In a paper published in 1939, Ernest Nagel described the role that projective duality had played in the reformulation of mathematical understanding through the turn of the nineteenth century, claiming that the discovery of the principle of duality had freed mathematicians from the belief that their task was to describe intuitive elements. While instances of duality in mathematics have increased enormously through the twentieth century, philosophers since Nagel have paid little attention to the phenomenon. In this paper I will argue (...) that a reassessment is overdue. Something beyond doubt is that category theory has an enormous amount to say on the subject, for example, in terms of arrow reversal, dualising objects and adjunctions. These developments have coincided with changes in our understanding of identity and structure within mathematics. While it transpires that physicists have employed the term ‘duality’ in ways which do not always coincide with those of mathematicians, analysis of the latter should still prove very useful to philosophers of physics. Consequently, category theory presents itself as an extremely important language for the philosophy of physics. (shrink)

Phenomena 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. (shrink)