In Chapter 1, I briefly explained how, in Husserl’s view, mathematicians are striving for a theory of theories as one of the goals of modern mathematics. In this chapter, I will investigate more closely the notion of “definite manifold” that Husserl identified as another goal guiding the development of modern mathematics. Husserl writes that, since Euclid, the notion of the definite manifold, to be worked out in this chapter, “has continually guided mathematics from within” (FTL, §31). He thinks that Hilbert tried to capture the same idea with his “completeness axiom,” which he (Hilbert) appended to his axiomatizations of geometry and arithmetic around the turn of the century (FTL, §31).