ABSTRACT
The word continuity has borne among philosophers, especially since the time of Hegel, a meaning totally unlike that given to it by Cantor. Thus Hegel says*: “Quantity, as we saw, has two sources: the exclusive unit, and the identification or equalization of these units. When we look, therefore, at its immediate relation to self, or at the characteristic of selfsameness made explicit by abstraction, quantity is Continuous magnitude; but when we look at the other characteristic, the One implied in it, it is Discrete magnitude.” When we remember that quantity and magnitude, in Hegel, both mean “cardinal number,” we may conjecture that this assertion amounts to the following: “Many terms, considered as having a cardinal number, must all be members of one class; in so far as they are each merely an instance of the class-concept, they are indistinguishable one from another, and in this aspect the whole which they compose is called continuous; but in order to their maniness, they must be different instances of the class-concept, and in this aspect the whole which they compose is called discrete.” Now I am far from denying—indeed I strongly hold—that this opposition of identity and diversity in a collection constitutes a fundamental problem of Logic—perhaps even the fundamental problem of philosophy. And being fundamental, it is certainly relevant to the study of the mathematical continuum as to everything else. But beyond this general connection, it has no special relation to the mathematical meaning of continuity, as may be seen at once from the fact that it has no reference whatever to order. In this chapter, it is the mathematical meaning that is to be discussed. I have quoted the philosophic meaning only in order to state definitely that this is not here in question; and since disputes about words are futile, I must ask philosophers to divest themselves, for the time, of their habitual associations with the word, and allow it no signification but that obtained from Cantor’s definition.
