  Hilbert thus realized that a direct consistency proof of analysis, i.e., one not based on reduction to another theory, was needed. He proposed the problem of finding such a proof as the second of his 23 mathematical problems in his address to the International Congress of Mathematicians in 1900  (1900a) and presented a sketch of such a proof in his Heidelberg talk  (1905).  Several factors delayed the further development of Hilbert’s foundational program. One was perhaps the criticism of  Poincaré (1906)  against what he saw as a viciously circular use of induction in Hilbert’s sketched consistency proof (see  Steiner 1975, Appendix). Hilbert also realized that axiomatic investigations required a well worked-out logical formalism. At the time he relied on a conception of logic based on the algebraic tradition, in particular, on Schröder’s work, which was not particularly suited as a formalism for the axiomatization of mathematics. (See  Peckhaus 1990  on the early development of Hilbert’s Program.)