The Form and Function of Duality in Modern Mathematics

International audiencePhenomena covered by the term duality occur throughout the history of mathematics in all of its branches, from the duality of polyhedra to Langlands duality. By looking to an “internal epistemology” of duality, we try to understand the gains mathematicians have found in exploiting dual situations. We approach these questions by means of a category theoretic understanding. Following Mac Lane and Lawvere-Rosebrugh, we distinguish between “axiomatic” or “formal” (or Gergonne-type) dualities on the one hand and “functional” or “concrete” (or Poncelet-type) dualities on the other. While the former are often used in the pursuit of a “two theorems by one proof”-strategy, the latter often allow the investigation of “spaces” by studying functions defined on them, which in Grothendieck's terms amounts to the strategy of proving a theorem by working in a dually equivalent framework where the corresponding proof is easier to find. We try to show by some examples that in the first case, dual objects tend to be more ideal (epistemologically more remote) than original ones, while this is not necessarily so in the second case