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.
Best Answer
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 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 relation: 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.