How to Say Things with Formalisms

The conception of formalism as uninterpreted, but interpretable, systems that was wavering into focus during the 1920s achieved clarity in Godel's 1931 paper. The historical irony here is that while this clarity fulfilled one of Hilbert's demands for firm foundations, the Godel results themselves have been seen as frustrating Hilbert's central epistemological desire. Due in part to Hilbert's vagueness in formulating his demands, and in part to the subtlety of the issues involved, there remains an ongoing debate as to whether the First Incompleteness Theorem scuttled Hilbert's Program, whether only the Second did, or whether neither did. Leaving detailed exegesis of Hilbert aside, I propose to investigate some very general issues that arise in the course of such debate. Hilbert wanted to save mathematics from Kronecker's constructivist critique, a critique made pointed by the antinomies of set theory. Hilbert, however, shared some of Kronecker's basic epistemological scruples and thus proposed, on one interpretation of his project, to salvage the nonconstructivist part of mathematics by making it meaningless but helpful. This instrumentalist reading of Hilbert is arguable. Or rather, "meaningless" might be taken only as Fregean meaningless. It is