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버트런드 러셀의 수리 수학적 철학의 기초적 서론.The Book of Introduction to Mathematical Philosophy, byBertrand Russell CONTENTS CHAP. PAGE PREFACE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii EDITOR’S NOTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v 1. THE SERIES OF NATURAL NUMBERS. . . . . . . . . . . . . . . . . . . . 1 2. DEFINITION OF NUMBER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3. FINITUDE AND MATHEMATICAL INDUCTION . . . . . . . . . . 24 4. THE DEFINITION OF ORDER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5. KINDS OF RELATIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6. SIMILARITY OF RELATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7. RATIONAL, REAL, AND COMPLEX NUMBERS. . . . . . . . . . . 77 8. INFINITE CARDINAL NUMBERS . . . . . . . . . . . . . . . . . . . . . . . . . . 95 9. INFINITE SERIES AND ORDINALS . . . . . . . . . . . . . . . . . . . . . . . . 110 10. LIMITS AND CONTINUITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 11. LIMITS AND CONTINUITY OF FUNCTIONS. . . . . . . . . . . . . . 132 12. SELECTIONS AND THE MULTIPLICATIVE AXIOM . . . . . . 145 13. THE AXIOM OF INFINITY AND LOGICAL TYPES . . . . . . . 163 14. INCOMPATIBILITY AND THE THEORY OF DEDUCTION 179 15. PROPOSITIONAL FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 16. DESCRIPTIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 17. CLASSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 18. MATHEMATICS AND LOGIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

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PREFACE This book is intended essentially as an “Introduction,” and does not aim at giving an exhaustive discussion of the problems with which it deals. It seemed desirable to set forth certain results, hitherto only available to those who have mastered logical symbolism, in a form offering the minimum of difficulty to the beginner. The utmost endeavour has been made to avoid dogmatism on such questions as are still open to serious doubt, and this endeavour has to some extent dominated the choice of topics considered. The beginnings of mathematical logic are less definitely known than its later portions, but are of at least equal philosophical interest. Much of what is set forth in the following chapters is not properly to be called “philosophy,” though the matters concerned were included in philosophy so long as no satisfactory science of them existed. The nature of infinity and continuity, for example, belonged in former days to philosophy, but belongs now to mathematics. Mathematical philosophy, in the strict sense, cannot, perhaps, be held to include such definite scientific results as have been obtained in this region; the philosophy of mathematics will naturally be expected to deal with questions on the frontier of knowledge, as to which comparative certainty is not yet attained. But speculation on such questions is hardly likely to be fruitful unless the more scientific parts of the principles of mathematics are known. A book dealing with those parts may, therefore, claim to be an introduction to mathematical philosophy, though it can hardly claim, except where it steps outside its province, to be actually dealing with a part of philosophy. It does deal, however, with a body of knowledge which, to those who accept it, appears to invalidate much traditional philosophy, and even a good deal of what is current in the present day. In this way, as well as by its bearing on still unsolved problems, mathematical logic is relevant to philosophy. For this reason, as well as on account of the intrinsic importance of the subject, some purpose may be served by a succinct account of the main results of 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

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