FOOTNOTES:

FOOTNOTES:[116]See infra, Axiom of Distance, in Sec.B.of this Chapter.[117]Thus on a cylinder, two geodesics,e.g.a generator and a helix, may have any number of intersections—a very important difference from the plane.[118]Cf. Cremona,Projective Geometry(Clarendon Press, 2nd ed. 1893) p. 50: "Most of the propositions in Euclid's Elements are metrical, and it is not easy to find among them an example of a purely descriptive theorem."[119]Op. cit. p. 226.[120]Some ground for this choice will appear when we come to metrical Geometry.[121]The straight lineσadenotes the straight line common to the planesσanda, the pointσadenotes the point common to the planeσand the straight linea, and similarly for the rest of the notation.[122]Cremona (op. cit. Chap.IX.p. 50) defines anharmonic ratio as a metrical property which is unaltered by projection. This, however, destroys the logical independence of projective Geometry, which can only be maintained by a purely descriptive definition.[123]There is no corresponding property ofthreepoints on a line, because they can be projectively transformed into any other three points on the same line. See§ 120.[124]Due to v. Staudt's"Geometrie der Lage."[125]See Cremona, op. cit. ChapterVIII.[126]The corresponding definitions, for the two-dimensional manifold of lines through a point, follow by the principle of duality.[127]It is important to observe that this definition of the Point introduces metrical ideas. Without metrical ideas, we saw, nothing appears to give the Point precedence of the straight line, or indeed to distinguish it conceptually from the straight line. A reference to quantity is therefore inevitable in defining the Point, if the definition is to be geometrical. A non-metrical definition would have to be also non-geometrical. SeeChap.IV.§§ 196–199.[128]§§ 163–175.[129]On this axiom, however, compare§ 131.[130]For the proof of this proposition, seeChap.III.Sec.B, Axiom of Dimensions.[131]The straight line and plane, in all discussions of general Geometry, are not necessarily Euclidean. They are simply figures determined, in general, by two and by three points respectively; whether they conform to the axiom of parallels and to Euclid's form of the axiom of the straight line, is not to be considered in the general definition.[132]That projective Geometry must have existential import, I shall attempt to prove in ChapterIV.[133]Logic, BookI.ChapterII.[134]Cf. Bradley'sLogic, p. 63. It will be seen that the sense in which I have spoken of space as a principle of differentiation is not the sense of a "principle of individuation" which Bradley objects to.[135]Chap.IV.§§ 186–191.[136]Chap.IV.§ 201 ff.[137]It is important to observe, however, that this way of regarding spatial relations is metrical; from the projective standpoint, the relation between two points is the whole unbounded straight line on which they lie, and need not be regarded as divisible into parts or as built up of points.[138]§§ 207,208. Cf. Hegel,Naturphilosophie, § 254.[139]SeeChap.IV.§§ 196–199.[140]See a forthcoming article on"The relations of number and quantity"by the present writer inMind, July, 1897.[141]Logic, Vol.II.Chap.VII.p. 211.[142]Real, as opposed to logical, diversity is throughout intended. Diverse aspects may coexist in a thing at one time and place, but two diverse real things cannot so coexist.[143]On the insufficiency of time alone, seeChapterIV.§ 191.[144]Geometrically, the axiom of the plane is, not that three points determine a figure at all, which follows from the axiom of the straight line, but that the straight line joining two casual points of the plane lies wholly in the plane. This axiom requires a projective method of constructing the plane,i.e.of finding all the triads of points which determine the same projective figure as the given triad. The required construction will be obtained if we can find any projective figure determined by three points, and any projective method of reaching other points which determine the same figure.LetO,P,Qbe the three points whose projective relation is required. Then we have given us the three straight linesPQ,QO,OP. Metrically, the relation between these points is made up of the area, and the magnitude of the sides and angles, of the triangleOPQ, just as the relation between two points is distance. But projectively, the figure is unchanged whenPandQtravel alongOPandOQ, or whenOPandOQturn aboutOin such a way as still to meetPQ. This is a result of the general principle of projective equivalence enunciated above (§§ 108,109). Hence the projective relation betweenO,P,Qis the same as that betweenO,p,qorO,P′,Q′; that is,p,qandP′,Q′lie in the planeOPQ. In this way, any number of points on the plane may be obtained, and by repeating the construction with fresh triads, every point of the plane can be reached. We have to prove that, when the plane is so constructed, the straight line joining any two points of the plane lies wholly in the plane.It is evident, from the manner of construction, that any point ofPQ,OP,OQ,OP′orOQ′lies in the plane. If we can prove that any point ofpqlies in the plane, we shall have proved all that is required, sincepqmay be transformed, by successive repetitions of the same construction, into any straight line joining two points of the plane. But we have seen that the same plane is determined byO,p,qand byO,P,Q. The straight linesPQ,pqhave, therefore, the same relation to the plane. ButPQlies wholly in the plane; thereforepqalso lies wholly in the plane. Hence our axiom is proved.[145]A detailed proof has been given above, Chap.I.3rd period. It is to be observed that any reference to infinitely distant elements involves metrical ideas.[146]Cf.Section A, §§ 115–117.[147]Contrast Erdmann, op. cit. p. 138.[148]Cf. Erdmann, op. cit. p. 164.[149]Strictly speaking, this method is only applicable where the two magnitudes are commensurable. But if we take infinite divisibility rigidly, the units can theoretically be taken so small as to obtain any required degree of approximation. The difficulty is the universal one of applying to continua the essentially discrete conception of number.[150]Cf. Erdmann, op. cit. p. 50.[151]Also called the axiom of congruence. I have taken congruence to be thedefinitionof spatial equality by superposition, and shall therefore generally speak of theaxiomas Free Mobility.[152]For the sense in which these figures are to be regarded as material, see criticism of Helmholtz,ChapterII.§§ 69 ff.[153]Op. cit. p. 60.[154]The view of Helmholtz and Erdmann, that mechanical experience suffices here, though geometrical experience fails us, has been discussed above,ChapterII.§§ 73,82.[155]ChapterII.§ 81.[156]ChapterII.§ 72.[157]ContrastDelbœuf,L'ancienne et les nouvelles géométries,II.Rev. Phil. 1894, Vol. xxxvii. p. 354.[158]Prolegomena, § 13. See Vaihinger'sCommentar,II.pp. 518–532 esp. pp. 521–2. The above was Kant's whole purpose in 1768, but only part of his purpose in the Prolegomena, where the intuitive nature of space was also to be proved.[159]On the subject of time measurement, cf. Bosanquet'sLogic, Vol. i. pp. 178–183. Since time, in the above account, is measured by motion, its measurement presupposes that of spatial magnitudes.[160]Cf. Stumpf.Ursprung der Raumvorstellung, p. 68.[161]As is Helmholtz's other axiom, that the possibility of superposition is independent of the course pursued in bringing it about.[162]Cf.§§ 129,130.[163]This deduction is practically the same as that in Sec. A, but I have stated it here with more special reference to space and to metrical Geometry.[164]The question: "Relations to what?" is a question involving many difficulties. It will be touched on later in this chapter, and answered, as far as possible, in the fourth chapter. For the present, in spite of the glaring circle involved, I shall take the relations as relations to other positions.[165]Wiss. Abh. Vol.II.p. 614.[166]Cp. Grassmann,Ausdehnungslehre von 1844, 2nd ed. p.XXIII.[167]Delbœuf, it is true, speaks of Geometries withm/ndimensions, but gives no reference (Rev. Phil. T. xxxvi. p. 450).[168]In criticizing Erdmann, it will be remembered, we saw that Free Mobility is a necessary property of his extents, though he does not regard it as such.[169]Cf. Riemann,Hypothesen welche der Geometrie zu Grunde liegen, Gesammelte Werke,p. 266; also Erdmann, op. cit. p. 154.[170]This is subject, in spherical space, to the modification pointed out below, in dealing with the exception to the axiom of the straight line. See§§ 168–171.[171]In speaking of distance at once as a quantity and as an intrinsic relation, I am anxious to guard against an apparent inconsistency. I have spoken of the judgment of quantity, throughout, as one of comparison; how, then, can a quantity be intrinsic? The reply is that, although measurement and the judgment of quantity express the result of comparison, yet the terms compared must exist before the comparison; in this case, the terms compared in measuring distances,i.e.in comparing theminter se, are intrinsic relations between points. Thus, although themeasurementof distance involves a reference to other distances, and its expression as a magnitude requires such a reference, yet its existence does not depend on any external reference, but exclusively on the two points whose distance it is.[172]See the end of the argument on Free Mobility,§ 155 ff.[173]In Frischauf's"Absolute Geometrie nach Johann Bolyai,"Anhang, there is a series of definitions, starting from the sphere, as the locus of congruent point-pairs when one point of the pair is fixed, and hence obtaining the circle and the straight line. From the above it follows, that the sphere so defined already involves a curve between the points of the point-pair, by which various point-pairs can be known as congruent; and it will appear, as we proceed, that this curve must be a straight line. Frischauf's definition by means of the sphere involves, therefore, a vicious circle, since the sphere presupposes the straight line, as the test of congruent point-pairs.[174]Nor in any argument which, like those of projective Geometry, avoids the notion of magnitude or distance altogether. It follows that the propositions of projective Geometry apply, without reserve, to spherical space, since the exception to the axiom of the straight line arises only on metrical ground.[175]Psychology, Vol.II.pp. 149–150.[176]This step in the argument has been put very briefly, since it is a mere repetition of the corresponding argument in Section A, and is inserted here only for the sake of logical completeness. See§ 137 ff.[177]Cf. Hannequin,Essai critique sur l'hypothèse des atomes, Paris, 1895, passim.

[116]See infra, Axiom of Distance, in Sec.B.of this Chapter.

[117]Thus on a cylinder, two geodesics,e.g.a generator and a helix, may have any number of intersections—a very important difference from the plane.

[118]Cf. Cremona,Projective Geometry(Clarendon Press, 2nd ed. 1893) p. 50: "Most of the propositions in Euclid's Elements are metrical, and it is not easy to find among them an example of a purely descriptive theorem."

[119]Op. cit. p. 226.

[120]Some ground for this choice will appear when we come to metrical Geometry.

[121]The straight lineσadenotes the straight line common to the planesσanda, the pointσadenotes the point common to the planeσand the straight linea, and similarly for the rest of the notation.

[122]Cremona (op. cit. Chap.IX.p. 50) defines anharmonic ratio as a metrical property which is unaltered by projection. This, however, destroys the logical independence of projective Geometry, which can only be maintained by a purely descriptive definition.

[123]There is no corresponding property ofthreepoints on a line, because they can be projectively transformed into any other three points on the same line. See§ 120.

[124]Due to v. Staudt's"Geometrie der Lage."

[125]See Cremona, op. cit. ChapterVIII.

[126]The corresponding definitions, for the two-dimensional manifold of lines through a point, follow by the principle of duality.

[127]It is important to observe that this definition of the Point introduces metrical ideas. Without metrical ideas, we saw, nothing appears to give the Point precedence of the straight line, or indeed to distinguish it conceptually from the straight line. A reference to quantity is therefore inevitable in defining the Point, if the definition is to be geometrical. A non-metrical definition would have to be also non-geometrical. SeeChap.IV.§§ 196–199.

[128]§§ 163–175.

[129]On this axiom, however, compare§ 131.

[130]For the proof of this proposition, seeChap.III.Sec.B, Axiom of Dimensions.

[131]The straight line and plane, in all discussions of general Geometry, are not necessarily Euclidean. They are simply figures determined, in general, by two and by three points respectively; whether they conform to the axiom of parallels and to Euclid's form of the axiom of the straight line, is not to be considered in the general definition.

[132]That projective Geometry must have existential import, I shall attempt to prove in ChapterIV.

[133]Logic, BookI.ChapterII.

[134]Cf. Bradley'sLogic, p. 63. It will be seen that the sense in which I have spoken of space as a principle of differentiation is not the sense of a "principle of individuation" which Bradley objects to.

[135]Chap.IV.§§ 186–191.

[136]Chap.IV.§ 201 ff.

[137]It is important to observe, however, that this way of regarding spatial relations is metrical; from the projective standpoint, the relation between two points is the whole unbounded straight line on which they lie, and need not be regarded as divisible into parts or as built up of points.

[138]§§ 207,208. Cf. Hegel,Naturphilosophie, § 254.

[139]SeeChap.IV.§§ 196–199.

[140]See a forthcoming article on"The relations of number and quantity"by the present writer inMind, July, 1897.

[141]Logic, Vol.II.Chap.VII.p. 211.

[142]Real, as opposed to logical, diversity is throughout intended. Diverse aspects may coexist in a thing at one time and place, but two diverse real things cannot so coexist.

[143]On the insufficiency of time alone, seeChapterIV.§ 191.

[144]Geometrically, the axiom of the plane is, not that three points determine a figure at all, which follows from the axiom of the straight line, but that the straight line joining two casual points of the plane lies wholly in the plane. This axiom requires a projective method of constructing the plane,i.e.of finding all the triads of points which determine the same projective figure as the given triad. The required construction will be obtained if we can find any projective figure determined by three points, and any projective method of reaching other points which determine the same figure.LetO,P,Qbe the three points whose projective relation is required. Then we have given us the three straight linesPQ,QO,OP. Metrically, the relation between these points is made up of the area, and the magnitude of the sides and angles, of the triangleOPQ, just as the relation between two points is distance. But projectively, the figure is unchanged whenPandQtravel alongOPandOQ, or whenOPandOQturn aboutOin such a way as still to meetPQ. This is a result of the general principle of projective equivalence enunciated above (§§ 108,109). Hence the projective relation betweenO,P,Qis the same as that betweenO,p,qorO,P′,Q′; that is,p,qandP′,Q′lie in the planeOPQ. In this way, any number of points on the plane may be obtained, and by repeating the construction with fresh triads, every point of the plane can be reached. We have to prove that, when the plane is so constructed, the straight line joining any two points of the plane lies wholly in the plane.It is evident, from the manner of construction, that any point ofPQ,OP,OQ,OP′orOQ′lies in the plane. If we can prove that any point ofpqlies in the plane, we shall have proved all that is required, sincepqmay be transformed, by successive repetitions of the same construction, into any straight line joining two points of the plane. But we have seen that the same plane is determined byO,p,qand byO,P,Q. The straight linesPQ,pqhave, therefore, the same relation to the plane. ButPQlies wholly in the plane; thereforepqalso lies wholly in the plane. Hence our axiom is proved.

LetO,P,Qbe the three points whose projective relation is required. Then we have given us the three straight linesPQ,QO,OP. Metrically, the relation between these points is made up of the area, and the magnitude of the sides and angles, of the triangleOPQ, just as the relation between two points is distance. But projectively, the figure is unchanged whenPandQtravel alongOPandOQ, or whenOPandOQturn aboutOin such a way as still to meetPQ. This is a result of the general principle of projective equivalence enunciated above (§§ 108,109). Hence the projective relation betweenO,P,Qis the same as that betweenO,p,qorO,P′,Q′; that is,p,qandP′,Q′lie in the planeOPQ. In this way, any number of points on the plane may be obtained, and by repeating the construction with fresh triads, every point of the plane can be reached. We have to prove that, when the plane is so constructed, the straight line joining any two points of the plane lies wholly in the plane.

It is evident, from the manner of construction, that any point ofPQ,OP,OQ,OP′orOQ′lies in the plane. If we can prove that any point ofpqlies in the plane, we shall have proved all that is required, sincepqmay be transformed, by successive repetitions of the same construction, into any straight line joining two points of the plane. But we have seen that the same plane is determined byO,p,qand byO,P,Q. The straight linesPQ,pqhave, therefore, the same relation to the plane. ButPQlies wholly in the plane; thereforepqalso lies wholly in the plane. Hence our axiom is proved.

[145]A detailed proof has been given above, Chap.I.3rd period. It is to be observed that any reference to infinitely distant elements involves metrical ideas.

[146]Cf.Section A, §§ 115–117.

[147]Contrast Erdmann, op. cit. p. 138.

[148]Cf. Erdmann, op. cit. p. 164.

[149]Strictly speaking, this method is only applicable where the two magnitudes are commensurable. But if we take infinite divisibility rigidly, the units can theoretically be taken so small as to obtain any required degree of approximation. The difficulty is the universal one of applying to continua the essentially discrete conception of number.

[150]Cf. Erdmann, op. cit. p. 50.

[151]Also called the axiom of congruence. I have taken congruence to be thedefinitionof spatial equality by superposition, and shall therefore generally speak of theaxiomas Free Mobility.

[152]For the sense in which these figures are to be regarded as material, see criticism of Helmholtz,ChapterII.§§ 69 ff.

[153]Op. cit. p. 60.

[154]The view of Helmholtz and Erdmann, that mechanical experience suffices here, though geometrical experience fails us, has been discussed above,ChapterII.§§ 73,82.

[155]ChapterII.§ 81.

[156]ChapterII.§ 72.

[157]ContrastDelbœuf,L'ancienne et les nouvelles géométries,II.Rev. Phil. 1894, Vol. xxxvii. p. 354.

[158]Prolegomena, § 13. See Vaihinger'sCommentar,II.pp. 518–532 esp. pp. 521–2. The above was Kant's whole purpose in 1768, but only part of his purpose in the Prolegomena, where the intuitive nature of space was also to be proved.

[159]On the subject of time measurement, cf. Bosanquet'sLogic, Vol. i. pp. 178–183. Since time, in the above account, is measured by motion, its measurement presupposes that of spatial magnitudes.

[160]Cf. Stumpf.Ursprung der Raumvorstellung, p. 68.

[161]As is Helmholtz's other axiom, that the possibility of superposition is independent of the course pursued in bringing it about.

[162]Cf.§§ 129,130.

[163]This deduction is practically the same as that in Sec. A, but I have stated it here with more special reference to space and to metrical Geometry.

[164]The question: "Relations to what?" is a question involving many difficulties. It will be touched on later in this chapter, and answered, as far as possible, in the fourth chapter. For the present, in spite of the glaring circle involved, I shall take the relations as relations to other positions.

[165]Wiss. Abh. Vol.II.p. 614.

[166]Cp. Grassmann,Ausdehnungslehre von 1844, 2nd ed. p.XXIII.

[167]Delbœuf, it is true, speaks of Geometries withm/ndimensions, but gives no reference (Rev. Phil. T. xxxvi. p. 450).

[168]In criticizing Erdmann, it will be remembered, we saw that Free Mobility is a necessary property of his extents, though he does not regard it as such.

[169]Cf. Riemann,Hypothesen welche der Geometrie zu Grunde liegen, Gesammelte Werke,p. 266; also Erdmann, op. cit. p. 154.

[170]This is subject, in spherical space, to the modification pointed out below, in dealing with the exception to the axiom of the straight line. See§§ 168–171.

[171]In speaking of distance at once as a quantity and as an intrinsic relation, I am anxious to guard against an apparent inconsistency. I have spoken of the judgment of quantity, throughout, as one of comparison; how, then, can a quantity be intrinsic? The reply is that, although measurement and the judgment of quantity express the result of comparison, yet the terms compared must exist before the comparison; in this case, the terms compared in measuring distances,i.e.in comparing theminter se, are intrinsic relations between points. Thus, although themeasurementof distance involves a reference to other distances, and its expression as a magnitude requires such a reference, yet its existence does not depend on any external reference, but exclusively on the two points whose distance it is.

[172]See the end of the argument on Free Mobility,§ 155 ff.

[173]In Frischauf's"Absolute Geometrie nach Johann Bolyai,"Anhang, there is a series of definitions, starting from the sphere, as the locus of congruent point-pairs when one point of the pair is fixed, and hence obtaining the circle and the straight line. From the above it follows, that the sphere so defined already involves a curve between the points of the point-pair, by which various point-pairs can be known as congruent; and it will appear, as we proceed, that this curve must be a straight line. Frischauf's definition by means of the sphere involves, therefore, a vicious circle, since the sphere presupposes the straight line, as the test of congruent point-pairs.

[174]Nor in any argument which, like those of projective Geometry, avoids the notion of magnitude or distance altogether. It follows that the propositions of projective Geometry apply, without reserve, to spherical space, since the exception to the axiom of the straight line arises only on metrical ground.

[175]Psychology, Vol.II.pp. 149–150.

[176]This step in the argument has been put very briefly, since it is a mere repetition of the corresponding argument in Section A, and is inserted here only for the sake of logical completeness. See§ 137 ff.

[177]Cf. Hannequin,Essai critique sur l'hypothèse des atomes, Paris, 1895, passim.

180.In the present chapter, we have to discuss two questions which, though scarcely geometrical, are of fundamental importance to the theory of Geometry propounded above. The first of these questions is this: What relation can a purely logical and deductive proof, like that from the nature of a form of externality, bear to an experienced subject-matter such as space? You have merely framed a general conception, I may be told, containing space as a particular species, and you have then shown, what should have been obvious from the beginning, that this general conception contained some of the attributes of space. But what ground does this give for regarding these attributes asà priori? The conception Mammal has some of the attributes of a horse; but are these attributes thereforeà prioriadjectives of the horse? The answer to this obvious objection is so difficult, and involves so much general philosophy, that I have kept it for a final chapter, in order not to interrupt the argument on specially geometrical topics.

181.I have already indicated, in general terms, the ground for regarding asà priorithe properties of any form of externality. This ground is transcendental,i.e.it is to be found in the conditions required for the possibility of experience. The form of externality, like Riemann's manifolds, is a general class-conception, including time as well as Euclidean and non-Euclidean spaces. It is not motived, however, like the manifolds, by aquantitativeresemblance to space, but by the fact that it fulfils, if it has more than one dimension, all those functions which, in our actual world, are fulfilled by space. But a formof externality, in order to accomplish this, must be, not a mere conception, but an actually experienced intuition. Hence the conception of such a form is the generalconception, containing under it every logically possibleintuitionwhich can fulfil the function actually fulfilled by space. And this function is, to render possible experience of diverse but interrelated things. Some form in sense-perception, then, whose conception is included under our form of externality, isà priorinecessary to experience of diversity in relation, and without experience of this, we should, as modern logic shows, have no experience at all. This still leaves untouched the relation of theà priorito the subjective: the form of externality is necessary to experience, but is not,on that account, to be declared purely subjective. Of course, necessity for experience can only arise from the nature of the mind which experiences; but it does not follow that the necessary conditions could be fulfilled, unless the objective world had certain properties. Thegroundof necessity, we may safely say, arises from the mind; but it by no means follows that thetruthof what is necessary depends only on the constitution of the mind. Where this is not the case, our conclusion, when a piece of knowledge has been declaredà priori, can only be: Owing to the constitution of themind, experience will be impossible unless theworldaccepts certain adjectives.

Such, in outline, will be the argument of the first half of this chapter, and such will be the justification for regarding asà priorithose axioms of Geometry, which were deduced above from the conception of a form of externality. For these axioms, and these only, are necessarily true of any world in which experience is possible.

182[178].The view suggested has, obviously, much in common with that of the Transcendental Aesthetic. Indeed the whole of it, I believe, can be obtained by a certain limitation and interpretation of Kant's classic arguments. But as it differs, in many important points, from the conclusions aimed at by Kant, and as the agreement may easily seem greater than it is, I will begin by a brief comparison, and endeavour, by referenceto authoritative criticisms, to establish the legitimacy of my divergence from him.

183.In the first place, the psychological element is much larger in Kant's thesis than in mine. I shall contend, it is true, that a form of externality, if it is to do its work, must not be a mere conception or a mere inference, but must be a given element in sense-perception—not, of course, originally given in isolation, but discoverable, through analysis, by attention to the object of sense-perception[179]. But Kant contended, not only that this element is given, but also that it is subjective. Space, for him, is, on the one hand, not conceptual, but on the other hand, not sensational. It forms, for him, no part of the data of sense, but is added by a subjective intuition, which he regards as not only logically, but psychologically, prior to objects in space[180].

This part of Kant's argument is wholly irrelevant for us. Whether a form of externality be given in sense, or in a pure intuition, is for us unimportant, since we neglect the question as to the connection of theà prioriand the subjective; while the temporal priority of space to objects in it has been generally recognized as irrelevant to Epistemology, and has often been regarded as forming no part of Kant's thesis[181]. If we call intuitional whatever is given in sense-perception, then we may contend that a form of externality must be intuitional; but whether it is a pure intuition, in Kant's sense, or not, is irrelevant to us, as is its priority to the objects in it.

That the non-sensational nature of space is no essential part of Kant'slogicalteaching, appears from an examination of his argument. He has made, in the introduction, the purely logical distinction of matter and form, but has given to this distinction, in the very moment of suggesting it, a psychological implication. This he does by the assertion that the form, in which the matter of sensations is ordered, cannot itself be sensational. From this assumption it follows, of course, that space cannot be sensational. But the assumption istotally unsupported by argument, being set forth, apparently, as a self-evident axiom; it has been severely criticized by Stumpf[182]and others[183], and has been described by Vaihinger as a fatalpetitio principii[184]; it is irrelevant to the logical argument, when this argument is separated, as we have separated it, from all connection with psychological subjectivity; and finally, it leaves us a prey to psychological theories of space, which have seemed, of late, but little favourable to the pure Kantian doctrine.

184.We have a right, therefore, in an epistemological inquiry, to neglect Kant's psychological teaching—in so far, at any rate, as it distinguishes spatial intuition from sensation—and attend rather to the logical aspect alone. That part of his psychological teaching, which maintains that space is not a mere conception, is, with certain limitations, sufficiently evident as applied to actual space; but for us, it must be transformed into a much more difficult thesis, namely, thatnoform of externality, which renders experience of diversity in relation possible, can be merely conceptual. This question, to which we must return later, is no longer psychological, but belongs wholly to Epistemology.

185.What, then, remains the kernel, for our purposes, of Kant's first argument for the apriority of space? His argument, in the form in which he gave it, is concerned with the eccentric projection of sensations. In order that I may refer sensations, he says, to something outside myself, I must already have the subjective space-form in the mind. In this shape, as Vaihinger points out (Commentar,II.pp. 69, 165), the argument rests on apetitio principii, for only if sensations are necessarily non-spatial does their projection demand a subjective space-form. But, further, is the logical apriority of space concerned with the externality of things to ourselves?

Spaceseemsto perform two functions: on the one hand, it reveals things, by the eccentric projection of sensations, as external to the self, while, on the other hand, it reveals simultaneously presented things as mutually external. Thesetwo functions, though often treated as coordinate and almost equivalent[185], seem to me widely different. Before we discuss the apriority of space, we must carefully distinguish, I think, between these two functions, and decide which of them we are to argue about.

Now externality to the Self, it would seem, must necessarily raise the whole question of the nature and limits of the Ego, and what is more, it cannot be derived from spatial presentation, unless we give the Self a definite position in space. But things acquire a position in space only when they can appear in sense-perception; we are forced, therefore, if we adopt this view of the function of space, to regard the Self as a phenomenon presented to sense-perception. But this reduces externality to the Self to externality to the body. The body, however, is a presented object like any other, and externality of objects to it is, therefore, a special case of the mutual externality of presented things. Hence we cannot regard space as giving, primarily at any rate, externality to the Self, but only the mutual externality of the things presented to sense-perception[186].

186.This, then, is the kind of externality we are to expect from space, and our question must be: Would the existence of diverse but interrelated things be unknowable, if there were not, in sense-perception, some form of externality? This is the crucial question, on which turns the apriority of our form, and hence of the necessary axioms of Geometry.

187.The converse argument to mine, the argument from the spatio-temporal element in perception to a world of interrelated but diverse things, is developed at length in Bradley's Logic. It is put briefly in the following sentence (p. 44, note): "If space and time are continuous, and if all appearance must occupy some time or space—and it is not hard to support both thesetheses—we can at once proceed to the conclusion, no mere particular exists. Every phenomenon will exist in more times or spaces than one; and against that diversity will be itself an universal[187]." The importance of this fact appears, when weconsider that, if anymereparticular existed, all judgment and inference as to that particular would be impossible, since all judgment and inference necessarily operate by means of universal. But all reality is constructed from theThisof immediate presentation, from which judgment and inference necessarily spring. Owing, however, to the continuity and relativity of space and time, noThiscan be regarded either as simple or as self-subsistent. EveryThis, on the one hand, can be analyzed intoThises, and on the other hand, is found to be necessarily related to other things, outside the limits of the given object of sense-perception. This function of space and time is presupposed in the following statement from Bosanquet's Logic (Vol.I.pp. 77–78): "Reality is given for meinpresent sensuous perception, andinthe immediate feeling of my own sentient existence that goes with it. The real world, as a definite organized system, isfor mean extension of this present sensation and self feeling by means of judgment, and it is the essence of judgment to effect and sustain such an extension.... The subject in every judgment of Perception is some given spot or point in sensuous contact with the percipient self. But, as all reality is continuous, the subject is notmerelythis given spot or point."

188.This doctrine of Bradley and Bosanquet is the converse of the epistemological doctrine I have to advocate. Owing to the continuity and relativity of space and time, they say, we are able to construct a systematic world, by judgment and inference, out of that fragmentary and yet necessarily complex existence which is given in sense-perception. My contention is, conversely, that since all knowledge is necessarily derived by an extension of theThisof sense-perception, and since such extension is only possible if theThishas that fragmentary and yet complex character conferred by a form of externality, therefore some form of externality, given with theThis, is essential to all knowledge, and is thus logicallyà priori. Bradley's argument, if sound, already proves this contention; for while, on the one hand, he uses no properties of space and time but those which belong to every form of externality, he proves, on the other hand, that judgment and inference require theThisto be neither single nor self-subsistent. But I willendeavour, since the point is of fundamental importance, to reproduce the proof, in a form more suited than Bradley's to the epistemological question.

189.The essence of my contention is that, if experience is to be possible, every sensationalThismust, when attended to, be found, on the one hand, resolvable intoThises, and on the other hand dependent, for some of its adjectives, on external reference. The second of these theses follows from the first, for if we take one of theThisescontained in the firstThis, we get a newThisnecessarily related to the otherThiseswhich make up the originalThis. I may, therefore, confine myself to the first proposition, which affirms that the object of perception must contain a diversity, not only of conceptual content, but of existence, and that this can only be known if sense-perception contains, as an element, some form of externality.

My premiss, in this argument, is that all knowledge involves a recognition of diversity in relation, or, if we prefer it, of identity in difference. This premiss I accept from Logic, as resulting from the analysis of judgment and inference. To prove such a premiss, would require a treatise on Logic; I must refer the reader, therefore, to the works of Bradley and Bosanquet on the subject. It follows at once, from my premiss, that knowledge would be impossible, unless the object of attention could be complex,i.e.not amereparticular. Now could the mental object—i.e., in this connection, the object of a cognition—be complex, if the object of immediate perception were always simple?

190.We might be inclined, at first sight, to answer this question affirmatively. But several difficulties, I think, would prevent such an answer. In the first place, knowledge must start from perception. Hence, either we could have no knowledge except of our present perception, or else we must be able to contrast and compare it with some other perception. Now in the first case, since the present perception, by hypothesis, is a mere particular, knowledge of it is impossible, according to our premiss. But in the second case, the other perception, with which we compare our first, must have occurred at some other time, and with time, we have at once a form of externality. But what is more, our present perception is nolonger a mere particular. For the power of comparing it with another perception involves a point of identity between the two, and thus renders both complex. Moreover, time must be continuous, and the present, as Bradley points out, is no mere point of time[188]. Thus our present perception contains the complexity involved in duration throughout the specious present: its mere particularity and its simplicity are lost. Its self-subsistence is also lost, for beyond the specious present, lie the past and the future, to which our present perception thus unavoidably refers us. Time at least, therefore, is essential to that identity in difference, which all knowledge postulates.

191.But we have derived, from all this, no ground for affirming a multiplicity of real things, or a form of externality of more than one dimension, which, we saw, was necessary for the truth of two out of our three axioms. This brings us to the question: Have we enough, with time alone as a form of externality, for the possibility of knowledge?

This question we must, I think, answer in the negative. With time alone, we have seen, our presented object must be complex, but its complexity must, if I may use such a phrase, be merely adjectival. Without a second form of externality, only one thing can be given at one moment[189], and this one thing, therefore, must constitute the whole of our world. The object of past perception must—since our one thing has nothing external to it, by which it could be created or destroyed—be regarded as the same thing in a different state. The complexity, therefore, will lie only in the changing states of our one thing—it will be adjectival, not substantival. Moreover we have the following dilemma: Either the one thing must be ourselves, or else self-consciousness could never arise. But the chief difficulty of such a world would lie in the changes of the thing. What could cause these changes, since we should know of nothing external to our thing? It would be like a Leibnitzian monad, without any God outside it to prearrange its changes. Causality, in such a world, could not be applied, and change would be wholly inexplicable.

Hence we require also the possibility of a diversity of simultaneously existing things, not merely of successive adjectives; and this, we have seen, cannot be given by time alone, but only by a form of externality for simultaneous parts of one presentation. We could never, in other words, infer the existence of diverse but interrelated things, unless the object of sense-perception could have substantival complexity, and for such complexity we require a form of externality other than time. Such a form, moreover, as was shown inChapterIII., Section A (§ 135), can only fulfil its functions if it has more than one dimension. In our actual world, this form is given by space; in any world, knowable to beings with our laws of thought, some such form, as we have now seen, must be given in sense-perception.

This argument may be briefly summed up, by assuming the doctrine of Bradley, that all knowledge is obtained by inference from theThisof sense-perception. For, if this be so, theThis—in order that inference, which depends on identity in difference, may be possible at all—must itself be complex, and must, on analysis, reveal adjectives having a reference beyond itself. But this, as was shown above, can only happen by means of a form of externality. This establishes the à priori axioms of Geometry, as necessarily having existential import and validity in any intelligible world.

192.The above argument, I hope, has explained why I hold it possible to deduce, from a mere conception like that of a form of externality, the logical apriority of certain axioms as to experienced space. The Kantian argument—which was correct, if our reasoning has been sound, in asserting that real diversity, in our actual world, could only be known by the help of space—was only mistaken, so far as its purely logical scope extends, in overlooking the possibility of other forms of externality, which could, if they existed, perform the same task with equal efficiency. In so far as space differs, therefore, from these other conceptions of possible intuitional forms, it is a mere experienced fact, while in so far as its properties are those which all such forms must have, it isà priorinecessary to the possibility of experience.

I cannot hope, however, that no difficulty will remain, forthe reader, in such a deduction, from abstract conceptions, of the properties of an actualdatumin sense-perception. Let us consider, for example, such a property as impenetrability. To suppose two things simultaneously in the same position in a form of externality, is a logical contradiction; but can we say as much of actual space and time? Is not the impossibility, here, a matter of experience rather than of logic? Not if the above argument has been sound, I reply. For in that case, we infer real diversity,i.e.the existence of different things, only from difference of position in space or time. It follows, that to suppose two things in the same point of space and time, is still a logical contradiction: not because we have constructed the data of sense out of logic, but because logic is dependent, as regards its application, on the nature of these data. This instance illustrates, what I am anxious to make plain, that my argument has not attempted to construct the living wealth of sense-perception out of "bloodless categories," but only to point out that, unless sense-perception contained a certain element, these categories would be powerless to grapple with it.

193.How we are to account for the fortunate realization of these requirements—whether by a pre-established harmony, by Darwinian adaptation to our environment, by the subjectivity of the necessary element in sense-perception, or by a fundamental identity and unity between ourselves and the rest of reality—is a further question, belonging rather to metaphysics than to our present line of argument. Theà priori, we have said throughout, is that which is necessary for the possibility of experience, and in this we have a purely logical criterion, giving results which only Logic and Epistemology can prove or disprove. What is subjective in experience, on the contrary, is primarily a question for psychology, and should be decided on psychological grounds alone. When these two questions have been separately answered, but not till then, we may frame theories as to the connection of theà prioriand the subjective; to allow such theories to influence our decision, on either of the two previous questions, is liable, surely, to confuse the issue, and prevent a clear discrimination between fundamentally different points of view.

194.I come now to the second question with which this chapter has to deal, the question, namely: What are we to do with the contradictions which obtruded themselves in ChapterIII., whenever we came to a point which seemed fundamental? I shall treat this question briefly, as I have little to add to answers with which we are all familiar. I have only to prove, first, that the contradictions are inevitable, and therefore form no objection to my argument; secondly, that the first step in removing them is to restore the notion of matter, as that which, in the data of sense-perception, is localized and interrelated in space.

195.The contradictions in space are an ancient theme—as ancient, in fact, as Zeno's refutation of motion. They are, roughly, of two kinds, though the two kinds cannot be sharply divided. There are the contradictions inherent in the notion of the continuum, and the contradictions which spring from the fact that space, while it must, to be knowable, be pure relativity, must also, it would seem, since it is immediately experienced, be something more than mere relations. The first class of contradictions has been encountered more frequently in this essay, and is also, I think, the more definite, and the more important for our present purpose. I doubt, however, whether the two classes are really distinct; for any continuum, I believe, in which the elements are not data, but intellectual constructions resulting from analysis, can be shown to have the same relational and yet not wholly relational character as belongs to space.

The three following contradictions, which I shall discuss successively, seem to me the most prominent in a theory of Geometry.

(1) Though the parts of space are intuitively distinguished, no conception is adequate to differentiate them. Hence arises a vain search for elements, by which the differentiation could be accomplished, and for a whole, of which the parts of space are to be components. Thus we get the point, or zero extension, as the spatial element, and an infinite regress or a vicious circle in the search for a whole.

(2) All positions being relative, positions can only be defined by their relations,i.e.by the straight lines or planesthrough them; but straight lines and planes, being all qualitatively similar, can only be defined by the positions they relate. Hence, again, we get a vicious circle.

(3) Spatial figures must be regarded as relations. But a relation is necessarily indivisible, while spatial figures are necessarily divisiblead infinitum.

196.(1)Points.The antinomy of the point—which arises wherever a continuum is given, and elements have to be sought in it—is fundamental to Geometry. It has been given, perhaps unintentionally, by Veronese as the first axiom, in the form: "There are different points. All points are identical" (op. cit.p. 226). We saw, in discussing projective Geometry, that straight lines and planes must be regarded, on the one hand as relations between points, and on the other hand as made up of points[190]. We saw again, in dealing with measurement, how space must be regarded as infinitely divisible, and yet as mere relativity. But what is divisible and consists of parts, as space does, must lead at last, by continued analysis, to a simple and unanalyzable part, as the unit of differentiation. For whatever can be divided, and has parts, possesses some thinghood, and must, therefore, contain two ultimate units, the whole namely, and the smallest element possessing thinghood. But in space this is notoriously not the case. After hypostatizing space, as Geometry is compelled to do, the mind imperatively demands elements, and insists on having them, whether possible or not. Of this demand, all the geometrical applications of the infinitesimal calculus are evidence[191]. But what sort of elements do we thus obtain? Analysis, being unable to find any earlier halting-place, finds its elements in points, that is, in zero quanta of space. Such a conception is a palpable contradiction, only rendered tolerable by its necessity and familiarity. A point must be spatial, otherwise it would not fulfil the function of a spatial element; but again it must contain no space, for any finite extension is capable of further analysis. Points can never be given in intuition, which has no concern with the infinitesimal: they are a purely conceptual construction, arisingout of the need of terms between which spatial relations can hold. If space be more than relativity, spatial relations must involve spatial relata; but no relata appear, until we have analyzed our spatial data down to nothing. The contradictory notion of the point, as a thing in space without spatial magnitude, is the only outcome of our search for spatial relata. Thisreductio ad absurdumsurely suffices, by itself, to prove the essential relativity of space.

197.Thus Geometry is forced, since it wishes to regard space as independent, to hypostatize its abstractions, and therefore to invent a self-contradictory notion as the spatial element. A similar absurdity appears, even more obviously, in the notion of a whole of space. The antinomy may, therefore, be stated thus: Space, as we have seen throughout, must, if knowledge of it is to be possible, be mere relativity; but it must also, ifindependentknowledge of it, such as Geometry seeks, is to be possible, be something more than mere relativity, since it is divisible and has parts. But we saw, inChap.III., Section A (§ 133)that knowledge of a form of externality must be logically independent of the particular matter filling the form. How then are we to extricate ourselves from this dilemma?

The only way, I think, is, not to make Geometry dependent on Physics, which we have seen to be erroneous[192], but to give every geometrical proposition a certain reference to matter in general. And at this point an important distinction must be made. We have hitherto spoken of space as relational, and of spatial figures as relations. But space, it would seem, is rather relativity than relations—itself not a relation, it gives the bare possibility of relations between diverse things[193]. As applied to a spatial figure, which can only arise by a differentiation of space, and hence by the introduction of some differentiating matter, the word relation is, perhaps, less misleading than any other; as applied to empty undifferentiated space, it seems by no means an accurate description.

But a bare possibility cannot exist, or be given in sense-perception! What becomes, then, of the arguments of thefirst part of this chapter? I reply, it is not empty space, but spatial figures, which sense-perception reveals, and spatial figures, as we have just seen, involve a differentiation of space, and therefore a reference to the matter which is in space. It is spatial figures, also, and not empty space, with which Geometry has to deal. The antinomy discussed above arises then—so it would seem—from the attempt to deal with empty space, rather than with spatial figures and the matter to which they necessarily refer.

198.Let us see whether, by this change, we can overcome the antinomy of the point. Spatial figures, we shall now say, are relations between the matter which differentiates empty space. Their divisibility, which seemed to contradict their relational character, may be explained in two ways: first, as holding of the figures considered as parts of empty space, which is itself not a relation; second, as denoting the possibility of continuous change in the relation expressed by the spatial figure. These two ways are, at bottom, the same; for empty space is a possibility of relations, and the figure, when viewed in connection with empty space, thus becomes apossiblerelation, with which other possible relations may be contrasted or compared. But the second way of regarding divisibility is the better way, since it introduces a reference to the matter which differentiates empty space, without which, spatial figures, and therefore Geometry, could not exist. It is empty space, then—so we must conclude—which gives rise to the antinomy in question; for empty space is a bare possibility of relations, undifferentiated and homogeneous, and thus wholly destitute of parts or of thinghood. To speak of parts of a possibility is nonsense; the parts and differentiations arise only through a reference to the matter which is differentiated in space.

199.But what nature must we ascribe to this matter, which is to be involved in all geometrical propositions? In criticizing Helmholtz (Chap.II.§ 73), it may be remembered, we decided that Geometry refers to a peculiar and abstract kind of matter, which is not regarded as possessing any causal qualities, as exerting or as subject to the action of forces. And this is the matter, I think, which we require for the needs of the moment. Not that we affirm, of course, that actual mattercan be destitute of the properties with which Physics is cognizant, but that we abstract from these properties, as being irrelevant to Geometry. All that we require, for our immediate purpose, is a subject of that diversity which space renders possible, or terms for those relations by which empty space, if space is to be studied at all, must be differentiated. But how must a matter, which is to fulfil this function, be regarded?

Empty space, we have said, is a possibility of diversity in relation, but spatial figures, with which Geometry necessarily deals, are the actual relations rendered possible by empty space. Our matter, therefore, must supply the terms for these relations. It must be differentiated, since such differentiation, as we have seen, is the special work of space. We must find, therefore, in our matter, that unit of differentiation, or atom[194], which in space we could not find. This atom must be simple,i.e.it must contain no real diversity; it must be aThisnot resolvable intoThises. Being simple, it can contain no relations within itself, and consequently, since spatial figures are mere relations, it cannot appear as a spatial figure; for every spatial figure involves some diversity of matter. But our atom must have spatial relations with other atoms, since to supply terms for these relations is its only function. It is also capable of having these relations, since it is differentiated from other atoms. Hence we obtain an unextended term for spatial relations, precisely of the kind we require. So long as we sought this term without reference to anything more than space, the self-contradictory notion of the point was the only outcome of our search; but now that we allow a reference to the matter differentiated by space, we find at once the term which was needed, namely, a non-spatial simple element, with spatial relations to other elements. To Geometry such a term will appear, owing to its spatial relations, as a point; but the contradiction of the point, as we now see, is a result only of the undue abstraction with which Geometry deals.

200.(2)The circle in the definition of straight lines and planes.This difficulty need not long detain us, since we have already, with the material atom, broken through the relativity which caused our circle. Straight lines, in the purely geometricalprocedure, are defined only by points, and points only by straight lines. But points, now, are replaced by material atoms: the duality of points and lines, therefore, has disappeared, and the straight line may be defined as the spatial relation between two unextended atoms. These atoms have spatial adjectives, derived from their relations to other atoms; but they have nointrinsicspatial adjectives, such as could belong to them if they had extension or figure. Thus straight lines and planes are the true spatial units, and points result only from the attempt to find, within space, those terms for spatial relations which exist only in a more than spatial matter. Straight lines, planes and volumes are the spatial relations between two, three or four unextended atoms, and points are a merely convenient geometrical fiction, by which possible atoms are replaced. For, since space, as we saw, is a possibility, Geometry deals not with actually realized spatial relations, but with the whole scheme of possible relations.

201.(3)Space is at once relational and more than relational.We have already touched on the question how far space is other than relations, but as this question is quite fundamental, asrelationis an ambiguous and dangerous word, as I have made constant use of the relativity of space without attempting to define a relation, it will be necessary to discuss this antinomy at length.

202.Now for this discussion it is essential to distinguish clearly between empty space and spatial figures. Empty space, as a form of externality, is not actual relations, but the possibility of relations: if we ascribe existential import to it, as the ground, in reality, of all diversity in relation, we at once have space as something not itself relations, though giving the possibility of all relations. In this sense, space is to be distinguished from spatial order. Spatial order, it may be said, presupposes space, as that in which this order is possible. Thus Stumpf says[195]: "There is no order or relation without a positive absolute content, underlying it, and making it possible to order anything in this manner. Why and how should we otherwise distinguish one order from another?... To distinguish different orders from one another, we must everywhere recognize aparticular absolute content, in relation to which the order takes place. And so space, too, is not a mere order, but just that by which the spatial order, side-by-sideness (Nebeneinander) distinguishes itself from the rest."

May we not, then, resolve the antinomy very simply, by a reference to this ambiguity of space? Bradley contends (Appearance and Reality, pp. 36–7) that, on the one hand, space has parts, and is therefore not mere relations, while on the other hand, when we try to say what these parts are, we find them after all to be mere relations. But cannot the space which has parts be regarded as empty space, Stumpf's absolute underlying content, which is not mere relations, while the parts, in so far as they turn out to be mere relations, are those relations which constitute spatial order, not empty space? If this can be maintained, the antinomy no longer exists.

But such an explanation, though I believe it to be a first step towards a solution, will, I fear, itself demand almost as much explanation as the original difficulty. For the connection of empty space with spatial order is itself a question full of difficulty, to be answered only after much labour.

203.Let us consider what this empty space is. (I speak of "empty" space without necessarily implying the absence of matter, but only to denote a space which is not a mere order of material things.) Stumpf regards it as given in sense; Kant, in the last two arguments of his metaphysical deduction, argues that it is an intuition, not a concept, and must be known before spatial order becomes possible. I wish to maintain, on the contrary, that it is wholly conceptual; that space is given only as spatial order; that spatial relations, being given, appear as more than mere relations, and so become hypostatized; that when hypostatized, the whole collection of them is regarded as contained in empty space; but that this empty space itself, if it means more than the logical possibility of space-relations, is an unnecessary and self-contradictory assumption. Let us begin by considering Kant's arguments on this point.

Leibnitz had affirmed that space was only relations, while Newton had maintained the objective reality of absolute space. Kant adopted a middle course: he asserted absolute space, butregarded it as purely subjective. The assertion of absolute space is the object of his second argument; for if space were mere relations between things, it would necessarily disappear with the disappearance of the things in it; but this the second argument denies[196]. Now spatial order obviously does disappear with matter, but absolute or empty space may be supposed to remain. It is this, then, which Kant is arguing about, and it is this which he affirms to be a pure intuition, necessarily presupposed by spatial order[197].

204.But can we agree in regarding empty space, the "infinite given whole," as really given? Must we not, in spite of Kant's argument, regard it as wholly conceptual? It is not required, in the first place, by the argument of the first half of this chapter, which required only that everyThisof sense-perception should be resolvable intoThises, and thus involved only an order amongThises, not anything given originally without reference to them at all. In the second place, Kant's two arguments[198]designed to prove that empty space is not conceptual, are inadequate to their purpose. The argument that the parts of space are not containedunderit, butinit, proves certainly that space is not a general conception, of which spatial figures are the instances; but it by no means follows that empty space is not a conception. Empty space is undifferentiatedand homogeneous;parts of space, or spatial figures, arise only by reference to some differentiating matter, and thus belong rather to spatial order than to empty space. If empty space be the pre-condition of spatial order, we cannot expect it to be connected with spatial relations as genus with species. But empty space may nevertheless be a universal conception; it may be related to spatial order as the state to the citizens. These are not instances of the state, but are contained in it; they also, in a sense, presuppose it, for a man can only become a citizen by being related to other citizens in a state[199].

The uniqueness of space, again, seems hardly a valid argument for its intuitional nature; to regard it as an argument implies, indeed, that all conceptions are abstracted from a series of instances—a view which has been criticized inChapter II. (§ 77), and need not be further discussed here[200]. There is no ground, therefore, in Kant's two arguments for the intuitional nature of empty space, which can be maintained against criticism.

205.Another ground for condemning empty space is to be found in the mathematical antinomies. For it is no solution, as Lotze points out (Metaphysik, Bk.II.Chap.I., § 106), to regard empty space as purely subjective: contradictions in a necessary subjective intuition form as great a difficulty as in anything else. But these antinomies arise only in connection with empty space, not with spatial order as an aggregate of relations. For only when space is regarded as possessed of some thinghood, can a whole or a true element be demanded. This we have seen already in connection with the Point. When space is regarded, so far as it is valid, as only spatial order, unbounded extension and infinite divisibility both disappear. What is divided is not spatial relations, but matter; and if matter, as we have seen that Geometry requires, consists of unextended atoms with spatial relations, there is no reason to regard matter either as infinitely divisible, or as consisting of atoms of finite extension.

206.But whence arises, on this view, the paradox that we cannot but regard space as having more or less thinghood, and as divisiblead infinitum? This must be explained, I think, as a psychological illusion, unavoidably arising from the fact that spatial relations are immediately presented. They thus have a peculiar psychical quality, as immediate experiences, by which quality they can be distinguished from time-relations or any other order in which things may be arranged. To Stumpf, whose problem is psychological, such a psychical quality would constitute an absolute underlying content, and would fully justify his thesis; to us, however, whose problem is epistemological, it would not do so, but would leave themeaningof the spatial element in sense-perception free from any implicationof an absolute or empty space[201]. May we not, then, abandon empty space, and say: Spatial order consists offeltrelations, andquâfelt has, for Psychology, an existence not wholly resolvable into relations, and unavoidablyseemingto be more than mere relations. But when we examine the information, as to space, which we derive from sense-perception, we find ourselves plunged in contradictions, as soon as we allow this information to consist of more than relations. This leaves spatial order alone in the field, and reduces empty space to a mere name for the logical possibility of spatial relations.

207.The apparent divisibility of the relations which constitute spatial order, then, may be explained in two ways, though these are at bottom equivalent. We may take the relation as considered in connection with empty space, in which case it becomes more than a relation; but being falsely hypostatized, it appears as a complex thing, necessarily composed of elements, which elements, however, nowhere emerge until we analyze the pseudo-thing down to nothing, and arrive at the point. In this sense, the divisibility of spatial relations is an unavoidable illusion. Or again, we may take the relation in connection with the material atoms it relates. In this case, other atoms may be imagined, differently localized by different spatial relations. If they are localized on the straight line joining two of the original atoms, this straight line appears as divided by them. But the original relation is not really divided: all that has happened is, that two or more equivalent relations have replaced it, as two compounded relations of father and son may replace the equivalent relation of grandfather and grandson. These two ways of viewing the apparent divisibility are equivalent: for empty space, in so far as it is not illusion, is a name for the aggregate of possible space-relations. To regard a figure in empty space as divided, therefore, means, if it means anything, to regard two or more other possible relations as substituted for it, which gives the second way of viewing the question.

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