The Principles of Mathematics (1903)

§ 109

It has been common in the past, among those who regarded numbers as definable, to make an exception as regards the number 1, and to define the remainder by its means. Thus 2 was 1 + 1, 3 was 2 + 1, and so on. This method was only applicable to finite numbers, and made a tiresome difference between 1 and other numbers; moreover the meaning of + was commonly not explained. We are able now-a-days to improve greatly upon this method. In the first place, since Cantor has shown how to deal with the infinite, it has become both desirable and possible to deal with fundamental properties of numbers in a way which is equally applicable to finite and infinite numbers. In the second place, the logical calculus has enabled us to give an exact definition of arithmetical addition; and in the third place, it has become as easy to define 0 and 1 as to define any other number. In order to explain how this is done, I shall first set forth the definition of numbers by abstraction; I shall then point out formal defects in this definition, and replace it by a nominal definition.(§ 109 ¶ 1)

Numbers are, it will be admitted, applicable essentially to classes. It is true that, where the number is finite, individuals may be enumerated to make up the given number, and may be counted one by one without any mention of a class-concept. But all finite collections of individuals form classes, so that what results is after all the number of a class. And where the number is infinite, the individuals cannot be enumerated, but must be defined by intension, i.e. by some common property in virtue of which they form a class. Thus when any class-concept is given, there is a certain number of individuals to which this class-concept is applicable, and the number may therefore be regarded as a property of the class. It is this view of numbers which has rendered possible the whole theory of infinity, since it relieves us of the necessity of enumerating the individuals whose number is to be considered. This view depends fundamentally upon the notion of all, the numerical conjunction as we agreed to call it (§ 59). All men, for example, denotes men conjoined in a certain way; and it is as thus denoted that they have a number. Similarly all numbers or all points denotes numbers or points conjoined in a certain way, and as thus conjoined numbers or points have a number. Numbers, then, are to be regarded as properties of classes.(§ 109 ¶ 2)

The next question is: Under what circumstances do two classes have the same number? The answer is, that they have the same number when their terms can be correlated one to one, so that any one term of either corresponds to one and only one term of the other. This requires that there should be some one-one relation whose domain is the one class and whose converse domain is the other class. Thus, for example, if in a community all the men and all the women are married, and polygamy and polyandry are forbidden, the number of men must be the same as the number of women. It might be thought that a one-one relation could not be defined except by reference to the number 1. But this is not the case. A relation is one-one when, if x and x′ have relation in question to y, then x and x′ are identical; while if x has the relation in question to y and y′, then y and y′ are identical. Thus it is possible, without the notion of unity, to define what is meant by a one-one relation. But in order to provide for the case of two classes which have no terms, it is necessary to modify slightly the above account of what is meant by saying that two classes have the same number. For if there are no terms, the terms cannot be correlated one to one. We must say: Two classes have the same number when, and only when, there is a one-one relation whose domain includes the one class, and which is such that the class of correlates of the terms of the one class is identical with the other class. From this it appears that two classes having no terms have always the same number of terms; for if we take any one-one relation whatever, its domain includes the null-class, and the class of correlates of the null-class is again the null-class. When two classes have the same number, they are said to be similar.(§ 109 ¶ 3)

Some readers may suppose that a definition of what is meant by saying that two classes have the same number is wholly unnecessary. The way to find out, they may say, is to count both classes. It is such notions as this which have, until very recently, prevented the exhibition of Arithmetic as a branch of Pure Logic. For the question immediately arises: What is meant by counting? To this question we usually get only some irrelevant psychological answer, as, that counting consists in successive acts of attention. In order to count 10, I suppose that ten acts of attention are required: certainly a most useful definition of the number 10! Counting has, in fact, a good meaning, which is not psychological. But this meaning is highly complex; it is only applicable to classes which can be well-ordered, which are not known to be all classes; and it only gives the number of the class when this number is finite--a rare and exceptional case. We must not, therefore, bring in counting where the definition of numbers is in question.(§ 109 ¶ 4)

The relation of similarity between classes has the three properties of being reflexive, symmetrical, and transitive; that is to say, if u, v, w be classes, u is similar to itself; if u be similar to v, v is similar to u; and if u be similar to v, and v to w, then u is similar to w. These properties all follow easily from the definition. Now these three properties of a relation are held by Peano and common sense to indicate that when the relation holds between two terms, those two terms have a certain common property, and vice versâ. This common property we call their number[75]. This is the definition of numbers by abstraction.(§ 109 ¶ 5)

§ 109 n. 1. Cf. Peano, F. 1901, § 32, ·0, Note.