FOOTNOTES:

The same reference to matter, then, by which the antinomy of the Point was solved, solves also the antinomy as to the relational nature of space. Space, if it is to be freed fromcontradictions, must be regarded exclusively as spatial order, as relations between unextended material atoms. Empty space, which arises, by an inevitable illusion, out of the spatial element in sense-perception, may be regarded, if we wish to retain it, as the bare principle of relativity, the bare logical possibility of relations between diverse things. In this sense, empty space is wholly conceptual; spatial order alone is immediately experienced.

208.But in what sense does spatial order consist of relations? We have hitherto spoken of externality as a relation, and in a sense such a manner of speaking is justified. Externality, when predicated of anything, is an adjective of that thing, and implies a reference to some other thing. To this extent, then, externality is analogous to other relations; and only to this extent, in our previous arguments, has it been regarded as a relation. But when we take account of further qualities of relations, externality begins to appear, not so much as a relation, but rather as a necessary aspect or element in every relation. And this is borne out by the necessity, for the existence of relations, of some given form of externality.

Every relation, we may say, involves a diversity between the related terms, but also some unity. Mere diversity does not give a ground for that interaction, and that interdependence, which a relation requires. Mere unity leaves the terms identical, and thus destroys the reference of one to another required for a relation. Mere externality, taken in abstraction, gives only the element of diversity required for a relation, and is thus more abstract than any actual relation. But mere diversity does not give that indivisible whole of which any actual relation must consist, and is thus, when regarded abstractly, not subject to the restrictions of ordinary relations.

But with mere diversity, we seem to have returned to empty space, and abandoned spatial order. Mere diversity, surely, is either complete or non-existent; degrees of diversity, or a quantitative measure of it, are nonsense. We cannot, therefore, reduce spatial order to mere diversity. Two things, if they occupy different positions in space, are necessarily diverse, but are as necessarily something more; otherwise spatial order becomes unmeaning.

Empty space, then, in the above sense of the possibility of spatial relations, contains only one aspect of a relation, namely the aspect of diversity; but spatial order, by its reference to matter, becomes more concrete, and contains also the element of unity, arising out of the connection of the different material atoms. Spatial order, then, consists of relations in the ordinary sense; its merely spatial element, however—if one may make such a distinction—the element, that is, which can be abstracted from matter and regarded as constituting empty space, is only one aspect of a relation, but an aspect which, in the concrete, must be inseparably bound up with the other aspect. Here, once more, we see the ground of the contradictions in empty space, and the reason why spatial order is free from these contradictions.

209.We have now completed our review of the foundations of Geometry. It will be well, before we take leave of the subject, briefly to review and recapitulate the results we have won.

In the first chapter, we watched the development of a branch of Mathematics designed, at first, only to establish the logical independence of Euclid's axiom of parallels, and the possibility of a self-consistent Geometry which dispensed with it. We found the further development of the subject entangled, for a while, in philosophical controversy; having shown one axiom to be superfluous, the geometers of the second period hoped to prove the same conclusion of all the others, but failed to construct any system free from three fundamental axioms. Being concerned with analytical and metrical Geometry, they tended to regard Algebra asà priori, but held that those properties of spatial magnitudes, which were not deducible from the laws of Algebra, must be empirical. In all this, they aimed as much at discrediting Kant as at advancing Mathematics. But with the third period, the interest in Philosophy diminishes, the opposition to Euclid becomes less marked, and most important of all, measurement is no longer regarded as fundamental, and space is dealt with by descriptive rather thanquantitative methods. But nevertheless, three axioms, substantially the same as those retained in 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 allactualmeasurement must be effected by means ofactualmatter, 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 anyà prioriargument 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 declaredà priori. In metrical Geometry, on the contrary, we found an empirical element, arising out of the alternatives of Euclidean and non-Euclidean space. Threeà prioriaxioms, 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'sTheory of Knowledge(Methuen 1896), PartI.ChapterII.[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'sCommentar,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'sCommentar,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. ChapII.passim; especially p. 51 ff. and pp. 70–1.[188]Logic, p. 51 ff.[189]For theThis, on such a hypothesis, has a purely temporal complexity, and is not resolvable into coexistingThises.[190]ChapterIII.Section A, (§ 131).[191]Cf. Hannequin,Essai critique sur l'hypothèse des atomes, Paris 1895, Chap.I.SectionIII.; especially p. 43.[192]SeeChapterII.§ 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'sCommentar,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'sCommentar,II.p. 218.[200]Cf. Vaihinger'sCommentar,II.p. 207.[201]Cf. James,Psychology, Vol.II., p. 148 ff.

[178]Compare, with the following paragraphs, the admirable discussion in Mr Hobhouse'sTheory of Knowledge(Methuen 1896), PartI.ChapterII.

[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'sCommentar,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'sCommentar,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. ChapII.passim; especially p. 51 ff. and pp. 70–1.

[188]Logic, p. 51 ff.

[189]For theThis, on such a hypothesis, has a purely temporal complexity, and is not resolvable into coexistingThises.

[190]ChapterIII.Section A, (§ 131).

[191]Cf. Hannequin,Essai critique sur l'hypothèse des atomes, Paris 1895, Chap.I.SectionIII.; especially p. 43.

[192]SeeChapterII.§ 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'sCommentar,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'sCommentar,II.p. 218.

[200]Cf. Vaihinger'sCommentar,II.p. 207.

[201]Cf. James,Psychology, Vol.II., p. 148 ff.

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TRANSCRIBER'S NOTEObvious typographical errors and punctuation errors have been corrected after careful comparison with other occurrences within the text and consultation of external sources.A minus sign is represented by the n-dash character "–". Date and number ranges also use the n-dash "–".Except for those changes noted below, all misspellings in the text, and inconsistent or archaic usage, have been retained. For example, co-exist, coexist; every-day, everyday; connexion; assertorial; apodeictic; premisses.§ 82, 'so Erdmannn confidently' replaced by 'so Erdmann confidently'.§ 150Footnote[157], 'Delboeuf' replaced by 'Delbœuf' for consistency.§ 152, 'one subdivison must' replaced by 'one sub-division must'.§ 159Footnote[167], 'Delboeuf' replaced by 'Delbœuf' for consistency.§ 204, 'and homogenous;' replaced by 'and homogeneous;'.

TRANSCRIBER'S NOTE

Obvious typographical errors and punctuation errors have been corrected after careful comparison with other occurrences within the text and consultation of external sources.

A minus sign is represented by the n-dash character "–". Date and number ranges also use the n-dash "–".

Except for those changes noted below, all misspellings in the text, and inconsistent or archaic usage, have been retained. For example, co-exist, coexist; every-day, everyday; connexion; assertorial; apodeictic; premisses.

§ 82, 'so Erdmannn confidently' replaced by 'so Erdmann confidently'.§ 150Footnote[157], 'Delboeuf' replaced by 'Delbœuf' for consistency.§ 152, 'one subdivison must' replaced by 'one sub-division must'.§ 159Footnote[167], 'Delboeuf' replaced by 'Delbœuf' for consistency.§ 204, 'and homogenous;' replaced by 'and homogeneous;'.

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