Abstract
In this paper, we argue that, contrary to the view held by most philosophers of mathematics, Bourbaki’s technical conception of mathematical structuralism is relevant to philosophy of mathematics. In fact, we believe that Bourbaki has captured the core of any mathematical structuralism.
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Notes
- 1.
Here is the original text in French: “Les mathématiciens, comme tous les scientifiques, ont besoin de visées téléologiques qui leur permettent de structurer et de légitimer leurs discours. Bourbaki avait réussi à donner une aura indiscutable à la pensée mathématique, en lui procurant un ressort—le structuralisme—et une finalité—la recherche d’une architecture aboutie et hiérarchisée de ses concepts et résultats.”.
- 2.
There is one important exception to this, namely the so-called structuralist school in philosophy of science of the 1970’s and 1980’s, led by Suppes, Suppe, Moulines and Balzer. Some of them explicitly based their work on Bourbaki’s technical notion of structure. See, for instance, Erhard Scheibe’s work in philosophy of physics.
- 3.
To be fair, although Dieudonné and Cartier, as well as others, have claimed that category theory provided a more convenient framework for structural mathematics, to use Dieudonné terminology, he never offered a proper and general analysis of the notion of abstract mathematical structure in the language of categories nor can we find a clear claim made by a member of Bourbaki that the latter provides foundations for mathematics.
- 4.
As we will see, this is not unfounded and unjustified.
- 5.
The main source here is his book: (Corry 1996). We agree with many claims made by Corry, in particular the claim that the volume on set theory stands apart in Bourbaki’s output, and not because of its clarity and rigor. We disagree on one particular, but important point about Corry’s approach, as we will make clear.
- 6.
When Corry uses the italics, he refers to the technical notion of structure.
- 7.
Note that this is an aesthetic evaluation. Elsewhere, Mac Lane called it “a cumbersome piece of pedantry.”(Lane 1996, 181). This is also an aesthetic judgment. It might nonetheless still be essentially correct, despite being ugly.
- 8.
Bourbaki’s axiomatic set theory was definitely odd, idiosyncratic and arguably inadequate. The details of Bourbaki’s set theory has no impact on our main claim. However, for more on Bourbaki’s set theory, see for instance (Mathias 2014).
- 9.
- 10.
It has to be pointed out that when Bourbaki writes and decides to systematically develop this standpoint, he finds resistance among contemporary mathematicians who believed that the axiomatic method was unable to produce any genuinely new concepts and results. At the risk of repeating ourselves, Bourbaki sees the axiomatic method as a creative method in mathematics. For a discussion of the creative role of axioms in mathematics, see also (Schlimm 2011).
- 11.
This is a point where the introduction of category theory would have changed the picture considerably. Indeed, a whole section should be written on the development of the axiomatic method in the language of categories, as was done by Grothendieck in his paper on homological algebra (Grothendieck 1957). It can be argued that Grothendieck’s way of doing mathematics is a natural extension of Bourbaki’s presentation in The Architecture of Mathematics. Thus, one aspect of Grothendieck’s style is not that surprising when seen in this light.
- 12.
We are not saying that this is original with Bourbaki. The idea, at least restricted to algebra, was already implicit in van der Waerden’s Moderne Algebra and explicit in a paper written by Helmut Hasse in 1931. See Hasse (1986). Since some members of Bourbaki have worked with Hasse and others of the German school in these years, it is not entirely ridiculous to believe that they had discussed these matters with them.
- 13.
This claim might sound silly, but it is fairly easy to explain how it happened by looking at the Bourbaki archives and the evolution of the project. It has to be kept in mind that Bourbaki had no logician among its members. Claude Chevalley was the only founding member who was interested in logic and metamathematics and there is clear evidence that he was responsible for the presence of logic and the metamathematical standpoint in the various versions of the notion. Other members, like Dieudonné and Weil, thought that logic and metamathematics were peripheral and secondary to the whole enterprise.
- 14.
The notion of mother structure is nowhere to be found in the official texts.
- 15.
We have to point out, as many have done, that Bourbaki’s presentation of logic and set theory is very idiosyncratic and it is difficult to understand why he clung to his vocabulary and axioms. One obvious example is his choice to talk about assemblage to designate what any other logician calls a formula. Most commentators would focus on his choice of the \(\tau \) operator and his axioms for set theory, and rightly so. He could easily have used standard notation and notions at that point, since after all, Kleene’s monumental Introduction to Metamathematics was published in 1952, to mention but the most famous textbook available at the time. Bourbaki was well aware of Hilbert and Ackermann’s book published in 1928, but he unfortunately did not adopt its conventions.
- 16.
We are not following Bourbaki’s conventions, which we find unnecessarily complicated and we simplify both the notation and the presentation.
- 17.
This can easily be translated into purely formal requirements of the usual kind.
- 18.
It is, indeed, very tempting to start the analysis in the category of sets and define the notion of species of structures directly there. That would yield a perfectly acceptable mathematical analysis of that latter notion. It is probably what Bourbaki would have done, had he agreed on a way to do it. It was done by Ehresmann in (Ehresmann 1965) and, more recently and in a different context, by Joyal in (Joyal 1981). Our goal, and we believe Bourbaki’s goal too, is to provide a genuine metamathematical analysis, something that is required to anchor a structuralist standpoint about the whole of abstract mathematics.
- 19.
Two specific and surprising cases have to be mentions: the notion of homeomorphism for topological spaces and the notion of equivalence of categories. In the case of topological spaces, mathematicians did not see immediately what the right notion was and there was some confusion in the literature for quite some time. See Moore (2007) for details. As for categories, Eilenberg and Mac Lane introduced the notion of isomorphism of categories in 1945, thinking that it was the proper criterion of identity for them. The right notion, namely the notion of equivalence, was introduced by Grothendieck in his paper in homological algebra in 1957, thus twelve years after the publication of Eilenberg and Mac Lane’s original paper. See Marquis (2009).
- 20.
Dieudonné does not give the definition of a topological space either, believing that “the degree of abstraction required for the formulation of the axioms of such a structure is decidedly greater that it was in the preceding examples; the character of the present article makes it necessary to refer interested readers to special treatises” (Bourbaki 1950, 227).
- 21.
Corry is very well aware of the fact that there were no general analysis of the notion of isomorphism before Bourbaki. Indeed, in a different paper, he says “None of these concepts, however, is defined in a general fashion so as to be a priori available for each of the particular algebraic systems [in van der Waerden’s textbook]. Isomorphisms for instance, are defined separately for groups and for rings and fields, and van der Waerden showed in each case that the relation “is isomorphic to” is reflexive, transitive and symmetric” (Corry 2001, 172).
- 22.
Notice the ambiguity here. We are still in metamathematics, but it is all too easy to fall back on a purely set theoretical reading of the definitions and notions given. We are talking about set theoretical formulas all along. We have to write down formulas such that, when they are interpreted in sets, then the \(f_i\)s are bijections.
- 23.
This is very surprising. Of course, Bourbaki does not literally mean that a species of structure is a text, for it has to be an interpretation of that text in a mathematical domain. There is no doubt that Bourbaki understood that, but this is what one reads. I suspect that the emphasis on the text was deliberate in order to underline the metamathematical nature of the analysis.
- 24.
See, for instance definition 3 in the first section of the volume on general topology. The reference is explicit. The definition given by Bourbaki of isomorphism of topological spaces is not the standard definition. Bourbaki then immediately shows that the definition that follows from the general notion of isomorphism of structure is equivalent to the standard definition of homeomorphism. Corry is absolutely correct to point out that “the verification of this simple fact (which is neither done nor suggested in the book) is a long and tedious (though certainly straightforward) formal exercice”(Corry 1992, 330). However, it misses the main point, which is essentially metamathematical.
- 25.
Thus, we disagree with Corry’s evaluation that by 1957, category theory had reached the status of an independent discipline that enabled generalized formulations of several recurring mathematical situations. Mac Lane had further developed some central ideas in his article on ‘duality’ (Corry 1992, 332). Category was not yet an independent discipline and although it did enabled generalized formulations of several recurring mathematical situations, these were restricted to algebraic topology and homological algebra. Mac Lane’s paper was not very influential and it is with hindsight that one sees into it some of the ideas that will become central after 1957, once they will be shown to be systematically related to central concepts of the theory.
- 26.
Makkai published only one “official” paper on FOLDS, namely (Makkai 1998). There is much more available on his web site: http://www.math.mcgill.ca/makkai/. For an informal presentation of FOLDS and some aspects of its motivation, see (Marquis 2018). Emily Riehl pointed out to me that the claim made in the footnote 37 of the latter that the simplex category with morphisms restricted to the injective functions is the FOLDS-signature for simplicial sets is wrong. It is the FOLDS-signature for semi-simplicial sets.
- 27.
It could be argued that homotopy type theory with the univalence axiom also provides a solution. See, HoTT (UFP 2013). We will not discuss HoTT here.
- 28.
One of the things that Bourbaki did not see is the importance of the relations between structures, in particular between algebraic and topological structures, which are made explicit and possible by category theory. The most striking example of that interaction that was available from early on is the equivalence between the category of affine schemes and the (opposite) of the category of commutative rings. It is this interaction that allowed Grothendieck to develop new foundations for algebraic geometry, foundations that are not given by the axiomatic method.
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Acknowledgments
The author gratefully acknowledge the financial support of the SSHRC of Canada while this work was done. This paper is part of a larger project on Bourbaki and structuralism which would not have seen the day without Michael Makkai’s influence and generosity. I want to thank him for the numerous discussions we had on the subject. I also want to thank Alberto Perruzzi and Silvano Zipoli for inviting me to present this work at the conference on Bourbaki, mother structures and category theory.
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Marquis, JP. (2020). Forms of Structuralism: Bourbaki and the Philosophers. In: Peruzzi, A., Zipoli Caiani, S. (eds) Structures Mères: Semantics, Mathematics, and Cognitive Science. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 57. Springer, Cham. https://doi.org/10.1007/978-3-030-51821-9_3
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