Page:The World and the Individual, First Series (1899).djvu/529

510 end at any given point, and by then saying, “A series taken thus as without end, may be called infinite.” We ourselves, so far in this discussion, have defined our infinite processes on the whole in a negative way. But the new definition of the infinity of our system uses positive rather than negative terms. The conception of a representation or of an imaging of one object by another, is wholly positive. This conception, if applied to the elements of a system A, with the proviso that A’, the image or the representation of A, shall form a constituent portion of A itself, remains still positive. But the system A, if defined as capable of this particular type of self-representation, proves, when examined, to contain, if it exists at all, an infinite number of elements. Whatever the metaphysical fate of the ideal object thus defined, the method of definition has a decided advantage over the older ones. It may be well at once to quote Dedekind’s original statement and illustration of the conception in question, in the passage cited in the note: —

“A System S is called ‘infinite’ when it is similar to a