the life is. The mathematician’s interests are not the philosopher’s. But neither of the two has a monopoly of the abstractions; and in the end each of them — and certainly the philosopher — can learn from the other. The metaphysic of the future will take fresh account of mathematical research.
The foregoing observation as to the parallelism between the structure of the number-series and the bare skeleton of the ideal Self, is due, then, in its present form, rather to Dedekind than to the idealistic philosophers proper.[1] It shall be briefly expounded in the form in which he has suggested it to me, although his discussion seems to have been written wholly without regard to any general philosophical consequences. And the present is the first attempt, so far as I know, to bring Dedekind’s research into its proper relation to general metaphysical inquiry.
The numbers have been so far taken as we find them. But how do we men come by our number-series? The usual answer is, by learning to count external objects. We see collections of objects, with distinguishable units, the “bare conjunctions” of Mr. Bradley once more. Their mysterious unity in diversity arouses our curiosity. We form the habit, however,
- ↑ Hegel indeed defines the positive Infinite as das Fürsichseiende, and sets it in opposition to the merely negative Infinitive, or das Schlecht-Unendliche. See the well-known discussion in the Logik, Werke, 2te Auflage, Bd. Ill, p. 148, sqq. Dr. W. T. Harris, in his Hegel (Chicago, 1890), and in other discussions, has ably defended and illustrated the Hegelian statements. They are applied to the problem of the quantitative Infinite by Hegel in the Logik, in the volume cited, p. 272 sqq. But near as Hegel thus comes to the full definition of the Infinite, his statement of the matter remains rather a postulate that the self-representative system shall be found, than a demonstration and exact explanation of its reality. The well-known Hegelian assertions that the only true image of the Infinite is the closed cycle (Logik, loc. cit., p. 156), that the quantitative infinite is a return to quality (loc. cit., p. 271), and that the rational fraction, taken as the equivalent of the endless decimal, is the one typical example of the completed quantitatively infinite process, — these, all of them valuable as emphasizing various aspects of the concept of the infinite, appear in the present day wholly inadequate to the complexity of our problem, and rather hinder than aid its final expression.
