An Essay on the Foundations of Geometry

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. Poincaré maintains that the question, whether Euclid or Metageometry should be accepted, is one of convenience and convention, not of truth; axioms are definitions in disguise, and the choice between definitions is arbitrary. This view has been discussed in Chapter I., in connection with Cayley's theory of distance, on which it depends. Lechalas is a philosophical disciple of Calinon. He is a rationalist of the pre-Kantian type, but a believer in the validity of Metageometry. He holds that Geometry can dispense with all purely spatial postulates, and work with axioms of magnitude alone[113], which, in his opinion, are purely analytic. The principle of contradiction, to him, is the sole and only test of truth; we make long chains of reasoning from our premisses to see if contradictions will emerge. It might be objected that this view, though it saves general Geometry from being logically empirical, leaves it only empirically logical; this must, in fact, be the fate of every piece of à priori knowledge, if M. Lechalas's were the only test of truth. However, he concludes that general Geometry is apodeictic, while the space of our actual world, like all other phenomena, is contingent. Delbœuf criticizes non-Euclidean space from an ultra-realist standpoint: he holds that real space is neither homogeneous nor isogeneous, but that conceived space, as abstracted from real space, has both these properties. He offers no justification for his real space, which seems to be maintained in the spirit of naïve realism, nor does he show how he has acquired his intimate knowledge of its constitution[114]. His arguments against Metageometry, in so far as they are not repetitions of Lotze, have been discussed above. Renouvier, finally, is a pure Kantian, of the most orthodox type. His views as to the importance, for Geometry, of the distinction between synthetic and analytic judgments, have been discussed, in connection with Kant, at the beginning of the present Chapter[115]. 101. Before beginning the constructive argument of the next Chapter, let us endeavour briefly to sum up the theories which have been polemically advocated throughout the criticisms we have just concluded. We agreed to accept, with Kant, necessity for any possible experience as the test of the à priori, but we refused, for the present, to discuss the connection of the à priori with the subjective, regarding the purely logical test as sufficient for our immediate purpose. We also refused to attach importance to the distinction of analytic and synthetic, since it seemed to apply, not to different judgments, but only to different aspects of any judgment. We then discussed Riemann's attempt to identify the empirical element in Geometry with the element not deducible from ideas of magnitude, and we decided that this identification was due to a confusion as to the nature of magnitude. For judgments of magnitude, we said, require always some qualitative basis, which is not quantitatively expressible. In criticizing Helmholtz, we decided that Mechanics logically presupposes Geometry, though space presupposes matter; but that the matter which space presupposes, and to which Geometry indirectly refers, is a more abstract matter than that of Mechanics, a matter destitute of force and of causal attributes, and possessed only of the purely spatial attributes required for the possibility of spatial figures. But we conceded that Geometry, when applied to mixed mathematics or to daily life, demands more than this, demands, in fact, some means of discovering, in the more concrete matter of Mechanics, either a rigid body, or a body whose departure from rigidity follows some empirically discoverable law. Actual measurement, therefore, we agreed to regard as empirical. Our conclusions, as regards the empiricism of Riemann and Helmholtz, were reinforced by a criticism of Erdmann. We then had an opposite task to perform, in defending Metageometry against Lotze. Here we saw that there are two senses in which Metageometry is possible. The first concerns our actual space, and asserts that it may have a very small space-constant; the second concerns philosophical theories of space, and asserts a purely logical possibility, which leaves the decision to experience. We saw also that Lotze's mathematical strictures arose from insufficient knowledge of the subject, and could all be refuted by a better acquaintance with Metageometry. Finally, we discussed the question of absolute magnitude, and found in it no logical obstacle to non-Euclidean spaces. Our conclusion, then, in so far as we are as yet entitled to a conclusion, is that all spaces with a space-constant are à priori justifiable, and that the decision between them must be the work of experience. Spaces without a space-constant, on the other hand, spaces, that is, which are not homogeneous throughout, we found logically unsound and impossible to know, and therefore to be condemned à priori. The constructive proof of this thesis will form the argument of the following chapter. FOOTNOTES: [67] The Critical Philosophy of Kant, Vol. I. p. 287. [68] For a discussion of Kant from a less purely mathematical standpoint, see Chap. IV. [69] Cf. Vaihinger's Commentar, II. pp. 202, 265. Also p. 336 ff. [70] E.g. second edition, p. 39: "So werden auch alle geometrischen Grundsätze, z. B. dass in einem Triangel zwei Seiten zusammen grösser sind als die dritte, niemals aus allgemeinen Begriffen von Linie und Triangel, sondern aus der Anschauung, und zwar à priori mit apodiktischer Gewissheit abgeleitet." [71] Cf. Bradley's Logic, Bk. III. Pt. I. Chap. VI.; Bosanquet's Logic, Bk. I. Chap. I. pp. 97–103. [72] Philosophie de la Règle et du Compas, Année Philosophique, II. pp. 1–66. [73] I have stated this doctrine dogmatically, as a proof would require a whole treatise on Logic. I accept the proofs offered by Bradley and Bosanquet, to which the reader is referred. [74] For a further discussion of this point, see Chaps. III. and IV. [75] See Chap. IV. for a discussion of this argument. [76] See Chap. IV. § 185. [77] An Otherness of substance, rather than of attribute, is here intended; an Otherness which may perhaps be called real as opposed to logical diversity. [78] This proposition will be argued at length in Chap. IV. [79] See Psychologie als Wissenschaft, I. Section III. Chap. VII.; II. Section I. Chap. III. and Section II. Chap. III. Compare also Synechologie, Section I. Chaps. II. and III. [80] On the influence of Herbart on Riemann, compare Erdmann, Die Axiome der Geometrie, p. 30. [81] I do not mean that measurement of colours is effected without reference to their relations, since all measurement is essentially comparison. But in colours, it is the elements which are compared, while in space, it is the relations between elements. [82] For a discussion of this point, see Chap. III. Sec. B, § 176. [83] The works of Helmholtz on geometrical philosophy comprise, in addition to the articles quoted in Chap. I., the following articles: "Ursprung und Sinn der geometrischen Axiome, gegen Land," Wiss. Abh. Vol. II. p. 640, 1878. (Also Mind, Vol. III.: an answer to Land in Mind, Vol. II.) "Ursprung und Bedeutung der geometrischen Axiome," 1870, Vorträge und Reden, Vol. II. p. 1. (Also Mind, Vol. I.) Two Appendices to "Die Thatsachen in der Wahrnehmung," entitled: II. "Der Raum kann transcendental sein, ohne dass es die Axiome sind"; and III. "Die Anwendbarkeit der Axiome auf die physische Welt," 1878, Vorträge und Reden, Vol. II. p. 256 ff. The two Appendices last mentioned