Mathematics and Its Logics
Philosophical Essays
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. Les mer
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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.
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Utgitt:
2021
Forlag: Cambridge University Press
Innbinding: Innbundet
Språk: Engelsk
ISBN: 9781108494182
Format: 24 x 16 cm
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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 (with Solomon Feferman); 8. Challenges to Predicative Foundations of Arithmetic (with
Solomon Feferman); 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.
Geoffrey Hellman is Professor of Philosophy at the University of Minnesota, Twin Cities. His publications include Mathematics
without Numbers: Towards a Modal-Structural Interpretation (1989), Varieties of Continua: From Regions to Points and Back
(with Stewart Shapiro, 2018), and Mathematical Structuralism, Cambridge Elements in Philosophy of Mathematics (with Stewart
Shapiro, Cambridge, 2018).