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 Tractatus 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 calculus view and the grammar view. 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.
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 Gödel’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
Biography
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).
