Mathematics plays a central role in much of contemporary science, but philosophers have struggled to understand what this role is or how significant it might be for mathematics and science. Pincock tackles this perennial question by asking how mathematics contributes to the success of our best scientific representations.

• 1 Introduction


• 1.1 A Problem


• 1.2 Classifying Contributions


• 1.3 An Epistemic Solution


• 1.4 Explanatory Contributions


• 1.5 Other Approaches


• 1.6 Interpretative Flexibility


• 1.7 Key Claims


• I Epistemic Contributions


• 2 Content and Confirmation


• 2.1 Concepts


• 2.2 Basic Contents


• 2.3 Enriched Contents


• 2.4 Schematic and Genuine Contents


• 2.5 Inference


• 2.6 Core Conceptions


• 2.7 Intrinsic and Extrinsic


• 2.8 Confirmation Theory


• 2.9 Prior Probabilities


• 3 Causes


• 3.1 Accounts of Causation


• 3.2 A Causal Representation


• 3.3 Some Acausal Representations


• 3.4 The Value of Acausal Representations


• 3.5 Batterman and Wilson


• 4 Varying Interpretations


• 4.1 Abstraction as Variation


• 4.2 Irrotational Fluids and Electrostatics


• 4.3 Shock Waves


• 4.4 The Value of Varying Interpretations


• 4.5 Varying Interpretations and Discovery


• 4.6 The Toolkit of Applied Mathematics


• 5 Scale Matters


• 5.1 Scale and ScientificRepresentation


• 5.2 Scale Separation


• 5.3 Scale Similarity


• 5.4 Scale and Idealization


• 5.5 Perturbation Theory


• 5.6 Multiple Scales


• 5.7 Interpreting Multiscale Representations


• 5.8 Summary


• 6 Constitutive Frameworks


• 6.1 A Different Kind of Contribution


• 6.2 Carnap's Linguistic Frameworks


• 6.3 Kuhn's Paradigms


• 6.4 Friedman on the Relative A Priori


• 6.5 The Need for Constitutive Representations


• 6.6 The Need for the Absolute A Priori


• 7 Failures


• 7.1 Mathematics and Scientific Failure


• 7.2 Completeness and Segmentation Illusions


• 7.3 The Parameter Illusion


• 7.4 Illusions of Scale


• 7.5 Illusions of Traction


• 7.6 Causal Illusions


• 7.7 Finding the Scope of a Representation


• II Other Contributions


• 8 Discovery


• 8.1 Semantic and Metaphysical Problems


• 8.2 A Descriptive Problem


• 8.3 Description and Discovery


• 8.4 Defending Naturalism


• 8.5 Natural Kinds


• 9 Indispensability


• 9.1 Descriptive Contributions and Pure Mathematics


• 9.2 Quine and Putnam


• 9.3 Against the Platonist Conclusion


• 9.4 Colyvan


• 10 Explanation


• 10.1 Explanatory Contributions


• 10.2 Inference to the Best Mathematical Explanation


• 10.3 Belief and Understanding


• 11 The Rainbow


• 11.1 Asymptotic Explanation


• 11.2 Angle and Color


• 11.3 Explanatory Power


• 11.4 Supernumerary Bows


• 11.5 Interpretation and Scope


• 11.6 Batterman and Belot


• 11.7 Looking Ahead


• 12 Fictionalism 413


• 12.1 Motivations


• 12.2 Literary Fiction


• 12.3 Mathematics


• 12.4 Models


• 12.5 Understanding and Truth


• 13 Facades


• 13.1 Physical and Mathematical Concepts


• 13.2 Against Semantic Finality


• 13.3 Developing and Connecting Patches


• 13.4 A New Approach to Content


• 13.5 Azzouni and Rayo


• 14 Conclusion: Pure Mathematics


• 14.1 Taking Stock


• 14.2 Metaphysics .


• 14.3 Structuralism


• 14.4 Epistemology


• 14.5 Peacocke and Jenkins


• 14.6 Historical Extensions


• 14.7 Non-conceptual Justification


• 14.8 Past and Future


• Appendices


• A Method of Characteristics


• B Black-Scholes Model


• C Speed of Sound


• D Two Proofs of Euler's Formula

Christopher Pincock received his Ph.D. in philosophy from the University of California, Berkeley in 2002. After eight years at Purdue University, he recently joined the philosophy department at the University of Missouri as an Associate Professor.