An Essay on the Foundations of Geometry

the second period, are still retained by all geometers. In the second chapter, we endeavoured, by a criticism of some geometrical philosophies, to prepare the ground for a constructive theory of Geometry. We saw that Kant, in applying the argument of the Transcendental Aesthetic to space, had gone too far, since its logical scope extended only to some form of externality in general. We saw that Riemann, Helmholtz and Erdmann, misled by the quantitative bias, overlooked the qualitative substratum required by all judgments of quantity, and thus mistook the direction in which the necessary axioms of Geometry are to be found. We rejected, also, Helmholtz's view that Geometry depends on Physics, because we found that Physics must assume a knowledge of Geometry before it can become possible. But we admitted, in Geometry, a reference to matter—not, however, to matter as empirically known in Physics, but to a more abstract matter, whose sole function is to appear in space, and supply the terms for spatial relations. We admitted, however, besides this, that all actual measurement must be effected by means of actual matter, and is only empirically possible, through the empirical knowledge of approximately rigid bodies. In criticizing Lotze, we saw that the most important sense, in which non-Euclidean spaces are possible, is a philosophical sense, namely, that they are not condemned by any à priori argument as to the necessity of space for experience, and that consequently, if they are not affirmed, this must be on empirical grounds alone. Lotze's strictures on the mathematical procedure of Metageometry we found to be wholly due to ignorance of the subject. Proceeding, in the third chapter, to a constructive theory of Geometry, we saw that projective Geometry, which has no reference to quantity, is necessarily true of any form of externality. Its three axioms—homogeneity, dimensions, and the straight line—were all deduced from the conception of a form of externality, and, since some such form is necessary to experience, were all declared à priori. In metrical Geometry, on the contrary, we found an empirical element, arising out of the alternatives of Euclidean and non-Euclidean space. Three à priori axioms, common to these spaces, and necessary conditions of the possibility of measurement, still remained; these were the axiom of Free Mobility, the axiom that space has a finite integral number of dimensions, and the axiom of distance. Except for the new idea of motion, these were found equivalent to the projective triad, and thus necessarily true of any form of externality. But the remaining axioms of Euclid—the axiom of three dimensions, the axiom that two straight lines can never enclose a space, and the axiom of parallels—were regarded as empirical laws, derived from the investigation and measurement of our actual space, and true only, as far as the last two are concerned, within the limits set by errors of observation. In the present chapter, we completed our proof of the apriority of the projective and equivalent metrical axioms, by showing the necessity, for experience, of some form of externality, given by sensation or intuition, and not merely inferred from other data. Without this, we said, a knowledge of diverse but interrelated things, the corner-stone of all experience, would be impossible. Finally, we discussed the contradictions arising out of the relativity and continuity of space, and endeavoured to overcome them by a reference to matter. This matter, we found, must consist of unextended atoms, localized by their spatial relations, and appearing, in Geometry, as points. But the non-spatial adjectives of matter, we contended, are irrelevant to Geometry, and its causal properties may be left out of account. To deal with the new contradictions, involved in such a notion of matter, would demand a fresh treatise, leading us, through Kinematics, into the domains of Dynamics and Physics. But to discuss the special difficulties of space is all that is possible in an essay on the Foundations of Geometry. FOOTNOTES: [178] Compare, with the following paragraphs, the admirable discussion in Mr Hobhouse's Theory of Knowledge (Methuen 1896), Part I. Chapter II. [179] I speak of sense-perception instead of sensation, so as not to prejudge the issue as to the sensational nature of space. [180] See Vaihinger's Commentar, II. pp. 86–7, 168–171. [181] See Caird, Critical Philosophy of Kant, Vol. I. p. 287. [182] Ursprung der Raumvorstellung, pp. 12–30. [183] See the references in Vaihinger's Commentar, II. p. 76 ff. [184] Commentar, II. p. 71 ff. [185] E.g. by Caird, op. cit. Vol. I. p. 286. [186] I have no wish to deny, however, that space is essential in the subsequent distinction of Self and not-Self. [187] See also Book I. Chap II. passim; especially p. 51 ff. and pp. 70–1. [188] Logic, p. 51 ff. [189] For the This, on such a hypothesis, has a purely temporal complexity, and is not resolvable into coexisting Thises. [190] Chapter III. Section A, (§ 131). [191] Cf. Hannequin, Essai critique sur l'hypothèse des atomes, Paris 1895, Chap. I. Section III.; especially p. 43. [192] See Chapter II. § 69 ff. [193] See third antinomy below, § 201 ff. [194] This atom, of course, must not be confounded with the atom of Chemistry. [195] Ursprung der Raumvorstellung, p. 15. [196] See Vaihinger's Commentar, II. pp. 189–190. [197] See ibid. p. 224 ff. for Kant's inconsistencies on this point. [198] The fourth and fifth in the first edition, the third and fourth in the second. [199] Cf. Vaihinger's Commentar, II. p. 218. [200] Cf. Vaihinger's Commentar, II. p. 207. [201] Cf. James, Psychology, Vol. II., p. 148 ff. Cambridge: PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS.