Defending the Axioms: On the Philosophical Foundations of Set Theory

Defending the Axioms

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Defending the Axioms: On the Philosophical Foundations of Set Theory Penelope Maddy

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Great Clarendon Street, Oxford ox2 6dp Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York # Penelope Maddy 2011 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2011 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by SPI Publisher Services, Pondicherry, India Printed in Great Britain on acid-free paper by MPG Books Group, Bodmin and King’s Lynn ISBN 978–0–19–959618–8 1 3 5 7 9 10 8 6 4 2

For the Cabal

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Contents Preface Introduction

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I. The Problem 1. An historical reversal 2. How applied mathematics became pure 3. Where we are now

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II. Proper Method 1. The meta-philosophy 2. Some examples from set-theoretic practice 3. Proper set-theoretic method 4. The challenge

38 38 41 52 55

III. Thin Realism 1. Introducing Thin Realism 2. What Thin Realism is not 3. Thin epistemology 4. The objective ground of Thin Realism 5. Retracing our steps

60 61 64 71 77 83

IV. Arealism 1. Introducing Arealism 2. Mathematics in application 3. What Arealism is not 4. Comparison with Thin Realism 5. Thin Realism/Arealism

88 88 89 96 99 103

viii conte nts V. Morals 1. Objectivity in mathematics 2. Robust Realism revisited 3. More examples from set-theoretic practice 4. Intrinsic versus extrinsic

113 114 117 123 131

Bibliography Index

138 147

Preface This question of how set-theoretic axioms are properly defended has been with me for some time. ‘Believing the axioms’ ([1988]) catalogs all the actual arguments I could find, in the literature or in conversation, as a preliminary step toward the project of determining which are cogent, which not, and why. Realism in Mathematics ([1990]) is an attempt to turn away the objection that questions independent of the standard axioms—which obviously includes all new axiom candidates—have no answers; there I propose a more naturalistic variant of Go¨del’s Robust Realism. Though some take me to task for apostasy, I soon despaired of this position, for three reasons: it relies on a Quine/ Putnam indispensability argument that I couldn’t continue to endorse; arguments for and against axiom candidates that seem compelling don’t fit well with the metaphysics; and most importantly, just as a fundamentally naturalistic perspective counts against criticizing a bit of mathematics on the basis of extra-mathematical considerations, it counts just as heavily against supporting a bit of mathematics on the basis of extra-mathematical considerations. Since then, Naturalism in Mathematics ([1997]) takes a more strictly naturalistic approach to the methodological question of how arguments for or against set-theoretic principles should be evaluated, attempting to separate that question from traditional philosophical issues of truth and existence. Second Philosophy ([2007]) lays out the broader philosophical background that seemed to me necessary for a return to those traditional questions; }IV.4 of that book contains a brief sketch of what the resulting answers might look like. The goal of the current book, then, is to fill in and develop those sketchy answers. An early version of Chapter I appeared in the inaugural issue of the Review of Symbolic Logic ([2008]). The Association for Symbolic Logic and Cambridge University Press have generously permitted its re-appearance here.

x pre face Finally, I’m grateful to many people for help of various kinds during the course of this project, among them Jeremy Avigad, John Burgess, Justin Clarke-Doane, Michael Ernst, Matthew Glass, Jeremy Heis, Juliette Kennedy, Peter Koellner, Michael Liston, David Malament, Patricia Marino, Tony Martin, Colin McLarty, Bennett McNulty, A. J. Packman, John Rapalino, Erich Reck, Brian Rogers, Jeffrey Roland, Stewart Shapiro, John Steel, Jamie Tappenden, Scott Tidwell, Clinton Tolley, Mark Wilson, and the members of my winter 2009 seminar. I’m also indebted, as always, to Peter Momtchiloff of Oxford University Press. And finally, my thanks again to David, for his friendship and encouragement along the way. P.M. Irvine, California May 2010

Introduction Mathematics, as we all know, depends on proofs. And proofs, as we all know, have to begin somewhere, from some fundamental assumptions. In contemporary pure mathematics, the axioms of set theory are particularly salient, for reasons traced in Chapter I. In this case, the Euclidean ideal of postulates that are simply obvious or selfevident can’t be the whole story, which raises two basic questions: what are the proper methods for defending set-theoretic axioms? and, why are these the proper methods? The first of these is the subject of Chapter II. Addressing the second requires engagement with the troublesome ontological and epistemological issues that have dogged the philosophy of mathematics from its beginnings. In Chapter III and IV, I describe and explore two apparently conflicting stands on these issues, not so much to recommend either one, but with an eye to suggesting that the question of which is correct has less bite than it might appear. In the end, my hope is to shift attention away from these elusive matters of truth and existence, and to direct it toward the type of mathematical objectivity emphasized in the opening section of Chapter V. (Though set theory is the focus in these pages, I believe the source of objectivity traced there is also at work in other branches of pure mathematics.) The concluding sections of Chapter V return, at last, to the question of set-theoretic method and draw some concrete morals for the project of defending the axioms.

I The Problem The subject of this book is contemporary set theory, its methods and its subject matter: what are set theorists doing? how are they managing to do it? To understand the nature and force of these questions, we first need to appreciate how we came to the point we now occupy, how pure mathematics arose out of applied mathematics and how set theory developed from there. From this perspective, we can then address the questions of how the progress of pure mathematics is guided, of which mathematical entities and proof techniques are legitimate, of what constraints our methods must properly satisfy. As a start, we need to recognize that the relationship between pure and applied mathematics hasn’t been static over the centuries, that the historical rise of pure mathematics has coincided with a gradual shift in our understanding of how mathematics works in application to the world. In some circles today, it’s held that historical developments of this sort simply represent changes in fashion, or in social arrangements, in governments, in power structures, or some such thing, but I resist the full force of this way of thinking, clinging to the old-school notion that we’ve gradually learned more about the world over time, that our opinions on these matters have improved, and that seeing how we reached the point we now occupy may help us avoid falling back into old philosophies that are now no longer viable. In that spirit, it seems to me that once we focus on the general question of how mathematics relates to science, one observation is immediate: the march of the centuries has produced an amusing reversal of philosophical fortunes. Let me begin there.

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1. An historical reversal In the beginning, that is, in Plato, mathematical knowledge was sharply distinguished from ordinary perceptual belief about the world. According to Plato’s metaphysics, mathematics is the study of eternal and unchanging abstract Forms1 while science is uncertain and changeable opinion about the world of mere becoming. Indeed, by Plato’s lights, of the two, only mathematics deserves to be called ‘knowledge’ at all! Of course if sense perception can’t give us knowledge, if mathematics is not about perceivable things, then Plato owes us an account of how we ordinary humans achieve this wonderful insight into the properties of the abstract world of Forms. Plato’s answer is that we don’t actually acquire mathematical knowledge at all; rather we recollect it from a time before birth, when our souls, unencumbered by physical bodies, were free to commune with the Forms, and not just the mathematical ones, either—also Truth, Beauty, Justice, The Good, and so on.2 Whatever appeal this position may have held for the ancient Greeks, it won’t begin to satisfy a contemporary, scientifically-minded philosopher. But in any case, Plato’s view on the relative standing of mathematics and science is unambiguous: mathematics is the highest form of knowledge; science is mere opinion. Of course ‘science’ for the Greeks wasn’t what we call ‘science’ today; between the ancients and ourselves looms the Scientific Revolution, the beginning of experimental natural science as we know it. Pioneers of the new science like Galileo took mathematics to be central to God’s design of the universe, as in this famous passage from 1610: [Nature] is written in that great book which ever lies before our eyes—I mean the universe—but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in the mathematical language, and the symbols are triangles, circles, and other geometric figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth. 1 In fact, for various reasons, Plato may posit a kind of intermediate abstract entity, called a ‘mathematical’, between the Forms and the physical world. For discussion, see Wedberg [1955]. 2 See e.g. Silverman [2003].

4 th e problem (as quoted by Kline [1972], pp. 328–329. See also Machamer [1998a], pp. 64–65, for a slightly different translation.)

In the case of free fall, for example, Galileo notes that many accounts of its causes have been proposed, but he rejects this inquiry: Such fantasies, and others like them, would have to be examined and resolved, with little gain. (Galileo [1638], p. 202)

His idea is that we should concentrate on finding and testing a purely mathematical description of motion, which he goes on to do: It suffices our Author that we understand him to want us to investigate and demonstrate some attributes of a motion so accelerated . . . whatever be the cause of its acceleration . . . that in equal times, equal additions of speed are made. (op. cit.)

The universe operates according to mathematical laws, which we can uncover by bringing mathematics to bear on our observations.3 Notice the dramatic shift here: mathematics isn’t placed above science; rather the two have become one. In the words of the mathematical historian Morris Kline: mathematics became the substance of scientific theories. . . . The upshot . . . was a virtual fusion of mathematics . . . and science. (Kline [1972], pp. 394–395)

The great thinkers of that time—from Descartes and Galileo to Huygens and Newton—did mathematics as science and science as mathematics without any effort to separate the two. Appealing as this picture may be, it’s not the way we tend to see things today. What with the various shocks dealt in the interim to our 3 It should be noted that this bold proposal, later adopted by Newton (see below), didn’t meet with universal approval. Cf. Kline [1972], pp. 333–334: ‘First reactions to this principle of Galileo are likely to be negative. Description of phenomena in terms of formulas hardly seems to be more than a first step. It would seem that the true function of science had really been grasped by the Aristotelians, namely, to explain why phenomena happened. Even Descartes [who reacted against the Aristotelians] protested Galileo’s decision to seek descriptive formulas. He said, “Everything that Galileo says about bodies falling in empty space is built without foundation: he ought first to have determined the nature of weight”. Further, said Descartes, Galileo should reflect on ultimate reasons’. Looking forward to Newton and his successors, Kline holds that ‘Galileo’s decision to aim for description was the deepest and most fruitful idea that anyone has had about scientific methodology’.

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intuitive sense of how an orderly mathematical universe should behave, a starker empiricism has come to replace the rationalistic tendencies of an earlier, simpler time. So, for example, early in the 20th century, the logical positivists of the famous Vienna Circle held that the meaning of any scientific assertion could be reduced in some way or other to the conditions under which it would be empirically verified. This idea did admirable service in eliminating much apparently empty philosophical talk as meaningless—Carnap classifies Heidegger’s claim that ‘the Nothing itself nothings’ as a pseudo-statement4—but it also threatened to do the same for mathematics, an outcome that certainly didn’t appeal to the self-described scientific philosophers of the Circle! Their solution was to view mathematics as purely linguistic, as true by the conventions of language, as telling us nothing contentful about the world.5 At the same time, those like Go¨del who continued to maintain that mathematics provides substantial information about properly mathematical objects were confronted with a variation of Plato’s problem: if the cognitive machinery of human beings works as we think it does, how can we gain knowledge of non-spatiotemporal, acausal entities? 6 In all this, experimental natural science is taken as the paradigm of well-grounded knowledge and mathematics is called into question when it doesn’t clearly measure up. This is the reversal of philosophical fortunes alluded to a moment ago: for Plato, mathematics is perfect knowledge and science is mere opinion; for the pioneers of the scientific revolution, mathematics and science are one; for many contemporary philosophers, science is the best knowledge we have and the status of mathematics is problematic. The movement from the cross-over point—when science and mathematics were identified—to our current state coincides roughly with the rise of pure mathematics, with the separation of mathematics from its worldly roots. Perhaps we can better understand where we are now if we reconsider how mathematics came to be peeled away from natural science in this way. This is a complex story, of course, but I hope to

4 5 6

See Carnap [1932], p. 69. See Carnap [1950]. See e.g. Go¨del [1964] and Benacerraf [1973].

6 th e problem draw out three of its individual strands, ranging from the more mathematical perspective to the more scientific.7

2. How applied mathematics became pure I think the first of these strands, primarily a mathematician’s-eye-view, is fairly familiar. Kline describes the situation this way: . . . Descartes, Newton, Euler, and many others believed mathematics to be the accurate description of real phenomena . . . they regarded their work as the uncovering of the mathematical design of the universe. (Kline [1972], p. 1028)

Over the course of the 19th century, this picture changed dramatically: . . . gradually and unwittingly mathematicians began to introduce concepts that had little or no direct physical meaning. (Kline [1972], p. 1029)

Citing the rise of negative numbers, complex numbers, n-dimensional spaces, and non-commutative algebras, he remarks that ‘mathematics was progressing beyond concepts suggested by experience’, but that ‘mathematicians had yet to grasp that their subject . . . was no longer, if it ever had been, a reading of nature’ (Kline [1972], p. 1030). By mid-century, the tide had turned: . . . after about 1850, the view that mathematics can introduce and deal with . . . concepts and theories that do not have immediate physical interpretation . . . gained acceptance. (Kline [1972], p. 1031)

This movement continued with the study, for example, of abstract algebras, pathological functions, and transfinite numbers. The heady new view of mathematics that accompanied this change is perhaps best expressed by Cantor: Mathematics is entirely free in its development . . . The essence of mathematics lies in its freedom. (as quoted in Kline [1972], p. 1031)

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I should admit that I’m no historian myself. As will become obvious, I draw heavily on the work of various real scholars in what follows.

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This sentiment appears in the thinking of many of the most innovative mathematicians of the late 19th century; today, it is standard orthodoxy. Mathematics progresses by its own lights, independent of ties to the physical world. Legitimate mathematical concepts and theories need have no direct physical interpretation. A careful, systematic analysis of these developments would explain how mathematicians gradually came to see themselves as free to investigate whatever concepts or structures or theories seem of sufficient mathematical interest or importance. Detailed historical studies would illustrate how particular mathematical inquiries are motivated by particular mathematical goals and values. To take just one example, consider the development of the concept of an abstract group.8 Though we now recognize the role of substitution groups and their subgroups in Galois’s work around 1830, Galois himself never isolated the concept; that was left to Cayley some 20 years later. The surprise is that Cayley’s version passed unnoticed, as did Dedekind’s a decade later still, simply because there weren’t enough examples of groups to make the notion useful. It wasn’t until the 1870s, when many diverse examples of groups had been identified—in Galois theory, number theory, geometry, and the theory of differential equations—that the idea of an abstract group caught on and flourished. Only at that point did it begin to serve a clear mathematical purpose: it calls attention to similarities between a broad range of otherwise quite dissimilar structures; it provides an elaborate and detailed general theory that can be applied in different contexts; and it produces illuminating diagnoses of the features responsible for particular phenomena (‘that x has feature y isn’t due to its idiosyncrasies z or v or w, but only to its group structure’). Incidentally, it wasn’t until the 1920s that group theory entered physics, where it is now a central theme. This, then, is the first strand to the story of how mathematics came to separate from natural science—the pursuit of various purely mathematical goals gradually led mathematicians to new studies not motivated by their immediate application to the world—but again I think this is only part of the story. Also worth recalling is a second familiar 8

For more, see Wussing [1969], Stillwell [2002], chapter 19, or the quick survey in [2007], }IV.3.

8 th e problem thread, most apparent in the evolution of attitudes toward geometry. In the beginning, it seems even Euclid found the parallel postulate less obvious than the rest of his fundamental assumptions; after him, generations of geometers attempted to prove it from the others. By 1800, several mathematicians held that the parallel postulate cannot be proved, that alternative geometries are logically consistent, but nevertheless that Euclidean geometry is the true theory of actual space. The pivotal figure in this story is Gauss, whose efforts to prove the parallel postulate eventually led him, in his words, ‘to doubt the truth of geometry itself’ (see Kline [1972], p. 872). We’ve probably all heard the sometimes-disputed tale of Gauss measuring the sum of the angles of a triangle formed by three mountain tops, intending to test Euclid, only to conclude that the disparity fell within the margins of experimental error. In any case, it’s beyond dispute that Gauss considered alternative geometries to be candidates for application to the physical world; the full flowering of this idea came when Gauss set the foundations of geometry as a topic for the qualifying exam of his student, Riemann. Of course it was Riemannian geometry that the mathematician Grossman recommended when Einstein consulted him in 1912. With the confirmation of General Relativity some years later, Euclidean geometry could no longer be regarded as true of physical space, but mathematicians were reluctant to classify it as straightforwardly false. Instead, they distinguished physical space from abstract mathematical space, or rather, from a full range of different abstract mathematical spaces, and Euclidean geometry was seen as true in some and false in others among these. (Resnik ([1997], p. 130) calls this sort of move a ‘Euclidean rescue’.) At that point, it became natural to regard mathematicians as providing a well-stocked warehouse of abstract structures from which the natural scientist is free to select whichever tool best suits his needs in representing the world.9 This, then, is the second strand in the story of how mathematics pulled away from science. This time applications are involved, as they weren’t in the first, purely mathematical strand, though the tale of Euclidean rescue is still one visible primarily from the mathematician’s

9

See [2007], }}IV.2.iii and IV.4 for further discussion and references.

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point of view: mathematical theories are protected from empirical falsification by positing a special realm of abstracta about which they remain true. The moral for the natural scientist, for the application of mathematics, is that we now have a wide variety of mathematical options and that it may take delicate empirical investigation to determine which works best for a given application. Nevertheless, the mathematics that is successfully applied might still be regarded as the native language of the Book of Nature, just as Galileo understood it so long ago. The third and final strand I’d like to touch on here, perhaps less familiar than the two rehearsed so far, originates directly from the point of view of the natural scientist. We’ve so far considered aspects of the historical rise of pure mathematics, but over roughly the same period there was a profound shift in the common understanding of how applied mathematics relates to the world. To get a feel for this, let’s return to the Scientific Revolution, to Galileo’s heir, Sir Isaac Newton. Like Galileo, Newton views the world as mathematical in design; following Galileo in method also, he sets out to give a mathematical description of gravitational force without concern for the mechanism that produces it.10 In a famous passage at the end of the Principia, he writes: I have explained the phenomena of the heavens and of our sea by the force of gravity, but I have not yet assigned a cause to gravity. . . . I do not feign hypotheses. For whatever is not deduced from the phenomena must be called a hypothesis; and hypotheses, whether metaphysical or physical, or based on occult quantities, or mechanical, have no place in experimental philosophy. . . . it is enough that gravity really exists and acts according to the laws that we have set forth. (Newton [1687], p. 943)

This move was especially liberating given the context: Newtonian gravity was not a natural fit for the prevailing Cartesian picture of action by contact forces, not to mention that the Cartesian doctrine of

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Cf. Kline [1972], p. 334: ‘we should note how completely Galileo’s program was accepted by giants such as Newton’.

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vortices in the plenum was highly problematic in its own right.11 Newton champions the mathematical description of the motions we can straightforwardly observe—what he calls ‘manifest qualities’— over hypothetical explanations of those motions in terms of hidden causes.12 Altogether, Newton paints a remarkable mathematical portrait of the universe: from the general laws of motion and the law of universal gravitation, he explains the motions of the planets, the action of the tides, the trajectories of comets, the shape of the earth, and more,13 and he does so while inventing the required mathematics. The only weakness in all this is his mathematical conservatism. Unlike Leibniz,14 who developed the rudiments of the calculus at roughly the same time, Newton didn’t think of his mathematical techniques as constituting a general theory or method: rather than devising all-purpose algorithms, he was content to solve one individual problem after another.15 Worse

11 See e.g. Slowik [2005], for more on Descartes’s physics. Cf. Smith [2002], pp. 141–142: ‘As Newton well realized . . . no hypothetical contact mechanism seems even imaginable to effect “attractive” forces among particles of matter generally. The Scholium [Newton [1687], pp. 588–589] thus occurs at the point where adherents to the mechanical philosophy would start viewing Newton’s reasoning as “absurd” (to use the word Huygens chose privately). The Scholium attempts to carry the reader past this worry, but not by facing the demand for a contact mechanism head-on. Instead, Newton warns that he is employing mathematically formulated theory in physics in a new way, with forces treated abstractly, independently of mechanism’. 12 Cf. Shapiro [2002], p. 228: ‘Newton believed that by formulating his theories phenomenologically, in terms of experimentally observed properties, or principles deduced from them, without any causal explanations (hypotheses) of those properties, he could develop a more certain science’. 13 Cf. Newton [1687], p. 382: ‘the basic problem of philosophy [i.e., natural science] seems to be to discover the forces of nature from the phenomena of motions and then to demonstrate the other phenomena from these forces. . . . we derive from celestial phenomena the gravitational forces by which bodies tend toward the sun and toward the individual planets. Then the motions of the plants, the comets, the moon, and the sea are deduced from these forces’. 14 For these methodological contrasts, see e.g. Cohen and Smith [2002a], pp. 20–22, Hall [2002], Kline [1972], pp. 378–380. 15 Cf. Truesdell [1981], p. 98: ‘Newton’s Principia . . . is a monument of human achievement; it deserves the admiration and esteem of everyone. Should an engineer study it with a view to using its contents to determine the motion of a capsule projected into space, he would be gravelled. Motions there are in abundance, but no general equations. Each motion furnishes a new problem and is treated by itself. Examples there are, but no algorism: towering concepts and a magnificent approach, certainly, but no method’.

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yet for the progress of British mathematics was Newton’s insistence on purely synthetic geometric methods. Though he sometimes used Descartes’s analytic techniques, especially early on, he came to regard them as merely heuristic; the justifications in the Principia are overwhelmingly geometric. Kline writes: Newton did not really believe that he had departed from Greek geometry. Though he used algebra and coordinate geometry, which were not to his taste, he thought his underlying methods were but natural extensions of pure geometry. (Kline [1972], p. 384)16

For Newton, as for Galileo, the language of the universe was synthetic geometry,17 and British mathematicians loyally, if ill-advisedly, continued in Newton’s footsteps.18

16 Incidentally, Stein notes Newton’s claim, in the preface to Principia, that the principles of geometry are ‘obtained from other fields’ and that ‘geometry is founded on mechanical practice’ (Newton [1687], p. 382) and remarks ‘I am not aware of any other mathematician or philosopher of the seventeenth century who expressed such a view . . . Gauss appears to have been the first mathematician of stature (after Newton) to have come—and only after a struggle—to hold seriously the view that the grounds of geometry are empirical’ (Stein [1990], pp. 30, 44). 17 Cf. Guicciardini [2002], p. 323: ‘According to the Galilean tradition the Book of Nature is written in geometric terms. Newton endorsed this tradition’. 18 See e.g. Kline [1972], pp. 380–381: Because of the priority dispute between Newton and Leibniz, ‘the English and Continental mathematicians ceased exchanging ideas. Because Newton’s major work and first publication on the calculus, the Principia, used geometrical methods, the English continued to use mainly geometry for about a hundred years after his death. The Continentals took up Leibniz’s analytical methods and extended and improved them. These proved to be far more effective; so not only did the English mathematicians fall behind, but mathematics was deprived of contributions that some of the ablest minds might have made’. When the British finally began to import Continental analysis in the early 19th century, the principles of Leibniz’s dy notation were called ‘d-ism’, as opposed to ‘dot-age’ for the use of Newton’s y (see Kline [1972], p. 622). See also Smith and Wise [1989], pp. 151–152: ‘The [British] reformers originally saw it as their mission to bring the most powerful techniques of mathematical analysis . . . to the moribund centres of mathematical nonlearning in Britain. Of greatest immediate importance was replacing the cumbersome system of dots in the Newtonian fluxional notation with the d’s of Leibnizian differentials. Symbols nearly as political as they were mathematical, the d’s represented youth and progress in the modern age. Babbage later claimed that in 1812, as the Memoirs of the Analytical Society neared publication, he had suggested a more apt title: “The Principle of pure D-ism in opposition to the Dot-age of the University”. His pun on deism suggests the ideology of natural law that he espoused and how it could be embedded in the symbols of mathematics. Dr Thomson, while no deist in the religious sense, agreed . . . that the “inferiority” of dot-age had been “a principle cause of the small progress made in later times by British mathematicians”’.

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Was a man of vision who thought in broad terms, like Descartes. He saw the long-term implications of the new ideas and did not hesitate to declare that a new science was coming to light. (Kline [1972], p. 384)

Leibniz’s algebraic, symbolic approach proved far more flexible and effective than Newton’s synthetic geometry, and his Continental followers were the ones to expand and improve the methods of the calculus. The historians Cohen and Smith write: It was left to individuals within the Leibnizian tradition to recast the Principia into the symbolic calculus. What became clear in this process was the superiority of purely symbolic methods . . . With this realization the fundamental step in problems of physics ceased being one of finding an adequate geometric representation of the quantities involved, and instead became one of formulating appropriate differential equations. (Cohen and Smith [2002a], p. 22)19

This was the job of the 18th century, carried out largely on the Continent. This time the key figure is Euler. What we know as Newton’s laws of motion—including the famous F=ma—were actually formulated by Euler in the mid-1700s, but that’s just the tip of the iceberg: Euler’s mathematical productivity is incredible. His major mathematical fields were the calculus, differential equations, analytic and differential geometry of curves and surfaces, the theory of numbers, series, and the calculus of variations. This mathematics he applied to the entire domain of physics. He created analytic mechanics (as opposed to the older geometrical mechanics) and the subject of rigid body mechanics. He calculated the perturbative effect of celestial bodies on the orbit of a planet and the paths of projectiles in resisting media. . . . He investigated the bending of beams and calculated the safety load of a column. . . . He was the first to treat the vibrations of light analytically and 19 Cf. Truesdell [1968], pp. 92–93: ‘Except for certain simple if important special problems, Newton gives no evidence of being able to set up differential equations of motion for mechanical systems. . . . As we shall see, a large part of the literature of mechanics for sixty years following the Principia searches various principles with a view to finding the equations of motion for the systems Netwon had studied and for other systems nowadays though of as governed by “Newtonian” equations’.

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to deduce the equation of motion taking into account the dependence on the elasticity and density of the ether . . . The fundamental differential equations for the motion of an ideal fluid are his. (Kline [1972], pp. 401–402)20

Euler’s collected works now run to nearly 90 volumes.21 In fact the 18th century was tremendously productive in all these areas; perhaps it’s not surprising that along the way the foundational difficulties experienced by Newton and Leibniz were only exacerbated as mathematical analysis expanded and deepened. The momentous shift away from Newton’s geometric methods toward the symbolic techniques of Leibniz not only opened the way to bold new developments at breathtaking speed, but also served to relax the level of rigor downwards from the high standard traditionally associated with the Greeks and their followers; Lacroix remarks ‘Such subtleties as the Greeks worried about we no longer need’.22 Kline describes the situation this way: Eighteenth-century thinking was certainly loose and intuitive. Any delicate questions of analysis, such as the convergence of series and integrals, the interchange of the order of differentiation and integration, the use of differentials of higher order, and questions of existence of integrals and solutions of

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See also Kline [1972], pp. 402–403: ‘Euler did not open up new branches of mathematics. But no one was so prolific or could so cleverly handle mathematics; no one could muster and utilize the resources of algebra, geometry, and analysis to produce so many admirable results. Euler was superbly inventive in methodology and a skilled technician. One finds his name in all branches of mathematics: there are formulas of Euler, polynomials of Euler, Euler constants, Euler integrals, and Euler lines’. 21 Several of these volumes include scholarly introductions by Truesdell who writes ([1968], p. 106): ‘Euler was the dominating theoretical physicist of the eighteenth century. His work is undervalued in the usual, vague historical works. . . . The great bulk of [his] publication is not the only impediment to a just historical estimate of what he did. He put most of mechanics into its modern form; from his books and papers, if indirectly, we have all learned the subject, and his way of doing things is so clear and natural as to seem obvious. In fact, it was he who made mechanics simple and easy, and for the straightforward it is unnecessary to give references. In return, the scientist of today who consults Euler’s later writings will find them perfectly modern, while other works of that period require effort and some historical generosity to be appreciated’. Much more could be said in Euler’s praise, but let me just add this stunning fact: ‘of the entire corpus of research on mathematics, theoretical physics, and engineering mechanics published from 1726 to 1800’ his writings ‘alone account for approximately one third’ (Calinger [1975], p. 211). 22 This appears in the 1810 preface to his three-volume compendium of 18th-century differential and integral calculus. See Kline [1972], p. 618, for the reference.

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differential equations, were all but ignored. That the mathematicians were able to proceed at all was due to the fact that the rules of operation were clear. Having formulated the physical problems mathematically, the virtuosos got to work, and new methodologies and conclusions emerged. . . . How could the mathematicians have dared merely to apply rules and yet assert the reliability of their conclusions? (Kline [1972], p. 617)

The answer lies in the virtual identification of mathematics and natural science inherited from Galileo and Newton: Their technical skill was unsurpassed; it was guided, however, not by sharp mathematical thinking but by intuitive and physical insights. (Kline [1972], p. 400) The physical meaning of the mathematics guided the mathematical steps and often supplied partial arguments to fill in nonmathematical steps. The reasoning was in essence no different from a proof of a theorem of geometry, wherein some facts entirely obvious in the figure are used even though no axiom or theorem supports them. Finally, the physical correctness of the conclusions gave assurance that the mathematics must be correct. (Kline [1972], p. 617)

Notice that if the result is a method that we don’t quite recognize as mathematical, it also isn’t what we normally think of as physical, either. For our purposes, though, these subtleties are less important than the clear line of influence tracing back to Galileo and Newton. Euler remarks: The generality I here take on . . . reveals to us the true laws of Nature in all their brilliance. (as quoted by Truesdell [1981], p. 113)

Kline summarizes that for the 18th century, ‘mathematics was simply unearthing the mathematical design of the universe’ (Kline [1972], p. 619). Clifford Truesdell, the great practitioner and historian of rational mechanics, remarks that its statements are called phenomenological, because they represent the immediate phenomena of experience, not attempting to explain them in terms of corpuscles or other inferred (or hypothesized) quantities. (Truesdell [1960], p. 22)

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Thus we find both key elements from Galileo and Newton continuing here: the conviction that mathematical theories truly represent the underlying mathematical structure of the world, and endorsement of these theories as describing phenomena directly, without appeal to theoretical hidden causes. This same world view carries forward into the 19th century as well, perhaps best exemplified by Fourier’s ground breaking work on the dynamics of heat. In the opening sentences of what has been called his great ‘mathematical poem’,23 first published in 1822, Fourier announces that Primary causes are unknown to us; but are subject to simple and constant laws, which may be discovered by observation, the study of them being the object of natural philosophy. Heat, like gravity, penetrates every substance of the universe, its rays occupy all of space. The object of our work is to set forth the mathematical laws which this element obeys. (Fourier [1822], p. 1)24

In the following paragraph he makes explicit his connection to Galileo and Newton.25 At the end he concludes: The chief results of our theory are the differential equations of the movement of heat in solid or liquid bodies, and the general equation which relates to the surface. The truth of these equations is not founded on any physical explanation of the effects of heat. In whatever manner we please to imagine the nature of this element[26] we shall always arrive at the same equations, since the hypothesis which we form must represent the general and simple facts from which the mathematical laws are derived. (Fourier [1822], p. 464)

23

See Smith and Wise [1989], p. 149, for references. See also Fourier [1822], p. 7: ‘Profound study of nature is the most fertile source of mathematical discoveries. . . . it is . . . a sure method of forming analysis itself ’. 25 The only other predecessor mentioned is Archimedes, whose name often appears in such contexts, e.g., Kline ([1972], p. 401) refers to Euler as ‘the man [of the eighteenth century] who should be ranked with Archimedes, Newton and Gauss’. Machamer [1998a] makes the case for Galileo’s debt to Archimedes. 26 ‘Whether we regard it as a distinct material thing which passes from one part of space to another, or whether we make heat consist simply in the transfer of motion’. 24

16

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Once again we see the conviction that nature is mathematical and that its laws can be directly observed and hold true whatever mechanisms may underlie them. As it happens, a lively controversy arose between Fourier and Poisson, whose rival theory of heat appeared in 1835.27 Poisson’s objections actually first appear in a 1808 paper of Laplace, written in response to an early version of Fourier’s work, where they trace to this Laplacian credo: I have wanted to establish that the phenomena of nature reduce in the final analysis to action ad distans from molecule to molecule, and that the consideration of these actions ought to serve as the basis of the mathematical theory of these phenomena. (As quoted by Smith and Wise [1989], p. 160)

Poisson took this approach to heat, positing a complex microstructure of molecules and caloric fluid,28 summing (integrating) over their interactions and eventually generating a differential equation that differs from Fourier’s by an extra term. This he embraced as representing a novel prediction of the theory—that conductivity varies with absolute temperature—subject to experimental test. Fourier also begins with molecules, but instead of devising a theory to test, he works directly from experiment in the first place: first the observation that a warmer body loses heat to a cooler body at a rate proportional to the temperature difference; second that radiant heat doesn’t penetrate a thin foil. He concludes that If two molecules of the same body are extremely near, and are at unequal temperatures, that which is the most heated communicates directly to the other during one instant a certain quantity of heat; which quantity is proportional to the extremely small difference of the temperatures. (Fourier [1822], pp. 456–457)

27

Here I follow Wise [1981], pp. 23–29, and Smith and Wise [1989], pp. 155–162. Cf. Smith and Wise [1989], p. 160: ‘an explicit model of the relation between ponderable molecules and caloric fluid in a solid, incorporating both the free caloric radiated from the molecules and responsible for temperature, and the bound or latent caloric involved in changes of phase. His model attributed to the radiating molecules the full complexity of observable objects, including radiation to finite distances, radiation rates between molecules proportional to finite temperature differences, and a correction factor depending on absolute temperature to account for possible non-linearity’. 28

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and The layers in contact are the only ones which communicate their heat directly . . . There is no direct action except between material points extremely near. (Fourier [1822], p. 460)

By a subtle shift of the term ‘molecule’ from a physical body to an infinitesimal volume, Fourier establishes his differential equation— which he regards as ‘rest[ing] on observations alone rather than on any hypothesis as to the true nature of radiated heat’ (Wise [1981], p. 25)—so for all his talk of ‘molecules’, Fourier in fact treats heat as a continuous flow. From Fourier’s perspective, Poisson’s novel prediction can be easily accommodated at a certain point in his derivation if the purported variability is in fact observed; until then, its presence in the equation is an inappropriate extension of theory beyond experimental fact. Given the stark contrast between their methodologies, Laplace and Poisson’s objections to Fourier are predictable. Most fundamentally, the charge is that Fourier is masking the underlying physics: the differential heat equation is not the literal truth, but an approximation (first a finite sum over the molecules involved is treated as an integral, then the integral equation is transformed into the familiar differential equation).29 Of course, for Fourier, the differential equation is fundamental, a direct representation of observed behavior. A similar disagreement concerns the transition between the object under consideration and its environment; the historian Norton Wise puts it this way: Because temperature to Laplace was a density of caloric, it could never change abruptly. Fourier, however, treated the boundary as a surface of no thickness across which the temperature jumped between internal and external values. [For Laplace,] an acceptable analysis would require a boundary of some thickness within which temperature changed gradually to the external value. (Wise [1981], p. 26)

29 Cf. Wise [1981], p. 28: ‘For Laplace and Poisson . . . the differential equations were not the fundamental representation of the physics; they were to be established only as transformations of integral equations, where the integrals represented physical sums over effects of isolated sources’.

18

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Given the unsatisfactory state of caloric theory, Fourier’s position may be seen as running parallel to Newton’s rejection of Descartes’s vortices. The denouement of this long story springs from a momentous scientific development that gathered energy over the course of the 19th century and culminated in the opening decades of the 20th. For all the talk of ‘corpuscles’ or ‘atoms’ or ‘molecules’ throughout the time period we’ve been reviewing, there was no responsible view of their nature—tiny infinitely hard ball-bearings,30 mathematical points,31 infinitesimal volumes,32?—and the hypothesis of underlying, invisible, discrete structure of matter richly deserved its Newtonian expulsion from experimental natural science. But this began to change with Dalton’s experimental work in the first decade of the 19th century. It’s often surprising to those of us with little background in the history of science that the discipline we now know as chemistry was such a relative late-comer. While Galileo was studying free fall and Newton was writing the Principia, chemistry was largely alchemy;33 the beginning of ‘modern’ chemistry is typically set around 1750, linked to the work of Lavoisier,34 and a viable atomic theory of chemical combination began only with Dalton, roughly contemporary with Fourier. Dalton proposed that a sample of an element consists of many identical atoms of constant weight, that the atoms of different elements are of different weight, that these atoms remain unchanged through chemical reactions, and that a chemical compound is composed of many identical molecules, each of which is composed of atoms of its 30 See e.g. Smith and Wise [1989], pp. 155–156: ‘Newton justified his programme through the “analogy of nature”, arguing that whatever held true for all observable objects has also to hold for their unobservable parts . . . Since all observable objects possessed the qualities of extension, hardness, impenetrability, mobility, inertia, and gravitation, so also did its parts; they were infinitely hard atoms of finite size that attracted one another with a force varying as the inverse square of the distance between their centres, like perfect planets or marbles’. Of course such a hypothesis would not appear in Newton’s official experimental science (see Shapiro [2002], p. 228). 31 As in Laplace, and before him, Boscovich. See Smith and Wise [1989], p. 156. 32 As we’ve seen in Fourier. 33 Boyle (1627–1691) was an exception (see Partington [1957], pp. 66–77). Newton was not; the so-called ‘other Newton’ spent much time and energy on alchemy (see Cohen and Smith [2002a], pp. 23–29). 34 See Idhe [1964], chapter 3; Partington [1957], chapter VII.

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constituent elements. This simple hypothesis immediately explained the known laws of chemical combination and then swept through chemistry during the first half of the 19th century; by 1860 stable atomic weights had been measured and confirmed, and ‘the atom [came] into general acceptance as the fundamental unit of chemistry’ (Idhe [1964], p. 257). Beginning in mid-century, atomic theory spread into physics with the kinetic theory of Maxwell and Boltzmann. By the end of the 1800s, the atomic theory was a well-developed scientific hypothesis with considerable empirical support.35 Meanwhile, the 19th century also saw the invention and development of classical thermodynamics by Carnot and Clausius; its familiar second law states that entropy never decreases in a closed system, for example, that spilt milk doesn’t spontaneously return to the glass. The physical chemist Jean Perrin sees Carnot as heir to Galileo, describing his ‘inductive’ method this way: Each of these principles [of thermodynamics] has been reached by noting analogies and generalising the results of experience, and our lines of reasoning and statements of results have related only to objects that can be observed and to experiments that can be performed. . . . in the doctrine . . . there are no hypotheses. (Perrin [1913], p. vii)

Another physical chemist of the period, Pierre Duhem, begins his account of the history with Newton, cites Fourier with approval, and concludes that late 19th-century physicists have been ‘led . . . gradually back to the sound doctrines Newton had expressed so forcefully’ (Duhem [1906], p. 53). Here we find a true successor to the purely phenomenological methods we’ve been tracing; the late 19th-century descendants of the Galileo/Newton/Euler/Fourier line regarded thermodynamics as ‘the very epitome of a scientific method of analogy and classification . . . the apex of inductively derived . . . science’ (Nye [1972], p. 34).36 Thus the familiar battle lines were drawn, between those who explain phenomena by appeal to hidden structures and those who describe what they see in terms of differential equations—except that For further discussion and references, see [1997], pp. 135–142, [2007], }IV.5. Cf. Einstein [1949], p. 33: ‘Classical thermodynamics is the only physical theory of universal content concerning which I am convinced that, within the framework of the applicability of its basic concepts, it will never be overthrown’. 35 36

20 the problem this time the hypothesis in question, atomic theory, is more highly developed and empirically successful than ever before. Those opposed to hypotheses raised the familiar objection that atomic theory goes beyond experience; indeed it was argued that atoms are in principle inaccessible to empirical test.37 To this general offense, one more specific was added, namely, that it conflicts with the favored theory, with classical thermodynamics: [atomic theory] robbed Carnot’s principle of its claim to rank as an absolute truth and reduced it to the mere expression of a very high probability. (Perrin [1913], p. 86)

According to kinetic theory, the spilt milk might spontaneously reassemble in the glass, though this is highly unlikely. The conflict between the atomists and their thermodynamical opponents was so acute that Einstein, in one of his remarkable series of papers in 1905, sets out to find facts which would guarantee as much as possible the existence of atoms of definite finite size. (Einstein [1949], p. 47)

He does so by re-deriving kinetic theory (because he was ‘not acquainted with earlier investigations of Boltzmann and Gibbs’ (op. cit.)) and predicting in mathematical detail the behavior of ‘bodies of microscopically-visible size suspended in a liquid’ (Einstein [1905], p. 1). For all his efforts, Einstein was skeptical that experiments of the required precision were possible,38 but Perrin, a brilliant experimentalist, was in fact able ‘to prepare spherules of measurable radius’ (Perrin [1913], p. 114) and to confirm Einstein’s predictions in a series of experiments on Brownian Motion carried out around 1910. Poincare´, along with other leading skeptics, was immediately converted, declaring to a 1912 conference that ‘the atom of the chemist is now a reality’.39 And this consensus has only grown stronger since. 37 Cf. Perrin [1913], p. 15: ‘It appeared to them more dangerous than useful to employ a hypothesis deemed incapable of verification’. 38 See Nye [1972], p. 135. 39 As quoted in Nye [1972], p. 157. The two well-known opponents of atomic theory who weren’t converted—Duhem and Mach—both died in 1916. Wilson [2006], pp. 356– 369, 654–659, argues that both were led to their anti-atomism in part because of real

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The upshot of this transformation, for our purposes, is an equally profound change in the understanding of how mathematics relates to the world. Before his conversion, Poincare´ mused on the consequences that would follow if the kinetic theory should turn out to be correct: Physical law will then take an entirely new aspect; it will no longer be solely a differential equation. (as quoted in Nye [1972], p. 38)

Much as Laplace and Poisson insisted—though decidedly not for the reasons they gave!—the differential equation for heat flow is actually an approximation, an idealization, what Richard Feynman describes as ‘a smoothed-out imitation of a really much more complicated microscopic world’ (Feynman et al [1964], p. 12–12). The same goes for all the wonderful episodes of applied mathematics developed by Euler and his successors, and indeed, contemporary applied mathematicians take great care to determine precisely when various idealizations and simplifications that underlie their central differential equations can be counted both beneficial and benign. To take just one example, consider the case of fluid dynamics. D. J. Tritton, the author of one recent textbook, observes: The equations concern physical and mechanical quantities, such as velocity, density, pressure, temperature, which will be supposed to vary continuously from point to point throughout the fluid. How do we define these quantities at a point? To do so we have to make what is known as the assumption of the applicability of continuum mechanics or the continuum hypothesis[40]. We suppose that we can associate with any volume of liquid, no matter how small, those macroscopic properties that we associate with the fluid in bulk. . . . Now we know that this assumption is not correct if we go right down to molecular scales. We have to consider why is it nonetheless plausible to formulate the equations on the basis of the continuum hypothesis. (Tritton [1988], p. 48)

difficulties in the foundations of classical mechanics. Duhem also had religious motivations: he held that religious revelation, not physical science, is the proper source of information about underlying metaphysics (see Duhem [1906], appendix). 40

Of course this is just an amusing terminological coincidence, not an unexpected appearance of the set-theoretic continuum hypothesis!

22 the problem Suppose, for example, that our model41 assigns a temperature to every point in a three-dimensional volume. In fact we know that temperature is an average energy over a group of molecules; the ‘temperature’ of our fluid point will function properly only if it successfully stands in for a small volume. Too small a volume will contain only a few molecules that come and go at random, so its average energy will be subject to large fluctuations; too large a volume will include areas of significantly different average energies. ‘The applicability of the continuum hypothesis depends on there being a significant plateau’ between these two extremes: One may regard [the intermediate volume] as being an infinitesimal distance so far as macroscopic effects are concerned, and formulate the equations (as differential equations implicitly involving the limit of small separations) ignoring the behavior on still smaller length scales. (Tritton [1988], p. 50)

There is more to it than that, but this gives the flavor of the applied mathematician’s task. But, even if the vaunted differential equations of Euler’s analytical mechanics and the observationally perfect laws of thermodynamics are now regarded as ‘smoothed-out imitation[s] of a really much more complicated microscopic world’ (Feynman et al [1964], p. 12–12), perhaps there remains room for literal description where continuum mathematics seems more fundamental. Unfortunately, there is little comfort from this quarter: consider, for example, the difficulties with the self-energy of a point particle in classical electrodynamics42 or the uncertainties about the small-scale structure of relativistic spacetime.43 It seems our best hope actually lies in the opposite direction, in the discrete44—in kinetic theory or statistical mechanics—where the underlying microstructure is taken seriously and the phenomenological principles of earlier theories are in some sense recovered from it. 41 Philosophers of science use the term ‘model’ in many senses; see Emch and Liu [2002], }1.3, for a bewildering survey. I use the term simply for an abstract mathematical object, ultimately (we might as well say) for a set (whose existence is presumably provable from the axioms of set theory). The ‘temperature’ assigned here is just a real number. 42 See [1997], pp. 147–149, for discussion and references. 43 See [1997], pp. 149–151, for discussion and references. 44 In [2007], Part III and }IV.2.ii, I argue that a rudimentary logic and elementary arithmetic are literally true of many aspects of the ordinary macro-world. The focus here is on the status of more advanced scientific theorizing.

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The first triumph of kinetic theory is the derivation of the Ideal Gas Law.45 Of course, this proof begins with a ‘gas’ made up of point masses that don’t interact with each other and engage only in perfectly elastic collisions with the walls of the container. This is clearly an abstract model, not a description of real gases, for which the Ideal Gas Law holds only under special conditions (at low densities and high temperatures) and even then only as a good approximation. The model can be improved by replacing the point masses with tiny hard spheres of finite radius and allowing for forces acting between them; the result is van der Waals’s equation.46 Still, a van der Waals gas remains an abstract model, if one somewhat closer in structure to a real gas than the ideal gas model. This process can be continued,47 but in the end the behavior of the gas elements is governed by quantum mechanics. Unfortunately, despite the stunning success of quantum mechanics as a predictive device, we still have no firm grasp of what worldly features underlie its various mathematical constructs.48 The promise of literal description is perhaps greater in elementary statistical mechanics, where the reasoning is almost purely combinatorial. Whatever the small discrete elements of an isolated49 mole of gas are actually like, they can be properly described as occupying the left or the right half of the box that contains them.50 Suppose I’m interested 45 For textbook treatment, see e.g. McQuarrie and Simon [1997], }16–1, or Engel and Reid [2006], }16.1. 46 For textbook treatment, see e.g. McQuarrie and Simon [1997], }16–2, 16–7, or Brown et al [2006], }10.9. 47 Cf. McQuarrie and Simon [1997], p. 648: ‘There are more sophisticated equations of state (some containing more than 10 parameters!) that can reproduce the experimental data to a high degree of accuracy over a large range of pressure, density and temperature’, or Engel and Reid [2006], p. 9: ‘there are other more accurate equations of state that are valid over a wider range than the van der Waals equation. Such equations of state include up to 16 adjustable substance-specific parameters’. 48 See [2007], }}III.4, III.6, for discussion and references. 49 Perfect thermal isolation is itself an idealization, of course, but let me set this aside. We might imagine that a small error factor has been included—so we only assume the box to be very well insulated—and that the error is too small to affect the main conclusions. 50 Determinate particle locations give way to probability densities over such locations in ordinary quantum theory, and further mysteries arise in attempts to formulate a relativistic quantum mechanics of particles (see Malament [1996] for discussion). Here I’m assuming that whatever our ultimate understanding of the structure of our mole of gas in the small, it will somehow reproduce enough particle structure to underwrite this reasoning from elementary statistical mechanics.

24 the problem in what portion of the Avogadro’s number of molecules is located in the left half of the box. I can straightforwardly calculate how many divisions of the individual molecules into one half or the other—how many ‘micro-states’—are configurations with, say, one-third of the molecules in the left half of the box. I can also calculate how many micro-states there are all together, and I can compute the ratio of the first number to the second. All this is literal description of the situation. If I now assume that all the micro-states are equally likely, this ratio I’ve computed is the probability that one-third of the molecules are now in the left half of the box. With a few more such calculations, I begin to realize that this is dramatically less likely than something closer to a 50/50 split. Assuming my probabilistic assumption is correct, this is still straightforwardly literal, a matter of combinatorial fact about any collection of this large but finite size. It’s comparable in status to 2þ2=4 as a description of the total number of fruits that result when we collect two apples and two oranges on the table.51 So far so good. But fairly soon I find myself wanting to compute the most likely state directly. To do this, I consider a function f from the set of configurations to the finite numbers, where f(x molecules in the left side of the box) = the number of micro-states that place x molecules in the left side of the box. The most likely state is the one where f reaches a peak, that is to say, where the derivative of f is 0. But applying the methods of the differential calculus to f doesn’t really make sense, because the domain of f is finite. To make this work, I treat the domain of f as a continuous variable—and now I’ve taken leave of literal representation. Ludwig Boltzmann, who pioneered this line of thought, apparently had finitistic leanings; he cautions his readers against forgetting that the underlying basis for his mathematical treatment lies in finite collections: The concepts of the integral and differential calculus, cut loose from any [finitary] atomic representation, are purely metaphysical. (Quoted with references and discussion in Emch and Liu [2002], p. 239)

In practical terms, the move from discrete sets to continuous mathematics actually takes place even earlier in our simple reasoning. The 51

See [2007], }IV.2.ii, for more on the status of elementary arithmetic.

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trouble is that determining the number of micro-states involves computing N!, where N is Avogadro’s number (approximately 6 x 1023); given that 100! is already 9.3 x 10157, the N! here is clearly out of feasible computational range. The standard solution is to work with ln(N!) and to use what’s called Stirling’s approximation: N x ln(N)  N. This move is justified because ln(N!) is the sum of the Rln(n)’s as n N varies from 1 to N, and if N is large, this is very close to 1 lnðxÞdx, 52 which yields Stirling’s approximation. So computational practicality counsels the move from large finite to continuous even before the need for derivatives. I should note that this phenomenon—finite collections that are treated more successfully with infinitary, indeed continuum mathematics—isn’t at all special to statistical mechanics: in garden-variety statistics, discrete phenomena like household incomes or numbers of correct responses are routinely treated as continuous variables for similar reasons.53 In any case, this style of reasoning in statistical mechanics leads to Boltzmann’s definition of entropy, the statistical version of the second law of thermodynamics, the statistical explanation of why real-world processes seem irreversible when the underlying mechanics is reversible,54 and more practically, to the physical chemist’s ability to predict the direction of chemical reactions.55 Efforts to justify the mathematical methods employed in these accounts involve departures from literal description of a more conceptual nature than those discussed so far: for example, in ergodic theory, one considers, among other things, the evolution of a system as time goes to infinity; in the theory of the thermodynamic limit, one takes a limit as the number of molecules in the system goes to infinity.56 So beyond its first baby-steps, statistical mechanics also departs substantially from literal description.

52 See e.g. McQuarrie and Simon [1997], pp. 809–815, or Arfken [1985], pp. 555–558, for textbook discussions. This method appears in introductory treatments, e.g., Halliday, Resnick, and Walker [2005], pp. 552–553, or Engel and Reid [2006], pp. 284–285. 53 For a textbook example, see Rice [2007], p. 60. 54 See e.g. Emch and Liu [2002], pp. 106–112. 55 For textbook treatments, see Brown et al [2006], chapter 19, or McQuarrie and Simon [1997], chapters 20–22. 56 See Uffink [2007], }}6.1 and 6.3, respectively.

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Under the circumstances, I think it’s fair to characterize the work of contemporary scientists as presenting mathematical models of physical systems much as Duhem describes them in this passage:57 When a physicist does an experiment, two very distinct representations of the instrument on which he is working fill his mind: one is the image of the concrete instrument that he manipulates in reality; the other is a schematic model of the same instrument, constructed with the aid of symbols supplied by theories; and it is on this ideal and symbolic instrument that he does his reasoning, and it is to it that he applies the laws and formulas of physics. (Duhem [1906], pp. 155–156)

A manometer, for example, is On the one hand, a series of glass tubes, solidly connected to one another . . . filled with a very heavy metallic liquid called mercury by the chemists; on the other hand, a column of that creature of reason called a perfect fluid in mechanics, and having at each point a certain density and temperature defined by a certain equation of compressibility and expansion. (Duhem [1906], pp. 156–157)

There would be little point to such talk if the relation between the model and the physical system were a straightforward isomorphism but the story just recounted shows that this is not true for the ubiquitous differential equations of applied mathematics and (with minor exceptions58) we don’t appear to be in a position to make any stronger claims about subsequent theories that use other types of mathematics. In fact, the exact structure of those relations varies from case-to-case, as does our level of understanding of them. When we represent a cannon ball as a perfect sphere, the lengths, times, angles and forces involved as real numbers, the local surface of the earth as flat, and so on, in order to

57 There is some irony in the appeal to Duhem here, as he was one of the last antiatomists. Obviously his embrace of Newton’s ‘sound doctrines’ didn’t include the idea that mathematics correctly describes reality; Duhem [1906], pp. 133–134, actually harkens back to Poisson’s objections to Fourier, e.g., ‘the body studied is geometrically defined; its sides are true lines without thickness’. In fact, Duhem is a fascinating case: he opposed those who formed hypotheses, but he also opposed those who took differential equations literally, perhaps for reasons of the sort noted in footnote 39. This drove him to an unappealing fictionalism about natural science (see below). 58 That is, for the likes of ‘2+2=4’ and elementary statistical mechanics.

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determine where a given ball, fired with a given force, will land, we have a fairly good idea of at least some of our departures from literal truth and why they are admissible. When we represent spacetime as a continuous manifold, we aren’t entirely sure whether or not this constitutes a literal truth, though our well-informed hunch is that even if it is an idealization, it’s a good one—much as Euclidean geometry is a good approximation to the truth in most ordinary cases. But the fact remains that the mathematics has been peeled away from the science; the actual claim the scientist makes about the world is that it is probably, at least approximately, similar in structure to the mathematical model in certain respects, and that the idealizations involved are beneficial and benign for the purposes at hand. We’ve now viewed the rise of pure mathematics from several vantage points. First, we’ve seen how the study of many pure mathematical concepts, structures and theories arose simply because mathematicians began to pursue a range of peculiarly mathematical goals with no immediate connection to applications. Second, we’ve seen how Euclidean geometry, once unblushingly regarded as the true theory of physical space, became the study of one among many abstract mathematical spaces. Third and finally, we’ve seen how our best mathematical accounts of physical phenomena aren’t the literal truths Newton took them for, but free-standing abstract models that resemble the world in ways that are complex and sometimes not fully understood. Paradoxical as it may sound, it now appears that even applied mathematics is pure.

3. Where we are now This story, the story of how applied mathematics became pure, has its morals for our understanding of mathematics in both its pure and applied forms. One clear moral for mathematics in application is that we aren’t in fact uncovering the underlying mathematical structures realized in the world; rather, we’re constructing abstract mathematical models and trying our best to make true assertions about the ways in which they do and don’t correspond to the physical facts. There are

28 the problem rare cases where this correspondence is something like isomorphism— we’ve touched on elementary arithmetic and the simple combinatorics of beginning statistical mechanics, and there are probably others, like the use of finite group theory to describe simple symmetries—but most of the time the correspondence is something more complex, and all too often it’s something we simply don’t yet understand: we don’t know the small-scale structure of spacetime or the physical structures that underlie quantum mechanics. And even this leaves out the additional approximations and accommodations required to move from the initial mathematical model to actual predictions. Stirling’s approximation is among the simpler of such ad hoc fixes.59 This sort of thing leads some philosophers to a despair, to something along the lines of this from Duhem: Our physical theories do not pride themselves on being explanations; our hypotheses are not assumptions about the very nature of material things. Our theories have as their sole aim the economical condensation and classification of experimental laws. (Duhem [1906], p. 219)[60] Agreement with experiment is the sole criterion of truth for a physical theory. (Duhem [1906], p. 21)

59

See Wilson [2006], p. 26, for his ‘lesson of applied mathematics’: ‘Why do predicates behave so perversely? . . . I believe the answer rests largely at the unwelcoming door of Mother Nature. The universe in which we have been deposited seems disinclined to render the practical description of the macroscopic bodies around us especially easy. . . . Insofar as we are capable of achieving descriptive successes within a workable language . . . we are frequently forced to rely upon unexpectedly roundabout strategies to achieve these objectives. It is as if the great house of science stands before us, but mathematics can’t find the keys to its front door, so if we are to enter the edifice at all, we must scramble up backyard trellises, crawl through shuttered attic windows and stumble along half-lighted halls and stairwells’. See also p. 452. 60 ‘The diverse principles or hypotheses of a theory are combined together according to the rules of mathematical analysis. . . . The magnitudes on which [the theorist’s] calculations bear are not claimed to be physical realities, and the principles he employs in his deductions are not given as stating real relations among those realities; therefore it matters little whether the operations he performs do or do not correspond to real or conceivable physical transformations. All that one has the right to demand of him is that his syllogisms be valid and his calculations accurate. . . . Thus a true theory is not a theory which gives an explanation of physical appearances in conformity with reality; it is a theory which represents in a satisfactory manner a group of experimental laws. A false theory is not an attempt at explanation based on assumptions contrary to reality; it is a group of propositions which do not agree with experimental laws’ (Duhem [1906], pp. 20–21). See also Duhem [1906], p. 266.

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Here the whole of theoretical science is regarded as a black box whose sole purpose and achievement is to catalog experimental outcomes. Just as the hope of a simple isomorphism between abstract model and worldly structure seems too optimistic, this fictionalist dismissal of any descriptive content seems too pessimistic, an over-reaction to the complexities of scientific modeling. Though we may well be at a complete loss as to how quantum mechanics relates to the world, this is hardly true of the many other cases we’ve touched on; in kinetic theory, in fluid dynamics, in practical chemistry, we have a fair idea of what our models are on to, of where they are deficient, and of why they work well nonetheless. Still, one chastening conclusion does seem warranted: it appears unlikely that any general, uniform account of how mathematics applies to the world could cover the wide variety of cases. To take just a few of the examples we’ve noted along the way, the point particle model of an ideal gas works effectively for dilute gases because the occupied volume is negligibly small compared to the total volume;61 the ‘continuum hypothesis’ works effectively in fluid dynamics because there is a suitable ‘plateau’ between volumes too small to have stable temperature and volumes too large to have uniform temperature; the billiard ball model of a van der Waal gas works effectively because actual molecules do have fairly stable ‘effective radii’.62 Textbook examples could be multiplied: We should discuss how good an approximation [a harmonic oscillator] is for a vibrating diatomic molecule. . . . Although the harmonic-oscillator potential may appear to be a terrible approximation to the experimental curve [of internuclear potential], note that it is, indeed, a good approximation in the region of the minimum. This region is the physically important region for many molecules at room temperature.[63] The harmonic oscillator will be a good approximation for vibrations with small amplitudes. (McQuarrie and Simon [1997], pp. 163–164)

61

For textbook treatment, see e.g. Engel and Reid [2006], pp. 149–150. For textbook discussion, see e.g. McQuarrie and Simon [1997], }16–7. 63 ‘Although the harmonic oscillator unrealistically allows the displacement to vary from 0 to +1, these large displacements produce potential energies so large that they do not often occur in practice.’ 62

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Treatments of applied mathematics also include careful analyses of when it’s appropriate to replace a discrete variable with a continuous one: The only difference between these last two equations is that the summation over discrete values of the variable . . . has been replaced by integration over the range of the variable. The preceding comparison demonstrates that the continuous approximation is close to the exact result given by summation. In general, if the differences between the values the function can assume are small relative to the domain of interest, then treating a discrete variable as continuous is appropriate. This issue will become critical when the various energy levels of an atom or molecule are discussed. Specifically, the approximation . . . will be used to treat translational and rotational states from a continuous perspective where direct summation is impractical. In the remainder of this text, situations in which the continuous approximation is not valid will be carefully noted. (Engel and Reid [2006], p. 290. See also p. 326)

Given the diversity of the considerations raised to delimit and defend these various mathematizations, anything other than a patient case-bycase approach would appear unpromising. But our primary focus here is on pure mathematics, and I think our historical survey teaches us an important methodological lesson about its pursuit. What we’ve traced is a more-or-less simultaneous rise of pure mathematics and re-evaluation of applied mathematics. Before all this, back in Newton’s or Euler’s day,64 the methods of mathematics and the methods of science were one and the same; if the goal is to uncover the underlying structure of the world, if mathematics is simply the language of that underlying structure, then the needs of celestial mechanics (for Newton) or rational mechanics (for Euler) are the needs of mathematics. From this perspective, the correctness of a new mathematical method—say the infinitary methods of the calculus

64

In fairness to Euler, I should note that Truesdell ([1981], pp. 120–121) writes: ‘Today we look upon classical physics as providing us with mathematical models for the behavior of physical objects. We use these models with great caution, for we are deeply aware of their limitations . . . The Bernoullis had no idea that they were dealing with models; like Galileo, they thought that nature herself spoke in mathematics. Euler in his middle life began to perceive how much the mathematician replaced nature by his own conceptions’. I don’t know what aspect of Euler’s thought Truesdell has in mind here.

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or the expanded notion of function65—is established by its role in application. In contrast, after the developments we’ve been tracing, mathematics has been freed to pursue inquiries without application, it’s encouraged to stock the warehouses with structures and leave the choices to the natural scientists, and even the mathematical constructs that do function in application do so with a new autonomy as freestanding abstract models. In this brave new world, where can the pure mathematician turn for guidance? How can we properly determine if a new sort of entity is acceptable or a new method of proof reliable? What constrains our methodological choices? These were among the pressing questions faced by the mathematical community at the end of the 19th century. In Kline’s words, The circle within which mathematical studies appeared to be enclosed at the beginning of the century was broken at all points, and mathematics exploded into a hundred branches. (Kline [1972], p. 1023)

This dramatic expansion brought with it the realization that for some time mathematics had ‘rested on an empirical and pragmatic basis’, that a ‘rapidly increasing mass of mathematics . . . rested on the loose foundations of the calculus’, that even in geometry mathematicians had overlooked ‘inadequacies in proofs’ and ‘been gullible and relied upon intuition’ (Kline [1972], pp. 1024–1025). All this produced a renewed emphasis on rigor, the central tool of which was axiomatization, along the lines of Hilbert’s axioms for geometry and Dedekind’s axioms for the real numbers. Kline continues, Mathematics . . . was by the end of the nineteenth century a collection of structures each built on its own system of axioms. (Kline [1972], p. 1038) . . . the axiomatic method . . . permitted the establishment of the logical foundations of many old and newer branches of mathematics, . . . revealed precisely what assumptions underlie each branch and made possible the comparison and clarification of the relationships of various branches. (Kline [1972], p. 1027) 65 In his treatment of the wave equation, D’Alembert required the initial shape of a vibrating string to be given by an analytic expression (so as to be twice differentiable), but Euler insisted that functions shouldn’t be limited in this way, because the most common initial conditions—for example, a plucked string—don’t satisfy this requirement. The result was a major revision of the notion of function, a firm push toward the modern set-theoretic conception. See [1997], pp. 116–123, for discussion and references.

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But it was widely appreciated that simply laying down a list of axioms isn’t enough to establish that they succeed in describing a genuine structure.66 So, for example, the coherence of the axioms for nonEuclidean geometries had been demonstrated by modeling them in Euclidean geometry, and Euclidean geometry itself could be modeled using the real numbers, but the buck has to stop somewhere. What emerged gradually—in the theory of functions, in Dedekind’s constructions of the reals, in the foundations of arithmetic and elsewhere—is that set theory provides a natural arena in which to interpret the myriad structural descriptions of mathematics, to settle which are and aren’t coherent.67 Unfortunately, early 20th-century set theory was itself subject to considerable controversy arising from the discovery of the paradoxes, Zermelo’s controversial proof of the well-ordering theorem, and a number of other uncertainties. Gregory Moore writes: In the wake of Russell’s paradox, published in 1903, it was even uncertain what constituted a set. Moreover, Zermelo’s proof itself raised a number of methodological questions: Was it legitimate to define a set . . . in terms of a totality of which [it] was a member . . . ? . . . Did the class . . . of all ordinals invalidate Zermelo’s proof and entangle it in Burali-Forti’s paradox? Most important of all, was his Axiom of Choice true? Was it a law of logic? Should one postulate simultaneous, independent choices in preference to successive,

66 Nowadays we’d tend to say that the axioms must be consistent, but our contemporary use of that term presupposes a developed account of syntax, semantics, and their interrelations that wasn’t available in 1900 (e.g., the completeness theorem wasn’t proved until Go¨del’s doctoral dissertation of 1930). Structuralists in the philosophy of mathematics tend to use the less precise-sounding word ‘coherent’ (see e.g. Shapiro [1997], pp. 95, 132–136, Parsons [2008], p. 114)—a structure exists if its description is coherent—and I follow them in the next sentence. 67 Following up the previous footnote, Shapiro ([1997], p. 136) allows that ‘in mathematics as practiced’, set theory is used to settle questions of which axiomatizations or implicit definitions are and aren’t coherent, and thus of which structures do and don’t exist. In contrast, Awodey ([2004], pp. 58, 60) suggests that ‘Every mathematical theorem is of the form “if such-and-such is the case, then so-and-so holds” . . . Of course many theorems do not literally have this form, but every theorem has some conditions under which it obtains . . . There is usually no question about whether such conditions are ever satisfied; rather, like axiomatic definitions, they serve to specify the range of application of the subsequent statement’. On this basis, he denies the need for a foundation of the sort set theory has been able to provide. Perhaps this sounds more appealing now, when the worries of 1900 have long since been quieted; one forgets the mathematical forces that led the mathematicians of 1900 to feel that need in the first place.

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dependent ones? Did the cardinality of the set of choices affect the validity of the Axiom, so that the Denumerable Axiom was true but not the Axiom of Choice in general? . . . All these questions echoed a broader problem which had rarely been enunciated explicitly: What methods were permissible in mathematics? Must such methods be constructive? If so, what constituted a construction? What did it mean to say that a mathematical object existed? (Moore [1982], p. 85)

Clearly set theory itself stood in need of the rigorizing benefits of axiomatization, which Zermelo set out to provide. In his 1908 presentation of the first system of axioms for set theory, Zermelo recognizes its special role: Set theory is that branch of mathematics whose task is to investigate mathematically the fundamental notions ‘number’, ‘order’ and ‘function’, taking them in their pristine, simple form, and to develop thereby the logical foundations of all arithmetic and analysis; it thus constitutes an indispensable component of the science of mathematics. (Zermelo [1908b], p. 200)

But despite its importance, The very existence of this discipline seems to be threatened by certain contradictions, or ‘antinomies’, that can be derived from its principles . . . to which no entirely satisfactory solution has yet been found. (op. cit.)

The simple motivating idea that any objects can be collected into a set ‘requires some restriction’, so he aims To seek out the principles required for establishing the foundations of this mathematical discipline . . . we must, on the one hand, restrict these principles sufficiently to exclude all contradictions and, on the other, take them sufficiently wide to retain all that is valuable in this theory. (op. cit.)

The result was a list of axioms that was eventually developed and extended into the now-standard system Zermelo-Fraenkel with Choice or ZFC. In the century since Zermelo’s first effort, set theory has solidified its role as the backdrop for classical mathematics. Questions of the form— is there a structure or a mathematical object like this?—are answered by finding an instance or a surrogate within the set-theoretic hierarchy. Questions of the form—can such-and-such be proved or disproved?—

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are answered by investigating what follows or doesn’t follow from the axioms of set theory. This isn’t to say that set theory shows us what numbers or functions really are; all set theory need claim is that certain sets can do all the mathematical jobs required of numbers or functions. It also isn’t to claim that we’re somehow more certain of the coherence of ZFC than, say, that of Peano Arithmetic, or that all mathematics could or should be done using exclusively set-theoretic methods.68 What set theory does is provide a generous, unified arena to which all local questions of coherence and proof can be referred. In this way, set theory furnishes us with a single tool that can give explicit meaning to questions of existence and coherence; make previously unclear concepts and structures precise; identify perfectly general fundamental assumptions that play out in many different guises in different fields; facilitate interconnections between disparate branches of mathematics now all uniformly represented; formulate and answer questions of provability and refutability; open the door to new strong hypotheses to settle old open questions; and so on. In this philosophically modest but mathematically rich sense, set theory can be said to found contemporary pure mathematics.69 The remaining question, then, is how set theory itself is properly done. What kinds of considerations count for or against a given assumption or method? Over the years, some have argued for a return to justifications based in natural science, for example, in the case of the Axiom of Choice. Those familiar with the history70 will recall that Zermelo’s introduction of the Axiom in 1904 set off a fierce debate between its proponents and its opponents. This dispute had many strands—some metaphysical, some practical—but one feature of the Axiom that particularly troubled many mathematicians was the so-called 68

e.g., it’s well-recognized that category-theoretic formulations are more natural than settheoretic for many mathematical purposes, but this doesn’t distinguish category theory from various other branches of mathematics with their own vocabularies and techniques. What does distinguish category theory is the claim that it can provide a foundation of the sort described in the text, an alternative to set-theoretic foundations, but it remains to be seen whether or not it can do this effectively for the whole of mathematics (including higher set theory). In any case, what matters here is that set theory was developed at least partly to do this job, and that this fact has consequences for its proper practice. 69 See [1997], }I.2, for more discussion of the nature of set-theoretic foundations. 70 See Moore [1982].

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Banach-Tarski paradox: using the Axiom, a sphere can be decomposed into finitely many parts and those parts reassembled into two spheres of the same size as the original.71 (The key is that the parts into which the sphere is cut are non-measurable, which is where Choice comes in.) This conclusion seems obviously absurd from a physical point of view, so Banach and Tarski’s result is sometimes taken as evidence against the Axiom of Choice, as something akin to a false prediction.72 If we’ve assimilated the lesson of the third strand of our story of how applied mathematics became pure, the strand that highlights the role of abstract models that resemble the world only partially and within certain limits, then the reply to this argument against Choice is straightforward: if physical regions aren’t literally modeled by sets of ordered triples of real numbers, then we can’t assume that all consequences of our mathematical theory of those sets will hold for those regions; therefore, we can’t conclusively draw our false empirical conclusion. The realization that our model departs from reality might lead us to modify our model, or it might, as in this case, simply lead us to apply our model with more care, making sure not to rely too heavily on its more esoteric aspects. What’s perhaps slightly less obvious is that the full force of the third thread isn’t needed to defend Choice here; the second thread is enough, the one leading to the well-stocked warehouses of abstract structures. To see this, ignore the third thread and suppose for the sake of argument that physical regions are literally modeled by subsets of ℝ3. Then the line of thought goes like this: working in a set theory with the Axiom of Choice, we perform the Banach-Tarski construction, which is physically ridiculous; we conclude that Choice has been empirically disconfirmed. But isn’t it at least as reasonable to conclude that the full power set of ℝ3 was a poor choice as a model for physical regions?73 Indeed, given the many internal mathematical considerations in favor of 71

Banach and Tarski proved this theorem in 1924. See Wagon [1985] for discussion. Fraenkel, Bar-Hillel, and Levy [1973], p. 83, give references to this sort of reaction from Borel and others. They also note that Hausdorff had a similarly ‘paradoxical’ result in 1914, ten years before Banach-Tarski. 73 Again, as above, I’m not suggesting that this is what we actually do in our full appreciation of the third strand. For most purposes, it isn’t worth complicating our model in this way; it’s more practical just to use it with care. 72

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the Axiom,74 wouldn’t it be considerably more reasonable to conclude that physical regions are more effectively modeled by measurable subsets of ℝ3? If, for example, our set theory includes sufficient large cardinals, we might count Banach-Tarski as a good reason to model physical space in L(ℝ), where all sets of reals are measurable.75 In any case, the suggestion is that what’s disconfirmed is the claim that regions are best modeled by P(ℝ3), not the Axiom of Choice.76 Once we have those well-stocked warehouses, any candidate for an empirical confirmation or disconfirmation of the mathematics is more reasonably viewed as confirming or disconfirming the claim that the best model has been identified. If this is right, then perhaps the added force of the third thread, where we relinquish the dream of literal modeling, is more consequential for the practice of science—cautioning us against regarding all questions about our mathematical models as real physical issues— than it is for the practice of mathematics. In any case, I think by now it’s clear that considerations from applications are quite unlikely to prompt mathematicians to restrict the range of abstract structures they admit. It’s just possible that as-yetunimagined pressures from science will lead to profound expansions of the ontology of mathematics, as with Newton and Euler, but this seems considerably less likely than in the past, given that contemporary set theory is explicitly designed to be as inclusive as possible.77 More likely, pressures from applications will continue to influence which parts of the set-theoretic universe we attend to, as they did in the case of Dirac’s delta function;78 in contemporary science, for example, the 74

See e.g. [1997], pp. 54–57, for summary and references. L(ℝ) is the smallest model of ZF containing all the real numbers; the existence of a supercompact cardinal implies that all its sets of reals are Lebesgue measurable. See Jech [2003], pp. 650–653, or Kanamori [2003], }32. ( Jech’s hypothesis is actually the existence of a superstrong cardinal, but the existence of a supercompact is considered the more natural hypothesis—a generalization of measurability—and it implies the existence of a superstrong. I come back to supercompacts in V.3.i.) For more, see II.2.iv. 76 See Urquhart [1990], p. 152, for a similar conclusion in the context of Field’s nominalization project. 77 For that matter, any hint of an important structure unrealized in the universe of sets would most likely motivate such an expansion, whether the structure originated in science or not! 78 The delta function takes the value zero except at x=0, but its overall integral is one. This is impossible, of course, because the so-called function is zero almost everywhere, 75

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needs of quantum field theory and string theory have both led to the study of new provinces of the set-theoretic universe.79 All this simply reinforces the conclusion that we should no longer expect science to provide the sort of methodological guidance for mathematics that it once did, and returns us to the challenge of isolating and understanding the proper methods of set theory. This question becomes especially acute when we discover that we cannot simply rest content with the time-honored axioms of ZFC. Powerful as it is—powerful enough to provide surrogates for all classical mathematical objects and to prove versions of all classical mathematical theorems—ZFC isn’t powerful enough to settle various natural questions that arise in set theory and the various branches of mathematics:80 are all projective point sets Lebesgue-measurable? are all Whitehead groups free? is every uncountable set of real numbers equinumerous with the set of all real numbers (the famous Continuum Problem)? To settle these matters, we must add new assumptions to Zermelo’s list, and the question of how to do so properly cannot be avoided. By this long and circuitous route, we’re brought at last to an appreciation of the mathematical importance of these two fundamental questions: what are the proper methods of set theory, and why? I touch on the first of these briefly in the next chapter, aiming to bring sharper focus to the special features and difficulties of the second. The remainder of the book then addresses that matter of ‘why’—what is set theory that these are the correct ways of going about it?81 which guarantees an integral of 0. Dirac noted that it could nevertheless be used ‘for practically all purposes of quantum mechanics without getting incorrect results’ (quoted by Pais in Pais et al [1998], p. 7). (This function actually played a role as early as the 1890s in Heaviside’s operational calculus. See van der Pol and Bremmer [1950], pp. 1–5, 62–66, for the history.) Efforts to rigorize the delta function eventually led to Schwartz’s theory of distributions or generalized functions (see van der Pol and Bremmer [1950], p. 84, or Zemanian [1965]). 79

See e.g. Emch [1972], Witten [1998]. For more on these questions, see [1997], }I.3, Eklof [1976]. 81 The exclusively methodological study of [1997] set this question aside (see pp. 200 –203). After an unavoidable detour through [2007], I now return to it. 80

II Proper Method We’re faced with a stark challenge: what are the proper methods for set theory and why? The goal of this chapter to address the first half of this challenge, and to refine our appreciation of what the second half requires of us. But first we need a clear sense of the philosophical perspective from which I propose to approach these matters.

1. The meta-philosophy Imagine a simple inquirer who sets out to discover what the world is like, the range of what there is and its various properties and behaviors. She begins with her ordinary perceptual beliefs, gradually develops more sophisticated methods of observation and experimentation, of theory construction and testing, and so on; she’s idealized to the extent that she’s equally at home in all the various empirical investigations, from physics, chemistry, and astronomy to botany, psychology, and anthropology. She believes that ordinary physical objects are made up of atoms, that plants live and grow by photosynthesis, that humans use language to describe the world to one another, that social groups tend to behave in certain ways, and so on. She also believes that she and her fellow inquirers are engaged in a highly fallible, but partly and potentially successful exploration of the world, and like anything else, she looks into the matter of how and why the methods she and others use in their inquiries work when they do and don’t work when they don’t; in these ways, she gradually improves her methods as she goes. In the course of these investigations, our inquirer begins to notice that logic and arithmetic are essential tools in her efforts to understand

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the world, and she eventually sees that the calculus, higher analysis, and much of contemporary pure mathematics are also invaluable for getting at the behaviors she studies and for formulating her explanatory theories. This gives her good reason to pursue mathematics herself, as part of her investigation of the world, but she also recognizes that it is developed using methods that appear quite different from the sort of observation, experimentation, and theory formation that guide the rest of her research. This raises questions of two general types. First, as part of her continual evaluation and assessment of her methods of investigation, she will want an account of the methods of pure mathematics; she will want to know how best to carry on this particular type of inquiry. Second, as part of her general study of human practices, she will want an account of what pure mathematics is: what sort of activity is it? what is the nature of its subject matter? how and why does it intertwine so remarkably with her empirical investigations? In this humdrum way, by entirely natural steps, our inquirer has come to ask questions typically classified as philosophical.1 She doesn’t do so from some special vantage point outside of science, but as an active participant, entirely from within. Meta-philosophical positions that advocate this sort of approach are often called ‘naturalistic’, but one cautionary note: our inquirer doesn’t believe as she does because ‘science says so’, as some naturalists would have it,2 but on perfectly ordinary grounds—this experiment, that well-confirmed theory. Though we might use the rough-and-ready term ‘scientific’ to describe her behavior, no specific characterization of what counts as ‘scientific’ is in play here, nothing beyond the perfectly ordinary story of beginning from observation, developing more refined methods, and so on. Indeed, any attempt at a once-and-for-all characterization of our inquirer’s methods would run counter to the ever-improving, open-ended

1 Earlier on, she would have paused to ask others, e.g., what is the nature of logical truth and elementary arithmetic? For discussion, see [2007], Part III and }IV.2.ii. 2 Compare Burgess and Rosen’s characterization of ‘naturalism’: ‘The naturalists’ commitment is at most to the comparatively modest proposition that when science speaks with a firm and unified voice, the philosopher is either obliged to accept its conclusions or to offer what are recognizably scientific reasons for resisting them’ (Burgess and Rosen [1997], p. 65). My inquirer doesn’t decide to place her faith in something called ‘science’; she is simply one of those speaking with a firm voice, on the basis of the evidence.

40 prope r method nature of her project. So I’m not advocating any meta-philosophical doctrine or principle to the effect that we should ‘trust only science’; I’m simply describing this inquirer, counting on you to get the hang of how she would approach the various traditionally philosophical questions we’re interested in, and hoping that you find this exercise as illuminating as I do. To round out this quick portrait, consider the contrast with philosophy understood as starting either before science begins or after all scientific evidence is in, that is, philosophy as an entirely independent enterprise. Notice that if such a philosophical undertaking intends to correct science, or even to justify it in some way, then it isn’t effectively separated from our inquirer’s sphere of interest: working without any litmus test for ‘science’ or ‘non-science’, she will view it as a potential part of her own project, out to revise or buttress her methods; faced with such a proposal, she will want to know the grounds on which the criticism or confirmation is based and to evaluate these grounds on her own terms.3 To be truly autonomous, a philosophical enterprise would grant that science is perfectly in order for scientific purposes, but insist that there are other, extra-scientific purposes for which different methods are appropriate. So, for example, Bas van Fraassen holds that familiar experimental evidence establishes the existence of unobservables (like atoms) for scientific purposes, but that, from a philosophical or epistemic standpoint, such beliefs can never be justified. Faced with such a challenge, our inquirer is simply baffled: all her good evidence has been declared irrelevant, ‘merely scientific’; she’s asked to justify her belief in atoms on other grounds for unfamiliar purposes, but she’s given no working understanding of what those purposes are and what methods are appropriate in their service. Philosophy undertaken in such complete isolation from science and common sense is often called ‘First Philosophy’, so I call our inquirer a Second Philosopher.4 Let’s now focus on the Second Philosopher’s investigation of mathematics. Since we’re mainly interested in her treatment of set theory, 3 e.g., she will test the merits of Descartes’s Method of Doubt, intended to found science more firmly (see [2007], }I.1). 4 For more, see [2007], especially Part I and }IV.1, and [2010].

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let’s skip lightly over the story of how her narrowly applied sense of the subject gradually gives way to the full pursuit of pure mathematics, recapitulating the dramatic developments surveyed in Chapter I. By that point, the great expansion in the range of mathematical structures studied and the variety of methods used left some uncertainty about which of these were legitimate, and as set theory developed, it gradually took on the role of providing a unified account of the world of classical mathematics and its fundamental assumptions. When our Second Philosopher is faced with contemporary set theory, we’ve seen that questions of two types arise. The first group is methodological: what are the proper grounds on which to introduce sets, to justify settheoretic practices, to adopt set-theoretic axioms? The second group is more traditionally philosophical: what sort of activity is set theory? how does set-theoretic language function? what are sets and how do we come to know about them? Given that the Second Philosopher will want to pursue set theory, along with her other inquiries, the most immediate problem will be the methodological one—how am I to proceed?—so it makes sense to begin there. To get a feel for the forces at work, let’s review some concrete examples.

2. Some examples from set-theoretic practice (i) Cantor’s introduction of sets In the early 1870s, Cantor was engaged in a straightforward project in analysis: generalizing a theorem on representing functions by trigonometric series.5 Having shown that such a representation is unique if the series converges at every point in the domain, Cantor began to investigate the possibility of allowing for exceptional points, where the series fails to converge to the value of the represented function. It turned out that uniqueness is preserved despite finitely many exceptional points, or even infinitely many exceptional points, as long as these are arranged around a single limit point, but Cantor realized that

See Dauben [1979], chapter 2, Ferreiro´s [2007], }}IV.4.3 and V.3.3, for historical context and references. 5

42 prope r method it extends even further. To get at this extension, he moved beyond the set of exceptional points and its limit points to what he called ‘the derived set’: It is a well determined relation between any point in the line and a given set P, to be either a limit point of it or no such point, and therefore with the pointset P the set of its limit points is conceptually co-determined; this I will denote P 0 and call the first derived point-set of P. (As translated and quoted in Ferreiro´s [2007], p. 143)

Once this new set, the first derived set, P 0 , is in place, the same operation can be applied again: with P 0 , the set of its limit points is ‘conceptually co-determined’; this P 00 is the second derived set of the original P; and so on. Cantor then proved that if the n-th derived set of the set of exceptional points is empty for some natural number n, then the representation is unique.6 Of course there had been talk of point sets before Cantor navigated this line of thought, but here for the first time a point set is regarded as an entity in its own right, susceptible to the operation of taking its derived set. In the words of the historian Jose´ Ferreiro´s, What is really original in this contribution is that Cantor does not consider limit points in isolation, so to say, as Weierstrass had done, but makes a step toward a set-theoretical perspective. As a result, ‘set derivation’ is conceived as an operation on sets. (Ferreiro´s [2007], p. 143) Cantor went beyond the customary approach to analysis within the Berlin school, with its close attention to explicit analytic representations . . . He presented the notion of limit point as an abstract one, and took the crucial steps of considering sets of limit points and forming the derived set of a point-set. In so doing, Cantor was turning toward an abstract approach to mathematics that employed the language of sets. (Ferreiro´s [2007], p. 159)

From a methodological point of view, what’s happened is that a new type of entity—a set—has been introduced as an effective means toward an explicit and concrete mathematical goal: extending our understanding of trigonometric representations. Ferreiro´s ([2007], p. 160) notes that uniqueness continues to hold if the Æ-th derived set is empty for some transfinite ordinal Æ, but Cantor apparently never made this extension. 6

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(ii) Dedekind’s introduction of sets Also in the 1870s, Dedekind made early use of what we now recognize as sets, this time in algebra, in his theory of ideals. One central strand of this methodologically rich story7 involves Dedekind’s decision to replace the ideal number of Kummer, which is never defined in its own right, but only as a divisor of actual numbers . . . by a noun for something which actually exists. (Dedekind writing in 1877; see Avigad [2006], p. 172, for translation and references)

This ‘something which actually exists’, the ideal number, Dedekind identifies with the set of numbers Kummer would have taken it to divide. In his illuminating survey of Dedekind’s thought during the development of his theory of ideals, Jeremy Avigad remarks:8 The insistence on treating [sets] of numbers . . . as objects in their own right [has] important methodological consequences: it encourages one to speak of ‘arbitrary’ [sets], and allows one to define operations on them in terms of their behavior as sets . . . in a manner that is independent of the way in which they are represented.

Among Dedekind’s goals were general arguments in representation-free terms that would then ‘ “explain” why calculations with and properties of the objects do not depend on these choices of representations’. By downplaying the underlying computational algorithms highlighted by Kummer, Dedekind Sends the strong message that such algorithms are not necessary, i.e. that one can have a fully satisfactory theory that fails to provide them. This paves the way to more dramatic uses of non-constructive reasoning.

Here again, sets are being introduced in service of explicit mathematical desiderata—representation-free definitions, abstract (non-constructive) reasoning—though Dedekind’s vision is broader than the above-cited example from Cantor:

7 See Avigad [2006], Ferreiro´s [2007], chapters III, IV and VII, for more on Dedekind’s thought. 8 All quotations in this paragraph come from Avigad [2006], pp. 173–174.

44 prope r method It is striking that Dedekind adopts the use of [infinite sets] without so much as a word of clarification or justification. That is, he simply introduced a style of reasoning that was to have decisive effects on future generations, without fanfare.

The mathematical fruitfulness of Dedekind’s innovations was dramatically demonstrated as abstract algebra went on to thrive in the hands of Noether and her successors.9 The same drive toward new numbers as actual objects with representation-free characterizations is on display in Dedekind’s theory of the real numbers. Here Dedekind’s goal is to provide a ‘perfectly rigorous foundation for the principles of infinitesimal analysis’,10 and in particular, to remove the ‘geometric evidence . . . [that] can make no claim to being scientific’. He takes (some version of ) the theorem—a non-empty set of reals with an upper bound has a least upper bound11— to be ‘a more or less sufficient foundation’ for the subject; since the calculus is held to deal with ‘continuous quantities’, he reasons, one would expect to find the proof of this theorem resting on ‘an explanation of this continuity’; instead one finds ‘appeal . . . to geometric representations or to representations suggested by geometry’. Dedekind sets out, therefore, ‘to discover [the] true origin’ of the theorem, and thereby ‘secure a real definition of the essence of continuity’. The result, of course, is his elegant definition of continuity in terms of cuts and his theory of real numbers, each an infinite set of rationals.12 As Ferreiro´s notes, The title and the whole exposition emphasize the fact that the definition of continuity, and the proof of the continuity of [the reals], constitute the core of the matter. This is a way of presenting the whole issue that differs from the expositions of Weierstrass, Heine and Cantor. Dedekind writes that only a precise definition of continuity will offer a sound foundation for ‘the investigation of all continuous domains’. (Ferreiro´s [2007], p. 133)

9

See McLarty [2006]. All quotations in this paragraph come from Dedekind [1872], p. 767. 11 Dedekind’s version is ‘every magnitude which grows continually, but not beyond all limits, must certainly approach a limiting value’ (Dedekind [1872], p. 767). 12 He had already defined the integers and rationals in terms of natural numbers (see Ferreiro´s [2007], p. 219). 10

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Weierstrass and Cantor employ infinitistic ‘constructs’ that were usual in analysis, series and sequences respectively; Dedekind chooses to rely on a new means for ‘construction’. The resulting theory is simpler in that every real number corresponds to only one, or at most two, cuts, while it corresponds to many . . . sequences . . . or . . . series. (Ferreiro´s [2007], p. 131)

So here again we see Dedekind preferring definitions that aren’t tied to particular representations (like series or sequences), while pursuing broader mathematical goals (a general theory of continuity). Another important mathematical goal, also clearly present in this work on real numbers, is the pursuit of rigor: ‘In science nothing capable of proof ought to be believed without proof ’ (Dedekind [1888], p. 790). This declaration opens Dedekind’s account of the natural numbers, a third venue for his appeal to sets. Here he officially lays out his background set theory and goes on to develop his account of the natural numbers. In all these cases, we find Dedekind introducing sets in the service of explicit mathematical goals: a representationfree, non-constructive abstract algebra; a rigorous characterization of continuity to serve as a foundation for analysis and a more general study of continuous structures; a rigorous characterization of the natural numbers and resulting foundation for arithmetic. (iii) Zermelo’s defense of his axiomatization Turning from the introduction of sets to the adoption of axioms about them, we find Zermelo in 1908 with a range of motives. Locally, he hopes to quiet the controversy over his proof of the well-ordering theorem from the Axiom of Choice by proving it again from a weaker and more precise list of assumptions.13 More globally (as noted in }I.3), he sees his efforts as part of the Hilbertian foundational project:14 Set theory is that branch of mathematics whose task is to investigate mathematically the fundamental notions ‘number’, ‘order’, and ‘function’, taking them in their pristine, simple form, and to develop thereby the logical foundations of all arithmetic and analysis; thus it constitutes an indispensable component of the science of mathematics. (Zermelo [1908b], p. 200)

13 14

See Moore [1982], pp. 143–160. See Ebbinghaus [2007], pp. 76–79.

46 prope r method He notes the poor prospects for a compelling and fruitful definition of ‘set’ that could found set theory—something comparable, say, to Dedekind’s definition of ‘continuity’ and its role in founding analysis—so he concludes that There is at this point nothing left for us to do but to proceed in the opposite direction and, starting from set theory as it is historically given, to seek out the principles required for establishing the foundations of this mathematical discipline. In solving the problem we must, on the one hand, restrict these principles sufficiently to exclude all contradictions and, on the other, take them sufficiently wide to retain all that is valuable in this theory. (Zermelo [1908b], p. 200)

He goes on to propose the first axiomatization of set theory, in seven axioms, including Choice; he then develops ‘the entire theory created by Cantor and Dedekind’ (op. cit.) on this basis and demonstrates that the known paradoxes cannot be so generated. Of particular interest for our purposes are his reflections on the proper methods for justifying axioms. Presumably their foundational success counts in favor of his axioms as a whole, but when pressed on the Axiom of Choice in particular, Zermelo distinguishes evidence of two sorts: That this axiom . . . has frequently been used, and successfully at that, in the most diverse fields of mathematics, especially in set theory, by Dedekind, Cantor, F. Bernstein, Schoenflies, J. Ko¨nig, and others is an indisputable fact . . . Such an extensive use of a principle can be explained only by its self-evidence . . . No matter if this self-evidence is to a certain degree subjective—it is surely a necessary source[15] of mathematical principles. (Zermelo [1908a], p. 187)

The claim is that Choice is ‘intuitively evident’ (op. cit.), as revealed in the informal practice of set theorists; we might now express this by saying it is part of the informal ‘concept of set’. But, as we’ve seen, Zermelo despairs of defining this concept with a precision adequate to the development of set theory. Instead he appeals to a second standard of evidence that can be ‘objectively decided’, namely ‘whether the principle is necessary for 15

Could Zermelo literally mean ‘source’ here, leaving open the question of justificatory force? Cf. }V.4, below.

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science’ (op. cit.). Here he lists various outstanding problems that can be resolved on the assumption of Choice, and concludes So long as the relatively simple problems mentioned here remain inaccessible [without Choice], and so long as, on the other hand, the principle of choice cannot be definitely refuted, no one has the right to prevent the representatives of productive science from continuing to use this ‘hypothesis’—as one may call it for all I care—and developing its consequences to the greatest extent, especially since any possible contradiction inherent in a given point of view can be discovered only in that way. . . . principles must be judged from the point of view of science, and not science from the point of view of principles fixed once and for all. (Zermelo [1908a], p. 189)

This mode of defense goes beyond the observation that his axioms allow the derivation of set theory as it currently exists and the foundational benefits thereof; Zermelo here counts the mathematical fruitfulness of his axioms, their effectiveness and promise, as points in their favor. Go¨del also recognized the importance of such evidence, for example, in this well-known passage: Even disregarding the intrinsic necessity of some new axiom, and even in case it has no intrinsic necessity at all, a probable decision about its truth is possible also in another way, namely, inductively by studying its ‘success’. Success here means fruitfulness in consequences, in particular in ‘verifiable’ consequences, i.e., consequences demonstrable without the new axiom, whose proofs with the help of the new axiom, however, are considerably simpler and easier to discover, and make it possible to contract into one proof many different proofs. . . . There might exist axioms so abundant in their verifiable consequences, shedding so much light upon a whole field, and yielding such powerful methods for solving problems . . . that, no matter whether or not they are intrinsically necessary, they would have to be accepted at least in the same sense as any well-established physical theory. (Go¨del [1964], p. 261)

It has become customary to describe these two rough categories of justification as ‘intrinsic’—self-evident, intuitive, part of the ‘concept of set’, and such like—and ‘extrinsic’—effective, fruitful, productive. (iv) The case for determinacy To round off this list of examples, we should consider a contemporary case. Determinacy hypotheses came in for serious study beginning in

48 prope r method the 1960s16 as part of a broader search for new principles that might settle the problems in analysis17 and set theory18 left open by the now-standard descendent of Zermelo’s system, Zermelo-Fraenkel with Choice (ZFC). Assuming Choice, as is now standard, not all sets of reals can be determined,19 but the assumption that all projective sets of reals20 are determined—Projective Determinacy (PD)—is an attractive fall back. Perhaps more natural is the assumption that the full Axiom of Determinacy21 holds in some appealing model of ZF; ADL(ℝ) asserts the determinacy of all sets of reals in L(ℝ), the smallest inner model containing all the real numbers. L(ℝ) includes the projective sets, so ADL(ℝ) implies PD. In his 1980 state-of-the-art compendium on the subject, Moschovakis observed that ‘no one claims direct intuitions . . . either for or against determinacy hypotheses’, that ‘those who have come to favor these hypotheses as plausible, argue from their consequences’ (Moschovakis [1980], p. 610). At that time, he concluded: At the present state of knowledge only few set theorists accept [ADL(ℝ)] as highly plausible and no one is quite ready to believe it beyond a reasonable doubt; and it is certainly possible that someone will simply refute [it] in ZFC. On the other hand, it is also possible that the web of implications involving determinacy hypotheses and relating them to large cardinals will grow steadily until it presents such a natural and compelling picture that more will succumb. (Moschovakis [1980], pp. 610–611)

See, e.g., Kanamori [2003], }27. e.g., the Lebesgue measurability of projective sets (see footnote 20). 18 e.g., of course, the Continuum Hypothesis (see footnote 40). 19 If A is a set of real numbers between 0 and 1 (imagine these uniquely represented as infinite sequences of 0s and 1s), then the game G(A) consists of two players alternately choosing a 0 or a 1; if the resulting real is in A, player I wins, otherwise II wins. A is said to be determined if one of the players has a winning strategy. 20 Borel sets are obtained from open sets of reals by taking complements and countable unions. An analytic set is the projection onto the x-axis of a Borel subset of the plane; a co-analytic set is the complement of an analytic set. (Borel sets can also be characterized as those that are both analytic and co-analytic.) The projective sets are obtained from the Borel sets by repeated application of projection and complementation. The determinacy of Borel sets is provable in ZFC, though this result, due to Martin, wasn’t established until 1974. 21 i.e., the assumption that all sets of reals are determined. 16 17

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Here Moschovakis displays impressive foresight, as more have succumbed in recent decades, on the basis of new discoveries. In telegraphic summary, the current evidence for determinacy falls roughly into four classes: 1. ADL(ℝ) generates a rich theory of definable sets of reals with many of the virtues identified by Go¨del.22 In Steel’s words, The theory . . . based on [ADL(ℝ)]23 contains answers to all the questions about projective sets from classical descriptive set theory. The theory of projective sets one gets in this way extends in a natural way the theory of low-level projective sets developed by the classical descriptive set theorists using only ZFC; indeed, in retrospect, much of the classical theory can be seen as based on open determinacy, which is provable in ZFC. Virtually nothing about sets in the projective hierarchy beyond the first few levels can be decided in ZFC alone, but . . . PD . . . yield[s] a deep and powerful extension of the classical theory to the full projective hierarchy . . . By placing the classical theory in this broader context, we have understood it better. (Steel [2000], p. 428)

Steel goes on to note that a rival theory of projective sets derived from Go¨del’s Axiom of Constructibility (V=L) ‘is certainly not the sort of theory that looks useful to Analysts’. Furthermore, the V=L theory isn’t lost if one adopts a theory containing PD, because it can be regarded as ‘a wonderful, useful part of the first-order theory of L’, a move that can’t be duplicated for the PD theory if one adopts V=L (Steel [2000], p. 429).24 2. Moschovakis’s ‘web of implications . . . relating [ADL(ℝ)] to large cardinals’ has indeed ‘grown steadily’. Large cardinal axioms themselves enjoy some intrinsic support, most fundamentally by reflecting the intuitive idea that the cumulative hierarchy of sets goes on forever, but it must be admitted that this sort of evidence is less compelling for the larger large cardinals central to the developments we’re tracing here.25 Still, the success of the inner model program provides evidence 22 And ADL(ℝ) is necessary for this theory: it’s actually implied by its consequences for definable sets (see Koellner [2006], pp. 170, 174). 23 Steel actually mentions large cardinal hypotheses that imply ADL(ℝ) and hence PD (see (2) below). 24 For an attempt to spell out an argument against V=L along these lines, see [1997], }III.6. 25 I come back to this point in }V.3.i. Indeed, the entire case for determinacy is re–visited in }V.3.

50 prope r method for the consistency of the large cardinals it has reached,26 and the hierarchy of large cardinals has emerged as a remarkably effective measure of the consistency strength of hypotheses going beyond ZFC.27 The relationship between large cardinals and determinacy was dramatically illuminated in the decade following Moschovakis’s book, when Martin, Steel and Woodin, building on work of Foreman, Magidor and Shelah, showed that ADL(ℝ) follows from the existence of large cardinals.28 Koellner describes the current situation this way: The case for axioms of definable determinacy is further strengthened by the fact that they are implied by large cardinal axioms . . . Conversely, definable determinacy implies (inner models of ) large cardinals . . . Ultimately, we see that definable determinacy is equivalent to the existence of certain inner models of large cardinals axioms. (Koellner [2006], pp. 170, 173)

Thus ADL(ℝ) inherits the intrinsic and extrinsic evidence for large cardinals, and large cardinals, in turn, gain extrinsic support by implying the determinacy-based account of projective sets. 3. Returning to the hierarchy of consistency strengths, a striking phenomenon has emerged. As Steel puts it, any natural theory of consistency strength at least that of PD actually implies PD. For example, the Proper Forcing Axiom [PFA] implies PD. So does the existence of a homogeneous saturated ideal on ø1. (Neither of these propositions has anything to do with PD on its surface.) (Steel [2000], p. 428)

26

See Steel [2000], p. 426: ‘these canonical inner models admit a systematic, detailed, “fine structure theory” much like Jensen’s theory of L. Such a thorough and detailed description of what a universe satisfying H might look like . . . provides good evidence that H is indeed consistent’. Also Martin and Steel [1988], p. 6583: ‘the full and detailed description of such a model gives some evidence of consistency . . . A hidden inconsistency . . . should emerge quickly in the theory of ’ the canonical inner model. 27 See Steel [2000], pp. 425–427: ‘these axioms have proved crucial to organizing and understanding the family of possible extensions of ZFC. Of course, there is nothing like a systematic classification of all the possible extensions of ZFC, but there is more order here than one might suspect . . . Thus it seems that the consistency strengths of all natural extensions of ZFC are wellordered, and the large cardinal hierarchy provides a sort of yardstick which enables us to compare these consistency strengths’. 28 See Kanamori [2003], }32, for discussion and references.

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On the same theme, Koellner cites ‘a method—Woodin’s core model induction’—which can be used to show that virtually every natural mathematical theory of sufficiently strong consistency strength actually implies ADL(ℝ). Here are two representative examples: . . . there is an ø1-dense ideal on ø1 . . . [and] PFA . . . . These two theories are incompatible and yet both imply ADL(ℝ). There are many other examples. For instance, the axioms of Foreman [1998] . . . also imply ADL(ℝ). Definable determinacy is inevitable in that it lies in the overlapping consensus of all sufficiently strong natural mathematical theories. (Koellner [2006], pp. 173–174)

Given the long-standing foundational goal of set theory and the openendedness of contemporary pure mathematics, we have good grounds to seek theories of ever-higher consistency strength. If all reasonable theories past a certain point imply ADL(ℝ), this constitutes a strong argument in its favor. 4. We’ve seen that sufficiently large cardinals, via determinacy, settle all questions the classical analysts thought to ask about projective sets, but we might wonder if other such questions are left unresolved. Our best method for demonstrating independence is forcing, and in the presence of large cardinals, forcing cannot succeed in showing a question about projective sets to be independent.29 This means that if anything is left unresolved, this can’t be shown by forcing; the independence involved would have to be a new and unfamiliar variety. It can also be shown, conversely, that (in the presence of very modest large cardinals) determinacy is required for this so-called ‘generic completeness’.30 Given that we want our theory of sets to be as decisive as possible, within the limitations imposed by Go¨del’s theorems, generic completeness would appear a welcome feature of determinacy theory. The current case for determinacy has blossomed so impressively that many would agree with Woodin’s assessment: ‘Projective determinacy is the correct axiom for the projective sets’ (Woodin [2001], p. 575). 29 If there is a proper class of Woodin cardinals, then L(ℝ) is elementarily equivalent to the L(ℝ) in any forcing extension. (See Koellner [2006], p. 171. Cf. Steel [2000], p. 430.) 30 If there is a proper class of inaccessibles, and L(ℝ) is elementarily equivalent to the L(ℝ) in any forcing extension, then ADL(ℝ). (See Koellner [2006], p. 173.)

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3. Proper set-theoretic method Assuming these examples are typical, the Second Philosopher hoping to undertake an investigation of sets has access to a rich array of methods, both for introducing them in the first place and for determining their extent and their properties thereafter. We’ve seen that sets were posited in the first place in the service of explicit mathematical goals: a broader condition on uniqueness of trigonometric representations, representation-free definitions for abstract algebra, an account of continuity to found analysis and launch a general study of continuous structures, a foundation for arithmetic. In broad overview, these goals range from relatively local problem-solving, to providing foundations, to more open-ended pursuit of promising mathematical avenues. The arguments subsequently offered for and against settheoretic hypotheses and axioms take many of the same forms: the Axiom of Choice solves an array of problems, helps found the theory of sets and promises more ‘productive science’ (Zermelo [1908a], p. 189); determinacy hypotheses settle the questions once raised by the early analysts and produce a rich and deep extension of the classical theory; V=L presents an unproductive alternative theory of projective sets and conflicts with set theory’s foundational role by restricting the universe of sets.31 And so on. Given what set theory is intended to do, relying on considerations of these sorts is a perfectly rational way to proceed: embrace effective means toward desired mathematical ends.32 Now actual discussions surrounding the introduction of mathematical objects or extensions of our assumptions about them often involve more speculative or typically philosophical material. To take just one example, Dedekind’s paper on the natural numbers includes his belief that they are ‘free creations of the human mind’ (Dedekind [1888], p. 791). Given the wide range of views mathematicians tend to hold on these matters, it seems unlikely that the many analysts, algebraists, and set theorists ultimately led to embrace sets would all agree on any single conception of the nature of mathematical objects in general, or of sets

31 32

This is the line of thought developed in [1997], }III.6. I set intrinsic justifications aside until Chapter V.

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in particular; the Second Philosopher concludes that such remarks should be treated as colorful asides or heuristic aides, but not as part of the evidential structure of the subject. What matters for her methodological purposes is that all concerned do feel the force of the kinds of considerations we’ve been focusing on here; these are the shared convictions that actually drive the practice.33 This study of actual set-theoretic methods also confirms the Second Philosopher’s initial impression that this is an inquiry governed by norms distinct from familiar observation, theory-formation and testing: for example, she isn’t accustomed to embracing new entities to increase her expressive powers (as in Cantor) or to encourage definitions of a certain desirable kind (as in Dedekind), or to rejecting a theory because it produces less interesting consequences (as with the alternative to determinacy’s theory of projective sets that results from V=L). She might reasonably wonder if her more familiar, tried-and-true methods could be called upon to underwrite, supplement or even correct these new approaches. On examination, though, she concludes that the answer here is no. Ordinary perceptual cognition is most likely involved in our grasp of elementary arithmetic,34 but she recognizes that this connection to the physical world has long since been idealized away in the infinitary structures of contemporary pure mathematics. Though Quine has argued that mathematical claims are empirically confirmed by a less direct route, this position appears to her to rest on accounts of science, mathematics, and the relations between them that don’t accurately reflect the true features of these practices.35 Though she appreciates that providing tools for empirical science remains one of the central goals of pure mathematics, she also realizes that science no longer shapes the ontology or fundamental assumptions of mathematics as it once did in the days of Newton or Euler. Finally, cases like group theory—which was considered useless and nearly dropped from the curriculum at Princeton just years before it entered physics as an 33 For more, see [1997], }III.4, also the terminological mea culpa of [2007], p. 349, footnote 12. 34 See [2007], IV.2.ii. 35 In [1997], }}II.6–7, I argue that Quine’s case is flawed (see also [2007], pp. 314–317). Chapter I above presents an account of the role of mathematics in application that runs starkly counter to Quine’s. This comes up again in }IV.2. See also }III.5, footnote 44.

54 prope r method essential tool36—such cases convince her that any effort to reign in the broad range of goals pursued by pure mathematicians would be unwise. So she’s faced with an array of new methods for justifying claims, methods that appear to be both rational and autonomous. If all she ultimately cared about were answering questions of the first type—what are the proper set-theoretic methods?—she’d now be done, but our Second Philosopher will also ask questions of the second type, beginning with the stark: are these methods reliable? Do they successfully track the existence of sets and their properties and relations? Of course she’s familiar with questions of this form: she has an ongoing investigation of how ordinary perception gives her information about the medium-sized objects in the world around her, of when it’s likely to fail her, of what she can do to increase the accuracy of her perceptual beliefs; she examines the efficacy of our instrumental means of detecting the small parts of matter; she devises double-blinds to reduce the risk of misleading experimental results, and so on. In all these familiar cases, she employs her usual methods to evaluate how humans, as described in biology, physiology, psychology, evolutionary theory, and so on, come to know the world, as described in physics, chemistry, geology, astronomy, and so on. The case of set theory is much the same: she’s begun by observing and lightly categorizing the methods set theorists actually use to justify the introduction of sets and the elaboration of our theory of them, and she’s asked whether or not these procedures lead to accurate beliefs. This is not the sort of question that can be answered within set theory or pure mathematics proper; even proof theory, our examination of the reliability of our proving techniques, is actually a piece of applied mathematics,37 and we’re now concerned with the less formal types of argumentation offered in support of the assumptions from which those proofs begin. Just as in other cases, we are, in effect, standing within empirical science, asking a question about a particular human practice: do its methods track the truth about its subject matter? We aren’t asking whether or not a certain statement of set theory is true in the sense of asserting that it holds in V;38 we’re asking whether set theory 36

See [2007], pp. 330–331, 347, for discussion and references. Cf. Burgess [1992]. 38 By ‘V ’ here, I intend the universe of sets that set theory is investigating, which we learn, in the course of that investigation, takes the form of stages VÆ, one for each ordinal Æ. 37

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as a whole should be viewed as a body of truths, alongside physics, astronomy, and botany. This raises a prior question: should set theory be understood as describing a subject matter, as attempting to deliver truths about it? As we’ve seen, the Second Philosopher differs from Quine in rejecting the idea that the mathematics used in application is justified by ordinary empirical evidence along with the physical theory in which it is embedded. If she’s to conclude that pure mathematics is a body of truths, her case for this will presumably rest more loosely on the way it is intertwined with empirical science. For now, let me leave a bookmark at this point, to return to it in Chapter IV. For now, let’s assume that the Second Philosopher is justified in regarding set theory as a body of truths, and since she sees no reason to take its existence claims at other than face value,39 she’s also justified in believing that sets exist. Though she’s viewing the practice from her external, scientific perspective, as a human activity, she sees no opening for the familiar tools of that perspective to provide supports, correctives, or supplements to the actual justificatory practices of set theory. She has no grounds to question the very procedures that do such a good job of delivering truths, so she concludes that the proper methods to employ, the operative supports and correctives, are the ones that set theory itself provides; she concludes that the methods of set theory are reliable guides to the facts about sets. Of course this doesn’t mean that every set-theoretic argument is correct, only that methods of this general sort are the right ones and that particular such arguments can be properly supported or critiqued only by more arguments of the same kind.

4. The challenge To this point, then, the Second Philosopher has settled the first group of questions to her satisfaction. She takes the proper methods for 39 I don’t have in mind here any general case for the reliability of surface syntax, e.g., of the sort proposed in Wright [1992] (see [2007], }II.5, for further discussion and references). It’s just that the Second Philosopher sees no reason to think that set-theoretic claims say anything other than what they appear to say. I touch on Wright’s minimalism in }III.2 below.

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introducing sets, for adding new axioms to our theory of them, to be methods of the sort we’ve been rehearsing: sets are legitimately posited as effective means toward various mathematical goals (in analysis, algebra, foundations, and elsewhere); axioms are defended by a careful balance of detailed considerations, both intrinsic and extrinsic. She’s also made a start on questions of the second sort: she’s concluded that set-theoretic methods are rational, autonomous, and generally reliable, that sets exist, and that set theory is (largely) a body of truths about them. But this is just the beginning; she still wonders: what sort of activity is set theory? how does set-theoretic language function? what are sets and why are these the proper methods for finding out about them? These questions would need answers in any case, but they can appear more pressing in the face of natural statements like the Continuum Hypothesis40 (CH) that can’t be settled on the basis of the current axioms. Is CH nevertheless a legitimate question with a determinate answer? This might seem to hinge on our account of the subject matter of set theory. It was this question, in fact, that inspired Go¨del to his well-known Platonistic metaphysics: The set-theoretic concepts and theorems describe some well-determined reality, in which Cantor’s conjecture must be either true or false. Hence its undecidability from the axioms being assumed today can only mean that these axioms do not contain a complete description of that reality. Such a belief is by no means chimerical, since it is possible to point out ways in which the decision of a question, which is undecidable from the usual axioms, might nevertheless be obtained. (Go¨del [1964], p. 260)

He goes on to describe intrinsic and extrinsic methods of justifying new axioms. The basic idea here—that the legitimacy of CH is to be defended by appeal to some sort of objective reality in which it is either true or false—this basic idea can be fleshed out in a number of different ways: the objective reality might be the cumulative hierarchy of sets (as in Go¨del [1964]), a set-theoretic structure (as in Shapiro [1997]),

40 Despite the amusing terminological convergence, this obviously isn’t the ‘continuum hypothesis’ from fluid dynamics touched on in }I.2, but (in one form) the claim that every infinite set of reals is either countable or of the same size as the full set of reals.

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the concept of set (as in Go¨del [1951]), some purely modal facts (as in Hellman [1989]), and there are doubtless other possibilities. An analogy may or may not be drawn between the set-theoretic reality described by set theory and the physical reality described by natural science (as in Go¨del [1944], [1964], or my [1990]). Let me call metaphysical positions of this general type ‘Robust Realism’. As noted in }I.1, there’s a well-known objection to views of this sort, familiar from Benacerraf [1973]: how can human beings, with the sorts of cognitive capacities we understand ourselves to enjoy, manage to gain reliable information about the world of sets? Benacerraf’s particular statement of the question rested on philosophical theories since defunct, but the issue has not evaporated. Burgess and Rosen trace its resilience to a ‘perennially powerful’ picture (Burgess and Rosen [2005], p. 534): Reality is . . . a system connected by causal relations and ordered by causal laws, containing entities ranging from the diverse inorganic creations and organic creatures that we daily observe and with which we daily interact, to the various unobservable causes of observable reactions that have been inferred by scientific theorists . . . [Robust Realists] hold that outside, above, and beyond all this (and here one gestures expansively to the circumambient universe) there is another reality, teaming with entities radically unlike concrete entities—and causally wholly isolated from them. . . . between [us] and the other world . . . there is a great gulf fixed . . . Surely [Robust Realists] owe us a detailed explanation of how anything we do here can provide us with knowledge of what is going on over there, on the other side of the great gulf. (Burgess and Rosen [1997], p. 29)

Robust Realists (including those noted above) have attempted to meet this challenge, but no one has managed to satisfy all parties involved. My own, second-philosophical concerns about Robust Realism begin even before this familiar epistemological challenge can take hold. To see how, consider any one of the set-theoretic arguments rehearsed above; for concreteness, let’s take Cantor’s case for positing sets in his study of trigonometric representations. Setting aside the murky talk of the ‘conceptual co-determination’ of P’s derived set— safely regarded as the sort of thing that’s not widely shared, or even understood, by the range of practitioners—we have a straightforward

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means-ends argument for this introduction of sets as objects: they allow Cantor to formulate and prove a stronger theorem on uniqueness. Of course this advantage could be outweighed by accompanying disadvantages—if these sets were difficult to work with or led to contradictions or some such thing—but no apparent downsides are in evidence here. Given the Second Philosopher’s evaluation of proper set-theoretic method, this would seem a clear example of good grounds for introducing sets, of good mathematical evidence for their existence. So far so good. But if the Robust Realist is right, if the goal of set theory is to describe an independently-existing reality of some kind, then it appears that Cantor’s evidence needs supplementation, and not supplementation of the same sort, like adding in Dedekind’s grounds and so on, but supplementation of an entirely different kind: we need an account of how the fact that sets serve this or that particular mathematical goal makes it more likely that they exist. Without this account we have no way of ruling out the possibility that reality is sadly uncooperative, that much as we’d like to use sets in our mathematical pursuits, they just don’t happen to exist. To the Second Philosopher, this hesitation seems misplaced: why should perfectly sound mathematical reasoning require supplementation? Hasn’t something gone wrong when rational mathematical methods are called into question in this way? The situation gets worse when we contemplate what the Robust Realist’s supplementation would require. If the world of sets that set theory hopes to describe is entirely objective, perhaps analogous to our familiar physical world, then it’s hard to see, for example, how the fact that ADL(ℝ) is implied by all natural hypotheses of sufficiently high consistency strength is evidence in its favor. Granted, we have good reason to pursue axioms with higher and higher consistency strength, so as to maximize the interpretive power of set theory and thus further its foundational goal, but how does our desire that set theory play this role make it any more likely that it’s capable of doing so? The same can be said about each of our many instances of preferring a set theory with the sorts of mathematical advantages we’ve been surveying. The physical world, famously, cares little for our theoretical preferences, as the failure of Euclidean Geometry and the mysteries of Quantum Mechanics

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59

amply demonstrate. Why should we expect the set-theoretic universe to be any more cooperative than the physical world? The familiar Benacerrafian challenge suggests that the abstract character of the objects of set theory poses a formidable obstacle to the sort of supplementary epistemological account the Robust Realist requires. In contrast, the Second Philosopher’s inclination is to think that no such supplementary account should be required in the first place: if Robust Realism questions the cogency of apparently sound mathematical reasoning, her guess is that the fault lies with Robust Realism, not the tried-and-true ways of set theory. The problem, then, is how to answer the second group of questions—about the nature of settheoretic activity, about its subject matter and the reliability of its methods—in a way that both respects the actual methods of set theory and preserves the legitimacy of the pursuit of new axioms and a solution to the continuum problem.

III Thin Realism What we’re after is a satisfying form of realism without the shortcomings of the Robust versions. Within the set-theoretic community we’ve been focused on, hints of this sort of position turn up in various remarks and observations of John Steel: Realism in set theory is simply the doctrine that there are sets . . . Virtually everything mathematicians say professionally implies there are sets. . . . As a philosophical framework, Realism is right but not all that interesting. (FOM posting 15 January 1998)1

I take what ‘mathematicians say professionally’ as an appeal to the ordinary set-theoretic methods we’ve been focused on here. Steel nods toward Robust Realism— Both proponents and opponents [of realism] sometimes try to present it as something more intriguing than it is, say by speaking of an ‘objective world of sets’. (FOM posting 15 January 1998)

—and he entertains the possibility that his mundane realism is something different Whether this is Go¨delian naı¨ve realism I don’t know. (FOM posting 30 January 1998)

1

All excerpts from Steel’s FOM postings are quoted with permission. As readers of [2005] or [2007], }IV.4, will know, the line of thought developed in this chapter was originally inspired by some remarks of Steel and John Burgess (see below). The position as it now stands can’t be attributed to either of them without putting quite a few words in their mouths, but I remain grateful to them both for pushing me in this direction when the only realism I could imagine was Robust. Apparently related thoughts turn up in Liston [2004] and Tait [1986] and [2001], but see footnotes 10 and 16, and 6, 17, 27, respectively.

thin realism 61 Speaking some years later, he concludes that it is not, and recommends A more sophisticated realism, one accompanied by some self-conscious, metamathematical considerations related to meaning and evidence in mathematics. (Steel [2004], p. 2)

This is exactly what our second-philosophical investigations have led us to hope for: a version of realism that genuinely accounts for the nature of set-theoretic language and practice, that respects the actual structure of set-theoretic justifications. The goal of this chapter is to develop such a position.

1. Introducing Thin Realism Recall (from }II.1) that the Second Philosopher faces two types of questions about set theory: first, what are its proper methods? second, what sort of activity is set theory, what is its subject matter, and why are these the proper methods? These aren’t strictly mathematical questions; she poses them in her capacity as an empirical scientist examining a particularly salient human practice. To this point, she has answered questions of the first type—the actual set-theoretic methods she’s cataloged are the proper methods, both rational and autonomous— and she’s made a start on questions of the second type: she’s concluded that sets exist, that set theory is a body of truths about them, and that set-theoretic methods are reliable guides in this inquiry.2 She’s now faced with the challenge of explaining what makes these methods reliable, of what sets must be like for this to be so. Under the circumstances, the Second Philosopher is naturally inclined to entertain the simplest hypothesis that accounts for the data: sets just are the sort of thing set theory describes; this is all there is to them; for questions about sets, set theory is the only relevant authority. From this narrow beginning we can draw some immediate consequences. For example, Steel points out that

Recall (from }II.3) that examination of the precise nature of her justification for the reliability, truth, and existence claims has been postponed to Chapter IV. 2

62 thin realism none of [what mathematicians say professionally] is about their causal relations to anything . . .

from which he concludes that sets do not depend causally on us (or anything else, for that matter). (FOM posting 15 January 1998)

Since set theory tells us nothing about sets being dependent on us as subjects, or enjoying location in space or time, or participating in causal interactions, it follows that sets are abstract in the familiar ways. John Burgess sums up this particular sentiment nicely, One can justify classifying mathematical objects as having all the negative properties that philosophers describe in a misleadingly positive-sounding way when they say that they are abstract [acausal, non-spatiotemporal, etc.]. But beyond this negative fact, and the positive things asserted by set theory, I don’t think there is anything more that can be or needs to be said about ‘what sets are like’.3

In contrast with the entities posited by the various rich metaphysical and epistemological theories of the Robust Realists—which all go well beyond ‘the positive things asserted by set theory’—these sets will seem rather insubstantial; this realism is ‘thin’, as opposed to robust. What’s happened here is that the second-philosophical Thin Realist, in the ordinary course of her investigations, has made a surprising discovery: in addition to the familiar concrete objects she’s been studying so far, there are also objective, non-spatiotemporal, acausal sets; not only the methods of set theory, but the things themselves are new. Of course the Robust Realist could say these same words; both our realists now take set theory to aim at describing the properties of an objectively-existing, non-spatiotemoral, acausal reality, so some care must be taken to distinguish the new version from the old. As a start, recall how Go¨del appeals to Robust Realism to justify his claim that CH has a determinate truth value, despite its independence from ZFC. How will our new realist view the case of CH? Her analysis is simpler: ‘CH or not-CH’ is a theorem, established by her best methods as a fact

3

Personal communication, 24 April 2002, quoted with permission.

thin realism 63 about V;4 therefore CH is either true or false there.5 For the Robust Realist, this appeal to classical logic isn’t enough; for him, without a guarantee that the logic tracks the metaphysics, the possibility remains open that this theorem is incorrect.6 In contrast, the Thin Realist holds the set-theoretic methods are the reliable avenue to the facts about sets, that no external guarantee is necessary or possible.7 So the fundamental diagnostic is this: the Robust Realist requires a non-trivial account of the reliability of set-theoretic methods, an account that goes beyond what set theory tells us; for the Thin Realist, set theory itself gives the whole story; the reliability of its methods is a plain fact about what sets are. The key here is that the second-philosophical Thin Realist begins from her confidence in the authority of set-theoretic methods when it comes to determining what’s true and false about sets, and from the observation that her more familiar methods appear irrelevant; she concludes that the call for a non-trivial, external epistemology is misplaced. But why couldn’t the Robust Realist, faced with Benacerraf-style challenges, make the same move? The answer is that the

4 Again, by ‘V ’ I mean the universe of sets that set theory is investigating, which we learn, in the course of that investigation, takes the form of stages VÆ, one for each ordinal Æ. Here I’m presupposing that the second-philosophical set theorist is moved by various mathematical considerations, conspicuously the desire for set theory to serve as a foundation (in the sense of }I.3), to seek a unified theory of this single universe V. (See e.g. [1997], pp. 208–9, [2007], p. 354.) 5 This isn’t to say there’s any guarantee that set-theoretic methods will eventually tell us which one it is. Set-theoretic methods are the only reliable source of information about sets; sets just are the sort of thing these methods tell us about; but they could have features that, for one reason or another, we miss, or even have no way of getting at. See footnote 40 below. 6 Tait appears to take up the new realist position—‘that CH is either true or false amounts to nothing more than the application of the law of excluded middle to it’ (Tait [2001], p. 96)—but he apparently loses the courage of his convictions two pages later, declaring that ‘until we determine it, CH is indeed indeterminate’. See footnote 17 below. 7 It’s conceivable that future mathematical developments could convince the Thin Realist that classical logic shouldn’t be applied in set theory in this way, for example, that set theory isn’t the study of a single universe of sets after all. If, to take a speculative example, we should eventually come upon conflicting, equally attractive theories of sets, and if we were unable in principle to regard them all as describing parts of a larger unified theory of sets, then we might well conclude that set theory has failed at its foundational goal: no one theory of sets is able to encompass all of contemporary mathematics. For now I think it’s fair to say there’s no compelling evidence that this is likely to be the outcome of set-theoretic investigation, so I ignore it in the text. (For more, see [2007], pp. 387–389, Koellner [2006], }5.2., [2009a], }5.)

64 thin realism Robust Realist wants more than Thin Realism can offer; for example, as Charles Parsons says of Go¨del, he wants an ‘objectively determinate answer’ for CH, ‘where this is to mean more than that our logic incorporates the law of excluded middle’.8 Of course the Thin Realist will protest that she does mean more—she takes the correctness of classical logic for set-theoretic inquiry to show that CH has an objective and determinate truth value—but for the Robust Realist, this is still too thin; he wants a full-bodied metaphysical theory of what sets are that will ratify CH in a more substantive way, and to get this, he needs a non-trivial epistemology.9 So the Thin Realist’s minimal theory to account for the methodological data—her simple assumption about what sets are like—will not satisfy the Robust Realist.

2. What Thin Realism is not As so far described, Thin Realism risks conflation with more familiar theories, so let me pause a moment to draw some contrasts. The close connection between sets and set theory, between sets and set-theoretic methods, has so far been treated as a brute fact. Of course we’d like a more fundamental explanation for this connection, but that impulse can easily lure us beyond the austerity of Thin Realism, beyond what set theory tells us. To illustrate, let me consider three typical examples. One way of thinking about the Second Philosopher’s approach to understanding the subject matter of set theory is that she’s asking: what must sets be like in order that we can know about them in these ways (that is, by the methods of set theory)? To the philosophical ear, this formulation recalls the iconic Kantian question: what must the world be like in order that we can know it as we do (that is, partly a priori)? Kant’s answer, famously, is that the world we experience is partly constituted by our human modes of cognition (the forms of intuition and the pure categories), so we can know a priori that this world will 8 Parsons [2004], p. 62. See also Parsons [1995], p. 71: ‘The widespread impression that Go¨del was not just affirming CH v ~CH, i.e., allowing the application of the law of the excluded middle here, seems to me correct’. 9 See }V.2 for examples.

thin realism 65 consist of spatiotemporal objects standing in causal relations. Among philosophers sympathetic to a position like Thin Realism, there’s a temptation to slip into an analogous stance: sets are constituted by our set-theoretic practices; that’s why we can be confident that those practices track the facts about sets.10 The Kantian story of the world of experience is unacceptable to the Second Philosopher because it involves an explicitly extra-scientific mode of inquiry.11 In Kant’s terms, space, time and causation are empirically real, but transcendentally ideal; this means that the empirical enquirer—that is, the scientist, the Second Philosopher—is correct to regard spatiotemporality and causality as objective features of the world, known a priori, but the transcendental inquirer—that is, the Kantian critical philosopher—sees that these features are contributions of our particular form of cognition, and thus, when regarded transcendentally, only subjective or ideal.12 This qualifies Kant as a First Philosopher as the term is used here: from the empirical viewpoint, these ordinary beliefs about the world are entirely in order; from the transcendental viewpoint, they aren’t. Once again, the Second Philosopher finds her beliefs challenged and all her usual forms of evidence set aside as irrelevant; once again, she doesn’t understand the higher purpose involved or the methods appropriate to it. Of course, to be fair, the Second Philosopher doesn’t feel the pressure Kant did to account for the a priori truth of geometry! But the situation with set theory is different, because the Second Philosopher does recognize a legitimate form of inquiry outside of set theory, namely, the ordinary empirical inquiry in which she begins her investigations. As we’ve seen, it’s from this point of view that she reaches her decision to undertake the study of sets, and from this point 10

In describing his ‘Thin-blooded Platonism’, Liston [2004], p. 154, cites with approval a passage from Stalnaker [1988], p. 119: ‘The existence of numbers is just constituted by the fact that there is a legitimate practice involving discourse with a certain structure, and that certain of the products of this discourse meet the standards of correctness that it sets’. (Stalnaker isn’t endorsing this idea.) 11 For more, see [2007], }I.4. 12 This transcendental psychology must be sharply distinguished from ordinary empirical psychology: the latter could at best tell us how we must perceive the world, not how the world must be; science all too often reveals that our natural ways of cognizing don’t match reality.

66 thin realism of view that she conducts her examination of proper set-theoretic method and the nature of sets and set theory. Presumably it would also be from this point of view that she would determine, if she were to determine, that sets are constituted by set-theoretic methods. The trouble with such a determination isn’t that it’s first-philosophical—it isn’t. Obviously it also isn’t the same as Thin Realism: sets are no longer objective, independent entities, but rather dim projections of our practices; there is considerably more to them than what set theory tells us. Recognizing that it’s different from Thin Realism, we should pause to ask if this Kantian suggestion might serve our Second Philosopher as a viable alternative. Recall that her discomfort with Robust Realism arose (in }II.4) from its demand that ordinary, apparently rational mathematical methods be supplemented with evidence tied to its non-trivial metaphysics, especially as it seems inevitable that the required supplementation will not in fact be forthcoming: from the Robust Realist’s perspective, the various considerations surveyed in }II.2 look like so much wishful thinking. Given that our proposed idealism views sets as somehow constituted by our methods, it would presumably claim to ratify many of the set-theoretic arguments that Robust Realism calls into question; though the Second Philosopher regards this kind of supplementation as unnecessary and even dubious, at least it wouldn’t immediately undercut what she regards as rational argumentation. Still, it seems likely that this idealism, however it’s spelled out, would have to face the possibility that set-theoretic methods will fail to settle CH, and in this way our idealist, like the Robust Realist, would want more than the Thin Realist’s straightforward confidence that CH is either true or false in V. Here, once again, a potential challenge to the cogency of an apparently rational set-theoretic method—in particular, use of excluded middle—would be grounded on an extra-mathematical metaphysics. The relevant ontology this time would be idealistic rather than realistic, but no less objectionable for that. Another tempting line of thought is to explain the fact that sets are what set theory describes by appeal to Carnapian13 themes: the 13 I have in mind here the Carnap of the popular imagination, especially as in Carnap [1950]. The real Carnap is considerably more complex and elusive (see [2007], }I.5, for discussion and references).

thin realism 67 connection between sets and set-theoretic methods is ‘analytic’ or ‘conceptual’.14 In the bold strokes of Carnap’s well-known ‘Empiricism, semantics and ontology’ (Carnap [1950]), the linguistic framework for Xs includes the vocabulary and the evidential rules necessary for talk of Xs; these framework principles fix the ‘meaning of X’ or delineate the ‘concept of X’; working within the framework, we establish what’s true and what’s false about Xs by confirming or disconfirming statements on the basis of the framework principles. In this way, the connection between Xs and the evidential methods enshrined as rules of the framework are deemed ‘analytic’ or ‘conceptual’; they are part of the language we use in order to talk about Xs at all. The general notion of a linguistic framework is extremely flexible: there’s a linguistic framework for ordinary physical objects, which Carnap calls ‘the thing language’; there’s one for numbers, one for sets, and presumably one for poltergeists, as well. The key to Carnap’s solution, or dissolution, of ontological questions is that matters of truth or falsity only make sense within a framework, where there are determinate evidential rules by means of which they can be settled, so one can’t coherently ask, prior to the adoption of a framework, whether or not the things it features exist. One can ask, however, whether or not it would be pragmatically useful to adopt the corresponding framework, to adopt the corresponding linguistic conventions. So, for example, it would be useful for many purposes to move from the thing language to the thing-and-number language. The mistake would be to imagine that this move is only allowed if one can first establish that numbers exist. From a second-philosophical point of view, this account is least compelling for the thing language. To begin with, we never consciously adopt it; as Carnap notes, ‘we all have accepted the thing language early in our lives as a matter of course’ (Carnap [1950], p. 243). He hopes to recover something of the flavor of the framework

14 Burgess seems sympathetic to some such line of thought, especially in his [2004a] and [2004b], but as far as I can tell, he doesn’t actually endorse it. I used the terminology of ‘conceptual truth’ myself (in [2005] and [2007], }IV.4), but now regret doing so, for reasons about to emerge.

68 thin realism story by suggesting that we do consciously decide not to switch to ‘a language of sense-data or other “phenomenal” entities’ (op. cit.), but given that no one has ever succeeded in devising such a language, this seems an empty gesture. At least as troubling is the status of the evidential rules; as part of the framework, they are themselves conceptual and known a priori by anyone working within the framework. The idea that what counts as good evidence for the existence of some physical object is a fact known a priori seems to cohere best with verification-based theories of meaning or a priori accounts of confirmation, both of which appealed to Carnap at various times in his career, but must appear problematic to a contemporary Second Philosopher. So let’s set empirical frameworks aside, return to our familiar secondphilosophical point of view—beginning from perception, corrected and confirmed by careful observation and experimentation, and so on—and see if we can understand the thin-realist embrace of set theory as the adoption of a new linguistic framework in something like Carnap’s sense. The answer is clearly no: the Thin Realist doesn’t regard the embrace of ‘the concept of set’, the adoption of the set-theoretic framework, as a purely pragmatic matter of linguistic convention; she takes Cantor and Dedekind to have discovered the existence of these mathematical objects, not to have come to a convenient decision. Just as sets aren’t constituted by our methods, as in the Kantian line, settheoretic truth isn’t constituted by the set-theoretic framework;15 its evidential rules allow us to get at the truth about sets, but that’s due to our simple hypothesis about what sets themselves are, not some special kind of truth. So this Carnapian idea is clearly not the same as Thin Realism, but again we should ask: might it be an attractive alternative for the Second Philosopher? Might the Second Philosopher regard set-theoretic practice as taking place within a specified linguistic framework that implicitly defines the concept of set? On this picture, presumably Cantor and 15 Liston seems tempted by this move: ‘truth is just correctness according to the standards set by the conception and the practice’ (Liston [2004], p. 155). Epistemic or pragmatic accounts of truth hold little appeal for the Second Philosopher (see [2007], }I.7). I’ve also argued that Thin Realism need not be linked with disquotational truth, that it’s compatible with a second-philosophical version of the correspondence theory (see [2007], }IV.4, pp. 370–377).

thin realism 69 Dedekind gave us reason to adopt this framework, and we’re now busily investigating within it. The problem comes when we try to specify what exactly would be enshrined in a linguistic framework for ‘the concept of set’. The simplest proposal would be to include the settheoretic axioms and classical logic. The trouble with this suggestion is that it fails to capture one of the elements of set-theoretic practice we’re most eager to describe and assess: the addition of new axioms. On this view, one axiom wouldn’t be selected over another for compelling set-theoretic reasons—these are all internal to the framework—but as a pragmatic, conventional decision to move from one linguistic framework to another.16 Obviously this is not a suitable path for the Second Philosopher. We might try broadening the characterization of the set-theoretic framework to include not just the current axioms, but all intrinsic justifications as well, since they are after all intended to spell out features implicit in the ‘concept of set’. But even if we could successfully corral what seems an open-ended range of intrinsic considerations, this approach would still exclude from set-theoretic practice all the justifications we tend to classify as extrinsic, and this is no more acceptable to the Second Philosopher than excluding intrinsic justifications. Finally, we could try to add even these to a still-looser characterization of a settheoretic framework, but then both intrinsic and extrinsic justifications would be lumped together as ‘conceptual’, obscuring a distinction important to our understanding of the practice. All in all, this seems an unpromising avenue for the Second Philosopher.17 16 Though he doesn’t mention Carnap, this seems to me to capture the essence of Tait’s position: ‘the question of mathematical truth or existence becomes well-defined only with the introduction of the axioms’ (Tait [2001], p. 89) and ‘the intuitions or dialectical considerations that may lead us to accept a new axiom’ (p. 98) don’t justify that axiom. This would explain the failure of nerve in footnote 6: viewed inside the framework, CH has a determinate truth value (as a consequence of classical logic); viewed outside the framework, it’s indeterminate (because the axioms of the framework are too weak to settle it). 17 Of course, the topic here is the pursuit of pure mathematics, and of set theory in particular. If we address instead a natural scientist’s adoption of a given bit of mathematics, the Second Philosopher will agree with Carnap that this decision need not wait on any question of abstract ontology or epistemology (see Chapter I and }IV.2, below). Still, she would hesitate to dismiss the whole process as purely pragmatic—there are descriptive goals involved—and absent the holism of Carnap and Quine, there’s no temptation to think any sort of ‘truth by convention’ is involved.

70 thin realism A third train of thought that might be confused with Thin Realism arises from Wright’s minimalism about truth.18 Like the Thin Realist, Wright’s minimalist believes in sets because he takes set-theoretic existence claims, understood at face value, to be true. The stark differences come out in the reasons the two offer for embracing the premises they share. Wright’s minimalist takes set theory to be a body of truths because it enjoys certain syntactic resources and displays wellestablished standards of assertion that our set-theoretic claims can be seen to meet; the idea is that a minimalist truth predicate can be defined for any such discourse in such a way that statements assertable by its standards come out true. In contrast, the Thin Realist take set theory to be a body of truths, not because of some general syntactic and structural features it shares with other discourses, but because of its particular relations with the defining empirical inquiry from which she begins.19 Similarly the minimalist takes set-theoretic claims at face value, rejecting efforts to re-interpet them as saying something other than they seem to say, because he rejects the general notion of covert logical structure, because he trusts surface syntax on principle. Here again the Thin Realist has no blanket position on the matter; she simply sees no grounds on which to argue that set-theoretic claims in particular say anything other than what they appear to say.20 So despite superficial similarities, the order of argument is quite different. The minimalist begins from the well-behaved syntax and agreed-upon methods of settheoretic discourse, moves by a general construction to an appropriate truth predicate and the truth of set-theoretic existence claims, and finally to the existence of sets via the general reliability of surface syntax. The Thin Realist begins from the role of mathematics in science, moves from there to the truth of set theory and its existence claims, and finally to the existence of sets via the lack of evidence that those existence claims stand in need of re-interpretation. So the Thin Realist is not arguing as a minimalist would. In fact, the minimalist’s

e.g., Wright [1992]. See [2007], }II.5, for further discussion and references. Again, I come back to this point in the next chapter. 20 Some would argue for nominalistic reinterpretations on epistemological grounds. See }IV.3 for a second-philosophical take on this line of thought. 18 19

thin realism 71 guiding ideas about truth and syntax are hardly congenial to the second-philosophical point of view. So idealist or conventionalist or minimalist elaborations like these hold little appeal for our second-philosophical Thin Realist. She’s left with the claim that sets simply are the things that set theory describes, and it’s hard to see how any further explanation could be given without introducing an external metaphysics that goes beyond what set theory tells us. Still, I think there is a bit more for the Thin Realist to say. Let me start with epistemology, then circle back to the metaphysics.

3. Thin epistemology So far, the only epistemology Thin Realism has offered is what follows from the bare claim that sets are known by set-theoretic methods. This might seem to bring any epistemological conversation to an abrupt halt, but in fact I think a few illuminating morals can be drawn. For the sake of contrast, let’s return for a moment to Kant, this time to the finer points of his theory of cognition. For Kant, any judgment arises from the cooperative efforts of two faculties: the passively receptive sensibility and the actively spontaneous understanding.21 What the sensibility passively receives is the so-called ‘matter’ of cognition; the sensibility automatically forms this inchoate matter spatiotemporally, into an intuition orderly enough for the understanding to apply its concepts. For our purposes, the key idea is that the contributions of two faculties are quite different: the sensibility achieves an immediate, direct connection to the object of cognition; the understanding relates to that object only mediately, by means of features or properties.22 Pure mathematics, on Kant’s account, is an investigation of precisely those spatiotemporal forms of intuition supplied by the sensibility. It 21 Hence the oft-quoted remark: ‘Without sensibility no object would be given to us, and without understanding none would be thought. Thoughts without content are empty, intuitions without concepts are blind’ (A51/B75). 22 Cf. A320/B376–7: ‘an objective perception is a cognition . . . [This] is either an intuition or a concept . . . The former is immediately related to the object and is singular; the latter is mediate, by means of a mark, which can be common to several things’.

72 thin realism isn’t entirely clear whether Kant thinks that pure mathematics is without sense until referred to an empirical object or that the close relationship between pure and empirical intuition makes pure intuition enough to ground a real cognition,23 but what’s beyond doubt is that some sort of conceptually unmediated input is required for a real judgment of any kind. Here our second-philosophical Thin Realist begins not from a theory of what judgment must be, but from her conviction that the way to find out about sets is to do set theory. For her, the fact that settheoretic methods give us knowledge of sets by appeal to an array of mathematical considerations, without a role for perception or any other such mechanism, is enough to establish that there are objects we can know without the sort of unmediated connection provided by Kant’s sensible intuition. (This is just to turn the familiar worry about Robust Realism on its head, to argue directly that, for example, the perception-like epistemology proposed in my [1990] is unnecessary.24) So set theory does tell us something more about epistemology: our settheoretic knowledge doesn’t require immediate cognition of sets. In fact, I think we can squeeze out a bit more along these lines. Recall Cantor and Dedekind’s grounds for introducing sets in the first 23

At least it’s not entirely clear to me. Kant says: ‘Sensible intuition is either pure intuition (space and time) or empirical intuition of that which, through sensation, is immediately represented as real in space and time. Through determination of the former we can acquire a priori cognitions of objects (in mathematics), but only as far as their form is concerned . . . whether there can be things that must be intuited in this form is still left unsettled. Consequently all mathematical concepts are not by themselves cognitions, except insofar as one presupposes that there are things that can be presented to us only in accordance with the form of that pure sensible intuition’ (B147). See also A239–240/B299: ‘Although all these principles [of mathematics], and the representation of the object with which this science occupies itself, are generated in the mind completely a priori, they would still not signify anything at all if we could not always exhibit their significance in appearances (empirical objects) . . . Mathematics fulfills this requirement by means of the construction of the figure’ or, in the case of number, ‘in the fingers, in the beads of an abacus, or in strokes and points that are placed before the eyes’. 24 Go¨del ([1964], p. 268) also speaks of ‘something like a perception . . . of the objects of set theory’, but he also denies that this need be ‘conceived of as a faculty giving an immediate knowledge of the objects concerned’. This may sound like a thin-realist denial that any immediate access is required, but Go¨del actually suggests that something else—other than ‘the objects concerned’—is immediately given. He appears to have in mind something akin to Kant’s pure categories, but now he’s strayed into precisely the features of Kant’s account that aren’t immediate. I won’t try to sort this out here.

thin realism 73 place; both rested on the effectiveness of sets in performing certain specific mathematical jobs in analysis, algebra, and foundations. As we’ve seen, grounds of this type are still offered for new sets. Of course any such effectiveness must be weighed against other desiderata: it would be overridden, for example, if the added sets rendered our overall mathematical theory inconsistent;25 less dramatically, its importance would be tempered if it somehow blocked the pursuit of other worthy goals. But whatever the outcome of this calculation, it remains true that the evidence for the existence of sets is all and only linked to their mathematical virtues, to the mathematical jobs they are able to perform. Now suppose that the overwhelming mathematical virtues of sets have been established, as I think we’d all agree that they have. Does there remain a remote possibility that sets don’t exist or that they’re wildly different than we think they are? Our analysis of the epistemological situation implies that we would never come to know this, that no corrective to set-theoretic method would be forthcoming. But I’m asking a metaphysical question about sets themselves: could it be a sad metaphysical fact that set-theoretic methods—however reasonable, however apparently reliable—are entirely wrong? Some writers sympathetic to positions in the vicinity of Thin Realism have drawn a parallel here with radical skepticism about the external world: any concern about the ultimate reliability of set-theoretic methods is no better founded, they claim, than the skeptic’s concern about the ultimate reliability of our perceptual beliefs.26 From a secondphilosophical point of view, this seems to me to under-estimate the strength of the set-theoretic case. To see this, consider for a moment the challenge posed by the external world skeptic: he asks me to defend 25

As noted in Chapter I, footnote 78, the delta function was used in physical science for some time, despite its obvious inconsistency. Indeed it’s often good policy to allow unrigorous methods to proceed for a time without too much hindrance, with the hope of domesticating them in the future. (Wilson [2006] calls this a period of ‘semantic agnosticism’.) This has happened in the history of mathematics as well, e.g., in the early days of the calculus, and it might conceivably happen even in set theory. Still, given that one of set theory’s leading motivations is to provide the sort of rigorous foundation that mathematics sorely needed in the early 20th century (as described in }I.3), the demand for immediate consistency is much stronger here than it is elsewhere. 26 See e.g. Burgess and Rosen [2005], pp. 522–523, Tait [1986], pp. 63–65.

74 thin realism my belief that there’s a rose in my garden when I’m looking straight at it under optimal conditions. I can answer this challenge, of course, if I’m allowed to appeal to my well-confirmed theories of optics, of my eye and my brain, of human belief-forming mechanisms, and so on; on this basis I can defend the reliability of my current perception. But this isn’t what the skeptic wants. By appeal to colorful hypotheses like the Evil Demon—who somehow fits me with a full range of false experiences of a non-existent external world—by entertaining such fanciful hypotheses the skeptic in effect challenges me to defend my belief in the rose without appealing to any of my usual ways of defending beliefs, because these are all undercut by the possibility of the Demon. This challenge is one I don’t know how to meet, but there’s nothing ill-formed about it; I wouldn’t mind having an answer to it myself, that is, a way of defending all my methods entirely from scratch, ex nihilo. So my second-philosophical response to the skeptic’s challenge as he intends it isn’t to argue that I can meet it directly or to argue that it’s in some way illegitimate; my response is rather that, because the challenge doesn’t arise naturally out of my ordinary ways of finding out about the world, my inability to meet it doesn’t undermine the reasonableness of my belief in that rose.27 Now compare this to the imagined radical challenge to the reliability of our set-theoretic methods. We’ve seen that sets are in fact introduced when they have important mathematical virtues and don’t produce inconsistencies or pre-empt any (or enough) other mathematical desiderata. The skeptic asks if it might not be—however unlikely it must inevitably seem to us—that all such evidence is misleading, that despite the positing of sets having all the payoffs we could ask, sets don’t exist or do exist but don’t have the properties we think they do. This isn’t a matter of our capacity to make mistakes about some particular piece of evidence, or even to be globally misguided about where the mathematical payoffs truly lie. The question is whether impeccable evidence about the mathematical merits of sets could still be entirely and undetectably unreliable, just as Evil

27

This is too quick, of course, but perhaps it can be taken on faith for present purposes. For more, see [2007], }I.2, and [2010].

thin realism 75 Demon-generated experiences would be entirely and undetectably unreliable.28 I think the Thin Realist’s answer to this question is that it couldn’t, that to suggest that set theory could enjoy all these virtues and sets still not exist or be radically different than they seem is to misunderstand the nature of set theory and its subject matter. Recall the Thin Realist’s credo: sets are the things set theory tells us about. Though set theory doesn’t now tell us the size of the continuum, and, for all we know, may never get around to settling that question, still, what’s needed to mount a skeptical challenge is far more radical than this: there has to be the Evil Demon-like possibility that virtually everything set theory tells us about sets is wrong. Not that its purported theorems are fallacious or its analyses of the mathematical benefits of sets are distorted, rather that this whole way of doing things is unreliable. But for the Thin Realist, sets simply are the sort of things we can find out about in these ways. From this point of view, there is no room for a radical epistemological gap between sets and set-theoretic methods; the skeptical challenge here, unlike the case of the external world, is simply ill-formed. I suspect this point is related to the Thin Realist’s discovery that sets can be known without any immediate cognition of them. That unmediated link to a subject matter is what allows us to identify ‘that which I’m linked to’ while possibly being all wrong in my beliefs about it.29 Kant provides a prime illustration: for him, that ‘matter’ involved in our judgments, which comes to us unbidden, is what makes our cognition into cognition of the world, even though, on his view, we can only know that world as it is experienced by knowers like us, not 28 In discussions of external world skepticism, it’s often assumed that we have incorrigible access to our own experiences, that I can’t be wrong about it appearing to me that there’s a rose in my garden; the issue is the inference from here to the existence of the rose. In the settheoretic case, there’s no temptation to think that we can’t be mistaken about the evidence— by getting the logical inferences wrong, by failing to notice the disadvantages, etc.—but again what’s at issue is the inference from the evidence: granting that we’re right about the payoffs of positing sets, could we still go wrong when we infer that they exist? 29 An aside: cognitive scientists note that human perceptual experience actually begins from a primitive system that simply tracks an object before any properties are attributed to it (think of ‘it’s a bird, it’s a plane, it’s Superman’). See [2007], pp. 255–257, for discussion and references.

76 thin realism as it is in itself. The corresponding idea for the Second Philosopher is that her experience could be entirely wrong; she can’t rule out the possibility of an Evil Demon arranging things so that ‘that which generates my experience’ is radically different from what she takes it to be. But for the second-philosophical Thin Realist, there is no perception-like direct access to sets and none is needed. This means there is no primitive connection with a subject matter that could underlie the possibility that these things (the ones with which I’m primitively connected) are radically different from all I think I know about them. This contrast between the external world case—where the radical skeptic’s challenge is coherent (though less damaging than he imagines)—and the set-theoretic case—where the radical skeptical challenge is incoherent—might serve as a criterion of the distinction between abstract objects and concrete ones.30 After all, the close connection between the practice of set theory and the nature of its objects is what makes sets so different from ordinary physical objects: though we’re capable of learning about physical things, we don’t regard this as a trivial consequence of what physical objects are; we give a detailed explanation of how we’re able to do this, beginning with an analysis of the general reliability and various shortcomings of our human perceptual apparatus. If this contrast is right, it might help undermine the ‘perennially powerful’ picture sketched by Burgess and Rosen ([2005], p. 534). The second-philosophical Thin Realist easily recognizes something not unlike their ‘cosmos’— a system . . . containing entities ranging from the diverse inorganic creations and organic creatures that we daily observe and with which we daily interact, to the various unobservable causes of observable reactions that have been inferred by scientific theorists (Burgess and Rosen [1997], p. 29)

—but she harbors no in-principle objection to expanding this ontology when the evidence points that way. Her investigations lead her to believe that there are ‘entities radically unlike concrete entities’, but, as we’ve seen, to deny that between us and these sets ‘there is a great gulf 30

This apparently differs from the various characterizations surveyed in Burgess and Rosen [1997], pp. 16–25.

thin realism 77 fixed’ (op. cit.). She thinks the Thin Realist has provided an account ‘of how anything we do here can provide us with knowledge of what is going on over there’ (op. cit.), an account that rests on what sets are, not on a direct cognition of sets ‘on the other side of the . . . great wall’ (op. cit.).

4. The objective ground of Thin Realism It’s time to take stock. The Thin Realism on offer guarantees the objectivity of set-theoretic truth and existence, respects the actual methods of set theory, recognizes a determinate truth value for CH, and raises no difficult epistemological problem. It not only squares with the Second Philosopher’s austere and hard-nosed scientism, it actually seems to arise naturally from it. It might be just as well to quit while we’re ahead, but I think it would be disingenuous to ignore a nagging worry that it’s all too easy, that it rests on some sleight of hand. Connecting sets and set-theoretic methods so intimately continues to invite the suspicion that sets aren’t fully real, that they’re a kind of shadow play thrown up by our ways of doing things, by our mathematical decisions. The position would be considerably more compelling if it offered some explanation of why sets are this way, but any step in that direction, in the direction of an underlying account of sets that explains this fact, seems to lead us inevitably beyond what set theory tells us about sets, into the realm of Robust Realism and the like. In fact, I think something can be offered that draws the sting from this nagging doubt, but it won’t take quite the form expected. What we want is a sense of what sets are that explains why these methods track them. What I think we can get, from the Thin Realist’s perspective, is a sense of an objective reality underlying both the methods and the sets that illuminates the intimate connection between them. Perhaps this will be enough. Let me come at the question by asking what objective reality underlies and constrains set-theoretic methods, what objective reality it is that set-theoretic methods track. The simple answer, of course, is that they track the truth about sets, but our goal is to find out more

78 thin realism about what sets are, without going beyond what set theory tells us, and our hope is that asking the question this way might help. So, what constrains our methods? Part of the answer lies in the ground of classical logic,31 but our interest here is in the mathematical features. To get at these, let me draw one last comparison to Kant. In his discussions of mathematical truth, Kant draws his now-familiar distinction between analytic and synthetic: in an analytic judgment, the predicate ‘is (covertly) contained in’ the concept of the subject; in a synthetic judgment, the predicate ‘lies . . . outside’ the concept of the subject (A6/B10). The concept of a triangle, for example, is defined by us; since we ‘deliberately made it up’ (A729/B757), we can know what belongs to it, that is, we can know trivial analytic truths like ‘all triangles are three-sided’. In contrast, no amount of meditating on the concept of triangle will reveal to us that the three interior angles of a triangle are equal to two right angles; for this we need to construct a triangle—in our imagination or on the page—draw a line through the apex parallel to the base and reason from there (cf. A716/B744). How does this process take us beyond the concept to something synthetic? Kant’s answer is that the constructions involved are shaped by the structure of our underlying spatial form of sensibility, either in pure intuition (when we construct in our visual imagination) or in empirical intuition (when we draw an actual diagram). Because of this ‘shaping’, the argument tracks more than just what’s built into the concept; the derivation is also constrained by the nature of space itself, which, as we know, Kant thought to be Euclidean. Of course this picture of geometric knowledge hasn’t survived subsequent progress in logic, mathematics, and natural science. I present it here because I think it provides a helpful analogy for what I want to suggest in the case of set theory. Think of it this way. Kant is out to explain what underlies the proof of this geometric theorem, what makes it a proof; his answer is: not just the concept of triangle, not just logical consequence, but also the nature of the underlying space. We’re out to explain what underlies the justificatory methods of set theory, what makes considerations like those sketched in }II.2 good

31

For discussion of the ground of logical truth, see [2007], Part III.

thin realism 79 reasons to believe what we believe. Part of the answer, for the intrinsic justifications, may be that they spell out what’s implicit in our ‘concept of set’, but the bulk of the justifications that interest us are extrinsic.32 What validates them? What takes us beyond mere logical connections and allows us to track something more? And what is this ‘something more’? We’re looking for the counterpart to Kant’s intuitive space. Before trying to answer these questions for set theory, let’s first consider another type of case in which we go beyond the logical, namely, in mathematical concept-formation. In the logical neighborhood of any central mathematical concept, say the concept of a group, there are innumerable alternatives and slight alterations that simply aren’t comparable in their mathematical importance. Logic does nothing to differentiate these one from another, assuming they are all consistently defined, but ‘group’ stands out from the crowd as getting at the important similarities between structures in widely differing areas of mathematics and allowing those similarities to be developed into a rich and fruitful theory. In ways that the historians of mathematics spell out in detail, ‘group’ effectively opens the door to deep mathematics in ways the others don’t.33 So what guides our conceptformation, beyond the logical requirement of consistency, is the way some logically possible concepts track deep mathematical strains that the others miss. Of course there are stark differences between group theory and set theory, because the two pursuits have different goals. Group theory aims to draw together a wide variety of diverse structures that share mathematically important features; it’d be counter-productive to require that all groups be commutative (or not), because there are deep structural similarities between commutative and non-commutative groups that it’s mathematically fruitful to trace. Set theory, on the other hand, aims at least in part to provide a single foundational arena for all classical mathematics, so it strives to develop a unified theory that’s as decisive as possible.34 This is why the set-theoretic theorem

I come back to intrinsic justifications in }}V.3 and V.4. e.g., see Wussing [1969], Stillwell [2002], chapter 19, or the quick survey in [2007], }IV.3. 34 See [2007], pp. 351–355. 32 33

80 thin realism ‘CH or not-CH’ has a different significance from the group-theoretic theorem ‘(x)(y)(x þ y=y þ x) or not-(x)(y)(x þ y=y þ x)’: the set theorist is describing the single structure V, and learns that one of CH or not-CH holds there;35 the group theorist is describing the features common to a wide range of structures, and learns that each is either commutative or not. Still, there are overarching similarities. Set-theoretic concepts are formed in response to set-theoretic goals just as the concept ‘group’ was formed in response to algebraic goals. In large cardinal theory, for example, we can trace the conceptual progression from the superstrong cardinal to the Shelah cardinal to the Woodin cardinal, which turned out to be the optimal notion for the purposes at hand,36 or the gradual migration of the concept of measurable cardinal from its origins in measure theory to the mathematically rich context of elementary embeddings.37 Of course the set-theoretic cases we’ve been concerned with involve not definitions but existence assumptions— like the introduction of sets in the first place or the addition of large cardinals—and new hypotheses—like determinacy—but in these cases, too, far more than consistency is at stake: these favored candidates differ from alternatives and near-neighbors in that they track what we might call the topography of mathematical depth. This topography stands over and above the merely logical connections between statements, and furthermore, it is entirely objective:38 just as it’s not up to us which bits of pure mathematics best serve the needs of natural science, just as it’s not up to us that it would be counterproductive to insist that all ‘groups’ be commutative, it’s also not up to us that appealing to sets and transfinite ordinals allows us to capture facts about the uniqueness of trigonometric representations, that the Axiom of Choice takes an amazing range of different forms and plays a fundamental role in many different areas, that large cardinals arrange themselves into a hierarchy that serves as an effective measure of 35

36 Though recall footnote 7. See Kanamori [2003], p. 461. See Kanamori [2003], }}2 and 5. 38 Perhaps it’s worth recalling (from [2007], Part III) that the Second Philosopher regards logic as robustly true in any situation with a certain minimal structuring, and that V enjoys this sort of structuring (see [2007], p. 382). So the network of logical implications that underlies the mathematically deep strains is itself objective. 37

thin realism 81 consistency strength, that determinacy is the root regularity property for projective sets and interrelates with large cardinals, and so on. These are the facts that play a role analogous to Kant’s Euclidean space, the facts that constrain our set-theoretic methods, and these facts, unlike Kant’s, are not traceable to ourselves as subjects. A generous variety of expressions is typically used to pick out the phenomenon I’m after here: mathematical depth, mathematical fruitfulness, mathematical effectiveness, mathematical importance, mathematical productivity, and so on. (I have been and will continue to use such terms more or less interchangeably.) One point worth emphasizing is that the notion in question is not being offered up as a candidate for conceptual analysis or some such thing. To begin with, I doubt that an attempt to give a general account of what ‘mathematical depth’ really is would be productive; it seems to me the phrase is best understood as a catch-all for the various kinds of special virtues we clearly perceive in our illustrative examples of concept-formation and axiom choice.39 But even if I’m wrong about this, even if something general can be said about what makes this or that bit of mathematics count as important or fruitful or whatever, I would resist the claim that this ‘something general’ would provide a more fundamental justification for the mathematics in question: our second-philosophical analysis strongly suggests that the context-specific justifications we’ve been considering so far are sufficient on their own, that they neither need nor admit supplementation from another source. It also bears repeating that judgments of mathematical depth are not subjective: I might be fond of a certain sort of mathematical theorem, but my idiosyncratic preference doesn’t make some conceptual or axiomatic means toward that goal into deep or fruitful or effective mathematics; for that matter, the entire mathematical community could be blind to the virtues of a certain method or enamored of a merely fashionable pursuit without changing the underlying facts of which is and which isn’t mathematically important.40 This is what

39 This is why I spend so much time rehearsing these various cases, to give the reader a feel for what ‘mathematical depth’ looks like. 40 In particular, we might never come to formulate or to appreciate the virtues of some settheoretic axiom that would settle CH. In this way, our Thin Realist might never come to know whether CH is true or false, despite its having a determinate truth value. (Cf. footnote 5 above.)

82 thin realism anchors our various local mathematical goals. Cantor may have wished to expand his theorem on the uniqueness of trigonometric representations, but if this theorem hadn’t formed part of a larger enterprise of real mathematical importance, his one isolated result wouldn’t have constituted such compelling evidence for the existence of sets; similarly the overwhelming case for Dedekind’s innovations depends in large part on the subsequent successes of the abstract algebra they helped produce. The key here is that mathematical fruitfulness isn’t defined as ‘that which allows us to meet our goals’, irrespective of what these might be; rather, our mathematical goals are only proper insofar as satisfying them furthers our grasp of the underlying strains of mathematical fruitfulness. In other words, the goals are answerable to the facts of mathematical depth, not the other way ’round.41 Our interests will influence which areas of mathematics we find most attractive or compelling, just as our interests influence which parts of natural science we’re most eager to pursue, but no amount of partiality or neglect from us can make a line of mathematics fruitful if it isn’t, or fruitless if it is.42 Thus we’ve answered our leading question: the objective ‘something more’ that our set-theoretic methods track is these underlying contours of mathematical depth. Of course the simple answer—they track sets— is also true, so what we’ve learned here is that what sets are, most fundamentally, is markers for these contours, what they are, most fundamentally, is maximally effective trackers of certain strains of mathematical fruitfulness. From this fact about what sets are, it follows that they can be learned about by set-theoretic methods, because settheoretic methods, as we’ve seen, are all aimed at tracking particular instances of effective mathematics. The point isn’t, for example, that ‘there is a measurable cardinal’ really means ‘the existence of measurable cardinals is mathematically fruitful in ways x, y, z (and this advantage isn’t outweighed by accompanying disadvantages)’; rather, the fact of measurable cardinals being mathematically fruitful in ways x, y, z (and

41

I’m grateful to Matthew Glass for pressing me to clarify this point. Here at last are grounds on which to reject the nihilism of footnote 9 on p. 198 of [1997], and even the tempered version in [2007], pp. 350–351. If mathematicians wander off the path of mathematical depth, they’re going astray, even if no one realizes it. 42

thin realism 83 these advantages not being outweighed by accompanying disadvantages) is evidence for their existence. Why? Because of what sets are: repositories of mathematical depth. They mark off a mathematically rich vein within the indiscriminate network of logical possibilities. Notice also that our conclusion about radical skepticism is reinforced. Any particular extrinsic justification may fail to meet its mark, for reasons ranging from a straightforward error in what follows from what to a deep misconception about the true mathematical values in play. We can be uncertain whether or not a given set-theoretic posit will pay off, and therefore uncertain about whether or not it exists, but if it does pay off, there’s no longer any room for doubt; we can be uncertain that we’re getting at the deepest and most fruitful theory of sets, and therefore uncertain about whether or not our axiom candidate is true, but if we are succeeding, there’s no further room to doubt that we’re learning about sets. This is what defeats an Evil Demonstyle concern: the Demon might somehow induce in me all the experiences I’d have if there were an external world without there actually being such a world, but he can’t present a set-theoretic posit that does a maximally-efficient job of tracking mathematical fruitfulness and yet doesn’t exist—because the posit just is the sort of thing that does this sort of job. So there is a well-documented objective reality underlying Thin Realism, what I’ve been loosely calling the facts of mathematical depth. The fundamental nature of sets (and perhaps all mathematical objects) is to serve as means for tapping into that well; this is simply what they are. And since set-theoretic methods are themselves tuned to detecting these same contours, they’re perfectly suited to telling us about sets; they lie beyond the reach of even the most radical skepticism. This, I suggest, is the core insight of Thin Realism.

5. Retracing our steps Before continuing on the Second Philosopher’s journey, I think it’s worth pausing for a moment to dissect the step-by-step structure of her progress to this point. In our schematic description, she first encounters mathematics as something of a black box that issues forth useful

84 thin realism expressive machinery and effective techniques for exploring and manipulating it.43 I propose to examine her train of thought from there by comparing it with a more transparent case. Let’s imagine the Second Philosopher in her lab, engaged in some cognitive investigation of chimpanzees. The work is going smoothly, but the chimps themselves don’t seem entirely well; they’re a bit lethargic, without good appetite. When our Second Philosopher airs her concern about this in the common lunch room, the local botanist asks where these chimps hail from. Given this information, she retires to her workroom and returns with a plastic bag of processed pellets. The Second Philosopher feeds these to her subjects and they thrive. Here botany is functioning as a black box, issuing forth useful advice. Given the all-encompassing curiosity of the Second Philosopher, she will want to know how this black box works. The botanist carefully explains how she and her fellows have collected plant samples from around the world, how they’ve studied and probed these to form an initial classification, how they experimented with growing conditions and hybridization, and so on. With her resulting expertise, our helpful botanist knew the prevalent plants in the chimps’ ancestral habitat, knew which of these their forebears were likely to have consumed, knew that one of the plants she herself was studying was a close relative, and guessed that feeding the pellets she’d prepared from them for other purposes might well appeal to the Second Philosopher’s chimps. Hearing about how the botanists have conducted their inquiries, the Second Philosopher can easily appreciate the structure of their discipline and the rationality of their methods of observation, experimentation, theory-formation, and testing. Based on her own experiences in other areas and in her more general methodological studies, she may even be able to offer some help to the botanist, perhaps some more refined thinking about experimental design or even, to stretch a bit, some information about subtle perceptual distortions that could throw off some of her observations.

43 A more direct approach would be to describe the Second Philosopher as developing the mathematics she needs as she goes along, as e.g. Newton did, then eventually finding her way into pure mathematics. I adopt the approach in the text to highlight the key points of novelty that arise in the case of mathematics.

thin realism 85 I hope all this seems entirely straightforward given our developed understanding of the Second Philosopher. Now let’s return to the case of mathematics. Impressed by the black box effectiveness of mathematics, the Second Philosopher begins as she did in the case of botany, by investigating the actual methods mathematicians have used to generate the relevant concepts, techniques, proofs, etc. As we saw in examples from Cantor, Dedekind, Zermelo, and the determinacy theorists, she’s able to appreciate that the methods used are rational given the goals being pursued, and that the goals themselves are natural and productive. Based on her previous efforts, can she go on to offer advice and corrections to the mathematician as she did with the botanist? We’ve seen (in }II.3) that the two cases diverge at this point, that for mathematics the answer is no: perception and experimentation are irrelevant to modern pure mathematics; its connections with applications do not provide the kind of methodological guidance they once did. The Second Philosopher concludes that pure mathematics, unlike botany, is autonomous. Now comes the matter of reliability, and here again the two cases diverge. The botanist’s goal is simply to learn about the world’s plant life, so the Second Philosopher’s assessment of the rationality of her methods involves a straightforward assessment of their reliability. Some might hold that the mathematician’s goal is simply to learn about a realm of mathematical things, but the various cases examined by the Second Philosopher don’t actually sound like this: Cantor wants to formulate a stronger theory on the uniqueness of trigonometric representations; Dedekind hopes for a fruitful abstract algebra; the determinacy theorists develop a theory of projective sets that’s mathematically richer than the alternative that follows from V=L. There’s no clear appeal to a mathematical reality in any of this—only to various mathematical benefits—so the Second Philosopher’s assessment of the means-ends rationality in terms of those goals doesn’t involve direct consideration of their reliability. When the issue of reliability is raised, it brings with it the prior question of whether or not set theory should be viewed as an attempt to correctly describe a subject matter at all. In }}II.3 and III.1, we temporarily took for granted that the close interconnection of mathematics with the Second Philosopher’s ongoing empirical studies gives

86 thin realism her good reason to regard it as a body of truths, and it’s a small step from the truth of its existential claims to the existence of a subject matter. The autonomous methods of set theory have produced this body of truths, and this provides persuasive evidence of their reliability. To explain why this is so, the second-philosophical Thin Realist forms her simple hypothesis about what sets are. Finally, she traces the truth of this hypothesis, the source of this fact about sets, to the strains of mathematical depth that the sets mark and the methods track (in }III.4). I think we can now see how the Second Philosopher is led to a form of realism so different from the familiar Robust variety. The key is her starting point, so firmly rooted in the practical details of the actual mathematics. Philosophers often begin from a more elevated perspective; rather than examining the day-to-day practices, they content themselves with classifying mathematics as a non-empirical, a priori discipline, concerning a robust abstract ontology, then begin to wonder how we could possibly come to know such things, how what mathematicians actually do could have any connection to the subject matter they’re attempting to describe. In this way, ‘the great gulf ’ is fixed. Roughly put, they begin with the metaphysics and are led to confusion about the methods. In contrast, the Second Philosopher begins with the methods, finds them good, then devises a minimal metaphysics to suit the case.44 Notice that this elevated brand of philosophizing sometimes makes an appearance—I would say an unwelcome appearance—in connection with those day-to-day practices. The way I’ve told the story here, the Second Philosopher follows a clear line of mathematical development that recapitulates the methodologically sound innovations of Cantor and Dedekind in the late 19th century. Historically, however, various lines of constructivist and predicativist thought developed around the same time. Now it nearly always makes good mathematical sense to investigate how much one can do with how few resources— such inquiries often generate useful insights and more precise techniques—so there’s every reason to pursue constructive and predicative analysis side-by-side with the more familiar classical approach, and all 44

This is the approach endorsed in [1997], pp. 200–202. (Here by the ‘metaphysics’ of a human linguistic practice, I just mean an account of its subject matter.)

thin realism 87 this can be understood as taking place within the vast arena afforded by set theory. But there are also less tolerant forms of constructivism and predicativism that go on to deny the legitimacy of the stronger methods of set theory. It’s hard to see a purely mathematical reason to take this stand, to reject the Cantor-Dedekind approach outright. Obviously the classical theory involves stronger hypotheses, of higher consistency strength, and in that sense it’s more risky, but this hardly gives us reason to decide ahead of time that this avenue is unworthy of investigation. What, then, could the motivation be? I suspect one answer45 is that some common features of set-theoretic method—like the notion of an arbitrary subset or the Axiom of Choice—are taken to derive their justification from some version of Robust Realism.46 Then, in light of the familiar epistemological objections to Robust Realism,47 the practice of set theory itself is called into question. On this approach, the justification—or lack of justification—for mathematical methods is based on a metaphysical account of its subject matter. From the Second Philosopher’s point of view, this gets things backwards: the order of justification goes the other way ‘round, from the math to the metaphysics, not the metaphysics to the math. From her point of view, metaphysical considerations of this sort shouldn’t be allowed to restrict the free pursuit of pure mathematics—and, in fact, they haven’t. Thus, Thin Realism. Let’s now approach questions of the second type—about the nature of set-theoretic activity and its subject matter— along an entirely different avenue. 45 Another might be that we should pursue only the mathematics that’s directly needed for natural science, and that non-constructive or impredicative mathematics is not. This sentiment is often based on a Quinean holism that sees mathematics in application as confirmed along with the rest of our overall web of belief, but leaves the remaining pure mathematics without justification. This picture of the relations between pure mathematics and natural science, mentioned in passing in }II.3, is undercut by the considerations of Chapter I. See [1997], }}II.6–7, [2007], pp. 314–317, for more direct engagement with Quine’s holism. The general suggestion that only applied mathematics is worthy comes up again in }IV.2. 46 See e.g. Feferman [1987], pp. 44–45. 47 Rehearsed in }II.4. These various schools of non-classical thought then replace Robust Realism with some other external metaphysics that supports only more restricted methods. One irony here is that Robust Realism doesn’t seem to support our actual set-theoretic methods (again see }II.4).

IV Arealism The discussion of Thin Realism in the last chapter was predicated on the assumption that the Second Philosopher has good reason to regard pure mathematics in general, and set theory in particular, as a body of truths. Along the way, I set aside the question of precisely what that good reason is. I come back to this central question below, but first I’d like to explore an alternative picture, one that does without that key assumption. After sketching such a view and contrasting it with near neighbors, I consider its relations to Thin Realism and, then, at last, take up the question of truth.

1. Introducing Arealism To return to the Second Philosopher’s starting point, she begins her investigation of the world with ordinary perception, graduates to more sophisticated forms of observation, theory-formation, and testing, improving her methods as she goes; eventually she turns to mathematical methods, and from there, to the pursuit of mathematics itself. Recapitulating the developments of the 19th century, she finds her mathematical inquiries broadening to include structures and methods without immediate application, which eventually leads her to set theory along the path of Cantor, Dedekind, Zermelo, and the rest. Now there’s no doubt that she has clear motivation to pursue pure mathematics, but the question before us is whether or not she has good grounds to regard it as a body of truths. When she notices that its methods are quite different, that its claims aren’t supported by her familiar observation, experimentation, theory-formation, and so on,

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but by the sorts of intrinsic and extrinsic arguments canvassed in }II.2, might she not simply conclude that whatever its merits, pure mathematics isn’t in the business of uncovering truths? But if he’s not uncovering truths, then what is the pure mathematician doing? For the case of set theory, we’ve got a sense of the answer: among many other things, Cantor is extending our grasp of trigonometric representations; Dedekind is pushing towards abstract algebra; Zermelo is providing an explicit foundation for a mathematically important practice; contemporary set theorists are trying to solve the continuum problem.1 Just as the concept of group is tailored to the mathematical tasks set for it, the development of set theory is constrained by its own particular range of mathematical goals, both local and global. Mightn’t the Second Philosopher rest content with this description? Set theory is the activity of developing a theory of sets that will effectively serve a concrete and ever-evolving range of mathematical purposes. Such a Second Philosopher would see no reason to think that sets exist or that set-theoretic claims are true—her well-developed methods of confirming existence and truth aren’t even in play here— but she does regard set theory, and pure mathematics with it, as a spectacularly successful enterprise, unlike any other.2 Let’s call this position Arealism.3

2. Mathematics in application Whatever reason the Thin Realist may have to count pure mathematics as true, it must rest somehow on the role of mathematics in empirical science, so we need to ask: can the Arealist account for the

1 And, lest we forget, much of pure mathematics is still consciously aimed at the goal of providing tools for empirical science. 2 In particular, its complex interrelations with natural science mark it off from other human endeavors—astrology, theology—whose methods also differ from those usual to the Second Philosopher. See [1997], pp. 203–205, [2007], pp. 345–347, and more below. 3 I noted earlier (}III.2, footnote 15) that Thin Realism doesn’t require a disquotational theory of truth. Perhaps it’s worth noting here that Arealism is compatible with disquotationalism: the Arealist isn’t straightforwardly asserting the claims of set theory, so isn’t committed to their truth.

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application of mathematics without regarding it as true? We’ve seen (in Chapter I) that contemporary pure mathematics works in application by providing the empirical scientist with a wide range of abstract tools; the scientist uses these as models—of a cannon ball’s path or the electromagnetic field or curved spacetime—which he takes to resemble the physical phenomena in some rough ways, to depart from it in others; indeed often enough, in fundamental theories, we aren’t sure exactly how the correspondence plays out in detail. The applied mathematician labors to understand the idealizations, simplifications and approximations involved in these deployments of his abstract structures; he strives as best he can to show how and why a given model resembles the world closely enough for the particular purposes at hand. In all this, the scientist never asserts the existence of the abstract model; he simply holds that the world is like the model is some respects, not in others. For this, the model need only be welldescribed, just as one might illuminate a given social situation by comparing it to a imaginary or mythological one, marking the similarities and dissimilarities. Michael Liston worries that this line of thought is not enough to account for the role of mathematics in physical science.4 His concern is that there is more to the use of mathematics in such cases than the description of an abstract model sufficiently similar to the physical phenomena. As we’ve just noted, scientists invariably pursue an account of how far the similarities extend and why, as part of assuring themselves that the model is well-suited to its job: some of these assurances will involve physical information, as the use of van der Waals’s equation is justified by the fact that actual molecules have stable ‘effective radii’; others will involve mathematics, as the reliability of Stirling’s approximation rests on the Robservation that the sum of ln(n), as n varies from 1 to N, approaches 1N ln(x)dx when N is large. In some cases, we still have no satisfactory answer to this type of question: for example, we suspect that our abstract quantum mechanics must

4 See Liston [2007], p. 4. This may be related to what Resnik [1997], chapter 3, has called his ‘pragmatic indispensability argument’, but I’m not sure. Liston himself cites Wilson [2006], presumably in connection with examples like the discussion of Euler’s Method (pp. 116–117, 163–165, 212–217, 573–575) (see below).

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resemble the world in the small in some way or other, because it makes such amazingly accurate predictions, but we have no account of what the underlying physical structures are like, which aspects of the model reflect them, and why this representation is so effective. Still, despite occasional setbacks, we always seek this type of account, very often with success, and we continue to do so even for quantum mechanics. Liston’s point is that giving these accounts—what he calls ‘reliability explanations’—often requires substantial mathematics, well beyond what goes into the original modeling. So, for example, Solutions to numerous important physical problems require the determination of a function satisfying a differential equation . . . Sometimes . . . the existence of a solution . . . can be established by directly solving the equations. . . . where direct methods fail, the existence of a solution must be established indirectly, generally by constructing a sequence of functions that converges to a limit function that satisfies the [equation]. Moreover, the solution . . . very often cannot be evaluated by analytic methods, and scientists must rely on discrete variable or finite element numerical methods to approximate the solution. . . . The numerical methods often provide our only way of extracting an actual solution. (Liston [2004], p. 146)

In such cases, these highly developed mathematical theories of indirect solution and approximation are essential to our treatment of the physical problem; they help us find solutions where this is possible, and ‘provide us with . . . valuable qualitative information about the solutions’ (op. cit.) where it isn’t. Liston observes that: Mathematical physicists rely on the theories presupposed in proving the existence of solutions and approximating them. It is difficult to see how they could do this while adding the . . . disclaimer, ‘But, you know, I don’t believe any of the mathematics I’m using’. (op. cit.)

For this reason, Liston argues, it isn’t enough that the Arealist can account for the role of abstract mathematical models as means of describing phenomena; she also needs to account for the role of mathematical reasoning in determining the features of those models.5

5

Liston’s actual target in these passages is Mark Balaguer’s fictionalism, but I assume he has something similar in mind in his [2007], p. 4.

92 arealism If she’s to believe her abstract models have certain properties, mustn’t she believe the mathematics used to establish those properties?6 Of course Liston is quite right that we aren’t and shouldn’t be satisfied to regard our mathematical model as a black box with a good track record, and to point out that some of our reliability explanations, like the case of Stirling’s approximation, involve mathematics. But let’s look a bit more closely, focusing on the kind of case he highlights. The original differential equation for our physical problem is proposed as an abstract model of some physical phenomenon: we know something about the rate of change of some quantity—the temperature, the location, the density—and we want to figure out the quantity itself—the temperature or location as a function of time, the density as a function of position. To get at this, we replace the actual situation with an abstract model: time is represented as a continuous real variable, space as ℝ3, the temperature, location and density as real-valued functions; what we know about the rate of change is then formulated as a differential equation. This mathematical structure—the abstract model satisfying the stated equation—doesn’t exist in a vacuum; it’s embedded in a rich mathematical universe—V if you like—and it has all the properties the advanced methods reveal—for example, it has a solution that can be approximated in a certain way—as part of its identity as a mathematical object. Liston’s challenge is this: we form a simple abstract model; why should we believe all the extra things that advanced mathematics tells us about our model? I’d put it somewhat differently: the abstract model has all those features to start with, as part of its mathematical pedigree; the question is why—given a model whose simple features track the world reasonably well—why should we expect its mathematically esoteric features to continue to track the world reasonably well? It seems what we need to know isn’t so much that the advanced mathematics is true, but that the more esoteric features it reveals will continue to be effective in modeling the world.

6 It’s worth noting that Liston takes the truth of the background mathematics to be necessary but not sufficient: ‘Robust Realism will not help: why should beliefs about a freefloating mathematical realm give us moral certainty of the practical success of our calculations?’ (Liston [2007], p. 4). Presumably the point is that, by itself, the truth of these mathematical beliefs wouldn’t make their applicability any less mysterious.

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This is undoubtedly an important question. The short answer is that the new features revealed by the full force of our pure mathematics often aren’t effective modelers: to recall an example from }I.3, though point sets in ℝ3 are well-known to be good stand-ins for spatial regions, we shouldn’t expect the Banach-Tarski construction to be physically realizable. So we need to focus more closely on the kinds of examples that interest Liston. Often the particular methods he discusses got their start in Euler’s day, when the relations between physics and mathematics were quite different than they are now (see }I.2): new mathematical structures were being added in response to the needs of science, and physical intuition was a central guide. Consider, for example, Euler’s Method, an approximation procedure familiar to readers of Mark Wilson: There is a venerable computational technique called Euler’s method of finite differences that will estimate our cannon ball’s instantaneous . . . deceleration using an averaged change of speed considered over, say, ¼ second stretches of time . . . This routine allows us to calculate a succession of numerical values which, if graphed and connected together by straight lines, generally provides a reasonable broken line facsimile to our cannon ball’s path. (Wilson [2006], p. 117)

This procedure is entirely natural from a physical point of view: The matrix of numerical data assembled by this syntactic routine provides us with an excellent stage by stage ‘image’ of our ball’s flight . . . Our symbolic calculations ‘walk along’ at discrete stages with our cannon ball . . . indeed, Euler’s procedure is commonly called a ‘marching method’ for that very reason. (Wilson [2006], p. 164)

The unwelcome surprise is that Euler’s Method doesn’t always work; if we represent the cannon ball with a slightly modified pair of differential equations, Euler’s Method tells us that the ball will never fall to the ground!7 Subsequent mathematical work uncovers the trouble: Euler’s Method provides a good approximation to the true solution if the differential equations in question satisfy the Lipschitz condition, and these modified equations do not.

7

See Wilson [2006], pp. 214–215.

94 arealism Here we have a clear illustration of what Liston means by a ‘reliability explanation’: we’ve proved that Euler’s Method is reliable when our equation meets the Lipschitz condition. Liston might imagine us building a bridge or some such thing. We model the situation with some differential equations but find we can’t solve them directly. Can we trust Euler’s Method to keep us close enough to the true solution; can we trust it enough to give the go-ahead to those poised to begin pouring concrete? We then confirm that our equations satisfy the Lipschitz condition. If we’re to allow this fact to underwrite our trust in the Method, don’t we have to believe the mathematics involved in proving the relevant theorems? Before answering yes, think again about what those theorems have done. Have they guaranteed that our computations will be physically reliable? No, what they’ve shown is something purely abstract; they’ve shown that if you have a differential equation with a certain nice feature, then Euler’s Method will generate outcomes fairly close to the actual function that satisfies your equation. This, as I’ve described it, is a property of the equation, as a mathematical object. In deciding whether or not to green-light the cement mixers, what we need to know is whether or not that equation, in all its Euler’s Methodratifying glory, is a good model for the relevant features of our building situation. If we trust Euler’s Method, go ahead, and the bridge falls down, it isn’t because the mathematics used to prove those theorems about Euler’s Method isn’t true; it’s because the original equation we selected to model our bridge wasn’t a good model, after all. What the reliability argument tells us isn’t that our original equation is a good model or that our bridge won’t fall down; it tells us that if the original equation is a good model, then the bridge won’t fall down (at least not because the use of Euler’s Method led us astray!). In ordinary engineering contexts these days, our ways of determining the appropriate equations have a long history, plausible physical underpinnings, and a strong track record, so it’s perfectly reasonable to trust Euler’s Method when the equation is of the right type. The case of the Axiom of Choice and the Banach-Tarski paradox is quite different. As noted in Chapter I, the set-theoretic methods involved, the set-theoretic goals in play, are largely the product of contemporary pure mathematics, a long way from the physically-inspired

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approaches employed by Euler. Though we could construct a bogus ‘reliability argument’ that tells us an object can be disassembled and recombined into two objects the same size as the original, of course that wouldn’t make it so. The trouble isn’t that the set theory used to prove the relevant theorems isn’t true; it’s that the theorems rely on aspects of point sets that don’t correspond well with physical regions. Unlike the case of Euler’s Method—which is well-grounded in our understanding of the physical situation and enjoys a long history of empirical success—what’s involved in the set-theoretic case is the finestructure of pathological sets of real numbers (that is, sets that don’t have a coherent ‘size’, even by the generous standards of Lebesgue measure). If we draw too heavily on these features of our mathematical model of space, we will be led astray. Now it might be tempting to suggest that this unfettered contemporary mathematics be sequestered in some way, set aside as unreliable—that we carve out from the vast reach of mathematical lore the sounder body of physically-inspired studies and restrict our serious attention, and perhaps our NSF grants and university professorships, to its pursuit. The trouble is that bits of unfettered pure mathematics have turned out to be profoundly applicable, and there’s no way to predict ahead of time which these will be.8 (As noted in }II.3, group theory is a famous example.) The practice of contemporary pure mathematics answers to many desiderata, including but not limited to foreseeable applications, but I think no honest observer, even one primarily interested in mathematics only for the sake of application, would think it prudent to rein it in. This raises an obvious question: how is it that mathematics pursued for purely mathematical reasons ends up serving the needs of empirical science? why should the theorems of group theory be reliable guides to the behavior of subatomic particles? why should some troublesome, mathematically esoteric aspect of an otherwise effective mathematical model turn out—most unexpectedly!—to have a physical correlate?! (Here the well-worn examples are radio waves and positrons.9) This, I think, is the sort of thing that’s really bothering Liston: just because 8 9

See [2007], pp. 329–343, for more on the topic of this paragraph and the two following. See [2007], pp. 332–333.

96 arealism the aspects of our model we’ve been attending to work well, why should we expect some newly revealed mathematical feature of that same model to continue to work well? There are good answers to this question for many familiar tools of applied mathematics, good reasons, for example, to regard an equation’s satisfying the Lipschitz condition as evidence that Euler’s Method will be reliable. But why should we expect these new features to work well when they arise from a notion of ‘correctness [that is] a function of furthering internal goals of pure mathematics’ (Liston [2007], p. 4)? What we have here is one sub-problem of Wigner’s famous ‘miracle of applied mathematics’: why does mathematics generated for mathematics’ sake end up being successfully applied?10 On closer examination of the contexts in which pure mathematics does and doesn’t work in applications, I’m not sure its successes are as miraculous as all that,11 but for present purposes it’s enough to notice that what’s at issue here, perplexing as it may be, doesn’t hinge on the truth (or not) of the mathematics involved, but on the fact that it is generated in pursuit of purely mathematical goals.

3. What Arealism is not Assuming then that the truth (or not) of mathematics is irrelevant to explaining its role in scientific application, it appears that Arealism is open to our Second Philosopher: she notes that mathematics is successful on its own terms and immensely useful to science, but since it isn’t confirmed by her usual methods, even by her need to explain the role it plays in her empirical theorizing, she concludes that she has no grounds on which to regard its objects as real or its claims as truths. In philosophical taxonomy, the standard term for someone who doesn’t believe in abstract objects is ‘nominalist’. If we limit attention to mathematical abstracta, the Arealist would seem to qualify, but, at least as ‘nominalism’ is usually conceived in contemporary philosophy 10

See Wigner [1960]. Again, see [2007], pp. 329–343. In his writings on ‘the miracle’, Liston [2000] seems to agree. See e.g. [2007], p. 340, footnote 58. 11

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of mathematics, this way of talking seems to me to invite misunderstanding. To see how, recall that contemporary nominalism began with Goodman and Quine’s annunciation of a philosophical intuition that cannot be justified by appeal to anything more ultimate . . .

namely, We do not believe in abstract entities. . . . We renounce them altogether. (Goodman and Quine [1947], p. 105)

In Burgess and Rosen’s characterization: Nominalism (as understood in contemporary philosophy of mathematics) arose toward the mid-century . . . It arose . . . among philosophers, and to this day is motivated largely by the difficulty of fitting orthodox mathematics into a general philosophical account of the nature of knowledge. (Burgess and Rosen [1997], p. vii)

What’s at work here is the picture of the ‘great gulf’ introduced in }II.4: to avoid nominalism, one must explain in detail how anything we do and say on our side of the great wall separating the cosmos of concreta from the heaven of abstracta can provide us with knowledge of the other side. (Burgess and Rosen [1997], p. 41)

Various familiar ideas on the nature of knowledge in concrete cases, like the causal theory of knowledge and its successors, are floated to highlight the severe obstacles that stand in the way of such an explanation. These elements provide the raw materials for a perfectly general, in-principle argument against abstracta of all kinds. I hope and trust it’s clear that this is not a portrait of the secondphilosophical Arealist. She doesn’t come to her investigations with any a priori prejudice against abstract objects or with any preconceptions about what knowledge must be like that would seem to rule out knowledge of sets. She doesn’t argue that set-theoretic knowledge is problematic or impossible on principle; she simply surveys the evidence at hand and concludes that it doesn’t confirm the existence of sets or the truth of our theory of them. So if Arealism is to be

98 arealism considered a version of nominalism, it certainly isn’t the ‘stereotypical’ variety (Burgess and Rosen [1997], p. 29). Another popular characterization of the view that there are no mathematical objects comes under the rubric of ‘fictionalism’. As Hartry Field has put it, ‘the sense in which’ a mathematical claim is true is pretty much the same as the sense in which ‘Oliver Twist lived in London’ is true: the latter is true only in the sense that it is true according to a certain well-known story, and the former is true only in that it is true according to standard mathematics. (Field [1989], p. 3)

Presumably the fictionalist would say that Cantor and Dedekind told the opening chapters in the story of set theory, that Zermelo organized it, that contemporary set theorists are extending it. Of course, as we’ve seen, the story line is closely constrained, not just by logical consequence and consistency, but (the fictionalist continues) the story line in good fiction-writing must also hew to demanding standards (say of realistic description and psychology). Drawing an analogy in this way between fiction-writing and the practice of set theory has the merit of providing a clear illustration of how one can legitimately make assertions, make truth and existence claims, ‘within a story’, while denying them in a broader context.12 Beyond this, perhaps further exploration of the similarities and dissimilarities will shed light on set theory—the Arealist has no cause or grounds to rule this out—but I confess to some skepticism. The central challenge is to delineate and defend the proper ways of extending what the fictionalist calls the ‘set-theoretic story’, but calling it that, rather that just ‘set theory’, doesn’t appear to advance our understanding on this point. It’s no doubt difficult to spell out what makes one way of extending a fictional story better than another, as the labors of literary critics demonstrate, but it seems unlikely that any insight that might be gleaned on that topic would translate in any useful way to set theory. Or vice versa. If this is right, then the value of the fictionalist analogy is limited, and keeping it at the forefront of our thinking about set theory might tempt us to impose categories and judgments foreign to our subject and to ignore important features without correlates in fiction. 12

Thanks to Patricia Marino for pressing this point.

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For reasons like these, it seems to me best to push all the popular analogies to the background—math is like science, math is like a game, along with math is like fiction—and to study set theory directly, on its own terms. Such an Arealist simply understands herself as developing a theory of sets, guided, as we’ve seen, by various concrete set-theoretic norms and goals and values. Finally, various strains of formalism or if-thenism also deny that mathematics is in the business of discovering truths about abstracta; they take their lead from mathematicians who say, ‘I’m just figuring out what follows from what’. Again, I hope it’s clear that the Arealist would say no such thing: though mathematicians are often engaged in proving one thing from another, they obviously don’t regard any starting point, even any consistent starting point, as equally worthy of investigation; if one characterizes set-theoretic practice as that of deriving theorems in one or another axiomatic setting, one ignores the very features of that practice that have been my Arealist’s focus, namely, the forces that shape the concepts and assumptions of the setting itself.13 A more sophisticated if-thenist would admit that mathematics is more than a matter of determining what follows from what, that mathematicians are also engaged in forming those concepts and selecting those assumptions, and would then assume responsibility for explaining how this process is constrained, what principles should guide it and why. While the term ‘if-thenism’ invites an overly narrow focus on logical connections, there doesn’t appear to be any difference of substance between this more subtle if-thenist and the Arealist.

4. Comparison with Thin Realism If Arealism doesn’t quite fit the standard profiles of nominalism or fictionalism or simple if-thenism, one position from which it would 13 Recall from }III.2 the related problem for an account of set-theoretic practice in terms of Carnapian linguistic frameworks: enshrining any fixed principles as implicitly defining ‘the concept of set’ seemed false to the open-ended nature of the practice. Essentially the same problem arises when the if-thenist decides what to put in the antecedent to the conditionals. Burgess [unpublished] argues persuasively that the same problem arises for many versions of structuralism.

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seem to stand unambiguously apart is Thin Realism. After all, the Thin Realist holds that sets exist and set theory is a body of truths, and the Arealist denies both. But despite their disagreements over truth and existence, the Thin Realist and the Arealist are indistinguishable at the level of method. On grounds like those that motivated Cantor and Dedekind, both would elect to introduce sets into their pursuit of pure mathematics; both would regard Zermelo’s defenses of his axioms as persuasive; both would follow the path of contemporary set theorists on determinacy and large cardinals. This methodological agreement reflects a deeper metaphysical bond: the objective facts that underlie these two positions are exactly the same, namely, the topography of mathematical depth brought to light in }III.4. For the Thin Realist, sets are the things that mark these contours; set-theoretic methods are designed to track them. For the Arealist, these same contours are what motivate and guide her elaboration of the theory of sets; she can go wrong as easily as the Thin Realist if she fails to detect the genuine mathematical virtues in play. For both positions, the development of set theory responds to an objective reality—and indeed to the very same objective reality. What separates the Arealist from the Thin Realist, then, doesn’t lie in their set-theoretic practices or what underlies them; in that respect, the two are indistinguishable. Where they differ is in their secondphilosophical reflections on the human undertaking called ‘set theory’. They would agree precisely on what counts as proper grounds for adding a new large cardinal axiom to the theory of sets; they would disagree only on the Thin Realist’s added assertion that these grounds confirm the existence of the large cardinal in question and the truth of the corresponding axiom. Notice that it isn’t an ordinary set-theoretic claim of existence or truth that’s at issue here: the Arealist like the Thin Realist will formulate the axiom in existential form and call it ‘true’ in the sense of holding in V. Their disagreement takes place not within set theory, but in the judgments they form as they regard set-theoretic language and practice from an empirical perspective and ask secondphilosophical versions of the traditional philosophical questions, questions in the second group we’ve been considering. At this point, we have two apparently second-philosophical positions in play; how is the Second Philosopher to adjudicate between

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Thin Realism and Arealism? This returns us at last to the problem we’ve set aside twice: on what grounds does the Thin Realist judge that set theory is a body of truths? Given that she rejects the usual Quinean arguments, given that she endorses the Arealist’s account of how mathematics works in application, the Second Philosopher’s case for Thin Realism will have to rest more loosely on the interconnections of mathematics with empirical science: she recognizes that pure mathematics arose out of a subject very closely tied to our study of the physical world; she regards the project of providing a rich array of structures for the contemporary scientist as one of the overarching goals of mathematical practice; she well appreciates that contemporary pure mathematics continues to find its way into scientific applications, sometimes along deliberately anticipated paths, and sometimes along wholly unexpected ones. Thus mathematics, whatever its idiosyncrasies, appears as an integral part of her overall enterprise (as opposed to astrology, theology, etc., which are idiosyncratic without playing a part in that enterprise).14 On this picture, the Second Philosopher pursues mathematics in a spirit continuous with her other inquiries: some of its methods, like logical deduction and means-ends reasoning, are familiar; others, like Cantor’s, Dedekind’s, Zermelo’s, and the determinacy theorists’, are unfamiliar, but taken to be rational and reliable along the lines we’ve been following. Thus the divergence between the second-philosophical Arealist and the second-philosophical Thin Realist comes down to this: as the Second Philosopher conducts her inquiry into the way the world is, beginning with her ordinary methods of perception and observation, theory-formation and testing, she’s eventually faced with the effectiveness of pure mathematics and elects to add it to her ever-growing list of investigations; she also recognizes that the appropriate methods are different and that the objects studied are different; the point at issue hinges on what she concludes from this. If the new objects seem a bit odd—non-spatiotemporal, acausal, etc.—but still enough like the old—singular bearers of properties, etc.—, if the new methods seem a bit odd, but still of-a-piece with the old, then she concludes that she’s

14

See [1997], pp. 203–205, [2007], pp. 345–347.

102 arealism made a surprising discovery, that the world includes abstracta as well as concreta. If, on the other hand, she regards the new methods and would-be objects as sharply discontinuous with what came before, she has no grounds for thinking pure mathematics is true, so she concludes that this new practice—valuable as it is—isn’t in the business of developing a body of truths. So, which is it? Is pure mathematics just another inquiry among many or it is a different sort of thing that’s immensely helpful to the others? Are the grounds cited by Cantor, Dedekind, Zermelo, and the determinacy theorists just more evidence of an unexpected sort, or are they the trademarks of a different sort of activity altogether? I think this understanding of the disagreement between the Thin Realist and the Arealist further illuminates the contrast between Arealism and more familiar forms of nominalism. To see this, consider David Lewis’s well-known credo: Renouncing [sets] means rejecting mathematics. That will not do. Mathematics is an established, going concern. Philosophy is shaky as can be. To reject mathematics for philosophical reasons would be absurd. If we philosophers are sorely puzzled by the [sets] that constitute mathematical reality, that’s our problem. We shouldn’t expect mathematics to go away to make our life easier. Even if we reject mathematics gently—explaining how it can be a most useful fiction . . . —we still reject it, and that’s still absurd. . . . How would you like the job of telling the mathematicians that they must change their ways . . . ? (Lewis [1991], pp. 58–59)

The Arealist doesn’t reject mathematics or recommend that mathematicians change their ways; as we’ve seen, the Arealist is indistinguishable from the Thin Realist as far as the practice of mathematics is concerned. Furthermore, as we’ve also seen, the Arealist isn’t denying the truth of mathematics on general philosophical grounds, as Lewis implies. She isn’t ‘puzzled’ by mathematical existence; she takes the Thin Realist’s position to be coherent, even in some ways attractive. She just doesn’t see that the evidence supports it. All this is familiar, but another element is suggested here: that the Arealist is disagreeing with what the mathematician says. In one sense, I’ve suggested that this isn’t right: the Arealist is speaking as a Second Philosopher proposing an account of human mathematical activity; she

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isn’t questioning the propriety of any aspect of the practice, including its assertions in the shape of existence claims. Of course, it’s also clear that mathematicians themselves have philosophical views about the nature of mathematics, about what those assertions mean—recall (from }II.3) Dedekind’s remark about ‘free creations of the human mind’— and this part of what mathematicians believe can come into conflict with views like Arealism and Thin Realism. The Arealist doesn’t disagree with what mathematicians say qua mathematicians, but when they branch out into questions of truth and existence external to mathematics proper—what is the nature of human mathematical activity? what is its subject matter and how do we come to know about it? and so on—then she reserves her right to differ. The larger point here is that the fine structure of what particular mathematicians mean or intend by their mathematical assertions—for example, whether or not they think like Dedekind or make their claims with ‘mental reservation or purpose of evasion’15—these psychological matters are no more relevant to the correctness or incorrectness of Thin Realism or Arealism than a physicist’s personal views—for example, that he’s discovering the acts of God or merely organizing the course of our sense-data—would be to the correctness or incorrectness of some analysis of the status of general relativity or quantum mechanics. What matters, from our second-philosophical point of view, isn’t what the practitioners think about these issues in their heart-of-hearts, but where the evidence leads. The Thin Realist gives one answer and the Arealist another.

5. Thin Realism/Arealism So who’s right, where does the evidence lead—to Thin Realism or Arealism? The way I’ve been describing the Second Philosopher’s intellectual journey, it apparently leaves the Thin Realist with the challenge of explaining exactly how the role of pure mathematics in science supports her view that it is a body of truths about objectively

15

Burgess [2004a], p. 54, brings this up in his discussion of fictionalism.

104 arealism existing things, exactly what justifies her assumption that the methods of pure mathematics, distinctive as they are, should be regarded as partand-parcel of her ever-evolving approach to finding out about the world. But there are other ways of describing that second-philosophical journey that might seem to reshape the debate. Consider, as a start, Burgess and Rosen’s characterization of the ‘stereotypical anti-nominalist’: We come to philosophy16 believers in a large variety of mathematical and scientific theories—not to mention many deliverances of everyday common sense—that are up to their ears in suppositions about entities nothing like concrete bodies we can see or touch, from numbers to functions and sets . . . To be sure, we also come . . . prepared to submit all our . . . beliefs to critical examination and to revise them if good reasons for doing so emerge. (Burgess and Rosen [1997], p. 34)

A Second Philosopher could be described along these lines. She begins in our contemporary world-view, where pure mathematics is ‘considered . . . the very model of a progressive and brilliantly successful cognitive endeavor’ (Burgess and Rosen [1997], p. 211). Burgess and Rosen’s thorough-going naturalist would take the fact that abstracta are customary and convenient for the mathematical (as well as other) sciences to be sufficient to warrant acquiescing in their existence. (Burgess and Rosen [1997], p. 212)

Our Second Philosopher may be a bit pickier than that—requiring, for example, the kinds of detailed support offered by Cantor, Dedekind and the rest—but the underlying idea is simple: this second-philosophical starting point already includes pure mathematics as a body of truths alongside physics, chemistry, botany, and so on. Of course some of the things we tend to believe, from the gambler’s fallacy to spiritualism, will succumb to critical examination, but what grounds would dictate that ‘the mathematical sciences . . . be expelled from the circle of “sciences” ’, what grounds could there be for ‘marginalizing some sciences (the mathematical) and privileging others

16

Speaking second-philosophically, we’d say: ‘to the examination of human mathematical activity . . . ’.

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(the empirical)’, for ‘abridging the roll of sciences’ (Burgess and Rosen [1997], pp. 211–212)? The stereotypical nominalist would offer such grounds, but we’ve seen that the Second Philosopher comes to the table with no prejudice against abstracta, no general theory of what knowledge must be, and thus apparently no reason to change her initial opinion about the truth of pure mathematics, her opinion that mathematical evidence is evidence. When she comes to inquire into the nature of the human mathematical practice, she elaborates Thin Realism as in the previous chapter. Since Chapter II, we’ve been investigating what the Second Philosopher would do or say in this situation or that, but we’ve now returned to the question of how Second Philosophy itself should be characterized. My rhetoric there, and subsequently, has been consistently of the bottom-up variety; the Second Philosopher begins with her ordinary perceptual beliefs and refines her methods from there, expanding her reach into all areas of what we might call ‘natural science’ and even ‘social science’: physics, chemistry, astronomy; biology, botany, mineralogy; psychology, linguistics, and the study of human inquiry itself. One of these extensions is into pure mathematics, where she may well wonder if its methods serve to establish truth and existence or actually do something else; this is what seemed to place a further explanatory demand on the Thin Realist and made Arealism appear as an attractive possibility. But if we imagine the Second Philosopher along these new lines inspired by Burgess and Rosen, starting with a complex body of beliefs and deciding which to reject, then pure mathematics is included from the start, and no piecemeal defense, beginning from perception and leading to mathematics, would seem to be required. From this point of view, Thin Realism may appear more natural than Arealism. It’s hard not to think that one must be right and the other wrong, that either sets exist or they don’t, that set theory is a body of truths or it isn’t, that either the considerations cited by Cantor, Dedekind, Zermelo, and the determinacy theorists are confirming evidence or they aren’t. But perhaps this tempting position is in fact incorrect, perhaps our strong conviction otherwise rests on what Mark Wilson calls, in his typically colorful style, ‘tropospheric complacency’: we tend to think that our concepts—in our case ‘true’, ‘exist’, ‘evidence’, ‘believe’, ‘know’—mark fully determinate features or attributes, that there is a determinate fact of

106 arealism the matter as to where they apply and where they don’t, that this is so even for questions we haven’t yet been able to settle one way or the other. Wilson’s case against this picture is one thread running through his massive Wandering Significance (Wilson [2006]); it rests largely on a series of fascinating and down-to-earth examples. To get a feel for how these examples go, let’s look at two of them. First, consider ice. Surely we all know what ice is—it’s frozen water—but Wilson takes us in for a closer look: Water, in fact, represents a notoriously eccentric substance, capable of forming into a wide range of peculiar structures. (Wilson [2006], p. 55)

He goes on to quote a recent textbook on the subject, which describes ‘ice cousins’, the clathrate hydrates . . . Like ice polymorphs, they are crystalline solids, formed by water molecules, but hydrogen-bonded in such a way that polyhedral cavities of different sizes are created that are capable of accommodating certain kinds of ‘guest’ molecules. (Quoted by Wilson [2006], p. 55)

Wilson remarks that The author doesn’t regard the clathrate structure as true ice . . . but is it clear that our everyday conception of ice requires—as opposed to accepts—this distinction? (I, for one, had never thought about such matters at all.) (Wilson [2006], pp. 55–56)

It gets worse: there are in fact more than a dozen ways that water can form into a solid. In one case, if one cools water quickly enough, the result lacks crystalline structure and more closely resembles ordinary glass. Wilson asks Should this glass-like stuff qualify as a novel form of ‘ice’ or not? Our chemist will presumably say ‘no’ because the stuff is not crystalline but many of us would perhaps put a higher premium on its apparent solidity. (Wilson [2006], p. 56)

In fact other chemists do happily call this ‘an amorphous type of ice’ (Caro [1992], p. 99).17 And so on.

17

Wilson doesn’t cite this passage in his discussion of ‘ice’, but he does quote Caro’s book when he treats the relations between ‘water’ and ‘H2O’ (Wilson [2006], pp. 428–429).

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Is there a right and a wrong answer here? Our everyday use of the word ‘ice’ clearly correlates with an objective feature of the world, the substance chemists call ‘ice Ih’ or hexagonal crystalline ice. So ‘ice’ definitely doesn’t apply to liquid water or to sand or to window glass. But does it apply to amorphous ice—is amorphous ice really ice? Wilson’s thought is that nothing in our ordinary use or understanding of the term ‘ice’, indeed nothing in the underlying chemical facts that we subsequently discover about the many ways water can form into a solid—in short, nothing in our heads, in our language, or in the world will force either answer to this question.18 And notice that this isn’t a version of the well-known Kripkesteinian challenge: what makes 1002 rather than 1004 the right continuation of þ2 after 1000? We have here not the hyperbolic doubt of a radical skeptic, but real life cases ‘where the underlying directivities seem genuinely unfixed’ (Wilson [2006], p. 39). A second example is more fanciful, but still quite compelling for all that. Imagine the inhabitants of an isolated island; imagine they’ve never seen an airplane until one passes overhead and crashes in their midst. They might quite naturally regard it as a bird, regard themselves as having learned, unexpectedly, that the world includes a type of bird very different from the ordinary birds they’re familiar with, a great silver bird made of metal. Now imagine the story again, except that this time the plane crashes undetected and the islanders discover it in the jungle with the stranded crew taking shelter in the fuselage. This time, the islanders might reasonably regard it as a house, might well regard themselves as having discovered a new and unusual type of house. Is there any temptation here to think that one group is wrong and the other right? It seems clear that nothing in their pre-airplane concepts of ‘bird’ and ‘house’ or the corresponding worldly resemblances is enough

18 A similar theme turns up in Austin [1940], pp. 67–68: ‘Suppose that I live in harmony and friendship for four years with a cat: and then it delivers a philippic. We ask ourselves, perhaps, “Is it a real cat? or is it not a real cat?” “Either it is, or it is not, but we cannot be sure which.” Now actually, that is not so: neither “It is real cat” nor “it is not a real cat” fits the facts semantically: each is designed for other situations than this one . . . Ordinary language breaks down in extraordinary cases . . . no doubt an ideal language would not break down, whatever happened . . . In ordinary language . . . words fail us. If we talk as though an ordinary [language] must be like an ideal language, we shall misrepresent the facts’.

108 arealism to determine this, that either option is open to them as a consistent and defensible extension of the earlier concepts, that their choice is determined by sheer historical contingency. But notice: neither set of alternative [islanders] has any psychological reason to suspect that they have not followed the preestablished conceptual contents of their words ‘bird’ and ‘house’. ( Wilson [2006], p. 36)

Here we see the psychological force of tropospheric complacency in its purest form.19 Could it be that a similar brand of complacency is at work in the case of the Second Philosopher faced with pure mathematics? The line of thought we’ve been following contrasts two ways of describing the Second Philosopher: if she starts from scratch, slowly accumulating true statements, gradually adding to the stock of her ontology, then she will be faced with justifying the leap to the existence of sets and the truth of set theory; if she starts from a body of accepted doctrine, and employs her critical faculties to eliminate those entities and statements that aren’t well-supported, she will find no clear grounds on which to remove her seal of approval from sets and set theory. Depending on her starting point, the Second Philosopher comes to these opposing conclusions, in each case equally convinced of their faithfulness to original concepts of ‘evidence’, ‘object’, ‘truth’, ‘existence’, and so on, but the difference of starting point is surely a contingency no deeper than the accident of how the islanders happen to first encounter the airplane. Appealing as it may be,20 the analogy here is imperfect. Consider this from Wilson, speaking of islanders: The key ingredient in our fictional tale lies in its attention to the enlargement of linguistic application: specifically, to the latitude displayed when a usage previously confined to a limited application silently expands into some wider domain. . . . we can profitably picture these circumstances as representing a circumstance where we prolong our usage from one neighborhood of local application into another. In the [islander] case, two competing continuations are

19 See Wilson [2006], pp. 34–37, for more on the islanders, or [2007], pp. 186–188, for a somewhat more complete summary. 20 It appealed to me in [2007], pp. 385–386.

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available whereby the old usage might plausibly enlarge to take proper account of aircraft. (Wilson [2006], p. 37)

Other examples also involve such linguistic extensions into new arenas, in our case, the extension of ‘ice’ to unusual temperatures and pressures. To treat our case analogously, we’d imagine ourselves with an established use of ‘true’, ‘exists’, and the rest in empirical contexts, now facing the question of how to properly extend it to the new arena of pure mathematics—but this is to presuppose the bottom-up starting point, to beg the question against the top-down point of view, from which there’s no call for any extending, because pure mathematics is already regarded as true. So the problem doesn’t appear to lend itself to Wilsonian dissolution, at least not in quite this way. We’ve been exploring the idea that the answer to our question—which position on pure mathematics emerges from the second-philosophical approach, Thin Realism or Arealism?—may hinge on whether we characterize Second Philosophy bottom-up or top-down. Indeed it may well be that something like this does lie behind a not-insignificant portion of the rhetoric of this debate. But in fact I think that the bottom-up/topdown contrast is a red herring, that the true point at issue is both less sophisticated and more fundamental. To see this, consider again the topdown story. We imagine ourselves as we are, immersed in a vast system of beliefs, adding to it and subtracting from it as we go. Some of those subtractions result from the kind of critical examination Burgess and Rosen acknowledge: we root out superstition, improve our experimental techniques, reject what’s ill-confirmed. Just as our investigations of optics, of the structure of the eye, and of cognitive processing assure us that perception is largely reliable under certain conditions, our cleareyed evaluation of astrological evidence finds it lacking. Now what happens when we turn our critical attention on the methods of pure mathematics? Burgess and Rosen’s stereotypical anti-nominalist responds primarily to their stereotypical nominalist’s in-principle epistemological case against abstracta, and the Second Philosopher is in agreement here that this case is ineffective. But this won’t be the end of her examinations, any more than the rejection of some superficial case against perception ends her efforts to assess its reliability. After setting

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aside the stereotypical nominalist’s general concerns, the Second Philosopher will still want to know whether or not the considerations cited by Cantor, Dedekind, Zermelo, and the determinacy theorists properly count as evidence for the truth of their claims and the existence of sets. In response, Burgess and Rosen’s stereotypical anti-nominalist observes that ‘abstracta are customary and convenient for the mathematical (as well as other) sciences’ and decries ‘the marginalizing [of ] some sciences (the mathematical) and privileging [of ] others (the empirical)’ (Burgess and Rosen [1997], pp. 211–212). This may seem to rely on a more substantive characterization of ‘science’ than we’ve got—the introduction of the Second Philosopher as a character was largely motivated, after all, by the lack of such necessary and sufficient conditions—but I don’t think it must. My bottom-up characterization of the Second Philosopher’s inquiries has repeatedly appealed to a suggestive list: in her hands, perceptual belief and ordinary common sense gradually lead to the pursuit of studies from physics, chemistry, and astronomy, to botany, psychology, and linguistics. I see no reason the top-down characterization couldn’t use the same device, in which case the claim needn’t be that pure mathematics is a science, but that it belongs on the list with physics, botany, and the rest. Which brings us to this question: given that the methods of pure mathematics differ from those of the other entries on that list far more than the other entries differ amongst themselves, on what grounds does the top-down Second Philosopher ratify her previously-unexamined judgment that it belongs in this company, nonetheless? As far as I can determine, the only available answer rests on the basic idea that the roots of pure mathematics in physics, astronomy, engineering, the way it now intertwines with these subjects and with others on the list, gives us reason to regard it as one of them. And here, quite unexpectedly, we find ourselves speaking in the precisely same terms as the bottom-up Second Philosopher when she defends Thin Realism with the same sort of appeal to the physical roots of pure mathematics and its continued interconnections with the empirical sciences. So the same considerations arise either way, whether we’re contemplating an extension of our usage or evaluating the propriety of an existing usage; the contrast of

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bottom-up with top-down characterizations of Second Philosophy turns out to be irrelevant. Thus we’re returned to an embarrassingly simple question: does the history and current practice of pure mathematics qualify it as just another item on the list with physics, chemistry, biology, sociology, geology, and so on? In fact, this is just another way of posing our original question: do honorifics like ‘true’, ‘exist’, ‘evidence’, ‘confirm’—indisputably at home in those other studies—belong in pure mathematics as well? We’ve examined how pure mathematics arose out of our empirical study of the world, how it remains intensely important as a tool for that study, even in parts that weren’t expressly developed for that purpose; we’ve noted how it continues to be inspired by the descriptive and inferential needs of the natural and social sciences. If all this is taken to establish it as a body of truths, we’ve seen how the Thin Realist explicates the ground of that truth and how mathematical evidence manages to track it. But we’ve also seen how the Arealist gives a plausible account of pure mathematics as a deep and vital undertaking that happens not to aim at producing truths. What I want to suggest now, indeed at last to claim, is that a path has opened to a simpler Wilsonian dissolution: our central questions—is pure mathematics of-a-piece with physics, astronomy, psychology, and the rest? is it a body of truths? do its methods confirm its claims?—these questions have no more determinate answers than ‘is amorphous ice really ice?’ Once we understand the various ways in which water can solidify, how these processes are affected by temperature, pressure, and other factors, how the various structures generated are similar and how they’re different, there’s nothing more to know; we can reflect these facts in either way of speaking, or, to put it the other way around, neither way of speaking comes into conflict with the facts. Some version of tropospheric complacency—our tendency to overestimate the determinateness of our concepts—might well leave us convinced of the exclusive correctness of one or the other— it must be ice because it’s solid! it can’t be ice because it’s not crystalline!—but we’ve seen that this psychological confidence is often baseless, and also largely harmless.

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Likewise, once we understand how pure mathematics developed, how it now differs from empirical sciences,21 once we understand the many ways in which it remains intertwined with those sciences, how its methods work and what they are designed to track—once we understand all these things, what else do we need to know? Or better, what else is there to know? Just as robins are birds and bungalows are houses, physics and botany are sciences, but this isn’t enough to settle the status of downed airplanes and pure mathematics. Just as amorphous ice can be classified as ice or as ice-like, mathematics can be classified as science or as science-like—and nothing in the world makes one way of speaking right and the other wrong. If this is right, then we, more self-aware than the islanders, should recognize that there is no substantive fact to which our decision between Thin Realism and Arealism must answer. The application of ‘true’ and ‘exists’ to the case of pure mathematics isn’t forced upon us—as it would be if Thin Realism were right and Arealism wrong— nor is it forbidden—as it would be if Arealism were right and Thin Realism wrong. Rather, the two idioms are equally well-supported by precisely the same objective reality: those facts of mathematical depth. These facts are what matter, what make pure mathematics the distinctive discipline that it is, and that discipline is equally well described as the Thin Realist does or as the Arealist does. Once we see this, we can feel free to employ either mode of expression, as we choose—even to move back and forth between them at will. The proposal, then, comes to this: Thin Realism and Arealism are equally accurate, second-philosophical descriptions of the nature of pure mathematics. They are alternative ways of expressing the very same account of the objective facts that underlie mathematical practice. 21 In case there’s any lingering doubt, I’m not assuming we have a characterization of ‘science’ or ‘empirical science’; I’m using the term as short-hand for the familiar list of activities we’ve been talking about.

V Morals At this point, I imagine that some readers sympathetic to the general themes of Thin Realism may recoil at the barren world of Arealism, and likewise that some hard-nosed Arealists may disapprove the ontological excesses of Thin Realism. Sadly for such partisans, our secondphilosophical analysis gives them little encouragement. To secure a Thin Realism unsullied by association with Arealism, one would need a sound argument for the truth of mathematics, and we’ve seen strong reasons for thinking that the most promising avenue to such an argument—the effectiveness of mathematics in application—isn’t conclusive. Likewise, the staunch advocate of an Arealism decoupled from Thin Realism would need an equally effective, and apparently equally elusive, argument against the truth of mathematical claims. Though the analysis also suggests that those preferring one idiom to the other are free to speak as they choose without fear of error, it denies them any exclusive rights. In the end, then, the Second Philosopher is led to a position that is neither a whole-hearted endorsement of Thin Realism nor an unequivocal assent to Arealism: she recognizes instead that there is no substantive difference between the two, that either way of describing the underlying constraints of the practice is admissible. In this final chapter, I hope to draw a few morals from this conclusion, with a look at a traditional line of thought on mathematical objectivity, a glance back at Robust Realism, and finally, a return to the original problem of defending the axioms, in particular, to the interrelations of intrinsic and extrinsic justifications.

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1. Objectivity in mathematics One way of posing perhaps the central philosophical question about mathematics arises from the pure phenomenology of the practice, from what it feels like to do mathematics. Anything from solving a homework problem to proving a new theorem involves the immediate recognition that this is not an undertaking in which anything goes, in which we may freely follow our personal or collective whims; it is, rather, an objective undertaking par excellence. Part of the explanation for this objectivity lies in the inexorability of the various logical connections,1 but that can’t be the whole story; an if-thenist effort to treat mathematics simply as a matter of what follows from what will capture the claim that the Peano axioms logically imply 2þ2=4, that some set-theoretic axioms imply the fundamental theorem of calculus, but miss 2þ2=4 and the fundamental theorem themselves. Another way of putting this is to say that we don’t form our mathematical concepts or adopt our fundamental mathematical assumptions willy-nilly, that these practices are highly constrained.2 But, one asks (as this whole book has asked), by what? Various versions of Robust Realism constitute perhaps the most popular response to this challenge: what constrains our practices here, what makes our choices right or wrong, is a world of abstracta that we’re out to describe. This idea is nicely expressed by Moschovakis: The main point in favor of the realistic approach to mathematics is the instinctive certainty of most everybody who has ever tried to solve a problem that he is thinking about ‘real objects’, whether they are sets, numbers, or whatever. (Moschovakis [1980], p. 605)

Often enough, this sentiment is accompanied by a loose analogy between mathematics and natural science: We can reason about sets much as physicists reason about elementary particles or astronomers reason about stars. (Moschovakis [1980], p. 606)3

1

For discussion of the ground of logical truth, see [2007], Part III. This is the problem touched on in }III.2 in connection with Carnap, and in }IV.3 in connection with if-thenism. 3 Cf. Go¨del [1944], p. 128: ‘It seems to me that the assumption of such objects [‘classes and concepts . . . conceived as real objects . . . existing independently of our definitions and 2

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In keeping with our close observation of the experience itself, it seems only right to admit that mathematics is, if anything, more tightly constrained than the physical sciences. We tend to think that mathematics doesn’t just happen to be true, it has to be true. Those wishing to avoid Robust Realism for one reason or another often appeal to a sentiment famously expressed by Kreisel—or perhaps I should say ‘apparently expressed’, as no clear published source is known to me.4 Dummett’s paraphrase goes like this: What is important is not the existence of mathematical objects, but the objectivity of mathematical statements. (Dummett [1981], p. 508)

Putnam casts the idea in terms of realism: The question of realism, as Kreisel long ago put it, is the question of the objectivity of mathematics and not the question of the existence of mathematical objects. (Putnam [1975], p. 70)

Shapiro makes the connection explicit: . . . there are two different realist themes. The first is that mathematical objects exist independently of mathematicians, and their minds, languages, and so on. Call this realism in ontology. The second theme is that mathematical statements have objective truth-values independent of the minds, languages, conventions, and so forth, of mathematicians. Call this realism in truth-value. . . . The traditional battles in philosophy of mathematics focused on ontology. . . . Kreisel is often credited with shifting attention toward realism in truthvalue, proposing that the interesting and important questions are not over

constructions’] is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence. They are in the same sense necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions’. Also Go¨del [1964], p. 268: ‘the question of the objective existence of the objects of mathematical intuition . . . is an exact replica of the question of the objective existence of the outer world’. 4 Dummett [1978], p. xxviii, identifies the source as something ‘Kreisel remarked in a review of Wittgenstein’, but if the passage in question is the one pinpointed by Linnebo [2009]—namely Kreisel [1958], p. 138, footnote 1—it’s hard not to agree with Linnebo that it ‘is rather less memorable than Dummett’s paraphrase’. (The relevant portion of the note in question reads: ‘Incidentally, it should be noted that Wittgenstein argues against a notion of a mathematical object . . . but, at least in places . . . not against the objectivity of mathematics’.)

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mathematical objects, but over the objectivity of mathematical discourse. (Shapiro [1997], p. 37)5

On this approach, our mathematical activities are constrained not by an objective reality of mathematical objects, but by the objective truth or falsity of mathematical claims, which traces in turn to something other than an abstract ontology (say to modality, to mention just one prominent example6). I bring this up because it seems to me that the Second Philosopher has arrived at a position that does Kreisel one better. If Thin Realism and Arealism are equally accurate, second-philosophical descriptions of the nature of pure mathematics, just alternative ways of expressing the very same account of the objective facts that underlie mathematical practice, then we have here a form of objectivity in mathematics that doesn’t depend on the existence of mathematical objects or the truth of mathematical statements, or even on the non-existence of mathematical objects or the rejection of mathematical claims. This form of objectivity is, as you might say, post-metaphysical. Though it doesn’t involve truths about a mathematical ontology, it does involve an array of facts like the sort of thing we roughly express by saying that the concept of group opens up a lot of deep mathematics. Indeed, it seems fair to say that the objectivity of such facts is more robust than thin: an Evil Demon could deceive us wholesale about what follows from what or about where the deepest, most fruitful strains lie. Still, the coherence of the radical skeptical challenge isn’t enough to revive Benacerraf-style worries: though it’s hard to see why our set-theoretic methods should track the truth about the Robust Realist’s ontology, they’re clearly welldesigned (the Demon aside) to track set-theoretic depth. To return to the phenomenology from which we began, I suggest that this account of the objective underpinning of mathematics—the phenomenon of mathematical fruitfulness—is closer to the actual constraint experienced by mathematicians than any sense of ontology, epistemology or semantics; what presents itself to them is the depth, the importance, the illumination provided by a given mathematical

5 6

See also Shapiro [2000], p. 29, and [2005], p. 6. See e.g. Hellman [1989].

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concept, theorem, or method. A mathematician may blanch and stammer, unsure of himself, when confronted with questions of truth and existence, but on judgments of mathematical importance and depth he brims with conviction. For this reason alone, a philosophical position that puts this notion center stage should be worthy of our attention. I certainly don’t pretend to have given a satisfactory account of mathematical depth—what I’ve said remains uncomfortably metaphorical—nor do I imagine that giving such an account would be an easy task, but if questions of ontology and truth are red herrings, as the present analysis suggests, then I can at least hope to shift attention away from a misplaced worry and to focus it instead on the challenge of understanding the phenomenon that in fact drives the practice of pure mathematics.

2. Robust Realism revisited Still, faced with the interchangeability of Thin Realism and Arealism, I suspect some may be tempted to throw up their hands and contemplate a return to the familiar terrain of Robust Realism. But now, what exactly makes a realism Robust rather than Thin? A few such contrasts emerge from discussions in Chapter III: the Robust Realist demands more than the Thin Realist’s appeal to classical logic to defend the legitimacy of the continuum problem; presumably the Robust Realist, unlike the Thin Realist, would be inclined to regard a radical skeptical challenge to our mathematical knowledge as running parallel to the more familiar challenge to our knowledge of the external world. Underlying observations like these, we’ve isolated the fundamental diagnostic: the Robust Realist requires a non-trivial account of the reliability of mathematical methods, or more precisely, an account that goes beyond what mathematics itself tells us.7 The Thin Realist, on the other hand, thinks that mathematics itself gives us the whole story. 7 On this generous characterization, some versions of truth-value realism (to use Shapiro’s term) would count as Robust despite their lack of ontology, providing the truths in question are understood in such a way as to require a non-trivial epistemology. e.g., Hale ([1996]) argues that Hellman’s modalism faces a serious epistemological challenge.

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This diagnostic easily assigns the Robust label to views that appeal to ‘perception-like’ mathematical intuition. One flagrant example is my own set-theoretic realism ([1990]), where the intuition involved is explicitly linked to ordinary perception, but there are others: for example, Shapiro ([1997]) proposes an epistemology that begins in human pattern recognition. In addition to more naturalistic views like these, the range of possibilities also includes descendants of the sort of intellectual intuition one finds in Descartes’s clear and distinct perception. Some readings of Go¨del’s ‘something like a perception . . . of the objects of set theory’ ([1964], p. 268) fall in this category, though these remarks of Go¨del’s are notoriously difficult to interpret (see e.g. Parsons [1995]). One clear contemporary example is James Brown’s platonistic position: We can intuit mathematical objects and grasp mathematical truths. Mathematical entities can be ‘seen’ or ‘grasped’ with ‘the mind’s eye’ . . . The main idea is that we have a kind of access to the mathematical realm that is something like our perceptual access to the physical realm. . . . My bold conjecture . . . is this: Some ‘pictures’ are not really pictures, but rather are windows to Plato’s heaven. . . . As telescopes help the unaided eye, so some diagrams are instruments (rather than representations) which help the unaided mind’s eye. (Brown [1999], pp. 13, 39)

Another advocate of ‘the mind’s eye’ (Katz [1998], p. 39) is Jerrold Katz: Our reason is an appropriate instrument for determining how things must be in [the realm of abstract objects] . . . our rationalist epistemology . . . posits basic ratiocinative knowledge of evident properties of abstract objects. . . . we have a case of seeing—though not with our eyes . . . intuition . . . is . . . an immediate, i.e., noninferential, purely rational apprehension of the structure of an abstract object. (Katz [1998], pp. 39, 42, 43, 44)

In all these cases, the realism on offer is clearly Robust. And for all of them, the Second Philosopher’s concerns remain as presented in }II.4:8 the Robust Realist’s epistemological stories seem to deny the autonomous authority of set-theoretic methods, to require external 8 Though the discussion in }II.4 takes place under the temporary assumption that the Second Philosopher has good reason to regard set theory as a body of truths, these concerns don’t rest on that premise.

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supplementation for mathematical reasoning that looks entirely rational on its own terms; in addition, there seems to be a mis-match between the focus of so many of those set-theoretic justifications on the concrete mathematical advantages of the hypotheses in question and the Robust Realist picture of tracking a set-theoretic reality potentially as uncooperative with our theorizing as the physical world. Still, not all versions of Robust Realism rely on a direct perceptual or intellectual connection with the objects of mathematics, and I think it’s worth examining one particularly intriguing example. The account I have in mind derives from Tyler Burge’s fascinating reconstruction of Frege’s epistemology.9 Frege holds that arithmetical judgments ultimately rest on logic, but what founds logic itself? He saw the foundation as consisting of primitive logical truths, which may be used as axioms . . . they are in need of no justification from any other principles. (Burge [1998], p. 317) Frege accepted the traditional rationalist account of knowledge of the relevant primitive truths . . . This account . . . maintained that the basic truths . . . of logic are self-evident. (Burge [1992], p. 299)

In agreement with his traditional forebears, Frege ‘did not regard selfevidence as subjective or psychological obviousness’ (Burge [2005], p. 61), nor did he regard it as infallible. Instead He took logical laws to be objectively self-evident and to be subjectively obvious only to a mind that adequately understands them. (Burge [2005], pp. 61–62)

Such understanding can be incomplete, which is how we can come to err. Where Burge’s Frege differs dramatically from the tradition is in his rejection of the idea that full understanding consists in a kind of immediate quasi-perceptual insight. In analyzing inferences Frege is concerned that appeals to self-evidence not be allowed to obscure the formal character of the inferences, which can be found only by rigorous logical analysis. This analysis is arrived at not primarily by 9 See Burge [1984], [1992], and especially [1998], plus pp. 61–68 of the introduction to the collection, Burge [2005], in which these papers are reprinted. For present purposes, it doesn’t matter if this is really Frege’s position.

120 morals consulting unaided intuition, but by surveying inferential patterns in actual scientific-mathematical reasoning. (Burge [1998], p. 340)

So the bedrock here is ordinary mathematical (and scientific) practice, which merits our confidence because All these concepts have been developed in science and have proved their fruitfulness . . . fruitfulness is the acid test of concepts, and scientific workshops the true field of study for logic. (Frege [1880/1], p. 33, cited in Burge [1998], p. 340)

In the case of pure mathematics, Justification derives from . . . considerations of simplicity, duration, fruitfulness, and power in pure mathematical practice . . . Frege emphasizes that pure mathematical practice works. It produces a community of agreement through finding some systems ‘better’, ‘simpler’, ‘more enduring’. (Burge [1998], p. 341)

With mathematics itself as a starting point, Frege’s method was to reason to logical structure by observing patterns of judgments and patterns of inferences—and then postulating formal structures that would account for these patterns. While this method makes use of intuitions about deductive validity, it has at least as much kinship to theory construction as to intuitive mathematical reflection. (Burge [1998], pp. 340–341)

By this process, Frege uncovers his logical system; he Repeatedly appeals to advantages, to simplicity, and to the power of his axioms in producing proofs of widely recognized mathematical principles, as recommendations for his logical axioms. (Burge [1998], p. 339)

Here Burge notes that Frege’s use of this extrinsic style of axiom defense anticipates Zermelo’s. So far so good, but what’s become of the original idea of selfevidence? As Burge understands him, Frege regards these extrinsic considerations as a ‘secondary, fallible, non-demonstrative’ (Burge [1998], p. 355) form of evidence: Our justification for believing [mathematical propositions] is partly pragmatic— we find their place in mathematical practice secure through long usage, through advantages of simplicity, plausibility and fruitfulness, and through applications to

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non-mathematical domains. A deeper justification for believing in these propositions lies in finding their place in a logicist proof structure, by understanding the grounds within this structure that support them (if they are non-basic) or by understanding the self-evidently true basic principles. (Burge [1998], p. 345)

Thus we seem to be back where we started, in search of an account of this elusive self-evidence. Frege is well aware that his purportedly selfevident logical laws are often not subjectively obvious to his colleagues: We find Frege recommending to those who are skeptical of his logical system that they get to know it from the inside. He thinks that familiarity with the proofs themselves will engender more confidence in his basic principles. (Burge [1998], p. 339) The basic principles gain something from our seeing what obviously correct consequences they have and from recognizing ‘advantages’ of simplicity, sharpness and the like. (Burge [1998], p. 344)

But this is no help, because once again we’ve moved away from self-evidence and toward extrinsic supports: What do the basic principles gain from our seeing their consequences and our realizing their various ‘advantages’? If they are indeed axioms, they can be recognized as true ‘independently of other truths’. The sort of justification that derives from understanding them and recognizing their truth through this understanding needs no further justificatory help from reflecting on their consequences or the advantages of the system in which they are embedded. (Burge [1998], p. 345)

So once again, the question is how we gain this particular sort of sharpened understanding that allows us to recognize the self-evidence of the basic laws. This is the point at which the true originality of Burge’s Frege comes into focus. Full understanding is not a product of a typically rationalistic, perception-like, clear-and-distinct insight; instead Frege offers his own original and deep conception of what goes into adequate understanding. This conception rests on his method of finding logical structure through studying patterns of inference. Coming to an understanding of logical structure is necessary to full understanding of a thought. And understanding logical structure derives from

122 morals seeing what structures are most fruitful in accounting for the patterns of inference that we reflectively engage in. (Burge [1998], p. 354)

So in addition to their decidedly inferior evidential contribution, extrinsic considerations help us achieve the requisite understanding. They serve not to justify the first principles (except in a secondary, inductive way which will be overshadowed, given full understanding) but to engender full understanding of them. One might recognize the truth of axioms independently of other truths only in so far as one fully understands the axioms. But understanding them depends not only on understanding Frege’s elucidatory remarks about the interpretation of his symbols, but also on understanding their logical structure—their power to entail other truths, and their reasongiving priority. This latter understanding is not independent of reasoning that connects them to other truths. All full understanding involves discursive elements, even if recognition of the truth of the axiom is, given sufficient understanding, ‘immediate’. (Burge [1998], pp. 354–355)

Thus for Burge’s Frege, the ultimate force of the extrinsic merits of basic axioms is not justification, but elucidation; they enable us to appreciate those axioms’ self-evidence. Now what would a set-theoretic version of this idea look like? Our efforts to justify or discredit an axiom candidate would involve careful examination of the very same kinds of considerations we’ve been focused on since Chapter II. The difference would be that the various welcome outcomes laid out by Cantor, Dedekind, Zermelo, Go¨del, and the contemporary supporters of determinacy and large cardinals—all these would not be regarded as evidence, or at least not as primary evidence, but as clarifications of the true nature of our set-theoretic claims, clarifications that ultimately allow us to appreciate their truth ‘immediately’, to recognize their self-evidence. Because it offers a non-trivial account of the reliability of set-theoretic methods that goes beyond what set theory tells us, this qualifies as a version of Robust Realism, but a version without a direct quasi-perceptual or intellectual access to mathematical objects. The Second Philosopher originally balked at Robust Realism because it takes the ordinary justifications of set theory to be incomplete, in need of supplementation, and because the nature of the supplementation required by the most familiar varieties—especially those based on

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some sort of analogy between mathematics and natural science, between the set-theoretic universe and the physical world, between mathematical intuition and sense perception—because these most familiar varieties seemed likely to come into conflict with the apparently reasonable methods in actual use. In this the Burge-inspired variety is different: the methods favored by practicing set theorists are happily and completely embraced, so no question of potential conflict arises. Still, significant supplementation is required, in the form of that non-trivial epistemology: we need an account of how the process of coming to appreciate the extrinsic merits of an axiom candidate manages to bring our subjective sense of obviousness into line with actual self-evidence, and thus, with the truth. Whatever the significance of this extra epicycle in the Fregean context,10 it’s hard to see what work the machinery of ‘full understanding’ and ‘self-evidence’ does in the set-theoretic context, so perhaps the Second Philosopher can be forgiven for thinking that the various mathematical advantages she identifies show just what they appear to show: that the concept or hypothesis or method in question is getting at some deep mathematics. The post-metaphysical Objectivist described in the previous section would view the evidence this way, allowing for both Thin Realist and Arealist modes of expression. It remains intriguing that at least one approach to providing a more nuanced version of Robust Realism ends up producing a position that in some ways resembles the Second Philosopher’s. Still, in addition to the fundamental disagreement over the need for a non-trivial epistemology, the two also diverge in their understanding of the relative importance of intrinsic and extrinsic evidence—the topic to which we now turn.

3. More examples from set-theoretic practice Finally, after this multi-chaptered excursion into the metaphysics and epistemology of set theory, with our post-metaphysical Objectivism 10 Frege is concerned with logical truth. For the record, I take Robust Realism to be appropriate in that case, which is to say that I take a non-trivial metaphysics with a non-trivial epistemology to be in order (see [2007], Part III).

124 morals now firmly on the table at last, I’d like to return in this section and the next to the mechanics of set-theoretic practice, in particular, to the rough classification of set-theoretic justifications into intrinsic and extrinsic, first raised in }II.2.11 The hope is that what we’ve learned in the interim might help us understand the varieties and the comparative virtues of the two sorts of considerations. In by-now-familiar second-philosophical form, we should begin with some examples. i. Intrinsic justifications When a principle is defended in terms customarily classified as intrinsic, various descriptors typically appear: the principle is intuitive, selfevident, obvious; it’s part of the meaning of the word ‘set’; it’s implicit in the very concept of set; and so on. Of course, each of these glosses raises its own suite of questions. These days, I think that the most common idea is the last-mentioned—implicit in the concept of set— and that the concept of set intended is the iterative conception. This well-known picture of the set-theoretic universe was introduced by Zermelo ([1930]) and subsequently entered into the fabric of set-theoretic pedagogy and practice; following Go¨del’s description,12 sets are understood as generated in a series of stages (‘a set is something obtainable . . . by iterated application of the operation “set of ”’); at each stage we take every possible combination of the available elements, ‘no matter whether we can define it in a finite number of words (so that random sets are not excluded)’; this process is iterated indefinitely into the transfinite (so, for example, ‘the totality of sets obtained by finite iteration is considered to be itself a set and a basis for further applications of the operation “set of”’). Of course the language of ‘generation’ and ‘application’ is regarded as metaphorical, merely a colorful way of describing the structure of the set-theoretic universe, V.

11 Terms like ‘justification’ and ‘evidence’ may seem better suited to the Thin Realist idiom, but the Arealist can also speak of justifying the addition of a new axiom or of giving evidence for its suitability. Likewise, the Objectivist can justify or give evidence for the mathematical effectiveness of the hypothesis. 12 See Go¨del [1964], p. 259.

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Some of the simpler set-theoretic axioms follow unproblematically from this concept. For example, if a and b are sets, then there must be stages sa and sb, respectively, at which a and b first appear; if s is the later of these two stages, then the pair set {a, b} will appear at the next stage after s. Here we have a straightforward intrinsic justification for the Pairing Axiom. The Axiom of Infinity is also contained in the stipulation that the series of stages extends into the transfinite. A slightly less immediate, but still typically intrinsic case concerns small large cardinals. For example, an uncountable inaccessible cardinal Œ is one that towers over its predecessors in two ways: you can’t climb up to Œ in fewer than Œ steps (unlike Aø, which can be reached in only A0 steps: A0, A1, A2 , . . . ); and if a set has fewer than Œ elements, then so does its power set (unlike A1: A0 is less than A1, but its power set has at least A1 members). If Œ is inaccessible, then VŒ is a model of ZFC— it’s big enough to satisfy the axioms of Power Set and Replacement— but we know from Go¨del’s incompleteness theorem that no consistent system of this sort can prove its own consistency, so ZFC (if consistent) can’t prove the existence of an inaccessible cardinal.13 But the iterative conception presumably involves the idea that the succession of stage after stage shouldn’t stop at some fixed point, and asserting the existence of an inaccessible is a way of saying it keeps going on and on, even after it exhausts the operations of Power Set and Replacement. This constitutes an intrinsic argument for the first large cardinal axiom: the Axiom of Inaccessibles. Intrinsic justifications for larger large cardinals involve less direct and less transparent connections with the concept of set.14 To get a feel for this, consider one of the large cardinal notions central to determinacy theory: the supercompact cardinal. As with any large cardinal, positing a supercompact can be viewed as a way of assuring that the stages go on and on; for example, below any supercompact cardinal Œ there are Œ measurable cardinals, and below any measurable cardinal º, there are 13 As Tony Martin remarks, this can actually be seen without appeal to Go¨del’s theorem: suppose ZFC implies that there is an inaccessible; let Œ be the smallest, then VŒ is a model of ZFC + ‘there is no inaccessible’; contradiction. 14 A customary mark of a ‘large’ large cardinal is inconsistency with V=L. Indeed implying that V6¼L is usually counted in favor of these cardinals, but this strong extrinsic evidence is beside the point here.

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º inaccessible cardinals. But in the case of inaccessibles (and other small large cardinals), this connection with the indefinite extendibility of the sequence of stages is direct: we consider some processes, like applications of Power Set and Replacement, and posit that there is a stage after all those generated by these processes. In the case of supercompacts (and measurables), the connection is less direct: they imply the existence of cardinals that have this intrinsically supported feature. Attempts to derive supercompacts by more direct intrinsic reasoning often appeal to other general principles regarded as implicit in the concept of set, for example, resemblance.15 The leading idea here is that the class of stages is so large as to be extraordinarily rich; indeed there are so many of them that some will be indistinguishable from others. It’s then argued that this resemblance between two stages can be spelled out in terms of a non-trivial elementary embedding of one stage into another, and that from there one can generate a supercompact.16 Arguments along these lines are vulnerable at various points, and their connection with the concept of set is more tenuous than in the earlier examples.17 Much of the case for supercompacts is actually extrinsic, arising from their role in determinacy theory. ii. Extrinsic justifications As we’ve seen (in }II.2.iii), the first extrinsic justification for a settheoretic axiom was Zermelo’s case for the Axiom of Choice, based on its effectiveness for solving problems in set theory and analysis. Zermelo himself lists seven such problems, but subsequent progress has uncovered many, many more.18 Those versed in set theory need only contemplate the theory of transfinite cardinals without Choice—to 15 This principle is sometimes linked to the more familiar idea of reflection—the settheoretic universe V is too large to be fully specifiable, so any description true of V is already true of some stage, VÆ (see Martin [1976], pp. 85–86, Solovay, Reinhardt, and Kanamori [1978], p. 75)—but the connection is somewhat attenuated. Koellner [2009b] argues that no true reflection principle can do this job. 16 See [1988], pp. 750–752, for a sketch of one such argument, with references. One might also express supercompactness directly in terms of an elementary embedding, as Magidor has shown (see Kanamori [2003], p. 302). 17 Cf. Martin [1976], p. 86: ‘as the axioms become stronger their link with the basic principle becomes more and more tenuous’. 18 See the definitive Moore [1982].

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begin with, they needn’t be linearly ordered—for a taste of the kinds of difficulties that arise in a Choice-less context. Choice in its many forms is now fundamental in analysis, topology, algebra, and other branches of the subject. A more recent example comes from Martin’s experience in the early days of determinacy hypotheses. In 1968, he proved a theorem of the form: if a set of Turing degrees is determined, then either it or its complement contains a cone. It’s not important for our purposes what these terms mean; what matters is Martin’s reaction: When I discovered the Cone Lemma, I became very excited. I was certain that I was about to achieve some notoriety within set theory by deducing a contradiction from AD. In fact I was pretty sure of refuting Borel determinacy. I had spent the preceding five years as a recursion theorist, and I knew many sets of degrees. I started checking them out, confident that one of them would . . . give me my contradiction. But this did not happen. For each set I considered, it was not hard to prove, from the standard ZFC axioms, that it or its complement contained a cone. (Martin [1998], p. 224)

Martin naturally sees this as a remarkable vindication: I take it to be intuitively clear that we have here an example of prediction and confirmation . . . The example seems fully analogous to striking instances of prediction and confirmation in empirical sciences. (Martin [1998], pp. 224–225)

And the confirmation arising from the prediction is more compelling for its initial implausibility. As sketched in }II.2.iv, the fully-developed case for determinacy is multifaceted and quite complex, and as we now note, largely extrinsic. The first component is the rich theory of projective sets of reals that arises from ADL(ℝ): the structure theory provable from ZFC for the early projective levels can be generalized to the full projective hierarchy. For example, with determinacy, an important classical property of co-analytic sets, first established by Luzin and Sierpin´ski in the 1920s, can be generalized to the higher levels of the projective hierarchy. The portion of the new determinacy-based proof that applies to co-analytic sets alone depends only on the determinacy of open sets, and this much determinacy is provable in ZFC, so we have here an example of what Go¨del calls a ‘verifiable’ consequence,

128 morals demonstrable without the new axiom, whose [proof] with the help of the new axiom, however, [is] considerably simpler and easier to discover, and [makes] it possible to contract into one proof many different proofs. (Go¨del [1964], p. 261)

Determinacy also yields ‘powerful methods for solving problems’ (op. cit.), for example, settling the questions Luzin thought would never be answered: all projective sets are Lebesgue-measurable and no uncountable projective set lacks a perfect subset. Furthermore, the theory based on ADL(ℝ) is clearly preferable to the alternative generated by V=L. As Steel notes, the latter lacks appeal for an analyst, by which I take him to mean that it lacks what our second-philosophical Objectivist would describe as the mathematical depth of its rival. And, in any case, anything derivable from V=L can be recovered in the determinacy context as part of the theory of L. In sum, then, this determinacy-based structure theory would certainly seem to qualify, to quote Go¨del again, as shedding so much light upon a whole field . . . that, no matter whether or not [it is] intrinsically necessary, [it] would have to be accepted at least in the same sense as any well-established physical theory. (op. cit.)

So this first component is paradigmatically extrinsic. The work that led to the remarkable interconnections between determinacy theory and large cardinal theory that make up the second component was originally motivated by the hope of generating intrinsic evidence: Because of the richness and coherence of its consequences, one would like to derive PD itself from more fundamental principles . . . whose justification is more direct. (Martin and Steel [1989], p. 72) The success of determinacy axioms led to a revised program for doing descriptive set theory based on large cardinal axioms: Show that large cardinal axioms imply determinacy axioms. (Martin and Steel [1988], p. 6582)

Martin and Steel went on, of course, to do just that: building on work of Foreman, Magidor, Shelah, and Woodin, they derived PD from the existence of a supercompact cardinal; Woodin then improved this result to the full ADL(ℝ).19 In this way, such intrinsic evidence as 19

The optimal hypothesis is weaker: countably many Woodin cardinals with a measurable above. See Kanamori [2003], }32.

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there is for supercompact cardinals transfers to determinacy, but as noted a moment ago, some support for supercompacts comes in the form of extrinsic evidence accruing from their implying determinacy! I think it’s fair to say that the gain in intrinsic justification for determinacy is significantly supplemented, and perhaps even somewhat overshadowed, by the extrinsic support accruing to both determinacy and large cardinals as a result of their interconnections. After the dramatic inference from large cardinals to determinacy, Woodin established a reverse implication: ADL(ℝ) implies the existence of inner models with large cardinals.20 Considering that determinacy and large cardinals arose in the course of such disparate, apparently unrelated contexts of mathematical inquiry, this ultimate equivalence is quite surprising and impressive: This sort of convergence of conceptually distinct domains is striking and unlikely to be an accident. (Koellner [2006], p. 174)

Our second-philosophical Objectivist understands the situation this way: the fact that two apparently fruitful mathematical themes turn out to coincide makes it all the more likely that they’re tracking a genuine strain of mathematical depth. The third and fourth components of the case for determinacy resemble each other in that they’re both predicated on attractive theoretical virtues. In the third component, the relevant theoretical virtue is high consistency strength: assuming we want our theory of sets to be as strong as possible, we will want theories of high consistency strength; all reasonable theories with consistency strength at least that of ADL(ℝ) actually imply ADL(ℝ); therefore our best theory of sets can be expected to include determinacy. This reasoning is cogent if we have good reason to want our theory to be as strong as possible; I think this can be traced to the foundational aspirations of set theory. Similarly, we want our set theory to be as decisive as possible, to settle as many questions as possible, so generic completeness (that is, immunity from independence arguments based on forcing) is a welcome feature; this drives the fourth component. The two cases are structurally 20

Specifically, an inner model with countably many Woodin cardinals. See Koellner [2006], p. 170.

130 morals different—in one determinacy itself enjoys the relevant theoretical virtue; in the other, determinacy is implied by virtuous theories— but they both turn on the same sort of fulcrum: we want our theory of sets to have certain features and we have good mathematical reasons for these preferences. At this point, even with this limited sampling, it should be clear that a number of different kinds of justifications are being collected together as ‘extrinsic’. We have some idea of what’s intended by ‘intrinsic’—here we’ve focused on ‘implicit in the concept of set’— but ‘extrinsic’ is being applied willy-nilly to any compelling justification that isn’t clearly intrinsic. I won’t attempt a complete taxonomy here, but a few observations are in order. Notice, for example, that a Robust Realist might find the considerations arising from Martin’s Cone Lemma especially convincing; particularly a Robust Realist impressed by an analogy between mathematics and natural science will see here a clear set-theoretic counterpart to the undoubtedly legitimate confirmation of a scientific hypothesis by successful predictions. To take the sharpest contrast, the purported justifications based on welcome theoretical features will be problematic from this same point of view. As remarked in }II.4, the fact that we prefer high consistency strength or generic completeness gives the Robust Realist no reason at all to suppose that such theories are more likely to be true.21 So the metaphysical perspective of Robust Realism requires careful distinctions between various purported justifications we’ve counted here as extrinsic: some are legitimate and some are not. The matter looks quite different to our second-philosophical Objectivist. Martin’s results clearly support the consistency of determinacy hypotheses; indeed it was the hypothesis of inconsistency that Martin actually entertained and disconfirmed. But there is more to Martin’s accomplishment than this:

21 As noted in [2007], pp. 358–359, 365–366, the case against V=L presented in [1997], }III.6, has a similar character: V=L is rejected because it’s restrictive, where restrictive theories are avoided to further set theory’s foundational goal. Here the Robust Realist should object that even if we don’t like restrictive theories, and for good reason, this doesn’t show that they’re likely to be false.

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What was predicted, moreover, was not just individual assertions. Though there had been much work on the structure of the degrees, no attention at all had been paid to the notion of a cone . . . Afterwards cones and calculations of ‘vertices’ of cones became significant in degree theory. In determinacy theory, the Cone Lemma became an important tool. What was predicted by the Cone Lemma was thus a whole phenomenon, not merely isolated facts. (Martin [1998], p. 224)

I think it’s fair to say that by this point the notion of prediction at work has become somewhat strained. The thrust of this passage is that viewing the theory of degrees from the perspective of determinacy hypotheses helps to isolate a mathematically fruitful concept—the cone—and in the process, sheds light on old questions of recursion theory, providing new proofs and situating them in a broader context, much as these same hypotheses illuminate classical descriptive set theory. This, too, our Objectivist applauds as further evidence that determinacy hypotheses are fruitful, are effective, and probably track something deep. The upshot is that for the Objectivist, and likewise for the Thin Realist and the Arealist, there is no call to rule against some classes of extrinsic justification on principle. This isn’t to say that all purported extrinsic justifications are sound or successful, but at least those we’ve been examining do aim at the right targets: effective, productive, important mathematics.

4. Intrinsic versus extrinsic One unmistakable theme that runs through almost all discussions of set-theoretic evidence is a strong preference for intrinsic over extrinsic justifications. I’d like to conclude here with a brief exploration of this preference, to see if our investigations might cast some light. Let me begin with a look at the motivations of those who reject extrinsic justifications altogether. Consider for illustration Solomon Feferman’s discussions of CH and the search for new axioms.22 Though he doesn’t explicitly rule out 22

See e.g. Feferman [1999], [2000].

132 morals extrinsic considerations, Feferman opens with the Oxford English Dictionary’s definition of ‘axiom’ as used in logic and mathematics . . . ‘a self-evident proposition requiring no formal demonstration to prove its truth, but received and assented to as soon as mentioned’. (Feferman [2000], p. 402, [1999], p. 100)

He remarks that I think it’s fair to say that something like this definition is the first thing we have in mind when we speak of axioms for mathematics . . . It’s surprising how far the meaning of axiom has become stretched from the ideal sense in practice. (op. cit.)

So for example, When the working mathematician speaks of axioms, he or she usually means those for some particular part of mathematics such as groups, rings, vector spaces, topological spaces, Hilbert spaces, and so on. (Feferman [2000], p. 403, [1999], p. 100)

These Feferman calls ‘structural axioms’, noting that they ‘have nothing to do with self-evident propositions’ (op. cit.). Of course Feferman introduces this idea of structural axioms in order to contrast them with something else: Axioms for such fundamental concepts as number, set and function that underlie all mathematical concepts. (Feferman [2000], p. 403, [1999], p. 100)

These he calls ‘foundational axioms’, whose role is ‘in the end to justify’ the ordinary practice of mathematics (Feferman [1999], p. 100). Here we require axioms in the OED’s ideal sense: statements whose own truth is self-evident, on which all other truths are based. Mary Tiles makes this connection explicit: extrinsic justifications are inappropriate to the conception of set theory as providing a logical foundation for mathematics. To claim this status for set theory it is necessary to claim an independent and intrinsic justification for the assertion of set-theoretic axioms. It would be circular indeed to justify the logical foundations by appeal to their logical consequences, i.e., by appeal to the propositions for which they are going to provide the foundations. (Tiles [1989], p. 208)

For this purpose, only intrinsic evidence will do.

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Whatever the attractions of this strong sort of foundational theory, I think it’s clear that set theorists today are not in the business of trying to provide one. We’ve seen that Zermelo didn’t insist that his axioms provide an entirely independent justification for classical mathematics; he appealed explicitly to their effectiveness for ‘productive science’, as did Go¨del and many others.23 Clearly no purely foundationalist epistemology is intended. We’ve seen (in Chapter I) that by the late 19th century, there was considerable confusion about which mathematical structures could safely be assumed to exist; this uncertainty ran through geometry with its newfound points at infinity and points with complex coordinates, through analysis with its pathological functions, through differential equations with its question of when they could be trusted to have solutions,24 and of course, as we’ve seen, through set theory itself. The goal of the axiomatization of set theory was to remove these ontological uncertainties, to provide a single ontological framework for classical mathematics. Obviously set theory doesn’t provide a foundation in certain truths, nor does it provide, given Go¨del’s incompleteness theorems, what MacLane calls ‘a security blanket’ (MacLane [1986], p. 406) against the risk of inconsistency, but despite these epistemic shortcomings, it still plays a profound unifying role, bringing all mathematical structures together in a single arena and codifying the fundamental assumptions of mathematical proof. There’s no doubting the mathematical value of a foundation in this sense.25 So this blanket rejection of extrinsic justifications rests on an overly strong understanding of the type of foundation set theory is intended to provide, but I suspect that a different source of disapproval is lurking nearby. As noted in passing in }III.5, set theory is sometimes regarded as an essentially Platonistic theory: Feferman, for example, refers to ‘the Platonistic philosophy of mathematics that is currently claimed

23 Cf. Russell [1907] on his ‘regressive method’. Echoing Tiles, Potter [2004], p. 35, writes, ‘the regressive method . . . seems powerless to justify a theory that aspires to be epistemically foundational’. 24 See Kline [1972], pp. 699–707. 25 See [1997], }I.2, for discussion and references.

134 morals to justify set theory’ (Feferman [1999], pp. 109–110). Given that Go¨del’s advocacy of extrinsic justifications sometimes intertwines with his Robust Realism—for example, when he speaks of axioms that ‘would have to be accepted . . . in the same sense as any wellestablished physical theory’ (Go¨del [1964], p. 261)—one might come to think that the viability of extrinsic justifications is grounded in some such metaphysics: set theory aims to describe the objective features of an independently existing world of mathematical abstracta; extrinsic justifications in set theory are understood to function on analogy with their counterparts in natural science. At that point, any and all objections to Platonism would apparently become objections to the use of extrinsic justifications, and if one further holds, as Feferman does, that ‘Platonism . . . is thoroughly unsatisfactory’ (op. cit.), then extrinsic justifications would fall with it.26 But we’ve seen that this line of thought is misguided: Robust Realism is not in fact well-suited to a defense of many extrinsic justifications; for that purpose, Thin Realism, Arealism and Objectivism are all more appropriate starting points. Still, even if we agree that many wholesale rejections of extrinsic justifications are ill-motivated (resting on an overly strong foundationalism or a mistaken association with Robust Realism) and ill-advised (disallowing productive avenues to mathematical fruitfulness), there remains a lingering sense that they are second-best, that intrinsic justifications are the gold standard against which extrinsic justifications are measured and often found wanting. In these final pages, I’d like to float the heretical suggestion that in fact intrinsic justifications are secondary to the extrinsic. As a first pass at what I’m after here, let’s return to the case of groups. We saw in }I.2 that despite Galois’s work around 1830 and Cayley’s formal definition around 1850, the notion of group only caught on in the 1870s, when there were finally enough known examples for it to begin to do real mathematical work. The detailed histories of this development show how the definition of a group evolved during this period.27 To take one example, Cayley’s 1854 definition required 26

Feferman doesn’t explicitly link Platonism to extrinsic justifications in particular. e.g., see Wussing [1969], Stillwell [2002], chapter 19, or the quick survey in [2007], }IV.3. 27

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associativity, an identity, and cancellation laws (if xy = xz, then y=z). As Cauchy had noted in his earlier work on the special case of finite permutation groups, the existence of inverses follows (for some n and m, xn = xm; if n<m, then by cancellation, 1=xm-n; so x(m-n)-1 is the inverse of x). When infinite groups entered the picture, largely from geometry, this inference broke down and inverses were no longer guaranteed; an explicit axiom requiring them was eventually added by Dyck in 1883. In this way, the concept of group was gradually shaped so as to best serve a particular set of mathematical goals. Now imagine that we’re back in the 1850s, with Cayley and the rest, and our definition of group posits cancellation laws but not inverses. Along come the geometers with their infinite groups, and Dyck proposes a change in the definition to require inverses directly. Given the goals of group theory, this is entirely reasonable; in other words, there’s an impeccable means-ends extrinsic justification for the change. Would it make any sense at this point to object that inverses aren’t part of our concept of group, that this intrinsic consideration trumps the extrinsic? I trust we’ll agree that it wouldn’t. Now let’s try the analogous thought experiment for set theory. Suppose someone proposes a new set-theoretic axiom that’s mathematically fruitful in the sorts of ways we’ve been considering, but which doesn’t follow from, or perhaps even appears to conflict with, our current concept of set. Would it be reasonable to reject the axiom on intrinsic grounds? This scenario may sound far-fetched, but at least one thread in the initial negative reaction to the Axiom of Choice had this character: insofar as the logical conception of a set as an extension was in the air, the existence of a choice set appeared problematic; there’s no specifiable property that picks out all and only its members. The Axiom was opposed on this score, but in the end, its extrinsic merits carried the day.28 So again perhaps the case of sets isn’t so different from that of groups after all.

28 See [1988], pp. 487–489, [1990], pp. 117–124, [1997], pp. 54–57, for discussion and references, or better Moore [1982]. Now that the so-called logical concept of collection—the extension of a property—has been replaced by the iterative concept of set, some regard the Axiom of Choice as intrinsically justified (e.g., see Shoenfield [1977], pp. 335–336; for the contrary view, see Boolos [1971], pp. 28–29).

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To push this one step further, consider for a moment why we value intrinsic supports, what it is we gain from them. First I think we’d all agree that it’s extremely useful to have a workable heuristic picture of the sort of thing we’re investigating mathematically. At a minimum, it can give us confidence in the consistency of our theory, especially if the picture is sufficiently detailed; we feel, reasonably enough, that an internal conflict would turn up there.29 More dramatically, a viable concept so effectively guides our thinking that, given the choice, we’re reluctant even to pursue a theory that lacks one (for example, Quine’s NF30). And once we have a concept that’s mathematically fruitful, it’s rational policy to exploit it further, to try to extend it in ways that seem ‘natural’ or harmonious with its leading intuitions, in hope of further gains;31 thus we explore Large Cardinal Axioms as a way of exploiting the ideas behind the iterative conception. What’s striking is that all these perfectly reasonable ways of proceeding are in fact grounded in their promise of leading to the realization of more of our mathematical goals, to the discovery of more fruitful concepts and theories, to the production of more deep mathematics. Ultimately we aim for consistent theories, for effective ways of organizing and extending our mathematical thinking, for useful heuristics for generating productive new hypotheses, and so on; intrinsic considerations are valuable, but only insofar as they correlate with these extrinsic payoffs. This suggests that the importance of intrinsic considerations is merely instrumental, that the fundamental justificatory force is all extrinsic. This casts serious doubt on the common opinion that intrinsic justifications are the grand aristocracy and extrinsic justifications the poor cousins. The truth may well be the reverse! 29

Cf. e.g. Kanamori [2003], p. 264: ‘The clear internal structure and striking global coherence of inner models of measurability . . . provide a forceful argument for the consistency of the theory: ZFC plus there is a measurable cardinal’. 30 See Fraenkel, Bar-Hillel and Levy [1973], p. 164: ‘there is no mental image of set theory which leads to [NF] and lends it credibility’. 31 This might be called ‘heuristic rationality’: extending or generalizing as a way of generating new concepts or hypotheses that could turn out to be justified in the stronger sense of successfully tracking depth. But the fact that a new concept or hypothesis arises from some rational heuristic isn’t enough to show that it’s justified; it’s just enough to make it worth a try. See [2001], }I.

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In any case, this is what the Objectivist would tell us. For her, the be-all and end-all of mathematics isn’t a remote metaphysics that we access through some rational faculty, but the entirely palpable facts of mathematical depth. She seeks concepts and assumptions that illuminate previously intractable problems, that reveal surprising interconnections, that open up new areas of mathematical understanding, and she does so using the familiar methods of mathematics itself, all carefully honed for just this undertaking. From this point of view, being part of our current concept only matters insofar as that concept is well-chosen; presented with a fruitful avenue that runs counter to current thinking, the Objectivist will happily throw the old concept over and embrace the new without regret. Indeed for her the often vexed distinction between finding out more about an existing concept and changing to a new one matters not at all. What does matter, all that really matters, is the fruitfulness and promise of the mathematics itself.

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Index abstracta 3, 8–9, 22n, 23, 27, 29, 31, 35–36, 59, 62, 69n, 76–77, 86, 90–92, 96–97, 99, 102, 104–105, 109–110, 114, 116, 118, 134 abstract algebra 6, 43–45, 52, 56, 73, 82, 85, 89, 127; see also, group ADL(ℝ), see determinacy applied mathematics Chapter I, 40–41, 53–55, 70, 85–86, 87n, }IV.2, 96, 101, 110–112, 113 Archimedes 15n Arealism Chapter IV, 113, 116–117, 123, 124n, 131, 134 Arfken, George 25n Aristotle 4n arithmetic 22n, 24n, 28, 32, 34, 38, 39n, 45, 52–53, 114, 119 atomic theory 18–20, 22–24, 26n, 29–30, 38, 40 Austin, J. L. 107n Avigad, Jeremy x, 43–44 Awodey, Steven 32n axioms 14, 31–32, 81, 131–132, 135 Axiom of Choice 32–36, 45–48, 52, 80, 87, 94, 126–127, 135 Axiom of Constructibility (V=L) 49, 52–53, 85, 125n, 128, 130n Axiom of Determinacy, see determinacy Axiom of Infinity 125 for logic 119–122 for set theory ix, 1, 22n, 32–34, 41, 45–47, 52, 56, 58–59, 69, 81, 83, 99–100, 113–114, 122–123, }}V.3-V.4; see also ZFC and entries for particular axioms large cardinal axioms, see large cardinals Pairing Axiom 125 Axiom of Power Set 125–126 Axiom of Replacement 125–126 Babbage, Charles 11n Balaguer, Mark 91n Banach, Stefan, see paradoxes, Banach-Tarski

Bar-Hillel, Yeoshua 35n, 136n Benacerraf, Paul 5n, 57, 59, 63, 116 Bernstein, Felix 46 Boltzmann, Ludwig 19, 20, 24, 25 Boolos, George 135n Borel, E´mile 35n Borel sets 48n, 127 Boscovich, Roger 18n Boyle, Robert 18n Bremmer, H. 37n Brown, James 118 Brown, Theodore 23n, 25n Burge, Tyler 119–123 Burgess, John x, 39n, 54n, 57, 60n, 62, 67n, 73n, 76, 97–98, 99n, 103n, 104–105, 109–110 Calinger, Ronald 13n Cantor, Georg 6, 41–46, 53, 56–58, 68, 72, 82, 85–87, 88–89, 98, 100–102, 104–105, 110, 122 Carnap, Rudolf 5, 66–69, 99n, 114n Carnot, Sadi 19–20 Caro, Paul 106 Clausius, Rudolf 19 category theory 34n Cauchy, Augustin-Louis 135 Cayley, Arthur 7, 134–135 CH, see continuum hypothesis Clarke-Doane, Justin x Cohen, Bernard 10n, 12, 18n Cone Lemma 127, 130–131 consistency strength 50–51, 58, 80–81, 87, 129–130 constructivism 33, 43, 45, 86–87 continuum hypothesis (fluid dynamics) 21–22, 29, 56n continuum hypothesis (set theory, CH) 21n, 37, 48n, 56, 59, 62–64, 66, 69n, 75, 77, 80, 81n, 89, 117, 131 D’Alembert, Jean-Baptiste 31n Dalton, John 18 Dauben, Joseph 41n

148 index Dedekind, Richard 7, 31–32, 43–46, 52–53, 58, 68–69, 72, 82, 85–87, 88–89, 98, 100–105, 110, 122 delta function 36, 37n, 73 depth see mathematical depth Descartes, Rene´ 4, 6, 9–10, 11–12, 18, 40n, 118 determinacy }II.1.iv, 52–53, 58, 80–81, 85, 100–102, 105, 110, 122, 125–131 Dirac, Paul 36, 37n Duhem, Pierre 19, 20n-21n, 26, 28 Dummett, Michael 115 Dyck, Walther von 135

general relativity 8, 22, 27–28, 90, 103 geometry 3, 7–9, 11–14, 27, 31–32, 44, 58, 65, 78–79, 81, 133 Gibbs, J. Willard 20 Glass, Matthew x, 82n Go¨del, Kurt ix, 5, 32n, 47, 49, 51, 56–57, 60, 62, 64, 72n, 114n-115n, 118, 122, 124–125, 127–128, 133–134 Goodman, Nelson 97 Grossman, Marcel 8 group 7, 28, 37, 53, 79–80, 89, 95, 116, 132, 134–135; see also abstract algebra Guicciardini, Niccolo 11n

Ebbinghaus, Heinz-Dieter 45n Eklof, Paul 37n Einstein, Albert 8, 19n, 20 Emch, Ge´rard 22n, 24, 25n, 37n Engel, Thomas 23n, 25n, 29n, 30 epistemology 1, 5, 57–59, 62–64, 66, 68, 69n, 70n, 71, }III.3, 77, 86–87, 97, 105, 109, }V.2, 116, 123, 133 Ernst, Michael x Euler, Leonhard 6, 12–14, 15n, 19, 21–22, 30, 31n, 36, 53, 90n, 93–96 existence ix, 1, 54–56, 58, 60–62, 65n, 67–68, 70, 73–75, 77, 82–83, 86, 89, 97–98, }}IV.4-IV.5, 114–117, 134; see also, ontology extrinsic justification 47, 50, 56, 69, 79, 83, 89, 113, 119–123, }}V.3-V.4

Hale, Bob 117n Hall, A. Rupert 10n Halliday, David 25n Hausdorff, Felix 35n Heaviside, Oliver 37n Heidegger, Martin 5 Heine, Eduard 44 Heis, Jeremy x Hellman, Geoffrey 57, 116n, 117n Hilbert, David 31, 45, 132 Huygens, Christiaan 4, 10n

Feferman, Solomon 87n, 131–134 Ferreiro´s, Jose´ 41n, 42, 43n, 44–45 Feynman, Richard 21–22 fictionalism 26n, 29, 91n, 98–99, 102, 103n Field, Hartry 36n, 98 First Philosophy 40, 65–66 fluid dynamics 21–22, 29, 56n Foreman, Matthew 50, 51, 128 formalism 99 foundations 13, 31–34, 44–47, 51–52, 56, 58, 63n, 73, 79, 89, 119, 129, 130n, 132–133 Fourier, Joseph 15–19, 26n Fraenkel, Abraham 33, 35n, 44, 136n Frege, Gottlob 119–123 function 6, 31–34, 37n, 45, 104, 132–133 Galileo Galilei 3–4, 9, 11, 14–15, 18–19, 30n Galois, E´variste 7, 134 Gauss, Carl Friedrich 8, 11n, 15n

ice 106–107, 109, 111–112 idealism 65–66, 71 Idhe, Aaron 18n, 19 if-thenism 32n, 99, 114 indispensability argument ix, 90n instrumentalism, see fictionalism intrinsic justification 47, 49–50, 52n, 56, 69, 79, 89, 113, 123, }}V.3-V.4 islanders 107–109, 112 iterative conception of set 124–125, 135n, 136 Jech, Thomas 36n Jenson, Ronald 50n Kant, Immanuel 64–66, 68, 71–72, 75–76, 78–79, 81 Kanamori, Akihiro 36n, 48n, 50n, 80n, 126n, 128n, 136n Katz, Jerrold 118 Kennedy, Juliette x kinetic theory, see atomic theory Kline, Morris 4, 6, 8, 9n, 10n, 11–14, 15n, 31, 133n Koellner, Peter x, 49n, 50–1, 63n, 126n, 129 Ko¨nig, Julius 46

index 149 Kreisel, Georg 115–116 Kummer, Ernst 43 Lacroix, Sylvestre Franc¸ois 13 Laplace, Pierre-Simon 16–17, 18n, 21 large cardinals 36, 48–51, 80–81, 100, 122, 125–126, 128–129, 136 inaccessible cardinal 51n, 125–126 measurable cardinal 36n, 80, 82, 125–126, 128n, 136n Shelah cardinal 80 supercompact cardinal 36n, 125–126, 128–129 superstrong cardinal 36n, 80 Woodin cardinal 51n, 80, 128n, 129n Lavoisier, Antoine 18 Lebesgue measure 35–37, 48n, 128 Leibniz, Gottfried Wilhelm 10, 11n, 12–13 Levy, Azriel 35n, 136n Lewis, David 102 Linnebo, ystein 115n Liston, Michael x, 60n, 65n, 68n, 90–96 Liu, Chuang 22n, 24, 25n logic 22n, 38, 39n, 63–64, 66, 69, 78–80, 83, 98–99, 101, 114, 117, 119–123 Luzin, Nikolai 127–128 Mach, Ernst 20n Machamer, Peter 4, 15n MacLane, Saunders 133 Malament, David x, 23n Magidor, Menachem 50, 126n, 128 Marino, Patricia x, 98n Martin, D. A. x, 48n, 50, 125n, 126n, 127–128, 130–131 mathematical depth 79–83, 86, 100, 112, 116–117, 128–129, 131, 134–137 mathematical nihilism 82n Maxwell, James Clerk 19 McLarty, Colin x, 44n McNulty, Bennett x McQuarrie, Donald 23n, 25n, 29 miracle of applied mathematics 95–96 Moore, Gregory H. 32–33, 34n, 45n, 126n, 135n Moschovakis, Yiannis 48–50, 114 Newton, Isaac 4, 6, 9–15, 18–19, 26n, 27, 30, 36, 53, 84n naturalism ix, 39–40 nihilism, see mathematical nihilism

Noether, Emmy 44 nominalism 70n, 96–99, 102, 105, 109–110 Nye, Mary Jo 19, 20n, 21 objectivity 1, 46, 56, 58–59, 60, 62, 64, 66, }III.4, 100, 103–104, 107, 112, 113, }V.1, 119, 134 Objectivism (post-metaphysical) }V.1, 123, 124n, 128–131, 134, 137 ontology 1, 36, 53, 66–68, 69n, 76, 86, 108, 113, 115–117; see also existence Packman, A. J. x Pais, Abraham 37n paradoxes 32–33, 46 Banach-Tarski 35–36, 93–94 Parsons, Charles 32n, 64, 118 Partington, J. R. 18n PD (projective determinacy), see determinacy Perrin, Jean 19–20 Plato 3, 5, 118 Platonism 56, 65n, 118, 133–134 Poincare´, Henri 20–21 Poisson, Sime´on 16–17, 21, 26n Potter, Michael 133n predicativism, see constructivism projective determinacy (PD), see determinacy projective sets 37, 48–53, 81, 85, 127–128 Putnam, Hilary ix, 115 quantum mechanics 23, 28–29, 37, 58, 90–91, 103 Quine, W. V. O. ix, 53, 55, 69n, 87n, 97, 101, 136 Rapalino, John x realism ix, 60–61, 66, 114–118, 133–134; see also Platonism Robust Realism ix, 57–59, 60, 62–64, 66, 72, 77, 86–87, 92n, 113–116, }V.2, 130, 134 set-theoretic realism ix, 57, 72, 118 stereotypical anti-nominalist 104, 109–110 Thin Realism Chapter III, 88–89, }}IV.4-IV.5, 113, 116–117, 123, 124n, 131, 134 real numbers 22n, 26, 31–32, 35–37, 44–45, 48–49, 52, 56n, 92–93, 95, 127

150

index

Reck, Erich x reflection 126n regressive method 133n Reid, Philip 23n, 25n, 29n, 30 Reinhardt, William 126n resemblance 126 Resnick, Robert 25n Resnik, Michael 8, 90n restrictiveness 130n Rice, John 25n, Riemann, Bernhard 8 Robust Realism see realism. Rogers, Brian x Roland, Jeffrey x Rosen, Gideon 39n, 57, 73n, 76, 97–98, 104–105, 109–110 Russell, Bertrand 32, 133n Schoenflies, Arthur 46 Schwartz, Laurent 37n Second Philosophy }II.1, 103–111 self-evidence 1, 46–47, 119–123, 124, 132; see also intrinsic justification Shapiro, Alan 10n, 18n Shapiro, Stewart x, 32n, 56, 115–116, 117n, 118 Shelah, Saharon 50, 80, 128 Shoenfield, J. R. 135n Sierpin´ski, Wacław 127 Silverman, Allan 3n Simon, John 23n, 25n, 29 skepticism 73–76, 83, 107, 116, 117 Slowik, Edward 10n Smith, Crosbie 11n, 15n, 16, 18n Smith, George 10n, 12, 18n Solovay, Robert 126n Stalnaker, Robert 65n statistical mechanics 22–25, 26n, 28 Steel, John x, 49–51, 60–62, 128 Stein, Howard 11n stereotypical nominalist, see nominalism Stirling’s approximation 25, 90, 92 Stillwell, John 7n, 79n, 134n structuralism 32n, 99n Tait, William 60n, 63n, 69n, 73n Tappenden, Jamie x

Tarski, Alfred see paradoxes, Banach-Tarski thermodynamics 19–20, 22, 25 Thin Realism see realism Thomson, William (Lord Kelvin) 11n Tidwell, Scott x Tiles, Mary 132, 133n Tolley, Clinton x transfinite numbers 6, 42n, 80, 126–127 trigonometric representations 41–42, 52, 57, 80, 82, 85, 89 Tritton, D. J. 21–22 tropospheric complacency 105–106, 108, 111 Truesdell, Clifford 10n, 12n, 13n, 14, 30n truth ix, 1, 54–56, 60–62, 64, 66–68, 69n, 70–71, 77–78, 81n, 83, 86, 88–99, }}IV.4-IV.5, 113, 115–117, 118, 123 Uffink, Jos 25n Urquhart, Alasdair 36n van der Pol, B. 37n van Fraassen, Bas 40 V=L, see Axiom of Constructibility Wagon, Stan 35n Walker, Jearl 25n Wedberg, Anders 3n Weierstrass, Karl 42, 44–45 Wigner, Eugene 96 Wilson, Mark x, 20n-21n, 28n, 73n, 90n, 93, 105–109, 111 Wise, Norton 11n, 15n, 16–17, 18n Witten, Edward 37n Wittgenstein, Ludwig 115n Kripkenstein 107 Woodin, Hugh 50–51, 80, 128–129 Wright, Crispin 55n, 70–71 Wussing, Hans 7, 79n, 134n Zemanian, A. H. 37n Zermelo, Ernst 32–34, 37, 45–48, 52, 85, 88–89, 98, 100–102, 105, 110, 120, 122, 124, 126, 133 ZFC 33–34, 37, 48–50, 62, 125, 127, 136n