Sébastien Gandon

Wittgenstein’s Color Exclusion and Johnson’s Determinable

The paper aims at comparing Wittgenstein’s discussion of color exclusion in his (1929) to Johnson’s doctrine of determinable and determinate expounded in his (1921). I first (Sects. §2–§4) summarize Wittgenstein’s developments about the incompatibility of elementary propositions and about the logic of color statements. In the second part (Sects. §5–§7), I present and discuss Johnson’s doctrine in relation to Wittgenstein’s development. In a third conclusive moment (Sect. §8), drawing on a early work of Prior, I argue that the distinction made by Wittgenstein and Johnson between predication and determination should be looked at from a long term historical perspective.

On Quantity and Number in Principia Mathematica: A Plea for an Ontological Interpretation of the Application Constraint

This paper has two distinct goals. The first is to offer an historical interpretation of Russell’s and Whitehead’s theory of rational and real numbers by comparing it to three other contemporary theories: Dedekind’s famous cut construction (Dedekind, 1872), Burali-Forti’s traditional conception of numbers as ratios of quantities (Burali-Forti, 1899), and Frege’s more balanced approach (Frege, 1903a). Russell and Whitehead had a thorough knowledge of the first two theories, and although there is nothing to show that they were familiar with the third, it is to Frege’s theory that Russell’s and Whitehead’s was the closest. The second goal is more philosophical. Frege’s theory has been recently resurrected by Bob Hale (2000, 2002), and his attempt has been discussed by Stewart Shapiro (2000), Crispin Wright (2000) and Vadim Batitsky (2002). The debate turns around the legitimacy of extending the so-called ‘application constraint’ (that is, the idea that a definition should account for the main applications of the concept that is defined) to the case of the real numbers. I will here attempt to show that Russell’s and Whitehead’s perspective, while having the same motivations as Hale’s neo-Fregean conception position, is a stronger option - in adopting an ontological instead of a structural perspective, it is more elaborate, more comprehensive and avoids many of the drawbacks of Hale’s theory.

Quantity in The Principles of Mathematics

Part III of PoM has not been widely studied. Certain aspects of Russell’s doctrine of quantity have been touched on when discussing Russell’s use of the abstraction principle1 or when examining the genesis of PoM.2 But, except Michell (1997; 1999), nobody, to my knowledge, has tried to present in detail Russell’s doctrine. There are some reasons for that. First, even if it is quite short (40 pages), PoM III is terribly complicated. Russell introduced there several idiosyncratic notions (for instance: kind of magnitude, divisibility, stretch, relational magnitude) that are nowhere precisely defined and which almost never occur again in the book. Second, Russell himself, at the beginning of Part III, warned his reader that ‘the whole of this part … is a concession to tradition; for quantity … is not properly a notion belonging to pure mathematics at all’ (p. 158). He is here alluding to the fact that, thanks to the works of Dedekind and Cantor, there was no need, in 1903, to refer to quantity for giving an account of the theory of real numbers and real analysis .3 Owing to these recent mathematical developments, Russell himself seemed to consider PoM III as a dispensable outgrowth, withdrawn into itself and unconnected with the rest of the book. No surprise then if scholars did not rush into this mire on the side.

Geometry, Logicism and ‘If-Thenism’

In the two previous chapters, we have dealt in some detail with Russell’s view of projective and metric geometry. But we have not yet explained how he viewed the articulation between geometry and logic. Russell is well known for having claimed that geometry (projective geometry at least) is a part of pure mathematics. But how did he proceed to reduce geometry to logic? As we began to see in Chapter 2, the standard answer is ‘if-thenism’: to regard a given theory as a part of logic, it is sufficient to show that the said theory is consistent (has a model) and can be axiomatized.1 This interpretation confronts us with a problem however. Russell included projective geometry within pure mathematics, but he did not believe that metrical geometry was a logical science. For all that, he admitted that metrical geometry was consistent and could be axiomatized — in at least two different ways: via the projective definition of metric and also in the direct Leibnizian way. Axiomatization and consistency cannot then be the sole criteria Russell used to characterize the sphere of logical science. More is needed — but what exactly?

Application Constraint in Principia Mathematica

In the previous chapter, we saw that Russell and Whitehead, like Frege, endorsed Applic, i.e. the principle according to which a good definition of a mathematical concept (rational and real numbers in the case in point) should account for its main applications. I emphasized that the way they implemented the constraint was different from Frege’s, but I have not yet explained why they endorsed Applic. Applic seems to be composed of two distinct theses: 1. The recognition that there is a distinction between the mathematical content of a concept and its extra-mathematical uses. 2. The thesis that a good definition must relate the concept to its main extra-mathematical uses.

Quantity in Principia Mathematica

In the Introduction, I quoted the letter Whitehead wrote to Russell on 14 September 1909: The importance of quantity grows upon further considerations — The modern arithmetization of mathematics is an entire mistake — of course a useful mistake, as turning attention upon the right points. It amounts to confining the proofs to the particular arithmetic cases whose deduction from logical premisses forms the existence theorem. But this limitation of proof leaves the whole theory of applied mathematics (measurement, etc.) unproved. Whereas with a true theory of quantity, analysis starts from the general idea, and the arithmetic entities fall into their place as providing the existence theorems. To consider them as the sole entities involves in fact complicated ideas by involving all sorts of irrelevancies — In short the old fashioned algebras which talked of ‘quantities’ were right, if they had only known what ‘quantities’ were — which they did not.

Russell’s Universalism and Topic-Specificity

A tension runs across all the six preceding chapters. On one hand, we have claimed that Russell wanted to provide his readers with more than a formally perfect substitute for the mathematical concept he considered; on the other hand, we have insisted on the fact that he never expound the additional criteria he used to select his favourite analysis from among the many possible ones. Russell’s decisions were not arbitrary; in each case, reasons were provided to justify the choices. But these justifications were always tied up to some local and topic-specific considerations, and they were not grounded on any general criteria. Thus, we saw in the last chapter that Applic, one of the best candidates to play the role of a criterion, was no more than a guiding rule of thumb. In other words, in the picture I have painted of PoM and PM so far, there is a maladjustment between Russell’s general characterization of logical analysis (as a formally perfect, or truth-preserving, translation of the mathematical theorems into the logical language) and his own practice of analysis (which takes into account non-formal aspects of the mathematics he analysed).