Philosophy of Mathematics: An Introduction

Content: 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.