Logic as Universal Science

Russell's Early Logicism and its Philosophical Context

A. Korhonen

• Publisher: Palgrave Macmillan
• Year: 2013
• Print ISBN: 9781349366859, 9780230577008
• eBook ISBN: 9781137304858
• Topic: Philosophie
• Subtopic: Philosophie analytique

Logic as Universal Science offers a detailed reconstruction of the underlying philosophy in The Principles of Mathematics showing how Russell sought to deliver a death blow to the dominant Kantian view that formal logic is a concise and dry science and unable to enlarge our understanding.

1

Russell’s Early Logicism: What Was It About?

Introduction

‘The nineteenth century, which prided itself upon the invention of steam and evolution’, Bertrand Russell wrote in an early essay, ‘might have derived a more legitimate title to fame from the discovery of pure mathematics’ (1901a, 366). Russell believed – and he was not alone in thinking this – that the discovery of pure mathematics was essentially a matter of finding out what is involved in mathematical reasoning. It is eminently reasonable to think that mathematical reasoning should be a matter of logic. But it was far from self-evident at the time Russell composed the essay. In the forms in which it existed before the late nineteenth century, formal logic had very few, if any, genuine applications. Insofar as one was concerned with discovering what was involved in mathematical reasoning, one could therefore set traditional Aristotelian logic aside, for it was simply irrelevant.

To bring logic to bear on the ‘analysis of mathematical reasoning’ – not to say anything about such extreme views as logicism, or the view that pure mathematics is just an extension of logic – one had to create that logic first. This is what Frege set out to accomplish in his Begriffsschrift. And when, on a few later occasions, he defended his concept script against critics – such as Ernst Schröder, who complained that Frege had done little that was genuinely new – he replied by arguing that although the logic of the Booleans could capture some logical forms, it was a failure if judged by the standard Frege had set himself from the start; what he had in mind was the expression of mathematical content.¹ It was for this purpose that Frege created his own concept script. Analysing conceptual content into function and object and introducing exact rules for reasoning involving multiple generality, he could analyse and reconstruct reasonings which went far beyond the expressive power of Boole’s logic. In the same spirit, Russell remarked that nineteenth-century logicians had invented ‘a new branch of logic, called the Logic of Relatives, ... to deal with topics that wholly surpassed the powers of the old logic’ (1901a, 367). Logicism is commonly seen as a paradigmatic reductive thesis, but taking into account the extent to which the birth of modern logic was a matter of invention, it would be more nearly correct to say that logicism was not so much a reduction of mathematics or parts thereof to logic as it was an extension of logic to include (parts of) mathematics.²

This observation raises an immediate question for a logicist – or any other advocate of the new logic: How should one go about justifying the claim that the discipline one is involved in really is logic rather than something else? This is a familiar question.³ It was raised early on by Henri Poincaré, a consistently hostile critic of what he saw as the pretensions of the new logic:

We see how much richer this new logic is than the classical logic. The symbols have been multiplied and admit of varied combinations, which are no longer of limited number. Have we any right to give this extension to the meaning of the word logic? It would be useless to examine this question and quarrel with Mr. Russell merely on the score of words. We will grant him what he asks; but we must not be surprised if we find that certain truths which had been declared to be irreducible to logic, in the old sense of the word, have become reducible to logic, in its new sense which is quite different. (Poincaré 1908, 162)

Russell’s logicism, Poincaré seems to be saying, involves no more than a simple extension in the meaning of ‘logic’. In this way, there really is no clash between the old and the new logic, for the simple reason that the two logics, traditional and Russell’s, are talking about (partly) different things. Whatever else may be involved in the passage, Poincaré raises at least the following legitimate question: in the absence of a further characterization of logic, an advocate of the new logic would not be entitled to draw distinctively philosophical consequences from his results. (Admittedly, he then withdraws the question, apparently seeing in it nothing but an idle quarrel about words; but making this move may involve just a piece of patronizing rhetoric rather than a philosophical point.)

Of course, logicism typically involved more than just a new usage for an old word. Russell argued that the new logic showed – indeed, proved – that Kant was wrong about the nature of mathematical reasoning: when properly understood, these reasonings do not presuppose an extralogical source, as in Kant’s theory of mathematics. It is clear that in arguing for these views, Russell did not consider himself to be redefining the term ‘logic’; he saw himself as revealing what is really involved in deductive reasoning in general and mathematical reasoning in particular. It is this ‘what is really involved’ that underlies Russell’s criticism of Kant, and at least from Russell’s point of view the disagreement was a genuine one and not just a matter of words.

In order, therefore, to gain understanding of Russell’s conception of logic, we can profitably turn to the philosophical use to which he put his logicism; evidently, this use is determined by how he understood logic. Our main interpretative question is therefore this: What must logic be like in order for logicism to have the importance that Russell took it to have?

We may begin to answer this question by considering first a familiar answer. In textbook presentations (and not only in these) the point of logicism used to be explained with the help of the notion of analyticity. On this view, the logicist reduction shows mathematics to be analytic, which in turn reveals something important about the nature of mathematics. This explanatory strategy could be applied to Russell’s case, too. As we shall see, however, the thesis that mathematics is analytic lacks independent explanatory power in the context of Russell’s logicism; what is important for him is not so much the distinction between analyticity and syntheticity but between logical truth and extralogical truth. Hence, we can do no better than tackle head on the issue of how Russell understood logicism and what philosophical lesson he derived from it.

Different logicisms

It is well known that the logicist Russell sought to undermine Kant’s theory of mathematics. This is what he himself said when looking back on his philosophical career.⁴ It is also made amply clear by the relevant texts from the early logicist period, where we find more than one stricture on the ‘Kantian edifice’, which, Russell argued, had been torn down by modern mathematics and modern logic.

But how, precisely, did Russell think his logicism would contribute to the collapse of Kantianism? Here it is worthwhile to compare Russell’s early logicism with two other philosophies of mathematics known by the same name: Frege’s thesis concerning the nature of arithmetic and the conception of mathematics which a number of philosophers, often known as logical empiricists, advocated in the 1920s and 1930s. Both Frege and Russell and the logical empiricists found it was philosophically enlightening to relate their own views on mathematics to Kant’s. Indeed, it used to be commonplace to subsume all these three logicisms under one and the same label as well as to formulate their (or its) philosophical point in terms of a consciously held opposition to Kant’s theory of mathematics. Closer inspection shows, however, that there were in fact as many logicisms as there were logicists.

The root of confusion here is the use of a familiar distinction, that between analyticity and syntheticity. The textbook characterization of logicism would attribute the following two points to it: first, the logicist reduction shows that mathematics, being reducible to logic, is analytic and (for that reason) a priori; second, this shows that Kant was mistaken in arguing that mathematical truths are synthetic and a priori.⁵

Analyticity and syntheticity

It is not difficult to find passages from the relevant authors that appear to suggest the attribution of just this pair of views to a ‘logicist’. Frege’s use of the terminology of analyticity in The Foundations of Arithmetic is well known (Frege 1884, §§88–9). In one of his retrospective essays Russell wrote that ‘he did not like the synthetic a priori’, which was why he ‘found Kant unsatisfactory’ in the philosophy of mathematics (1944, 12). Among logical empiricists, A. J. Ayer’s Language, Truth and Logic gives a particularly powerful expression to one of their leading ideas; namely, that the analyticity of mathematical truth explains its necessity and apriority and thereby enables one decisively to undermine a particularly annoying case of the Kantian synthetic a priori (see Ayer 1936, chapter 4).⁶ Given such passages, the conclusion lies at hand that there was one philosophy of mathematics that can appropriately be labelled ‘logicism’, the point of which was to show that since mathematics is reducible to logic, it is analytic and a priori, contrary to what Kant in particular had thought. The fact is, however, that there never was just one logicism but distinct theories different in their motivation, content and scope.

Kant. For Kant, the distinction between analytic and synthetic judgments was important for epistemological reasons. He had made the observation that there is a class of judgments which are not only knowable a priori but whose apriority is unproblematic; namely, judgments he referred to as ‘analytic’. A judgment is analytic, he explained, if ‘the predicate B belongs to the subject A, as something that is (covertly) contained in this concept A’ (KrV, A6/B10).⁷ As the quotation shows, conceptual containment comes in two kinds: explicit, as in ‘all amphibious animals are animals’, and implicit, as in ‘all bachelors are men’. A judgment of the second kind is one which adds ‘nothing through the predicate to the concept of the subject, but merely [breaks] it up into those constituent concepts that have all along been thought in it, although confusedly’ (ibid., A7/B11). This explanation leads to the second characterization of analyticity, one that is based, in effect, on the sort of proof that is appropriate for analytical truths. Since every analytical truth is either explicitly or implicitly of the form ‘(every) AB is B’, our knowledge of analytical truths is either immediate (‘all amphibious animals are animals’) or else mediated by a proof of the sort that leads from ‘all bachelors are men’ via a simple substitution to ‘all unmarried men are men’. This analysis of analyticity leads Kant to say that all analytical truths are grounded in the principle of contradiction, which is at the same time the principle sufficient for analytical knowledge (ibid., A150–1/B190–1).

It is this feature of analytical truths that makes them epistemologically unproblematic. There is no problem about how we can know their content a priori. We need not consult experience to come to possess the piece of knowledge that all bachelors are unmarried, for we cannot fail to recognize the truth of this judgment if only we possess the relevant concepts.⁸

Kant then pointed out that analytical judgments by no means exhaust the class of truths that are knowable a priori. For reflection on the relevant instances shows that there is a further class of judgments which shares with the first one the property of being a priori but has the additional property of being epistemologically problematic in the sense that no straightforward explanation of apriority is forthcoming in this case.

For Kant, then, the importance of the distinction between analytic and synthetic judgments was in the first instance epistemological. In particular, he intended the notion of analyticity to single out a class of judgments for which mere analysis of content would yield as a corollary an explanation of their epistemology. When turning to consider logicists, it is precisely on this point that we can discern important differences.

Logical empiricism. Logical empiricists were followers of Kant in the use to which they put the notion of analyticity. In both cases the importance of analyticity stems from the fact that it is intended to play an explanatory role. That a truth is analytic in the logical empiricist sense – true solely in virtue of meaning and hence true, somehow, in virtue of linguistic rules and, for that very reason, devoid of what they called ‘factual content’ – is meant to explain why the truth is knowable independently of experience in precisely the way that Kant meant his notion of analyticity to explain the apriority of ‘merely explicative’ truths. Of course, the logical empiricist path from analyticity to apriority is more convoluted than the Kantian one, because the former comprises much more than those trifling truths that can be covered by the Kantian conceptual containment model. Having in this way extended the scope of analytical truths, logical empiricists found themselves in a position to argue for a decidedly anti-Kantian conclusion: to explain why mathematical truth is knowable independently of experience, nothing beyond a grasp of the rules of the relevant language – the language in which the truth has been formulated – is needed. And this was meant as a partial answer to Kant’s epistemological question about the source of a priori knowledge.⁹ As logical empiricists saw it, there was no need to assume with Kant a distinctive type of truth – that which is synthetic and yet knowable a priori – nor to postulate a special source of knowledge (viz., intuition) to guarantee access to and validity of these truths. In spite of their reaching radically different conclusions, the point of introducing analyticity is precisely the same in the two cases: to wit, to explain how a proposition is recognizable as true merely by entertaining its conceptual content and without having to rely on any empirical input.

Frege and Russell. Although deceptively similar, Frege’s use of analyticity must be distinguished from the logical empiricist theory. Frege does speak about analyticity, by which he means reducibility to logic via explicit definitions.¹⁰ He argues, it is true, that his intention is not to assign any new meaning to the terms ‘analytic’ and ‘synthetic’ but that he wants only to state precisely what earlier authors, Kant in particular, have had in mind (1884, §3). Yet it is no less true that Frege has very little to say about the epistemological problems which had exercised Kant and were to exercise logical empiricists. Frege, it seems, was (most of the time, anyway) content with the assumption that there is knowledge that qualifies as a priori and did not bother to explain what this knowledge is like and how we could have it.

Consider, finally, Russell’s early logicism. There are conspicuous differences between Russell’s and Frege’s logicisms. Nevertheless, when it comes to the issue of analyticity, their views are importantly similar. As in Frege’s case, Russell’s motives, too, must be distanced from the epistemological underpinnings of logical empiricism. The fact is that Russell has very little use for analyticity.¹¹ The notion of analytic truth or the distinction between analyticity and syntheticity is given no role in PoM. It receives no extended discussion and is not put to any use. Indeed, he almost fails to mention it, and when he once mentions it, he does so only to put it aside as being of no concern to him. Moreover, what little he says distinguishes him firmly from logical empiricism. This is what Russell has to say about the notion.

Kant never doubted for a moment that the propositions of logic are analytic, whereas he rightly perceived that those of mathematics are synthetic. It has since appeared that logic is just as synthetic as all other kinds of truths: but this is a purely philosophical question, which I shall here pass by. (PoM, §434)

Once we take this passage into account, we cannot very well apply the standard characterization of logicism to the early Russell. If logic has turned out to be synthetic, logicism cannot be used to show that mathematics is analytic. On ...

Table of contents

• Cover
• Title
• Introduction
• 1  Russells Early Logicism: What Was It About?
• 2  Kant and Russell on the Mathematical Method
• 3  Russell on Kant and the Synthetic a priori
• 4  Russells Ontological Logic
• 5  Russell and the Bolzanian Conception of Logic
• Notes
• Bibliography
• Index