An Essay on the Foundations of Geometry

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[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 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[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 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[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'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[182] and others[183], and has been described by Vaihinger as a fatal petitio 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, 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[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 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[187]." 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