The essays in this volume present a case for pluralism in mathematics and its logics, largely supporting coexistence despite apparent contradictions between different systems. In addition, the volume further develops Hellman's modal-structuralist account of mathematics, recognizing indefinite extendability of models and stages at which sets occur.
The essays in this volume present a case for pluralism in mathematics and its logics, largely supporting coexistence despite apparent contradictions between different systems. In addition, the volume further develops Hellman's modal-structuralist account of mathematics, recognizing indefinite extendability of models and stages at which sets occur.
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).
Inhaltsangabe
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.
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.
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