Marc Lange is a philosopher of science. He serves as Chair of the Philosophy Department at the University of North Carolina at Chapel Hill, where he is the Theda Perdue Distinguished Professor. His previous books include Laws and Lawmakers (OUP 2009), An Introduction to the Philosophy of Physics: Locality, Fields, Energy, and Mass (2002), and Natural Laws in Scientific Practice (OUP 2000).

This is a tremendous book. It brings together and synthesizes Marc Lange's highly original work over the past decade on non-causal explanation in science and mathematics. Like much of Lange's oeuvre, it represents naturalistic metaphysics of science that draws inspiration and support from a wealth of detailed, carefully researched examples from the sciences, going back to the early nineteenth century and beyond. ... The way in which these examples are coupled with open-minded

• 0. Preface0.1 Welcome0.2 What this book is not about0.3 Coming attractions Part 1: Scientific Explanations by Constraint 1. What Makes a Scientific Explanation Distinctively Mathematical?1.1 Distinctively mathematical explanations in science as non-causal scientific explanations1.2 Are distinctively mathematical explanations set apart by their failure to cite causes?1.3 Mathematical explanations do not exploit causal powers1.4 How these distinctively mathematical explanations work1.5 Elaborating my account of distinctively mathematical explanations1.6 Conclusion 2. "There Sweep Great General Principles Which All The Laws Seem To Follow"2.1 The task: to unpack the title of this chapter2.2 Constraints versus coincidences2.3 Hybrid explanations2.4 Other possible kinds of constraints besides conservation laws2.5 Constraints as modally more exalted than the force laws they constrain2.6 My account of the difference between constraints and coincidences2.7 Accounts that rule out explanations by constraint 3. The Lorentz Transformations and the Structure of Explanations by Constraint3.1 Transformation laws as constraints or coincidences3.2 The Lorentz transformations given an explanation by constraint3.3 Principle versus constructive theories3.4 How this non-causal explanation comes in handy3.5 How explanations by constraint work3.6 Supplying information about the source of a constraint's necessity3.7 What makes a constraint Appendix: A purely kinematical derivation of the Lorentz transformations 4. The Parallelogram of Forces and the Autonomy of Statics4.1 A forgotten controversy in the foundations of classical physics4.2 The dynamical explanation of the parallelogram of forces4.3 Duchayla's statical explanation4.4 Poisson's statical explanation4.5 Statical explanation under some familiar accounts of natural law4.6 My account of what is at stake Part 2: Two Other Varieties of Non-Causal Explanation in Science 5. Really Statistical Explanations and Genetic Drift5.1 Introduction to Part 25.2 RS (Really Statistical) explanations5.3 Drift 6. Dimensional Explanations6.1 A simple dimensional explanation6.2 A more complicated dimensional explanation6.3 Different features of a derivative law may receive different dimensional explanations6.4 Dimensional homogeneity6.5 Independence from some other quantities as part of a dimensional explanans Part 3. Explanation in Mathematics7. Aspects of Mathematical Explanation: Symmetry, Salience, and Simplicity7.1 Introduction to proofs that explain why mathematical theorems holds7.2 Zeitz's biased coin: A suggestive example of mathematical explanation7.3 Explanation by symmetry7.4 A theorem explained by a symmetry in the unit imaginary number7.5 Geometric explanations that exploit symmetry7.6 Generalizing the proposal7.7 Conclusion 8. Mathematical Coincidences and Mathematical Explanations That Unify8.1 What is a mathematical coincidence?8.2 Can mathematical coincidence be understood without appealing to mathematical explanation?8.3 A mathematical coincidence's components have no common proof8.4 A shift of context may change a proof's explanatory power8.5 Comparison to other proposals8.6 Conclusion 9 Desargues' Theorem as a Case Study of Mathematical Explanation, Existence, and Natural Properties9.1 Introduction9.2 Three proofs - but only one explanation - of Desargues' theorem in two-dimensional Euclidean geometry9.3 Why Desargues' theorem in two-dimensional Euclidean geometry is explained by an exit to the third dimension9.4 Desargues' theorem in projective geometry: unification and existence in mathematics9.5 Desargues' theorem in projective geometry: explanation and natural properties in mathematics9.6 Explanation by subsumption under a theorem9.7 Conclusion Part 4: Explanations in Mathematics and Non-Causal Scientific Explanations -- Together 10 Mathematical Coincidence and Scientific Explanation10.1 Physical coincidences that are no mathematical coincidence10.2 Explanations from common mathematical form10.3 Explanations from common dimensional architecture10.4 Targeting new explananda 11 What Makes Some Reducible Physical Properties Explanatory?11.1 Introduction11.2 Centers of mass and reduced mass11.3 Reducible properties on Strevens's account of scientific explanation11.4 Dimensionless quantities as explanatorily powerful reducible properties11.5 My proposal11.6 Conclusion: all varieties of explanation as species of the same genus ReferencesIndex

Marc Lange is a philosopher of science. He serves as Chair of the Philosophy Department at the University of North Carolina at Chapel Hill, where he is the Theda Perdue Distinguished Professor. His previous books include Laws and Lawmakers (OUP 2009), An Introduction to the Philosophy of Physics: Locality, Fields, Energy, and Mass (2002), and Natural Laws in Scientific Practice (OUP 2000).

This is a tremendous book. It brings together and synthesizes Marc Lange's highly original work over the past decade on non-causal explanation in science and mathematics. Like much of Lange's oeuvre, it represents naturalistic metaphysics of science that draws inspiration and support from a wealth of detailed, carefully researched examples from the sciences, going back to the early nineteenth century and beyond. ... The way in which these examples are coupled with open-minded

• 0. Preface0.1 Welcome0.2 What this book is not about0.3 Coming attractions Part 1: Scientific Explanations by Constraint 1. What Makes a Scientific Explanation Distinctively Mathematical?1.1 Distinctively mathematical explanations in science as non-causal scientific explanations1.2 Are distinctively mathematical explanations set apart by their failure to cite causes?1.3 Mathematical explanations do not exploit causal powers1.4 How these distinctively mathematical explanations work1.5 Elaborating my account of distinctively mathematical explanations1.6 Conclusion 2. "There Sweep Great General Principles Which All The Laws Seem To Follow"2.1 The task: to unpack the title of this chapter2.2 Constraints versus coincidences2.3 Hybrid explanations2.4 Other possible kinds of constraints besides conservation laws2.5 Constraints as modally more exalted than the force laws they constrain2.6 My account of the difference between constraints and coincidences2.7 Accounts that rule out explanations by constraint 3. The Lorentz Transformations and the Structure of Explanations by Constraint3.1 Transformation laws as constraints or coincidences3.2 The Lorentz transformations given an explanation by constraint3.3 Principle versus constructive theories3.4 How this non-causal explanation comes in handy3.5 How explanations by constraint work3.6 Supplying information about the source of a constraint's necessity3.7 What makes a constraint Appendix: A purely kinematical derivation of the Lorentz transformations 4. The Parallelogram of Forces and the Autonomy of Statics4.1 A forgotten controversy in the foundations of classical physics4.2 The dynamical explanation of the parallelogram of forces4.3 Duchayla's statical explanation4.4 Poisson's statical explanation4.5 Statical explanation under some familiar accounts of natural law4.6 My account of what is at stake Part 2: Two Other Varieties of Non-Causal Explanation in Science 5. Really Statistical Explanations and Genetic Drift5.1 Introduction to Part 25.2 RS (Really Statistical) explanations5.3 Drift 6. Dimensional Explanations6.1 A simple dimensional explanation6.2 A more complicated dimensional explanation6.3 Different features of a derivative law may receive different dimensional explanations6.4 Dimensional homogeneity6.5 Independence from some other quantities as part of a dimensional explanans Part 3. Explanation in Mathematics7. Aspects of Mathematical Explanation: Symmetry, Salience, and Simplicity7.1 Introduction to proofs that explain why mathematical theorems holds7.2 Zeitz's biased coin: A suggestive example of mathematical explanation7.3 Explanation by symmetry7.4 A theorem explained by a symmetry in the unit imaginary number7.5 Geometric explanations that exploit symmetry7.6 Generalizing the proposal7.7 Conclusion 8. Mathematical Coincidences and Mathematical Explanations That Unify8.1 What is a mathematical coincidence?8.2 Can mathematical coincidence be understood without appealing to mathematical explanation?8.3 A mathematical coincidence's components have no common proof8.4 A shift of context may change a proof's explanatory power8.5 Comparison to other proposals8.6 Conclusion 9 Desargues' Theorem as a Case Study of Mathematical Explanation, Existence, and Natural Properties9.1 Introduction9.2 Three proofs - but only one explanation - of Desargues' theorem in two-dimensional Euclidean geometry9.3 Why Desargues' theorem in two-dimensional Euclidean geometry is explained by an exit to the third dimension9.4 Desargues' theorem in projective geometry: unification and existence in mathematics9.5 Desargues' theorem in projective geometry: explanation and natural properties in mathematics9.6 Explanation by subsumption under a theorem9.7 Conclusion Part 4: Explanations in Mathematics and Non-Causal Scientific Explanations -- Together 10 Mathematical Coincidence and Scientific Explanation10.1 Physical coincidences that are no mathematical coincidence10.2 Explanations from common mathematical form10.3 Explanations from common dimensional architecture10.4 Targeting new explananda 11 What Makes Some Reducible Physical Properties Explanatory?11.1 Introduction11.2 Centers of mass and reduced mass11.3 Reducible properties on Strevens's account of scientific explanation11.4 Dimensionless quantities as explanatorily powerful reducible properties11.5 My proposal11.6 Conclusion: all varieties of explanation as species of the same genus ReferencesIndex