Wittgenstein on Mathematics

<p>This book offers a detailed account and discussion of Ludwig Wittgenstein’s philosophy of mathematics. In Part I, the stage is set with a brief presentation of Frege’s logicist attempt to provide arithmetic with a foundation and Wittgenstein’s criticisms of it, followed by sketches of Wittgenstein’s early views of mathematics, in the <i>Tractatus</i> and in the early 1930s. Then (in Part II), Wittgenstein’s mature philosophy of mathematics (1937-44) is carefully presented and examined. Schroeder explains that it is based on two key ideas: the <i>calculus view</i> and the <i>grammar view</i>. On the one hand, mathematics is seen as a human activity — calculation — rather than a theory. On the other hand, the results of mathematical calculations serve as grammatical norms. The following chapters (on mathematics as grammar; rule-following; conventionalism; the empirical basis of mathematics; the role of proof) explore the tension between those two key ideas and suggest a way in which it can be resolved. Finally, there are chapters analysing and defending Wittgenstein’s provocative views on Hilbert’s Formalism and the quest for consistency proofs and on Gödel’s incompleteness theorems.</p> <p>Preface viii</p><p>List of Abbreviations xii</p><p>PART I</p><p>Background 1</p><p>1 Foundations of Mathematics 3</p><p>2 Logicism 9</p><p>2.1 Frege’s Logicism 9</p><p>2.2 The Class Paradox and Russell’s Theory of Types 12</p><p>2.3 Tractatus Logico-Philosophicus<i>: Logicism Without Classes 13</i></p><p>3 Wittgenstein’s Critique of Logicism 15</p><p>3.1 Can Equality of Number Be Defined in Terms of One-to-One Correlation? 15</p><p>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</p><p>3.3 Platonism 22</p><p>3.4 Russell’s Reconstructions of False Equations Are Not Contradictions 26</p><p>3.5 Frege’s and Russell’s Formalisation of Sums as Logical Truths Cannot Be Foundational as It Presupposes Arithmetic 27</p><p>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</p><p><b>4 The Development of Wittgenstein’s Philosophy of</b></p><p>Mathematics: <i>Tractatus </i>to <i>The Big Typescript </i>35</p><p>4.1 Tractatus Logico-Philosophicus <i>35</i></p><p>4.2 Philosophical Remarks <i>(MSS 105–8: 1929–30) to </i>The Big Typescript <i>(TS 213: 1933) 36</i></p><p>PART II</p><p>Wittgenstein’s Mature Philosophy of Mathematics</p><p>(1937–44) 55</p><p>5 The Two Strands in Wittgenstein’s Later Philosophy of Mathematics 57</p><p>6 Mathematics as Grammar 59</p><p>7 Rule-Following 78</p><p>7.1 Rule-Following and Community 88</p><p>8 Conventionalism 93</p><p>8.1 Quine’s Circularity Objection 95</p><p>8.2 Dummett’s Objection That Conventionalism Cannot Explain Logical Inferences 101</p><p>8.3 Crispin Wright’s Infinite Regress Objection 103</p><p>8.4 The Objection to ‘Moderate Conventionalism’ From Scepticism About Rule-Following 105</p><p>8.5 The Objection From the Impossibility of a Radically Different Logic or Mathematics 109</p><p>8.6 Conclusion 124</p><p>9 Empirical Propositions Hardened Into Rules 126</p><p>Synthetic A Priori 134</p><p>10 Mathematical Proof 141</p><p>10.1 What Is a Mathematical Proof? 142</p><p>(a) Proof That a0 = 1 150</p><p>(b) Skolem’s Inductive Proof of the Associative Law of Addition 150</p><p>(c) Cantor’s Diagonal Proof 151</p><p>(d) Euclid’s Construction of a Regular Pentagon 158</p><p><i>(e) Euclid’s Proof That There Is No Greatest Prime Number 160</i></p><p>(f) Proof (Calculation) in Elementary Arithmetic 166</p><p>Proof and experiment 169</p><p>10.2 What Is the Relation Between a Mathematical Proposition and Its Proof? 171</p><p>10.3 What Is the Relation Between a Mathematical Proposition’s Proof and Its Application? 181</p><p>11 Inconsistency 189</p><p>12 Wittgenstein’s Remarks on Gödel’s First Incompleteness Theorem 203</p><p>12.1 Wittgenstein Discusses Godel’s Informal Sketch of His Proof 206</p><p>12.2 ‘A Proposition That Says About Itself That It Is Not Provable in P’ 207</p><p>12.3 The Difference Between the Godel Sentence and the Liar Paradox 209</p><p>12.4 Truth and Provability 210</p><p>12.5 Godel’s Kind of Proof 213</p><p>12.6 Wittgenstein’s First Objection: A Useless Paradox 216</p><p>12.7 Wittgenstein’s Second Objection: A Proof Based on Indeterminate Meaning 218</p><p>13 Concluding Remarks: Wittgenstein and Platonism 220</p><p>Bibliography 226</p><p>Index 234</p>