An essay on the foundations of geometry

by Bertrand Russell

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.