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.

In a similar vein, we tried also to narrow the range of philosophical issues discussed in the selection ones that could most easily be recognized as concerning the philosophy of mathematics. Both of these admittedly loose principles served as guidelines only; but any attempt to observe them inevitably constrains the range of literature available for consideration.

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.

Seminal articles in the philosophy of mathematics by Russell, Quine, G??del and other major thinkers.

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.