Were Bourbaki committed to set-theoretical reductionism?

Question

A set-theoretical reductionist holds that sets are the only abstract objects, and that (e.g.) numbers are identical to sets. (Which sets? A reductionist is a relativist if she is (e.g.) indifferent among von Neumann, Zermelo, etc. ordinals, an absolutist if she makes an argument for a priviledged reduction, such as identifying cardinal numbers with equivalence classes under equipotence). Contrasting views: classical platonism, which holds that (e.g.) numbers exist independently of sets; and nominalism, which holds that there are no abstract particulars.

I'm interested in the relationship between "structuralism" as it is understood by philosophers of science and mathematics and the structuralist methodology in mathematics for which Bourbaki is well known. A small point that I'm hung up on is the place of set theory in Bourbaki structuralism. I'm weighing two readings.

• (1) conventionalism: Bourbaki used set theory as a convenient "foundation", a setting in which models of structures may be freely constructed, but "structure" as understood in later chapters is not essentially dependent on the formal theory of structure developed in Theory of Sets,
• (2) reductionism: sets provide a ground floor ontology for mathematics; mathematicians study structures in the realm of sets.

In favor of conventionalism:

• (a) Leo Corry's arguments in “Nicolas Bourbaki and the Concept of Mathematical Structure” that the formal structures of Theory of Sets are to be distinguished from and play only a marginal role in the subsequent investigation of mathematical structure,
• (b) ordered pairs: definitions reducing pairs to sets like Kuratowski's bring "baggage" (i.e., extra structure) and Bourbaki used primitive ordered pairs in the first edition of Theory of Sets, showing no excess concern for complete reduction,
• (c) statements of Dieudonne to the Romanian Institute indicating chs. 1 and 2 are mostly to satisfy bothersome philosophers (like me I suppose) before getting on to topics of greater interest,
• (d) the discussion of axiomatics and structure in "The Architecture of Mathematics", placing no special emphasis on sets,
• (e) this interpretation serves my selfish philosophical agenda.

In favor of reductionism:

• (a) linear ordering of texts suggests perceived logical dependence on Theory of Sets,
• (b) reductionism makes sense of unity of mathematics,
• (c) 1970 edition includes Kuratowski pairs,
• (d) makes sense of controversies over category theory,
• (e) makes sense of some outsider criticisms (e.g., Mac Lane in "Mathematical Models" that Bourbaki was dogmatic and stifling),
• (f) I fear that in leaning towards conventionalism I'm self-deceiving to serve my selfish philosophical agenda.

Apologies: not sure this is MO appropriate, any answers may be anachronistic, probably no univocality of opinion among Bourbaki members, my views are based on popular expositions, interviews, and secondary literature and not close study of the primary texts.

Discussion related to this question has recently occurred at n-category cafe, occasioned by Manin's recent claim that Bourbaki provided "pragmatic foundations". The conventionalist interpretation, I think, helps make sense of Manin's claim and would show some criticisms levelled toward Bourbakism to misapprehend their intention (if not their impact). I have Borel's "Twenty-Five Years With Bourbaki" which discusses Grothendieck and the controversy over the direction following the first six books. Corry makes the claim that the Theory of Sets approach had limitations in dealing with category theory. I would especially appreciate references or answers that help me better understand these issues in particular, which are accessible to a philosopher with some grad coursework in mathematics and with only a self taught rudimentary understanding of categories.

Answer 1

First, most mathematicians don't really care whether all sets are "pure" -- i.e., only contain sets as elements -- or not. The theoretical justification for this is that, assuming the Axiom of Choice, every set can be put in bijection with a pure set -- namely a von Neumann ordinal.

I would describe Bourbaki's approach as "structuralist", meaning that all structure is based on sets (I wouldn't take this as a philosophical position; it's the most familiar and possibly the simplest way to set things up), but it is never fruitful to inquire as to what kind of objects the sets contain. I view this as perhaps the key point of "abstract" mathematics in the sense that the term has been used for past century or so. E.g. an abstract group is a set with a binary law: part of what "abstract" means is that it won't help you to ask whether the elements of the group are numbers, or sets, or people, or what.

I say this without having ever read Bourbaki's volumes on Set Theory, and I claim that this somehow strengthens my position!

Namely, Bourbaki is relentlessly linear in its exposition, across thousands of pages: if you want to read about the completion of a local ring (in Commutative Algebra), you had better know about Cauchy filters on a uniform space (in General Topology). In places I feel that Bourbaki overemphasizes logical dependencies and therefore makes strange expository choices: e.g. they don't want to talk about metric spaces until they have "rigorously defined" the real numbers, and they don't want to do that until they have the theory of completion of a uniform space. This is unduly fastidious: certainly by 1900 people knew any number of ways to rigorously construct the real numbers that did not require 300 pages of preliminaries.

However, I have never in my reading of Bourbaki (I've flipped through about five of their books) been stymied by a reference back to some previous set-theoretic construction. I also learned only late in the day that the "structures" they speak of actually get a formal definition somewhere in the early volumes: again, I didn't know this because whatever "structure-preserving maps" they were talking about were always clear from the context.

Some have argued that Bourbaki's true inclinations were closer to a proto-categorical take on things. (One must remember that Bourbaki began in the 1930's, before category theory existed, and their treatment of mathematics is consciously "conservative": it's not their intention to introduce you to the latest fads.) In particular, apparently among the many unfinished books of Bourbaki lying on the shelf somewhere in Paris is one on Category Theory, written mostly by Grothendieck. The lack of explicit mention of the simplest categorical concepts is one of the things which makes their work look dated to modern eyes.

Answer 2

Adrian Mathias has written a number of excellent essays criticising various aspects of Bourbaki's logical foundations, and I encourage you to follow the link and read them. He writes supremely well, and some of these essays are simply riotous, when he exposes particularly ridiculous aspects of the Bourbaki systems, such as the last item below.

Probably the main reference you should look at is:

• Adrian Mathias, The Ignorance of Bourbaki (a commentary on the foundational stance of the Bourbaki group. In: Mathematical Intelligencer 14 (1992) 4--13 MR 94a:03004b, and also in Physis Riv. Internaz. Storia Sci (N.S.) 28 (1991) 887--904, MR 94a:03004a. A translation by Andras Racz into Hungarian is available, under the title Bourbaki tevutjai, in A Termeszet Vilaga, 1998, III. kulonszama.)

In this essay, he is highly critical of the Bourbaki stance on set theoretic foundations, and seems to view it as taking place in a bizarre historical vacuum. Although they discuss various historical contributions to set theory, they do not mention Goedel and his towering contributions. Their set theoretic system, amounting essentially to the Zermelo axioms with choice, is strangely weak, insufficient for many mathematical constructions. (For example, in ZC, you cannot prove that the ordinal ω+ω exists, or that there are any sets of cardinality Aleph_ω.)

Mathias has several follow up articles on his web page, continuing the discussion of this topic, and his essays now form a dialogue with various writers defending Bourbaki. For example, he has interesting articles engaging with Mac Lane and with Mac Lane's set theory, which shares similarities to that of Bourbaki.

Finally, there is his charming essay lampooning the Bourbaki formal system, while also giving a thorough logical analysis of it

• Adrian Mathias, A Term of Length 4,523,659,424,929, Synthese 133 (2002) 75--86.

He describes it thus:

A calculation of the number of symbols required to give Bourbaki's definition of the number 1; to which must be added 1,179,618,517,981 disambiguatory links. The implications for Bourbaki's philosophical claims and the mental health of their readers are discussed.

(I mentioned these essays also in this related question.)

Answer 3

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.

Answer 4

The difficulties in formalizing categorical reasoning in set theory are actually pretty simple to understand -- it's just an annoying incompatibility in how the notion of size is used in practice in category theory versus set theory. In category theory, it's common to talk about categories like the category of groups, and categories of functors into this category, and so on. And for reasons akin to Russell's paradox, we need to distinguish between small and large categories.

In set theory, this corresponds to the distinction between sets and proper classes. But when we interpret categories in sets, we don't want to identify small and large with set and class -- we want it to be a relative distinction, so that as we form new categories we can "change our mind" about what's small and what's large. Otherwise, we can't perform many natural categorical constructions, such as forming functor categories, as soon as the source and target are large. (The reason is that we can't take exponentials/powers of proper classes.)

Grothendieck handled this by introducing universes, which are a nested family of sets closed under all the set-forming operations of ZFC. (Postulating their existence corresponds to a large cardinal axiom.) So now, we're "really" working ambiguously at some level in the universe hierarchy, and when we need to form a functor category between large categories, we move our point of view up a universe level, so that the two particular categories we want to construct a functor category of, are small again. In this way, proper classes are never used to interpret categories, and so all the set-forming operations are always available.

Whether or not this is essential or not is a matter of furious debate (though I am told Grothendieck universes are a relatively mild large cardinal axiom).

Answer 5

Geoffrey Hellman has written something on structuralism that compares Bourbaki structuralism with category theoretic structuralism. Here. His take seems to be that they were being reductionist.