Is There a “Hilbert Thesis”?

In his introductory paper to first-order logic, Jon Barwise writes in the Handbook of Mathematical Logic (1977):[T]he informal notion of provable used in mathematics is made precise by the formal notion provable in first-order logic. Following a sug[g]estion of Martin Davis, we refer to this view as Hilbert’s Thesis.This paper reviews the discussion of (different variations of) Hilbert’s Thesis in the literature. In addition to the question whether it is justifiable to use Hilbert’s name here, the arguments for this thesis are compared with those for Church’s Thesis concerning computability. This leads to the question whether one could provide an analogue for proofs of the concept of partial recursive function.