Introduction to Mathematical Philosophy

1 Introduction to MathematicalPhilosophybyBertrand RussellOriginally published byGeorge 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.[Russell s blurb from the original dustcover:]This book is intended for those who have no previ-ous acquaintance with the topics of which it treats,and no more knowledge of mathematics than canbe acquired at a primary school or even at Eton. Itsets forth in elementary form the logical definitionof number, the analysis of the notion of order, themodern doctrine of the infinite, and the theory ofdescriptions and classes as symbolic fictions. Themore controversial and uncertain aspects of the sub-ject are subordinated to those which can by now beregarded as acquired scientific knowledge. Theseare explained without the use of symbols, but insuch a way as to give readers a general understand-ing of the methods and purposes of mathematicallogic, which, it is hoped, will be of interest not onlyto those who wish to proceed to a more serious studyof the subject, but also to that wider circle who feel adesire to know the bearings of this important mod-ern.

2 IvPreface ..viEditor s Note ..ixI. The Series of Natural Numbers .. II. Definition of Number .. III. Finitude and Mathematical Induction IV. The Definition of Order.. V. Kinds of Relations .. VI. Similarity of Relations .. VII. Rational, Real, and Complex Numbers VIII. Infinite Cardinal Numbers .. IX. Infinite Series and Ordinals .. X. Limits and Continuity.. XI. Limits and Continuity of Functions . XII. Selections and the Multiplicative Ax-iom .. ivXIII. The Axiom of Infinity and LogicalTypes .. XIV. Incompatibility and the Theory of De-duction .. XV. Propositional Functions .. XVI. Descriptions .. XVII. Classes .. XVIII. mathematics and Logic .. Index .. Appendix: Changes to Online Edition .. PrefaceThisbook is intended essentially as an Introduc-vtion, and does not aim at giving an exhaustivediscussion of the problems with which it deals. Itseemed desirable to set forth certain results, hith-erto only available to those who have mastered log-ical symbolism, in a form offering the minimum ofdifficulty to the beginner.

3 The utmost endeavourhas been made to avoid dogmatism on such ques-tions as are still open to serious doubt, and thisendeavour has to some extent dominated the choiceof topics considered. The beginnings of mathemati-cal logic are less definitely known than its later por-tions, but are of at least equal philosophical of what is set forth in the following chaptersis not properly to be called Philosophy , thoughthe matters concerned were included in philosophyso long as no satisfactory science of them nature of infinity and continuity, for example,belonged in former days to Philosophy , but belongsnow to mathematics . Mathematicalphilosophy, inthe strict sense, cannot, perhaps, be held to includesuch definite scientific results as have been obtainedin this region; the Philosophy of mathematics willnaturally be expected to deal with questions on thefrontier of knowledge, as to which comparative cer-tainty is not yet attained.

4 But speculation on suchquestions is hardly likely to be fruitful unless themore scientific parts of the principles of mathemat-ics are known. A book dealing with those parts may,therefore, claim to be anintroductionto mathemati-cal Philosophy , though it can hardly claim, exceptwhere it steps outside its province, to be actuallydealing with a part of Philosophy . It does deal,|however, with a body of knowledge which, to thoseviwho accept it, appears to invalidate much tradi-tional Philosophy , and even a good deal of what iscurrent in the present day. In this way, as well as byits bearing on still unsolved problems, mathemati-cal logic is relevant to Philosophy . For this reason, aswell as on account of the intrinsic importance of thesubject, some purpose may be served by a succinctaccount of the main results of Mathematical logic ina form requiring neither a knowledge of mathemat-ics nor an aptitude for Mathematical , however, as elsewhere, the method is moreimportant than the results, from the point of viewof further research; and the method cannot well beexplained within the framework of such a book asthe following.

5 It is to be hoped that some readersmay be sufficiently interested to advance to a studyof the method by which Mathematical logic can bemade helpful in investigating the traditional prob-lems of Philosophy . But that is a topic with whichthe following pages have not attempted to s Note[The note below was written by J. H. Muirhead, , editor of the Library of Philosophy series inwhichIntroduction to Mathematical Philosophywasoriginally published.]Thosewho, relying on the distinction between Math-ematical Philosophy and the Philosophy of Math-ematics, think that this book is out of place in thepresent Library, may be referred to what the au-thor himself says on this head in the Preface. It isnot necessary to agree with what he there suggestsas to the readjustment of the field of philosophyby the transference from it to mathematics of suchproblems as those of class, continuity, infinity, inorder to perceive the bearing of the definitions anddiscussions that follow on the work of traditionalphilosophy.

6 If philosophers cannot consent to rel-egate the criticism of these categories to any of thespecial sciences, it is essential, at any rate, that theyshould know the precise meaning that the science ofmathematics, in which these concepts play so large apart, assigns to them. If, on the other hand, there bemathematicians to whom these definitions and dis-cussions seem to be an elaboration and complicationof the simple, it may be well to remind them fromthe side of Philosophy that here, as elsewhere, ap-parent simplicity may conceal a complexity whichit is the business of somebody, whether philosopheror mathematician, or, like the author of this volume,both in one, to IThe Series of Natural NumbersMathematicsis a study which, when we start from its most familiar portions, may be pursued in eitherof two opposite directions. The more familiar direc-tion is constructive, towards gradually increasingcomplexity: from integers to fractions, real num-bers, complex numbers; from addition and multipli-cation to differentiation and integration, and on tohigher mathematics .

7 The other direction, which isless familiar, proceeds, by analysing, to greater andgreater abstractness and logical simplicity; insteadof asking what can be defined and deduced fromwhat is assumed to begin with, we ask instead whatmore general ideas and principles can be found, interms of which what was our starting-point can bedefined or deduced. It is the fact of pursuing thisopposite direction that characterises mathematicalphilosophy as opposed to ordinary it should be understood that the distinctionis one, not in the subject matter, but in the stateof mind of the investigator. Early Greek geome-ters, passing from the empirical rules of Egyptianland-surveying to the general propositions by whichthose rules were found to be justifiable, and thenceto Euclid s axioms and postulates, were engagedin Mathematical Philosophy , according to the abovedefinition; but when once the axioms and postulateshad been reached, their deductive employment, aswe find it in Euclid, belonged to mathematics in the|ordinary sense.

8 The distinction between mathe- matics and Mathematical Philosophy is one whichdepends upon the interest inspiring the research,and upon the stage which the research has reached;not upon the propositions with which the researchis may state the same distinction in anotherway. The most obvious and easy things in mathe-matics are not those that come logically at the begin-ning; they are things that, from the point of view oflogical deduction, come somewhere in the as the easiest bodies to see are those that areneither very near nor very far, neither very small norvery great, so the easiest conceptions to grasp arethose that are neither very complex nor very simple(using simple in alogicalsense). And as we needtwo sorts of instruments, the telescope and the mi-croscope, for the enlargement of our visual powers,so we need two sorts of instruments for the enlarge-ment of our logical powers, one to take us forwardto the higher mathematics , the other to take us back-ward to the logical foundations of the things that weare inclined to take for granted in mathematics .

9 Weshall find that by analysing our ordinary mathemat-ical notions we acquire fresh insight, new powers,and the means of reaching whole new mathematicalsubjects by adopting fresh lines of advance after ourbackward journey. It is the purpose of this book toexplain Mathematical Philosophy simply and un-technically, without enlarging upon those portionswhich are so doubtful or difficult that an elemen-tary treatment is scarcely possible. A full treatmentwill be found inPrincipia Mathematica; the treat-ment in the present volume is intended merely asan the average educated person of the presentday, the obvious starting-point of mathematics Cambridge University Press, vol. i., ; vol. ii., ; , . By Whitehead and be the series of whole numbers, , , , , ..etc.|Probably only a person with some Mathematical knowledge would think of beginning with insteadof with , but we will presume this degree of knowl-edge; we will take as our starting-point the series.

10 N, n+ , ..and it is this series that we shall mean when wespeak of the series of natural numbers. It is only at a high stage of civilisation that wecould take this series as our starting-point. It musthave required many ages to discover that a brace ofpheasants and a couple of days were both instancesof the number : the degree of abstraction involvedis far from easy. And the discovery that is a num-ber must have been difficult. As for , it is a veryrecent addition; the Greeks and Romans had nosuch digit. If we had been embarking upon mathe-matical Philosophy in earlier days, we should havehad to start with something less abstract than theseries of natural numbers, which we should reachas a stage on our backward journey. When the log-ical foundations of mathematics have grown morefamiliar, we shall be able to start further back, atwhat is now a late stage in our analysis. But forthe moment the natural numbers seem to representwhat is easiest and most familiar in though familiar, they are not few people are prepared with a definition ofwhat is meant by number, or , or.