Mathematics As a Science of Patterns

Cover......Page 1
Title Page......Page 5
Preface......Page 9
Contents......Page 13
Part One: Problems and Positions......Page 17
1. Introduction......Page 19
1. To Characterize Realism…......Page 26
2. Immanent Truth......Page 30
2.1. Truth Vehicles, Truth-Theories, and Some Conceptions of Truth......Page 31
2.2. Disquotational Biconditionals for Truth......Page 35
2.3. Immanent vs. Transcendent Truth......Page 42
3. Realism and Immanent Truth......Page 46
3.1. Demands from Metaphysics......Page 48
3.2. Demands from Philosophy of Science......Page 50
3.3. Demands from the Theory of Cognition......Page 52
3.4. Demands from Philosophy of Language......Page 53
4. Some Concluding Remarks......Page 55
1. The Prima Facie Case for Realism......Page 57
2. The Quine-Putnam View of Applied Mathematics......Page 59
3. Indispensability Arguments for Mathematical Realism......Page 60
4. Indispensability and Fictionalism about Science......Page 65
5. Conclusion......Page 66
4. Recent Attempts at Blunting the Indispensability Thesis......Page 68
1. Synthetic Science: Field......Page 69
2.1. Chihara\'s Constructibility Theory......Page 75
2.2. Kitcher\'s Idealizations......Page 80
3. An Intermediate Approach: Hellman\'s Modal-Structuralism......Page 83
3.1. Number Theory and Analysts According to Modal-Structuralism......Page 84
3.2. The Modal-Structuralist Account of Applied Mathematics......Page 86
4. What Has Introducing Modalities Gained?......Page 91
5. Conclusion......Page 97
1. How Can We Know Mathematical Objects?......Page 98
2. How Can We Refer to Mathematical Objects?......Page 103
3. The Incompleteness of Mathematical Objects......Page 105
4. Some Morals for Realists......Page 108
5. An Aside: Penelope Maddy\'s Perceivable Sets......Page 109
Part Two: Neutral Epistemology......Page 113
Introduction to Part Two......Page 115
6. The Elusive Distinction between Mathematics and Natural Science......Page 117
1.1. Quantum Particles......Page 118
1.2. Undetectable Physical Objects......Page 122
2. Some Other Attempts to Distinguish Mathematical From Physical Objects......Page 123
3. Our Epistemic Access to Space-Time Points......Page 124
4. Morals for the Epistemology of Mathematics......Page 126
7. Holism: Evidence in Science and Mathematics......Page 128
1. The Initial Case for Holism......Page 130
2. Objections to Holism......Page 134
3. Testing Scientific and Mathematical Models......Page 137
4. Global and Local Theories......Page 140
5. Revising Logic and Mathematics......Page 146
8. The Local Conception of Mathematical Evidence: Proof, Computation, and Logic......Page 153
1. Some Norms of Mathematical Practice......Page 154
2. Computation and Mathematical Empiricism......Page 164
2.1. Computation and Mathematical Reasoning......Page 165
2.2. From Empirical Premisses to Formal Conclusions......Page 166
2.3. What Do Computational Inferences Show Us about the Nature of Mathematical Knowledge?......Page 169
3. Mathematical Proof, Logical Deduction, and Apriority......Page 171
3.1. Mathematical Proof and Logical Deduction......Page 172
3.2. Wide Reflective Equilibrium and the ‘Epistemology’ of Logic......Page 174
3.3. Against Logical Realism......Page 177
3.4. In Favour of Logical Non-Cognitivism......Page 182
3.5. The Apriority of Logic......Page 187
4. Summary......Page 188
1. Introduction......Page 191
2. A Quasi-Historical Account......Page 193
3. Mathematical Positing Naturalized?......Page 198
4. Positing and Knowledge......Page 200
5. Postulational Epistemologies and Realism......Page 204
Part Three: Mathematics as a Science of Patterns......Page 213
Introduction to Part Three......Page 215
1. Introduction......Page 217
2. Patterns and their Relationships......Page 218
3. Patterns and Positions: Entity and Identity......Page 225
4. Composite and Unified Mathematical Objects......Page 229
5. Mathematical Reductions......Page 232
6. Reference to Positions in Patterns......Page 236
7. Concluding Remarks on Reference and Reduction......Page 238
1. Introduction......Page 240
2. From Templates to Patterns......Page 242
3. From Proofs to Truth......Page 248
4. From Old Patterns to New Patterns......Page 256
2. On ‘Facts of the Matter’......Page 259
3. Patterns as Mathematical Objects......Page 262
4. Structural Relativity......Page 266
5. Structuralist Formulations of Mathematical Theories?......Page 270
6. The Status of Structuralism......Page 273
7. Structuralism, Realism, and Disquotationalism......Page 277
8. Epistemic vs. Ontic Structuralism: Structuralism All the Way Down......Page 281
9. A Concluding Summary......Page 286
Bibliography......Page 291
Index......Page 299