In the [December 2009 issue](http://www.ems-ph.org/journals/newsletter/pdf/2009-12-74.pdf) of the [newsletter of the European Mathematical Society](http://www.ems-ph.org/journals/all_issues.php?issn=1027-488X) there is a very interesting interview with Pierre Cartier. In page 33, to the question



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> What was the ontological status of categories for Grothendieck?
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he responds



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



Is "all categorical reasoning formally contradictory"?
Note: I am not a historian. I'm just guessing as to what prompted the comments.


Here's my guess: if you do set theory naively, in the old-fashioned "anything is a set" way, then you run into Russell's paradox; the set consisting of all sets that aren't elements of themselves gives you trouble. So you then decide set theory needs formalising (I'm talking about 100 years ago here of course) and you write down some axioms, and the ones that "won" are ZFC, where only "small" collections of things are sets, and "big" things like "all groups" aren't sets. Of course there's nothing contradictory about considering all groups, or quantifying over groups (i.e. saying "every group has an identity element"), but you can't quantify over the *set* of all groups.


And now because it's the 50s or 60s and you want to do homological algebra and take derived functors and do spectral sequences and stuff in some abstract way, you are now feeling the pressure a bit, because you want to define "functions" from the category of all G-modules (G a group) to the category of abelian groups called "group cohomology", but "H^n(G,-)" isn't a function, because its domain and range aren't sets. So you call it a "functor", which is fine, and press on.


And as time goes on, and you start composing derived functors, you know in your heart that it's all OK. 


And then Grothendieck comes along, and probably other people too, and raise the issue that one really should be a bit more careful, because we don't want another Frege (who wrote a huge treatise on set theory but allowed big sets and his axioms were contradictory because of Russell's paradox). So Grothendieck tried to tame these beasts and "go back to basics"---but in some sense he "failed"---or, more precisely, realised that there were fundamental problems if he really wanted to treat categories as sets. "Sod it all", thought Grothendieck, "this is not really the main point". So he said "let's just assume there's a universe, i.e. (basically) a set where all the axioms of set theory hold" (it was a bit more complicated than that but still). This assumption (a) fixed all his problems but was (b) unprovable from the axioms of ZFC (because of Goedel).


So there's a guess. Cartier is perhaps going over the top with "contradictory"---the statement "Russell's paradox is a paradox" is true but the statement "any mathematical manipulation with collections of objects that don't form a set is formally contradictory" is much stronger and surely false.


