An Essay on the Foundations of Geometry

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[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, 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[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 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[194], 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[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 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