In the [December 2009 issue][1] of the [newsletter of the European Mathematical Society][2] there is a very interesting interview with Pierre Cartier.  In page 33, to the question

> What was the ontological status of categories for Grothendieck?

he responds

> 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.

Could someone explain what this actually means?  (Please feel free to retag.)




  [1]: http://www.ems-ph.org/journals/newsletter/pdf/2009-12-74.pdf
  [2]: http://www.ems-ph.org/journals/all_issues.php?issn=1027-488X