Philosophy of mathematics: an introduction

1.Plato versus Aristotle

A. Plato

The Socratic background

The theory of recollection

Platonism in mathematics

Retractions: the divided line in Republic VI (509d-511e)

B. Aristotle

The overall position

Idealisations

Complications

Problems with infinity

C. Prospects

2. From Aristotle to Kant

Medieval times

Descartes

Locke, Berkeley, Hume

A remark on conceptualism

Kant: the problem

Kant: the solution

3.Reactions to Kant

Mill on geometry

Mill versus Frege on arithmetic

Analytic truths

Concluding remarks

4.Mathematics and its Foundations

Geometry

Different kinds of number

The calculus

Return to Foundations

Infinite numbers

Foundations again

5.Logicism

Frege

Russell

Borkowski/Bostock

Set theory

Logic

Definition

6.Formalism

Hilbert

Gödel

Pure formalism

Structuralism

Some comments

7.Intuitionism

Brouwer

Intuitionist logic

The irrelevance of ontology

The attack on classical logic

8.Predicativism

Russell and the VCP

Russell's ramified theory and the axiom of reducibility

Predicative theories after Russell

9.Realism versus Nominalism

A. Realism

Gödel

Neo-Fregeans

Quine and Putnam

B. Nominalism

Reductive nominalism

Fictionalism

Concluding remarks.