mathematical logic in a form requiring neither a knowledge of mathematics nor an aptitude for mathematical symbolism. Here, however, as elsewhere, the method is more important than the results, from the point of view of further research; and the method cannot well be explained within the framework of such a book as the following. It is to be hoped that some readers may be sufficiently interested to advance to a study of the method by which mathematical logic can be made helpful in investigating the traditional problems of philosophy. But that is a topic with which the following pages have not attempted to deal. BERTRAND RUSSELL. [Pg vii] EDITOR'S NOTE THOSE who, relying on the distinction between Mathematical Philosophy and the Philosophy of Mathematics, think that this book is out of place in the present Library, may be referred to what the author himself says on this head in the Preface. It is not necessary to agree with what he there suggests as to the readjustment of the field of philosophy by the transference from it to mathematics of such problems as those of class, continuity, infinity, in order to perceive the bearing of the definitions and discussions that follow on the work of "traditional philosophy." If philosophers cannot consent to relegate the criticism of these categories to any of the special sciences, it is essential, at any rate, that they should know the precise meaning that the science of mathematics, in which these concepts play so large a part, assigns to them. If, on the other hand, there be mathematicians to whom these definitions and discussions seem to be an elaboration and complication of the simple, it may be well to remind them from the side of philosophy that here, as elsewhere, apparent simplicity may conceal a complexity which it is the business of somebody, whether philosopher or mathematician, or, like the author of this volume, both in one, to unravel. [Pg viii] CONTENTS CHAP. PREFACE EDITOR'S NOTE 1. THE SERIES OF NATURAL NUMBERS 2. DEFINITION OF NUMBER 3. FINITUDE AND MATHEMATICAL INDUCTION 4. THE DEFINITION OF ORDER 5. KINDS OF RELATIONS 6. SIMILARITY OF RELATIONS 7. RATIONAL, REAL, AND COMPLEX NUMBERS 8. INFINITE CARDINAL NUMBERS 9. INFINITE SERIES AND ORDINALS 10. LIMITS AND CONTINUITY 11. LIMITS AND CONTINUITY OF FUNCTIONS 12. SELECTIONS AND THE MULTIPLICATIVE AXIOM 13. THE AXIOM OF INFINITY AND LOGICAL TYPES 14. INCOMPATIBILITY AND THE THEORY OF DEDUCTION 15. PROPOSITIONAL FUNCTIONS 16. DESCRIPTIONS 17. CLASSES 18. MATHEMATICS AND LOGIC INDEX [Pg ix] INTRODUCTION TO MATHEMATICAL PHILOSOPHY CHAPTER I THE SERIES OF NATURAL NUMBERS