, Volume 190, Issue 12, pp 2141–2164

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


DOI: 10.1007/s11229-011-9962-0

Cite this article as:
Marquis, JP. Synthese (2013) 190: 2141. doi:10.1007/s11229-011-9962-0


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.


Philosophy of mathematicsAlgebraic geometryCategory theoryHomotopy theory

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© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Département de philosophieUniversité de MontréalMontrealCanada