Preface viii

List of Abbreviations xii

PART I

Background 1

1 Foundations of Mathematics 3

2 Logicism 9

2.1 Frege’s Logicism 9

2.2 The Class Paradox and Russell’s Theory of Types 12

2.3 Tractatus Logico-Philosophicus: Logicism Without Classes 13

3 Wittgenstein’s Critique of Logicism 15

3.1 Can Equality of Number Be Defined in Terms of One-to-One Correlation? 15

3.2 Frege’s (and Russell’s) Definition of Numbers as Equivalence Classes Is Not Constructive: It Doesn’t Provide a Method of Identifying Numbers 21

3.3 Platonism 22

3.4 Russell’s Reconstructions of False Equations Are Not Contradictions 26

3.5 Frege’s and Russell’s Formalisation of Sums as Logical Truths Cannot Be Foundational as It Presupposes Arithmetic 27

3.6 Even If We Assumed (for Argument’s Sake) That All Arithmetic Could Be Reproduced in Russell’s Logical Calculus, That Would Not Make the Latter a Foundation of Arithmetic 31

4 The Development of Wittgenstein’s Philosophy of

Mathematics: Tractatus to The Big Typescript 35

4.1 Tractatus Logico-Philosophicus 35

4.2 Philosophical Remarks (MSS 105?8: 1929?30) to The Big Typescript (TS 213: 1933) 36

PART II

Wittgenstein’s Mature Philosophy of Mathematics

(1937?44) 55

5 The Two Strands in Wittgenstein’s Later Philosophy of Mathematics 57

6 Mathematics as Grammar 59

7 Rule-Following 78

7.1 Rule-Following and Community 88

8 Conventionalism 93

8.1 Quine’s Circularity Objection 95

8.2 Dummett’s Objection That Conventionalism Cannot Explain Logical Inferences 101

8.3 Crispin Wright’s Infinite Regress Objection 103

8.4 The Objection to ‘Moderate Conventionalism’ From Scepticism About Rule-Following 105

8.5 The Objection From the Impossibility of a Radically Different Logic or Mathematics 109

8.6 Conclusion 124

9 Empirical Propositions Hardened Into Rules 126

Synthetic A Priori 134

10 Mathematical Proof 141

10.1 What Is a Mathematical Proof? 142

(a) Proof That a0 = 1 150

(b) Skolem’s Inductive Proof of the Associative Law of Addition 150

(c) Cantor’s Diagonal Proof 151

(d) Euclid’s Construction of a Regular Pentagon 158

(e) Euclid’s Proof That There Is No Greatest Prime Number 160

(f) Proof (Calculation) in Elementary Arithmetic 166

Proof and experiment 169

10.2 What Is the Relation Between a Mathematical Proposition and Its Proof? 171

10.3 What Is the Relation Between a Mathematical Proposition’s Proof and Its Application? 181

11 Inconsistency 189

12 Wittgenstein’s Remarks on Godel’s First Incompleteness Theorem 203

12.1 Wittgenstein Discusses Godel’s Informal Sketch of His Proof 206

12.2 ‘A Proposition That Says About Itself That It Is Not Provable in P’ 207

12.3 The Difference Between the Godel Sentence and the Liar Paradox 209

12.4 Truth and Provability 210

12.5 Godel’s Kind of Proof 213

12.6 Wittgenstein’s First Objection: A Useless Paradox 216

12.7 Wittgenstein’s Second Objection: A Proof Based on Indeterminate Meaning 218

13 Concluding Remarks: Wittgenstein and Platonism 220

Bibliography 226

Index 234