欧几里得比例与比值的范畴论方法
A Categorical Approach to Euclidean Ratios and Proportions
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中文总结 AI 辅助
本文提出一种基于范畴论的图解语法，用于直观呈现欧几里得《几何原本》第五、六卷的非度量几何论证，并证明其植根于亚里士多德传统，能解决古代数学谜题。
中文摘要 AI 辅助
本文提出了一种对欧几里得《几何原本》第五卷和第六卷中非度量几何的范畴论方法。具体而言，我们引入了一种图解语法，该语法可以直接叠加在他的图形上，从而在直观呈现与忠实于欧几里得论证之间架起桥梁。这种语法使得像V.5这样复杂的定义，以及贯穿第五卷和第六卷的论证（包括关于相似图形的论证）变得直观清晰。我们在附录中展示了这种语法可以用来解决一个关于古代数学的谜题。最后，我们提供证据表明，这种处理欧几里得图形的方法植根于亚里士多德传统本身，并且在古代，类似的语法已被用于处理相关的数值与比例问题。因此，这种语法很可能忠实于欧几里得自己的思想世界，而非外部强加。
英文摘要
A categorial approach to the non-metric geometry in Books V and VI of Euclid's \textit{Elements} is presented. Specifically, we introduce a diagrammatic syntax that can be overlaid immediately on his diagrams, thus bridging intuitive presentation with fidelity to Euclid's arguments. This syntax makes complicated definitions like V.5, and indeed the arguments throughout books V and VI, including arguments about similar figures, intuitively clear. We show in an appendix that this syntax can be used to solve a puzzle regarding ancient mathematics. Finally, we offer evidence that this approach to Euclidean diagrams is rooted in the Aristotelian tradition itself, and that a similar syntax was utilized, in antiquity, for related questions of numeric and proportions. Thus the syntax is plausibly faithful to Euclid's own thought-world, and not an outside-imposition.
发表机构
- Department of Mathematics, Whitman College（惠特曼学院数学系）
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