Accommodationism in Mathematics

Mathematics presents a fundamental puzzle: we possess extraordinary confidence in mathematical knowledge while struggling to explain how such knowledge is possible, especially in light of traditional realist conceptions of mathematics. This book solves this puzzle by resolving the tension between mathematical knowledge and mathematical realism through the development of an innovative accommodationist theory of reference in mathematics. The book provides an account of mathematical reference utilizing detailed case studies of mathematical induction, aspects of computability theory, and compactness phenomena in mathematics and logic. It answers two of the most significant and persistent open questions in the philosophy of mathematics: the Benacerraf-Field Challenge, which defies us to explain the reliability of mathematicians' mathematical beliefs, and the question of mathematical realism. By adapting Richard Boyd's influential theory of scientific reference to mathematics, the book shows how mathematical terms successfully refer through accommodation between mathematical practices and stable features of mathematical phenomena, while preserving traditional views of mathematical knowledge as both a priori and foundationalist. Accommodationism in Mathematics will appeal to researchers and graduate students working in philosophy and foundations of mathematics, philosophy of science, philosophy of language, and epistemology, as well as foundationally-minded mathematicians.

Jeffrey W. Roland is Associate Professor of Philosophy at Louisiana State University, USA. He specializes in the philosophy of mathematics, epistemology, and the philosophy of science with a special interest in the nature and scope of philosophical naturalism and the prospects for a realist philosophy of mathematics. His publications include articles in Philosophy and Phenomenological Research, The British Journal for the Philosophy of Science, The Australasian Journal of Philosophy, Erkenntnis, and Pacific Philosophical Quarterly.

Introduction 1. Rethinking Causal-Historical Theories of Reference 1.1 Reference Fixing and Reference "Fixing" 1.2 Reference as an Epistemic Notion 2. Accommodationism in Science 3. Interlude: Causal Theories of Knowledge and Reliabilism 4. Stability in Mathematics 4.1 Mathematical Induction 4.2 Dovetailing Computations 4.3 Compactness Phenomena 4.4 Two Hierarchies and Post's Theorem 5. Accommodationism in Mathematics 6. Answering the Benacerraf-Field Challenge 7. Defending Mathematical Realism 8. Preserving Apriority and Foundationalism 8.1 Apriority 8.2 Foundationalism 9. Concluding Thoughts