List of Figures ................................................ xi
Preface ...................................................... xiii
Mathematical Notation .......................................... xv
1 Introduction ................................................. 3
1.1 A Problem ............................................... 3
1.2 Classifying Contributions ............................... 5
1.3 An Epistemic Solution ................................... 8
1.4 Explanatory Contributions .............................. 12
1.5 Other Approaches ....................................... 15
1.6 Interpretive Flexibility ............................... 18
1.7 Key Claims ............................................. 21
PART I Epistemic Contributions
2 Content and Confirmation .................................... 25
2.1 Concepts ............................................... 25
2.2 Basic Contents ......................................... 27
2.3 Enriched Contents ...................................... 29
2.4 Schematic and Genuine Contents ......................... 31
2.5 Inference .............................................. 33
2.6 Core Conceptions ....................................... 35
2.7 Intrinsic and Extrinsic ................................ 36
2.8 Confirmation Theory .................................... 38
2.9 Prior Probabilities .................................... 41
3 Causes ...................................................... 45
3.1 Accounts of Causation .................................. 45
3.2 A Causal Representation ................................ 48
3.3 Some Acausal Representations ........................... 51
3.4 The Value of Acausal Representations ................... 60
3.5 Batterman and Wilson ................................... 63
4 Varying Interpretations ..................................... 66
4.1 Abstraction as Variation ............................... 66
4.2 Irrotational Fluids and Electrostatics ................. 68
4.3 ShockWaves ............................................. 74
4.4 The Value of Varying Interpretations ................... 78
4.5 Varying Interpretations and Discovery .................. 80
4.6 The Toolkit of Applied Mathematics ..................... 82
5 Scale Matters ............................................... 87
5.1 Scale and Scientific Representation .................... 87
5.2 Scale Separation ....................................... 88
5.3 Scale Similarity ....................................... 93
5.4 Scale and Idealization ................................. 96
5.5 Perturbation Theory ................................... 104
5.6 Multiple Scales ....................................... 105
5.7 Interpreting Multiscale Representations ............... 113
5.8 Summary ............................................... 120
6 Constitutive Frameworks .................................... 121
6.1 A Different Kind of Contribution ...................... 121
6.2 Carnaps Linguistic Frameworks ......................... 122
6.3 Kuhns Paradigms ....................................... 126
6.4 Friedman on the Relative A Priori ..................... 131
6.5 The Need for Constitutive Representations ............. 137
6.6 The Need for the Absolute A Priori .................... 138
7 Failures ................................................... 141
7.1 Mathematics and Scientific Failure .................... 141
7.2 Completeness and Segmentation Illusions ............... 142
7.3 The Parameter Illusion ................................ 146
7.4 Illusions of Scale .................................... 153
7.5 Illusions of Traction ................................. 155
7.6 Causal Illusions ...................................... 161
7.7 Finding the Scope of a Representation ................. 163
PART II Other Contributions
8 Discovery .................................................. 169
8.1 Semantic and Metaphysical Problems .................... 169
8.2 A Descriptive Problem ................................. 175
8.3 Description and Discovery ............................. 179
8.4 Defending Naturalism .................................. 183
8.5 Natural Kinds ......................................... 187
9 Indispensability ........................................... 190
9.1 Descriptive Contributions and Pure Mathematics ........ 190
9.2 Quine and Putnam ...................................... 190
9.3 Against the Platonist Conclusion ...................... 196
9.4 Colyvan ............................................... 200
10 Explanation ................................................ 203
10.1 Explanatory Contributions ............................. 203
10.2 Inference to the Best Mathematical Explanation ........ 210
10.3 Belief and Understanding .............................. 217
11 The Rainbow ................................................ 221
11.1 Asymptotic Explanation ................................ 221
11.2 Angle and Color ....................................... 223
11.3 Explanatory Power ..................................... 228
11.4 Supernumerary Bows .................................... 229
11.5 Interpretation and Scope .............................. 236
11.6 BattermanandBelot ..................................... 239
11.7 Looking Ahead ......................................... 242
12 Fictionalism ............................................... 243
12.1 Motivations ........................................... 243
12.2 Literary Fiction ...................................... 244
12.3 Mathematics ........................................... 250
12.4 Models ................................................ 256
12.5 Understanding and Truth ............................... 261
13 Facades .................................................... 264
13.1 Physical and Mathematical Concepts .................... 264
13.2 Against Semantic Finality ............................. 265
13.3 Developing and Connecting Patches ..................... 268
13.4 A New Approach to Content ............................. 275
13.5 Azzouni and Rayo ...................................... 278
14 Conclusion: Pure Mathematics ............................... 279
14.1 Taking Stock .......................................... 279
14.2 Metaphysics ........................................... 280
14.3 Structuralism ......................................... 284
14.4 Epistemology .......................................... 285
14.5 Peacocke and Jenkins .................................. 290
14.6 Historical Extensions ................................. 293
14.7 Nonconceptual Justification ........................... 295
14.8 Past and Future ....................................... 297
Appendix A: Method of Characteristics ......................... 301
AppendixB: Black-Scholes Model ................................ 303
Appendix C: Speed of Sound .................................... 305
Appendix D: Two Proofs of Euler's Formula ..................... 307
Bibliography .................................................. 309
Index ......................................................... 317
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