索书号：O1-0 /B456（2）（MIT） Philosophy of mathematics: Selected readings, SECOND EDITTION Contents

Preface to the second edition vii Introduction 1 Part I. The foundation of mathematics Symposium on the foundations of the mathematics 41 1. The logicist foundation of mathematics 41 2. The intuitionist foundations of mathematics 52 3. The formalist foundations of mathematics 61 Disputation 66 Intuitionism and formalist 77 Consciousness, philosophy, and mathematics 90 The philosophical basis of intuitionistic logic 97 The concept of number 130 Selections form Introduction to Mathematical Philosophy 160 On the infinite 183 Remarks on the definition and nature of mathematics 202 Hilbert’s programme 207 Part II. The existence of mathematical objects Empiricism, semantics, and ontology 241 On Platonism in mathematics 258 What numbers could not be 272 Mathematics without foundations 295 Part III. Mathematical truth The priori 315 Truth by convention 329 Carnap and logical truth 355 On the nature of mathematical truth 377 On the nature of mathematical reasoning 394 Mathematical truth 403 Models and reality 421 Part IV. The concept of set Russell’s mathematical logic 447 What is Cantor’s continuum problem? 470 The iterative concept of set 486 What is the iterative conception of set? 503 The concept of set 530 Abstract

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.