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A New Fundamental Notion: Riemann’s Manifolds

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.

Keywords

Riemann Surface Abelian Function Mathematical Context Traditional Logic Geometrical Intuition 
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Copyright information

© Birkhäuser Verlag AG 2007

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