  Hilbert’s work on the foundations of mathematics has its roots in his work on geometry of the 1890s, culminating in his influential textbook Foundations of Geometry  (1899)  (see  19th Century Geometry).  Hilbert believed that the proper way to develop any scientific subject rigorously required an axiomatic approach. In providing an axiomatic treatment, the theory would be developed independently of any need for intuition, and it would facilitate an analysis of the logical relationships between the basic concepts and the axioms. Of basic importance for an axiomatic treatment are, so Hilbert, investigation of the independence and, above all, of the consistency of the axioms. For the axioms of geometry, consistency can be proved by providing an interpretation of the system in the real plane, and thus, the consistency of geometry is reduced to the consistency of analysis. The foundation of analysis, of course, itself requires an axiomatization and a consistency proof. Hilbert provided such an axiomatization in  (1900b),  but it became clear very quickly that the consistency of analysis faced significant difficulties, in particular because the favored way of providing a foundation for analysis in Dedekind’s work relied on dubious assumptions akin to to those that lead to the paradoxes of set theory and  Russell’s Paradox  in Frege’s foundation of arithmetic.