Button, T; (2011) The Metamathematics of Putnam's Model-Theoretic Arguments. Erkenntnis , 74 (3) pp. 321-349. 10.1007/s10670-011-9270-6.

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Abstract

Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted models. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I examine the metamathematics of all three model-theoretic arguments, and I argue against Bays (2001, 2007) that Putnam is largely immune to metamathematical challenges. Hilary Putnam famously attempted to use model theory to draw metaphysical conclusions. Specifically, he attacked metaphysical realism, a position characterised by the following credo: "[T]he world consists of a fixed totality of mind-independent objects." (Putnam 1981 p. 49; cf. 1978, p. 125). "Truth involves some sort of correspondence relation between words or thought-signs and external things and sets of things." (1981, p. 49; cf. 1989, p. 214) "[W]hat is epistemically most justifiable to believe may nonetheless be false." (1980, p. 473; cf. 1978, p. 125) To sum up these claims, Putnam characterised metaphysical realism as an “externalist perspective” whose “favorite point of view is a God’s Eye point of view” (1981, p. 49). Putnam sought to show that this externalist perspective is deeply untenable. To this end, he treated correspondence in terms of model-theoretic satisfaction. This enabled him to deploy results from model theory against metaphysical realism. In particular, he presented two famous model-theoretic arguments: his Skolemisation argument and his permutation argument. In this paper, I will investigate the metamathematical underpinnings of both arguments. Since both arguments require only extremely weak model-theoretic resources, it would seem that metaphysical realists cannot reasonably object to Putnam’s metaphysical conclusions on purely metamathematical grounds. Timothy Bays, however, has raised a challenge against Putnam on exactly those grounds. Bays’ main target is Putnam’s less famous constructivisation argument, which seeks to establish that any empirical evidence is compatible with the Axiom of Constructibility. However, Bays thinks that his challenge applies equally well against the Skolemisation argument. I agree that Bays’ challenge poses considerable problems for the constructivisation argument. However, I shall show that it has no impact at all on either the Skolemisation or the permutation arguments. Perhaps Putnam’s arguments can be refuted on other grounds; but the metamathematics at the heart of Putnams model-theoretic arguments is completely secure.

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