Philosophy of mathematics: an introduction

Abstract

Introduction. Part I: Plato versus Aristotle: . A. Plato . 1. The Socratic Background. 2. The Theory of Recollection. 3. Platonism in Mathematics. 4. Retractions: the Divided Line in Republic VI (509d-511e). B. Aristotle . 5. The Overall Position. 6. Idealizations. 7. Complications. 8. Problems with Infinity. C. Prospects . Part II: From Aristotle to Kant: . 1. Medieval Times. 2. Descartes. 3. Locke, Berkeley, Hume. 4. A Remark on Conceptualism. 5. Kant: the Problem. 6. Kant: the Solution. Part III: Reactions to Kant: . 1. Mill on Geometry. 2. Mill versus Frege on Arithmetic. 3. Analytic Truths. 4. Concluding Remarks. Part IV: Mathematics and its Foundations: . 1. Geometry. 2. Different Kinds of Number. 3. The Calculus. 4. Return to Foundations. 5. Infinite Numbers. 6. Foundations Again. Part V: Logicism: . 1. Frege. 2. Russell. 3. Borkowski/Bostock. 4. Set Theory. 5. Logic. 6. Definition. Part VI: Formalism: . 1. Hilbert. 2. Godel. 3. Pure Formalism. 4. Structuralism. 5. Some Comments. Part VII: Intuitionism: . 1. Brouwer. 2. Intuitionist Logic. 3. The Irrelevance of Ontology. 4. The Attack on Classical Logic. Part VIII: Predicativism: . 1. Russell and the VCP. 2. Russell's Ramified Theory and the Axiom of Reducibility. 3. Predicative Theories after Russell. 4. Concluding Remarks. Part IX: Realism versus Nominalism: . A. Realism . 1. Godel. 2. Neo-Fregeans. 3. Quine and Putnam. B. Nominalism . 4. Reductive Nominalism. 5. Fictionalism. 6. Concluding Remarks. References. Index