Mathematics, foundations of

3. Logicism

The view of the logicist Frege (and, to some extent, of Dedekind) accepted Kant’s synthetic a priori conception of geometry but maintained arithmetic to be analytic. Russell, another logicist, rejected Kant’s views of both geometry and arithmetic (and also of pure mechanics) and maintained the analyticity of both. (See Logicism.)

Frege’s logicism differed sharply from intuitionism. First, it differed in the place in mathematical reasoning it assigned to logic. Frege (1884: preface, III–IV) maintained that reasoning is essentially the same everywhere and that even an inference pattern such as that of mathematical induction, which appears to be peculiar to mathematics, is, at bottom, purely logical. Second, it differed in its conception of geometry. Like the early intuitionists, Frege regarded the discovery of non-Euclidean geometries as revealing an important asymmetry between arithmetic and geometry. Unlike them, however, he did not see this as grounds for rejecting Kant’s synthetic a priori conception of geometry (1873: 3; 1884: §89), but rather as indicating a fundamental difference between geometry and arithmetic. Frege believed the fundamental concept of arithmetic – magnitude – to be both too pervasive and too abstract to be the product of Kantian intuition (1874: 50). It figured in every kind of thinking and so must, he reasoned, have a basis in thought deeper than that of intuition. It must have its basis in the very core of rational thought itself; the laws of logic.

The problem was to account for the cognitive richness of arithmetic on such a view. How could the ‘great tree of the science of number’ (1884: §16) have its roots in bare logical or analytical ‘identities’? Frege responded by offering new accounts of both the objectivity and the informativeness of arithmetic. The former he attributed to its subject matter – the so-called ‘logical objects’ (§§26, 27, 105). The latter he derived from a new theory of content which allowed concepts to contain (tacit) content that was not needed for their grasp. On this view, analytic judgments could have content that was not required for the mere understanding of the concepts contained in them. Consequently, they could yield more than knowledge of transparent logical identities (§§64–66, 70, 88, 91).

Unlike Kant, then, Frege maintained an important epistemic asymmetry between geometry and arithmetic – an asymmetry based upon his belief that arithmetic is more basic to human rational thought than is geometry. In addition, he departed from Kant in maintaining a realistic conception of arithmetic knowledge despite its analytic character. He saw it as being about a class of objects – so-called ‘logical objects’ – that are external but intimately related to the mind and therefore not the mere expression of standing traits of human cognition. The differences between arithmetical and geometric necessity were to be accounted for by separating the relationship the mind has to the objects of arithmetic from that which it has to the objects of geometry.

Russell’s logicism differed from Frege’s. Perhaps most importantly, Russell did not regard the existence of non-Euclidean geometries as evidence of an epistemological asymmetry between geometry and arithmetic. Rather, he saw the ‘arithmetization’ of geometry and other areas of mathematics as evidence of an epistemological symmetry between arithmetic and the rest of mathematics. Russell thus extended his logicism to the whole of mathematics. The basic components of his logicism were a general methodological ideal of pursuing each science to its greatest level of generality, and a conception of the greatest level of generality in mathematics as lying at that point where all its theorems are of the form ‘p implies q‘, all their constants are logical constants and all their variables of unrestricted range. Theorems of this sort, Russell maintained, would rightly be regarded as logical truths.

Russell’s logicism was thus motivated by a view of mathematics which saw it as the science of the most general formal truths; a science whose indefinables are those constants of rational thought (the so-called logical constants) that have the most ubiquitous application, and whose indemonstrables are those propositions that set out the basic properties of these indefinables (Russell 1903: 8). In his opinion, such a view provided the only precise description of what philosophers have had in mind when they have described mathematics as a necessary or a priori science.

Russell thus accounted for the necessity of mathematics by pointing to its logical character. He accounted for its richness principally by invoking a new definition of syntheticity that allowed all but the most trivial logical truths and inferences to be counted as informative or synthetic. Mathematical truths would thus be logical truths, but they would not, for all that, be analytic truths. Similarly for inferences. An inference would count as synthetic so long as its conclusion was a different proposition from its premises. Cognitive richness was conceived primarily as the production of new propositions from old, and, on Russell’s conception (supposing the criterion of propositional identity to be sufficiently strict), even purely logical inference could produce a bounty of new knowledge from old.

Russell was thus able to account for both the necessity and the cognitive richness of mathematics while making mathematics part of logic. What had kept previous generations of thinkers and, in particular, Kant from recognizing the possibilities of such a view was the relatively impoverished state of logic before the end of the nineteenth century. The new logic, with its robust stock of new forms, its functional conception of the proposition and the ensuing fuller axiomatization of mathematics which it made possible, changed all this forever and provided for the final refutation of Kant. Such, at any rate, was Russell’s position.