Duality - an archetype of mathematical thinking

二元性在现代数学中以多种方式和不同的背景出现。人们可以谈论数学思维的基本模式(原型)(类似于类比或对称)；没有对该术语的有用的一般定义和对其用法的系统审查。拟议的研究项目旨在对这一原型进行历史研究。对偶性在多面体理论(欧几里得的《元素》第十五卷，开普勒的《和声芒迪》第二册和第五册)和球面几何(极三角形的形状)中早期就出现了(通常被称为极性或互易性)。从1810年起，对偶性在射影几何领域变得相当突出，发展成为一种普遍现象，特别是在20世纪(见下文)。除了著名的平行问题之外，对偶性的基础和如何确保它们的问题是发展元数学的早期和重要的初始问题。尽管二元性如此重要，但对其多种形式的二元性及其与物理和静力学等其他领域的相互作用，仍然没有全面的历史研究。我们的目标是在拟议的项目中弥合这一差距。特别是，应结合下列领域的发展来研究和比较对偶性的发生和作用：1多面体；2球面和射影几何；3逻辑和布尔代数；4代数拓扑学(Poincaré，Alexander)；5向量空间理论和泛函分析；6群论(Pontryagin)；7范畴理论和应用，例如在代数几何中。大致分为三个时期(约1880(1，2)，约1880-1945(3-6)，约1945(7))，在我们看来，这三个时期对应于数学的三个不同发展阶段，每个阶段都有独特的方法和基本概念，但也有典型的知识传播形式，我们想结合“二元性”这一中心概念来考察这三个阶段。拟议的项目具有很强的国际性，其成果将以若干篇论文、一卷集、一本资料书和一个数据库的形式提出。该项目的创新之处首先在于设想的总体观点的跨学科性质，从术语和文献计量问题的历史发展到认识论方面。此外，该项目的特点是，它不是关于数学理论、概念或类似的；因此，不能像以前对数学概念史的大多数研究那样，将调查局限于选定的数学子学科的背景。最常见的情况是，二元性被模糊地定义为一种“原则”。相反，我们更愿意称它为数学思维的原型；这样，我们希望为数学史和数学哲学的研究以及对一般数学的反思开辟新的视野。

Duality occurs in modern mathematics in multiple ways and different contexts. One could speak about a basic pattern (archetype) of mathematical thinking (comparable to analogy or symmetry); a useful general definition of the term and a systematic review of its uses are not available. The proposed research project aims to study this archetype historically. Dualities occured (often referred to as polarity or reciprocity) early in the theory of polyhedra (Euclid's "Elements" Book XV, Kepler's "Harmonices Mundi" Books II and V) and in spherical geometry (in the shape of the polar triangle). Becoming rather prominent in the field of projective geometry from 1810 on, duality developed into a widespread phenomenon, especially during the 20th century (see below). The problem of the foundations of duality and the question how to ensure them were - besides the famous problem of the parallels - an early and important initial problem for the development of metamathematics. Despite this importance, there is still no comprehensive historical study of duality in its many forms and its interaction with other fields like physics and statics. We aim to close this gap in the proposed project. In particular, the occurrence and role of duality shall be investigated and compared with respect to the development of the following fields: 1 polyhedra; 2 spherical and projective geometry; 3 Logic and Boolean algebra; 4 algebraic topology (Poincaré, Alexander); 5 vector space theory and functional analysis; 6 group theory (Pontryagin); 7 category theory and applications e.g. in algebraic geometry. Roughly, these fall into three periods (up to about 1880 (1, 2), ca. 1880-1945 (3-6), from ca. 1945 (7)), which in our opinion correspond to three different stages of development of mathematics, each with characteristic methods and basic concepts, but also typical forms of knowledge dissemination, that we would like to examine in relation to the central notion of "duality". The proposed project is strongly international in orientation, and its results will be presented in the form of several dissertations, a collective volume, a source book and a database. The project is innovative first of all by the interdisciplinarity of the envisaged overall view, ranging from aspects of the historical development of terminology and bibliometric issues to epistemological aspects. Furthermore, the project is characterized by the fact that it is not about a mathematical theory, a concept, or the like; thus, the investigation cannot, unlike most previous studies on the conceptual history of mathematics, be restricted to the context of a selected mathematical subdiscipline. Most often, duality is termed somewhat blurredly a "principle". In contrast, we prefer to call it an archetype of mathematical thinking; in this way we hope to open new horizons for the research on the history and the philosophy of mathematics and on the reflection on mathematics in general.