黎曼的几何思想萌芽

The Rudiments of the Idea of Riemann for Geometry

在分析黎曼的几何思想和其他学科关系基础上,指出了其几何思想的三个来源:数学、物理和哲学。这其中,19世纪前半叶几何学对黎曼有着直接影响;物理研究也是黎曼的兴趣所在,尤其是他一度试图建立各种作用力的统一理论,促使黎曼用无穷小分析研究流形,并触发了黎曼对空间几何基础的讨论;哲学上,黎曼对空间认识论的思考从赫尔巴特那里得到了灵感,并且在流形概念上可能也借鉴了赫尔巴特的研究。最后指出:研究黎曼几何思想不能把各学科分开考虑,必须全面考虑其间关系。

Based upon studying its relationships with other subjects, this paper points out three origins of the idea of Riemann for geometry：mathematics, physics and philosophy. The geometry in the first half of 19th century had a direct influence on Riemann＇s thinking. The research of physics, especially grand unifield theories, was also Riemann＇s interest. That made him study manifolds by in- finitesimal analysis and review the foundation of space geometry. Herbart＇s study in philosophy was the source of inspiration for Riemann＇s knowledge about space, and probably led to the concept of manifold. The paper points out finally that the study of the idea of Riemann for geometry should be considered thoroughly.