Mathematical Knowledge

Preface......Page 6
List of contributors......Page 8
Contents......Page 10
1 Benacerraf’s worry......Page 12
2 Mathematical knowers and mathematical knowledge......Page 18
1 What is distinctive about the mathematical case?......Page 27
2 Implicationism all the way down?......Page 28
3 Mother theories......Page 30
4 Understanding and truth......Page 32
5 Why externalism?......Page 33
6 The route to knowledge......Page 35
7 Benacerraf’s problem......Page 36
8 Benacerraf’s problem generalized......Page 38
9 The real problem......Page 40
1 Introduction......Page 44
2 The problem of induction in mathematics......Page 45
3 The seemingly inappropriate use of modal language by mathematicians......Page 46
4 Explicating informal mathematical terminology......Page 48
5 Memory......Page 49
6 Direct memory, competence, and generated memory......Page 52
7 Two proofs and how one remembers them......Page 54
8 Mental arithmetic and the concept of width......Page 60
9 How does the notion of width apply to proofs?......Page 65
10 Concluding remarks......Page 69
1 Introduction......Page 70
2 Enumerative induction and discovery......Page 71
3 The descriptive question: Two case studies......Page 72
4 Hume’s problem of induction......Page 75
5 The normative question: Is enumerative induction in mathematics rationally justified?......Page 76
6 Re-examining the descriptive question......Page 79
7 Conclusions......Page 83
Marinella CAPPELLETTI & Valeria GIARDINO:
The cognitive basis of mathematical knowledge......Page 85
1 Numerical cognition is innate and distinct from other cognitive skills......Page 86
2 Numbers and language......Page 90
3 Different uses of the same numeral......Page 92
4 Limits of cognitive science......Page 93
5 Concluding remarks......Page 94
Mary LENG:
What’s there to know?......Page 95
1 Trading ontology for modality......Page 97
2 Fictionalism and nominalism......Page 101
3 A new Benacerraf problem?......Page 105
4 A new indispensability argument?......Page 114
5 Defending the consistency of ZFC and PA......Page 115
6 Conclusion......Page 118
1 Empiricism in the philosophy of mathematics......Page 120
2 An empiricist account of mathematical knowledge......Page 121
3 Unapplied mathematics as mathematical recreation......Page 123
4 Is all mathematics recreation?......Page 128
5 Empiricism revisited......Page 133
2 Preliminaries......Page 134
3 The pragmatic and indifference objections......Page 141
4 Weak and strong scientific platonism......Page 144
5 For the indifference objection......Page 147
6 The publication test......Page 149
7 General principles of scientific method......Page 151
8 Scientific grounds and the actual content of mathematics......Page 157
9 Conclusion......Page 160
Crispin WRIGHT:
On quantifying into predicate position......Page 161
1 Basic idea and project outline......Page 164
2 Fixing the meanings of the quantifiers......Page 166
3 Extreme neutralism......Page 170
4 A neutralist heuristic......Page 171
5 Comprehension......Page 175
6 Incompleteness......Page 177
7 Impredicativity......Page 180
8 Appendix: Abstractionist Mathematical Theories......Page 182
Bibliography......Page 186
Index......Page 195