An Essay on the Foundations of Geometry

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[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 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[198] 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[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 in Chapter 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 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[201]. 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