笛卡尔圆定理、伊丽莎白公主与旋量
Descartes' Circle Theorem, Princess Elizabeth, and Spinors
- University of Lyon（里昂大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文探讨笛卡尔圆定理的起源、推广至旋量形式化的历程，并联系伊丽莎白公主的通信，揭示真理通过归纳推广而扩展的价值。
AI中文摘要：
在本文中，关于笛卡尔著名的“圆定理”，我们首先尝试解释这位哲学家如何可能得出他在1643年11月写给波希米亚的帕拉蒂尼公主伊丽莎白的信中未加证明就提出的方程（据我们所知，这是一个此前未发表的证明）。然后，我们报告了该定理后续推广的一些阶段，其克利福德代数的风味，通过二次形式与线性形式的平方的等价性，至今仍激励着数学家们。这一命题，随着时间推移经历了不同的推广，包括欧几里得和非欧几里得的，最终找到了旋量形式化。我们从中看到了我们可称之为，以巴什拉风格，真理的“归纳价值”的证明，它通过连续的推广而扩展。我们最后更简要地得出结论，关于伊丽莎白与笛卡尔的接触，以及这些关于“相切圆”（盎格鲁-撒克逊国家如此称呼）的思考在她们关于灵魂与身体以及激情问题的通信背景中可能具有的象征意义。
英文摘要：
In this article, on Descartes' famous "circle theorem," we first attempt to explain how the philosopher could have arrived at the equation that he presents without demonstration in a letter to Princess Palatine Elizabeth of Bohemia in November 1643 (a previously unpublished proof, to our knowledge). Then we report some stages of the subsequent generalization of this theorem, whose Cliffordian flavor, via the equivalence of a quadratic form and the square of a linear form, still mobilizes mathematicians today. This statement, which, over time, has undergone different extensions, Euclidean and non-Euclidean, has found, {\it in fine}, a spinor formalization. We see there the proof of what we may call, in a Bachelardian style, an "inductive value" of the truth, which extends by successive generalizations. We conclude, more briefly, with the contacts between Elizabeth and Descartes, and the perhaps symbolic meaning of these considerations on "kissing circles," as they are called in Anglo-Saxon countries, in the context of their correspondence on the soul and the body, and the question of passions.
