whereby he directs his investigation of the relations of phenomena, which relations he then generalizes by induction and enunciates as laws. To classify these laws in turn, theories are necessary. The more and better defined the laws, the fewer theories, and ideally but one should be tenable. However, no science is so rich in laws but that contradictory principles may be held. Thus, as to warmth, Mayer holds that it is not, Joule that it is, a mode of motion. Both reach the same equations. In the whole progress of modern science, then, the marked feature is the enunciation of these equations, superseding thereby explanations based on hypothetical properties of matter. Science advances to a purely formal and algorithmic explanation of things. H. C. Howe.
The physicist seems to be compelled to believe in a magnitude having parts, and not infinitely divisible; the mathematician, to postulate a magnitude continuous and infinitely divisible. The opposition is that of magnitude and number. A magnitude (say a line AB) must be divisible, for it can be supposed greater or less, not infinitely divisible, for there could be no particle greater than zero, which, infinitely multiplied, would not exceed any assignable number. Doubtless the sensible line is continuous, but we must pass to the reality, which is discontinuous. The understanding, starting with the phenomenal, goes toward the subjective and infinite; the reason starts at the same point, but proceeds to the objective and finite. Mathematics is of the understanding, physics of the reason. The atom, which rules in the physical world, is the very negation of infinite divisibility; but the mathematician admits in each finite magnitude, infinite divisibility. Mathematicians oppose number to magnitude as the discontinuous to the continuous. Back of this, however, we are compelled to see the necessary conflict of the potential or subjective (number) with the actual or objective (magnitude), the former infinite as nothing in thought, which creates it, creates a limit; the latter finite, for it is a datum. Idealism sees only the infinite, realism only the finite. Now a pure mathematics would deal only with number. Geometry and the like are mixed sciences, which, applying number to magnitude, impose upon the latter properties not its own. Hence the self-contradictory opinion among mathematicians that magnitude itself is infinitely divisible.
