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

Written by A.C. 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 RussellÕs 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.

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.

I take the second point first, concerning the relation of a priori judgments to the branches of experience which presuppose them. Moore seems to be confused about what Russell is claiming here. He seems to take Russell's claim to be that unless such judgments are true, judgements about the subject-matter in question must be false (taking the modalities seriously). Moore thus reads 'inaccessible to cognition' as implying that there is something about which we are not in a position to make judgments. His realist commitment to the independence of judgments from knowledge of their truth or falsity accordingly portrays this as a straightforward mistake. But Russell is not saying this; his claim is the familiar Kantian one that there could be no such branch of experience as the one in question unless we know a priori the judgments that make it possible. He put the point by saying that the only thing that could make such a priori judgments false is if the branch of experience in question were impossible; again taking the incorporated modalities seriously, his talk of the 'inaccessibility to cognition' of a branch of experience is not to be read as implying or (as Moore sees it) conceding that there is something to be known if only we could get at it; this is Moore's mistake; rather it says that if nothing constitutes the a priori condition of possibility for there being such cognition, there could be no such subject-matter.

Russell's point here in fact concerns the very nature of transcendental arguments. He offers a novel and interesting way of capturing what is essential about such arguments, which, contrary to what is often thought, are not in the least logically peculiar or special, but are distinguished from other argumentative strategies by a certain distinctive aim, which is to establish conceptual title to a principle or claim which, accepted as true, licenses our activity in some region of judgment (and I intend that to be a terminological variant for 'makes a certain kind of experience possible'.) Kant is somewhat to blame for leading some commentators–in this connection I have Griffin in mind, who in his discussion of Moore on Russell's EFG expects more from transcendental arguments without quite saying what–to think that transcendental arguments have to take us beyond their premisses, which concern the nature of a certain kind of experience (or thought), and establish something not already implicit in its character and conditions. But the attempt to show that we have a title to some principle or claim proceeds exactly by showing that we could not do something we in fact do–enjoy spatial sense-perception for example–unless a condition for doing so were satisfied; from which it follows that the principle is satisfied. For Kant as for Russell in EFG the spotlight of attention is on the conditions, because once the deduction of title (the legal metaphor was consciously intended by Kant) has been achieved, the next step is to note that, since the judgment–whose acceptance as true is a condition for the experience in question–could not itself have been derived from that experience but is logically anterior to it, it is a priori: and part of their transcendental task was to identify what has to be known a priori as a ground for that experience.

Put in the most schematic way, transcendental arguments state that there would not be A unless there were B, and that since there is A, there is B. Familiarly and prosaically, the underlying move is a statement of necessary conditionality: B is a necessary condition for A, and since A is the case, so therefore is B. (If one said: therefore B has to be the case too, the 'has to' has to be understood purely conditionally.) Arguments of this form are very common; they only cause a stir when applied in what might be called Kantian contexts. Russell in EFG succeeded in capturing their character by casting them as portrayals of presuppositionality in terms of truth-value, partially (but only partially: see below) anticipating later variants of the move. We see this by noting Moore's mistake in taking Russell to have asserted that there is a presuppositional relation between A and B such that if judgments as to B are false, those as to A must be necessarily false. This makes the relation very peculiar, for it issues in the necessary falsehood of A-judgments when B-judgments are false–which it is not clear how one might motivate. But the mistake takes us close to what Russell intended. In later debate, presupposition is more familiarly (if no less problematically) taken as a relation obtaining between judgments such that a given presupposing judgment has a truth-value only if a given presupposed judgment is true (I am adhering here to the Russell-Moore terminology of the day: mutatis mutandis, the same points can be made in more careful ways, thus bringing out the fact that it is as a semantic relation that the notion is now standardly understood–rather than an epistemic one between, on the one hand, an asserter or judger seeking to assert or judge a content p and, on the other, another content q required for p's being assertable or judgeable as true or false). This is only part of what Russell intends, for the good reason that it is only part of what is implied in a transcendental argument, in which a stronger claim is at stake, namely, that the falsity of a given judgment–a B-judgment, say–renders impossible even the circumstances for making or entertaining a would-be A-judgment, a matter far antecedent to the would-be A-judgment's having a truth-value. So on the Kantian view B's being true makes A either true or false; but B's being false makes it impossible to make or even entertain A-type judgements. Moore took it that Russell meant that B's being false makes A-judgements necessarily false. But the two claims are of course very different, and it is unclear whether what Moore imputes to Russell is even coherent. Russell, however, gives us an insight into the style of argument at stake.

In the foregoing there is no suggestion that either Moore or Russell had anticipated exactly what has come to be meant by talk of 'presupposition' since Pears and Strawson; rather, their debate illustrates the sense in which that later debate itself captures something close to but weaker than the relation which a transcendental argument asserts to hold between a given kind of experience (or conceptual practice) and what makes it possible. But in the uncertain oscillation between talk of experience and talk of judgments as we find it in Russell and Moore, it is easy to see that in one mode the appropriate locution is 'ground of possibility' and in the other, talk about the condition for possession by a judgment of truth-value as lying in the truth-value of another judgment. The two jargons are not the same in meaning, but once that is recognised there are no irreducible difficultes in straying between them as our protagonists do.

It is worth remarking at this juncture that this account of Moore's attack does not agree with the accounts given by Hylton and Griffin. On the reading we each give of Moore there might be a little latitude for interpretation because of the unclarities in Moore's presentation, but not that much: and Griffin, as noted, misidentifies the character of the transcendental strategy in general and Russell's in particular, chiefly by asking too much of it. I defer an itemised comparison here. But the crucial respect in which I hope that this supplements their discussions is in recognising that the more significant argument of Moore's is the one directed not against the style of argument Russell uses, but against the argument itself: this occurs as the first point criticised by Moore above, together with its all-important ancillary requirement.

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