A new fundamental notion

Riemann's manifolds

pp. 39-80

in: José Ferreirós, Labyrinth of thought, Basel, Birkhäuser, 2007

Abstract

In this chapter we trace back the first influential appearance of a set-theoretical viewpoint to the work of Riemann. Of course, by speaking of "a set-theoretical viewpoint" I do not mean to suggest that Riemann reached technical results that we would classify today as belonging to set theory-only that he introduced set language substantially in his treatment of mathematical theories and regarded sets as a foundation for mathematics. This comes out in a public lecture given in 1854, on the occasion of his Habilitation as a professor at Göttingen, when he proposed a general notion of manifold-the famous "On the Hypotheses upon which Geometry is Founded," published posthumously by Dedekind in 1868. We shall refer to it as Riemann's Habilitationsvortrag.