 The case of Bolzano suggests that a liberation from metric concepts (which came with the development of theories of projective geometry and especially of topology) was to have a crucial role in making possible the abstract viewpoint of set theory. Bernhard Riemann proposed visionary ideas about topology, and about basing all of mathematics on the notion of set or “manifold” in the sense of class (Mannigfaltigkeit), in his celebrated inaugural lecture “On the Hypotheses which lie at the Foundations of Geometry” (1854/1868a). Also characteristic of Riemann was a great emphasis on conceptual mathematics, particularly visible in his approach to complex analysis (which again went deep into topology). To give but the simplest example, Riemann was an enthusiastic follower of Dirichlet’s idea that a function has to be conceived as an arbitrary correspondence between numerical values, be it representable by a formula or not; this meant leaving behind the times when a function was defined to be an analytic expression. Through this new style of mathematics, and through his vision of a new role for sets and a full program for developing topology, Riemann was a crucial influence on both Dedekind and Cantor (see Ferreirós 1999).