sentative system, in the sense here in question, or else, being but a part of the real world, it is a more or less arbitrarily selected, or an empirically given portion or constituent of such a system, — a portion whose reality, apart from that of the whole system, is unintelligible.
Far from lacking totality, then, in the way in which the infinite, or rather the indefinite, multitude of such accounts as Mr. Bosanquet’s is said to lack totality,[1] those genuinely self-representative systems, whose images are portions of their own objects, are the only ones which can be said to possess any totality whatever. It is they alone that are wholly positive in their definition. Finite systems are either capable only of negative definition, or, at all events, have positive characters only by virtue of their relation to their inclusive infinite, or, in our present sense, self-representative systems.[2] Or, again, as we have already begun to see, only the processes of recurrent thought make explicit the true unity of the One and the Many. But these very processes express themselves in systems of the type now in question.
To make these matters clearer, it will be necessary to consider each of the just-mentioned illustrations more in detail. First, then, as to the simple case of the number-system, whose logical genesis we for the moment leave out of consideration, and whose general constitution we assume as known. The whole numbers first form what Cantor calls a wohl-definirte Menge, — or exactly defined multitude. That is, you can precisely distinguish between any conceived or presented object that is not a whole number (as, for example, one-half, or the moral law, or the odor of a rose), and an object that is a whole number, abstract or concrete (e.g. ten, or ten thousand, or the
- ↑ See Bosanquet’s Logic, loc. cit. et sq.
- ↑ Mr. Charles Peirce, as noted above, has indeed given a perfectly positive and exact definition of a finite system; but in order to set that definition to work you have first to suppose your Many externally given, while, in order to define the Gedankenwelt, or the Self, or, as we shall later see, the Real World, you have only to presuppose a single, and unavoidable, internal meaning. The infinity then follows of itself.
