Library of Philosophy EDITED BY J. H. MUIRHEAD, LL.D. INTRODUCTION TO MATHEMATICAL PHILOSOPHY By the same Author. PRINCIPLES OF SOCIAL RECONSTRUCTION. 3rd Impression. Demy 8vo. 7s. 6d. net. "Mr Russell has written a big and living book."—The Nation. ROADS TO FREEDOM: SOCIALISM, ANARCHISM, AND SYNDICALISM. Demy 8vo. 7s. 6d. net. An attempt to extract the essence of these three doctrines, first historically, then as guidance for the coming reconstruction. London: George Allen & Unwin, Ltd. [Pg iii] INTRODUCTION TO MATHEMATICAL PHILOSOPHY BY BERTRAND RUSSELL LONDON: GEORGE ALLEN & UNWIN, LTD. NEW YORK: THE MACMILLAN CO. [Pg iv] First published May 1919 Second Edition April 1920 [All rights reserved] [Pg v] 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, [Pg vi] 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