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# An essay on the Foundations of Geometry: Chapter IV - Philosophical Consequences

An Essay on the Foundations of Geometry, by Bertrand Russell is part of HackerNoon’s Book Blog Post series. The Table of Links for this book can be found here. Chapter IV: Philosophical Consequences

## CHAPTER IV. PHILOSOPHICAL CONSEQUENCES.

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 as à priori? The conception Mammal has some of the attributes of a horse; but are these attributes therefore à priori adjectives 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 as à priori the 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 a quantitative resemblance 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 form of 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 general conception, containing under it every logically possible intuition which 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, is à priori necessary 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 the à priori to 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. The ground of necessity, we may safely say, arises from the mind; but it by no means follows that the truth of 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 declared à priori, can only be: Owing to the constitution of the mind, experience will be impossible unless the world accepts certain adjectives.

Such, in outline, will be the argument of the first half of this chapter, and such will be the justification for regarding as à priori those 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. 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 reference to 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. 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.

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 the à priori and 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. 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's logical teaching, 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 is totally unsupported by argument, being set forth, apparently, as a self-evident axiom; it has been severely criticized by Stumpf and others, and has been described by Vaihinger as a fatal petitio principii; 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, that no form 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 a petitio 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?

Space seems to 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. These two functions, though often treated as coordinate and almost equivalent, 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. 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 these theses—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." The importance of this fact appears, when we consider that, if any mere particular 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 the This of immediate presentation, from which judgment and inference necessarily spring. Owing, however, to the continuity and relativity of space and time, no This can be regarded either as simple or as self-subsistent. Every This, on the one hand, can be analyzed into Thises, 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 me in present sensuous perception, and in the immediate feeling of my own sentient existence that goes with it. The real world, as a definite organized system, is for me an 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 not merely this 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 the This of sense-perception, and since such extension is only possible if the This has that fragmentary and yet complex character conferred by a form of externality, therefore some form of externality, given with the This, is essential to all knowledge, and is thus logically à 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 the This to be neither single nor self-subsistent. But I will endeavour, 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 sensational This must, when attended to, be found, on the one hand, resolvable into Thises, 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 the Thises contained in the first This, we get a new This necessarily related to the other Thises which make up the original This. 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 a mere particular. 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 no longer 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. 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, 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 in Chapter III., 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 the This of sense-perception. For, if this be so, the This—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 is à priori necessary to the possibility of experience.

I cannot hope, however, that no difficulty will remain, for the reader, in such a deduction, from abstract conceptions, of the properties of an actual datum in 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. The à 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 the à priori and 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 Chapter III., 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 planes] through 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 divisible ad 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. 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. 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, arising out 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. This reductio ad absurdum surely 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, if independent knowledge 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, in Chap. 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, 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. 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 the first 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 a possible relation, 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 matter can 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, which in space we could not find. This atom must be simple, i.e. it must contain no real diversity; it must be a This not resolvable into Thises. 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 geometrical procedure, 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 no intrinsic spatial 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, as relation is 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: "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 a particular 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, but regarded 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. 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.

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 every This of sense-perception should be resolvable into Thises, and thus involved only an order among Thises, not anything given originally without reference to them at all. In the second place, Kant's two arguments designed to prove that empty space is not conceptual, are inadequate to their purpose. The argument that the parts of space are not contained under it, but in it, 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 undifferentiated and 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.

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 in Chapter II. (§ 77), and need not be further discussed here. 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 divisible ad 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 the meaning of the spatial element in sense-perception free from any implication of an absolute or empty space. May we not, then, abandon empty space, and say: Spatial order consists of felt relations, and quâ felt has, for Psychology, an existence not wholly resolvable into relations, and unavoidably seeming to 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.

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 from contradictions, 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.

Conclusion

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 as à 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 than quantitative 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 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.

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