What is Really Involved in Rigour

The point behind Russell’s early logicism is best understood in the light of Coffa’s semantic sense of rigour, rather than the epistemo-logical programme of providing mathematical knowledge with an epistemically unshakeable foundation. This comes out very forcefully in the early sections of the Principles. In the Preface to that work Russell makes it clear that what he is after is a non-arbitrary answer to the question regarding the nature of pure mathematics. In the course but can never be proved. The disproof will consist in pointing out contradic-tions and inconsistencies; but the absence of these can never amount to a proof. All depends, in the end, upon immediate perception; and philosophi-cal argument, strictly speaking, consists mainly of an endeavour to cause the reader to perceive what has been perceived by the author. The argument, in short, is not of the nature of proof, but of exhortation”.

of answering this question, he also comes up with answers to a whole host of other similar-sounding problems: the nature of number, of infinity, of space, time and motion, and of mathematical inference (1903, §2). These are all problems that are semantic in Coffa’s sense:

in each case Russell professes to give a “clear account of the basic notions of a discipline” (Coffa 1991, p. 27). Furthermore, these questions or problems are alike in an important respect. In earlier times they had been subject to endless philosophical debate and the uncertainty that is typical of such controversies. As Russell sees it, this unsatisfactory state of affairs had been brought to an end through mathematicians’ efforts. The problems addressed in the Principles are no longer specifically philosophical; instead, they are trans-formed into problems that can be given a mathematical treatment.

This change has important consequences. Minimally, it follows that philosophers, unless they are willing to take the risk of being rele-gated to the category of the scientifically ignorant, must take into account the definitive results of mathematical research in treating any topic on which mathematics can be brought to bear. As Russell puts it elsewhere:

In the whole philosophy of mathematics, which used to be at least as full of doubt as any other part of philosophy, order and certainty have re-placed the confusion and hesitation which formerly reigned. Philoso-phers, of course, have not yet discovered this fact, and continue to write on such subjects in the old way. But mathematicians [—] have now the power of treating the principles of mathematics in an exact and masterly manner, by means of which the certainty of mathematics extends also to mathematical philosophy. Hence many of the topics which used to be placed among the great mysteries for example, the natures of infinity, of continuity, of space, time and motion are now no longer in any de-gree open to doubt or discussion. Those who wish to know the nature of these things need only read the works of such men as Peano or Georg Cantor; they will there find exact and indubitable expositions of all these quondam mysteries (1901a, p. 369)

The contrast to which Russell alludes in this passage – one between endless and apparently fruitless philosophising and progressive

mathematical research capable of yielding definitive results – could not be made any clearer. It shows, above all, that the import of his logicism should not be seen in the lights of epistemic logicism. As he sees it, mathematics does not need philosophy to safeguard its truth or certainty, which would be in jeopardy in the absence of support from philosophy. On the contrary, it is mathematics which should show the way for philosophers. It is a distinctive gain if a topic can be given a mathematical treatment, since that resolves at once the uncer-tainty and doubt characteristic of philosophy: “[f]or the philosophers there is [...] nothing left but graceful acknowledgment” (1901a, p.

367), once mathematicians have taken over a topic and shown how to treat it with all the precision and exactitude that mathematics is capable of.

This attitude has two consequences, one of them constructive, the other critical. The first has to do with the status Russell assigns to logicism. As he sees it, logicism is not something extraneous to actual or real-life mathematics. In asserting his own version of the logicist thesis, he does not regard himself as advancing a philosophical and hence at least potentially controversial thesis about of the nature of mathematics; on the contrary, logicism is presented as an integral part of the “rigorous mathematics” which was being developed by such mathematicians as Dedekind, Weierstrass, Cantor and Peano. Logi-cism, that is to say, stands closer to mathematics than it does to traditional philosophical accounts of the nature of mathematics. That this is Russell’s attitude is borne out by what he writes in section 3 of the Principles. There he gives a brief description of the benefits which accrue to the philosophy of mathematics from the logicist reduction:

The philosophy of mathematics has been hitherto as controversial, ob-scure and unprogressive as the other branches of philosophy. Although it was generally agreed that mathematics is in some sense true, philoso-phers disputed as to what mathematical propositions really meant: al-though something was true, no two people were agreed as to what it was that was true, and if something was known, no one knew what it was that was known. So long, however, as this was doubtful, it could hardly be said that any certain and exact knowledge was to be obtained in

mathematics. [...] This state of things, it must be confessed, was thor-oughly unsatisfactory. Philosophy asks Mathematics: What does it mean?

Mathematics in the past was unable to answer; and Philosophy answered by introducing the totally irrelevant notion of mind. But now mathemat-ics is able to answer, so far at least as to reduce the whole of its proposi-tions to certain fundamental noproposi-tions of logic. At this point the discus-sion must be resumed by Philosophy. I shall endeavour to indicate what are the fundamental notions involved, to prove at length that no others occur in mathematics, and to point out briefly the philosophical difficul-ties involved in the analysis of these notions. A complete treatment of these difficulties would involve a treatise on logic, which will not be found on the following pages.

The general tone of this passage is already familiar. The point to be emphasised now is that Russell presents logicism as the answer that mathematics gives to philosophical queries about the nature of mathe-matics (and related subjects). The importance of logicism, that is to say, lies in the fact that it shows how the hitherto obscure subject known as philosophy of mathematics can be replaced by a new discipline one that could be called “mathematical philosophy”

which differs in no way from mathematics as far as its exactness or finality is concerned.⁵⁰ What logicism effects is the transformation of philosophical problems into problems of logic. As Russell also indi-cates in the above quotation, the “fundamental notions” to which the problems of the philosophy of mathematics are reduced present fresh difficulties of their own, so that the outline of logicism presented in the Principles is characterised by a certain incompleteness; in addition to the obvious incompleteness which results from the fact that the derivation of “pure mathematics” from logic is not given in detail, there remains the additional task of presenting a detailed account of

50 What justifies the application of the word “philosophy” to logicism is the fact that logicism deals with the foundations of mathematics, or those principles which underlie mathematical sciences. These are topics which were formerly left to philosophers. There is thus a “topical continuation”

between traditional philosophical theories of the nature of mathematics (like the one that Kant developed) and Russell’s logicism.

logic itself, a task that Russell shuns in the book. This latter incom-pleteness, however, does not undermine the significance that he attaches to logicism.