constituent (or proper) part of itself; in the contrary case S is called a ‘finite’ system.
“Theorem. — There exist infinite systems.[1]
“Proof. — My own realm of thoughts (meine Gedankenwelt), i.e. the totality S, of all things that can be objects of my thought, is infinite. For if s is an element of S, it follows that the thought s’, viz., the thought, That s can be object of my thought, is itself an element of S. If one views s’ as the image (or representative) of the element s, the representation S’ of the system S, which is hereby defined, has the character that the representation S’ is a constituent portion (echter Theil) of S, since there are elements in S (for example, my own Ego) (?) which are different from every such thought s’, and which are, therefore, not contained in S’. Finally, it is plain that if ɑ and b are different elements of S, their images, ɑ’ and b’ are also different, so that the representation of S is distinct (deutlich) and similar. It follows that S [by definition], is infinite.”
Here, as we observe, the infinity of an ideal system is defined, and in a special case proved, without making any explicit reference to the number of its elements. That this number, negatively viewed, turns out to be no finite number, that is, to be that of a multitude with no last term, is for Dedekind a result to be later proved, — a secondary consequence of the infinity as first defined. The proof that my Gedankenwelt is infinite, is thus not my negative powerlessness to find the last term, but my positive power to image each of my thoughts s, by a new and reflective thought s’. It is the finite, and not the infinite, that here appears as the object negatively definable. For a finite system is one that cannot be adequately represented through a one-to-one correspondence with one of its own constituent parts.[2] In any
- ↑ Es giebt unendliche Systeme. Es giebt, is of course here used to express existence within the realm of consistent mathematical definitions. The conception of Being in question is the Third Conception of our own list.
- ↑ That the finite and infinite here quite change places is pointed out in an interesting way by Professor Franz Meyer, in his Antrittsrede at Tübingen entitled Zur Lehre vom Unendlichen (Tubingen, 1889). The same observation is made by Kerry in his comments upon Dedekind (in Kerry’s before-cited Theorie der Grenzbegriffe, p. 49). Bolzano, who, in his Paradoxien des Unendlichen had much earlier reached a position in many ways near to that of Dedekind, proves the existence of the infinite in a closely similar, but less exact way. Schroeder, in his very elaborate essay in the Abhandlungen der Leopold. Carolinischen Akad. d. Naturforscher for 1898, entitled Ueber Zwei Definitionem der Endlichkeit, insists indeed that this whole distinction between positive and negative definitions is, from the point of view of formal Logic, vain, and that Mr. Charles Peirce’s definition of finite systems, given in the American Journal of Mathematics, Vol. 7, p. 202, while it is the polar opposite of Dedekind’s definition of the Infinite, is, logically speaking, at once equivalent to Dedekind’s definition, and yet as positive as the latter, although Mr. Peirce, in the passage in question, starts from the finite, and not from the infinite. Schroeder seems to me quite right in regarding the distinction between essentially positive and essentially negative definitions as one for which a purely formal Logic has no place. But as a fact, the distinction in question, between what is positive and what is negative, has an import wholly metaphysical. Our interest in it here lies in the fact that if you begin, in Dedekind’s way, with the positive concept of the Infinite, you need not presuppose the “externally given” Many, but may develope the multitude out of the internal meaning of a single purpose. Mr. Charles Peirce, in his parallel definition of finite systems, has first to presuppose them as given facts of experience. We, however, are seeking to develope the Many out of the One.
