Mathematics and Scientific Representation

Christopher Pincock

Oxford Studies in Philosophy of Science

Considers many different case studies that show the different ways mathematics contributes to the success of science
Brings together philosophy of mathematics and philosophy of science in a fruitful way.

Mathematics and Scientific Representation

Christopher Pincock

Oxford Studies in Philosophy of Science

Description

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.

Mathematics and Scientific Representation

Christopher Pincock

Oxford Studies in Philosophy of Science

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

Mathematics and Scientific Representation

Christopher Pincock

Oxford Studies in Philosophy of Science

Author Information

Christopher Pincock, Associate Professor of Philosophy, University of Missouri

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.

Mathematics and Scientific Representation

Christopher Pincock

Oxford Studies in Philosophy of Science

Reviews and Awards

"a rare and fairly comprehensive philosophical account of the success of mathematics in science and after reading it you may be left with the impression that something like this should have been published years ago. This book is a major contribution to an otherwise underdeveloped area in the philosophy of science and is most likely to be well referenced ... this book is at the cutting-edge." - Stuart Rowlands, Science & Education

"Mathematics and Scientific Representation is an engaging piece of contemporary philosophy of mathematics and science. Its deeply science-informed approach and focus on applied mathematics, with an aim to seriously tackle also more traditional issues in philosophy of mathematics, exemplify exciting and fertile scholarly 'border-hopping'." - Juha Saatsi, Notre Dame Philosophical Reviews

"Pincocks writing style is engaging, and the book is structured in a way that makes it easy to follow the contours of the main lines of argumentan...an impressive book and one that repays detailed reading and re-reading." - Alan Baker, The British Journal for the Philosophy of Science

"Pincock's book is an excellent analysis of some of the most important topics in philosophy of science and philosophy of mathematics, and is well worth a read for any philosopher interested in the issue of mathematical application." - Ashley Graham Kennedy, International Studies in the Philosophy of Science