Plato's problem. An introduction to mathematical platonism

✍ Scribed by Panza, Marco; Sereni, Andrea

• Publisher: Palgrave Macmillan
• Year: 2013
• Tongue: English
• Leaves: 323
• Category: Library

✦ Synopsis

What is mathematics about? And if it is about some sort of mathematical reality, how can we have access to it? This is the problem raised by Plato, which is still today the subject of lively philosophical disputes. This book traces the history of the problem, from its origins to its contemporary treatment. It discusses the answers given by Aristotle, Proclus and Kant, through Frege's and Russell's versions of logicism, Hilbert's formalism, Gödel's platonism, up to the current debate on Benacerraf's dilemma and the indispensability argument. Through the consideration of themes in the philosophy of language, ontology, and the philosophy of science, the book aims at offering an historically informed introduction to the philosophy of mathematics, approached through the lenses of its most fundamental problem. Book jacket.  Read more...

Abstract:

What is mathematics about? And how can we have access to the reality it is supposed to describe? The book tells the story of this problem, first raised by Plato, through the views of Aristotle, Proclus, Kant, Frege, Goedel, Benacerraf, up to the most recent debate on mathematical platonism.  Read more...

✦ Table of Contents

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.

✦ Subjects

Mathematics -- Philosophy.