I will discuss the first objection in connection with Delbœuf's articles referred to above. The second, which has not, to my knowledge, been widely used in criticism, will be better deferred to Chapter III.
Delbœuf.
98. M. Delbœuf's four articles in the Revue Philosophique contain much matter that has already been dealt with in the criticism of Lotze, and much that is irrelevant for our present purpose. The only point, which I wish to discuss here, is the question of absolute magnitude, as it is called—the question, that is, whether the possibility of similar but unequal geometrical figures can be known à priori[109].
In discussing this question, it is important, to begin with, to distinguish clearly the sense in which absolute magnitude is required in non-Euclidean Geometry, from another sense, in which it would be absurd to regard any magnitude as absolute. Judgments of magnitude can only result from comparison, and if Metageometry required magnitudes which could be determined without comparison, it would certainly deserve condemnation. But this is not required. All we require is, that it shall be impossible, while the rest of space is unaffected, to alter the magnitude of any figure, as compared with other figures, while leaving the relative internal magnitudes of its parts unchanged. This construction, which is possible in Euclid, is impossible in Metageometry. We have to discuss whether such an impossibility renders non-Euclidean spaces logically faulty.
M. Delbœuf's position on this axiom—which he calls the postulate of homogeneity[110]—is, that all Geometry must presuppose it, and that Metageometry, consequently, though logically sound, is logically subsequent to Euclid, and can only make its constructions within a Euclidean "homogeneous" space (Rev. Phil. Vol. XXXVII., pp. 380–1). He would appear to think, nevertheless, that homogeneity (in his sense) is learnt from experience, though on this point he is not very explicit. (See Vol. XXXVIII., p. 129.) No à priori proof, at any rate, is offered in his articles. As a result of experience, every one would admit, similarity is known to be possible within the limits of observation; but the fact that this possibility extends to Ordnance maps, which deal with a spherical surface, should make us chary of inferring, from such a datum, the certainty of Euclid for large spaces. Moreover if homogeneity be empirical, Metageometry, which dispenses with it, is not necessarily in logical dependence upon Euclid, since homogeneity and isogeneity are logically separable. I shall assume, therefore, as the only contention which can be interesting to our argument, that homogeneity is regarded as à priori, and as logically essential to Geometry.
99. Now we saw, in discussing Erdmann's views of the judgment of quantity, that in non-Euclidean space, as in Euclidean, a change of all spatial magnitudes, in the same ratio, would be no change at all; the ratios of all magnitudes to the space-constant would be unchanged, and the space-constant, as the ultimate standard of comparison, cannot, in any intelligible sense, be said to have any particular magnitude. The absolute magnitudes of Metageometry, therefore, are absolute only as against any other particular magnitude, not as against other magnitudes in general. If this were not the case, the comparative nature of the judgment of magnitude would be contradicted, and metrical Metageometry would become absurd. But as it is, the difference from Euclid consists only in this: that in Metageometry we have, while in Euclid we have not, a standard of comparison involved in the nature of our space as a whole, which we call the space-constant. We have to discuss whether the assertion of such a standard involves an undue reification of space.
I do not believe that this is the case. For an undue reification of space would only arise, if we were no longer able to regard position as wholly relative, and as geometrically definable only by departure from other positions. But the relativity of position, as we have abundantly seen, is preserved by all spaces of constant curvature—in all of these, positions can only be defined, geometrically, by relations to fresh positions[111]. This series of definitions may lead to an infinite regress, but it may also, as in spherical space, form a vicious circle, and return again to the position from which it started. No reification of space, no independent existence of mere relations, seems involved in such a procedure. The whole of Metageometry, in short, is a proof that the relativity of position is compatible with absolute magnitude, in the only sense required by non-Euclidean spaces. We must conclude, therefore, that there is nothing incompatible, in a denial of homogeneity (in Delbœuf's sense), either with the relational nature of space, or with the comparative nature of magnitude. This last à priori objection to Metageometry, therefore, cannot be maintained, and the issue must be decided on empirical grounds alone.
100. The foundations of Geometry have been the subject of much recent speculation in France, and this seems to demand some notice. But in spite of the splendid work which the French have done on the allied question of number and continuous quantity, I cannot persuade myself that they have succeeded in greatly advancing the subject of geometrical philosophy. The chief writers have been, from the mathematical side, Calinon and Poincaré, from the philosophical, Renouvier and Delbœuf; as a mediator between mathematics and philosophy, Lechalas.
Calinon, in an interesting article on the geometrical indeterminateness of the universe, maintains that any Geometry may be applied to the actual world by a suitable hypothesis as to the course of light-rays. For the earth only is known to us otherwise than by Optics, and the earth is an infinitesimal part of the universe. This line of argument has been already discussed in connection with Lotze, but Calinon adds a new suggestion, that the space-constant may perhaps vary with the time. This would involve a causal connection between space and other things, which seems hardly conceivable, and which, if regarded as possible, must surely destroy Geometry, since Geometry depends throughout on the irrelevance of Causation[112]. Moreover, in all operations of measurement, some time is spent; unless we knew that space was unchanging throughout the operation, it is hard to see how our results could be trustworthy, and how, consequently, a change in the parameter could be discovered. The same difficulties would arise, in fact, as those which result from supposing space not homogeneous.