Philosophy of Mathematics

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 Godel. In the...

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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 Godel. 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 Godel 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.

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;
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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 Gödel;
24. What in Cantor's continuum problem? Kurt Gödel;
25. The iterative concept of set George Boolos;
26. The concept of set Hao Wang;
Bibliography.
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