The Unity of Wittgenstein's Philosophy: Necessity, Intelligibility, and Normativity

4. The “Unbridgeable Gulf” between Rule and Application

4 THE “UNBRIDGEABLE GULF” BETWEEN RULE AND APPLICATION The style of my sentences is extraordinarily strongly influenced by Frege. And if I wanted to, I could establish this influence where at first sight no one would see it. (Zettel §712) As discussed in the previous chapter, the core idea in Wittgenstein’s view of meaning in the early 1930s is that the rules for the use of a sign are constitutive of its meaning.108 Wittgenstein’s conception of symbolic rules as constitutive of meaning seems to have been influenced by formalist accounts of geometry and arithmetic. Wittgenstein’s claim that the meaning of a sign is given by an entire system of rules (e.g., Lectures 1932–35, p. 3) echoes Hilbert’s thesis that the meaning of a geometrical sign is defined by a “whole axiom-structure,” in which “each axiom contributes something to the definition.”109 More explicitly, Wittgenstein’s semantic view in the early 1930s draws on the formalist analogy between arithmetic and the game of chess: “The truth in formalism is that every syntax can be conceived of as a system of rules of a game. I have been thinking about what Weyl may mean when he says that a formalist conceives of the axioms of mathematics as like chess-rules” (WWK p. 103). The semantic import of this analogy was explained by Thomae110 as follows: “For the formalist, arithmetic is a game with signs, which [. . .] have no other content (in the calculating game) than they are assigned by their behaviour with respect to rules of combination (rules of the game)” (quoted by Frege in BLA II §88). It is against the background of formalist views such as Hilbert’s and Thomae’s that Wittgenstein uses the laws of geometry and arithmetic as paradigmatic examples of grammatical rules. Following the formalists, Wittgenstein conceives of the rules of grammar as arbitrary symbolic conventions that tell us what we can and cannot do with signs, just as the rules of chess tell us what we can and cannot do with chess pieces (cf. e.g. PG §§11–13). 83 84 The “Unbridgeable Gulf” In the early 1930s Wittgenstein gives special attention to Thomae’s formulation of the analogy between arithmetic and chess and to Frege’s criticisms of it (cf. WWK pp. 105, 138, and 150ff; and PG II pp. 289–95). Insofar as Wittgenstein’s view of the relation between symbolic rules and their applications is based on Thomae’s analogy, Frege’s criticisms of Thomae’s view of arithmetical laws have also critical force against Wittgenstein ’s view of grammatical rules. In this chapter I will take a detour to examine how the debate between Frege and the formalists shaped Wittgenstein ’s view of the relationship between rules and their application. I will try to show that Frege’s critique of formalism sets the agenda of Wittgenstein’s rule-following discussion in the early 1930s. Thomae’s analogy between the laws of arithmetic and the rules of chess is one of the central targets of Frege’s critique of formalism in Grundgesetze (BLA II §§88–96). Frege’s main argument against Thomae is that his formalist approach cannot account for the applicability of the laws of arithmetic (cf. esp. BLA II §91). Frege argues that when arithmetic is conceived as a game with signs that have no other content than “their behaviour with respect to rules of combination,” the propositions of arithmetic become detached from their applications and lose the importance they have in science and everyday life: “How could we possibly apply an equation which expressed nothing and was nothing more than a group of figures, to be transformed into another group of figures in accordance with certain rules?” (BLA II §91). According to Frege, the laws of arithmetic must indeed fix the applicability of arithmetical terms in ordinary language and in the language of the different sciences. But when arithmetical laws are conceived as arbitrary rules for the manipulation of signs, they cannot explain the use of arithmetical terms in science and ordinary language. For arithmetical terms are used in inferences, and the correct application of a term in an inference cannot be justified simply by appealing to “arbitrarily stipulated” rules (cf. BLA II §91). To see the kind of applications that, according to Frege, are left out of account by the formalist, let’s take as an example the application of the equation “5 + 2 = 7” in the following inference: “If we...