Pierre Cartier

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The axioms of set theory are inconsistent, but the proof of inconsistency is too long for our physical universe.
[quoted by Ruelle, Chance and Chaos]
Nowadays, one of the most interesting points in mathematics is that, although all categorical reasonings are formally contradictory, we use them and we never make a mistake. Grothendieck provided a partial foundation in terms of universes but a revolution of the foundations similar to what Cauchy and Weierstrass did for analysis is still to arrive. In this respect, he was pragmatic: categories are useful and they give results so we do not have to look at subtle set-theoretic questions if there is no need. Is today the moment to think about these problems? Maybe . . .
Interview in Newsletter of the European Mathematical Society, January 2010.