POINCARE AND THE PHILOSOPHY OF MATHEMATICS

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1 POINCARE AND THE PHILOSOPHY OF MATHEMATICS

2 Poincare and the Philosophy of Mathematics' Janet Folina Assistant Professor of Philosophy MacA laster College. St. Paul Palgrave Macmillan

3 ISBN ISBN (ebook) DOI / Scots Philosophical Club 1992 Softcover reprint of the hardcover 1st edition 1992 All rights reserved. For infonnation, write: Scholarly and Reference Division, S1. Martin's Press, Inc., 175 Fifth Avenue, New York, N.Y First published in the United States of America in 1992 ISBN Library of Congress Cataloging-in-Publication Data Folina, Janet, Poincare and the philosophy of mathematics / Janet Folina. p. cm. Includes bibliographical references and index. ISBN Mathematics-Philosophy. 2. Poincare, Henri, I. Title. QA8.4.F O'.I-dc CIP

4 Contents Preface Introduction ix xii 1 KANT AND MATHEMATICS, AN OUTLINE The Basic Distinctions 2 2 The Synthetic A Priori 5 3 The Synthetic A Priori Instantiated: Geometry 10 4 The Synthetic A Priori Instantiated: Arithmetic 12 5 The Foundation of the Theory of the Synthetic A Priori 16 6 Re-examination of How the Theory Works: A Precarious Analogy 19 7 The Key to the Synthetic Aspect of Arithmetic: Induction 23 2 INTRODUCTION TO POINCARE'S THEORY OF THE SYNTHETIC A PRIORI The Synthetic A Priori and Time 33 2 The Synthetic A Priori and Space 36 3 DEFENDING MATHEMATICAL APRIORISM Certainty and A Priori Knowledge 43 2 The Definition of 'A Priori Warrant' and Some Counterexamples 45 3 Critique of Kitcher's Account 49 4 Alternative Ways to Look at the Relationship Between Apriority and Defeasibility 52 5 Defending Mathematical Apriorism by Defending the Traditional Account of A Priori Knowledge 58 6 Conclusion 66 v

5 VI Contents 4 LOGIC AND INTUITION Poincare's Conception of Logic: is it a Mere Misconception? 74 2 Russell's Logicism Does Not Refute Kant 80 3 Intuitions and Poincare's Theory of 'Glossing Over' 84 5 THE A IT ACK ON LOGICISM Analysis of the Principle of Induction 93 2 The Problem of Induction for the Logicists 96 3 Some Attempts to A void the Circle The Second Order Principle Non-Inductive Arithmetic The Synthetic A Priori Nature of Arithmetic Intuition SET THEORY AND THE CONTINUUM Poincare's Theory of Definitions Axioms and Intuitions Epistemology and the Characterisation Problem The Limits of the Arithmetisation of the Continuum Sets as Contained Collections The Crucial Importance of Cantor's Result for Poincare's Theory of the Continuum The Argument for the Existence of Geometric Intuition The Argument that Continuity is Foundational Physical Discontinuity and Mathematical Continuity POINCARE'S THEORY OF PREDICATIVITY Poincare and Platonism Analysis of the Concept of Impredicativity 147

6 Contents Vll 3 The Emergence of the Concept Poincare's Account Russell's Account The Objection to Zermelo's Solution Poincare's Diagnosis and Solution of the Paradoxes Circles, Vicious Circles, and Two Types of Definitions Poincare's Conception of Sets as Constructed Entities, and his 'True Solution' POINCARE'S THEORY OF MEANING Poincare's Criterion of Meaningfulness How Poincare Employs the Notion A Generalised Version of the Criterion Why Poincare's Theory of Meaning is not Intuitionistic Potential Infinity and the Domain Argument Blocked Why the Strict Finitist Objects to Potential Infinity Poincare Against the Finitist: the Metaphor of Construction CONCLUSION Summary of Poincare's 'Middle Position' Between Intuitionism and Platonism 190 Bibliography 193 Author Index 200 Subject Index 201

7 Preface This book is an elaboration, extension, and hopefully an improvement upon my doctoral dissertation, submitted in May 1986, to St Andrews University, St Andrews, Scotland. I became interested in Poincare during my first year of graduate study. I knew he was considered to be some sort of conventionalist in science, and especially with regard to applied geometry. But this label seemed inappropriate to his views concerning the philosophy of number-theoretic mathematics. During this first year I wrote a short M.Phil. thesis concerning the controversy between Poincare and Russell over the status of mathematical induction. Yet I was still mystified as to what sort of 'ism' to attach to Poincare's views on the philosophy and foundations of mathematics. To me, he seemed to be neither platonist, conventionalist, nor even intuitionist (in the formal revisionary sense of, say, Brouwer and Heyting). As I learned more about his philosophy, I became defensive of his views. Though he was and is only praised for his mathematical achievements, his philosophical writings are in general not highly regarded. Indeed his various remarks are often represented as idiosyncratic, polemical and even inconsistent. Although I was aware of the apparent inconsistency of some of his views (see Introduction, below), I was intrigued by his convictions concerning intuition, and convinced that beneath the surface of apparent inconsistency there was a consistent and coherent underlying core. I thus undertook a Ph.D thesis on Poincare's philosophy of mathematics, with the aim of investigating whether such a core exists, and if so, to defend it as a viable alternative in the contemporary philosophy of mathematics. My approach to Poincare was therefore pretty single-mindedly defensive. It is true that very often I found myself frustrated by trying to make sense of his remarks, which are notoriously scattered and unsystematic. But in general I was self-consciously a sympathetic reader. One of the goals for this book has been to make my presentation of Poincare's less myopic than the dissertation. With this in mind I have expanded it to try to make it more accessible and more interesting to a wider audience by (i) setting Poincare a little more firmly in his historical framework, and (ii) being a little more critical of Poincare's views in general. I still regard this as a defence of Poincare, in that it is an attempt to rationally reconstruct his philosophical position (with the emphasis on 'rational'). There are two reasons for preserving the 'defensive' aspect of the thesis. First, though it IX

8 x Preface seems much 'safer' and somewhat easier to criticise than to sensibly defend a position, I believe one comes away from the latter endeavour with a far deeper understanding of the real issues. As Crispin Wright has said, 'To understand any philosophical view involves knowing what best can be said on its behalf.' Of course this does not warrant ignoring the problems with a position; and I have here tried to include relevant criticisms of Poincare as well as (when possible) arguments for his views. The other reason I have kept this book sympathetic is that I think a defence of Poincare is due: he has had more than his share of critics. Since the main body of this text is reworked from material I put together for my dissertation, I must re-thank people at St Andrews. First and foremost I must thank Peter Clark, my supervisor. Many of the ideas in this book are his - although he was kind enough at the time to make me think they were mine! Peter gave me the confidence to do graduate work in philosophy, when I had very little experience in philosophy; and he led me to a great topic which has proved both intellectually rich and inherently interesting. I must also thank Crispin Wright, whose influence has been very powerful in shaping my approach to philosophy in general, and Mike Hallett, the external reader for my Ph.D., for providing many helpful and constructive criticisms. With regard to particular chapters, I'd like to thank Leslie Stevenson for comments on Chapter 1, Arthur Fine for comments on Chapter 4, and Jim Page for help with Chapter 5 (in addition to his generous moral support). I also learned a great deal from Steve Read, who taught me to appreciate logic(s), and Roger Squires, who taught me about intellectual honesty. I would also like to thank my family for their encouragement, the secretaries at the Dept. of Logic and Metaphysics for being surrogate moms, and all my friends who were at St Andrews - the 'physics crowd', my flatmates, and especially Eileen and Andy - for providing emotional support and lots of fun in between the frustration and the hard work (and for proofreading the dissertation!). In addition I am grateful to St Andrews University for providing financial support for my graduate work; and to the editors of the series MacMillan Studies in Contemporary Philosophy for allowing me to take much longer than the initial time estimate to complete the proposed revisions. One final word of caution about this book: it is a philosophy book and not a mathematics book. In trying to understand and account for Poincare's philosophy of mathematics, I have looked almost exclusively at his philosophical papers, and have not tackled the hundreds of mathematical results which he published. There are two reasons for this. First, although I agree that it is important to know something about what Poincare accomplished

9 Preface Xl mathematically in order to assess his philosophical views about mathematics, the major focus of this book is the existence of a core underlying his philosophical remarks. Hopefully this core will cohere with his mathematical practice, but I have not done an extensive survey of his technical works to see if the coherence is complete. (Though much of his mathematical practice would seem to indicate a classical realist point of view, I believe it all coheres with the weak anti-realist point of view which I attribute to him.) His philosophical comments are almost exclusively concerned with basic number theory, set theory and logic (including induction), for which attention to some logic and very little mathematics will suffice. Second, I do not agree with those who regard the sole interest of a view (in the philosophy of mathematics) in terms of what it technically enables us to do. Thus I have not concentrated very much on setting out the details of how much mathematics we can 'get' out of the Poincare view, but instead on the question of its internal coherence and philosophical attractiveness, in addition to whether or not it agrees with our informal intuitions about the nature of mathematics. Thus, this book will undoubtedly disappoint those who seek technical approaches, technical argumentation, and technical results. Nevertheless, the approach I have chosen makes this, on the whole, accessible to most philosophers (students or otherwise) with some logic and some interest in the foundations of mathematics. If the book kindles some interest in Poincare and/or in an anti-realist alternative to intuitionism in the philosophy of mathematics, then it will have served a useful purpose.

10 Introduction Jules Henri Poincare ( ), the Gauss of modem mathematics, was a 'universal' mathematician whose contributions were seminal in the development of contemporary pure mathematics, in mathematical physics, and in the philosophical foundations of mathematics. The first two claims, concerning his influence in the technical areas, are uncontestable. Poincare was the greatest practitioner of mathematics of his time, and it is with justice that he is credited for this. In contrast, his contribution to the philosophy of mathematics is, in general, profoundly underestimated (or discreetly overlooked). In fact he is best known in philosophy for his theory of the role of conventions in science; and he is thought of as one of the first' conventionalists'. In the philosophy of mathematics, however, his writings are regarded as idiosyncratic and based upon a misunderstanding of the logicist tradition which he criticised. This interpretation is not, moreover, entirely unfounded, for there are passages, especially in his earlier publications, in which he is clearly mistaken concerning the nature of modem formal logic, as well as the status of mathematical induction. And at first glance, his writings seem glib, and at times based on a polemical reaction to the work of Russell, Zermelo, Peano, and Couturat. Many of his published papers are based on reports he gave, as chair of mathematical physics at the University of Paris ( ), concerning the current state of physics, to somewhat general audiences. This results in writings which are conversational in tone, and very amusing to read; but it is sometimes quite hard to get to their philosophical 'bottom'. Reading them now, therefore, out of context, it is even harder to figure out what Poincare really believed; for it is easy to mistake sarcasm and wit for paradox and contradiction. However, it is important not to allow the manner in which Poincare expresses himself to obscure the potential depth and philosophical import of his ideas. Since his philosophical work was always quite clearly secondary to his work in mathematics itself, he never attempted to expound his ideas in a structured, systematic presentation. However, this does not mean that there is no general philosophical core underpinning his various insightful and controversial comments. It just means one must take care to interpret his writings in terms of his views concerning very general philosophical theses, and in terms of the era in which he lived. The era in which Poincare lived was exciting and disturbing in the foundations of mathematics. It was exciting because the need for increased xii

11 Introduction xiii precision in mathematical definitions and axiomatic foundations was being realised. However, at the same time it was disturbing to Poincare (and others, such as Borel and Lebesgue) because the means of accomplishing the worthy aim of precision brought problems of its own. For example, logicism (Frege's logicism, which Poincare knew about via Russell and Couturat, even if not by Frege's name) brought contradiction; and even the consistent axiomatic advances in set theory brought impredicative definitions and the long rejected concept of the actually infinite. Poincare took it upon himself to comment on each of these 'problems' (not all were problems according to the set theorist or realist) at various times. In general, he was worried about undue emphasis on formalisation at the expense of intuition, and about the impact of formalisation on the progress and teaching of mathematics. That is, Poincare was concerned about the future of mathematics, with the direction in which it was progressing. It seemed to him that it was progressing too far from the real world and too far from our intuitions which form the connection between pure mathematics and the real world - that is, which make applied mathematics possible. We see this concern in almost every page of his writing in the philosophy of mathematics. Unfortunately, sufficient care is not always taken when interpreting Poincare's writings, and this can result in an unfair representation of Poincare. For instance, in the context of a brief survey of the emergence of the concept of impredicativity, Kneale and Kneale (1962) comment on Poincare's view that there is a relation between the set-theoretic paradoxes and the attempt to treat infinities as completed wholes. Poincare did hold such a view; but this was not his complete view. Kneale and Kneale go on to cite a short passage by Poincare to support their claim.l However, the passage they quote is misleading out of context; and Poincare follows a similar passage on the previous page with an explanation of his view. 2 Later they again comment that Poincare suggested that the paradoxes of the theory of sets were due to the fundamental mistake of assuming actually infinite aggregates. He did not explain in detail why... 3 It is true that Poincare's 'explanations' are hard to find; and the relation between actual infinity and the paradoxes, in which he believed, is subtle. However, I maintain that his theory of predicativity exists and is coherent; and I attempt to show this in Chapter 7. Chihara, in his book Ontology and the Vicious-Circle Principle (1973), devotes a chapter to a better interpretation of Poincare's philosophy, in its

12 xiv Introduction historical and philosophical context. However, he also seems to have missed the point of Poincare's connection between the belief in actual infinity and the contradictions. The connection is interpreted by Chihara as causal: that the belief in actual infinity simply causes the paradoxes. 4 Whereas in fact, Poincare's view was that the belief in actual infinity was symptomatic of an approach which easily leads to paradoxes, that is, a (misplaced) realist approach. In addition, Chihara concludes that Poincare is a 'nominalist' with regard to mathematics, citing Poincare's remarks that the continuum is a mere 'system of symbols', and that mathematics can 'give to the physicist only a convenient language'.5 Again, it is true that Poincare was deeply concerned with applied mathematics and with the potential applicability of all mathematics in general. But interpreting him as a nominalist does not ring quite true to his views concerning the synthetic a priori status of mathematics, such as his belief that the continuum is an object of intuition. Given Poincare's theory concerning the epistemologically foundational nature of the continuum (which is the subject of Chapter 6), it seems clear that his remark that the continuum is 'only' a system of symbols is not intended to be taken in a straightforward way. The 'language' the mathematician provides for the scientist has - contra-chihara - not to do with the trivial fact that physical laws are expressed in terms of mathematical symbols and notation. Rather, it is a consequence of the strong Kantian nature of Poincare's philosophy: mathematics expresses what is necessarily common to all thinking beings; and the best science can do, insofar as discovering 'true relations', is to discover mathematical relations which survive the inevitable changes in background theory and conventions. 6 This book is in general the attempt to show that interpreting Poincare as a neo-kantian, as a defender of Kant's (general) epistemological position, is not only defensible, it is the only interpretation which enables us to make sense of Poincare's various claims in the philosophy of mathematics. There is thus no 'only' in Poincare's view of the language which mathematics provides for science; the use of the term 'only' is therefore not 'misleading' (as Chihara claims), but surely intended in sarcasm, as a criticism of any formalist account of the continuum. Perhaps most surprising of all the (mis)interpretations of Poincare's philosophy, is a claim made by Parsons that Poincare is an intuitionist, but not a Kantian, because he seems 'quite uninfluenced' by Kant's notion of pure intuition. This is particularly astonishing given that Poincare repeatedly refers to 'mathematical intuition' and 'synthetic a priori intuition', it seems consciously adopting Kant's terminology. It is true that the details of Poincare's theory of intuition are different from Kant's. But he regarded

13 Introduction xv space and time as 'the frames in which nature seems enclosed.' And though Poincare thought some properties of these 'frames' to be purely conventional (and therefore not necessary), he did regard some as 'a form preexistent in our mind' and necessary for the conceptualisability and organisability of experience. 7 It seems that these aspects of time and space match exactly Kant's conception of 'pure intuition'. Chapters 2 and 4 are devoted to explaining Poincare's theory of intuition, and how it differs from Kant's theory. The other purpose of Chapter 4, in conjunction with Chapter 5, is to argue that some such theory of intuition is necessary in order to make sense of Poincare's very general claims against both the logicist and the set theorist. There is no doubt that it is not easy to make sense of Poincare's remarks, which are often, at first blush, paradoxical and even trite. For example, he appears both to condone and to oppose the formalisation of mathematics. He devotes pages to extolling the virtues of the new precise methods, for example those which are involved in making the concepts of continuity and limit rigorous. Yet never far from such praise is a corresponding criticism of formal methods. His fear appears to have been that the benefits of exactness were being bought at the cost of purging our mathematical concepts of all intuitive content. Poincare wants both precision and intuition to be considered a bona fide part of mathematics. From his own experience he knows that 'creative intuition' is hardly a formal matter. The relation between our formal characterisations and our intuitive concepts was a tension which he sought to resolve. There are also prima-facie difficulties in coming to grips with his views on set theory. Although he was one of the first mathematicians to employ Cantor's theory of sets, and thus one of the first to reap the benefits of the theory, he explicitly rejected its fundamental theorem in its standard interpretation - that of a proof of the existence of an uncountable set. And there is an apparent outrageous inconsistency in his attitude towards the continuum. Time and again he stresses that all infinity is potential, that 'there is no actual infinity', that ineliminably impredicative specifications must be rejected - and so, that the greatest cardinal number is XO' And yet he seems to retain a classical notion of continuity and the continuum: for instance in his mathematical work his employment of variables which range over all the points in a continuous interval is classical; and he accepts the standard account of least upper bound, which is impredicative. (Though he does re-work the definition when its impredicativity is pointed out to him.) Indeed, the notion of continuity is one of the most central to his creative thinking; and his greatest theoretical achievements in the development of 'analysis situs' occurred when he

14 xvi Introduction considered what happens if certain parameters are allowed to vary continuously. As I explain in Chapter 6, all the points on the line exist - since the continuum is apprehended a priori as an intuition; but there is no cardinal number of all the points on the line. That is, according to Poincare, the continuum cannot be treated as a mere set. The fact that the question of the 'correct' account of the continuum is an open philosophical and mathematical matter to this day in some sense vindicates Poincare's sceptical attitude towards the concept of 'set' as foundational for mathematics in general, but especially when applied to the continuum and to even higher order infinities. Poincare was a diverse and global thinker; and this is revealed by noting his influence on the foundations of a variety of schools of thought. For instance, the theory of meaning he employed - the criterion of (weak) 'verifiability in principle' - became foundational in intuitionism, and it is important for anti-realist semantics in general. This theory is the subject of Chapter 8. In addition, his theory of impredicativity and vicious circles, examined in Chapter 7, led to studies in predicative set theory and predicative analysis (as, for example, is found in Feferman (1964) and, more recently, in S. Shapiro (1985)), and is relevant to contemporary questions concerning computability. Poincare's contributions also led to a critical reassessment of metamathematics. In Chapter 4 I discuss how, in Poincare's view, intuitions must be epistemologically prior to any significant formal structure (and, indeed, to any systematic thinking); and in Chapter 5, I focus, with a view to the same end, on Poincare's theory of induction. The question of the apriority of mathematics (and not only its synthetic, vs analytic, character) also stands in need of a defence. And the subject of Chapter 3 is a defence of mathematical apriorism against the recent empiricist challenge made by Philip Kitcher (1984). My project has been to determine whether there is a general philosophical core which underpins Poincare's scattered, diverse, yet often profound and insightful remarks. Is there a foundation which makes even his apparently paradoxical views cohere? I believe that there is. Though not without its difficulties, Poincare's philosophy is coherent. My general thesis is that the fundamental key to an appropriate understanding of his philosophy on the whole is not to underestimate the legacy of Kant in his views. Poincare adopts Kant's view that mathematics is synthetic a priori. Yet he adapts it as well. For instance, on his account (in opposition to Kant) the metric geometry we employ is not synthetic a priori; in fact, it is conventional. His general philosophical position can be described as 'neo-kantian'. (Chapter 2 is a brief introduction to Poincare's theory

15 Introduction xvii of the synthetic a priori: what distinguishes him as both Kantian but different from Kant.) The main body of this text can be seen as a description of the way in which Poincare adapts the Kantian thesis, with a view to defending Kant from the 'Leibnizian' logicist arguments of the time - those of Russell, Zermelo, Peano, and indirectly (via Couturat) Frege. My first task will thus be to outline Kant's position on mathematics; to this, Chapter 1 is devoted. Throughout, it will be important to bear in mind that, from the point of view of one of the greatest practitioners of classical mathematics since Gauss, Poincare's philosophical work can be regarded as possibly being motivated by a desire to steer a middle course between the 'Scylla' of triviality and the 'Charybdis' of contradiction; in fact, to steer a middle course between strict constructivism and set theory. The interest of his philosophy of mathematics as I reconstruct it, is that, since it is a non-intuitionistic middle position (between classical set-theoretic mathematics and strict finitism), it might provide a stable, anti-realist alternative to intuitionism - an alternative which is both philosophically and mathematically sensible. Notes 1. See Kneale and Kneale [1962] pp Poincare [1952a] p Kneale and Kneale [1962] pp See Chihara p Although in the later sections of the chapter he makes more sense of Poincare's views, for example on impredicativity. (For a very detailed and careful historical account of the development of the concept of predicativity, see G. Heinzmann [1985].) 5. Chihara [1973] pp See Giedymin [1982] for a good account of the Kantian element in Poincare's 'conventionalism'. 7. Poincare [1958] pp. 13 and 26; [1963] p. 44.

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