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PDF Version 4 Introduction to Mathematical Philosophy by Bertrand Russell Originally published by George Allen & Unwin, Ltd., London. May . Online Corrected Edition version . (February , ), based on the “second edition” (second printing) of April , incorporating additional corrections, marked in green. ii Introduction to Mathematical Philosophy [Russell’s blurb from the original dustcover:] CONTENTS This book is intended for those who have no previous acquain- tance with the topics of which it treats, and no more knowledge of mathematics than can be acquired at a primary school or even at Eton. It sets forth in elementary form the logical definition of number, the analysis of the notion of order, the modern doctrine Contents.............................. iii of the infinite, and the theory of descriptions and classes as sym- Preface............................... iv bolic fictions. The more controversial and uncertain aspects of the Editor’s Note............................ vi subject are subordinated to those which can by now be regarded I. The Series of Natural Numbers............ as acquired scientific knowledge. These are explained without the II. Definition of Number.................. use of symbols, but in such a way as to give readers a general un- III. Finitude and Mathematical Induction........ derstanding of the methods and purposes of mathematical logic, IV. The Definition of Order................ which, it is hoped, will be of interest not only to those who wish V. Kinds of Relations.................... to proceed to a more serious study of the subject, but also to that VI. Similarity of Relations................. wider circle who feel a desire to know the bearings of this impor- VII. Rational, Real, and Complex Numbers........ tant modern science. VIII. Infinite Cardinal Numbers............... IX. Infinite Series and Ordinals.............. X. Limits and Continuity................. XI. Limits and Continuity of Functions.......... XII. Selections and the Multiplicative Axiom....... XIII. The Axiom of Infinity and Logical Types....... XIV. Incompatibility and the Theory of Deduction.... XV. Propositional Functions................ XVI. Descriptions....................... XVII. Classes.......................... XVIII. Mathematics and Logic................. Index................................ Appendix: Changes to Online Edition............. iii Preface v tually dealing with a part of philosophy. It does deal, vihowever, 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 PREFACE its bearing on still unsolved problems, mathematical logic is rel- evant 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 ap- titude for mathematical symbolism. Here, however, as elsewhere, v This book is intended essentially as an “Introduction,” and does the method is more important than the results, from the point of not aim at giving an exhaustive discussion of the problems with view of further research; and the method cannot well be explained which it deals. It seemed desirable to set forth certain results, within the framework of such a book as the following. It is to hitherto only available to those who have mastered logical sym- be hoped that some readers may be sufficiently interested to ad- bolism, in a form offering the minimum of difficulty to the begin- vance to a study of the method by which mathematical logic can ner. The utmost endeavour has been made to avoid dogmatism be made helpful in investigating the traditional problems of phi- on such questions as are still open to serious doubt, and this en- losophy. But that is a topic with which the following pages have deavour has to some extent dominated the choice of topics con- not attempted to deal. sidered. The beginnings of mathematical logic are less definitely BERTRAND RUSSELL. known than its later portions, but are of at least equal philosoph- ical interest. Much of what is set forth in the following chapters is not properly to be called “philosophy,” though the matters con- cerned were included in philosophy so long as no satisfactory sci- ence 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, can- not, 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 ac- iv EDITOR’S NOTE CHAPTER I THE SERIES OF NATURAL NUMBERS vii [The note below was written by J. H. Muirhead, LL.D., editor of the Mathematics is a study which, when we start from its most fa- Library of Philosophy series in which Introduction to Mathematical miliar portions, may be pursued in either of two opposite direc- Philosophy was originally published.] tions. The more familiar direction is constructive, towards grad- ually increasing complexity: from integers to fractions, real num- Those who, relying on the distinction between Mathematical Phi- bers, complex numbers; from addition and multiplication to dif- losophy and the Philosophy of Mathematics, think that this book ferentiation and integration, and on to higher mathematics. The is out of place in the present Library, may be referred to what the other direction, which is less familiar, proceeds, by analysing, to author himself says on this head in the Preface. It is not neces- greater and greater abstractness and logical simplicity; instead of sary to agree with what he there suggests as to the readjustment asking what can be defined and deduced from what is assumed to of the field of philosophy by the transference from it to mathe- begin with, we ask instead what more general ideas and principles matics of such problems as those of class, continuity, infinity, in can be found, in terms of which what was our starting-point can order to perceive the bearing of the definitions and discussions be defined or deduced. It is the fact of pursuing this opposite di- that follow on the work of “traditional philosophy.” If philoso- rection that characterises mathematical philosophy as opposed to phers cannot consent to relegate the criticism of these categories ordinary mathematics. But it should be understood that the dis- to any of the special sciences, it is essential, at any rate, that they tinction is one, not in the subject matter, but in the state of mind should know the precise meaning that the science of mathematics, of the investigator. Early Greek geometers, passing from the em- in which these concepts play so large a part, assigns to them. If, pirical rules of Egyptian land-surveying to the general proposi- on the other hand, there be mathematicians to whom these defini- tions by which those rules were found to be justifiable, and thence tions and discussions seem to be an elaboration and complication to Euclid’s axioms and postulates, were engaged in mathematical of the simple, it may be well to remind them from the side of phi- philosophy, according to the above definition; but when once the losophy that here, as elsewhere, apparent simplicity may conceal a axioms and postulates had been reached, their deductive employ- complexity which it is the business of somebody, whether philoso- ment, as we find it in Euclid, belonged to mathematics in the pher or mathematician, or, like the author of this volume, both in ordinary sense. The distinction between mathematics and math-| one, to unravel. ematical philosophy is one which depends upon the interest in- spiring the research, and upon the stage which the research has vi Introduction to Mathematical Philosophy Chap. I. The Series of Natural Numbers reached; not upon the propositions with which the research is con- series: cerned. , , , , ... n, n + ,... We may state the same distinction in another way. The most and it is this series that we shall mean when we speak of the “series obvious and easy things in mathematics are not those that come of natural numbers.” logically at the beginning; they are things that, from the point of It is only at a high stage of civilisation that we could take this view of logical deduction, come somewhere in the middle. Just as series as our starting-point. It must have required many ages to the easiest bodies to see are those that are neither very near nor discover that a brace of pheasants and a couple of days were both very far, neither very small nor very great, so the easiest concep- instances of the number : the degree of abstraction involved is tions to grasp are those that are neither very complex nor very sim- far from easy. And the discovery that is a number must have ple (using “simple” in a logical sense). And as we need two sorts been difficult. As for , it is a very recent addition; the Greeks and of instruments, the telescope and the microscope, for the enlarge- Romans had no such digit. If we had been embarking upon math- ment of our visual powers, so we need two sorts of instruments ematical philosophy in earlier days, we should have had to start for the enlargement of our logical powers, one to take us forward with something less abstract than the series of natural numbers, to the higher mathematics, the other to take us backward to the which we should reach as a stage on our backward journey.
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