Mathematics and Its Logics: Philosophical Essays

Description

In these essays Geoffrey Hellman presents a strong case for a healthy pluralism in mathematics and its logics, supporting peaceful coexistence despite what appear to be contradictions between different systems, and positing different frameworks serving different legitimate purposes. The essays refine and extend Hellman's modal-structuralist account of mathematics, developing a height-potentialist view of higher set theory which recognizes indefinite extendability of models and stages at which sets occur. In the first of three new essays written for this volume, Hellman shows how extendability can be deployed to derive the axiom of Infinity and that of Replacement, improving on earlier accounts; he also shows how extendability leads to attractive, novel resolutions of the set-theoretic paradoxes. Other essays explore advantages and limitations of restrictive systems - nominalist, predicativist, and constructivist. Also included are two essays, with Solomon Feferman, on predicative foundations of arithmetic. Cover Half-title page Title page Copyright page Contents Acknowledgements Introduction Part I Structuralism, Extendability, and Nominalism 1 Structuralism without Structures 2 What Is Categorical Structuralism? 3 On the Significance of the Burali-Forti Paradox 4 Extending the Iterative Conception of Set: A Height-Potentialist Perspective 5 On Nominalism 6 Maoist Mathematics?: Critical Study of John Burgess and Gideon Rosen, A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics (Oxford, 1997) Part II Predicative Mathematics and Beyond 7 Predicative Foundations of Arithmetic 8 Challenges to Predicative Foundations of Arithmetic Appendix Realizing Dummett’s Approach in EFSC 9 Predicativism as a Philosophical Position 10 On the Gödel–Friedman Program Part III Logics of Mathematics 11 Logical Truth by Linguistic Convention 12 Never Say “Never”!: On the Communication Problem between Intuitionism and Classicism 13 Constructive Mathematics and Quantum Mechanics: Unbounded Operators and the Spectral Theorem 14 If “If-Then” Then What? 15 Mathematical Pluralism: The Case of Smooth Infinitesimal Analysis Index