Mathematical Forms and Forms of Mathematics: Leaving the Shores of Extensional Mathematics

Synthese 190 (12):2141-2164 (2013)
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
Keywords Philosophy of mathematics  Algebraic geometry  Category theory  Homotopy theory
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