Project Details
Projekt
Duality - an archetype of mathematical thinking
Applicants
Professor Dr. Ralf Krömer; Professor Dr. Klaus Volkert
Subject Area
History of Science
Term
from 2016 to 2019
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 279002986
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.
DFG Programme
Research Grants
International Connection
Canada, France, Italy, Switzerland, United Kingdom
