Page:The World and the Individual, First Series (1899).djvu/539

520 paper, will especially interest us. In this type a system is capable of standing in an exact one-to-one correspondence with one of its own constituent portions. We are to be interested throughout this paper in cases of self-representation, such as Self-consciousness, and the relation between thought and Reality, and all the problems of Reflection, bring to our notice. And in all these cases, as we shall see, the system before us will combine the characters of selfhood and internal unity of nature, with the character of being also internally manifold, self-dirempted, Other than Self, and that in most complex and highly antithetic fashion. The relational systems of the type of the number-system especially exemplify — of course in a highly abstract fashion — the sort of unity in contrast, and of exact self-representation, which we are to learn to comprehend. Hence, the stress here to be laid upon one type of self-representative system.

Yet, mathematically regarded, this is indeed only one of several possible types of self-representation.

In the work by Dedekind already cited, the general name, Kette, is given to any self-representative system, whether of the present type or any other self-representative type. In the most general terms, a Kette is formed when a system is made to correspond, whether exactly, and element for element, or in any other way, either to the whole, or to a part of itself. The correspondence might be summary and inexact in type, if to many elements of the original system a single element of the representation or image were made to correspond, as, in a summary account or diagram, a single item or stroke can be made, at pleasure, to correspond to a whole series of facts in the original object which the account or the diagram represents. In this way, for instance, the one word prime can be made to correspond, in a given discussion, to all the prime numbers. If, in case of a Kette, the correspondence of the whole to the part is of this inexact type, the Kette need not be endless, but may even consist of the original object, and a single one of its constituent parts. Then all the later members of the Kette, the A’’, the A’’’, etc., of the previous account,