fixed character, as a finished or finally given quantity, nor any character which could be defined as a determinately real somewhat, apart from our defining thought, and apart from the conditions of a given problem. The Calculus is deeply interested in computing results of such variation without limit; but as a branch of mathematics, it is, in fact, not at all directly interested in our present problem about the actually infinite.[1]
Now, this result of the whole experience of the students of the Calculus with the logic of their own science, — this outcome of the modern critical restudy of the bases of the science of the continuously variable quantities, — tends of itself to indicate (as one may say, and as objectors to the actually Infinite have often said) that the conception of the actually Infinite, formerly confounded with the conceptions lying at the bases of the Calculus, is, as a fact, not only in this region, but everywhere, scientifically superfluous; while the conception of the Infinite merely in potentia, originally defended by Aristotle, thus triumphs in the very realm where, for a time, its rival seemed to have found a firm foothold.[2]
Yet it has indeed to be observed that, from the mathematical point of view, not the questions of the Calculus, but certain decidedly special problems of the Theory of Numbers, and of the modern Theory of Functions, have given the mathematical basis for these newer efforts towards an exact and positive definition of the Infinite. As a fact, in our foregoing statement of the merely prima facie case for the recent definition of the positively Infinite, we have deliberately re-
- ↑ All this is not only admitted, but insisted upon by Cantor himself, as a preliminary to his own discussion of das Eigentlich-Unendliche, which he sharply distinguishes from such Uneigentlicher concept of the Infinite as has to be used in the Calculus. See his separately published Grundzüge einer allgemeinen Mannigfaltigkeitslehre (Leipzig, 1883), p. 1, sqq. Compare the statement in Professor Franz Meyer’s lecture, before cited, to the same effect.
- ↑ This line of argument against the Infinite has often been used, — most recently perhaps by F. Evellin, in his two articles directed against the metaphysical use of Cantor’s theories, in the Revue Philosophique for February and November, 1898.
