A New Foundational Crisis in Mathematics, Is It Really Happening?
The article reconsiders the position of the foundations of mathematics after the discovery of the homotopy type theory HoTT. Discussion that this discovery has generated in the community of mathematicians, philosophers and computer scientists might indicate a new crisis in the foundation of mathematics. By examining the mathematical facts behind HoTT and their relation with the existing foundations, we conclude that the present crisis is not one. We reiterate a pluralist vision of the foundations of mathematics.
The article contains a short survey of the mathematical and historical background needed to understand the main tenets of the foundational issues.
Many thanks to the organisers of the FOMUS conference in July 2016 for their invitation to give a talk and to participate in the panel discussion. I would also like to thank Peter Aczel, Andrej Bauer, Mark Bezem, Thierry Coquand, Laura Crossila, Deborah Kant, Angus Mcintyre, Marco Panza, Duško Pavlović, Michael Rathjen, Christian Rosendal and Andrés Villaveces, as well as to the anonymous referee, for very useful and interesting discussions about various parts of this paper. My thanks equally go to the audiences in Paris, Nancy, Oxford, Teheran and Mexico City who have listened and contributed by their comments to the talk that accompanied the development of this paper.
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