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A Defence of Predicativism As a Philosophy of Mathematics

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A Defence of Predicativism as a Philosophy of Mathematics Timothy James STORER Trinity College, University of Cambridge ¿is dissertation is submitted for the degree of Doctor of Philosophy ċ 2010 ċ 2 Preface Declaration & Statement of length ¿is dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except where specically indicated in the text. No part of this dissertation has been submitted for any other degree or qualic- ation. ¿is dissertation conforms to the word limits set by the Degree Committee of Philosophy. It is approximately 75,000 words in length. Acknowledgements & Dedication I am very grateful for the support and guidance of my supervisors: my primary supervisor, Michael Potter, who put up with a great deal; Arif Ahmed, my shadow supervisor, for his helpful suggestions in the earlier stages of my research; and Peter Smith, who very generously gave to me a huge amount of his time, advice, enthusiasm, and encouragement in the later stages. I would also like to thank Brian King for his kindness in reading and comment- ing on an earlier dra of this thesis, and Lesley Lancaster for her help with the administrative side of things. I am grateful for the nancial support of the AHRC. My greatest debts are to my friends and family for their love and support: most of all, to my father, John Storer; my step-mother, Vivi Paxton; and — of course — to my friend Diktynna Warren, without whom not. ¿is thesis is dedicated to my teachers. 3 4 Abstract A Defence of Predicativism as a Philosophy of Mathematics T. J. Storer A specication of a mathematical object is impredicative if it essentially involves quantication over a domain which includes the object being specied (or sets which contain that object, or similar). ¿e basic worry is that we have no non-circular way of understanding such a specication. Predicativism is the view that mathematics should be limited to the study of objects which can be specied predicatively. ¿ere are two parts to predicativism. One is the criticism of the impredicative aspects of classical mathematics. ¿e other is the positive project, begun by Weyl in Das Kontinuum, to reconstruct as much as possible of classical mathematics on the basis of a predicatively acceptable set theory, which accepts only countably innite objects. ¿is is a revisionary project, and certain parts of mathematics will not be saved. Chapter 2 contains an account of the historical background to the predicativist project. ¿e rigorization of analysis led to Dedekind’s and Cantor’s theories of the real numbers, which relied on the new notion of abitrary innite sets; this became a central part of modern classical set theory. Criticism began with Kronecker; continued in the debate about the acceptability of Zermelo’s Axiom of Choice; and was somewhat claried by Poincaré and Russell. In the light of this, chapter 3 examines the formulation of, and motivations behind the predicativist position. Chapter 4 begins the critical task by detailing the epistemological problems with the classical account of the continuum. Explanations of classicism which appeal to second-order logic, set theory, and primitive intuition are examined and are found wanting. Chapter 5 aims to dispell the worry that predicativism might collapses into math- 5 6 ematical intuitionism. I assess some of the arguments for intuitionism, especially the Dummettian argument from indenite extensibility. I argue that the natural numbers are not indenitely extensible, and that, although the continuum is, we can nonetheless make some sense of classical quantication over it. We need not reject the Law of Excluded Middle. Chapter 6 begins the positive work by outlining a predicatively acceptable ac- count of mathematical objects which justies the Vicious Circle Principle. Chapter 7 explores the appropriate shape of formalized predicative mathematics, and the question of just how much mathematics is predicatively acceptable. My conclusion is that all of the mathematics which we need can be predicativist- ically justied, and that such mathematics is particularly transparent to reason. ¿is calls into question one currently prevalent view of the nature of mathematics, on which mathematics is justied by quasi-empirical means. Contents 1 Introduction 11 1.1 Overview . 11 1.2 Plan of the rest of the thesis . 14 2 History 17 2.1 ¿e back-story . 18 2.1.1 Beginnings: Irrationals, the innite, and the development ofnaiveanalysis ......................... 18 2.1.2 ¿e rigorization of analysis . 20 2.1.3 ¿e arithmetization of the reals — Dedekind . 23 2.1.4 ¿e arithmetization of the reals — Kronecker’s programme 25 2.2 Cantor’s theory of the innite . 26 2.3 A er the paradoxes . 31 2.3.1 ¿e Axiom of Choice . 31 2.3.2 Poincaré on the paradoxes . 34 2.3.3 Das Kontinuum and Brouwer’s ‘revolution’ . 37 2.4 Conclusions . 40 3 Predicativism 43 3.1 Motivations . 45 3.2 Open-endedness and quantication . 48 3.2.1 ‘Any’ and ‘all’ . 48 3.2.2 Indenite extensibility . 50 3.3 ¿e indenite extensibility of P(N) . 52 3.3.1 Denitionism . 52 3.3.2 Richard’s Paradox . 53 7 8 CONTENTS 3.3.3 König and Cantor on Richard’s paradox . 54 3.3.4 Ramication . 55 3.3.5 Uncountability . 56 3.3.6 ¿e arbitrary innite . 58 3.4 Predicativism . 60 3.4.1 ‘Given the natural numbers’ . 62 3.4.2 Conclusion . 64 4 Classicism about the continuum 65 4.1 Preliminaries . 65 4.1.1 Taxonomy: three routes to classicism . 67 4.1.2 Taxonomy: rst-order and second-order . 69 4.1.3 ¿e Continuum Hypothesis . 70 4.2 Second-orderism . 73 4.2.1 ¿e meaning of the second-order quantiers . 74 4.2.2 Second-order logic as logic . 76 4.2.3 Arbitrary concepts . 78 4.2.4 ¿e plural alternative . 81 4.2.5 Arithmetic and the ancestral . 83 4.2.6 ¿e Full Induction Axiom . 85 4.3 Cantorianism . 86 4.3.1 Coming to grasp P(N) . 87 4.3.2 V=L . 89 4.3.3 Countable models of set theory . 91 4.3.4 Wright’s Skolemism . 93 4.4 Intuitively based classicism . 99 4.4.1 Cantor again . 99 4.4.2 Postulating or intuiting the classical continuum . 100 4.4.3 Arithmetic and geometric continua . 103 5 ¿e stability of the predicativist position 107 5.1 Arguments for intuitionism . 108 5.1.1 Brouwer: Mathematical objects as mental constructions . 108 5.1.2 Dummett: Indenite extensibility . 111 5.2 Are the natural numbers indenitely extensible? . 117 CONTENTS 9 5.2.1 Characterizing the natural numbers . 117 5.2.2 ¿e alleged impredicativity of the natural numbers . 120 5.2.3 ¿e negative translation . 123 5.3 ¿e indenite extensibility of the continuum . 126 5.3.1 Indenitely extensibility again . 126 5.3.2 ¿e continuum . 132 5.3.3 Predicativist analysis . 133 5.3.4 LEM and analysis . 136 5.3.5 Conclusion . 139 6 ¿e metaphysics behind predicativism 141 6.1 Russell and ramication . 143 6.2 Russell and Reducibility . 147 6.3 Weyl’s account of properties . 152 6.4 Sets as dependent objects . 157 6.5 Quantication over open domains . 161 6.6 ¿e dependence of sets on their dening properties . 164 6.7 Conclusion . 167 7 Predicative mathematics and its scope 169 7.1 What predicativists can get . 170 7.1.1 Reverse mathematics . 170 7.1.2 Justifying ACA0 .......................... 171 7.1.3 Believing in ACA0; the metamathematics of ACA0 . 172 7.1.4 Predicative mathematics: ACA0 and Das Kontinuum . 173 7.1.5 ¿e Law of Excluded Middle . 175 7.1.6 Delimiting the scope of predicativism . 178 7.1.7 Going beyond ACA0 . 180 7.1.8 Alternatives? . 181 7.1.9 ¿e predicative continuum . 182 7.2 What we all need — indispensability arguments . 183 7.2.1 Science — Quine–Putnam . 184 7.2.2 Mathematics — Gödel–Friedman . 187 8 Conclusion 189 10 CONTENTS Chapter 1 Introduction 1.1 Overview ¿is thesis presents a case for predicativism as a philosophy of mathematics. By predicativism, I will mean what is sometimes called predicativism given the natural numbers. ¿e position can be briey characterized as the acceptance of the natural numbers as a foundation for mathematics, and the rejection of impredicative meth- ods in building up from that foundation. ¿e view was rst clearly put forward by Hermann Weyl in his short book of 1918, Das Kontinuum. As its title suggests, the main object of Weyl’s book was to develop a satisfactory account of the arithmetical continuum, i.e., the real-number system. ¿ere were two familiar accounts of the real numbers available to Weyl, which are essentially equivalent: the reals can be explained in terms either of Cauchy sequences (certain sequences of rational numbers) or in terms of Dedekind cuts (certain sets of rational numbers). Given that sequences can be implemented as sets, the question of what real numbers there are boils down either way to a question about what innite sets there are. So what theory of innite sets should we adopt? ¿e dominant account when Weyl was writing — and to this day — was a full-blooded Cantorian set theory. Weyl found such such an account unsatisfactory because of its use of a particular concept of set which was, he maintained, ill-dened. In its place, Weyl proposed a foundation based on the natural numbers, and a non-Cantorian ‘logical’ (i.e. predicative) concept of sets of natural numbers. More specically, Weyl’s objection to Cantorian set theory, and to the classical 11 12 CHAPTER 1.
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