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

这个项目是关于焦散曲线和曲面的历史，作为一面镜子，从长期的角度理解数学和物理之间复杂和不断演变的相互作用的历史。焦散，定义为沿曲线或曲面的高密度光线，在日常生活中很常见，但也是理解引力透镜等不太明显的现象的关键模式。自17世纪以来，焦散现象在不同层面上一直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学到20世纪的灾变理论。它还与介于微粒论和波动理论之间的光的物理理论的对比和矛盾密切相关。它的经验研究需要不同的实验星座，从玻璃球和燃烧的镜子到星系和黑洞的引力透镜。如今，焦散现象也有一个跨学科的维度，因为同样的数学理论在从生物学到社会学和经济学的各种知识领域都有应用。尽管焦散曲线和曲面与许多领域相关，但其研究和概念化的背景在历史文献中却鲜有涉及。对这种边界对象及其发展的分析，为从历史上理解数学和物理在其特定的历史背景和具体实践中的复杂关系提供了极好的挑战。为了填补历史文献在这方面的空白，该项目的一个目标是开发一个一般的历史-哲学框架，在概念构造、数学描述和实验实践之间相互作用地处理这类对象。最后但并非最不重要的一点是，该项目还打算在数学和物理教学论领域发挥作用，特别是通过具体的历史背景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.