Christopher Pincock

Mathematics and scientific representation

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 Carnaps Linguistic Frameworks 6.3 Kuhns 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 Eulers Formula

A Revealing Flaw in Colyvan's Indispensability Argument*

Mark Colyvan uses applications of mathematics to argue that mathematical entities exist. I claim that his argument is invalid based on the assumption that a certain way of thinking about applications, called ‘the mapping account,’ is correct. My main contention is that successful applications depend only on there being appropriate structural relations between physical situations and the mathematical domain. As a variety of non‐realist interpretations of mathematics deliver these structural relations, indispensability arguments are invalid.

Overextending Partial Structures: Idealization and Abstraction

The partial structures program of da Costa, French and others offers a unified framework within which to handle a wide range of issues central to contemporary philosophy of science. I argue that the program is inadequately equipped to account for simple cases where idealizations are used to construct abstract, mathematical models of physical systems. These problems show that da Costa and French have not overcome the objections raised by Cartwright and Suárez to using model‐theoretic techniques in the philosophy of science. However, my concerns arise independently of the more controversial assumptions that Cartwright and Suárez have employed.

Towards a Philosophy of Applied Mathematics

If there is any trend in contemporary philosophy of mathematics worthy of the label “New Wave”, it is surely the call to turn our attention to the practices, priorities and developments that are prized by working mathematicians. Most actual mathematicians, unsurprisingly, have little interest in the questions that have dominated the philosophy of mathematics since the 1960s. Few mathematicians, for example, are likely to be troubled by Benacerraf’s argument in “What Numbers Could Not Be” that numbers are not objects for the simple reason that they typically do their mathematics without worrying about what numbers might be. Similarly, the extended debates between platonists and nominalists, and the associated epistemo-logical worries about our knowledge of ordinary mathematics, have little impact on what is usually called “mathematical practice”.

Mathematics, Science, and Confirmation Theory

This article begins by distinguishing intrinsic and extrinsic contributions of mathematics to scientific representation. This leads to two investigations into how these different sorts of contributions relate to confirmation. I present a way of accommodating both contributions that complicates the traditional assumptions of confirmation theory. In particular, I argue that subjective Bayesianism does best in accounting for extrinsic contributions, while objective Bayesianism is more promising for intrinsic contributions.

Mathematical Structural Realism

Unrestricted or global scientific realism is the view that we should take seriously the whole content of empirically successful scientific theories. This attitude requires us to believe that the theoretical claims of the theory are true, or approximately true, and that scientific progress consists in increasing the scope and accuracy of these theories. A series of devastating objections to this position has been developed based on an examination of both the history and practice of science. On the history side, it is arguable that a majority of empirically successful scientific theories are not anywhere near approximately true as we now have evidence that the entities they posited do not exist. The practice of contemporary science raises different and more subtle concerns. Here we find scientists engaging in a wide array of seemingly ad hoc techniques of idealization and approximation. This suggests that we cannot explain the success of our theories by appeal to their truth as the assumptions deployed in the application of these theories have little bearing on the truth of the theoretical claims made by the theory.

Proof and other dilemmas: mathematics and philosophy, edited by Bonnie Gold & Roger A. Simons, Spectrum Series, MAA, 2008 346 pages, hardcover

This book is made up of 16 newly commissioned essays by prominent philosophers of mathematics along with some mathematicians and mathematics educators who have reflected on philosophical aspects of mathematics. As Bonnie Gold, one of the editors, explains in her introduction, the point of the volume is “to increase the level of interest among mathematicians in the philosophy of mathematics” (p. xiii). This goal seems to have led the editors to shy away from contributions

Accounting for the Unity of Experience in Dilthey, Rickert, Bradley and Ward

I. The history of philosophy differs from other kinds of history mainly in its attempts to understand historical change exclusively through the examination and evaluation of philosophical arguments. Of course, nobody writing the history of philosophy is likely to deny that philosophical arguments are given by people and that the context and aspirations of the philosopher will shape her arguments. Despite this concession it remains common practice to ignore such contributions in the reconstruction of a historical figure’s philosophical views. Reflecting an extreme version of this approach, Scott Soames has recently written in response to criticism of his Philosophical Analysis in the Twentieth Century, if progress [in philosophy] is to be made, there must at some point emerge a clear demarcation between genuine accomplishments that need to be assimilated by later practitioners, and other work that can be forgotten, disregarded, or left to those whose interest is not in the subject itself, but in history for its own sake. The aim of my volumes was to contribute to making that demarcation (Soames 2006, 655).