math philosophy - Were Bourbaki committed to set-theoretical reductionism?

I'm really upset because I wrote up a whole answer to your question and lost it because my connection dropped and it never registered =(.

Okay, Bourbaki's structuralism is effectively using categories, but only restricting yourself to concrete categories. One must remember that at the beginning of the writing, category theory had not yet been discovered, and by the time the first two chapters had been published, the work of Grothendieck and Lawvere hadn't even begun to discover topos theory. In terms of formal mathematics, set theories were the only game in town for formal exposition (and are still very much the prevailing model). That is, without first constructing a theory of metamathematics (chapter 1 section 1), logic (chapter 1), a proof calculus (chapter 1), and set theory (chapter 2), one was unable to be completely formal.

Bourbaki's global choice operator $tau$ allows you to find a distinguished object satisfying a proposition unless no object satisfies in which case it returns any object (this is by axiom scheme S7 of Bourbaki also called the axiom scheme of epsilon extensionality by Hilbert and his school). This effectively lets us talk about objects that are identical in terms of some structure, without worrying about the underlying set.

As for Bourbaki's reductionism in the later version of the book (I've read the older version, in fact [this is the source of the english translation]), I can say, having read the older version of the book, that the newer definition of an ordered pair is much easier to use to define the first and second projections, which is an exercise in painful tautology in the first edition (I just found and read the section in a copy of the french second edition, and the discussion is easier to understand even though I don't speak French). However, even in that book, the kuratowski structure is used once and then thrown away, never to be seen again. I would say that the change between editions was merely to make the page easier to read. Here is the reason why: The axiom of the ordered pair was redundant, since the ordered pair provably exists. Perhaps one could have defined the ordered pair (x,y) to be any object satisfying the axiom of the ordered pair (axiom 3 in Bourbaki Theorie des Ensembles 1. ed.), but this is really an unimportant point, and if you've read the book before, no time is wasted on unimportant details.

My conclusion on their reductionism in this case is that it was for simplicity of exposition and parsimony, because, as I've said above, why would one take as an axiom what one can prove?

[I have edited the following paragraph to maintain a positive tone and make clear that certain pronouncements are opinions rather than facts. -- Pete L. Clark]

[I edited it a little more because I didn't like the style, but the paragraph below is my opinion -- Harry Gindi]

Also, I find Mac Lane's criticism is a little strong. Bourbaki is a standard reference on elementary abstract algebra and general topology (if one wants to find the most general version of a theorem known to date in one of those subjects, a good place to start is Algebre or Topologie General by Bourbaki). One of the best places to learn about uniform spaces (which have come up on MO a striking number of times in the past few months) is in Bourbaki. Bourbaki proofs are also incredibly clear and really wonderful to read (once you have the mathematical maturity to do so). Again, Bourbaki on Topological Vector Spaces is again a standard reference on topological vector spaces. Their book on integration theory may only include Radon measures, but their section on the Haar measure is a standard reference on the subject. Their commutative Algebra book is one of the most in-depth books on commutative algebra currently around (rivaled, I would say, only by Matsumura [not so old] and Zariski-Samuel [which is really ancient]), and don't forget about the masterpiece that is Lie Groups and Lie Algebras, which is the only Bourbaki book that I've seen assigned as a class text rather than a reference. Anyone who's read SGA will see that Bourbaki actually wrote a number of sections (who participated isn't exactly clear, but the citation is to Bourbaki). Mac Lane has made great contributions to the world of mathematics, but I respectfully disagree with his assessment of Bourbaki. Bourbaki was a landmark in the style of mathematical exposition, with its emphasis on formalism, rigour, and clarity, in a way, ignoring the words of Goedel, and taking Hilbert's program of formalism as far as it could go.