BY LONDON: GEORGE ALLEN & UNWIN, LTD. NEW YORK: THE MACMILLAN CO. First published May 1919 Second Edition April 1920 [All rights reserved]...

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 top

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 definitio

MATHEMATICS is a study which, when we start from its most familiar portions, may be pursued in either of two opposite directions. The more familiar direction is constructive, towards gradually increasing complexity: from integers to fractions, real numbers, complex numbers; from addition and multiplication to differentiation and integration, and on to higher mathematics. The other direction, which is less familiar, proceeds, by analysing, to greater and greater abstractness and logical simplicit

THE question "What is a number?" is one which has been often asked, but has only been correctly answered in our own time. The answer was given by Frege in 1884, in his Grundlagen der Arithmetik . [3] Although this book is quite short, not difficult, and of the very highest importance, it attracted almost no attention, and the definition of number which it contains remained practically unknown until it was rediscovered by the present author in 1901. [3] The same answer is given more fully and wit

THE series of natural numbers, as we saw in Chapter I., can all be defined if we know what we mean by the three terms "0," "number," and "successor." But we may go a step farther: we can define all the natural numbers if we know what we mean by "0" and "successor." It will help us to understand the difference between finite and infinite to see how this can be done, and why the method by which it is done cannot be extended beyond the finite. We will not yet consider how "0" and "successor" are to

WE have now carried our analysis of the series of natural numbers to the point where we have obtained logical definitions of the members of this series, of the whole class of its members, and of the relation of a number to its immediate successor. We must now consider the serial character of the natural numbers in the order 0, 1, 2, 3,.... We ordinarily think of the numbers as in this order , and it is an essential part of the work of analysing our data to seek a definition of "order" or "series

A great part of the philosophy of mathematics is concerned with relations , and many different kinds of relations have different kinds of uses. It often happens that a property which belongs to all relations is only important as regards relations of certain sorts; in these cases the reader will not see the bearing of the proposition asserting such a property unless he has in mind the sorts of relations for which it is useful. For reasons of this description, as well as from the intrinsic interes

WE saw in Chapter II. that two classes have the same number of terms when they are "similar," i.e. when there is a one-one relation whose domain is the one class and whose converse domain is the other. In such a case we say that there is a "one-one correlation" between the two classes. In the present chapter we have to define a relation between relations, which will play the same part for them that similarity of classes plays for classes. We will call this relation "similarity of relations," or

WE have now seen how to define cardinal numbers, and also relation-numbers, of which what are commonly called ordinal numbers are a particular species. It will be found that each of these kinds of number may be infinite just as well as finite. But neither is capable, as it stands, of the more familiar extensions of the idea of number, namely, the extensions to negative, fractional, irrational, and complex numbers. In the present chapter we shall briefly supply logical definitions of these variou

THE definition of cardinal numbers which we gave in Chapter II. was applied in Chapter III. to finite numbers, i.e. to the ordinary natural numbers. To these we gave the name "inductive numbers," because we found that they are to be defined as numbers which obey mathematical induction starting from 0. But we have not yet considered collections which do not have an inductive number of terms, nor have we inquired whether such collections can be said to have a number at all. This is an ancient prob

AN "infinite series" may be defined as a series of which the field is an infinite class. We have already had occasion to consider one kind of infinite series, namely, progressions. In this chapter we shall consider the subject more generally. The most noteworthy characteristic of an infinite series is that its serial number can be altered by merely re-arranging its terms. In this respect there is a certain oppositeness between cardinal and serial numbers. It is possible to keep the cardinal numb

THE conception of a "limit" is one of which the importance in mathematics has been found continually greater than had been thought. The whole of the differential and integral calculus, indeed practically everything in higher mathematics, depends upon limits. Formerly, it was supposed that infinitesimals were involved in the foundations of these subjects, but Weierstrass showed that this is an error: wherever infinitesimals were thought to occur, what really occurs is a set of finite quantities h

IN this chapter we shall be concerned with the definition of the limit of a function (if any) as the argument approaches a given value, and also with the definition of what is meant by a "continuous function." Both of these ideas are somewhat technical, and would hardly demand treatment in a mere introduction to mathematical philosophy but for the fact that, especially through the so-called infinitesimal calculus, wrong views upon our present topics have become so firmly embedded in the minds of

IN this chapter we have to consider an axiom which can be enunciated, but not proved, in terms of logic, and which is convenient, though not indispensable, in certain portions of mathematics. It is convenient, in the sense that many interesting propositions, which it seems natural to suppose true, cannot be proved without its help; but it is not indispensable, because even without those propositions the subjects in which they occur still exist, though in a somewhat mutilated form. Before enuncia

THE axiom of infinity is an assumption which may be enunciated as follows:— "If be any inductive cardinal number, there is at least one class of individuals having terms." If this is true, it follows, of course, that there are many classes of individuals having terms, and that the total number of individuals in the world is not an inductive number. For, by the axiom, there is at least one class having terms, from which it follows that there are many classes of terms and that is not the number of

WE have now explored, somewhat hastily it is true, that part of the philosophy of mathematics which does not demand a critical examination of the idea of class . In the preceding chapter, however, we found ourselves confronted by problems which make such an examination imperative. Before we can undertake it, we must consider certain other parts of the philosophy of mathematics, which we have hitherto ignored. In a synthetic treatment, the parts which we shall now be concerned with come first: th

WHEN, in the preceding chapter, we were discussing propositions, we did not attempt to give a definition of the word "proposition." But although the word cannot be formally defined, it is necessary to say something as to its meaning, in order to avoid the very common confusion with "propositional functions," which are to be the topic of the present chapter. We mean by a "proposition" primarily a form of words which expresses what is either true or false. I say "primarily," because I do not wish

We dealt in the preceding chapter with the words all and some ; in this chapter we shall consider the word the in the singular, and in the next chapter we shall consider the word the in the plural. It may be thought excessive to devote two chapters to one word, but to the philosophical mathematician it is a word of very great importance: like Browning's Grammarian with the enclitic , I would give the doctrine of this word if I were "dead from the waist down" and not merely in a prison. We have a

IN the present chapter we shall be concerned with the in the plural: the inhabitants of London, the sons of rich men, and so on. In other words, we shall be concerned with classes . We saw in Chapter II. that a cardinal number is to be defined as a class of classes, and in Chapter III. that the number 1 is to be defined as the class of all unit classes, i.e. of all that have just one member, as we should say but for the vicious circle. Of course, when the number 1 is defined as the class of all

MATHEMATICS and logic, historically speaking, have been entirely distinct studies. Mathematics has been connected with science, logic with Greek. But both have developed in modern times: logic has become more mathematical and mathematics has become more logical. The consequence is that it has now become wholly impossible to draw a line between the two; in fact, the two are one. They differ as boy and man: logic is the youth of mathematics and mathematics is the manhood of logic. This view is res