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Article ID: CBB466745331

“Are the Genre and the Geschlecht One and the Same Number?” an Inquiry into Alfred Clebsch's Geschlecht (2020)

Lê, François (Author)

Historia Mathematica
Volume: 53
Pages: 71-107

DOI: 10.1016/j.hm.2020.04.002

URI: https://www.sciencedirect.com/science/article/pii/S0315086020300239

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This article is aimed at throwing new light on the history of the notion of genus, whose paternity is usually attributed to Bernhard Riemann while its original name Geschlecht is often credited to Alfred Clebsch. By comparing the approaches of the two mathematicians, we show that Clebsch's act of naming was rooted in a projective geometric reinterpretation of Riemann's research, and that his Geschlecht was actually a different notion than that of Riemann. We also prove that until the beginning of the 1880s, mathematicians clearly distinguished between the notions of Clebsch and Riemann, the former being mainly associated with algebraic curves, and the latter with surfaces and Riemann surfaces. In the concluding remarks, we discuss the historiographic issues raised by the use of phrases like “the genus of a Riemann surface”—which began to appear in some works of Felix Klein at the very end of the 1870s—to describe Riemann's original research. Résumé Cet article a pour but d'apporter une lumière nouvelle sur l'histoire de la notion de genre, dont la paternité est habituellement attribuée à Bernhard Riemann, et l'appellation originale Geschlecht à Alfred Clebsch. En comparant les approches de ces deux mathématiciens, nous montrons que l'introduction de cette appellation par Clebsch s'ancre dans une réinterprétation, en géométrie projective, des travaux de Riemann, et que son Geschlecht désigne en fait une notion différente de celle de Riemann. Nous montrons également que jusqu'au début des années 1880, les mathématiciens faisaient une nette distinction entre les notions de Clebsch et de Riemann, la première étant principalement associée aux courbes algébriques, et la deuxième aux surfaces et surfaces de Riemann. Dans les remarques de conclusion, nous discutons des problèmes historiographiques soulevés par l'utilisation d'expressions comme « le genre d'une surface de Riemann » — expressions qui commencèrent à apparaître dans des travaux de Felix Klein à la toute fin des années 1870 — pour décrire les recherches de Riemann.

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