Hilbert and Schröder’s mathematical logic

I examine (for the first time in detail) the relationship between David Hilbert’s early mathematical logic (of 1905) and the work of Ernst Schröder. The evidence confirms the hypothesis widely shared among scholars that Schröder was a key reference point for Hilbert. Contrary to what scholars have assumed, however, it turns out that Hilbert relied not on Schröder’s opus magnum, the Lectures on the Algebra of Logic (1890–1905), but rather on Schröder’s early The Circle of Operations of the Logical Calculus (1877). I show furthermore how Hilbert’s use of this booklet as a source can explain certain peculiar features and remarks in Hilbert’s early logical work, which have either a parallel or a plausible source in Schröder’s booklet (but not in Schröder’s Lectures) and are hard to account for otherwise. I discuss, in particular, Hilbert’s early conception of proofs “as factors” (of the proposition they establish) at some length. Finally, to provide a comprehensive overview of the historical background of Hilbert’s early logic, I trace certain relevant features of Schröder’s work back further to the work of Robert Grassmann and Peirce, and argue that Frege played no significant role in the development of Hilbert’s mathematical logic.