The Principles of Mathematics (1903)

§ 130

Let us return to the notion of the numerical conjunction. It is plain that it is of such objects as A and B, A and B and C, that numbers other than one are to be asserted. We examined such objects, in Part I, in relation to classes, with which we found them to be identical. Now we must investigate their relation to numbers and plurality.(§ 130 ¶ 1)

The notion to be now examined is the notion of a numerical conjunction or, more shortly, a collection. This is not to be identified, to begin with, with the notion of a class, but is to receive a new and independent treatment. By a collection I mean what is conveyed by A and B or A and B and C, or any other enumeration of definite terms. The collection is defined by the actual mention of the terms, and the terms are connected by and. It would seem that and represents a fundamental way of combining terms, and it might be urged that just this way of combination is essential if anything is to result of which a number other than 1 is to be asserted. Collections do not presuppose numbers, since they result simply from the terms together with and: they could only presuppose numbers in the particular case where the terms of the collection themselves presupposed numbers. There is a grammatical difficulty which, since no method exists of avoiding it, must be pointed out and allowed for. A collection, grammatically, is one, whereas A and B, or A and B and C, are essentially many. The strict meaning of collection is the whole composed of many, but since a word is needed to denote the many themselves, I choose to use the word collection in this sense, so that a collection, according to the usage here adopted, is many and not one.(§ 130 ¶ 2)

As regards what is meant by the combination indicated by and, it gives what we called before the numerical conjunction. That is A and B is what is denoted by the concept of a class of which A and B are the only terms, and is precisely A and B denoted in the way which is indicated by all. We may say, if u be the class-concept corresponding to a class of which A and B are the only terms, that all u's is a concept which denotes the terms A, B combined in a certain way, and A and B are those terms combined in precisely that way. Thus A and B appears indistinguishable from the class, though distinguishable from the class-concept and from the concept of the class. Hence if u be a class of more than one term, it seems necessary to hold that u is not one, but many, since u is distinguished both from the class-concept and from the whole composed of the terms of u[91]. Thus we are brought back to the dependence of numbers upon classes; and where it is not said that the classes in question are finite, it is practically necessary to begin with class-concepts and the theory of denoting, not with the theory of and which has just been given. The theory of and applies practically only to finite numbers, and gives to finite numbers a position which is different, at least psychologically, from that of infinite numbers. There are, in short, two ways of defining particular finite classes, but there is only one practicable way of defining particular infinite classes, namely by intension. It is largely the habit of considering classes primarily from the side of extension which has hitherto stood in the way of a correct logical theory of infinity.(§ 130 ¶ 3)

§ 130 n. 1. A conclusive reason against identifying a class with the whole composed of its terms is, that one of these terms may be the class itself, as in the case class is a class, or rather, classes are one among classes. The logical type of the class class is of an infinite order, and therefore the usual objection to xx does not apply to this case.