Wittgenstein on Mathematics

Severin Schroeder is Associate Professor of Philosophy at the University of Reading. He has published three monographs on Wittgenstein: Wittgenstein: The Way Out of the Fly Bottle (2006), Wittgenstein Lesen (2009), and Das Privatsprachen-Argument (1998). He is the editor of Wittgenstein and Contemporary Philosophy of Mind (2001) and Philosophy of Literature (2010).

• Preface viiiList of Abbreviations xiiPART IBackground 11 Foundations of Mathematics 32 Logicism 92.1 Frege’s Logicism 92.2 The Class Paradox and Russell’s Theory of Types 122.3 Tractatus Logico-Philosophicus: Logicism Without Classes 133 Wittgenstein’s Critique of Logicism 153.1 Can Equality of Number Be Defined in Terms of One-to-One Correlation? 153.2 Frege’s (and Russell’s) Definition of Numbers as Equivalence Classes Is Not Constructive: It Doesn’t Provide a Method of Identifying Numbers 213.3 Platonism 223.4 Russell’s Reconstructions of False Equations Are Not Contradictions 263.5 Frege’s and Russell’s Formalisation of Sums as Logical Truths Cannot Be Foundational as It Presupposes Arithmetic 273.6 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 314 The Development of Wittgenstein’s Philosophy ofMathematics: Tractatus to The Big Typescript 354.1 Tractatus Logico-Philosophicus 354.2 Philosophical Remarks (MSS 105–8: 1929–30) to The Big Typescript (TS 213: 1933) 36PART IIWittgenstein’s Mature Philosophy of Mathematics(1937–44) 555 The Two Strands in Wittgenstein’s Later Philosophy of Mathematics 576 Mathematics as Grammar 597 Rule-Following 787.1 Rule-Following and Community 888 Conventionalism 938.1 Quine’s Circularity Objection 958.2 Dummett’s Objection That Conventionalism Cannot Explain Logical Inferences 1018.3 Crispin Wright’s Infinite Regress Objection 1038.4 The Objection to ‘Moderate Conventionalism’ From Scepticism About Rule-Following 1058.5 The Objection From the Impossibility of a Radically Different Logic or Mathematics 1098.6 Conclusion 1249 Empirical Propositions Hardened Into Rules 126Synthetic A Priori 13410 Mathematical Proof 14110.1 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 166Proof and experiment 16910.2 What Is the Relation Between a Mathematical Proposition and Its Proof? 17110.3 What Is the Relation Between a Mathematical Proposition’s Proof and Its Application? 18111 Inconsistency 18912 Wittgenstein’s Remarks on Gödel’s First Incompleteness Theorem 20312.1 Wittgenstein Discusses Godel’s Informal Sketch of His Proof 20612.2 ‘A Proposition That Says About Itself That It Is Not Provable in P’ 20712.3 The Difference Between the Godel Sentence and the Liar Paradox 20912.4 Truth and Provability 21012.5 Godel’s Kind of Proof 21312.6 Wittgenstein’s First Objection: A Useless Paradox 21612.7 Wittgenstein’s Second Objection: A Proof Based on Indeterminate Meaning 21813 Concluding Remarks: Wittgenstein and Platonism 220Bibliography 226Index 234

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