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Russell's Transcendental Argument in: An Essay on the Foundations of Geometry

The chief argument is that qualitative relations must be prior to quantitative ones. There are four fundamental qualitative principles: (1) all parts of space are homogeneous, that is, are qualitatively similar, and all are relative, that is, lie outside one another; (2) space is continuous and infinitely divisible, with the point as the limit of infinite divisibility; (3) two points determine a straight line, three points not on a line determine a plane, and so on for higher figures; and (4) the dimension of space must be finite.

Certain refinements of these constitute further a priori principles required for metrical geometry–required because measurement presupposes them. Homogeneity of space becomes free mobility (analytically equivalent to the constant curvature of space); and the 'two points-straight line' principle becomes an axiom about distance. (These principles are presupposed by measurement, and turn out to lie in the domain of metageometry, and therefore to apply to physical space.) Russell concluded that because these geometries are the only mathematically-possible ones whose spaces are homogeneous, they are the only ones that can apply to physical space. Therefore physical space must be one of Euclidean, Lobatchevskyan, Riemannian or Kleinian. On empirical grounds Russell said that it is Euclidean.

Two developments subsequent to the writing of EFG rendered its views, as Russell says, obsolete. One, the development of topology–which generalises on projective as projective had generalised on Euclidean geometry–imposes an obligation on any Kantians staying the course to review afresh the question of what, if any, geometrical principles are a priori. But much more seriously, every feature of the four dimensional, non-Euclidean, nonhomogeneous (not having a constant curvature) space of the General Theory of Relativity had been effectively or explicitly denied by Russell, who had not registered Riemann's point that a belief in the constant curvature of space depends upon ignoring the existence of matter. When matter is taken into account, homogeneity disappears, as the General Theory states (matter is absorbed into the geometry of space-time which therefore varies regionally according to the matter in it ).

The fourth and final chapter of EFG contains the discussion of the philosophical aspects of these views which interest us here, because it is these that Moore attacks. Here Russell argues that the a priori axioms of geometry can be deduced from the form of externality as a transcendental ground of experience–that is, the condition of the possibility of experience (see esp. section 189 EFG). Russell's view differs in significant ways from Kant's, especially in the interesting respect that it requires the mutual externality of things presented in sense-perception rather than (to begin with anyway) the externality of things to the Self. This and other points in Russell's account are independently interesting and perhaps important for theories of perceptual representation, which makes them worth pursuing on their own account.

Russell defined the a priori as that which is logically presupposed in experience, where (as Hylton reminds us ) the force of 'logically' is the Kantian transcendental one in which questions about the conditions of the possibility of experience are at stake. But whereas for Kant these were synthetic judgments only–for him, analytic ones follow from the principle of contradiction alone–for Russell this division will not do, and along with other post-Kantians he rejected it . But he also objected to the conflation of the a priori with subjectivity, on the grounds that it places a priori truth at the mercy of empirical psychology , and so a second import of Russell's use of 'logically' is its marking a refusal to accept that the validity of Euclid waits upon empirical facts about human spatial intuition .

Russell's argument goes as follows. Knowledge starts from sense experience, the objects of sense experience are complex, whatever is complex has parts, parts have to be mutually external to one another, and therefore a form of externality is logically prior to experience. This form of externality cannot be purely temporal, for the reason–among others–that things given in experience must be 'various' or 'diverse' to allow for complexity, and one crucial way in which they are so is by occupying different positions in space–hence space as the form of externality required. The notion of a form of externality is an essentially relative one; nothing can be external to itself, and so for any one thing there must be another thing to which it is external; the externality is of course mutual, and there have to be yet other positions from which the positions they occupy in turn differ. (The second main contention of EFG is that geometry contains contradictions: this is the Hegelian aspect of the thesis. I leave this aside; see Hylton ).

Moore took himself to have two fatal objections to Russell's project. One is that the most that could be established by an argument of this kind is something about what is presupposed to the kind of experience we in fact have, and that therefore the argument is philosophically valueless because it tells us only about certain psychological contingencies.

The other is less easy to state briefly. Russell said that an a priori judgment is one whose truth-value is insensitive to empirical considerations, and can only be rendered false 'by a change which should render some branch of experience formally impossible, i.e. inaccessible to our methods of cognition' (EFG 60). Moore seems to have taken Russell to be saying that there is something–a subject matter of some sort–to which cognitive access can be had only if a certain a priori judgment is true; and that the judgment's being rendered false would be the effect of the necessary falsehood of judgments about that subject matter; to which Moore responded, 'that which is "inaccessible to our methods of cognition" would seem only to mean that which we cannot know; it cannot imply that the judgments in question cannot be true' (Moore 398). Moore labels the conflation of questions about what is true with what can be known the 'Kantian fallacy'. In his view it is for psychology to answer questions about what and how we know, so such questions are philosophically irrelevant. The crucial commitment in this view is to the independence of judgments from thought: hence the reason for Russell's citing Moore's other 1899 paper, 'The Nature of Judgment' (hereafter NJ) , as the engine in their break with the Kantian and Hegelian traditions (in 'The Nature of Judgment' Moore specifically addressed himself to Bradley's views).

There is much to contest here. The two points Moore addresses are intimately connected in Russell's strategy, in a way that Moore fails to see. He misunderstands the second of them–the one about the presuppositional relationship–and opposes to it a familiar realist claim the argument for which, offered in NJ, is inadequate (there might be better arguments for it, but Moore does not give them). He does however see that the first part of Russell's argument requires a particular supplementation, the satisfaction of an ancillary requirement, one which involves a break with Kant on an important matter and which, on the face of it, seems impossibly difficult to give. Russell evidently took Moore's argument on this point to be conclusive in view of his complete abandonment of the Kantian enterprise.

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