Duality - an archetype of mathematical thinking

对偶在现代数学中以多种方式和不同的背景出现。我们可以说数学思维的基本模式（原型）（类似于类比或对称）；目前还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从历史上研究这一原型。对偶性（通常被称为极性或互易性）早在多面体理论（欧几里得的“元素”第十五册，开普勒的“世界和谐”第二册和第五册）和球面几何（极三角形的形状）中就出现了。从1810年开始，对偶在射影几何领域变得相当突出，发展成为一种广泛的现象，特别是在20世纪（见下文）。对偶的基础问题和如何确保它们的问题——除了著名的平行问题——是元数学发展的一个早期和重要的初始问题。尽管具有这种重要性，但仍然没有对二元性的多种形式及其与物理和静力学等其他领域的相互作用进行全面的历史研究。我们的目标是在拟议的项目中缩小这一差距。特别地，对偶的发生和作用应在下列领域的发展方面进行调查和比较：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（poincar<e:1>，亚历山大）；5 .向量空间理论与泛函分析；6群论（Pontryagin）；范畴论及其在代数几何中的应用。粗略地说，这些可以分为三个时期（大约到1880年（1,2），大约1880-1945年（3-6），大约1945年(7)），在我们看来，这对应于数学发展的三个不同阶段，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，我们想要检查与“对偶”的中心概念有关。拟议的项目具有很强的国际性，其结果将以几篇论文、一本书、一本源书和一个数据库的形式呈现。该项目的创新之处首先在于设想的整体观点的跨学科性，从术语和文献计量学问题的历史发展方面到认识论方面。此外，这个项目的特点是它不是关于数学理论、概念或类似的东西；因此，调查不能，不像大多数以前的研究数学的概念史，被限制在一个选定的数学分支学科的背景下。大多数情况下，二元性被模糊地称为“原则”。相反，我们更愿意称之为数学思维的原型；通过这种方式，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.