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“Are the genre and the Geschlecht one and the same number?” An inquiry into Alfred Clebsch's Geschlecht
Historia Mathematica ( IF 0.5 ) Pub Date : 2020-11-01 , DOI: 10.1016/j.hm.2020.04.002
François Lê

Abstract 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.

中文翻译:

“流派和 Geschlecht 是同一个数字吗?” 对阿尔弗雷德·克莱布施 (Alfred Clebsch) 的 Geschlecht 的调查

摘要 本文旨在阐明属概念的历史,其父系通常归因于伯恩哈德·黎曼 (Bernhard Riemann),而其原名 Geschlecht 通常归功于阿尔弗雷德·克莱布施 (Alfred Clebsch)。通过比较两位数学家的方法,我们表明 Clebsch 的命名行为植根于对黎曼研究的投影几何重新解释,并且他的 Geschlecht 实际上与黎曼的概念不同。我们还证明,直到 1880 年代初,数学家们才清楚地区分 Clebsch 和 Riemann 的概念,前者主要与代数曲线相关,后者主要与曲面和黎曼曲面相关。在结束语中,
更新日期:2020-11-01
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