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

AC Grayling

Russell was generous in attributing the sources of his inspiration to others, and never more so than in explaining what he described as the 'revolution' in his philosophical thought which occurred in the closing years of the nineteenth century. At Cambridge he had been made to feel the influence of Kant and Hegel, and especially of the latter, with whom he sided whenever he encountered disagreement between them. His great plan for two series of books effecting a synthesis of philosophy and science was Hegelian in inspiration, and his Fellowship dissertation, An Essay on the Foundations of Geometry, was Kantian not just in inspiration but in aim and, to a significant degree, content also.

Russell gave the credit for the revolution in his thought to Moore: 'Moore led the way but I followed in his footsteps' . Several times he cites Moore's paper 'The Nature of Judgment' as the key document in this change . Commenting on its influence, he says that its most important doctrine is its realist commitment to the independence of fact from experience. But each of them pursued differently-emphasised routes from this agreement: Moore was concerned to refute idealism, while Russell was more interested in refuting monism. Nevertheless Russell took these two -isms to be connected through the doctrine of relations. In his view monism arises from commitment to the view that all relations are grounded in their terms; the application to idealism is that relations between thought or experience and their objects are asserted to be internal likewise, rendering them interdependent in ways that make what we pretheoretically take to be objective relata in some sense mental or grounded in the mental.

So much is familiar enough. But there is reason to think that another paper Moore published in 1899 had as big an effect on Russell from the viewpoint of what one might call its reach into Russell's later philosophical work. This is Moore's Critical Notice of Russell's Essay on the Foundations of Geometry (hereafter EFG). One is tempted to compare it in character and effect to Frege's celebrated conversion of Husserl from psychologism by his review of Husserl's Philosophie der Arithmetik . This would not be gathered from MPD, where Russell dismisses the argument of EFG on the grounds that the General Theory of Relativity made it obsolete. But when one considers what Russell says in Human Knowledge, Its Scope and Limits about Kant and about the postulates of scientific inference, one sees that the revolution in his philosophical views as profoundly rooted out the temptations of transcendental philosophy as it did monism–but, it might be argued, hardly with so positive an effect, because Russell ended by believing that there can be no ultimate appeal to a priori knowledge in the constitution of knowledge in general: hence the tottering version of fallibilism in Human Knowledge, with its ultimate reliance on supposed contingencies about evolution and animal habits. The flight from sophisticated psychologism had, so to say, crash-landed in crude biologism. Note however that Russell says in a letter to Moore of 18 July 1899: 'I had not written to you about your review, because on all important points I agreed with it' Thereafter he made no more use of Kantian strategies and took to calling him by Cantor's disagreeable label for him, 'Yon sophistical Philistine'.

Yet in EFG Russell not only employs a transcendental argument of great interest, but gives a characterisation of the nature of transcendental arguments which is of even greater interest. On the first head: it is noteworthy that espousing or rejecting some version of the transcendentalist strategy in something like Kant's sense (one need not accept much else of the Kantian luggage; compare Strawson's selectivity in The Bounds of Sense) is a matter quite independent of espousal or rejection of either or both of pluralism and realism, the two commitments which mark Russells philosophy after EFG. If Russell had not lumped everything Kantian together for wholesale rejection, but had made use of transcendental arguments as he subtly understood them in EFG, he might have spared himself the later epistemological insecurities he variously suffered. On the second head: as is characteristic of Russell, his account of the nature of transcendental arguments in EFG anticipates later revived interest in the strategy, not only as Strawson and some others of us have deployed them, but in variant forms; as for example in the generalised notion of presuppositions, whose role in the solution of certain semantic problems was later to return to haunt Russell.

Russell's insight into the transcendental strategy is well brought out by Moore's interesting failure to understand it, which is why I enter his account via Moore's devastating-seeming attack in the Critical Notice of EFG . In recent major studies of Russell both Nick Griffin and Peter Hylton give it attention, the former as part of his detailed account of Russell on geometry and the latter more briefly as constituting an attack on psychologism, which in important part it was indeed intended to be; but Hylton leaves aside questions about the merits of Moore's attack, and whether Russell should have capitulated to it so entirely–bearing in mind that, as Russell saw and indeed insisted, the philosophical consequences of theories of geometry are not confined to choice of geometry for physical theory, but impinge significantly on our theories of perceptual experience, representation and indexical thought . On this question Griffin takes the view that the jury remains out on whether Russellšs way with the Kantian strategy–in particular, his reworking of a transcendental argument to the necessity for experience of a 'form of externality'–is in any degree successful . I am inclined to think that it indeed has something to offer: but in proceeding by way of a discussion of the merits of Moore's attack on EFG I shall, except in relation to one matter, only obliquely indicate part of why that is so.

It is useful to have a reminder of what Russell was attempting in EFG. His aim was to survey the foundations of geometry in the light of the revolutionary advances in that science since Kant. Kant had claimed that space is the form of outer sensibility, and that Euclidean geometry describes it; but 19th century mathematicians called into question both the belief that space is Euclidean and the claim that a Euclidean form of space is necessary to outer experience. Moreover they showed that Euclidean (space has zero curvature), Lobatchevskyan (with Gauss and Bolyai: hyperbolic geometry–space has negative curvature) Riemannian (spherical or double elliptic–space has positive curvature) and Kleinian (single elliptic) geometries can be derived as special cases of projective geometry, which deals with the qualitative (descriptive) properties of space, whereas Euclidean and the other non-Euclidean basic geometries deal with its quantitative (metric) properties. So a set of properties not recognised in Euclidean geometry, namely, the qualitative ones, had been shown to be logically prior to Euclidean properties. The question of what if anything constitutes the a priori foundation of geometrical knowledge therefore needed to be considered afresh, and this task Russell undertook in EFG.

Russell accepted the Kantian view that there must be such a thing as a 'form of externality' as a condition of possibility for spatial experience. In an interesting modification of Kant's thesis he argued that the possibility of such experience rests not just on the constitution of sensibility but on the world's receptiveness to the adjectives we impose on it. But he locates the properties of the form of externality not in Euclidean but in projective geometry, its transcendental status–carefully disentangled from the question of the subjectivity of a priori elements in experience–consisting in its applying to all spaces independently of experience of any of them.

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