The Middle Wittgenstein’s Critique of Frege

ABSTRACT

This article aims to analyse Wittgenstein’s 1929–1932 notes concerning Frege’s critique of what is referred to as old formalism in the philosophy of mathematics. Wittgenstein disagreed with Frege’s critique and, in his notes, outlined his own assessment of formalism. First of all, he approvingly foregrounded its mathematics-game comparison and insistence that rules precede the meanings of expressions. In this article, I recount Frege’s critique of formalism and address Wittgenstein’s assessment of it to show that his remarks are not so much a critique of Frege as rather a defence of the formalist anti-metaphysical investment.

KEYWORDS:

• Wittgenstein
• Frege
• formalism
• meaning
• rules

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. David Hilbert used a similar notion of Formelspiel (game with formulas), which was also known to Wittgenstein (cf. Waismann 1979, 119).

2. The critique of formalism coincides here with his critique of psychologism.

3. Frege writes: ‘Why can no application be made of a configuration of chess pieces? Because, obviously, it does not express a thought. If it did, and if a move in chess in accordance with the rules corresponded to a transition from one thought to another, then applications of chess would be conceivable (…) Now, it is applicability alone which elevates arithmetic above a game to the rank of a science’ (Frege 2013, § 91, 100).

4. That is why Wolfgang Kienzler seems to miss the point when he objects that when Frege links the scientific rank of arithmetic to its applicability, he ‘makes his philosophical analyses of arithmetic and numbers dependent on something external to it, on its applicability in geodesy, physics or astronomy’ (Kienzler 1997, 203).

5. Michael Dummett (1991, 257–259) maintains that Frege understands the applicability of mathematics as exemplifying applicability of more general logical truths. He also discusses the internal and external applicability of arithmetic some differences in Frege’s concepts in the Grundlagen and the Grundgesetze.

6. Cf. Frege (2013, § 93, 101–102).

7. Cf. Dummett (1991, 255).

8. According to Kienzler (1997, 205), Frege’s main objection to formalism was not the problem of distinguishing between the game itself and the theory of the game, but rather the impossibility of putting together a really complete list of rules, which would be necessary to prevent any imaginable contradiction among them.

9. Marion refers to Wittgenstein’s early position as ‘logicism without classes’ (Marion 1998, 26). For criticism of the logistic interpretation, see Redych (1995) and Wringley (1998).

10. This view is upheld, for example, by Dummett. He claims that we cannot simply oppose the notion of use to Frege’s notion of sense (Dummett 1981, 34).

11. For this reason, Mühlhölzer calls Wittgenstein’s entire philosophy (and not only his philosophy of mathematics) a kind of ‘formalism’ (Mühlhölzer 2008, 116).

12. Mühlhölzer is of a similar opinion. He insists that Frege took rules of use into account as constitutive of the sign. What he indeed failed to notice was the full potential of the alternative embraced by Wittgenstein, whose assessment should be taken exactly as pointing to this potential. According to Mühlhölzer, Wittgenstein’s entire philosophy of mathematics develops this potential (cf. Mühlhölzer 2010, 32). Kienzler claims that Frege did notice the third option but dismissed it so firmly that Wittgenstein could offer it as a new idea (cf. Kienzler 1997, 201).

13. Sören Stenlund (Stenlund 2018, 75–86) seeks to explain how signs acquire content through use by referring to the distinction of the Tractatus makes between sign and symbol. Following 3.32, the sign is what is perceptible by the senses in the symbol. As opposed to signs, symbols have meanings, yet not in the form of objects but through their use and function in symbolism. However, Stenlund passes over theses 3.321–3.325, in which Wittgenstein explicitly states what he needs this distinction for. Namely, he needs it to show that it happens in everyday language that signs (words) signify in various ways, i.e. they belong to different symbols (‘Green is green’), or that two words which signify in different ways are used in a sentence in the same way, as is the case with the word ‘is.’ In mathematics, signs are not used in different ways, and symbolism which employs the same sign, such as ‘3+1,’ in different ways is not used (26). In the first sentence of the passage from the conversations with Schlick quoted above, Wittgenstein – like most logicians and mathematicians – uses the terms sign and symbol as synonymous. Given this, I believe that Stenlund may miss the point when he asserts that Frege failed to recognize the difference between sign and symbol and, consequently that his critique of Thomae’s formalism is off the mark.

14. This also happened in The Blue Book.

15. Frege himself addressed this issue when commenting on his context principle in the Grundlagen: ‘If the second principle [context principle] is not observed, one is almost forced to take as the meanings of words mental pictures or acts of the individual mind’ (Frege 1953, xxii).

16. Not only in his later philosophy in fact, for as early as in the Tractatus Wittgenstein identified an error in Russell’s ‘theory of types’ in that he must speak of the reference of a given type of signs. In 3.331, Wittgenstein states, ‘Russel must be wrong, because he had to mention the meaning (Bedeutung) of signs when establishing the rules for them’ (Wittgenstein 2017, 28).

17. ‘I’m almost inclined to say: In a game there is (to be sure) no “true” or ‘false, but then again in arithmetic there is no “winning” and “losing”’ (BT, 374).

18. This is how the problem of grasping a thought is interpreted by, for example, Dummett (1993, 101).

19. That in Frege thought is independent of language is highlighted by Tyler Burge (1992, 633–650).

20. See also TLP 6.21.

21. The example may be a reminder of the discussion of Leibniz’ proof of 2+2=4 criticized by Frege in Grundlagen der Arithmetik §6.

22. For this notion, see Glock (1996).