New Waves in Philosophy of Mathematics (New Waves in Philosophy)

Cover......Page 1
Title Page......Page 5
Contents......Page 7
Series Editors’ Foreword......Page 9
Acknowledgements......Page 10
List of Contributors......Page 11
The beginning of modern philosophy of mathematics......Page 13
The recent past......Page 14
The contemporary debate......Page 15
Part I. Reassessing the orthodoxy in the philosophy of mathematics......Page 16
Part II. The question of realism in mathematics......Page 17
Part III. Mathematical practice and the methodology of mathematics......Page 18
Part IV. Mathematical language and the psychology of mathematics......Page 19
Part V. From philosophical logic to the philosophy of mathematics......Page 20
Note......Page 21
Part I. Reassessing the Orthodoxy in the Philosophy of Mathematics......Page 23
1 Motivations philosophical and personal......Page 25
2 Fregean philosophy of mathematics......Page 27
3 The Neo-Fregean picture......Page 33
Notes......Page 42
References......Page 45
1 Reductionism......Page 47
1.3 Partial-denotation reductionism......Page 48
2 Reductionism’s response......Page 49
2.1 Context-independent objectual reductionism......Page 50
2.3 Structural reductionism......Page 51
3.1 Semantic objections......Page 52
3.2 Slippery slope objection......Page 56
3.3 Epistemological objection......Page 57
4 Meaning analysis versus explication......Page 58
Notes......Page 63
References......Page 66
Part II. The Question of Realism in Mathematics......Page 69
1 Introduction: Platonism and nominalism......Page 71
3.1 Fictionalism and nominalism......Page 75
3.2.1 The crucial idea......Page 76
3.2.2 Meeting the desiderata......Page 78
3.3.1 The crucial point......Page 82
3.3.2 On the existence of mathematical objects......Page 83
3.3.3 Meeting the desiderata......Page 85
Notes......Page 88
References......Page 90
4 Truth in Mathematics: The Question of Pluralism......Page 92
1 The emergence of pluralism......Page 93
1.2 Kant......Page 94
1.3 Reichenbach......Page 95
1.4 Carnap......Page 96
2.1.1 Some key terminology......Page 97
2.1.2 The analytic/synthetic distinction......Page 98
2.1.3 Criticism #1: The argument from free parameters......Page 100
2.1.4 Criticism #2: The argument from assessment sensitivity......Page 101
2.2 Radical pluralism......Page 102
2.3 Philosophy as logical syntax......Page 104
2.3.1 Conclusion......Page 106
3 A new orientation......Page 108
4 The initial stretch: First- and second-order arithmetic......Page 110
4.2 The problem of selection for second-order arithmetic......Page 111
5.1 An initial pass......Page 116
5.2 A more promising approach......Page 117
Notes......Page 122
References......Page 127
5 “Algebraic” Approaches to Mathematics......Page 129
1 “Assertory” views of mathematics and Benacerraf’s problems......Page 130
2 “Algebraic” views and Benacerraf’s problems......Page 132
3.1 Modal structuralism......Page 133
3.2 Ante rem structuralism......Page 134
3.3 Full-blooded Platonism......Page 135
3.4 Fictionalism......Page 136
4.1 The problem of modality......Page 137
4.2 The problem of mixed claims......Page 139
Notes......Page 143
References......Page 145
Part III. Mathematical Practice and the Methodology of Mathematics......Page 147
1 Introduction......Page 149
1.2 Explanation in mathematics (intertheoretic)......Page 150
1.4 Explaining the role of mathematics in science......Page 151
2 Mathematical coincidences......Page 152
3 From mathematical coincidences to accidental generalizations......Page 154
3.2 Natural kinds......Page 155
3.3 Explanation......Page 156
4.1 The Goldbach conjecture......Page 157
4.2 The Four-Color Theorem......Page 159
5 Mathematical accidents defined......Page 160
6.1 The end of explanation......Page 162
6.2 Axiom choice......Page 164
7 Intertheoretic mathematical accidents......Page 165
8 Conclusions......Page 167
8.2 Explanatory basis......Page 168
Notes......Page 169
Bibliography......Page 170
1 Introduction......Page 172
2 Indispensability of inconsistent mathematical objects......Page 173
3 A philosophical account of applied mathematics......Page 175
Notes......Page 180
References......Page 182
I......Page 185
II......Page 186
III......Page 189
IV......Page 196
V......Page 200
A1. From the Navier-Stokes equations to the Euler equations......Page 202
A2. From the Navier-Stokes equations to the boundary layer equations......Page 203
Notes......Page 204
References......Page 205
Part IV. Mathematical Language and the Psychology of Mathematics......Page 207
1 How can we do better?......Page 209
2 Formal tools......Page 210
3 Two projects in the philosophy of mathematics......Page 212
4 Mathematical activity......Page 213
5 Mathematical language......Page 214
5.1 Syntax......Page 215
5.2 Number words......Page 216
5.3 Quantifiers......Page 219
6 Inferential relations......Page 220
7 The status of axioms......Page 222
8 An epistemic role?......Page 224
9 Pluralism......Page 225
10 Conclusion......Page 229
Notes......Page 230
References......Page 231
1 Introduction......Page 232
2 Individuation and criteria of identity......Page 233
3 The individuation of the natural numbers......Page 235
4 Against the cardinal conception......Page 237
4.1 The objection from special numbers......Page 238
4.2 The objection from the philosophy of language......Page 239
4.4 Alleged advantages of the cardinal conception......Page 240
5.1 Refining the criterion of identity......Page 241
5.2 Justifying the axioms of Dedekind-Peano Arithmetic......Page 243
6 The metaphysical status of natural numbers......Page 245
Notes......Page 247
References......Page 249
1 Against conventional wisdom......Page 251
2.1 Intelligibility......Page 260
2.2 Intelligibility and identity......Page 261
2.3 Knowledge......Page 266
References......Page 271
Part V. From Philosophical Logic to the Philosophy of Mathematics......Page 273
12 On Formal and Informal Provability......Page 275
1 Preliminary remarks: Provable, proof, and proving......Page 277
2 Formal versus informal provability: A conceptual distinction......Page 279
3 A Gödelian perspective on informal provability......Page 285
4 The logic of informal provability......Page 295
5 Are there true but informally unprovable statements?......Page 299
Notes......Page 304
References......Page 308
2 The problem of absolute generality......Page 312
3 Indefinite extensibility without a domain......Page 317
3.1 Two candidate restrictions of plural comprehension......Page 319
3.2 Quantification without a domain......Page 323
4 The costs of the restriction......Page 325
4.1 Second-order ZFC(U)......Page 326
4.2 Proper classes......Page 327
4.3 Reflection......Page 328
5 Conclusion......Page 331
Notes......Page 332
References......Page 334
Index......Page 336