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The following are two closely related questions.

What was Bourbaki's position on the ontological status of mathematical objects? Were they some kind of Realist/Platonist or were they Formalist?

Dieudonne, a Bourbaki, provides a confusing answer stating that they "act" like realists. It is unclear whether they actually adhere to that position. Is the realist position, or some variant, Bourbaki's philosophical position?

Furthermore, Pierre Deligne a major successor of the school claims that he uses the Axiom of Choice for convenience but does not believe in it. What does he mean by that statement? For example, does it mean to reject certain results based on AC like the embedding of the p-adic field onto the complex plane because they are counterintuitive? Are there any sources where he elaborates this position? Does this imply that he is not a Realist/Platonist?

I hope my questions make sense. Thank you so much.

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  • The short answer is that Bourbaki did not care about the ontological status. It was a diverse group and what unified them was logical systematization of mathematical results, not ontology. As Manin put it in his interview:"They created “pragmatic foundations”, adopted for many decades by all working mathematicians, as opposed to “normative foundations” that logicists or constructivists tried to impose upon us." Deligne's remark is in the same vein, he does not reject any results, he just does not believe in a platonic heaven.
    – Conifold
    Nov 19, 2023 at 8:33
  • Welcome to SE. Your second paragraph asks two questions. I can see that they are probably linked, but you would improve things if you combined the two questions so that the relationship between them is clear.
    – Ludwig V
    Nov 19, 2023 at 10:47
  • @Conifold, are you sure that an ontological perspective can't be deducted from their approach? (even if they haven't accounted for) Nov 19, 2023 at 19:57
  • @IoannisPaizis Pretty sure. See e.g. a discussion of their "structuralism" being contrasted with categorical "structuralism". Mathematicians would call this "philosophy", but in the folk sense of practical approach to or methodology of solving problems and building theories. It does not track ontological structuralism. Some of them might have had ontological beliefs, but they were kept out of the Bourbaki corpus.
    – Conifold
    Nov 20, 2023 at 5:10

2 Answers 2

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  1. Concerning your first question see Nicolaus Bourbaki The architecture of mathematics. The paper is a self-presentation of Bourbaki from 1950, on request I can send a copy.

    Reading the paper confirms @Conifold’s comment that the ontology of mathematics was not the point of Bourbaki.

    Instead, Bourbaki’s topic is “structure” as the fundamental concept for mathematic:

    […] all of these are decisive instances of mathematical progress, of turning points at which a stroke of genius brought about a new orientiation of a theory, by revealing the existence in it of a structure which did not a priori seem to play a priori in it.

    Bourbaki’s three basic structures are algebraic structures, structures defined by an order, and topological structures.

  2. Concerning your second question whether Pierre Deligne is a mathematical realist, possibly you may want to ask him in person by an email to [email protected]

    Added: If he replies, please let us know his answer.

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Bourbaki have insisted that they are interested in the way mathematicians do their work rather than in foundations, and there are indications that their philosophy of mathematics is not carefully thought out. However, this does not necessarily mean that one cannot discern a background philosophy of mathematics in their writings. A useful case in point is the Bourbaki member Roger Godement. There is a 100-page chapter on Set Theory opening his voluminous text on algebra (the 1963 Cours d'Alg`ebre).

Adrian Mathias, who is a professional logician, comments on the dilettantish approach to logic and set theory in Bourbaki's work, and more specifically analyses Godement's comments on logical languages:

We were told on page 21 that assemblies are built up from fundamental signs and letters, which suggests that a letter is a symbol, but at the top of page 30 we read

Let R be a relation, A a mathematical object, and x a letter (i.e. a “totally indeterminate” mathematical object),

so by page 30 a letter is not a symbol: Godement is slipping away from treating his system as an uninterpreted calculus, and moving towards an informal Platonism. (Mathias, "Hilbert, Bourbaki and the scorning of logic", pp. 28-29).

Thus Bourbaki's background philosophy may well incorporate aspects of Platonism. This was also my personal impression when I read their comments on the construction of the natural numbers: absent is any distinction between the metalanguage and the object-language. In general, axioms are treated in a lackadaisical fashion, as argued by Mathias.

There is a number of discussions of Bourbaki's dilettantism (in the area of logic and set theory) under the tag https://mathoverflow.net/questions/tagged/bourbaki

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