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| 0. Preface
| xi
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| 0.1 Welcome
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| 0.2 What this book is not about
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| 0.3 Coming attractions
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Part 1: Scientific Explanations by Constraint
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| 1. What Makes a Scientific Explanation Distinctively Mathematical?
| 3
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| 1.1 Distinctively mathematical explanations in science as non-causal scientific explanations
| 3
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| 1.2 Are distinctively mathematical explanations set apart by their failure to cite causes?
| 12
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| 1.3 Mathematical explanations do not exploit causal powers
| 22
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| 1.4 How these distinctively mathematical explanations work
| 25
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| 1.5 Elaborating my account of distinctively mathematical explanations
| 32
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| 1.6 Conclusion
| 44
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| 2. "There Sweep Great General Principles Which All The Laws Seem To Follow"
| 46
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| 2.1 The task: to unpack the title of this chapter
| 46
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| 2.2 Constraints versus coincidences
| 49
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| 2.3 Hybrid explanations
| 58
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| 2.4 Other possible kinds of constraints besides conservation laws
| 64
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| 2.5 Constraints as modally more exalted than the force laws they constrain
| 68
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| 2.6 My account of the difference between constraints and coincidences
| 72
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| 2.7 Accounts that rule out explanations by constraint
| 86
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| 3. The Lorentz Transformations and the Structure of Explanations by Constraint
| 96
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| 3.1 Transformation laws as constraints or coincidences
| 96
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| 3.2 The Lorentz transformations given an explanation by constraint
| 100
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| 3.3 Principle versus constructive theories
| 112
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| 3.4 How this non-causal explanation comes in handy
| 123
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| 3.5 How explanations by constraint work
| 128
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| 3.6 Supplying information about the source of a constraint's necessity
| 136
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| 3.7 What makes a constraint "explanatorily fundamental"?
| 141
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| Appendix: A purely kinematical derivation of the Lorentz transformations
| 145
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| 4. The Parallelogram of Forces and the Autonomy of Statics
| 150
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| 4.1 A forgotten controversy in the foundations of classical physics
| 150
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| 4.2 The dynamical explanation of the parallelogram of forces
| 154
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| 4.3 Duchayla's statical explanation
| 159
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| 4.4 Poisson's statical explanation
| 167
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| 4.5 Statical explanation under some familiar accounts of natural law
| 173
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| 4.6 My account of what is at stake
| 178
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Part 2: Two Other Varieties of Non-Causal Explanation in Science
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| 5. Really Statistical Explanations and Genetic Drift
| 189
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| 5.1 Introduction to Part 2
| 189
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| 5.2 RS (Really Statistical) explanations
| 190
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| 5.3 Drift
| 196
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| 6. Dimensional Explanations
| 204
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| 6.1 A simple dimensional explanation
| 204
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| 6.2 A more complicated dimensional explanation
| 209
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| 6.3 Different features of a derivative law may receive different dimensional explanations
| 215
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| 6.4 Dimensional homogeneity
| 219
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| 6.5 Independence from some other quantities as part of a dimensional explanans
| 221
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Part 3. Explanation in Mathematics
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| 7. Aspects of Mathematical Explanation: Symmetry, Salience, and Simplicity
| 231
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| 7.1 Introduction to proofs that explain why mathematical theorems holds
| 231
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| 7.2 Zeitz's biased coin: A suggestive example of mathematical explanation
| 234
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| 7.3 Explanation by symmetry
| 238
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| 7.4 A theorem explained by a symmetry in the unit imaginary number
| 239
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| 7.5 Geometric explanations that exploit symmetry
| 245
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| 7.6 Generalizing the proposal
| 254
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| 7.7 Conclusion
| 268
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| 8. Mathematical Coincidences and Mathematical Explanations That Unify
| 276
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| 8.1 What is a mathematical coincidence?
| 276
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| 8.2 Can mathematical coincidence be understood without appealing to mathematical explanation?
| 283
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| 8.3 A mathematical coincidence's components have no common proof
| 287
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| 8.4 A shift of context may change a proof's explanatory power
| 298
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| 8.5 Comparison to other proposals
| 304
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| 8.6 Conclusion
| 311
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| 9 Desargues' Theorem as a Case Study of Mathematical Explanation, Existence, and Natural Properties
| 314
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| 9.1 Introduction
| 314
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| 9.2 Three proofs - but only one explanation - of Desargues' theorem in two-dimensional Euclidean geometry
| 315
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| 9.3 Why Desargues' theorem in two-dimensional Euclidean geometry is explained by an exit to the third dimension
| 323
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| 9.4 Desargues' theorem in projective geometry: unification and existence in mathematics
| 327
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| 9.5 Desargues' theorem in projective geometry: explanation and natural properties in mathematics
| 335
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| 9.6 Explanation by subsumption under a theorem
| 341
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| 9.7 Conclusion
| 345
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Part 4: Explanations in Mathematics and Non-Causal Scientific Explanations -- Together
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| 10 Mathematical Coincidence and Scientific Explanation
| 349
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| 10.1 Physical coincidences that are no mathematical coincidence
| 349
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| 10.2 Explanations from common mathematical form
| 350
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| 10.3 Explanations from common dimensional architecture
| 361
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| 10.4 Targeting new explananda
| 368
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| 11 What Makes Some Reducible Physical Properties Explanatory?
| 371
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| 11.1 Some Reducible Properties Are Natural
| 371
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| 11.2 Centers of mass and reduced mass
| 378
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| 11.3 Reducible properties on Strevens's account of scientific explanation
| 381
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| 11.4 Dimensionless quantities as explanatorily powerful reducible properties
| 384
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| 11.5 My proposal
| 386
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| 11.6 Conclusion: all varieties of explanation as species of the same genus
| 394
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| Notes
| 401
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| References
| 461
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| Index
| 483
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