爱因斯坦反对奇点：分析 versus 几何
Einstein Against Singularities: Analysis versus Geometry
AI总结：
本文从数学方法史角度分析爱因斯坦与后来相对论者对奇点的分歧，指出爱因斯坦以解析表达式优先、坚持消除奇点，并揭示其消除任意性的物理哲学动机。
摘要：
爱因斯坦识别出时空中的奇点，例如在史瓦西半径处，而后来的相对论者在那里只发现一个坐标系将多个值赋予单个时空事件。这些不同的判断源于数学方法的差异。后来的相对论者运用几何结构来修正解析表达式中使用的坐标系的反常之处。爱因斯坦则把解析表达式视为首要的，而把几何结构仅仅视为启发式工具，如果物理假设需要，这些几何结构可以被推翻。爱因斯坦的非几何方法在数学方法史上有着坚实的基础。它们延续了克里斯托费尔、里奇和列维-奇维塔的非几何取向。爱因斯坦坚持必须消除奇点，这标志着对早期容忍奇点态度的背离。这一坚持建立在他长期致力于从基础物理理论中消除任意性的计划之上。然而，如果奇点是当时可用的任意性最小的方法，爱因斯坦也愿意暂时借助奇点进行理论构建。
英文摘要：
Einstein identified singularities in spacetimes, such as at the Schwarzschild radius, where later relativists only find a coordinate system assigning multiple values to a single spacetime event. These differing judgments derive from differences in mathematical methods. Later relativists employ geometrical structures to correct anomalies in the coordinate systems used in analytic expressions. Einstein took the analytic expressions to be primary and the geometrical structures as mere heuristics that could be overruled if physical assumptions required it. Einstein's non-geometric methods had a firm base in the history of mathematical methods. They continued the non-geometric orientation of Christoffel, Ricci and LeviCivita. Einstein's insistence that singularities must be eliminated marked a departure from earlier tolerance of singularities. It was founded upon his longterm project of eliminating arbitrariness from fundamental physical theories. However, Einstein was willing to theorize with singularities only temporarily if they were the least arbitrary approach then available.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
