AI 中文总结
本文探讨罗巴切夫斯基几何观，介绍其非欧几何发现的背景、方法、相关理论及被误解情况，指出其认知观点对当前数学与物理结合讨论仍具意义。
AI 中文摘要
罗巴切夫斯基发现非欧几何，是其改革传统欧几里得式几何基础这一更宏大计划的副产品。该计划与达朗贝尔（d'Alembert）、孔狄亚克（Condillac）面向实践的数学愿景一致，旨在弥合纯数学与自然科学、工程学之间被感知的鸿沟。罗巴切夫斯基遵循达朗贝尔的思路，刻意避开几何理论的公理化发展，转而将一些初步的直观综合构造与其严格的分析处理相结合。除非欧几何外，罗巴切夫斯基的这一计划还催生了早期形式的维数理论。本文描述了罗巴切夫斯基著作的局部背景，包括指向19世纪初俄罗斯几何教科书的若干线索；总结了促使他取得数学成就的哲学观点，并说明这些动机在20世纪初被亨利·庞加莱（Henri Poincaré）和恩斯特·卡西尔（Ernst Cassirer）误解和曲解；最后论证，在当前关于数学与物理重新结合的讨论背景下，罗巴切夫斯基的认知观点仍具相关性。附录包含八篇罗巴切夫斯基不同篇幅和性质的文献的英文译文，这些文献表达了他关于数学、尤其是几何的一般认知观点。
英文摘要
Lobachevsky's discovery of Non-Euclidean geometry was a byproduct of his more ambitious plan of reforming the traditional Euclid-style foundations of geometry. This project aligned with d'Alembert's and Condillac's practically-oriented visions of mathematics and aimed at closing the perceived gap between the pure mathematics, on the one hand, and the natural sciences and engineering, on the other hand. Following d'Alembert Lobachevsky deliberately avoided the axiomatic development of his geometrical theory and combined instead some preliminary intuitive synthetic constructions with their rigorous analytic treatment. In addition to Non-Euclidean geometry, Lobachevsky's project brought about an early form of Dimension theory. The paper comprises a description of the local context of Lobachevsky's works, including some pointers to Russian geometry textbooks of the beginning of the 19th century. We summarise Lobachevsky's philosophical views which motivated his mathematical achievements and show how these motivations were misunderstood and misinterpreted in the beginning of the 20th century by Henri Poincaré and Ernst Cassirer. Finally we argue that Lobachevsky's epistemic views remain pertinent in the context of the ongoing discussion about the reunion of mathematics and physics. The \emph{Appendix} comprises English translations of eight Lobachevsky's documents of various length and character where he expresses his general epistemic views on mathematics and, in particular, geometry.
Comments102 pages, 2 figures
