The axioms of set theory are inconsistent, but the proof of inconsistency is too long for our physical universe.

[quoted by Ruelle,

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.

JOC/EFR December 2013

The URL of this page is:

http://www-history.mcs.st-andrews.ac.uk/Quotations/Cartier.html