Mathematics and Scientific Representation

by Pincock, Christopher

• ISBN13: 9780199757107
• ISBN10: 0199757100
• Format: Hardcover
• Copyright: 2012-01-13
• Publisher: Oxford University Press

Summary

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. In this book Christopher Pincock tackles this perennial question in a new way by asking how mathematics contributes to the success of our best scientific representations. In the first part of the book this question is posed and sharpened using a proposal for how we can determine the content of a scientific representation. Several different sorts of contributions from mathematics are then articulated. Pincock argues that each contribution can be understood as broadly epistemic, so that what mathematics ultimately contributes to science is best connected with our scientific knowledge. In the second part of the book, Pincock critically evaluates alternative approaches to the role of mathematics in science. These include the potential benefits for scientific discovery and scientific explanation. A major focus of this part of the book is the indispensability argument for mathematical platonism. Using the results of part one, Pincock argues that this argument can at best support a weak form of realism about the truth-value of the statements of mathematics. The book concludes with a chapter on pure mathematics and the remaining options for making sense of its interpretation and epistemology. Thoroughly grounded in case studies drawn from scientific practice, this book aims to bring together current debates in both the philosophy of mathematics and the philosophy of science and to demonstrate the philosophical importance of applications of mathematics.

Author Biography

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.

Table of Contents

List of Figures	p. xi
Preface	p. xiii
Mathematical Notation	p. xv
Introduction	p. 3
A Problem	p. 3
Classifying Contributions	p. 5
An Epistemic Solution	p. 8
Explanatory Contributions	p. 12
Other Approaches	p. 15
Interpretive Flexibility	p. 18
Key Claims	p. 21
Epistemic Contributions
Content and Confirmation	p. 25
Concepts	p. 25
Basic Contents	p. 27
Enriched Contents	p. 29
Schematic and Genuine Contents	p. 31
Inference	p. 33
Core Conceptions	p. 35
Intrinsic and Extrinsic	p. 36
Confirmation Theory	p. 38
Prior Probabilities	p. 41
Causes	p. 45
Accounts of Causation	p. 45
A Causal Representation	p. 48
Some Acausal Representations	p. 51
The Value of Acausal Representations	p. 60
Batterman and Wilson	p. 63
Varying Interpretations	p. 66
Abstraction as Variation	p. 66
Irrotational Fluids and Electrostatics	p. 68
Shock Waves	p. 74
The Value of Varying Interpretations	p. 78
Varying Interpretations and Discovery	p. 80
The Toolkit of Applied Mathematics	p. 82
Scale Matters	p. 87
Scale and Scientific Representation	p. 87
Scale Separation	p. 88
Scale Similarity	p. 93
Scale and Idealization	p. 96
Perturbation Theory	p. 104
Multiple Scales	p. 105
Interpreting Multiscale Representations	p. 113
Summary	p. 120
Constitutive Frameworks	p. 121
A Different Kind of Contribution	p. 121
Carnap's Linguistic Frameworks	p. 122
Kuhn's Paradigms	p. 126
Friedman on the Relative A Priori	p. 131
The Need for Constitutive Representations	p. 137
The Need for the Absolute A Priori	p. 138
Failures	p. 141
Mathematics and Scientific Failure	p. 141
Completeness and Segmentation Illusions	p. 142
The Parameter Illusion	p. 146
Illusions of Scale	p. 153
Illusions of Traction	p. 155
Causal Illusions	p. 161
Finding the Scope of a Representation	p. 163
Other Contributions
Discovery	p. 169
Semantic and Metaphysical Problems	p. 169
A Descriptive Problem	p. 175
Description and Discovery	p. 179
Defending Naturalism	p. 183
Natural Kinds	p. 187
Indispensability	p. 190
Descriptive Contributions and Pure Mathematics	p. 190
Quine and Putnam	p. 190
Against the Platonist Conclusion	p. 196
Colyvan	p. 200
Explanation	p. 203
Explanatory Contributions	p. 203
Inference to the Best Mathematical Explanation	p. 210
Belief and Understanding	p. 217
The Rainbow	p. 221
Asymptotic Explanation	p. 221
Angle and Color	p. 223
Explanatory Power	p. 228
Supernumerary Bows	p. 229
Interpretation and Scope	p. 236
Batterman and Belot	p. 239
Looking Ahead	p. 242
Fictionalism	p. 243
Motivations	p. 243
Literary Fiction	p. 244
Mathematics	p. 250
Models	p. 256
Understanding and Truth	p. 261
Facades	p. 264
Physical and Mathematical Concepts	p. 264
Against Semantic Finality	p. 265
Developing and Connecting Patches	p. 268
A New Approach to Content	p. 275
Azzouni and Rayo	p. 278
Conclusion: Pure Mathematics	p. 279
Taking Stock	p. 279
Metaphysics	p. 280
Structuralism	p. 284
Epistemology	p. 285
Peacocke and Jenkins	p. 290
Historical Extensions	p. 293
Nonconceptual Justification	p. 295
Past and Future	p. 297
Method of Characteristics	p. 301
Black-Scholes Model	p. 303
Speed of Sound	p. 305
Two Proofs of Euler's Formula	p. 307
Bibliography	p. 309
Index	p. 317
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