Why Rigour?

This brief charting of the background of Russell’s logicism does not as yet explain the sense that he attached to rigour. As was pointed out above, “rigour” is not a self-explanatory term. In order correctly to understand the point behind Russell’s logicism, we must acknowledge that the term is ambiguous in a way that relates to the very point and purpose of such foundational programmes as logicism.²⁹

The terminology of rigour involves an important and often unac-knowledged ambiguity which relates to the very point and purpose of such foundational programs as logicism. In one sense, rigour is an epistemological notion. In this sense, the search for a firm or rigorous foundation is a search for an epistemically secure or unshakeable ground. In the second sense, the pursuit of rigour is a broadly semantic enterprise that relates to the improvement of mathematical understand-ing, rather than mathematical knowledge. It is somewhat less easy to give a brief characterisation of this second sense than it is to describe the importance that an epistemically secure foundation might have for someone. Following Coffa (1991, p. 27), we may say that this second sense involves “a search for a clear account of the basic notions of a discipline”.

We can see the semantic sense of rigour in use when we consider Russell’s view that rigorous definitions of such notions as infinity,

29 The following discussion draws on Coffa (1991, pp. 26-8).

continuity and limit had made it possible to form a correct concep-tion of such philosophically pregnant topics as space, time, moconcep-tion and change.³⁰ Given the mathematical results established by Weier-strass, Cantor and Dedekind, earlier philosophers’ theories of these matters (and here Russell has Kant and other idealists in mind) were capable of being conclusively refuted for the simple reason that they had not really known what they were talking about. To illustrate, consider the case of infinity. If it is accepted as an evident truth that a whole is always greater than a part (as many philosophers had done), then the infinite – and anything in which it is involved – must be condemned and rejected as contradictory. Where philosophers went wrong, Russell explains, is that they never asked what infinity is, and their actual treatments of the topic show that had they ever been asked the crucial question, they “might have produced some unintel-ligible rigmarole, but [they] would certainly not have been able to give a definition that had any meaning at all” (1901a, p. 372). Mathemati-cians, by contrast, not only posed this question to themselves but also gave “a perfectly precise definition of infinite number or an infinite collection of things” (ibid.) It follows from these developments that modern mathematics had deprived philosophers’ views on infinity and related notions of whatever plausibility they may have possessed before mathematicians gave their exact definitions and notions with perfectly precise meanings.

Rigour is important, furthermore, not only because it enhances our understanding of such topic as infinity. Even more important for Russell was the fact that mathematicians pursuing rigour had in fact changed the whole of their science: “one of the chief triumphs of modern mathematics”, he wrote, “consists in having discovered what mathematics really is” (1901a, p. 366). From a philosophical point of view, then, the pursuit of rigour put a philosopher of mathematics for

30 This charge is stock-in-trade of Russell’s. For an early formulation see Russell (1901a, pp. 372-3). Elsewhere he goes so far as to say that “[t]he solution of the difficulties which formerly surrounded the mathematical infinite is probably the greatest achievement of which our age has to boast”

(1907a, p. 66).

the first time in a position to give a non-arbitrary answer to the question of the nature of mathematics. Russell’s logicism is an at-tempt to give a detailed answer to this question.

Even though the semantic sense is evidently present in Russell’s writings – and is arguably what gives his logicism its real bite, a claim which will be developed in detail below – later interpreters have in general tended to interpret his interest in the foundations of mathe-matics in the light of what Coffa calls the epistemological sense of rigour. According to a common conception, Russell’s logicism was a special case of his more general preoccupation with such issues as scepticism, certainty and the possibility of knowledge; on this view Russell wanted to find good reasons to believe that mathematics gives us genuine – that is, certain, indubitable – knowledge.³¹ It was for this reason, it is argued, that he attempted to demonstrate that mathematics is just logic: if that turns out to be the case, then we can be assured that there is certainty and genuine knowledge.³²

31 For this view, see Andersson (1994).

32 This very same ambiguity affects also one’s reading of Frege. He was clearly concerned with rigour in the semantic sense when he laments, in the Preface to Grundlagen, arithmeticians’ ignorance of number one (that is their inability to give an acceptable answer to the question, what number one is), and says that, consequently, we “hardly succeed in finally clearing up nega-tive numbers, or fractional or complex numbers, so long as our insight into the foundation of the whole structure of arithmetic is still defective” (1884, p. ii). In order to rectify the situation, he thought, it was necessary to provide arithmetic with genuine foundations, which would, among other things, tell us what the number one is. Nevertheless, Frege’s logicism is often inter-preted in a way that connects his insistence on rigour with skepticism and a search of certainty. This reading of Frege is found, for instance, in Currie (1982, pp. 28-9).