Behandlung zu unterwerfenden Objekten, welche durch eine Anzahl Charaktere sich voneinander unterscheiden lassen; die Charaktere sind die natürliche Zahlen. Durch das symbolische Verfahren, das jene Objekte durch ihre Charaktere ersetzt, ist der Anschluß an die reine Mathematik und ihre Konstruktionen erreicht. So liegt der Punktgeometrie auf der geraden Linie das System der oben erwähnten Dualintervalle zugrunde, die wir durch zwei ganzzahlige Charaktere kennzeichnen konnten.’ [238, pp. 58–59]. 80

‘Die aus diesen allgemeinen Sätzen zu gewinnenden eigentlichen Urteile entstehen dadurch, daß . . . für die in freier Entwickelung begriﬀene Wahlfolge aber ein Gesetz φ, das eine einzelne Zahlfolge ins Unendliche hinaus bestimmt [eingesetzt wird].’ [238, p. 58]. 81

[157, p. 94].

82

[238, p. 50].

83

[238, p. 58].

84

[238, p. 52].

85

[238, p. 58].

86

‘ Es kann sich ereignen, daß es im Wesen einer werdenden Folge liegt, einer Folge, in welcher jeder einzelne Wahlschritt völlig frei ist, daß sie die Eigen¯ [i.e., not-E] besitzt. Wie derartige Wesenseinsichten zu gewinnen sind, schaft E das auseinanderzusetzen ist hier nicht der Ort. Nur sie liefert uns einen Rechtsgrund dafür, daß wir, so uns jemand ein Gesetz φ vorlegt, ihm ohne Prüfung

142

Notes

auf den Kopf zu sagen können: Die durch dieses Gesetz ins Unendliche hinaus bestimmte Folge hat nicht die Eigenschaft E.’ [238, p. 52, original emphasis]. 87

‘Es sei aber noch einmal betont, daß in den Theoremen der Mathematik von Fall zu Fall einzelne bestimmte derartige Funktionen auftreten, niemals aber allgemeine Sätze über sie aufgestellt werden. Die allgemeine Formulierung dieser Begriﬀe ist deshalb auch nur erforderlich, wenn man sich über Sinn und Verfahren der Mathematik Rechenschaft gibt; für die Mathematik selber, den Inhalt und ihrer Lehrsätze, kommt sie gar nicht in Betracht . . . Denn diese Sätze sind, sofern sie sich selbst genügen und nicht rein individuelle Urteile sind, allgemeine Aussagen über Zahlen oder Wahlfolgen von Zahlen, nicht aber über “Funktionen”.’ [238, p. 66]. 88

‘Mengen aber von Funktionen und Mengen von Mengen wollen wir uns ganz aus dem Sinne schlagen. Kein Platz ist da in unserer Analysis für eine allgemeine Mengenlehre, so wenig wie für generelle Aussagen über Funktionen . . . In den hier gezogenen radikalen Konsequenzen stimme ich, soviel ich verstehe, nicht mehr ganz mit Brouwer überein. Beginnt er doch sogleich mit einer allgemeinen Funktionenlehre (unter den Namen “Menge” tritt bei ihm auf, was ich hier als functio continua bezeichne), betrachtet Eigenschaften von Funktionen, Eigenschaften von Eigenschaften usf. und wendet auf sie das Identitätsprinzip an. (Mit vielen seiner Aussagen gelingt es mir nicht, einen Sinn zu verbinden.)’ [238, p. 70]. 89

‘Arithmetik und Analysis enthalten lediglich allgemeine Aussagen über Zahlen und frei werdende Folgen; keine allgemeine Funktionen- und Mengenlehre von selbständigem Inhalt!’ [238, p. 71]. 90

‘Die aus diesen allgemeinen Sätzen zu gewinnenden eigentlichen Urteile entstehen dadurch, daß . . . für die in freier Entwickelung begriﬀene Wahlfolge aber ein Gesetz φ, das eine einzelne Zahlfolge ins Unendliche hinaus bestimmt [eingesetzt wird].’ [238, p. 58, original emphasis]. 91

It is also for this reason that I will not discuss here classical approximations to various notions of choice sequence, a topic on which Joan Moschovakis has worked; e.g. [164]. 92

‘In der Einschränkung des Objektes der Mathematik sind Sie tatsächlich radikaler als ich.’ [57, p. 167]. 93 94

[157, p. 106], [238, p. 66].

‘Durch den Schluss des zweiten Absatzes von S. 34 scheint mir den ganzen Zweck ihrer Schrift gefährdet zu werden. Nachdem sie eben den schlafenden aufgeruttelt haben sagt er sich hier: “Also der Verf[asser] gibt zu, dass die wirklichen mathematischen Sätze von seinen Darlegungen nicht berührt werden? Dann soll er mich weiter nicht stören!” und wendet sich ab und schläft weiter.

Notes

143

Damit tun Sie aber unserer Sache Unrecht, denn mit dem Existenszsatz des Verdichtungspunktes einer unendlichen Punktmenge wird doch gleichzeitig manchem klassischen Existenztheorem einer Minimalfunktion wie auch dem Existenzialsatze der geodätischen Linie bei Fortlassung der zweiten Diﬀerenzierbarkeitsbedingungen der Boden entzogen!’ [57, p. 167]. 95 ‘Der Begriﬀ der Folge schwankt, je nach der logischen Verbindung, in welcher er auftritt, zwischen “Gesetz” und “Wahl”, “Sein” und “Werden”.’ [238, p. 71]. 96

‘Für mich ist “werdende Folge” weder das eine noch das andere; man betrachte die Folgen vom Standpunkte eines machtlosen Zuschauers, der nichts darüber weiss, in wiefern die Ergänzung frei gewesen ist.’ The reprint, with Brouwer’s marginalia on it, is in the Brouwer Archive in Utrecht. 97

‘Eine Menge ist ein Gesetz, auf Grund dessen, wenn immer wieder ein willkürlicher Ziﬀerkomplex der Folge ζ [i.e., the natural number sequence] gewählt wird, jede dieser Wahlen entweder ein bestimmtes Zeichen, oder nichts erzeugt, oder aber die Hemmung des Prozesses und die deﬁnitive Vernichtung seines Resultates herbeiführt, wobei für jedes n nach jeder ungehemmten Folge von n − 1 Wahlen wenigstens ein Ziﬀernkomplex angegeben werden kann, der, wenn er als n-ter Ziﬀernkomplex gewählt wird, nicht die Hemmung des Prozesses herbeiführt. Jede in dieser Weise von der Menge erzeugte Zeichenfolge (welche also im allgemeinen nicht fertig darstellbar ist) heisst ein Element der Menge. Die gemeinsame Entstehungsart der Elemente einer Menge M werden wir ebenfalls kurz als die Menge M bezeichnen.’ 98

‘Ein Gesetz, das aus einer werdenden Zahlfolge eine von dem Ausfall der Wahlen abhängige Zahl n erzeugt, ist notwendig solcher Art, daß die Zahl n festgelegt ist, sobald ein gewisser endlicher Abschnitt der Wahlfolge fertig vorliegt, und sie bleibt dieselbe, wie sich nun auch die Wahlfolge weiter entwickeln möge.’ [238, p. 51]. 99

‘Nach diesem Gesetz tritt bei einer werdenden Folge, sie mag sich entwickeln wie sie will, stets einmal der Augenblick ein, wo sie eine Zahl aus sich gebiert. Das ist das Merkmal, welches allein wesentlich ist für den Begriﬀ der f[unctiones].m[ixtae].’ [238, p. 64]. 100

‘Ein Gesetz, das jedem Elemente g von C ein Element h von A zuordnet, muss nämlich das Element h vollständig bestimmt haben nach dem Bekanntwerden eines gewissen Anfangssegmentes α der Folge von Ziﬀernkomplexen von g. Dann aber wird jedem Elemente von C, welches α als Anfangssegment besitzt, dasselbe Element h vom A zugeordnet.’

144

Notes

101

This is an application of Ramsey’s maxim:

In such cases it is a heuristic maxim that the truth lies not in one of the two disputed views but in some third possibility which has not yet been thought of, which we can only discover by rejecting something assumed as obvious by both the disputants. [177, pp. 115-116]. 102

Robert Tragesser informed me that in 1966, Kreisel told him the following. He had got the idea expressed in this quotation from Gödel, who said that he was inspired to it by reading Husserl. It seems to me there are two likely places in Husserl that Gödel may have been referring to, sections 18–23 of Ideas I, and section 16 of the Britannica article, which had become available in 1962; see note 104 below. 103

Compare also Tragesser’s formulation of this idea [217, pp. 293–294].

104

In the Britannica article, one can ﬁnd a passage that comes very close to what Kreisel writes. Husserl there speaks of oppositions such as between rationalism (Platonism) and empiricism, relativism and absolutism, subjectivism and objectivism, ontologism and transcendentalism, psychologism and anti-psychologism, positivism and metaphysics, teleological and causal interpretations of the world. and adds the comment Throughout all of these, [one ﬁnds] justiﬁed motives, but throughout also half-truths or impermissible absolutizing of only relatively and abstractively legitimate one-sidedness. [143, p. 319, trl. modiﬁed] (‘Gegensätze wie die zwischen Rationalismus (Platonismus) und Empirismus, Relativismus und Absolutismus, Ontologismus und Transzendentalismus, Psychologismus und Antipsychologismus, Positivismus und Metaphysik, teleologischer und kausalistischer Weltauﬀassung. Überall berechtigte Motive, überall aber Halbheiten oder unzulässige Verabsolutierungen von nur relativ und abstraktiv berechtigten Einseitigkeiten.’ [103, p. 300]) 105 ‘Die Philosophie darf den tatsächlichen Gebrauch der Sprache in keiner Weise antasten, sie kann ihn am Ende also nur beschreiben. Denn sie kann ihn auch nicht begründen. Sie läßt alles wie es ist. Sie läßt auch die Mathematik wie sie ist, und keine mathematische Entdeckung kann sie weiterbringen. Ein “führendes Problem der mathematischen Logik” ist für uns ein Problem der Mathematik, wie jedes andere.’ [243, I.124].

Notes

145

106 ‘Das Ziel, die Mathematik sicher zu begründen, ist auch das meinige; ich möchte der Mathematik den alten Ruf der unanfechtbaren Wahrheit, der ihr durch die Paradoxien der Mengenlehre verloren zu gehen scheint, wiederherstellen; aber ich glaube, daß dies bei voller Erhaltung ihres Besitzstandes möglich ist. Die Methode, die ich dazu einschlage, ist keine andere als die axiomatische.’ [92, p. 160]. 107 The thesis that genetic phenomenology (usually not recognised as such) performs crucial functions in the conception and revisions of any science, and that analysis of a science should beneﬁt from being aware of these functions, has been formulated sharply and programmatically before by Rota [185]. The present argument, although written independently, in eﬀect works out that program for the particular case of mathematics, and reﬁnes it by adding the strong-weak distinction. 108

‘Sie will dem Spezialforscher nicht ins Handwerk pfuschen, sondern nur über Sinn und Wesen seiner Leistungen in Beziehung auf Methode und Sache zur Einsicht kommen . . . Erst die philosophische Forschung ergänzt die wissenschaftlichen Leistungen des Naturforschers und Mathematikers so, daß sich reine und echte theoretische Erkenntnis vollendet. Die ars inventiva des Spezialforschers und die Erkenntniskritik des Philosophen, das sind ergänzende wissenschaftliche Betätigungen, durch welche erst die volle, alle Wesensbeziehungen umspannende theoretische Einsicht zustande kommt.’ [113, pp. 255–6] Lohmar [152, p. 197n. 21] recalls the discussion of the ‘ars inventiva’ by Descartes and Leibniz. 109

‘Alles, was uns die Wissenschaften von den Onta, die rationalen und empirischen Wissenschaften (im erweiterten Sinn können sie alle “Ontologien” heißen, sofern es sich zeigt, daß sie auf Einheiten der “Konstitution” gehen) darbieten, “löst sich in Phänomenologisches auf” . . . Was hier zu leisten ist . . . , was wir hier unter den Titel “Rückgang auf das konstituierende absolute Bewußtsein” im Auge haben, setzt voraus eine transzendentale Phänomenologie.’ [99, p. 78]. 110

These were written at the other end of a period during which Husserl reconceived and developed phenomenology, and accordingly the simultaneous use of these early and late texts may need some justiﬁcation. Therefore I mention the following details. Husserl’s stenographic manuscript of Ideas III is from 1912; Edith Stein transcribed it in 1916; Ludwig Landgrebe typed her transcription in 1924–1925. Contrary to the manuscript of Ideas II, Husserl hardly made any changes or additions to the manuscript during these years [98, pp. xv, xvi]. This accords with the judgement one forms from reading these texts, that Husserl had not essentially altered his ideas on this speciﬁc subject by the time he wrote the Britannica article and Formal and Transcendental Logic.

146 111

Notes

All versions, and a discussion of the textual diﬀerences, are in [103].

112

1.1. ‘Die transzendentale Phänomenologie ist die Wissenschaft von allen erdenklichen transzendentalen Phänomenen.’; 1.2. ‘Alles Seiende schöpft sein Seinssinn aus intentionaler Konstitution.’; 1.3. Therefore, ‘Diese Phänomenologie [ist] eo ipso die absolute universale Wissenschaft von allem Seienden’; 1.3*. ‘Unter Heranziehung und Erweiterung des traditionellen Ausdrucks können wir auch sagen: die transzendentale Phänomenologie ist die wahre, die wirklich universale Ontologie.’ [103, pp. 519–520]. 113

‘Schon Leibniz hatte die fundamentale Einsicht, daß für eine echte theoretische Erkenntnis und Wissenschaft die Erkenntnis der Möglichkeiten derjenigen der Wirklichkeiten vorangehen muß. Demgemäß fordert er für jederlei reale und ideale Seinssphäre zugehörige apriorische Wissensschaften als solche der reinen Möglichkeiten (z.B. auch eine reine Grammatik, reine Rechtslehre usw.).’ [103, p. 520]. 114

‘Das Prinzipielle der mathematischen Methode [hat sich] als unzulänglich, die viel bewunderte mathematische Evidenz sich als eine der Kritik und der methodischen Reform bedürftige herausgestellt . . . Der Kampf um die “Paradoxien”, um die rechtmäßige oder Scheinevidenz der Grundbegriﬀe der Mengenlehre, der Arithmetik, der Geometrie, der reinen Zeitlehre usw. . . . hat es zutage gebracht, daß alle diese Wissenschaften ihrem ganzen Methodentypus nach noch nicht als Wissenschaften im vollen und echten Sinn gelten können: als Wissenschaften, die bis ins letzte methodisch durchsichtig sind und daher zu volkommenster Rechtfertigung jedes methodischen Schrittes befähigt und bereit.’ [103, pp. 520–521]. 115

‘Überall beobachten wir, wie bei der erkenntnistheoretischen Problematik sonst, die schon wiederholt erwähnte Verkehrtheit, daß man die Wissenschaften als etwas nimmt, das schon ist; als ob Grundlagenforschung nur eine nachkommende Klärung oder allenfalls eine diesen Wissenschaften selbst nicht wesentlich ändernde Besserung bedeuten sollte. In Wahrheit sind Wissenschaften, die Paradoxien haben, die mit Grundbegriﬀen operieren, die nicht aus der Arbeit der Ursprungsklärung und Kritik geschaﬀen sind überhaupt keine Wissenschaften, sondern bei aller ingeniösen Leistung bloß theoretische Techniken.’ [112, p. 189]. 116

‘Es gilt, die Wissenschaften auf ihren Einsicht und strenge Geltung verlangenden Ursprung zurückzuführen und sie in Systeme einsichtiger Erkenntnis zu verwandeln durch klärende, verdeutlichende, letzt-begründende Arbeit, die Begriﬀe und Sätze auf in der Intuition faßbare begriﬄiche Wesen selbst und die sachlichen Gegebenheiten selbst zurückzuführen, denen sie angemessenen Ausdruck geben, soweit sie wirklich Wahr sind.’ [99, pp. 96–97].

Notes

147

117 ‘So bedeutet ursprüngliche Besinnung ineins Näherbestimmung der bloß vage unbestimmten Vorzeichnung, Abhebung der aus assoziativen Überschiebungen herstammenden Vorurteile und Durchstreichung der mit der besinnlichen Erfüllung streitenden; also mit einem Wort Kritik der Echtheit und Unechtheit.’ [112, p. 14]. 118 The demand that all sciences be grounded in transcendental phenomenology raises the obvious question in what that phenomenology is to be grounded. Grounding is a matter of obtaining evidence, and the science of evidence is precisely transcendental phenomenology itself. So it can only be grounded in itself. Husserl is very much aware of this task, and came to see this need for reﬂexivity as the one crucial methodological demand. See, for example, Formal and Transcendental Logic [112, pp. 294:35–295:6]. A convincing account how transcendental phenomenology indeed is reﬂexive in the required sense, is in Tito’s chapter 5 (g) [215]. 119

‘Das Ziel der Klärung kann man im Sinn des schon Ausgeführten auch dahin fassen, daß sie den vorgegebenen Begriﬀ gewissermaßen von neuem schaffen, ihn aus der Urquelle der begriﬄichen Geltung, der Anschauung, speisen und ihm innerhalb der Anschauung die Teilbegriﬀe geben will, die zu seinem originären Wesen gehören.’ [99, p. 102]. 120

‘Sie geht aus von den theoretischen Gebilden, die uns in der Überschau die historische Erfahrung in die Hand gibt, also von dem, was ihren traditionellen Gehalt ausmacht, und versetzt sie zurück in die lebendige Intention der Logiker, aus der sie als Sinngebilde entsprangen.’ [112, p. 14]. 121

Although written independently of Tragesser’s ‘On the phenomenological foundations of mathematics’ [217], my argument for 2.5 can be read as spelling out the details of an argument found there (section 10). 122

If ‘square’ and ‘circle’ are taken as deﬁned rather than as primitive terms, a formal contradiction probably would show up. But a deﬁnition of these terms (with the intended interpretation in mind) would depend on conceptual analysis of squares and circles, so the appearance of a formal contradiction would also in this case depend on more than only logic. 123

‘Sehr vieles ist logisch möglich, was nicht wesensmöglich ist. Die Logik muß mit “unmöglichen Gegenständen” rechnen, d.h. mit wesensunmöglichen (viereckiger Kreis); je nachdem, was sie an vorgegebenen Merkmalen (etwa des Kreises) anerkennt, sind solche Gegenstände für sie möglich oder nicht. Die “Unmöglichkeit” solcher Gegenstände ist eben eine Seinsunmöglichkeit (etwa eine geometrische), nicht eine logische.’ Compare note 122. 124

‘[Wirklichkeit und Unwirklichkeit] spielen keine eigene Rolle im idealen Sein neben der Möglichkeit des Seins und Nichtseins. Sie sind mit ihr gesetzt,

148

Notes

sind ein selbstverständliches, besagen nicht “mehr” als das Seinkönnen und Nichtseinkönnen.’ 125

‘Denn Erkenntnis kann vom Sein ihres Gegenstandes aus überhaupt nicht impliziert werden. Das ideale Sein ist an sich nicht weniger indiﬀerent gegen die Idealerkenntnis, als das reale Sein gegen die Realerkenntnis.’ 126

It should be kept in mind that, phenomenologically, mathematical existence is not shown by proving a formal existential statement, but by obtaining evidence for the objects of the intended interpretation of that statement. So there is no worry that the completeness theorem (which is equivalent to ‘Every consistent theory has a model’) trivialises 2.5, for such a trivialisation would be an equivocation of the formal and the phenomenological senses of existence. The models exhibited in proofs of the completeness theorem hardly ever bear any resemblance to our intended interpretation. 127

‘Alle Existenzialurteile der Mathematik als apriorische Existenzialurteile sind in Wahrheit Existenzialurteile von Möglichkeiten.’ [124, p. 450]. 128

‘Hieraus erwächst eine universale Wissenschaftsidee, die einer formalen Mathematik im voll umfassenden Sinne, deren Universalgebiet sich fest umgrenzt als Umfang des obersten Formbegriﬀes Gegenstand-überhaupt oder des in leerster Allgemeinheit gedachten Etwas-überhaupt, mit allen in diesem Feld apriori erzeugbaren und daher erdenkbaren Ableitungsgestalten, die in immer neuer iterativer Konstruktion immer neue Gestalten ergeben. Solche Ableitungen sind neben Menge und Anzahl (endliche und unendliche), Kombination, Relation, Reihe, Verbindung, Ganzes und Teil, usw.’ [112, p. 82]. 129

‘Jedes sachhaltige Apriori . . . fordert zur kritischen Herstellung der echten Evidenz den Rückgang auf exemplarische Anschauung von Individuellem . . . Die Evidenz analytisch apriorischer Gesetze bedarf solcher bestimmten individuellen Anschauungen nicht, sondern nur irgendwelcher Exempel von Kategorialien, evtl. mit unbestimmt allgemeinen Kernen (wie wenn Sätze über Zahlen als Beispiele dienen), die zwar auf Individuelles intentional zurückweisen mögen, aber in dieser Hinsicht nicht weiter befragt und ausgelegt werden müssen.’ [112, p. 221]. 130

Two remarks:

1. Rosado Haddock arrives, via a diﬀerent route (a discussion of analyticity) at essentially the same conclusion as 2.5: ‘Actually, the laws of the possibility of constitution of categorial objectualities are precisely the laws of mathematical existence’ [184, p. 97]. He does not connect it to the issue of revisionism. (And whether these ‘laws of possibility’ are of a constructivistic nature (in the sense in which intuitionism is), is another question.) His remark is a special case of one that Husserl makes in the sixth Logical

Notes

149

Investigation: ‘The ideal conditions of categorial intuition in general are, correlatively regarded, the conditions of the possibility of the objects of categorial intuition, and of the possibility of categorial objects simpliciter.’ [111, p. 822] (‘Die idealen Bedingungen der Möglichkeit kategorialer Anschauung überhaupt sind korrelativ die Bedingungen der Möglichkeit der Gegenstände kategorialer Anschauung und der Möglichkeit von kategorialen Gegenständen schlechthin’ [122, pp. 718–719]). 2. Similarly, Gödel writes that one of the distinguishing features of mathematical objects is that they can be known (in principle) without using the senses (that is, by reason alone) for this very reason, that they don’t concern actualities about which the senses (the inner sense included) inform us, but possibilities and impossibilities. [74, p. 312n. 3] This remark dates from 1951, which presumably precedes Gödel’s study of Husserl by eight years [233, p. 121]. 131

‘Alle mathematischen Existentialsätze haben diesen modiﬁzierten Sinn: “es gibt” Dreiecke, Vierecke, Polygone aller weiter aufsteigenden Zahlen; “es gibt” regelmäßige Polyeder von 56, aber nicht von allen Zahlen von Seitenﬂächen. Der wahre Sinn ist nicht schlechthin ein “es gibt”, sondern: es ist a priori möglich, das es gibt.’ [124, p. 450]. 132 ‘Negative Existenzialsätze haben die Funktion, die ungültigen Begriﬀe, die wesenlosen Ausdrücke auszuscheiden.’ [99, p. 83]. 133

This objection was brought to my attention by Richard Tieszen.

134

‘Bei logischen Bedeutungen sehen wir nun, daß das Gedachte als solches (logische Bedeutung im noematischen Sinn) “widersinnig” sein kann, es, das doch innerhalb der Seinskategorie “logische Bedeutung” und, allgemeiner, “Noema” “existiert”, sein wirkliches Sein hat wie z.B. die Denkbedeutung “rundes Viereck” . . . Das Wesen ist etwas anderes als die Bedeutung. Das Wesen “rundes Viereck” gibt es nicht; aber um das urteilen zu können, ist vorausgesetzt, das “rundes Viereck” eine in dieser Einheitlichkeit seiende Bedeutung ist . . . Bedeutungen setzen und Gegenstände setzen ist zweierlei.’ [99, pp. 85–86, 89]. 135

‘gelten können die Wortbedeutungen als logische Wesen nur dann, wenn nach idealer Möglichkeit das sie in sich aktualisierende “logische Denken” einem “entsprechenden Anschauen” anpaßbar ist; bzw. wenn es ein entsprechendes durch intuition erfaßbares Wesen als entsprechendes Noema gibt, das durch den logischen Begriﬀ seinen getreuen “Ausdruck” ﬁndet.’ [99, p. 26, original emphasis].

150

Notes

136 ‘Ein geometrisches Urteil gilt nur, wenn die Idee, das Wesen Raum und Raumgestalt ist, oder, dem Umfang nach gesprochen, wenn eine Raumgestalt möglich ist.’ [99, p. 82; original emphasis]. 137

‘daß es im Raum seinem Wesen nach ein diesem Gestalt-begriﬀ (einer frei gebildeten logischen Bedeutung) entsprechenden geometrisches Wesen wahrhaft gibt . . . Jedes gültige geometrische Urteil setzt eidetische Einzelheiten (was äquivalent ist mit einer Setzung entsprechender Wesen als Gegenstände), die insgesamt das durch die gültig gesetzte regionale Idee umgrenzte Gebiet der Ontologie ausmachen.’ [99, pp. 82–83; original emphasis]. 138

‘Solche Kritik ist schöpferische Konstitution der betreﬀenden Gegenständlichkeiten in der Einheit einstimmiger Selbstgegebenheit und Schöpfung ihrer Wesen und Wesensbegriﬀe.’ [112, p. 188]. 139

‘Soll man sich etwa in der Beurteilung der Mathematik, deren Gesamtsinn ganz und gar von diesen [Grund]begriﬀen abhängt, an einen Hilbert halten oder an einen Brouwer, oder an wen sonst? Ist es so sicher, trotzdem gerade das heutzutage communis opinio ist, daß nicht die klassische Mathematik und ebenso Physik besser beraten war? Aber da werden wir nicht besser fahren. Sie war nie ein Fertiges sondern selbst im Werden, und so wiederholt sich die Schwierigkeit, die Unmöglichkeit einer eindeutigen Auswahl, die uns normbestimmend sein könnte. Indessen es zeigt sich bald, daß es in der Tat auf eine solche Auswahl durch Entscheidung für irgend eine Partei oder irgend einen maßgebenden Forscher wenig ankommt. Jeder, der Mathematik studiert hat, besitzt das allgemeine Phänomen, das da heißt Mathematik – Mathematik als diese exakte Wissenschaft, die in jeder Zeit im Werden ist und im Lauf der Gesamtzeit, in der sie war und noch ist, die eine im Werden jede Gegenwart gewordene und von Gegenwart zu Gegenwart doch einheitliche im Fortwerden, unerachtet aller nie fehlenden Diskrepanzen der forschenden Personen und ihrer begriﬄichen, ihrer theoretischen Leistungsgebilde.’ The existing transcription from the Gabelsberger was kindly corrected by Karl Schuhmann. 140

‘Untersuchungen, die über die wissenschaftsinterne Begründung, wie die intuitionistische Mathematik sie impliziert, hinausgehen und auf eine Klärung der Subjektivität als solcher abzielen.’ 141

At the time of Formal and Transcendental Logic [112, §77], Husserl has come to recognise the idealisations involved in the principle of the excluded middle – the classical mathematician’s ‘ﬁsts to box with’, as Hilbert once put it [93, p. 80]. That acclaimed law, Husserl says, is not grounded in a subjective law of evidence. But it is not clear that Husserl actually considered the idealisations involved unjustiﬁed.

Notes

151

142 Robert Sokolowski [200] has argued, against interpretations put forward by Mohanty, Gurwitsch and Føllesdal, that there is a diﬀerence between noematic analysis and meaning analysis as normally understood. Namely, the former would require the performance of the phenomenological reduction while the latter does not. For the present argument, the outcome of that discussion does not matter, as both alternatives show the same contrast with the critical eidetic analysis. (Richard Cobb-Stevens told me that Bachelard would have equated noematic analysis and meaning analysis.) 143 But Bachelard sees what Husserl does not, namely, that intuitionism is, in a sense, concerned with intentionality in mathematics. 144

Dummett’s project [65] is to justify intuitionistic mathematics on purely meaning-theoretical grounds, thus leaving out intuition altogether. From a phenomenological point of view, such an approach must be mistaken, but I will not work out the speciﬁc argument here. An argument, independent of phenomenology, to the eﬀect that a meaning-theoretical approach cannot make intuition dispensable, has been put forward by Parsons [169]. 145

I cannot agree with Schmit when he gives the following as one of his reasons for not discussing transcendental phenomenology in his work on Husserl’s philosophy of mathematics: In contrast to reﬂection immanent to science, which has a concrete inﬂuence on the methods of science, transcendental phenomenology in no way prejudges the content and methods of mathematics. This explains why transcendental philosophy is not taken into account in this work, which is ﬁrst of all concerned with Husserl’s concept of mathematics in the pregnant sense. [189, p. 144, trl. mine] (‘Im Gegensatz zur wissenschaftsimmanenten Reﬂexion, die einen positiven Einﬂuß auf die Methoden der Wissenschaft hat, präjudiziert die transzendentale Konstitution in keiner Weise über den Inhalt und die Methoden der Mathematik. Dies erklärt, weshalb die Transzendentalphilosophie keine Berücksichtigung in dieser Arbeit ﬁndet, die sich in erster Linie mit Husserls Mathematikbegriﬀ im prägnanten Sinn beschäftigt.’) I argue that, because of the entirely formal nature of pure mathematics, transcendental phenomenology does have an a priori contribution to make. 146

[181] is a historical study of this and other discussions between the pretranscendental Husserl and various other members of the school of Brentano. 147

‘In den Logischen Untersuchungen schreibt Husserl dem, was er als Ideales von Realen unterscheidet, eine zeitlose Existenz zu, während das Reale zeitlich sei. Ich kann auch eine Unterscheidung dieser Art nicht berechtigt ﬁnden. Alles, so scheint mir, was wir als seiend denken, denken wir zeitlich. Was

152

Notes

schon daraus hervorgeht, daß wir es entweder als bestehend und beharrend oder als sich verändernd denken. Das im strengen Sinne Zeitlose wäre weder das eine noch das andere.’ [158, p. 328]. 148

‘Ja dabei habe ich doch nicht Martys Begriﬀ des Realen ?!’ [181, p. 232n. 1].

149

‘Aber weder, wo die Zeitbestimmung indeﬁnit bleibt, noch wo wir ausdrücklich jede Beschränkung der Temporalbestimmung aufheben, scheint es mir passend von einer zeitlosen Existenz zu sprechen. Was ist, ohne zu werden und zu vergehen, ist darum noch nicht zeitlos sondern eben zu jeder Zeit.’ 150

‘Zeitlosigkeit des Idealen bestritten’ [181, p. 232n. 2].

151

‘Es ist auf alle Zeiten bezogen, oder auf welche auch immer bezogen, immerfort absolut dasselbe’, in manuscript L II 13 from 1917/18, used for [124, p. 311]. 152

‘Eine solche Irrealität [hat] das zeitliche Sein der Überzeitlichkeit, der Allzeitlichkeit, die doch ein Modus der Zeitlichkeit ist.’, in manuscript F I 39 from 1920/21, used for [124, p. 313]. 153

Neither in work published during his lifetime, nor in the posthumous Experience and Judgement, nor in the full research manuscripts on which the relevant section of Experience and Judgement, 64c, was based. These manuscripts are L II 13 (1917/18), F I 39 (1920/21), and B III 5 (1929) [154, p. 65], [190, p. 69]. 154

‘Eben diese Identität als Korrelat einer in oﬀen endloser und freier Wiederholung zu vollziehenden Identiﬁzierung macht den prägnanten Begriﬀ des Gegenstandes aus.’ [124, p. 64]. 155

‘Der Gegenstand des Bewußtseins in seiner Identität mit sich selbst während des strömenden Erlebens kommt nicht von außen her in dasselbe hinein, sondern liegt in ihm selbst als Sinn beschlossen, und das ist als intentionale Leistung der Bewußtseinssynthesis.’ [126, p. 44]. 156 Internal time consciousness is not consciousness of physical or objective time. The latter is founded on the former, and depends on the constitution of an objective, outside world. Consciousness of internal time is just the awareness co-present in every intentional act, that other acts preceded it and that others will follow. A key question in the phenomenology of time is how the ﬂow of internal time is itself constituted [104], [82]. (The same distinction one ﬁnds in Brouwer’s thesis as well, [23, p. 99n. 1]). 157 ‘Die Leistungen der Synthesis im inneren Zeitbewußtsein sind die untersten, die alle anderen notwendig verknüpfen. Das Zeitbewußtsein ist die

Notes

153

Urstätte der Konstitution von Identitätseinheit überhaupt.’ [124, pp. 75–76] See also [126, p. 43]. 158

‘Gegebenheitszeit’ [124, pp. 303–305].

159

‘Wesenszeit’ [124, pp. 303–305].

160

‘eine Zeitform gehört zu ihm als der noematische Modus seiner Gegebenheitsweise’ [124, p. 309]. 161

‘eigentlich hat es keine Dauer als zu seinem Wesen gehörige Bestimmung’ [124, p. 311]. 162

In section 97 of Ideas I [115], Husserl still held that the noema is not ‘reell’, but as Küng [150] has pointed out, Husserl changed his mind on this. There are still diﬀerent interpretations of the noema, but I am convinced by Küng’s account: it seems more accurate descriptively. 163

Jahrbuch für Philosophie und phänomenologische Forschung. Eleven volumes appeared, between 1913 and 1930. 164

This reprint can be found in Husserl’s library. There are no marks or notes on it. Also in his possession was a reprint of Weyl’s paper from 1925, ‘The current epistemological situation in mathematics’ [239]. There are no notes written on that either, but many passages are marked with pencil lines in the margin; these marks may very well be Husserl’s. 165

‘Wie intensiv mein Freiburger Kreis für ihre Arbeiten interessiert ist, zeigt die nun fertig gewordene und der Fakultät vorgelegte Habilitationsschrift Dr. Becker’s, die ich eingehend studiert und höchst anerkennend recensiert habe.’ The Husserl–Weyl correspondence is in volume VIII of the Briefwechsel [128]. Van Dalen [54] and Tonietti [216] have commented on part of this exchange. 166 ‘Wirklich studieren kann ich das Werk jetzt noch nicht . . . , aber ich freue mich schon auf die Ferien u. lasse mir sogleich von einem ausgez[eichneten] math[ematischen] Schüler Referate erstatten u. spreche dann die Gedanken mit ihm durch.’ 167

‘Herrn Brouwer, den ich kennen zu lernen mich freue, werde ich freilich enttäuschen. Denn mir liegt z.Z[t]. das philosophisch-Mathematische etwas fern u. ich möchte darüber nicht vom Katheder sprechen, ich würde zu viel Zeit brauchen mich wieder hineinzuﬁnden – so viel ich darüber früher gearbeitet habe.’ 168

According to the editors of the Briefwechsel, a planned article ‘Questions of logical principle in foundational reseach in mathematics’ by Kaufmann for Husserl’s Jahrbuch.

154

Notes

169 ‘Da die Arbeit ins Mathematische geht, bitte ich sie hinsichtl[ich] dessen von einem Fachmathematiker genau durchlesen zu lassen. Das kleinste Versehen, selbst schon Abweichung von dem üblichen Darstellungsstil giebt bei den math[ematischen] Lesern ein großes Geschrei u. verdirbt dann mittelbar eine gute allgem[eine] Aufnahme . . . Da seit Jahren College Becker die Redaktionsgeschäfte für mich besorgt, kann ich nicht anders als ihm auch diese Arbeit vorzulegen, u. auch er ist Fachmathematiker.’ 170 ‘Vergessen Sie nicht für Ihre math[ematische] Abh[andlung] fachmäßige Nachprüfung von Seiten eines Mathematikers. Mit den Mathem[atikern] ist nicht gut Streit zu bekommen.’ 171

Husserl’s letter to Weyl quoted on p. 72, tells that Husserl studied Becker’s Habilitationsschrift intensively. That appears not to have been the case with Mathematical Existence. In the period in between these two works, Becker shifted his position from Husserlian to Heideggerian phenomenology. (Letter from Husserl to Heidegger, May 24, 1927: ‘Have you also read Becker’s work? Direct application of the Heideggerian ontology’ (‘Haben Sie Beckers Arbeit mitgelesen? Direkte Anwendung der Heid[eggersche]n Ontologie’ [128, IV:p. 143, trl. mine].) In a letter to Ludwig Landgrebe of October 1, 1931 [128, IV:p. 269], Husserl expresses his regret about this shift on his former assistant’s part. Was this why Husserl read the second half of Mathematical Existence only 10 years after its appearance [191, p. 484]? 172

‘Es ist eben gar nicht derselbe Begriﬀ der Folge, der einmal in willkürlicher Allgemeinheit und das andere Mal in eindeutig bestimmte Gesetzlichkeit verwandt werden kann.’ 173

However, in contrast to Weyl in 1921, Becker in 1927 was well aware that there is a wide variety of choice sequences in between lawlike and lawless [11, pp. 758–759]. 174

Recently, van Dalen gave an overview [59], and van Stigt presented a translation of Brouwer’s seminal paper ‘Leven, Kunst en Mystiek’ [48], [208]. 175

The demand for reﬂexivity and the question whether it can be satisﬁed were much debated among Kant’s contemporaries in reaction to his critical project; see [13]. There is a familiar argument to the eﬀect that no system can account for itself, based on a trilemma sometimes associated with the name of Münchhausen: either a vindication of the system appeals to its own principles, but that would be circular; or it appeals to other principles, but then these other principles come from outside and are unsystematic; or it appeals to no ﬁnal principles and never terminates, thus leading to an inﬁnite regress. In each of the three cases, the system fails to account for itself.

Notes

155

An interpretation of Kant showing his modest approach to this trilemma is given by O’Neill [168]. Tito [215, ch. 5] shows that within the context of Husserl’s phenomenology, the trilemma is false and the inﬁnite regress is averted. 176

For this objection, I am thankful to an anonymous reader.

177

In Brouwer’s philosophical expositions before ‘Consciousness, philosophy, and mathematics’, i.e. [22, 23, 35, 37], he has a somewhat wider concept of mathematics. There, mathematics is the study of all exact thought; the Dutch word for mathematics, ‘wiskunde’, literally means ‘the art of that which is certain’. It comprises causal attention and mathematical abstraction; in ‘Consciousness, philosophy, and mathematics’, Brouwer reserves ‘mathematics’ for the latter. The description of consciousness as a whole, however, seems no diﬀerent across ‘Consciousness, philosophy, and mathematics’ and the earlier papers. 178

‘wijsheid heft de splitsing op in subject en iets anders’, [24, p. 3].

179

‘in wijsheid [wordt] de verschijning van den tijd niet langer aanvaard.’ [24, p. 3]. 180

‘In wijsheid, die de splitsing opheft in subject en iets anders, is geen wiskundig intelligeeren’ [24, p. 3; original emphasis]. 181

Here I take it that for Brouwer, ‘wisdom’ and ‘mysticism’ are synonyms. Should it turn out that they are not, that would not weaken my criticism of Brouwer’s philosophy, as I employ this identiﬁcation to see if it yields a more charitable reading of Brouwer. 182

‘En misschien is de beste qualiﬁcatie van mystiek een gebruik van de taal, onafhankelijk van de wiskundige systemen der verstandhouding, maar ook onafhankelijk van directe dierlijke aandoeningen van vrees of begeerte. Kleedt zij zich zodanig in, dat het lezen van voorstellingen van de beide zooeven genoemde groepen, onmogelijk is, dan kunnen misschien die contemplatieve gedachten, waarvan de in het wiskundig systeem levende, de wiskundige vereenzijdigingen zijn, weer ongetroebeld doorbreken, daar er geen wiskundig systeem is, dat ze verwringt.’ [46, p. 28]. 183

The ambiguity in Brouwer’s use of ‘sensation’ in the whole paragraph between sensation as a state of consciousness and sensation as content of consciousness is harmless, I think, as being in the status of sensation is equivalent to having sensation(s) in the second sense. 184

‘Es kann jedermann die innere Erfahrung machen, daß man nach Willkür entweder sich ohne zeitliche Einstellung und ohne Trennung zwischen Ich und

156

Notes

Anschauungswelt verträumen, oder die letztere Trennung aus eigener Kraft vollziehen . . . kann.’ [35, p. 154]. 185

‘We kunnen nog verder gaan, en zeggen dat de schepping van den tijd als matrix van momenten, een vrije daad van onszelf is.’ [46, p. 31]. 186

Given that time is fundamental to Husserl’s conception of constitution, and that the reduction is in some sense the converse of constitution, Brouwer’s claim in these two quotes seem related to Husserl’s repeated assertion that the reduction is performed in complete freedom, as an act of the will [115, section 31], [102, section 28]. For discussion of Husserl’s assertion, see Luft [155] and, in particular, Kim [142]. 187

‘Anstelle der realen Praxis – sei es also der handelnden oder die empirischen Möglichkeiten bedenkenden, die es mit wirklichen und real-möglichen empirischen Körpern zu tun hat – haben wir jetzt eine ideale Praxis eines “reinen Denkens”, das sich ausschließlich im Reiche reiner Limesgestalten hält . . . Aber in diesen mathematischen Praxis erreichen wir, was uns in der empirischen versagt ist: “Exaktheit”; denn für die idealen Gestalten ergibt sich die Möglichkeit, sie in absoluter Identität zu bestimmen.’ [100, pp. 23–24, original emphasis]. 188

Martin-Löf and Sambin have developed, within the context of constructive type theory, formal (or ‘pointless’) topology, e.g. [188]. There one starts with only the discrete as primitive notion, not in the form of discrete points but of constructive sets and a covering relation; points are then deﬁned as particular ﬁlters of neighbourhoods. This reverses the conceptual order from classical topology, and one arrives at an analysis of the continuum such that it is not made up out of atoms. However, from the Brouwerian point of view, on that construal continuity is thematised too late; continuity here is (only) a deﬁned notion, whereas a primitive notion of continuity already comes in when the complementary notion of the discrete is accepted and used. A discrete continuum equipped with a deﬁnition of continuity remains, ontologically, discrete, and hence distinct from the intuitive continuum. (I thank Per Martin-Löf for bringing up the topic of formal topology in our conversations on Brouwer.) 189

‘Hier sind die Teile der Teile in genau derselben Weise Teile des Ganzen wie die ursprünglichen Teile.’ [121, p. 276]. 190

For a more elaborate phenomenological description of the continuum, see

[6]. 191

Brouwer already had the concept of causal sequence, and conceived of choice sequences as based on causal sequences (treated in section 5.5). The question for a motivation here is the question what made Brouwer realise that such sequences can be used to analyse the continuum at all.

Notes

157

192 This makes it puzzling that Becker [10, p. 426], after correctly remarking that

Accordingly, for Brouwer the fundamental relation is that between whole and part (more precisely, in Husserl’s terminology: between ‘extensive whole’ and ‘piece’), not that between set and element (i.e., a part that cannot be divided further, an ‘atom’). The operation of dividing is thought of as repeatable without limit. (‘Dementsprechend tritt bei Brouwer als Grundbeziehung auf die Relation zwischen Ganzem und Teil (genauer in Husserls Terminologie: zwischen “extensivem Ganzen” und “Stück”), nicht diejenige zwischen Menge und Element, (d.h. einem nicht weiter Teilbaren Teil, einen “Atom”). Die Operation des Teilens wird unbegrenzt fortsetzbar gedacht.’) immediately adds, The ‘point’ appears as the limit of this inﬁnite process (‘Der “Punkt” erscheint als der Limes dieses unendlichen Prozesses’) and continues (while an atom would be reached after ﬁnitely many divisions) (‘(während ein Atom nach endlich vielen Teilungen erreicht werden würde).’) [Emphasis mine] Taking limits would amount to an atomistic conception of the continuum again. This is why Brouwer, in contrast to Becker, identiﬁes the point with the unﬁnished sequence. 193

‘Es gibt im extensiven Ganzen keine an sich erste Teilung und auch keine festbegrenzte Gruppe von Teilungen als eine erste Teilungsstufe; es gibt von einer gegebenen Teilung aus keinen durch die Natur der Sache bestimmten Fortschritt zu einer neuen Teilung, bzw. Teilungsstufe. Mit jeder Teilung können wir beginnen, ohne einen inneren Vorzug zu mißachten.’ [121, pp. 276-277]. 194

‘Die Beliebigkeit dieses unter Erhaltung der Fortsetzbarkeitsmöglichkeit einer endlichen Wahlfolge zugeordneten “Verengerungszusatzes” erteilt dieser Wahlfolge, mithin auch ihren Fortsetzungen eine neue Willkür.’ 195

In the meantime, I have learned that the possibility of provisional conditions was probably ﬁrst suggested by van Dalen in 1968 [52]: ‘One can restrict the choices by some law, which, however, need not be predetermined (changing one’s mind is allowed).’ This possibility is not exploited there. 196

Strictly speaking, a quotient of a spread, just as in the case of the reals.

158 197

Notes

See the quotation from Brouwer’s lecture from 1951 on p. 91.

198

Passive synthesis diﬀers from active synthesis in that it occurs without a deliberate turning-toward of an already constituted ego [126, §38]. Typical cases of passive synthesis are sense data and sedimentations of earlier active syntheses; such are the ‘givens’ that active synthesis presupposes. The terms active and passive are no absolutes [124, p. 119]. The ego-as-constituted at a genetically higher level will ﬁnd at hand, passively, what was constituted by the ego-as-constituted at a genetically lower level. For example, sense data are passively given for the ego that is a being in the world, but are actively constituted in the ﬂow of time by the transcendental ego. 199

The other parts of the full noema, besides the nucleus, are (1) a pure substrate of all predicates, ‘the pure X in abstraction from all predicates’ (‘das pure X in Abstraktion von allen Prädikaten’ [115, section 131]), as all predications are predications of something; (2) a quality or thetic character that determines the mode of givenness of the object, i.e., whether the object is given to us as perceived, or remembered, or constructed, etc. [115, pp. 260, 298]. 200

Husserl uses ‘kategoriale Gegenstand’ and ‘Verstandesgegenständlichkeit’ as synonyms [124, p. 285]; an ‘Urteilsform’ is a particular kind of categorial object. 201

‘Es tritt hier die neue Bestimmungsform des “und so weiter” auf, eine Grundform in der Urteilssphäre. Das “Undsoweiter” geht in die Urteilsgestalten ein oder nicht, je nachdem, wie weit das thematische interesse an S reicht; es schaﬀt also Diﬀerenzen in den Urteilsformen selbst.’ [124, p. 259]. 202

‘Jede gegliederte geistige Bewegung, in gleichmäßigen Stile fortschreitend, führt einen solchen oﬀenen Horizont mit sich; nicht ein nächstes Glied ist als einziges vorgezeichnet, sondern Fortgang des Prozesses, der somit immer die intentionale Charakteristik eines oﬀenen Prozesses hat.’ [124, p. 258]. 203

‘Waar men zegt “en zoo voort”, bedoelt men het onbepaald herhalen van eenzelfde ding of operatie, ook al is dat ding of die operatie tamelijk complex gedeﬁnieerd.’ 204

‘Der Begriﬀ der successiven Anwendung der Operation ist äquivalent mit dem Begriﬀ “und so weiter”.’ [242, 5.2523] Sundholm [212, p. viii] and Wang [233, pp. 245–246] elaborate this comparison with Wittgenstein. 205

Further examples showing how choice sequences are diﬀerent from, yet dependent on, their generating processes, are given by Troelstra [222, p. 215], and by Troelstra and van Dalen [226, p. 647–648]. See also note 20 on p. 130.

Notes

159

206 ‘Der Ausdruck “momentanes Bewußtseinsganzes” soll nicht so sehr auf solche Inhalte beziehen, die tatsächlich nur einen Moment währen (deren Existenz höchst fraglich ist), sondern ganz allgemein auf solche, die, abgesehen von ihr etwaigen Dauer, in jedem Moment vollständig sind, d.h. alle zusammengehörigen bzw. zur Erzeugung der Auﬀassung nötigen Elemente isochron erhalten, so daß in der zeitlichen Ausdehnung kein integrierender Faktor gegeben ist.’ 207 Another analogy between intuitionism and music is noted by von Imhof [132, p. 155n. 43]:

Brouwer mathematics reminds us of art in a further sense: the mathematician is like the rare composer who composes entire symphonies in his mind, and only eventually takes to writing them down. 208

At some point between 1922 and 1926, Husserl asked about real objects: ‘Being actual in the mode of eternity, in an “eternal duration”. To what extent is that conceivable? An eternal past. And future? And so on.’ [105, p. 417n. 1, trl. mine] (‘Aktuellsein in der Weise der Ewigkeit, in einer “ewigen Dauer”. Wiefern ist das denkbar? Ewige Vergangenheit. Und Zukunft? etc.’) Because he held that ideal objects are omnitemporal, he could not ask this question of the possibility of inﬁnite temporality in only one direction for ideal objects. 209

A key text of Husserl here is ‘Consciousness (Flow), Appearance (Immanent Object), and Object’ [125, esp. 379] (‘Bewußtsein (Fluß. Erscheinung (immanentes Objekt) und Gegenstand)’ [104, pp. 368–9]). 210

[104, p. 7]; the famous footnote on that page dates from after the 1905 lectures. 211

Eidetically – they are not purely formal (‘empty’).

212

Lawlike sequences, on the other hand, can; in a constructivist idiom, one might say that they are ‘reproducable’ or that they can be ‘cloned’. 213

I used to think that the fact that I cannot give my sequence to you could be expressed by saying that this sequence is incommunicable, as in this argument by van Dalen [55, p. 37]: The fact that a lawless sequence cannot be communicated to another person is a direct consequence of the continuity property of lawless sequences . . . : suppose A wants to communicate the lawless sequence α to B, i.e., he wants to induce enough information in B so that B can recreate α. The communication is a ﬁnite act and so A can, as far as the values of α are concerned, pass only a ﬁnite number of values.

160

Notes

Hence on the basis of this information B cannot recreate the sequence α in isolation, but he has to allow for all sequences that share the same given ﬁnite initial values. But here it is misleading to say that the sequence cannot be communicated, as that suggests that A does know what the whole sequence is but for some reason cannot tell B. And that is not the case: A has no more information than can be shared with B, and A cannot determine the whole sequence α from that either. (This was emphasised to me by Charles Parsons and Richard Tieszen.) 214

This objection and my reply are closely related to the issue of translations, see section 4.3.4. 215

Dummett has repeatedly ascribed to Kreisel the following dictum, ‘The point is not the existence of mathematical objects, but of the objectivity of mathematical truth’ (e.g. [67, p. 228]), and refers to a review by Kreisel of Wittgenstein’s Remarks on the Foundations of Mathematics. Sundholm [213, p. 144n. 24] points out that this attribution may not be correct, for all Kreisel wrote there is that ‘Wittgenstein argues against a mathematical object (presumably: substance), but, at least in places . . . not against the objectivity of mathematics, espcially through the recognition of formal facts’. 216

It is an analogy, because Kant applied the terms ‘phenomenal’ and ‘noumenal’ to physical objects, to refer to, respectively, the physical object as experienced by us, and as it is in itself [137, B307–8]. Here I extend their use to abstract objects, to render the meaning of the objection; whether this distinction really applies to abstract objects is a further question. Kant would say it does not, trivially, for he would not accept the applicability of the extension to begin with. He acknowledges that intuitions do not contain only matter but also form (of space and time), but denies that humans have a faculty of intellectual intuition, that is, intuition of non-sensuous or abstract objects [137, e.g., B308]. In his view, mathematics does not describe independent abstract objects, but constructions that are a priori possible in our perceptual space and our perceptual time [136, section 10]. Husserl’s answer would be negative, too: although he acknowledges, contrary to Kant, that we do have such a non-sensuous intuition (purely categorial intuition), he rejects the phenomenon-noumenon distinction for any kind of object, abstract or not, as mentioned further in the text. 217

The ‘thing in itself’ has also been interpreted idealistically, e.g., as Idea that expresses the ultimate goal of knowledge (Cohen) [141, p. 120]. But idealistic interpretations do not suit the objection under consideration, for that objection requires that certain objects must be constituted by us as having property P while in reality they do not.

Notes

161

218 ‘Prinzipiell entspricht (im Apriori der unbedingten Wesensallgemeinheit) jedem “wahrhaft seienden” Gegenstand die Idee eines möglichen Bewußtseins, in welchem der Gegenstand selbst originär und dabei volkommen adäquat erfaßbar ist. Umgekehrt, wenn diese Möglichkeit gewährleistet ist, ist eo ipso der Gegenstand wahrhaft seiend.’ [115, p. 329]. 219 ‘Andere Bedenken richten sich wiederum auf die Ansicht, daß die Gegenstände der Mathematik im Handeln des Mathematikers erzeugt werden. So verständlich dies als Gegenreaktion zum Platonismus ist und so klarsichtig hiermit auf den Anteil an Handlungsaktivität hingewiesen wird, der in der Konstitution mathematischer Gegenständlichkeiten enthalten ist, so wird damit doch überdeckt, daß Erkennen und Selbstgegebenheit mathematischer Zusammenhänge auf etwas beruht, das sich in der Aktivität des Heranführens passiv einstellt. Auch in der Mathematik gilt, daß wir nur an den Punkt heranführen können an dem sich Erkennen einstellt oder nicht.’ 220

In addition, Albert Visser has shown that, given an enumeration, WC-N! does not hold either (WC-N! is WC-N with antecedent ∀α∃!x). For the proof, see [5]. 221

Weyl announced this in his ‘Grundlagen’ paper [238, p. 76], but, as van Dalen [57, pp. 160–161] explains, this is not really the same strong result as Brouwer’s. Weyl deﬁned real functions in such a way that they are continuous by deﬁnition, i.e., via mappings of the intervals of the choice sequence determining the argument to intervals of the image sequence. This way, the function type is reduced from R → R to N → N (initial segments to initial segments). Brouwer, on the other hand, established the continuity of functions from choice sequences to choice sequences. 222

For a detailed explanation of the bar theorem and its proof, and also for the fan theorem mentioned later, see [3, ch. 4]. 223

This proof is diﬀerent from Veldman’s direct (i.e., not appealing to continuity of functions) one; it is van Dalen’s presentation [60, p. 386] of a proof by Heyting [89]. 224

It does work for the universe of just the lawless sequences.

225

Dialogue in the context of choice sequences is used by Brouwer too, e.g., in a course in 1933, quoted in the notes to the published Cambridge Lectures [45, p. 17]. 226

Letter Kreisel to the author, January 10, 2005. For more on informal rigour, see [148] and [223]. 227

‘De vrijheid van voortzetting der betrokken keuzenrij (eventueel in den kiezer opgelegde afhankelijkheid van gebeurtenissen in de wiskundige

162

Notes

gedachtenwereld van den kiezer) kan na iedere keuze willekeurig (b.v. tot volledige bepaaldheid, of ook volgens een spreidingswet) worden beperkt. Het willekeurig karakter dezer bij iedere nieuwe keuze onder behoud der voortzettingsmogelijkheid geoorloofde beperking der voortzettingsvrijheid is een essentieel onderdeel der vrije wording van het spreidingselement, evenals de mogelijkheid om aan iedere keuze een beperking der vrijheid van verdere vrijheidsbeperkingen te koppelen, enzovoort.’ [221, p. 483]. 228 The conception of choice sequences given here is more liberal than the one Brouwer later adopts, e.g., in 1952 [40, p. 142]. There the freedom is limited by demanding that revisions only narrow down earlier restrictions; I do not see the necessity of this demand. Also, in the same paper higher-order restrictions are rejected because they are ‘not justiﬁed by close introspection’ and moreover ‘endanger the simplicity and rigour of future developments’. Although it is easy to agree with the point about simplicity, that cannot be suﬃcient reason to reject higher-order restrictions. But I do not see the justiﬁcation from introspection; here I agree with Heyting [44, p. 607]. 229

As formulated by Michael Dummett, letter to the author, April 3, 2001.

230

This was overlooked in earlier versions of this manuscript and, subsequently, also in [5]. 231

‘und sodann durch eine unbegrenzte Folge von Wahlen von λ-Intervallen in solcher Weise einen Punkt 2 des Einheitskontinuums konstruieren, daß wir einstweilen für jede schon ins Auge gefaßte natürliche Zahl n die ersten n Intervallen mit den ersten n Intervallen von 1 identisch wählen, uns aber die Freiheit vorbehalten, jederzeit, nachdem das erste, zweite, . . . , (m − 1)-te und m-te Intervall gewählt worden sind, die Wahl aller weiteren (d.h. des (m + 1)ten, (m + 2)-ten usw.) Intervalle in der Weise festzulegen, daß entweder [2 mit 1 zusammenfallt oder beide örtlich verschieden sind].’ [33, p. 62]. 232

As A(α, x) is required to be graph-extensional, in particular the presence (or absence) of provisional restrictions on α cannot be exploited in the deﬁnition of A. 233

Brentano considered his descriptive psychology to be ‘empirical’; by this he meant that its basic concepts are to be derived from either inner or outer intuition, that is, from mental experience. See [182, p. 258], and also the points on terminology in [165, p. 67]. 234 235

[210, p. v].

Incidentally, this is an analysis of the type indicated by the early Husserl in ‘On the logic of signs (semiotic)’ [127, p. 49n. 4], [107, p. 371n].

Notes

163

236 Hence (early) Brouwer’s references to Kant, as well as the name, ‘intuitionism’; on the many diﬀerences between Kant and Brouwer, see [207, p. 151] and [3, ch. 6]. 237

[131, p. 357], [107, p. 339].

238

The notion was made explicit in his paper from 1933, ‘Volition, knowledge, language’ [37], but had been implicit all along. 239

‘Indien nu op grond van redelijke bezinning de exactheid der wiskunde, in den zin van uitgeslotenheid van fouten en misverstand, niet langs linguïstischen weg kan worden verzekerd, rijst de vraag, of deze verzekering dan wel langs anderen weg kan geschieden. Op deze vraag luidt het antwoord, dat de door de zelfontvouwing der oerintuïtie ontstaande taallooze constructies, uit kracht van hun in de herinnering aanwezig zijn alleen, exact en juist zijn, dat echter het menschelijk herinneringsvermogen, dat deze constructies heeft te overzien, ook als het linguïstische teekens te hulp roept, uit den aard der zaak beperkt en feilbaar is. Voor een met onbeperkt herinneringsvermogen uitgerusten menschelijken geest zou dus de in eenzaamheid beoefende en geen linguïstische teekens gebruikende zuivere wiskunde exact zijn, een exactheid die echter bij wiskundige gedachtenwisseling tusschen menschen met onbeperkt herinneringsvermogen, die immers op de taal als middel tot verstandhouding blijven aangewezen, weer verloren zou moeten gaan.’ [37, p. 14, original emphasis]. 240

See section 3 of [224].

241

Chomsky holds that ‘Linguistic theory is concerned primarily with an ideal speaker-listener, in a completely homogeneous speech-community, who knows its language perfectly and is unaﬀected by such gramatically irrelevant conditions as memory limitations, distractions, shifts of attention and interest, and errors (random or characteristic) in applying his knowledge of the language in actual performance.’ [50, p. 3]. 242

The use of Chomsky’s terms here is not meant to imply further resemblances between his philosophy and Brouwer’s. For example, Brouwer would certainly not have been favourably disposed towards Chomsky’s speculation that ‘we might think of the human number faculty as essentially an “abstraction” from human language, preserving the mechanism of discrete inﬁnity and eliminating the other special features of language’ [51, p. 169]. For Brouwer, the structure that language exhibits would rather be a sign that language arises as an application of mathematical thinking to speech. 243

For recent discussion of Turing’s analysis of computability, see [171] and [195].

164

Notes

244 This is the sense in which intuitionism, for all the idealisations it makes, remains ﬁrmly rooted in our actual experiences (and thereby remains ‘empirical’ in Brentano’s sense); in particular, this distinguishes it from forms of realism that deﬁne themselves in terms of even further idealisations and thereby lose their empirical ground. (See, for example, Brouwer’s comments on Cantor’s deﬁnition of the second number class [44, p. 81].) 245 In his footnote 12 on the same page, Parsons mentions that ‘such a view is usually attributed to Brouwer, not unreasonably in spite of the strongly naturalistic tendencies in his philosophy’. It has been argued that Brouwer’s notion of the subject is that of Kant’s transcendental subject; in [3, ch. 6], I explain why I think Husserl’s notion that goes by the same name comes much closer to what Brouwer seems to have had in mind. That explanation is part of a defence of intuitionism against charges of psychologism. 246

For the full text of the letter, see [3, ch. 6].

247

Mitchell [161] develops a response to ﬁnitistically inspired criticisms of Dummett’s intuitionism (such as have been voiced by Crispin Wright, Alexander George, and Saul Kripke) along the same lines. I note that it just as much shows that Brouwer’s intuitionism is not strict ﬁnitism either. Moreover, Mitchell argues, as Dummett has done before but in a diﬀerent way, that strict ﬁnitism is incoherent. The one point where I disagree with Mitchell is where he says that intuitionism describes the abilities of the subject ‘in a way that is independent of virtually every physical predicate except time’ [161, p. 448]. Brouwer explicitly says that intuitionism is not founded on time as understood in physics: ‘We mean here intuitive time which must be distinguished from scientiﬁc time. By means of experience and very much a posteriori it appears that scientiﬁc time can be introduced for the cataloguing of phenomena’ [44, p. 61, original emphasis] (‘Natuurlijk wordt hier bedoeld de intuïtieve tijd, wel te onderscheiden van de wetenschappelijke tijd, die, wel zeer a posteriori, eerst door de ervaring blijkt, als met een eenledige groep voorziene eendimensionale coördinaat geschikt te kunnen worden ingevoerd tot het catalogiseren der verschijnselen.’ [23, p. 99, original emphasis]). This correction actually strengthens Mitchell’s case. 248

‘Eine Erweiterung der Vorstellungsfähigkeit, die sie in den Stand setzen würde, Mengen von hundert, tausend, Millionen Elementen in eigentlicher Betätigung kollektiv zu befassen, ist sehr wohl denkbar. Und so bietet denn unsere Intention, welche der symbolischen Vorstellung so großer Mengen zugrunde liegt, keinen Anlaß zu logischen Bedenken; sie geht auf die wirkliche Vorstellung von Kollektionen, die, wenn nicht in den Bereich unserer, so doch einer idealisierten menschlichen Erkenntnisfähigkeit fallen.’ [107, pp. 218–219]. 249

‘Aus der Möglichkeit, die sukzessive Setzung von Einheiten beliebig weit fortzusetzen, zieht Brix nämlich den sehr gewagten Schluß . . . daß auf solchem

Notes

165

Wege beliebig große Zahlen gebildet werden könnten. Aber das bloße Nacheinander der wiederholten Setzungen gewährleistet doch noch keine Synthesis, ohne welche die kollektive Einheit der Zahl undekbar ist. Eben an der Unfähigkeit des wirklichen Vollzuges solcher Synthesis scheitert tatsächlich, wie wir noch erörtern werden, jeder Versuch, die höheren Mengen und Zahlen in eigentlicher Vorstellung zu bilden. Durch einfachsten Versuch hätte sich Brix davon überzeugen können’ [107, p. 30n. 1]. 250 ‘Wie nämlich die Anzahlen sich auf Mengen beziehen, so die Ordinalzahlen auf Reihen. Reihen sind aber geordnete Mengen . . . Achtet man auf die gemeinüblichen Bedeutungen derselben, dann bleibt es zu Recht bestehen, daß der Begriﬀ der Ordinalzahl den der Anzahl einschließt, also voraussetzt, wie die Bildungsweise der Namen dies richtig ausdrückt.’ [107, p. 11]. 251

In particular, [17, section 85] and [18, section 7].

252

‘In jedem Fall ist somit eine Analyse des Anzahlbegriﬀes ein wichtiges Vorerfordernis für eine Philosophie der Arithmetik; und sie ist deren ersten Erfordernis, falls nicht etwa dem Ordinalzahlbegriﬀe, wie von anderer Seite behauptet wurde, die logische Priorität zukommt. Die Möglichkeit einer von diesem Begriﬀe völlig absehenden Analyse des Anzahlbegriﬀes wird den besten Beweis für die Unzulässigkeit dieser Ansicht liefern.’ [107, pp. 12–13]. 253

Willard in his introduction to ‘On the Concept of Number: Psychological Analyses’, [118, p. 90]. 254

Lohmar [152, p. 86] holds, as did Husserl, that ordinal numbers are founded on cardinal numbers, but, as he still ﬁnds reason not to accept Husserl’s speciﬁc analysis of cardinal numbers [pp. 75–78], he presents an alternative analysis [ch. 4]. 255

See also appendix 1 to the 1905 lectures in [104].

256

This paragraph is meant as the beginning of an answer to a question raised by Sundholm [212, pp. 269–270]. 257

Compare, in Brouwer’s deﬁnition of his notion of set [29, p. 3]: ‘We will refer to the common mode of generation of the elements of a set M as, for short, the set M as well.’ (trl. Dirk van Dalen and me). 258

On this point, see also Tieszen’s analysis [214, ch. 5].

259 Brouwer to Korteweg, November 7, 1906 [49, pp. 15–16]. An English translation of that letter can be found in [207, pp. 491–493]. 260

In his dissertation, Brouwer deﬁned the concept of a ‘denumerably unﬁnished set’, which is in some sense in between a denumerably inﬁnite set given by an inductive deﬁnition and an actually inﬁnite set:

166

Notes

We call a set denumerably unﬁnished if it has the following properties: we can never construct in a well-deﬁned way more than a denumerable subset of it, but when we have constructed such a subset, we can immediately deduce from it, following some previously deﬁned mathematical process, new elements which are counted to the original set. But from a strictly mathematical point of view this set does not exist as a whole, nor does its power exist; however we can introduce these words here as an expression for a known intention. [44, p. 82] (‘We verstaan dan onder een aftelbaar onaﬀe verzameling een, waarvan niet anders dan een aftelbare groep welgedeﬁnieerd is aan te geven, maar waar dan tevens dadelijk volgens een of ander vooraf gedeﬁnieerd wiskundig proces uit elke zodanige aftelbare groep nieuwe elementen zijn af te leiden, die gerekend worden eveneens tot de verzameling in kwestie te behoren. Maar streng wiskundig bestaat die verzameling als geheel niet; evenmin haar machtigheid; we kunnen deze woorden echter invoeren als willekeurige uitdrukkingswijzen voor een bekende bedoeling.’ [23, p. 148]) This is a special case of the notion of indeﬁnitely extensible concept, the idea of which was introduced by Russell in his paper ‘On some diﬃculties in the theory of transﬁnite numbers and order types’ from 1906: ‘the contradictions result from the fact that . . . there are what we may call self-reproductive processes and classes. That is, there are some properties such that, given any class of terms all having such a property, we can always deﬁne a new term also having the property in question.’ (As quoted in [69, p. 317n. 5]). According to Brouwer’s reading list as compiled in [149, pp. 327–335], Brouwer was not acquainted with that paper of Russell’s. 261

E.g. [29, p. 5].

262

Richard Tieszen, in a personal communication, commented that Husserl’s view in Philosophy of Arithmetic seems to require viewing sequences (or functions) as extensions or graphs, not as rules or intentions, and that Brouwer is clearer about this diﬀerence and its importance. 263

I am much obliged to Richard Tieszen for providing me with a copy of their transcriptions. 264 265

[159, p. 39]. The same point is made by Chomsky [51, p. 167].

266

See also [107, pp. 216–217], and in particular the footnote to the last line of section 51 of the 6th Logical Investigation [122]. 267 Even the formalist Hilbert adopted, in lectures from 1905 titled ‘Logical principles of mathematical thought’, an ‘axiom of the existence of mind’, and

Notes

167

pointed out the central role of introspection or reﬂection: ‘I have the ability to think things, and to designate them by simple signs (a, b, . . . X, Y, . . .) in such a completely characteristic way that I can always recognize them again without doubt. My thinking operates with these designated things in certain ways, according to certain laws, and I am able to recognize these laws through self-observation, and to describe them perfectly.’ (As quoted in [172, p. 150].) 268

This paragraph has been adapted from [3, ch. 6].

269

See also [152, p. 139] and [214, p. 137].

270

Tieszen [214, pp. 101–102] too points out that Husserl did not make the connection between the temporal structure of consciousness and the genesis or constitution of the ordinal numbers.

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Name and Citation Index

Aristotle, 32, 33 [1], 33 van Atten, M., [3], 111, 129, 135, 163, 165, 166, 169 [4], 26 van Atten, M. and van Dalen, D., [5], 163 van Atten, M., van Dalen, D., and Tieszen, R., [6], 158 van Atten, M. and Kennedy, J., [7], 129 [8], 26, 104 Bachelard, S., 8, 65, 66, 71, 153 [9], 8, 65, 66 Baire, R., 35 Becker, O., 6, 7, 42, 72–75, 90, 131, 137, 155, 156, 159 [10], 7, 72, 73, 86, 137, 159 [11], 6, 7, 73–75, 90, 156 Beeson, M., 104 [12], 104 Beiser, F., [13], 156 Bell, J., 140 [14], 140 Bernays, P., 26 Beth, E., 88 Biemel, M., 55 Bishop, E., 130 [16], 130

Boethius, 133 Bolzano, B., 119 [17], 167 [18], 167 Borel, E., 35 [19], 35 Brentano, F., 28–31, 113–115, 126, 132, 138, 139, 153, 164, 166 [20], 138 [21], 113 Brouwer, L.E.J., [22], 6, 157 [23], 22, 35, 91, 120, 137, 140, 154, 157, 166, 168 [24], 78, 157 [25], 35 [26], 35 [27], 81, 141 [28], 35 [29], 49, 50, 104, 167, 168 [30], 81, 82 [31], 104 [32], 88 [33], 104, 164 [34], 130 [35], 65, 114, 135, 157, 158 [36], 35, 36, 141 [37], 157, 165 [38], 50, 114, 120 [39], 6, 14, 20, 21, 62, 76–78, 80–83, 123, 135

181

182

Name and Citation Index

Brouwer, L.E.J. (continued ) [40], 13, 23, 24, 35, 36, 54, 82, 116, 119, 120, 135, 164 [41], 82 [42], 136 [43], 14 [44], 23, 33, 78, 79, 116, 120, 124, 164, 166, 168 [45], 91, 120, 125, 135, 163 [46], 157, 158 [47], 12 [48], 156 [49], 167 Cantor, G., 23, 35, 36, 119, 120, 137 Chomsky, N., 113, 116, 165, 168 [50], 165 [51], 165, 168 Cobb-Stevens, R., 153 Cohen, H., 162 van Dalen, D., IX, 34, 74, 105, 127, 129, 131, 132, 142, 155, 156, 159–161, 163 [52], 159 [53], 88 [54], 155 [55], 161 [56], 131 [57], 43, 48–50, 134, 142, 144, 145, 163 [58], 141 [59], 156 [60], 13, 35, 72, 141, 163 [61], 36 [62], 130 [63], 5, 130, 141 van Dantzig, D., 21 Dedekind, R., 35 Descartes, R., 147 Dummett, M., 75, 129, 141, 142, 153, 162, 164, 166 [65], 153 [66], 130 [67], 162 [68], 21 [69], 168 [70], 142 [71], 88, 96

Elsenhans, T., 32 [72], 32 Fechner, G., 121 Føllesdal, D., 153 Fraenkel, A., 36, 143 [73], 143 Gentzen, G., 26 George, A., 166 Gödel, K., IX, 26, 27, 37, 43, 51, 52, 100, 103, 104, 106, 129, 133, 146, 151 [74], 151 [75], 26 [77], 26, 88 Grattan-Guinness, I., [78], 143 Gurwitsch, A., 122, 153 Ha, B.-H., 131 Hamlet, 95 Hardy, G.H., 38 [79], 38 Hartmann, N., 59, 60 [80], 59–61 Heidegger, M., 5, 7, 156 van Heijenoort, J., [81], 108 Held, K., [82], 154 Herbrand, J., 133 Hersh, R., [83], 28 Hesseling, D., 133 [84], 133 Heyting, A., 6, 34, 42, 43, 75, 127, 129, 131, 142, 143, 163, 164 [85], 6 [86], 131 [87], 6, 75 [88], 6 [89], 163 [90], 35 Hilbert, D., 15, 16, 22, 23, 26, 64, 133, 137, 143, 152, 168 [91], 137 [92], 133, 147 [93], 152

Name and Citation Index van der Hoeven, G., [94], 110 Husserl, E., [95], 64, 65 [96], 138, 139 [97], 55, 56 [98], 147 [99], 55, 66, 147–149, 151, 152 [100], 25, 86, 134, 158 [101], 140 [102], 158 [103], 5, 55, 56, 58, 146, 148 [104], 69, 92, 93, 118, 154, 161, 167 [105], 161 [106], 15, 40, 57, 58, 61, 62, 64, 134 [107], 29, 117, 121, 122, 164–168 [108], 19, 86 [111], 31, 55, 87, 88, 151 [109], 69 [110], 12, 61, 62, 68–70, 90, 91 [112], 7, 17, 19, 56, 58, 61, 64, 66, 131, 133, 134, 137, 142, 148– 150, 152 [113], 6, 29, 55, 96, 147 [114], 28, 139 [115], 56, 61, 69, 70, 75, 89, 106, 133, 137, 142, 155, 158, 160, 163 [116], 30, 31, 139, 140 [117], 55, 58, 62, 63 [118], 167 [119], 114 [120], 15, 40, 100 [121], 60, 67, 140, 158, 159 [122], 62, 151, 168 [123], 94 [124], 6, 11, 15, 69, 71, 87, 89, 98, 126, 131, 132, 150, 151, 154, 155, 160 [125], 161 [126], 7, 25, 56, 154, 155, 160 [127], 30, 32, 164 [128], 5, 30, 31, 56, 72, 73, 155, 156 [129], 56, 57 [131], 117–119, 165 von Imhof, F., [132], 161

183

Jervell, H., [134], 141 Johnstone, H.W., Jr., 10, 134 [135], 10, 134 Kant, I., 21, 22, 33, 34, 59, 74, 83, 99, 136, 141, 156, 157, 162, 165, 166 [136], 162 [137], 21, 59, 136, 137, 162 [138], 33, 136, 137 Kaufmann, F., 7, 38, 39, 73, 74, 131, 155 [139], 7, 38, 142 [140], 38, 39 Kern, I., 19, 22 [141], 6, 19, 21, 22, 32, 162 Kim, H.-B., [142], 158 Kockelmans, J., [143], 146 Korteweg, D., 167 Kraus, O., 138 Kreisel, G., 26, 27, 41–43, 51, 96, 99, 106, 143, 146, 162, 163 [144], 51 [145], 106 [146], 41–43 [147], 26, 45, 52, 137 [148], 163 Kripke, S., 88, 166 Kronecker, L., 28, 36, 141 Kuiper, J., [149], 168 Kummer, E., 28 Küng, G., 155 [150], 155 Lambert, J., 97 Landgrebe, L., 156 Laugwitz, D., 140 [151], 140 Lebesgue, H., 35 Leibniz, G.W., 4, 57, 147, 148 Lohmar, D., 8, 15, 51, 100, 101, 131, 137, 147 [152], 8, 15, 100, 101, 137, 147, 167, 169 [153], 8, 51 [154], 131, 154 Lotze, H., 31, 121, 139

184

Name and Citation Index

Luft, S., 135 [155], 135, 158 Münchhausen, 156 Mac Lane, S., 89 [156], 89 Mancosu, P., [157], 15, 20, 34, 43–47, 49, 50, 54, 134, 143, 144 Markov, A., 17 Martin-Löf, P., 158 Marty, A., 67, 68, 70, 154 [158], 67, 68, 154 McKenna, W., 122 [159], 168 Meinong, A., 93, 94, 132 Mensch, J., 124 [160], 124 Meyer, F., 120 Mitchell, S., 166 [161], 166 Mohanty, J.N., 22, 131, 153 [162], 131 [163], 22 Moschovakis, J., 144 [164], 144 Mulligan, K., [165], 164 Myhill, J., 42 [166], 42 Natorp, P., 30, 31 Niekus, J., 135 [167], 135 O’Neill, O., 157 [168], 157 Parsons, C., 22, 117, 134, 135, 153, 162, 166 [169], 21, 22, 134, 153 [170], 54 [171], 117, 165 Peano, G., 23, 137 Peckhaus, V., [172], 169 Philo Judaeus, 133 Picker, B., 138 [173], 138

Placek, T., 34 [174], 34, 74, 111, 136 Plato, 133 Plotinus, 133 Plutarchus, 133 Poincaré, H., 35, 36, 141 Pos, H., 56, 72, 131 Post, E., 133 Posy, C., 22, 33 [175], 18, 22, 74 [176], 33 Ramsey, F.P., 145 [177], 146 Rang, B., [178], 118 Rescher, N., [179], 37 Roberts, J., 135 [180], 135 Rollinger, R., 68 [181], 29, 31, 67, 68, 138, 153, 154 [182], 115, 164 Rosado Haddock, G.E., 8, 150 [183], 8 [184], 150 Rostand, J., 42 Rota, G.-C., 147 [185], 147 Rota, G.-C., Sharp, D., and Sokolowski, R., [187], 42 Russell, B., 23, 137, 168 Sambin, G., 158 [188], 158 Schmit, R., 8, 65, 153 [189], 8, 65, 153 Schopenhauer, A., 34, 141 Schuhmann, K., 73, 133, 152 [190], 64, 154 [191], 28, 156 Seebohm, T., 29, 131 [192], 131 [193], 29 Shakespeare, W., 119 Sieg, W., [195], 165

Name and Citation Index Skolem, T., 42 [196], 42 Sokolowski, R., 69, 153 [199], 69 [200], 153 Sorabji, R., 133 [202], 133 Spiegelberg, H., [203], 29 Stecher, J., [204], 110 Stern, L., 94 [205], 94 van Stigt, W., 156 [206], 79, 81 [207], 165, 167 [208], 156 Ströker, E., [209], 86 Stumpf, C., 113, 126 [210], 164 Sundholm, B.G., 131, 160, 162, 167 [211], 131 [212], 131, 160, 167 [213], 162 de Swart, H., 88 Tennant, N., 141 Tieszen, R., 8, 111, 123, 131, 151, 162, 168, 169 [214], 8, 111, 123, 167, 169 Tito, J.M., 131, 157 [215], 131, 149, 157 Tonietti, T., 155 [216], 155 Tragesser, R., 8, 17, 85, 141, 146, 149 [217], 85, 146, 149 [218], 8, 17 Troelstra, A., 34, 41, 42, 48, 96, 106, 110, 116, 141, 160 [219], 42 [220], 41, 47, 96, 106 [221], 36, 50, 107, 133, 141, 164 [222], 48, 88, 106, 160

185

[223], 163 [224], 165 Troelstra, A. and van Dalen, D., [225], 110 [226], 34, 36, 108, 160 Turing, A., 116, 117, 165 [227], 116 Twardowski, K., 29, 30 Tymoczko, T., 27 [228], 27 Valéry, P., 36 [229], 36 Veldman, W., 88, 104, 105, 163 [230], 104, 105 Vesley, R., 42 [231], 42 Visser, A., 163 Vongehr, T., 130, 131 Wang, H., 26, 54, 100, 103, 160 [233], 151, 160 [234], 100 [235], 103 Weierstraß, K., 28, 138 Weyl, H., 6, 7, 20, 27, 33, 36, 42–51, 72– 75, 86, 131, 137, 140–143, 155, 156, 163 [237], 140 [238], 6, 7, 20, 42, 45, 72, 73, 75, 140, 143–145, 163 [239], 6, 7, 42, 140, 155 [240], 33 Willard, D., 113, 119, 126, 167 [241], 113, 126 Wittgenstein, L., 27, 53, 54, 91, 160, 162 [242], 160 [243], 146 [244], 54 [245], 91 [246], 27 Wright, C., 53, 54, 166 [247], 53, 54 Yuting, S., 26

Subject Index

anti-reductionism, 48 aporetic cluster, 37, 51 atemporality, 8, 14, 16, 40, 67, 68, 70, 71, 131 axiom of choice, 103 bar theorem, 105, 161 bounded (object), see object, bounded choice sequence analogy with melody, 93 approaches to analytic, 48 ﬁgure of speech, 48 holistic, 48 categorial form, 91, 99 as empirical object, 27, 37, 45, 95 hesitant sequence, 108 individuation of, 36, 48, 49, 71, 92 initial segment of, 1, 36, 46, 50, 87, 90–92, 97–99, 103, 105–107, 109, 110, 161 lawless, 13, 41–47, 50, 87, 89, 90, 96, 97, 100, 106–108, 110, 131, 154, 159, 161 axiomatisations of, 42 cannot follow lawlike sequence, 46, 47 captured by second-order restriction, 43 continuity principle for (open data), 50 deﬁned contextually, 41

187

elimination of, 41 individuated in time, 15 LS, 41 not closed under most operations, 45 not explained away, 41 primordial notion of choice sequence, 42, 87 quantiﬁcation over, 42, 47 term suggested by Gödel, 43 translations, 41, 96 Weyl’s conception, 43–45 lawlike, 12, 43–47, 49, 50, 73, 74, 85, 92, 94, 99, 100, 106, 108, 154, 159 compared to melodies, 95 conceived of as omnitemporal, 73 conceived of as static, 15, 16, 90 isochronous, 94 no essential binding to time, 90, 92, 96, 97 motivation for, 36, 85 non-lawlike, 36, 43, 47, 49–51, 73, 74, 98 compared to melodies, 95 contrast with ﬁctional objects, 95 diﬀerent from concept, 89 do not form a set, 36 in ﬁnitary mathematics, 26 free but not arbitrary, 100 implicitly rejected by Husserl, 15 indeterminateness of, 17

188

Subject Index

choice sequence (continued ) non-lawlike (continued ) individuated in time, 15, 71 inexhaustibility of the continuum represented by, 87 necessarily intratemporal, 92, 94, 97 ‘noumenally lawlike’, 99 open-ended, 36 owned, 98 rejected by Weyl, 47, 73 unﬁnished, 15, 46, 98 objections to ‘does not obey PEM’, 14 Husserl’s (implicit), 6 ‘introduces arbitrariness into mathematics’, 100 ‘noumenally lawlike’, 99 ‘proofs need only ﬁnite sequences’, 99 vs process of choosing, 91, 92, 158 restrictions on, 14, 15, 43, 82, 85, 87, 88, 91–93, 95, 98, 99, 107–110, 162 advantage of, 88 arbitrariness of, 107 decidability of, 107 deﬁnitive, 88, 107–110 diﬀerent orders of, 107, 162 ﬁrst-order, 36, 43, 109 freely chosen, 88 higher-order, 110 lawless sequences and, 107 lifting of, 88, 93, 108–110 once limited by Brouwer to ﬁrst order, 162 provisional, 88, 106–110, 157, 162 revision of, 93 clariﬁcation, 57–59, 65 competence (theory of), 116–118 constitution, 7, 47, 55–57, 59, 62, 64, 67, 69, 74, 86, 93, 100, 101, 152 of categorial objects, 62, 85, 148 of choice sequences, 88, 93, 96, 101, 103, 125, 133 construction and, 25, 26 of ordinal numbers, 122, 167 reduction and, 156

strict, 51, 56, 57, 59, 62, 64, 66, 69, 85 time and, 70, 71, 92, 98, 156 construction, 15, 17, 22, 26, 39, 44, 45, 49, 50, 95, 100, 105, 111, 114, 116, 119, 120, 123, 131, 133, 160 of cardinal numbers, 123 constitution and, 25, 26 of continua, 33, 35 creation and, 100 as deduction, 15 diverging senses of, 131 ﬁnite, 14, 105 idealisation and, 117 languageless, 41, 116 protention and, 125 provisional properties and, 109 of spreads, 34 continuity principle, 50, 88, see also GWC-N; WC-N; WC-N! Gödel’s use of, 27, 103 continuity theorem, 105 uniform, 105 continuum, 1–4, 6, 27, 33–35, 37, 44–46, 48, 73, 86–88, 99, 105, 108, 119–121, 128, 139, 141, 156, see also intuition, of the continuum; two-ity arithmetic, 38 atomistic conception of, 33, 37, 138, 156, 157 Borel on, 35 homogeneity of, 86–88 inexhaustibility of, 42, 86–88 intuitive, 24, 33–36, 38, 40, 42, 43, 73, 85–88, 99, 100 measurable, 35 non-discreteness of, 35, 36, 42, 86–88 practical, 35 reduced, 35 semi-intuitionist, 36 unsplittability of, 105 creating subject, 21, 95, 104, 108, 133, see also ideal mathematician descriptive psychology, 28–32, 113, 114, 118, 121, 162

Subject Index determinate (object), see object, determinate distributed (object), see object, distributed dynamic (object), see object, dynamic ego, 120, 125, 158 transcendental, 19–21, 25, 56, 132, 133, 158 fan theorem, 105, 161 ﬁnished (object), see object, ﬁnished genesis, 22, 23, 25, 29, 124, 125, 136, 167 of intentionality, 124 genetic accounts, 22, 23, 25, 26 analysis, 22, 23, 35, 37, 58, 69 functions, 22, 25 order, 23 phenomenology, 69, 145 precedence of ordinals over cardinals, 122, 123 psychologism, 29, 31 schema, 23 uniformity, 39 GWC-N, 104–106, 109 ideal mathematician, 21, 82, 95, 115–118, 130, see also creating subject idealisation, 17, 21, 95, 115–117, 131, 133, 150, 163, 164 indeterminate (object), see object, indeterminate indistributed (object), see object, indistributed intentionality, 6, 19, 21, 28–31, 56, 58, 66, 69, 89, 91–93, 117, 119, 124, 125, 147, 148, 151, 152, 164, 166 intersubjectivity, 21, 95, 99, 101, 134 intratemporality, 16, 17, 37, 39, 51, 52, 73, 90, 96, 97 intuition, 6, 22, 23, 33, 37, 48, 58, 62, 63, 75, 114, 116, 120, 125, 134, 140, 149, 151, 160 of abstract objects, 22 Brentano on, 162 Brouwer on, 24, 33, 34, 77, 114, 120, see also two-ity

189

categorial, 101, 149, 160 of the continuum, 33, 34, 36, 40 Husserl on, 40, 160 Kant on, 22, 134, 160 intuitionism, IX, 6, 8, 23, 24, 27, 34, 35, 42, 43, 48, 49, 54, 65, 66, 75, 82, 91, 100, 111, 117, 128, 135, 148, 151, 164 empirical in Brentano’s sense, 163 ﬁnitism and, 117, 164 ﬁrst act of, 24, 42, 66, 91 Husserl’s informants on, 73, 129 Husserl’s silence on, 65 intersubjectivity and, 21, 134 Kant and, 163 music and, 95, 159 not psychologism, 21, 117 object of, 116 phenomenology and, 6, 21, 25, 111 revisionism and, 54, 66 as a schematic description, 116 second act of, 24, 42, 49, 66 theory of competence, 116, 117 mathematics Brouwer’s conception of, 6, 114, 116, 155 Husserl’s conception of, 6, 62, 65 noetic-noematic correlation, 106 non-revisionism, see revisionism number cardinal, 61, 118, 119, 121–123, 165 ﬁnite, 113, 118 natural, 12, 44, 45, 47, 50, 60, 89, 103, 120, 121 ordinal, 23, 118–123, 126, 165, 167 rational, 34 real, 12, 33, 34, 39, 43, 85, 105, 140 whole, 33 object bounded, 16 determinate, 12–14, 48 dynamic, 5–9, 15–17, 37, 39–41, 49, 51, 52, 111, 129 object (continued ) ﬁnished, 6, 12–16, 71, 94, 97, 125 indeterminate, 12–14, 85

190

Subject Index

open-ended, 6, 15, 36, 92, 97, 98, 105 static, 6, 9, 15, 16, 39, 41, 67, 71, 90, 97, 129 temporally distributed, 94 temporally indistributed, 94 unbounded, 16, 17, 37, 51, 52 unﬁnished, 12–15, 36, 46, 71, 85, 87, 90, 91, 94, 95, 97, 157 omnitemporality, 6–8, 15, 16, 37, 39, 40, 47, 48, 51, 66–69, 70, 71, 73, 74, 89, 90, 92, 96–98, 101, 129, 131, 159 ontological region, 8, 17, 26, 39, 61, 69 ontology, 9, 20, 47, 55–57, 61, 63, 65, 66, 71, 73, 74, 83, 103, 111, 114, 115, 154 open-ended (object), see object, openended PEM, 13, 14, 17, 114 performance (theory of), 113, 116–118, 163 possibility a priori, 160 conceptual, 60, 63 formal, 60 logical, 59, 60 material, 60 synthetic, 60 transcendental, 59–61, 85, 95 primordial impression, 125 principle of open data, 50 process (of choosing), 49, 90–94, 100, 101, 107, 108, 131, 134, 157 vs choice sequence, 91–93, 95, 158 freedom in, 106 as object, 92 object and, 91 open-ended, 89, 91 revisions and, 91 protention, 125 psychologism, 21, 27–29, 37, 86, 96, 111, 137, 144, 164 descriptive, 29 genetic, 29, 31 reductionism, 28, 37, 40

representation, 30, 113–115, 117, 118, 122, 134 retention, 92, 93, 119, 125, 126 revisionism, 51, 53–55, 58, 59, 64, 148 non-revisionism, 53, 54 strong, 53, 54, 59, 61, 64–67 weak, 53, 54, 57, 59, 64, 66 solipsism, 75, 111 species, 24, 34 spread, 12, 34, 35, 49, 50, 88, 107, 121, 157 static (object), see object, static temporality, 16, 31, 68, 71, 92, 97, see also atemporal; intratemporal; omnitemporal early vs later Husserl, 130 inﬁnite, 97, 159 time, 2, 11, 13–16, 18, 25, 28, 31, 57, 65, 67–71, 76, 80, 81, 89, 92, 94, 96–98, 111, 114, 116, 117, 120– 126, 130, 156, 164 Brouwer on, 14, 24, 62, 77, 78, 80, 81, 119, 120, 124, 125, 132, 164, see also two-ity cardinal numbers and, 122 characterises diﬀerence Brouwer– Husserl, 17 choice sequences and, 2, 3, 6, 15, 36, 38, 39, 89–91, 94–98 ﬁnished objects and, 13 ﬂow of (Husserl), 69–71, 89, 93, 94, 98, 119, 120, 124, 158, see also time, move of Husserl on, 11, 12, 31, 67, 94, see also time, internal identity and, 69, 70 internal, 70, 93, 98, 118, 122, 125, 133, 152 Kant on, 22, 33, 160 Kaufmann on, 39 Marty on, 67, 68 Meinong on, 93, 94 move of (Brouwer), 80, 81, 121, see also time, ﬂow of noema and, 71, 90 objective, 11, 31, 70, 152 ordinal numbers and, 122

Subject Index phenomenology of, 152 Stern on, 94 transcendental idealism, 48, 56, 59–61, 66, 99, 131 transcendental phenomenology, 7, 9, 28, 32, 40, 51, 55, 56, 58, 103, 129, 147, 151 two-ity, 24, 34, 39, 41, 62, 77, 79, 119–126

unbounded (object), see object, unbounded unﬁnished (object), see object, unﬁnished WC-N, 50, 103–106, 109, 161 WC-N!, 161

191

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11. 12. 13. 14.

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20. 21. 22. 23.

J. M. Boch´enski, A Precis of Mathematical Logic. Translated from French and German by O. Bird. 1959 ISBN 90-277-0073-7 P. Guiraud, Probl`emes et m´ethodes de la statistique linguistique. 1959 ISBN 90-277-0025-7 H. Freudenthal (ed.), The Concept and the Role of the Model in Mathematics and Natural and Social Sciences. 1961 ISBN 90-277-0017-6 E. W. Beth, Formal Methods. An Introduction to Symbolic Logic and to the Study of Effective Operations in Arithmetic and Logic. 1962 ISBN 90-277-0069-9 B. H. Kazemier and D. Vuysje (eds.), Logic and Language. Studies dedicated to Professor Rudolf Carnap on the Occasion of His 70th Birthday. 1962 ISBN 90-277-0019-2 M. W. Wartofsky (ed.), Proceedings of the Boston Colloquium for the Philosophy of Science, 1961–1962. [Boston Studies in the Philosophy of Science, Vol. I] 1963 ISBN 90-277-0021-4 A. A. Zinov’ev, Philosophical Problems of Many-valued Logic. A revised edition, edited and translated (from Russian) by G. K¨ung and D.D. Comey. 1963 ISBN 90-277-0091-5 G. Gurvitch, The Spectrum of Social Time. Translated from French and edited by M. Korenbaum and P. Bosserman. 1964 ISBN 90-277-0006-0 P. Lorenzen, Formal Logic. Translated from German by F.J. Crosson. 1965 ISBN 90-277-0080-X R. S. Cohen and M. W. Wartofsky (eds.), Proceedings of the Boston Colloquium for the Philosophy of Science, 1962–1964. In Honor of Philipp Frank. [Boston Studies in the Philosophy of Science, Vol. II] 1965 ISBN 90-277-9004-0 E. W. Beth, Mathematical Thought. An Introduction to the Philosophy of Mathematics. 1965 ISBN 90-277-0070-2 E. W. Beth and J. Piaget, Mathematical Epistemology and Psychology. Translated from French by W. Mays. 1966 ISBN 90-277-0071-0 G. K¨ung, Ontology and the Logistic Analysis of Language. An Enquiry into the Contemporary Views on Universals. Revised ed., translated from German. 1967 ISBN 90-277-0028-1 R. S. Cohen and M. W. Wartofsky (eds.), Proceedings of the Boston Colloquium for 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 C. D. Broad, Induction, Probability, and Causation. Selected Papers. 1968 ISBN 90-277-0012-5 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 N. Rescher, Topics in Philosophical Logic. 1968 ISBN 90-277-0084-2 R. S. Cohen and M. W. Wartofsky (eds.), Proceedings of the Boston Colloquium for the Philosophy of Science, 1966–1968, Part I. [Boston Studies in the Philosophy of Science, Vol. IV] 1969 ISBN 90-277-0014-1 R. S. Cohen and M. W. Wartofsky (eds.), Proceedings of the Boston Colloquium for the Philosophy of Science, 1966–1968, Part II. [Boston Studies in the Philosophy of Science, Vol. V] 1969 ISBN 90-277-0015-X J. W. Davis, D. J. Hockney and W. K. Wilson (eds.), Philosophical Logic. 1969 ISBN 90-277-0075-3 D. Davidson and J. Hintikka (eds.), Words and Objections. Essays on the Work of W. V. Quine. 1969, rev. ed. 1975 ISBN 90-277-0074-5; Pb 90-277-0602-6 P. Suppes, Studies in the Methodology and Foundations of Science. Selected Papers from 1951 to 1969. 1969 ISBN 90-277-0020-6 J. Hintikka, Models for Modalities. Selected Essays. 1969 ISBN 90-277-0078-8; Pb 90-277-0598-4

SYNTHESE LIBRARY 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39.

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N. Rescher et al. (eds.), Essays in Honor of Carl G. Hempel. A Tribute on the Occasion of His 65th Birthday. 1969 ISBN 90-277-0085-0 P. V. Tavanec (ed.), Problems of the Logic of Scientific Knowledge. Translated from Russian. 1970 ISBN 90-277-0087-7 M. Swain (ed.), Induction, Acceptance, and Rational Belief. 1970 ISBN 90-277-0086-9 R. S. Cohen and R. J. Seeger (eds.), Ernst Mach: Physicist and Philosopher. [Boston Studies in the Philosophy of Science, Vol. VI]. 1970 ISBN 90-277-0016-8 J. Hintikka and P. Suppes, Information and Inference. 1970 ISBN 90-277-0155-5 K. Lambert, Philosophical Problems in Logic. Some Recent Developments. 1970 ISBN 90-277-0079-6 R. A. Eberle, Nominalistic Systems. 1970 ISBN 90-277-0161-X P. Weingartner and G. Zecha (eds.), Induction, Physics, and Ethics. 1970 ISBN 90-277-0158-X E. W. Beth, Aspects of Modern Logic. Translated from Dutch. 1970 ISBN 90-277-0173-3 R. Hilpinen (ed.), Deontic Logic. Introductory and Systematic Readings. 1971 See also No. 152. ISBN Pb (1981 rev.) 90-277-1302-2 J.-L. Krivine, Introduction to Axiomatic Set Theory. Translated from French. 1971 ISBN 90-277-0169-5; Pb 90-277-0411-2 J. D. Sneed, The Logical Structure of Mathematical Physics. 2nd rev. ed., 1979 ISBN 90-277-1056-2; Pb 90-277-1059-7 C. R. Kordig, The Justification of Scientific Change. 1971 ISBN 90-277-0181-4; Pb 90-277-0475-9 ˇ M. Capek, Bergson and Modern Physics. A Reinterpretation and Re-evaluation. [Boston Studies in the Philosophy of Science, Vol. VII] 1971 ISBN 90-277-0186-5 N. R. Hanson, What I Do Not Believe, and Other Essays. Ed. by S. Toulmin and H. Woolf. 1971 ISBN 90-277-0191-1 R. C. Buck and R. S. Cohen (eds.), PSA 1970. Proceedings of the Second Biennial Meeting of the Philosophy of Science Association, Boston, Fall 1970. In Memory of Rudolf Carnap. [Boston Studies in the Philosophy of Science, Vol. VIII] 1971 ISBN 90-277-0187-3; Pb 90-277-0309-4 D. Davidson and G. Harman (eds.), Semantics of Natural Language. 1972 ISBN 90-277-0304-3; Pb 90-277-0310-8 Y. Bar-Hillel (ed.), Pragmatics of Natural Languages. 1971 ISBN 90-277-0194-6; Pb 90-277-0599-2 S. Stenlund, Combinators, γ Terms and Proof Theory. 1972 ISBN 90-277-0305-1 M. Strauss, Modern Physics and Its Philosophy. Selected Paper in the Logic, History, and Philosophy of Science. 1972 ISBN 90-277-0230-6 M. Bunge, Method, Model and Matter. 1973 ISBN 90-277-0252-7 M. Bunge, Philosophy of Physics. 1973 ISBN 90-277-0253-5 A. A. Zinov’ev, Foundations of the Logical Theory of Scientific Knowledge (Complex Logic). Revised and enlarged English edition with an appendix by G. A. Smirnov, E. A. Sidorenka, A. M. Fedina and L. A. Bobrova. [Boston Studies in the Philosophy of Science, Vol. IX] 1973 ISBN 90-277-0193-8; Pb 90-277-0324-8 L. Tondl, Scientific Procedures. A Contribution concerning the Methodological Problems of Scientific Concepts and Scientific Explanation. Translated from Czech by D. Short. Edited by R.S. Cohen and M.W. Wartofsky. [Boston Studies in the Philosophy of Science, Vol. X] 1973 ISBN 90-277-0147-4; Pb 90-277-0323-X N. R. Hanson, Constellations and Conjectures. 1973 ISBN 90-277-0192-X

SYNTHESE LIBRARY 49. 50. 51. 52. 53. 54.

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56. 57. 58. 59.

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K. J. J. Hintikka, J. M. E. Moravcsik and P. Suppes (eds.), Approaches to Natural Language. 1973 ISBN 90-277-0220-9; Pb 90-277-0233-0 M. Bunge (ed.), Exact Philosophy. Problems, Tools and Goals. 1973 ISBN 90-277-0251-9 R. J. Bogdan and I. Niiniluoto (eds.), Logic, Language and Probability. 1973 ISBN 90-277-0312-4 G. Pearce and P. Maynard (eds.), Conceptual Change. 1973 ISBN 90-277-0287-X; Pb 90-277-0339-6 I. Niiniluoto and R. Tuomela, Theoretical Concepts and Hypothetico-inductive Inference. 1973 ISBN 90-277-0343-4 R. Fraiss´e, Course of Mathematical Logic – Volume 1: Relation and Logical Formula. Translated from French. 1973 ISBN 90-277-0268-3; Pb 90-277-0403-1 (For Volume 2 see under No. 69). A. Gr¨unbaum, Philosophical Problems of Space and Time. Edited by R.S. Cohen and M.W. Wartofsky. 2nd enlarged ed. [Boston Studies in the Philosophy of Science, Vol. XII] 1973 ISBN 90-277-0357-4; Pb 90-277-0358-2 P. Suppes (ed.), Space, Time and Geometry. 1973 ISBN 90-277-0386-8; Pb 90-277-0442-2 H. Kelsen, Essays in Legal and Moral Philosophy. Selected and introduced by O. Weinberger. Translated from German by P. Heath. 1973 ISBN 90-277-0388-4 R. J. Seeger and R. S. Cohen (eds.), Philosophical Foundations of Science. [Boston Studies in the Philosophy of Science, Vol. XI] 1974 ISBN 90-277-0390-6; Pb 90-277-0376-0 R. S. Cohen and M. W. Wartofsky (eds.), Logical and Epistemological Studies in Contemporary Physics. [Boston Studies in the Philosophy of Science, Vol. XIII] 1973 ISBN 90-277-0391-4; Pb 90-277-0377-9 R. S. Cohen and M. W. Wartofsky (eds.), Methodological and Historical Essays in the Natural and Social Sciences. Proceedings of the Boston Colloquium for the Philosophy of Science, 1969–1972. [Boston Studies in the Philosophy of Science, Vol. XIV] 1974 ISBN 90-277-0392-2; Pb 90-277-0378-7 R. S. Cohen, J. J. Stachel and M. W. Wartofsky (eds.), For Dirk Struik. Scientific, Historical and Political Essays. [Boston Studies in the Philosophy of Science, Vol. XV] 1974 ISBN 90-277-0393-0; Pb 90-277-0379-5 K. Ajdukiewicz, Pragmatic Logic. Translated from Polish by O. Wojtasiewicz. 1974 ISBN 90-277-0326-4 S. Stenlund (ed.), Logical Theory and Semantic Analysis. Essays dedicated to Stig Kanger on His 50th Birthday. 1974 ISBN 90-277-0438-4 K. F. Schaffner and R. S. Cohen (eds.), PSA 1972. Proceedings of the Third Biennial Meeting of the Philosophy of Science Association. [Boston Studies in the Philosophy of Science, Vol. XX] 1974 ISBN 90-277-0408-2; Pb 90-277-0409-0 H. E. Kyburg, Jr., The Logical Foundations of Statistical Inference. 1974 ISBN 90-277-0330-2; Pb 90-277-0430-9 M. Grene, The Understanding of Nature. Essays in the Philosophy of Biology. [Boston Studies in the Philosophy of Science, Vol. XXIII] 1974 ISBN 90-277-0462-7; Pb 90-277-0463-5 J. M. Broekman, Structuralism: Moscow, Prague, Paris. Translated from German. 1974 ISBN 90-277-0478-3 N. Geschwind, Selected Papers on Language and the Brain. [Boston Studies in the Philosophy of Science, Vol. XVI] 1974 ISBN 90-277-0262-4; Pb 90-277-0263-2 R. Fraiss´e, Course of Mathematical Logic – Volume 2: Model Theory. Translated from French. 1974 ISBN 90-277-0269-1; Pb 90-277-0510-0 (For Volume 1 see under No. 54)

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A. Grzegorczyk, An Outline of Mathematical Logic. Fundamental Results and Notions explained with all Details. Translated from Polish. 1974 ISBN 90-277-0359-0; Pb 90-277-0447-3 F. von Kutschera, Philosophy of Language. 1975 ISBN 90-277-0591-7 J. Manninen and R. Tuomela (eds.), Essays on Explanation and Understanding. Studies in the Foundations of Humanities and Social Sciences. 1976 ISBN 90-277-0592-5 J. Hintikka (ed.), Rudolf Carnap, Logical Empiricist. Materials and Perspectives. 1975 ISBN 90-277-0583-6 ˇ M. Capek (ed.), The Concepts of Space and Time. Their Structure and Their Development. [Boston Studies in the Philosophy of Science, Vol. XXII] 1976 ISBN 90-277-0355-8; Pb 90-277-0375-2 J. Hintikka and U. Remes, The Method of Analysis. Its Geometrical Origin and Its General Significance. [Boston Studies in the Philosophy of Science, Vol. XXV] 1974 ISBN 90-277-0532-1; Pb 90-277-0543-7 J. E. Murdoch and E. D. Sylla (eds.), The Cultural Context of Medieval Learning. [Boston Studies in the Philosophy of Science, Vol. XXVI] 1975 ISBN 90-277-0560-7; Pb 90-277-0587-9 S. Amsterdamski, Between Experience and Metaphysics. Philosophical Problems of the Evolution of Science. [Boston Studies in the Philosophy of Science, Vol. XXXV] 1975 ISBN 90-277-0568-2; Pb 90-277-0580-1 P. Suppes (ed.), Logic and Probability in Quantum Mechanics. 1976 ISBN 90-277-0570-4; Pb 90-277-1200-X H. von Helmholtz: Epistemological Writings. The Paul Hertz / Moritz Schlick Centenary Edition of 1921 with Notes and Commentary by the Editors. Newly translated from German by M. F. Lowe. Edited, with an Introduction and Bibliography, by R. S. Cohen and Y. Elkana. [Boston Studies in the Philosophy of Science, Vol. XXXVII] 1975 ISBN 90-277-0290-X; Pb 90-277-0582-8 J. Agassi, Science in Flux. [Boston Studies in the Philosophy of Science, Vol. XXVIII] 1975 ISBN 90-277-0584-4; Pb 90-277-0612-2 S. G. Harding (ed.), Can Theories Be Refuted? Essays on the Duhem-Quine Thesis. 1976 ISBN 90-277-0629-8; Pb 90-277-0630-1 S. Nowak, Methodology of Sociological Research. General Problems. 1977 ISBN 90-277-0486-4 J. Piaget, J.-B. Grize, A. Szemin´sska and V. Bang, Epistemology and Psychology of Functions. Translated from French. 1977 ISBN 90-277-0804-5 M. Grene and E. Mendelsohn (eds.), Topics in the Philosophy of Biology. [Boston Studies in the Philosophy of Science, Vol. XXVII] 1976 ISBN 90-277-0595-X; Pb 90-277-0596-8 E. Fischbein, The Intuitive Sources of Probabilistic Thinking in Children. 1975 ISBN 90-277-0626-3; Pb 90-277-1190-9 E. W. Adams, The Logic of Conditionals. An Application of Probability to Deductive Logic. 1975 ISBN 90-277-0631-X M. Przełe¸cki and R. W´ojcicki (eds.), Twenty-Five Years of Logical Methodology in Poland. Translated from Polish. 1976 ISBN 90-277-0601-8 J. Topolski, The Methodology of History. Translated from Polish by O. Wojtasiewicz. 1976 ISBN 90-277-0550-X A. Kasher (ed.), Language in Focus: Foundations, Methods and Systems. Essays dedicated to Yehoshua Bar-Hillel. [Boston Studies in the Philosophy of Science, Vol. XLIII] 1976 ISBN 90-277-0644-1; Pb 90-277-0645-X

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108. 109. 110. 111. 112. 113. 114.

J. Hintikka, The Intentions of Intentionality and Other New Models for Modalities. 1975 ISBN 90-277-0633-6; Pb 90-277-0634-4 W. Stegm¨uller, Collected Papers on Epistemology, Philosophy of Science and History of Philosophy. 2 Volumes. 1977 Set ISBN 90-277-0767-7 D. M. Gabbay, Investigations in Modal and Tense Logics with Applications to Problems in Philosophy and Linguistics. 1976 ISBN 90-277-0656-5 R. J. Bogdan, Local Induction. 1976 ISBN 90-277-0649-2 S. Nowak, Understanding and Prediction. Essays in the Methodology of Social and Behavioral Theories. 1976 ISBN 90-277-0558-5; Pb 90-277-1199-2 P. Mittelstaedt, Philosophical Problems of Modern Physics. [Boston Studies in the Philosophy of Science, Vol. XVIII] 1976 ISBN 90-277-0285-3; Pb 90-277-0506-2 G. Holton and W. A. Blanpied (eds.), Science and Its Public: The Changing Relationship. [Boston Studies in the Philosophy of Science, Vol. XXXIII] 1976 ISBN 90-277-0657-3; Pb 90-277-0658-1 M. Brand and D. Walton (eds.), Action Theory. 1976 ISBN 90-277-0671-9 P. Gochet, Outline of a Nominalist Theory of Propositions. An Essay in the Theory of Meaning and in the Philosophy of Logic. 1980 ISBN 90-277-1031-7 R. S. Cohen, P. K. Feyerabend, and M. W. Wartofsky (eds.), Essays in Memory of Imre Lakatos. [Boston Studies in the Philosophy of Science, Vol. XXXIX] 1976 ISBN 90-277-0654-9; Pb 90-277-0655-7 R. S. Cohen and J. J. Stachel (eds.), Selected Papers of L´eon Rosenfield. [Boston Studies in the Philosophy of Science, Vol. XXI] 1979 ISBN 90-277-0651-4; Pb 90-277-0652-2 R. S. Cohen, C. A. Hooker, A. C. Michalos and J. W. van Evra (eds.), PSA 1974. Proceedings of the 1974 Biennial Meeting of the Philosophy of Science Association. [Boston Studies in the Philosophy of Science, Vol. XXXII] 1976 ISBN 90-277-0647-6; Pb 90-277-0648-4 Y. Fried and J. Agassi, Paranoia. A Study in Diagnosis. [Boston Studies in the Philosophy of Science, Vol. L] 1976 ISBN 90-277-0704-9; Pb 90-277-0705-7 M. Przełe¸cki, K. Szaniawski and R. W´ojcicki (eds.), Formal Methods in the Methodology of Empirical Sciences. 1976 ISBN 90-277-0698-0 J. M. Vickers, Belief and Probability. 1976 ISBN 90-277-0744-8 K. H. Wolff, Surrender and Catch. Experience and Inquiry Today. [Boston Studies in the Philosophy of Science, Vol. LI] 1976 ISBN 90-277-0758-8; Pb 90-277-0765-0 K. Kos´ık, Dialectics of the Concrete. A Study on Problems of Man and World. [Boston Studies in the Philosophy of Science, Vol. LII] 1976 ISBN 90-277-0761-8; Pb 90-277-0764-2 N. Goodman, The Structure of Appearance. 3rd ed. with an Introduction by G. Hellman. [Boston Studies in the Philosophy of Science, Vol. LIII] 1977 ISBN 90-277-0773-1; Pb 90-277-0774-X K. Ajdukiewicz, The Scientific World-Perspective and Other Essays, 1931-1963. Translated from Polish. Edited and with an Introduction by J. Giedymin. 1978 ISBN 90-277-0527-5 R. L. Causey, Unity of Science. 1977 ISBN 90-277-0779-0 R. E. Grandy, Advanced Logic for Applications. 1977 ISBN 90-277-0781-2 R. P. McArthur, Tense Logic. 1976 ISBN 90-277-0697-2 L. Lindahl, Position and Change. A Study in Law and Logic. Translated from Swedish by P. Needham. 1977 ISBN 90-277-0787-1 R. Tuomela, Dispositions. 1978 ISBN 90-277-0810-X H. A. Simon, Models of Discovery and Other Topics in the Methods of Science. [Boston Studies in the Philosophy of Science, Vol. LIV] 1977 ISBN 90-277-0812-6; Pb 90-277-0858-4

SYNTHESE LIBRARY 115. R. D. Rosenkrantz, Inference, Method and Decision. Towards a Bayesian Philosophy of Science. 1977 ISBN 90-277-0817-7; Pb 90-277-0818-5 116. R. Tuomela, Human Action and Its Explanation. A Study on the Philosophical Foundations of Psychology. 1977 ISBN 90-277-0824-X 117. M. Lazerowitz, The Language of Philosophy. Freud and Wittgenstein. [Boston Studies in the Philosophy of Science, Vol. LV] 1977 ISBN 90-277-0826-6; Pb 90-277-0862-2 118. Not published 119. J. Pelc (ed.), Semiotics in Poland, 1894–1969. Translated from Polish. 1979 ISBN 90-277-0811-8 120. I. P¨orn, Action Theory and Social Science. Some Formal Models. 1977 ISBN 90-277-0846-0 121. J. Margolis, Persons and Mind. The Prospects of Nonreductive Materialism. [Boston Studies in the Philosophy of Science, Vol. LVII] 1977 ISBN 90-277-0854-1; Pb 90-277-0863-0 122. J. Hintikka, I. Niiniluoto, and E. Saarinen (eds.), Essays on Mathematical and Philosophical Logic. 1979 ISBN 90-277-0879-7 123. T. A. F. Kuipers, Studies in Inductive Probability and Rational Expectation. 1978 ISBN 90-277-0882-7 124. E. Saarinen, R. Hilpinen, I. Niiniluoto and M. P. Hintikka (eds.), Essays in Honour of Jaakko Hintikka on the Occasion of His 50th Birthday. 1979 ISBN 90-277-0916-5 125. G. Radnitzky and G. Andersson (eds.), Progress and Rationality in Science. [Boston Studies in the Philosophy of Science, Vol. LVIII] 1978 ISBN 90-277-0921-1; Pb 90-277-0922-X 126. P. Mittelstaedt, Quantum Logic. 1978 ISBN 90-277-0925-4 127. K. A. Bowen, Model Theory for Modal Logic. Kripke Models for Modal Predicate Calculi. 1979 ISBN 90-277-0929-7 128. H. A. Bursen, Dismantling the Memory Machine. A Philosophical Investigation of Machine Theories of Memory. 1978 ISBN 90-277-0933-5 129. M. W. Wartofsky, Models. Representation and the Scientific Understanding. [Boston Studies in the Philosophy of Science, Vol. XLVIII] 1979 ISBN 90-277-0736-7; Pb 90-277-0947-5 130. D. Ihde, Technics and Praxis. A Philosophy of Technology. [Boston Studies in the Philosophy of Science, Vol. XXIV] 1979 ISBN 90-277-0953-X; Pb 90-277-0954-8 131. J. J. Wiatr (ed.), Polish Essays in the Methodology of the Social Sciences. [Boston Studies in the Philosophy of Science, Vol. XXIX] 1979 ISBN 90-277-0723-5; Pb 90-277-0956-4 132. W. C. Salmon (ed.), Hans Reichenbach: Logical Empiricist. 1979 ISBN 90-277-0958-0 133. P. Bieri, R.-P. Horstmann and L. Kr¨uger (eds.), Transcendental Arguments in Science. Essays in Epistemology. 1979 ISBN 90-277-0963-7; Pb 90-277-0964-5 134. M. Markovi´c and G. Petrovi´c (eds.), Praxis. Yugoslav Essays in the Philosophy and Methodology of the Social Sciences. [Boston Studies in the Philosophy of Science, Vol. XXXVI] 1979 ISBN 90-277-0727-8; Pb 90-277-0968-8 135. R. W´ojcicki, Topics in the Formal Methodology of Empirical Sciences. Translated from Polish. 1979 ISBN 90-277-1004-X 136. G. Radnitzky and G. Andersson (eds.), The Structure and Development of Science. [Boston Studies in the Philosophy of Science, Vol. LIX] 1979 ISBN 90-277-0994-7; Pb 90-277-0995-5 137. J. C. Webb, Mechanism, Mentalism and Metamathematics. An Essay on Finitism. 1980 ISBN 90-277-1046-5 138. D. F. Gustafson and B. L. Tapscott (eds.), Body, Mind and Method. Essays in Honor of Virgil C. Aldrich. 1979 ISBN 90-277-1013-9 139. L. Nowak, The Structure of Idealization. Towards a Systematic Interpretation of the Marxian Idea of Science. 1980 ISBN 90-277-1014-7

SYNTHESE LIBRARY 140. C. Perelman, The New Rhetoric and the Humanities. Essays on Rhetoric and Its Applications. Translated from French and German. With an Introduction by H. Zyskind. 1979 ISBN 90-277-1018-X; Pb 90-277-1019-8 141. W. Rabinowicz, Universalizability. A Study in Morals and Metaphysics. 1979 ISBN 90-277-1020-2 142. C. Perelman, Justice, Law and Argument. Essays on Moral and Legal Reasoning. Translated from French and German. With an Introduction by H.J. Berman. 1980 ISBN 90-277-1089-9; Pb 90-277-1090-2 ¨ 143. S. Kanger and S. Ohman (eds.), Philosophy and Grammar. Papers on the Occasion of the Quincentennial of Uppsala University. 1981 ISBN 90-277-1091-0 144. T. Pawlowski, Concept Formation in the Humanities and the Social Sciences. 1980 ISBN 90-277-1096-1 145. J. Hintikka, D. Gruender and E. Agazzi (eds.), Theory Change, Ancient Axiomatics and Galileo’s Methodology. Proceedings of the 1978 Pisa Conference on the History and Philosophy of Science, Volume I. 1981 ISBN 90-277-1126-7 146. J. Hintikka, D. Gruender and E. Agazzi (eds.), Probabilistic Thinking, Thermodynamics, and the Interaction of the History and Philosophy of Science. Proceedings of the 1978 Pisa Conference on the History and Philosophy of Science, Volume II. 1981 ISBN 90-277-1127-5 147. U. M¨onnich (ed.), Aspects of Philosophical Logic. Some Logical Forays into Central Notions of Linguistics and Philosophy. 1981 ISBN 90-277-1201-8 148. D. M. Gabbay, Semantical Investigations in Heyting’s Intuitionistic Logic. 1981 ISBN 90-277-1202-6 149. E. Agazzi (ed.), Modern Logic – A Survey. Historical, Philosophical, and Mathematical Aspects of Modern Logic and Its Applications. 1981 ISBN 90-277-1137-2 150. A. F. Parker-Rhodes, The Theory of Indistinguishables. A Search for Explanatory Principles below the Level of Physics. 1981 ISBN 90-277-1214-X 151. J. C. Pitt, Pictures, Images, and Conceptual Change. An Analysis of Wilfrid Sellars’ Philosophy of Science. 1981 ISBN 90-277-1276-X; Pb 90-277-1277-8 152. R. Hilpinen (ed.), New Studies in Deontic Logic. Norms, Actions, and the Foundations of Ethics. 1981 ISBN 90-277-1278-6; Pb 90-277-1346-4 153. C. Dilworth, Scientific Progress. A Study Concerning the Nature of the Relation between Successive Scientific Theories. 3rd rev. ed., 1994 ISBN 0-7923-2487-0; Pb 0-7923-2488-9 154. D. Woodruff Smith and R. McIntyre, Husserl and Intentionality. A Study of Mind, Meaning, and Language. 1982 ISBN 90-277-1392-8; Pb 90-277-1730-3 155. R. J. Nelson, The Logic of Mind. 2nd. ed., 1989 ISBN 90-277-2819-4; Pb 90-277-2822-4 156. J. F. A. K. van Benthem, The Logic of Time. A Model-Theoretic Investigation into the Varieties of Temporal Ontology, and Temporal Discourse. 1983; 2nd ed., 1991 ISBN 0-7923-1081-0 157. R. Swinburne (ed.), Space, Time and Causality. 1983 ISBN 90-277-1437-1 158. E. T. Jaynes, Papers on Probability, Statistics and Statistical Physics. Ed. by R. D. Rozenkrantz. 1983 ISBN 90-277-1448-7; Pb (1989) 0-7923-0213-3 159. T. Chapman, Time: A Philosophical Analysis. 1982 ISBN 90-277-1465-7 160. E. N. Zalta, Abstract Objects. An Introduction to Axiomatic Metaphysics. 1983 ISBN 90-277-1474-6 161. S. Harding and M. B. Hintikka (eds.), Discovering Reality. Feminist Perspectives on Epistemology, Metaphysics, Methodology, and Philosophy of Science. 1983 ISBN 90-277-1496-7; Pb 90-277-1538-6 162. M. A. Stewart (ed.), Law, Morality and Rights. 1983 ISBN 90-277-1519-X

SYNTHESE LIBRARY 163. D. Mayr and G. S¨ussmann (eds.), Space, Time, and Mechanics. Basic Structures of a Physical Theory. 1983 ISBN 90-277-1525-4 164. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. I: Elements of Classical Logic. 1983 ISBN 90-277-1542-4 165. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. II: Extensions of Classical Logic. 1984 ISBN 90-277-1604-8 166. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. III: Alternative to Classical Logic. 1986 ISBN 90-277-1605-6 167. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. IV: Topics in the Philosophy of Language. 1989 ISBN 90-277-1606-4 168. A. J. I. Jones, Communication and Meaning. An Essay in Applied Modal Logic. 1983 ISBN 90-277-1543-2 169. M. Fitting, Proof Methods for Modal and Intuitionistic Logics. 1983 ISBN 90-277-1573-4 170. J. Margolis, Culture and Cultural Entities. Toward a New Unity of Science. 1984 ISBN 90-277-1574-2 171. R. Tuomela, A Theory of Social Action. 1984 ISBN 90-277-1703-6 172. J. J. E. Gracia, E. Rabossi, E. Villanueva and M. Dascal (eds.), Philosophical Analysis in Latin America. 1984 ISBN 90-277-1749-4 173. P. Ziff, Epistemic Analysis. A Coherence Theory of Knowledge. 1984 ISBN 90-277-1751-7 174. P. Ziff, Antiaesthetics. An Appreciation of the Cow with the Subtile Nose. 1984 ISBN 90-277-1773-7 175. W. Balzer, D. A. Pearce, and H.-J. Schmidt (eds.), Reduction in Science. Structure, Examples, Philosophical Problems. 1984 ISBN 90-277-1811-3 176. A. Peczenik, L. Lindahl and B. van Roermund (eds.), Theory of Legal Science. Proceedings of the Conference on Legal Theory and Philosophy of Science (Lund, Sweden, December 1983). 1984 ISBN 90-277-1834-2 177. I. Niiniluoto, Is Science Progressive? 1984 ISBN 90-277-1835-0 178. B. K. Matilal and J. L. Shaw (eds.), Analytical Philosophy in Comparative Perspective. Exploratory Essays in Current Theories and Classical Indian Theories of Meaning and Reference. 1985 ISBN 90-277-1870-9 179. P. Kroes, Time: Its Structure and Role in Physical Theories. 1985 ISBN 90-277-1894-6 180. J. H. Fetzer, Sociobiology and Epistemology. 1985 ISBN 90-277-2005-3; Pb 90-277-2006-1 181. L. Haaparanta and J. Hintikka (eds.), Frege Synthesized. Essays on the Philosophical and Foundational Work of Gottlob Frege. 1986 ISBN 90-277-2126-2 182. M. Detlefsen, Hilbert’s Program. An Essay on Mathematical Instrumentalism. 1986 ISBN 90-277-2151-3 183. J. L. Golden and J. J. Pilotta (eds.), Practical Reasoning in Human Affairs. Studies in Honor of Chaim Perelman. 1986 ISBN 90-277-2255-2 184. H. Zandvoort, Models of Scientific Development and the Case of Nuclear Magnetic Resonance. 1986 ISBN 90-277-2351-6 185. I. Niiniluoto, Truthlikeness. 1987 ISBN 90-277-2354-0 186. W. Balzer, C. U. Moulines and J. D. Sneed, An Architectonic for Science. The Structuralist Program. 1987 ISBN 90-277-2403-2 187. D. Pearce, Roads to Commensurability. 1987 ISBN 90-277-2414-8 188. L. M. Vaina (ed.), Matters of Intelligence. Conceptual Structures in Cognitive Neuroscience. 1987 ISBN 90-277-2460-1

SYNTHESE LIBRARY 189. H. Siegel, Relativism Refuted. A Critique of Contemporary Epistemological Relativism. 1987 ISBN 90-277-2469-5 190. W. Callebaut and R. Pinxten, Evolutionary Epistemology. A Multiparadigm Program, with a Complete Evolutionary Epistemology Bibliograph. 1987 ISBN 90-277-2582-9 191. J. Kmita, Problems in Historical Epistemology. 1988 ISBN 90-277-2199-8 192. J. H. Fetzer (ed.), Probability and Causality. Essays in Honor of Wesley C. Salmon, with an Annotated Bibliography. 1988 ISBN 90-277-2607-8; Pb 1-5560-8052-2 193. A. Donovan, L. Laudan and R. Laudan (eds.), Scrutinizing Science. Empirical Studies of Scientific Change. 1988 ISBN 90-277-2608-6 194. H.R. Otto and J.A. Tuedio (eds.), Perspectives on Mind. 1988 ISBN 90-277-2640-X 195. D. Batens and J.P. van Bendegem (eds.), Theory and Experiment. Recent Insights and New Perspectives on Their Relation. 1988 ISBN 90-277-2645-0 ¨ 196. J. Osterberg, Self and Others. A Study of Ethical Egoism. 1988 ISBN 90-277-2648-5 197. D.H. Helman (ed.), Analogical Reasoning. Perspectives of Artificial Intelligence, Cognitive Science, and Philosophy. 1988 ISBN 90-277-2711-2 198. J. Wolen´ski, Logic and Philosophy in the Lvov-Warsaw School. 1989 ISBN 90-277-2749-X 199. R. W´ojcicki, Theory of Logical Calculi. Basic Theory of Consequence Operations. 1988 ISBN 90-277-2785-6 200. J. Hintikka and M.B. Hintikka, The Logic of Epistemology and the Epistemology of Logic. Selected Essays. 1989 ISBN 0-7923-0040-8; Pb 0-7923-0041-6 201. E. Agazzi (ed.), Probability in the Sciences. 1988 ISBN 90-277-2808-9 202. M. Meyer (ed.), From Metaphysics to Rhetoric. 1989 ISBN 90-277-2814-3 203. R.L. Tieszen, Mathematical Intuition. Phenomenology and Mathematical Knowledge. 1989 ISBN 0-7923-0131-5 204. A. Melnick, Space, Time, and Thought in Kant. 1989 ISBN 0-7923-0135-8 205. D.W. Smith, The Circle of Acquaintance. Perception, Consciousness, and Empathy. 1989 ISBN 0-7923-0252-4 206. M.H. Salmon (ed.), The Philosophy of Logical Mechanism. Essays in Honor of Arthur W. Burks. With his Responses, and with a Bibliography of Burk’s Work. 1990 ISBN 0-7923-0325-3 207. M. Kusch, Language as Calculus vs. Language as Universal Medium. A Study in Husserl, Heidegger, and Gadamer. 1989 ISBN 0-7923-0333-4 208. T.C. Meyering, Historical Roots of Cognitive Science. The Rise of a Cognitive Theory of Perception from Antiquity to the Nineteenth Century. 1989 ISBN 0-7923-0349-0 209. P. Kosso, Observability and Observation in Physical Science. 1989 ISBN 0-7923-0389-X 210. J. Kmita, Essays on the Theory of Scientific Cognition. 1990 ISBN 0-7923-0441-1 211. W. Sieg (ed.), Acting and Reflecting. The Interdisciplinary Turn in Philosophy. 1990 ISBN 0-7923-0512-4 212. J. Karpin´ski, Causality in Sociological Research. 1990 ISBN 0-7923-0546-9 213. H.A. Lewis (ed.), Peter Geach: Philosophical Encounters. 1991 ISBN 0-7923-0823-9 214. M. Ter Hark, Beyond the Inner and the Outer. Wittgenstein’s Philosophy of Psychology. 1990 ISBN 0-7923-0850-6 215. M. Gosselin, Nominalism and Contemporary Nominalism. Ontological and Epistemological Implications of the Work of W.V.O. Quine and of N. Goodman. 1990 ISBN 0-7923-0904-9 216. J.H. Fetzer, D. Shatz and G. Schlesinger (eds.), Definitions and Definability. Philosophical Perspectives. 1991 ISBN 0-7923-1046-2 217. E. Agazzi and A. Cordero (eds.), Philosophy and the Origin and Evolution of the Universe. 1991 ISBN 0-7923-1322-4

SYNTHESE LIBRARY 218. M. Kusch, Foucault’s Strata and Fields. An Investigation into Archaeological and Genealogical Science Studies. 1991 ISBN 0-7923-1462-X 219. C.J. Posy, Kant’s Philosophy of Mathematics. Modern Essays. 1992 ISBN 0-7923-1495-6 220. G. Van de Vijver, New Perspectives on Cybernetics. Self-Organization, Autonomy and Connectionism. 1992 ISBN 0-7923-1519-7 221. J.C. Ny´ıri, Tradition and Individuality. Essays. 1992 ISBN 0-7923-1566-9 222. R. Howell, Kant’s Transcendental Deduction. An Analysis of Main Themes in His Critical Philosophy. 1992 ISBN 0-7923-1571-5 223. A. Garc´ıa de la Sienra, The Logical Foundations of the Marxian Theory of Value. 1992 ISBN 0-7923-1778-5 224. D.S. Shwayder, Statement and Referent. An Inquiry into the Foundations of Our Conceptual Order. 1992 ISBN 0-7923-1803-X 225. M. Rosen, Problems of the Hegelian Dialectic. Dialectic Reconstructed as a Logic of Human Reality. 1993 ISBN 0-7923-2047-6 226. P. Suppes, Models and Methods in the Philosophy of Science: Selected Essays. 1993 ISBN 0-7923-2211-8 227. R. M. Dancy (ed.), Kant and Critique: New Essays in Honor of W. H. Werkmeister. 1993 ISBN 0-7923-2244-4 228. J. Wole´nski (ed.), Philosophical Logic in Poland. 1993 ISBN 0-7923-2293-2 229. M. De Rijke (ed.), Diamonds and Defaults. Studies in Pure and Applied Intensional Logic. 1993 ISBN 0-7923-2342-4 230. B.K. Matilal and A. Chakrabarti (eds.), Knowing from Words. Western and Indian Philosophical Analysis of Understanding and Testimony. 1994 ISBN 0-7923-2345-9 231. S.A. Kleiner, The Logic of Discovery. A Theory of the Rationality of Scientific Research. 1993 ISBN 0-7923-2371-8 232. R. Festa, Optimum Inductive Methods. A Study in Inductive Probability, Bayesian Statistics, and Verisimilitude. 1993 ISBN 0-7923-2460-9 233. P. Humphreys (ed.), Patrick Suppes: Scientific Philosopher. Vol. 1: Probability and Probabilistic Causality. 1994 ISBN 0-7923-2552-4 234. P. Humphreys (ed.), Patrick Suppes: Scientific Philosopher. Vol. 2: Philosophy of Physics, Theory Structure, and Measurement Theory. 1994 ISBN 0-7923-2553-2 235. P. Humphreys (ed.), Patrick Suppes: Scientific Philosopher. Vol. 3: Language, Logic, and Psychology. 1994 ISBN 0-7923-2862-0 Set ISBN (Vols 233–235) 0-7923-2554-0 236. D. Prawitz and D. Westerst˚ahl (eds.), Logic and Philosophy of Science in Uppsala. Papers from the 9th International Congress of Logic, Methodology, and Philosophy of Science. 1994 ISBN 0-7923-2702-0 237. L. Haaparanta (ed.), Mind, Meaning and Mathematics. Essays on the Philosophical Views of Husserl and Frege. 1994 ISBN 0-7923-2703-9 238. J. Hintikka (ed.), Aspects of Metaphor. 1994 ISBN 0-7923-2786-1 239. B. McGuinness and G. Oliveri (eds.), The Philosophy of Michael Dummett. With Replies from Michael Dummett. 1994 ISBN 0-7923-2804-3 240. D. Jamieson (ed.), Language, Mind, and Art. Essays in Appreciation and Analysis, In Honor of Paul Ziff. 1994 ISBN 0-7923-2810-8 241. G. Preyer, F. Siebelt and A. Ulfig (eds.), Language, Mind and Epistemology. On Donald Davidson’s Philosophy. 1994 ISBN 0-7923-2811-6 242. P. Ehrlich (ed.), Real Numbers, Generalizations of the Reals, and Theories of Continua. 1994 ISBN 0-7923-2689-X

SYNTHESE LIBRARY 243. G. Debrock and M. Hulswit (eds.), Living Doubt. Essays concerning the epistemology of Charles Sanders Peirce. 1994 ISBN 0-7923-2898-1 244. J. Srzednicki, To Know or Not to Know. Beyond Realism and Anti-Realism. 1994 ISBN 0-7923-2909-0 245. R. Egidi (ed.), Wittgenstein: Mind and Language. 1995 ISBN 0-7923-3171-0 246. A. Hyslop, Other Minds. 1995 ISBN 0-7923-3245-8 247. L. P´olos and M. Masuch (eds.), Applied Logic: How, What and Why. Logical Approaches to Natural Language. 1995 ISBN 0-7923-3432-9 248. M. Krynicki, M. Mostowski and L.M. Szczerba (eds.), Quantifiers: Logics, Models and Computation. Volume One: Surveys. 1995 ISBN 0-7923-3448-5 249. M. Krynicki, M. Mostowski and L.M. Szczerba (eds.), Quantifiers: Logics, Models and Computation. Volume Two: Contributions. 1995 ISBN 0-7923-3449-3 Set ISBN (Vols 248 + 249) 0-7923-3450-7 250. R.A. Watson, Representational Ideas from Plato to Patricia Churchland. 1995 ISBN 0-7923-3453-1 251. J. Hintikka (ed.), From Dedekind to G¨odel. Essays on the Development of the Foundations of Mathematics. 1995 ISBN 0-7923-3484-1 252. A. Wi´sniewski, The Posing of Questions. Logical Foundations of Erotetic Inferences. 1995 ISBN 0-7923-3637-2 253. J. Peregrin, Doing Worlds with Words. Formal Semantics without Formal Metaphysics. 1995 ISBN 0-7923-3742-5 254. I.A. Kiesepp¨a, Truthlikeness for Multidimensional, Quantitative Cognitive Problems. 1996 ISBN 0-7923-4005-1 255. P. Hugly and C. Sayward: Intensionality and Truth. An Essay on the Philosophy of A.N. Prior. 1996 ISBN 0-7923-4119-8 256. L. Hankinson Nelson and J. Nelson (eds.): Feminism, Science, and the Philosophy of Science. 1997 ISBN 0-7923-4162-7 257. P.I. Bystrov and V.N. Sadovsky (eds.): Philosophical Logic and Logical Philosophy. Essays in Honour of Vladimir A. Smirnov. 1996 ISBN 0-7923-4270-4 ˚ 258. A.E. Andersson and N-E. Sahlin (eds.): The Complexity of Creativity. 1996 ISBN 0-7923-4346-8 259. M.L. Dalla Chiara, K. Doets, D. Mundici and J. van Benthem (eds.): Logic and Scientific Methods. Volume One of the Tenth International Congress of Logic, Methodology and Philosophy of Science, Florence, August 1995. 1997 ISBN 0-7923-4383-2 260. M.L. Dalla Chiara, K. Doets, D. Mundici and J. van Benthem (eds.): Structures and Norms in Science. Volume Two of the Tenth International Congress of Logic, Methodology and Philosophy of Science, Florence, August 1995. 1997 ISBN 0-7923-4384-0 Set ISBN (Vols 259 + 260) 0-7923-4385-9 261. A. Chakrabarti: Denying Existence. The Logic, Epistemology and Pragmatics of Negative Existentials and Fictional Discourse. 1997 ISBN 0-7923-4388-3 262. A. Biletzki: Talking Wolves. Thomas Hobbes on the Language of Politics and the Politics of Language. 1997 ISBN 0-7923-4425-1 263. D. Nute (ed.): Defeasible Deontic Logic. 1997 ISBN 0-7923-4630-0 264. U. Meixner: Axiomatic Formal Ontology. 1997 ISBN 0-7923-4747-X 265. I. Brinck: The Indexical ‘I’. The First Person in Thought and Language. 1997 ISBN 0-7923-4741-2 266. G. H¨olmstr¨om-Hintikka and R. Tuomela (eds.): Contemporary Action Theory. Volume 1: Individual Action. 1997 ISBN 0-7923-4753-6; Set: 0-7923-4754-4

SYNTHESE LIBRARY 267. G. H¨olmstr¨om-Hintikka and R. Tuomela (eds.): Contemporary Action Theory. Volume 2: Social Action. 1997 ISBN 0-7923-4752-8; Set: 0-7923-4754-4 268. B.-C. Park: Phenomenological Aspects of Wittgenstein’s Philosophy. 1998 ISBN 0-7923-4813-3 269. J. Pa´sniczek: The Logic of Intentional Objects. A Meinongian Version of Classical Logic. 1998 Hb ISBN 0-7923-4880-X; Pb ISBN 0-7923-5578-4 270. P.W. Humphreys and J.H. Fetzer (eds.): The New Theory of Reference. Kripke, Marcus, and Its Origins. 1998 ISBN 0-7923-4898-2 271. K. Szaniawski, A. Chmielewski and J. Wolen´ski (eds.): On Science, Inference, Information and Decision Making. Selected Essays in the Philosophy of Science. 1998 ISBN 0-7923-4922-9 272. G.H. von Wright: In the Shadow of Descartes. Essays in the Philosophy of Mind. 1998 ISBN 0-7923-4992-X 273. K. Kijania-Placek and J. Wolen´ski (eds.): The Lvov–Warsaw School and Contemporary Philosophy. 1998 ISBN 0-7923-5105-3 274. D. Dedrick: Naming the Rainbow. Colour Language, Colour Science, and Culture. 1998 ISBN 0-7923-5239-4 275. L. Albertazzi (ed.): Shapes of Forms. From Gestalt Psychology and Phenomenology to Ontology and Mathematics. 1999 ISBN 0-7923-5246-7 276. P. Fletcher: Truth, Proof and Infinity. A Theory of Constructions and Constructive Reasoning. 1998 ISBN 0-7923-5262-9 277. M. Fitting and R.L. Mendelsohn (eds.): First-Order Modal Logic. 1998 Hb ISBN 0-7923-5334-X; Pb ISBN 0-7923-5335-8 278. J.N. Mohanty: Logic, Truth and the Modalities from a Phenomenological Perspective. 1999 ISBN 0-7923-5550-4 279. T. Placek: Mathematical Intiutionism and Intersubjectivity. A Critical Exposition of Arguments for Intuitionism. 1999 ISBN 0-7923-5630-6 280. A. Cantini, E. Casari and P. Minari (eds.): Logic and Foundations of Mathematics. 1999 ISBN 0-7923-5659-4 set ISBN 0-7923-5867-8 281. M.L. Dalla Chiara, R. Giuntini and F. Laudisa (eds.): Language, Quantum, Music. 1999 ISBN 0-7923-5727-2; set ISBN 0-7923-5867-8 282. R. Egidi (ed.): In Search of a New Humanism. The Philosophy of Georg Hendrik von Wright. 1999 ISBN 0-7923-5810-4 283. F. Vollmer: Agent Causality. 1999 ISBN 0-7923-5848-1 284. J. Peregrin (ed.): Truth and Its Nature (if Any). 1999 ISBN 0-7923-5865-1 285. M. De Caro (ed.): Interpretations and Causes. New Perspectives on Donald Davidson’s Philosophy. 1999 ISBN 0-7923-5869-4 286. R. Murawski: Recursive Functions and Metamathematics. Problems of Completeness and Decidability, G¨odel’s Theorems. 1999 ISBN 0-7923-5904-6 287. T.A.F. Kuipers: From Instrumentalism to Constructive Realism. On Some Relations between Confirmation, Empirical Progress, and Truth Approximation. 2000 ISBN 0-7923-6086-9 288. G. Holmstr¨om-Hintikka (ed.): Medieval Philosophy and Modern Times. 2000 ISBN 0-7923-6102-4 289. E. Grosholz and H. Breger (eds.): The Growth of Mathematical Knowledge. 2000 ISBN 0-7923-6151-2

SYNTHESE LIBRARY 290. G. Sommaruga: History and Philosophy of Constructive Type Theory. 2000 ISBN 0-7923-6180-6 291. J. Gasser (ed.): A Boole Anthology. Recent and Classical Studies in the Logic of George Boole. 2000 ISBN 0-7923-6380-9 292. V.F. Hendricks, S.A. Pedersen and K.F. Jørgensen (eds.): Proof Theory. History and Philosophical Significance. 2000 ISBN 0-7923-6544-5 293. W.L. Craig: The Tensed Theory of Time. A Critical Examination. 2000 ISBN 0-7923-6634-4 294. W.L. Craig: The Tenseless Theory of Time. A Critical Examination. 2000 ISBN 0-7923-6635-2 295. L. Albertazzi (ed.): The Dawn of Cognitive Science. Early European Contributors. 2001 ISBN 0-7923-6799-5 296. G. Forrai: Reference, Truth and Conceptual Schemes. A Defense of Internal Realism. 2001 ISBN 0-7923-6885-1 297. V.F. Hendricks, S.A. Pedersen and K.F. Jørgensen (eds.): Probability Theory. Philosophy, Recent History and Relations to Science. 2001 ISBN 0-7923-6952-1 298. M. Esfeld: Holism in Philosophy of Mind and Philosophy of Physics. 2001 ISBN 0-7923-7003-1 299. E.C. Steinhart: The Logic of Metaphor. Analogous Parts of Possible Worlds. 2001 ISBN 0-7923-7004-X 300. P. G¨ardenfors: The Dynamics of Thought. 2005 ISBN 1-4020-3398-2 301. T.A.F. Kuipers: Structures in Science Heuristic Patterns Based on Cognitive Structures. An Advanced Textbook in Neo-Classical Philosophy of Science. 2001 ISBN 0-7923-7117-8 302. G. Hon and S.S. Rakover (eds.): Explanation. Theoretical Approaches and Applications. 2001 ISBN 1-4020-0017-0 303. G. Holmstr¨om-Hintikka, S. Lindstr¨om and R. Sliwinski (eds.): Collected Papers of Stig Kanger with Essays on his Life and Work. Vol. I. 2001 ISBN 1-4020-0021-9; Pb ISBN 1-4020-0022-7 304. G. Holmstr¨om-Hintikka, S. Lindstr¨om and R. Sliwinski (eds.): Collected Papers of Stig Kanger with Essays on his Life and Work. Vol. II. 2001 ISBN 1-4020-0111-8; Pb ISBN 1-4020-0112-6 305. C.A. Anderson and M. Zel¨eny (eds.): Logic, Meaning and Computation. Essays in Memory of Alonzo Church. 2001 ISBN 1-4020-0141-X 306. P. Schuster, U. Berger and H. Osswald (eds.): Reuniting the Antipodes – Constructive and Nonstandard Views of the Continuum. 2001 ISBN 1-4020-0152-5 307. S.D. Zwart: Refined Verisimilitude. 2001 ISBN 1-4020-0268-8 308. A.-S. Maurin: If Tropes. 2002 ISBN 1-4020-0656-X 309. H. Eilstein (ed.): A Collection of Polish Works on Philosophical Problems of Time and Spacetime. 2002 ISBN 1-4020-0670-5 310. Y. Gauthier: Internal Logic. Foundations of Mathematics from Kronecker to Hilbert. 2002 ISBN 1-4020-0689-6 311. E. Ruttkamp: A Model-Theoretic Realist Interpretation of Science. 2002 ISBN 1-4020-0729-9 312. V. Rantala: Explanatory Translation. Beyond the Kuhnian Model of Conceptual Change. 2002 ISBN 1-4020-0827-9 313. L. Decock: Trading Ontology for Ideology. 2002 ISBN 1-4020-0865-1

SYNTHESE LIBRARY 314. O. Ezra: The Withdrawal of Rights. Rights from a Different Perspective. 2002 ISBN 1-4020-0886-4 315. P. G¨ardenfors, J. Wole´nski and K. Kijania-Placek: In the Scope of Logic, Methodology and Philosophy of Science. Volume One of the 11th International Congress of Logic, Methodology and Philosophy of Science, Cracow, August 1999. 2002 ISBN 1-4020-0929-1; Pb 1-4020-0931-3 316. P. G¨ardenfors, J. Wole´nski and K. Kijania-Placek: In the Scope of Logic, Methodology and Philosophy of Science. Volume Two of the 11th International Congress of Logic, Methodology and Philosophy of Science, Cracow, August 1999. 2002 ISBN 1-4020-0930-5; Pb 1-4020-0931-3 317. M.A. Changizi: The Brain from 25,000 Feet. High Level Explorations of Brain Complexity, Perception, Induction and Vagueness. 2003 ISBN 1-4020-1176-8 318. D.O. Dahlstrom (ed.): Husserl’s Logical Investigations. 2003 ISBN 1-4020-1325-6 319. A. Biletzki: (Over)Interpreting Wittgenstein. 2003 ISBN Hb 1-4020-1326-4; Pb 1-4020-1327-2 320. A. Rojszczak, J. Cachro and G. Kurczewski (eds.): Philosophical Dimensions of Logic and Science. Selected Contributed Papers from the 11th International Congress of Logic, Methodology, and Philosophy of Science, Krak´ow, 1999. 2003 ISBN 1-4020-1645-X 321. M. Sintonen, P. Ylikoski and K. Miller (eds.): Realism in Action. Essays in the Philosophy of the Social Sciences. 2003 ISBN 1-4020-1667-0 322. V.F. Hendricks, K.F. Jørgensen and S.A. Pedersen (eds.): Knowledge Contributors. 2003 ISBN Hb 1-4020-1747-2; Pb 1-4020-1748-0 323. J. Hintikka, T. Czarnecki, K. Kijania-Placek, T. Placek and A. Rojszczak † (eds.): Philosophy and Logic In Search of the Polish Tradition. Essays in Honour of Jan Wole´nski on the Occasion of his 60th Birthday. 2003 ISBN 1-4020-1721-9 324. L.M. Vaina, S.A. Beardsley and S.K. Rushton (eds.): Optic Flow and Beyond. 2004 ISBN 1-4020-2091-0 325. D. Kolak (ed.): I Am You. The Metaphysical Foundations of Global Ethics. 2004 ISBN 1-4020-2999-3 326. V. Stepin: Theoretical Knowledge. 2005 ISBN 1-4020-3045-2 327. P. Mancosu, K.F. Jørgensen and S.A. Pedersen (eds.): Visualization, Explanation and Reasoning Styles in Mathematics. 2005 ISBN 1-4020-3334-6 328. A. Rojszczak (author) and J. Wolenski (ed.): From the Act of Judging to the Sentence. The Problem of Truth Bearers from Bolzano to Tarski. 2005 ISBN 1-4020-3396-6 329. A. Pietarinen: Signs of Logic. Peircean Themes on the Philosophy of Language, Games, and Communication. 2005 ISBN 1-4020-3728-7 330. A. Aliseda: Abductive Reasoning. Logical Investigations into Discovery and Explanation. 2005 ISBN 1-4020-3906-9 331. B. Feltz, M. Crommelinck and P. Goujon (eds.): Self-organization and Emergence in Life Sciences. 2005 ISBN 1-4020-3916-6 332. 333. L. Albertazzi: Immanent Realism. An Introduction to Brentano. 2006 ISBN 1-4020-4201-9 334. A. Keupink and S. Shieh (eds.): The Limits of Logical Empiricism. Selected Papers of Arthur Pap. 2006 ISBN 1-4020-4298-1 335. M. van Atten: Brouwer meets Husserl. On the Phenomenology of Choice Sequences. 2006 ISBN 1-4020-5086-0 Previous volumes are still available.

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