Focusing between Mathematics and Physics. A Longterm History of Caustics from the 17th to the 20th Century

这个项目是关于苛性曲线和曲面的历史，作为一个从长远角度对数学和物理之间复杂和不断发展的相互作用的历史理解的镜子。焦散被定义为光线沿曲线或表面的高度集中，在日常生活中很常见，但也是理解引力透镜等不太明显现象的关键模式。自17世纪以来，焦散现象在不同层面上都是一个具有挑战性的问题。它的数学理解需要复杂和不断变化的几何和分析框架，从微积分前的射线几何到20世纪的突变理论。它也与光的对比和矛盾的物理理论密切相关，位于微粒和波动理论概念之间。它的实证研究需要不同的实验星座，从玻璃球和燃烧镜到星系和黑洞的引力透镜。如今，苛性现象也具有跨学科的维度，因为相同的数学理论在各种知识领域都有应用，从生物学到社会学和经济学。尽管在许多领域都有相关性，但在历史文献中，苛性曲线和曲面的主题及其调查和概念化的背景几乎没有被处理过。对这些边界对象及其发展的分析，为理解数学和物理在其特定的历史背景和情境实践中令人烦恼的关系提供了一个极好的挑战。为了填补这方面历史文献的空白，该项目的目标是发展一个一般的历史哲学框架，以处理概念结构、数学描述和实验实践之间的相互作用。最后但并非最不重要的是，该项目也打算与数学和物理教学领域相关，特别是通过具体的历史背景，重点介绍数学和物理概念。

This project is about the history of caustic curves and surfaces as a looking glass for a historical understanding of the complex and evolving interactions between mathematics and physics from a long-term perspective. Caustics, defined as the high concentration of light rays along curves or surfaces, are common in everyday life, but are also a key pattern to understand less evident phenomena like gravitational lensing. Ever since the 17th century, the phenomenon of caustics has been a challenging problem on different levels. Its mathematical understanding required sophisticated and changing geometric and analytical frameworks, from precalculus ray geometry to 20th century catastrophe theory. It is also intimately linked to contrasting and contradictory physical theories of light, situated between corpuscular and wave theoretical conceptions. And its empirical investigation called for different experimental constellations from glass spheres and burning mirrors to gravitational lensing by galaxies and black holes. Nowadays, caustic phenomena have also a transdisciplinary dimension, since the same mathematical theory has applications in a variety of knowledge areas, ranging from biology to sociology and economics. Despite its relevance in many fields, the topic of caustic curves and surfaces and the contexts of their investigation and conceptualization has hardly been treated in the historical literature. The analysis of such boundary objects and their development offer an excellent challenge for a historical understanding of the vexing relations between mathematics and physics in their specific historical contexts and situated practices. Aiming at filling a gap in the historical literature in this respect, it is a goal of the project to develop a general historical-philosophical framework for dealing with this kind of objects at the interplay between conceptual constructions, mathematical descriptions, and experimental practices. Last but not least, the project intends to be relevant in the field of didactics of mathematics and physics as well, specifically by focusing on the introduction of mathematical and physical concepts through their concrete historical contexts.