Applying Mathematics: Immersion, Inference, Interpretation

Description

How is that when scientists need some piece of mathematics through which to frame their theory, it is there to hand? What has been called 'the unreasonable effectiveness of mathematics' sets a challenge for philosophers. Some have responded to that challenge by arguing that mathematics is essentially anthropocentric in character, whereas others have pointed to the range of structures that mathematics offers. Otavio Bueno and Steven French offer a middle way, which focuses on the moves that have to be made in both the mathematics and the relevant physics in order to bring the two into appropriate relation. This relation can be captured via the inferential conception of the applicability of mathematics, which is formulated in terms of immersion, inference, and interpretation. In particular, the roles of idealisations and of surplus structure in science and mathematics respectively are brought to the fore and captured via an approach to models and theories that emphasize the partiality of the available information: the partial structures approach. The discussion as a whole is grounded in a number of case studies drawn from the history of quantum physics, and extended to contest recent claims that the explanatory role of certain mathematical structures in scientific practice supports a realist attitude towards them. The overall conclusion is that the effectiveness of mathematics does not seem unreasonable at all once close attention is paid to how it is actually applied in practice. Cover Applying Mathematics: Immersion, Inference, Interpretation Copyright Dedication Preface Contents Acknowledgements List of Illustrations 1: Just How Unreasonable is the Effectiveness of Mathematics? 1.1 Introduction 1.2 Mystery Mongering 1.3 Mathematical Optimism 1.4 Mathematical Opportunism 2: Approaching Models: Formal and Informal 2.1 Introduction 2.2 One Extreme: The Structuralists 2.3 The Other Extreme: Giere's Account of Models 2.4 Surplus Structure 2.5 Partial Structures 2.6 Problems with Isomorphism 2.7 The Inferential Conception 2.8 Conclusion. 3: Scientific Representation and the Application of Mathematics3.1 Introduction 3.2 General Challenge: Reducing all Representations to the Mental 3.3 Specific Challenges to the Formal Account 3.3.1 Necessity 3.3.2 Sufficiency 3.3.3 Logic 3.3.4 Mechanism 3.3.5 Style 3.3.6 Misrepresentation 3.3.7 Ontology 3.4 The Formal Account Defended 3.4.1 Preamble: Informational vs Functional Accounts 3.4.2 Necessity 3.4.3 Logic and Sufficiency 3.4.4 Mechanism 3.4.5 Style 3.4.6 Misrepresentation 3.4.7 Ontology 4: Applying New Mathematics: Group Theory and Quantum Mechanics. 4.1 Introduction4.2 The Historical Context 4.3 Applying Group Theory to Atoms 4.3.1 A (Very) Brief History of Quantum Statistics 4.3.2 The 'Wigner Programme' 4.3.3 The 'Weyl Programme' 4.4 Applying Group Theory to Nuclei 4.5 Conclusion 5: Representing Physical Phenomena: Top-Down and Bottom-Up 5.1 Introduction 5.2 From the Top: The Applicability of Mathematics 5.3 Bose-Einstein Statistics and Superfluidity 5.3.1 The Liquid Degeneracy of Helium 5.3.2 The Application of Bose-Einstein Statistics 5.4 The Autonomy of London's Model 5.5 Conclusion. 6: Unifying with Mathematics: Logic, Probability, and Quantum States6.1 Introduction 6.2 Group Theory, Hilbert Spaces, and Quantum Mechanics 6.3 Logic and Empiricism 6.4 The 1937 Manuscript: Logics and Experience 6.5 The Status of Mathematics 7: Applying Problematic Mathematics, Interpreting Successful Structures: From the Delta Function to the Positron 7.1 Introduction 7.2 Dirac and the Delta Function 7.2.1 Introducing the Delta Function 7.2.2 Dispensing with the Delta Function 7.2.3 The Status of the Delta Function 7.3 The Pragmatic and Heuristic Role of Mathematics in Physics. 7.4 The Discovery of Antimatter7.5 Conclusion 8: Explaining with Mathematics? From Cicadas to Symmetry 8.1 Introduction 8.2 The Strong Claim and Indispensability 8.3 The Enhanced Indispensability Argumentand Explanation 8.3.1 Indexing and Representing 8.3.2 Explaining 8.4 The Weak Claim and the Hybridity of Spin 8.5 Conclusion 9: Explaining with Mathematics?: Idealization, Universality, and the Criteria for Explanation 9.1 Introduction 9.2 Immersion, Inference, and Partial Structures 9.3 Idealization and Surplus Structure 9.4 The Rainbow. 9.5 Accommodating Process and Limit Operations.