Russell’s conception of the status of mathematics-cum-logicism vis-à-vis (traditional) philosophy has an important critical function as well, as it makes the former a very effective means of philosophical criti-cism. Suppose that some philosophical theory turns out to conflict with some part of established mathematics (and, Russell argued time and again, what idealists had to say about mathematics and related subjects offered any number of illustrations of this clash). This could happen in either of the following two ways: either the theory is in straightforward contradiction with an established mathematical result or else it cannot make room for some mathematical phenomenon.51 Either way, Russell’s conclusion would be that it is the philosophers’
views which must be rejected or at least modified so as to bring them into harmony with acknowledged mathematical facts. Furthermore, those philosophies which he knew were, by and large, indifferent to what had taken place in mathematics itself. This was either because they were tied up with pre-nineteenth century mathematics or else because they simply ignored what mathematicians had discovered
51 A an example of the former case is the idealist Russell’s rejection of Cantor’s transfinite arithmetic (see Russell 1896a, p. 37; 1896b, pp. 50-2, 1897a; cf. Griffin 1991, pp. 240-3; G. H. Moore 1995, Levine 1998, pp. 96-8). If, as Russell did at the time, numbers are looked on as something that results from counting or synthesis, it follows immediately that a “com-pleted” or actual infinity stands condemned, since, as he explains, “absolute infinity is merely the negation of possible synthesis, and thus the negation of number” (1896b, p. 50). An example of the latter case is Kant’s theory of geometry, which, if unmodified, comes to conflict the mathematical fact that there are several internally consistent but mutually inconsistent geometries (this claim, though in my opinion correct, is of course controversial; for a forceful defence of it, see Friedman (1992, ch. 2).
about their own discipline.52 From his newly acquired perspective, all these philosophies had been rendered obsolete by progress in mathematics.
For Russell, Kant’s theory of mathematics occupied a special place among these philosophies. His own gradual progress towards the logicist position could be described roughly as a transition, both mathematically and philosophically, from an essentially pre-nineteenth century understanding of mathematics to a full apprecia-tion of mathematical rigour. Initially, his knowledge of mathematics was based on what he had learned as a Cambridge undergraduate in mathematics, and his earliest attempts within the philosophy of mathematics were conceived within a broadly idealistic framework which owed a great deal to Kant.53 In the course of his subsequent
52 Thus, for instance, mathematics was no longer describable as the science of “discrete and continuous quantity”, which had been the standard characterisation up until the early nineteenth century (and remained so at least for many philosophers for decades to come (as Russell (1901a, p. 376-7) in fact points out). As Bolzano wrote in 1810: “in all modern textbooks of mathematics the definition is put forward: mathematics is the science of quantities (1810, §1). Bolzano, of course, went on to criticise variants of this definition for being either too wide (sloppy or off-hand definitions of “quantity” did not give a distinguishing characteristic of mathematics) or else too narrow (the quantity view excluded important regions of mathematical research).
53 Griffin (1991) is the first detailed study of Russell’s idealism. It con-tains, among other things, an elaborate description of Russell’s mathematical development in the 1890s. For the evolution of Russell early logicism, see Rodríguez-Consuegra (1991). Though brief, Levine (1998) is very informa-tive and illuminating on transitional period in Russell’ development from idealism, through Moorean logic and metaphysics, to Peano and the Principles of Mathematics. Kant’s influence on Russell is best seen in Russell’s Essay on the Foundations of Geometry of 1897, the substance of which consists in an endeavour to bring a broadly Kantian position in line with the existence of non-Euclidean geometries. In 1898, while working on a manuscript entitled An Analysis on Mathematical Reasoning, he could still write to Louis Couturat:
“I am asking the question from the Prolegomena, ‘Wie ist reine Mathematik möglich?” I am preparing a work of which this question could be the title,
development Russell absorbed more and more of the work which was being done on the continent and which he associated with such names as Weierstrass, Dedekind, Cantor and Peano all of them advocates of “rigorous mathematics”. With his growing realisation that the mathematics which was being done on the Continent was very far from the mathematics which he had been taught at Cam-bridge, he also came to see that he had been looking in the wrong direction for a philosophical account of mathematics. These devel-opments in Russell’s view have been very well summed up by Nicho-las Griffin: “[o]ne of the things which makes Russell’s development in the 1890s so interesting is that, within the space of seven years, he moves from a full-blooded Kantian position, such as might have been widely accepted at the beginning of the century, to a complete rejection of Kant, a position which was not common even among the advanced mathematicians of the time”.54 The facts about Russell’s own development that Griffin mentions in this passage, to wit, that Russell’s own efforts in the philosophy of mathematics had been inspired by Kant, and that he gradually fought himself free from all even remotely Kantian elements in his own philosophy, make it easy to understand why he was, once he had formulated his logicist posi-tion, so eager to spell out the consequences of “modern mathemat-ics” for Kant’s philosophy.
The logicist Russell’s interest in Kant’s theory of mathematics is thus a natural outcome of his own development. Kant’s theory, he came to think, was tied up with an outmoded conception of mathe-matics in exactly the same way that his own efforts had been. Under-stood in this way, Kant’s position is vulnerable to an obvious and immediate charge. Russell was not slow to make use of it (and his own development certainly made him very sensitive to it); if Kant’s theory of mathematics reflects the mathematics of his day, it is unlikely to stand unscathed if mathematics itself changes. And
cer-and in which the results will, I think, be for the most part purely Kantian”
(letter to Louis Couturat, 3 June 1898; quoted in Russell 1990, p. 157).
54 Griffin (1991, p. 99).
tainly it was not far-fetched for Russell, or for anyone else, to think that the whole science of mathematics had been revolutionised in the course of the nineteenth century. If, then, as Russell said, “pure mathematics” was only discovered in the nineteenth century (1901a, p. 366), all earlier attempts – Kant’s theory included – were marred by sheer ignorance as to the subject with which they thought they were dealing with:
This science [sc. pure mathematics], like most others, was baptised long before it was born; and thus we find writers before the nineteenth cen-tury alluding to what they called pure mathematics. But if they had been asked what thus subject was, they would only have been able to say that it consisted of Arithmetic, Algebra, Geometry, and so on. As to what these studies had in common, and as to what distinguished them from applied mathematics, our ancestors were completely in the dark. (ibid.) Russell’s criticism of Kant is a straightforward application of this charge.
Russell’s attitude can be illustrated briefly and in a preliminary fashion by the special case of geometry. On the face of it, traditional or pre-nineteenth century formulations of Euclidean geometry were radically different from the geometry of the late nineteenth century.
The “second birth” of mathematics put Russell in a position to argue that the distinctive features of the traditional geometrical practice, in particular its notorious “reliance on figures” in the proofs of theo-rems, were simply due to the fact that geometry had not been devel-oped with full rigour. This in turn had immediate consequences for Kant’s theory of geometry. The latter was centrally concerned with providing an explanation of the possibility of geometry. Given the inadequacies of the mathematics of his day, of which the peculiarities of Euclid’s axiomatisation provided one instance, Kant was bound to be led astray in his purported explanation. That is, Russell thought that it could be demonstrated, thanks to rigorous mathematicians, that Kant’s diagnosis of the conceptual situation was simply an error due to ignorance. This diagnosis, furthermore, could be easily dis-posed of by looking at the most advanced and carefully formulated
mathematical theories of the time. Thus, Russell wrote about Peano and his disciples: “I liked the way in which they developed geometry without the use of figures, thus displaying the needlessness of Kant’s Anschauung (1959, p. 72). In general, Russell argued that changes in mathematics called for a new philosophy as well; this new philoso-phy, furthermore, would be radically incompatible with anything that Kant had ever said or taught.
Russell’s criticism of Kant’s theory of mathematics naturally focuses on the notion of Anschauung or intuition. As the secondary literature on Kant amply demonstrates, there are considerable diffi-culties in saying what this notion amounts to.55 Whatever one’s preferred answer is to this matter, the important point now is that in arguing for his positive theory, Kant was able to draw on several mathematical practices of his time. From a modern point of view, or from that perspective which Russell came to know in the late 1890s, one may feel the temptation, as Coffa has put it, “to compare Kant’s discovery of pure intuition [...] with Leverrier’s discovery of Vulcan”
(1981, p. 33).56 Nevertheless, as Coffa goes on to point out, this
“comparison is unfair to Kant in that there were mathematical data that could well be interpreted as justifying Kant’s conclusions” (ibid.) Examples – and these play a major role in Kant’s own expositions of his theory – are provided by the Newtonian understanding of the calculus, the proof-structure of Euclid’s geometry, and accounts of arithmetic which attempt, in one way or another, to build that science on calculation. From the standpoint of the late 19th century, all these data belongs to a superseded stage of mathematical research. Hence, Russell argues, Kant’s theory of mathematics with its notion of pure intuition has become equally outdated; with the data gone, pure intuition (i.e., Kant’s explanation for the data) is also dispensed with.
Russell’s case against Kant thus rests on the charge that the latter’s theory of mathematics had been grounded in a mathematical practice
55 This issue is dealt with, e.g., in several of the essays collected in Posy (1992).
56 That is, a discovery of what is not there in the first place.
which was insufficiently explicit about its own foundations; this practice, that is to say, had neglected the task of analysing its con-cepts, judgments and reasonings. When nineteenth century mathema-ticians carried out this conceptual and methodological revision, thereby producing a completely new picture of mathematics, it began to seem as if Kant’s theory of mathematics had been conditioned by sheer ignorance (“ignorance” in the somewhat peculiar sense that Kant had failed to anticipate the major mathematical and logical discoveries made by later generations).
Logicism was not just an exhortation to do mathematics in a rigorous manner; it was also, and above all, a substantial thesis con-cerning the true nature of mathematics. Kant had tried to make sense of that nature by developing a conception of mathematical method in which the notion of pure intuition played the central role. Further-more, when he explained how his method could yield apriori knowl-edge, he ended up in transcendental idealism. The logicist Russell subjected these results to a criticism in which the following elements can be discerned. Firstly, his immediate target was Kant’s theory of mathematics and the notion of pure intuition. And the philosophical point of Russell’s logicism lies precisely here; if successful, it shows that and why there is no room for anything like Kantian intuitions in the proper treatment of pure mathematics. Secondly, considerations pertaining to mathematics paved the way for a more general attack on transcendental idealism, or those far-reaching consequences which Kant had drawn from what he thought was needed to understand mathematics and mathematical knowledge.
The distinctive features of Kant’s transcendental idealism are connected with the introduction of a new category of judgments, the synthetic apriori, together with his explanation of how there can be such judgments. The underlying theme of Russell’s anti-Kantianism, of which logicism is the cornerstone, is his attempt to undermine Kant’s purported explanation of the synthetic apriori; in those cases where Kant claims we are dealing with synthetic apriori judgments and which, for that reason, involve pure intuition, Russell seeks to show that what is really involved is something purely logical. To draw
substantial and non-arbitrary conclusions from this contrast, he had to have a fairly sophisticated conception of logic. This conception can thus be elucidated by scrutinising what Russell has to say about Kant’s theory of mathematics and his transcendental idealism.