The Principles of Mathematics (1903)

§ 111

There are two ways in which we may attempt to remedy this defect. One of these consists in defining as the number of a class the whole class of entities, chosen one from each of the above sets of entities, to which all classes similar to the given class (and no others) have some many-one relation or other. But this method is practically useless, since all entities, without exception, belong to every such class, so that every class will have as its number the class of all entities of every sort and description. The other remedy is more practicable, and applies to all the cases in which Peano employs definition by abstraction. This method is, to define as the number of a class the class of all classes similar to the given class. Membership of this class of classes (considered as a predicate) is a common property of all the similar classes of all the similar classes and of no others; moreover every class of the set of similar classes has to the set a relation which it has to nothing else, and which every class has to its own set. Thus the conditions are completely fulfilled by this class of classes, and it has the merit of being determinate when a class is given, and of being different for two classes which are not similar. This, then, is an irreproachable definition of the number of a class in purely logical terms.(§ 111 ¶ 1)

To regard a number as a class of classes must appear, at first sight, a wholly indefensible paradox. Thus Peano (F. 1901, § 32) remarks that we cannot identify the number of [a class] a with the class of classes in question [i.e. the class of classes similar to a], for these objects have different properties. He does not tell us what these properties are, and for my part I am unable to discover them. Probably it appeared to him immediately evident that a number is not a class of classes. But something may be said to mitigate the appearance of paradox in this view. In the first place, such a word as couple or trio obviously does denote a class of classes. Thus what we have to say is, for example, that two men means logical product of class of men and couple, and there are two men means there is a class of men which is also a couple. In the second place, when we remember that a class-concept is not itself a collection, but a property by which a collection is defined, we see that, if we define the number as the class-concept, not the class, a number is really defined as a common property of a set of similar classes and of nothing else. This view removes the appearance of paradox to a great degree. There is, however, a philosophical difficulty in this view, and generally in the connection of classes and predicates. It may be that there are many predicates common to a certain collection of objects and to no others. In this case, these predicates are all regarded by Symbolic Logic as equivalent, and any one of them is said to be equal to any other. Thus if the predicate were defined by the collection of objects, we should not obtain, in general, a single predicate, but a class of predicates; for this class of predicates we should require a new class-concept, and so on. The only available class-concept would be predicability of the given collection of terms and of no others. But in the present case, where the collection is defined by a certain relation to one of its terms, there is some danger of a logical error. Let u be a class; then the number of u, we said, is the class of classes similar to u. But similar to u cannot be the actual concept which constitutes the number of u; for, if v be similar to u, similar to v defines the same class, although it is a different concept. Thus we require, as the defining predicate of the class of similar classes, some concept which does not have any special relation to one or more of the constituent classes. In regard to every particular number that may be mentioned, whether finite or infinite, such a predicate is, as a matter of fact, discoverable; but when all we are told about a number is that it is the number of some class u, it is natural that a special reference to u should appear in the definition. This, however, is not the point at issue. The real point is, that what is defined is the same whether we use the predicate similar to u or similar to v, provided u is similar to v. This shows that it is not the class-concept or defining predicate that is defined, but the class itself whose terms are the various classes which are similar to u or to v. It is such classes, therefore, and not predicates such as similar to u, that must be taken to constitute numbers.(§ 111 ¶ 2)

Thus, to sum up: Mathematically, a number is nothing but a class of similar classes: this definition allows the deduction of all the usual properties of numbers, whether finite or infinite, and is the only one (so far as I know) which is possible in terms of the fundamental concepts of general logic. But philosophically we may admit that every collection of similar classes has some common predicate applicable to no entities except the similar classes in question, and if we can find, by inspection, that there is a certain class of such common predicates, of which one and only one applies to each collection of similar classes, then we may, if we see fit, call this particular class of predicates the class of numbers. For my part, I do not know whether there is any such class of predicates, and I do know that, if there be such a class, it is wholly irrelevant to Mathematics. Wherever Mathematics derives a common property from a reflexive, symmetrical, and transitive relation, all mathematical purposes of the supposed common property are completely served when it is replaced by the class of terms having the given relation to a given term; and this is precisely the case presented by cardinal numbers. For the future, therefore, I shall adhere to the above definition, since it is at once precise and adequate to all mathematical uses.(§ 111 ¶ 3)