The Manifold Meanings of Nineteenth-Century Mathematics: Bernhard Riemann’s Construction of the Manifold

This paper scrutinizes the revisions of mathematician Bernhard Riemann’s (1822-1862) 1854 habilitation lecture at the University of Göttingen. It argues that the lecture is a reflection of how mathematicians developed non-Euclidean geometries in the nineteenth century, breaking with long-standing professional conventions and philosophical convictions in order to do so. Riemann’s concept of the “manifold,” which he presented in this lecture, was one of the most widespread non-Euclidean frameworks in the nineteenth century, and endures as a foundational concept in mathematics today. This paper argues that, while the manifold (and non-Euclidean geometry) was a “rupture,” it was also continuous with the mathematical practices that came before it, including the study of minimal surfaces. More broadly, Riemann’s papers reveal surprising aspects of mathematical practice, at exactly the moment when mathematics purportedly became abstract, immaterial, and unempirical. Riemann’s mathematical research directly addressed questions of religion and metaphysics: he argued that the “world manifold” was the mechanism connecting human souls to the “world soul.” And Riemann described mathematics as though it could act, and act against him; he frequently was so captivated by his research that he could not pull himself away until he became physically ill. By using Riemann’s revisions to temporally reconstruct the creation of the manifold, this paper challenges two narratives, one historiographical and one cultural: the myth of non-Euclidean geometry as a total rupture, and the notion that mathematics is immaterial and disembodied.