FREGE, HILBERT, AND THE CONCEPTUAL STRUCTURE OF MODEL-THEORY

This paper attempts to confine the preconceptions that prevented Frege from appreciating Hilbert's Grundlagen der Geometrie to two: (i) Freg e's reliance on what, following Wilfrid Hodges, I call a Frege-Peano l anguage, and (ii) Frege's view that the sense of an expression wholly determines its reference. I argue that these two preconceptions preven ted Frege from achieving the conceptual structure of model theory, whe reas Hilbert, at least in his practice, was quite close to the model-t heoretic point of view. Moreover, the issues that divided Frege and Hi lbert did not revolve around whether one or the other allowed metalogi cal notions. Frege, e.g., succeeded in formulating the notion of logic al consequence, at least to the extent that Bolzano did; the point is rather that even though Frege had certain semantic concepts, he did no t articulate them model-theoretically, whereas, in some limited sense, Hilbert did.