, Volume 190, Issue 12, pp 2141-2164
Date: 10 Jun 2011

Mathematical forms and forms of mathematics: leaving the shores of extensional mathematics

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Abstract

In this paper, I introduce the idea that some important parts of contemporary pure mathematics are moving away from what I call the extensional point of view. More specifically, these fields are based on criteria of identity that are not extensional. After presenting a few cases, I concentrate on homotopy theory where the situation is particularly clear. Moreover, homotopy types are arguably fundamental entities of geometry, thus of a large portion of mathematics, and potentially to all mathematics, at least according to some speculative research programs.