Philosophy of Mathematics: Selected Readings

by Edited by Paul Benacerraf , Hilary Putnam

• ISBN13: 9780521296489
• ISBN10: 052129648X
• Edition: 2nd
• Format: Paperback
• Copyright: 1984-01-27
• Publisher: Cambridge University Press

Summary

The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox (Russell's Paradox), a challenge to 'classical' mathematics from a world-famous mathematician (the 'mathematical intuitionism' of Brouwer), a new foundational school (Hilbert's Formalism), and the profound incompleteness results of Kurt Gödel. In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably (but in different ways) with Bertrand Russell, W. V. Quine, and Gödel himself, and which remains at the focus of Anglo-Saxon philosophical discussion. The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field.

Table of Contents

Preface to the second edition

Introduction

Part I. The Foundations of Mathematics: 1. The logicist foundations of mathematics Rudolf Carnap

2. The intuitionist foundations of mathematics Arend Heyting

3. The formalist foundations of mathematics Johann von Neumann

4. Disputation Arend Heyting

5. Intuitionism and formalism L. E. J. Brouwer

6. Consciousness, philosophy, and mathematics L. E. J. Brouwer

7. The philosophical basis of intuitionistic logic Michael Dummett

8. The concept of number Gottlob Frege

9. Selections from Introduction to Mathematical Philosophy Bertrand Russell

10. On the infinite David Hilbert

11. Remarks on the definition and nature of mathematics Haskell B. Curry

12. Hilbert's programme Georg Kreisel

Part II. The Existence of Mathematical Objects: 13. Empiricism, semantics, and ontology Rudolf Carnap

14. On Platonism in mathematics Paul Bernays

15. What numbers could not be Paul Benacerraf

16. Mathematics without foundations Hilary Putnam

Part III. Mathematical Truth: 17. The a priori Alfred Jules Ayer

18. Truth by convention W. V. Quine

19. On the nature of mathematical truth Carl G. Hempel

20. On the nature of mathematical reasoning Henri Poincaré

21. Mathematical truth Paul Benacerraf

22. Models and reality Hilary Putnam

Part IV. The Concept of Set: 23. Russell's mathematical logic Kurt Gö

del

24. What in Cantor's continuum problem? Kurt Gö

del

25. The iterative concept of set George Boolos

26. The concept of set Hao Wang

Bibliography.