The Principles of Mathematics (1903)

§ 148

We may now sum up the results obtained in Part II. In the first four chapters, the modern mathematical theory of cardinal integers, as it results from the joint labours of arithmeticians and symbolic logicians, was briefly set forth. Chapter XI explained the notion of similar classes, and showed that the usual formal properties of integers result from defining them as classes of similar terms. In Chapter XII, we showed how arithmetical addition and multiplication both depend upon logical addition, and how both may be defined in a way which applies equally to finite and infinite numbers, and to finite and infinite sums and products, and which moreover introduces nowhere any idea of order. In Chapter XIII, we gave the strict definition of an infinite class, as one which is similar to a class resulting from taking away one of its terms; and we showed in outline how to connect this definition with the definition of finite numbers by mathematical induction. The special theory of finite integers was discussed in Chapter XIV, and it was shown how the primitive propositions, which Peano proves to be sufficient in this subject, can all be deduced from our definition of finite cardinal integers. This confirmed us in the opinion that Arithmetic contains no indefinables or indemonstrables beyond those of general logic.(§ 148 ¶ 1)

We then advanced, in Chapter XV, to the consideration of philosophical questions, with a view of testing critically the above mathematical deductions. We decided to regard both term and a term as indefinable, and to define the number 1, as well as all other numbers, by means of these indefinables (together with certain others). We also found it necessary to distinguish a class from its class-concept, since one class may have several different class-concepts. We decided that a class consists of all the terms denoted by the class-concept, denoted in a certain indefinable manner; but it appeared that both common usage and the majority of mathematical purposes would allow us to identify a class with the whole formed of the terms denoted by the class-concept. The only reasons against this view were, the necessity of distinguishing a class containing only one term from that one term, and the fact that some classes are members of themselves. We found also a distinction between finite and infinite classes, that the former can, while the latter cannot, be defined extensionally, i.e. by actual enumeration of their terms. We then proceeded to discuss what may be called the addition of individuals, i.e. the notion involved in A and B; and we found that a more or less independent theory of finite integers can be based upon this notion. But it appeared finally, in virtue of our analysis of the notion of class, that this theory was really indistinguishable from the theory previously expounded, the only difference being that it adopted an extensional definition of classes.(§ 148 ¶ 2)

Chapter XVI dealt with the relation of whole and part. We found that there are two indefinable senses of this relation, and one definable sense, and that there are two correspondingly different sorts of wholes, which we called unities and aggregates respectively. We saw also that, by extending the notion of aggregates to single terms and to the null-class, we could regard the whole of the traditional calculus of Symbolic Logic as an algebra specifically applicable to the relations of wholes and parts in the definable sense. We considered next, in Chapter XVII, the notion of an infinite whole. It appeared that infinite unities, even if they be logically possible, at any rate never appear in anything accessible to human knowledge. But infinite aggregates, we found, must be admitted; and it seemed that all infinite wholes which are not unities must be aggregates of terms, though it is by no means necessary that the terms should be simple. (They must, however, owing to the exclusion of infinite unities, be assumed to be of finite complexity.)(§ 148 ¶ 3)

In Chapter XVIII, finally, we considered ratios and fractions: the former were found to be somewhat complicated rleations of finite integers, while the latter were relations between the divisibilities of aggregates. These divisibilities being magnitudes, their further discussion belongs to Part III, in which the general nature of quantity is to be considered.(§ 148 ¶ 4)