 Hilbert viewed the axiomatic method as the crucial tool for mathematics (and rational discourse in general). In a talk to the Swiss Mathematical Society in 1917, published the following year as Axiomatisches Denken (1918), he articulates his broad perspective on that method and presents it “at work” by considering, in detail, examples from various parts of mathematics and also from physics. Proceeding axiomatically is not just developing a subject in a rigorous way from first principles, but rather requires, for advanced subjects, their deeper conceptual organization and serves, for newer investigations, as a tool for discovery. In his talk Hilbert reflects on his investigations of the arithmetic of real numbers and of Euclidean geometry from before 1900. We emphasize the particular form of his axiomatic formulations; they are not logical formulations, but rather mathematical ones: he defines Euclidean space in a similar way as other abstract notions like group or field; that’s why we call it structural  axiomatics.[1]  However, Hilbert turns from the past and looks to the future, pointing out a programmatic direction for research in the foundations of mathematics; he writes: