Plato's Problem: An Introduction to Mathematical Platonism

Content: 1 The Origins 16 --
1.1 Plato as a Platonist? 17 --
1.2 Aristotle Between Platonism and Anti-platonism 27 --
1.3 Proclus: The Neoplatonic Interpretation of Euclid's Geometry 32 --
1.4 Kant: The Transcendental Interpretation of Classical Arithmetic and Geometry 36 --
2 From Frege to Gödel (Through Hilbert) 45 --
2.1 Frege's Logicist Platonism 45 --
2.2 Russell and the Separation of Logicism and Platonism 66 --
2.3 Set Theory 69 --
2.4 The Problem of Foundations 73 --
2.5 Gödel's Platonism and the Rise of Mathematical Intuition 90 --
3 Benacerraf's Arguments 99 --
3.1 What Natural Numbers Could Not Be (According to Benacerraf) 101 --
3.2 Benacerraf's Dilemma 107 --
3.3 A Map of Responses to Benacerraf's Dilemma: Contemporary Solutions to Plato's Problem 110 --
4 Non-conservative Responses to Benacerraf's Dilemma 112 --
4.1 Field's Nominalism: Mathematics Without Truth and Science Without Numbers 112 --
4.2 Mathematics as Fiction: Field and Yablo 125 --
4.3 Eliminative Structuralism and its Modal Version 136 --
4.4 Maddy and the Cognitive Origins of Set Theory 144 --
5 Conservative Responses to Benacerraf's Dilemma 149 --
5.1 Neo-logicism: A Revised Version of Frege's Programme 149 --
5.2 Linsky, Zalta and 'Object Theory': Mathematics and Logic (or Metaphysics) of Abstract Objects 165 --
5.3 A First Version of Non-eliminative Structuralism: Ante Rem Structuralism 177 --
5.4 A Second Version of Non-eliminative Structuralism: Parsons and the Role of Intuition 187 --
6 The Indispensability Argument: Structure and Basic Notions 196 --
6.1 Four Versions of IA 197 --
6.2 The Quine-Putnam Argument and Colyvan's Argument 201 --
6.3 (In)dispensability 203 --
6.4 Quine's Criterion of Ontological Commitment 210 --
6.5 Naturalism 212 --
6.6 Confirmational Holism 214 --
6.7 The Dispensability of Naturalism and Confirmational Holism 215 --
7 The Indispensability Argument: The Debate 217 --
7.1 Against Indispensability 217 --
7.2 Against Ontological Commitment 224 --
7.3 Against Naturalism and Scientific Realism 235 --
7.4 Against Confirmational Holism 241.