An essay on the foundations of geometry

by Bertrand Russell

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.