Note 81 — Introductory material

After this protest against the initial assumptions in Erdmann's deduction of space, let us return to consider the manner, in which this deduction is carried out. Here there will be less ground for criticism, as the deduction, given its presuppositions, is, I think, as good as such a deduction can be. To define space as a magnitude, he says, let us start with two of its most obvious properties, continuity and the three dimensions. Tones and colours afford other instances of a manifold with these two properties, but differ from space in that their dimensions are not homogeneous and interchangeable. To designate this difference, Erdmann introduces a useful pair of terms: in the general case, he calls a manifold n-determined (n-bestimmt); in the case where, as in space, the dimensions are homogeneous, he calls the manifold n-extended (n-ausgedehnt). Manifolds of the latter sort he calls extents (Ausgedehntheiten). That the difference between the two kinds is one of quality, not of quantity, he seems not to perceive; he also overlooks the fact that, in the second kind, from its very definition, the axiom of Congruence must hold, on account of the qualitative similarity of different parts. In spite of this fact, he defines space as an extent, and then regards Congruence as empirical, and as possibly false in the infinitesimal. This is the more strange, as he actually proves (p. 50) that measurement is impossible, in an extent, unless the parts are independent of their place, and can be carried about unaltered as measures. In spite of this, he proceeds immediately to discuss whether the measure of curvature is constant or variable, without investigating how, in the latter case, Geometry could exist. We cannot know, he says, from geometrical superposition, that geometrical bodies are independent of place, for if their dimensions altered in motion according to any fixed law, two bodies which could be superposed in one place could be superposed in any other. That such a hypothesis involves absolute position, and denies the qualitative similarity of the parts of space, which he declares (p. 171) to be the principle of his theory of Geometry, is nowhere perceived. But what is more, his notion that magnitude is something absolute, independent of comparison, has prevented him from seeing that such a hypothesis is unmeaning. He says himself that, even on this hypothesis, a geometrical body can be defined as one whose points retain constant distances from each other, for, since we have no absolute measure, measurement could not reveal to us the change of absolute magnitude (p. 60). But is not this a reductio ad absurdum? For magnitude is nothing apart from comparison, and the comparison here can only be effected by superposition; if, then, as on the above hypothesis, superposition always gives the same result, by whatever motion it is effected, there is no sense in speaking of magnitudes as no longer equal when separated: absolute magnitude is an absurdity, and the magnitude resulting from comparison does not differ from that which would result if the dimensions of bodies were unchanged in motion. Therefore, since magnitude is only intelligible as the result of comparison, the dimensions of bodies are unchanged in motion, and the suggested hypothesis is unmeaning. On this subject I shall have more to say in Chapter III.[96]