contradict the axiom as to the whole and the part.[1] These arguments can be illustrated by an endless list of examples, drawn from the realm of discrete collections of objects, as well as from cases where limitless extended lines, surfaces, or volumes are in question, and from cases where limitless divisibility is to be exemplified. The variety of the examples, however, need not confuse one as to the main issue. What is brought out, in every case, is that the infinite collections or multitudes, if real at all, must be in paradoxical contrast to all finite multitudes, and must also be in such contrast as to seem, at first sight, either quite indeterminate or else hopelessly incomplete, and, in either case, incapable of reality.
Upon a somewhat different basis rest a series of arguments which have more novelty, just because they are due to the experience of the modern exact sciences. In the seventeenth century one of the greatest methodical advances ever made in the history of descriptive science occurred, when the so-called Infinitesimal Calculus was invented. The Newtonian name, Fluxions, used for the objects to whose calculation the new science was devoted, indicated better than much of the more recent terminology, that one principal purpose of this advance in method, was to enable mathematical exactness to be used in the description of continuously varying quantities. But the generalization which was made when the Calculus appeared had been the outcome of a long series of studies of quantity, both temporal and spatial. And the Calculus brought under one method of treatment, not only the problems about continuous processes of actual change, such as motions, or other continuous physical alterations, but also problems regarding the properties, the relations, the lengths, and the areas of curves, and
- ↑ Couturat, in his dialectical discussion between the “finitist” and the “infinitist,” in L’Infini Mathematique, p. 443 sqq., gives full room to a statement of these arguments of his opponents. Our account of the Ketten has discounted them in advance. Dedekind’s Definition of the Infinite deliberately makes naught of them. If infinite multitudes corresponding to his definition can be proved real, these paradoxes will be simply obvious properties of such multitudes.
