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• CommentRowNumber1.
• CommentAuthorThe User
• CommentTimeAug 27th 2012
From the Bourbaki article: “Mathematically, the biggest single credit to Bourbaki is for decisively defining a general notion of structure”

What are the reasons for this statement? This notion of a structure is actually nearly never used in mathematics, even Bourbaki himself did not use it in later volumes and he covered only very few topics of universal algebra and no model theory at all. On the other hand “he” invented for example uniform structures and proved essential properties, which have a lot of applications in functional analysis and the theory of topological groups.
• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeAug 27th 2012

I agree this sounds a very odd claim to me. Shouldn’t one say that their definition of structure stood in the way of an effective uptake of category theoretic ideas?

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeAug 27th 2012
• (edited Aug 27th 2012)

We are speaking about the $n$Lab entry Bourbaki, right?

I bet what is meant there is not Bourbaki’s formalized notion of structure, but the practice which they introduced of defining mathematical structures – notably algebraic structures – as sets with structure. As in “a group is a set equipped with … such that …”.

But that’s just what gather from hearsay. I never much looked into this. But I did link at Bourbaki to a text Nicholas Bourbaki: Theory of structures, which might be relevant.

I suggest whoever feels like an expert on this topic should consider expanding the entry Bourbaki where need be.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeAug 27th 2012
• (edited Aug 27th 2012)

By the way, earlier this year there had been related discussion here.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeAug 27th 2012

Maybe those interested in this might look into expanding on the $n$Lab entry structure and maybe creating the entry structure after Bourbaki that is projected there.

I would also like to know to which extent the entry structure in model theory should really be independent of model.

• CommentRowNumber6.
• CommentAuthorZhen Lin
• CommentTimeAug 27th 2012

Corry’s 1992 article, Nicolas Bourbaki and the concept of mathematical structure, argues that Bourbaki’s notion of structure was never really taken seriously – not even by themselves! He says credit should perhaps instead go to the early algebraists like van der Waerden…

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeAug 27th 2012

I agree with Urs #3: the only quasi-defensible interpretation of that remark that I can think of would be a reference to the idea of mathematics as the study of structure, rather than the (arguably misguided) attempt at a material-set-theoretic formal notion of “structure”. On the other hand, I certainly wouldn’t claim myself that even the idea of structure was Bourbaki’s “single biggest credit”!

• CommentRowNumber8.
• CommentAuthorDavid_Corfield
• CommentTimeAug 27th 2012

I made a change to say Bourbaki developed the Gottingen structural approach. There’s probably much more to be said. I think it was Weil who stood in the way of category theory being adopted. Then we ought to say something about Grothendieck’s relation to them.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeAug 27th 2012

Okay, so I have tweaked the paragraph still a bit further. Check:

A central achievement of Bourbaki is the emphasis of mathematics as the study of structure, as in the approach developed by Göttingen mathematicians in the 1920s and 1930s, such as Emmy Noether and Emil Artin, and written up by van der Waerden in Moderne Algebra (see Corry). This approach was very influential in the mainstream mathematics of the second half of the 20th century.

However, Bourbaki did not embrace category theory, which may be thought of as being the essence of that structural approach, though some of the universal properties treated in category theory in fact first appeared in early editions of Bourbaki. Instead, Bourbaki proposed its own formalization of the notion of “structure”, which however neither caught on nor seemed to have taken very seriously by the group itself.

• CommentRowNumber10.
• CommentAuthorThe User
• CommentTimeAug 27th 2012
@David
Did they really “stood in the way”? When they started with the Théorie des ensembles there was no category theory at all. And even if they would have started a few years later and would have put more emphasize on categories, there would not have been category theoretical foundations yet. It would still have appeared natural to introduce categories as special structures. Their formal definition of structures actually reflects quite accurately the mathematical intuition of structures (sets+structure) and their homomorphisms.
• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeAug 27th 2012
• (edited Aug 27th 2012)

Their formal definition of structures actually reflects quite accurately the mathematical intuition of structures (sets+structure) and their homomorphisms.

What is their formal definition of structure? I never looked it up. Zoran seemed to have planned to write a note about this at structure after Bourbaki, but never got around to. Maybe you have a minute to spare for this?

• CommentRowNumber12.
• CommentAuthorMike Shulman
• CommentTimeAug 27th 2012

Their formal definition of structures actually reflects quite accurately the mathematical intuition of structures (sets+structure) and their homomorphisms.

Perhaps it reflects the structures themselves, but I think it isn’t really adequate for talking generally about morphisms except maybe in the algebraic case. There was a discussion about this on the categories mailing list a little bit ago.

• CommentRowNumber13.
• CommentAuthorTobyBartels
• CommentTimeAug 27th 2012

I can’t find the book itself, but drafts are available at the Bourbaki archives; the last draught there of the chapter on structures pdf gives the idea.

It is not pretty. Much better to say that a structure (on sets) is a functor from the groupoid of sets (and bijections) to itself. Bourbaki certainly could have said this (although necessarily in less concise language), and they kind of did, but there’s also this surrounding logical framework that serves no purpose that I can understand.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeAug 27th 2012
• (edited Aug 27th 2012)

Thanks!

If you have the minute to say all this here, please consider saying it at structure after Bourbaki or otherwise please try to deal with the grey link to this term that somebody left at structure. That would be great. For currently the entry structure looks much more like a mess than it deserves to.

In fact I will re-iterate what I said elsewhere:

I feel the layout of that entry is mixed up. It should have a Definition-section with essentially just a commented pointer to stuff, structure, property (maybe also a pointer to structure after Bourbaki, with due caveats)…

… and then structure in model theory should at best be in the Examples-section, as an example of a structure on something (not a notion of structure as such).

Would you agree?

• CommentRowNumber15.
• CommentAuthorThe User
• CommentTimeAug 27th 2012
• (edited Aug 27th 2012)
Let me call them “heterogeneous higher order relational structures”. You have multiple (thus “heterogeneous”) basic sets and multiple fixed auxiliary sets (for example for measure spaces you need the reals as fixed auxiliary set, while your basic set could be anything, for a given type of a structure the auxiliary sets are always the same). The additional structure are relations, but not only relations on the basic and auxiliary sets, there can also be relations on relations (in measure spaces: the basic sigma algebra is a set of relations on the basic set) etc. and even relations “mixing” multiple levels (a measure is a relation between certain subsets of the basic set (kinda second order) and the reals (plain first order)). Thus you can describe the type of complicated structures like Hilbert spaces, measure spaces, uniform structures, enriched categories etc. And by simple restrictions you get the usual notions of higher order models, first order models, algebraic structures etc. That is clear enough?

Notable observation: The description of it is quite pedantic, but actually not formal in a strict sence, i.e. it is metamathematical. He explicitly states that. Thus types of structures are not mathematical objects themselves, but metamathematical notions, metamathematical guidelines. I do not understand why he did it that way, it would not have been a problem to treat the type of a structure including the axioms as mathematical objects, maybe he wanted to avoid model theory or a boring section full of sentences like “the formula ‘∀x…’ is fulfilled iff for all elements…” .
• CommentRowNumber16.
• CommentAuthorDavid_Corfield
• CommentTimeAug 28th 2012

Re #10, it would require us to look at some history to establish Bourbaki’s changing relationship to category theory. It’s during the 50s when the pressure to use category theory builds up. There was certainly some resistance, understandably as it would have required a lot of rewriting and new work:

Here’s Colin McLarty on the subject:

Bourbaki’s first publication was

Bourbaki, N. [1939]: Th{'e}orie des ensembles, Fascicules de r{'e}sultats, Paris: Hermann, Paris.

It is very sketchy on “structures,” and uses no notion of mapping between structures except isomorphisms. Their actual theory of structures first appeared in Bourbaki, N. [1957]: Th{'e}orie des ensembles}, Chapter 4, Paris: Hermann.

That theory was a rear-guard action meant to give an alternative to category theory. As i mentioned before, Weil was citing the categorical idea, and thinking about finding an in-house alternative to it, already in 1951. By 1957 Grothendieck, and Cartier, and Chevalley, probably Dieudonne, and others, all saw that category theory was more agile than these structure, simpler, and more to the point, plus it had a natural “higher order” aspect in the theory of functors which was actually more useful in practice than categories alone.

Cartier has justly said it would have been a huge job to formulate all Bourbaki’s ideas in terms of categories and functors. It would have called for a lot of ideas which were only invented in the coming years.

It was relatively easy to give Bourbaki’s theory of structures – because it never really worked at all even for Bourbaki’s purposes (as Corry documents in detail). Naturally it is easier to give an unusable theory of structures than to work out the ways categories and functors would actually be used.

Mac Lane gave a nationalistic reason

“Categorical ideas might well have fitted in with the general program of Nicolas Bourbaki for the systematic presentation of mathematics. However, his first volume on the notion of mathematical structure was prepared in 1939 before the advent of categories. It chanced to use instead an elaborate notion of an dchelle de structure which has proved too complex to be useful. Apparently as a result, Bourbaki never took to category theory. At one time, in 1954, I was invited to attend one of the private meetings of Bourbaki, perhaps in the expectation that I might advocate such matters. However, my facility in the French language was not sufficient to categorize Bourbaki. Perhaps the explanation for his resistance is the hard fact that categories were not made in France. Even Eilenberg’s later membership in Bourbaki did not serve to overcome Bourbaki’s disinclination. It may be that the circulation of new ideas is not always unhindered.” (Applied Categorical Structures, Vol. 4, No. 2-3 (1996), 129-136)

The full discussion is here.

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeAug 28th 2012
• (edited Aug 28th 2012)

Thanks, David. I have added that to the entry.

By the way and on the other hand: I find that restricting attention to _iso_morphisms first is not an outright bad idea. Typically it is the objects and their isomorphisms that really characterize a notion of structure, while the definition of the non-isomorphic morphisms between these structures amounts to a further choice. (A basic example being Set and Rel).

Indeed, I think, in a historical vein, there is another story to be told of a simple prejudice about the formalization of structure blocking progress of a whole field: if the old-school higher category theory had earlier abandoned the emphasis on non-invertible higher morphisms, the present progress might have come much sooner.

• CommentRowNumber18.
• CommentAuthorTodd_Trimble
• CommentTimeAug 28th 2012

By the way and on the other hand: I find that restricting attention to _iso_morphisms first is not an outright bad idea. Typically it is the objects and their isomorphisms that really characterize a notion of structure, while the definition of the non-isomorphic morphisms between these structures amounts to a further choice. (A basic example being Set and Rel).

Another example of this can be seen in the query box discussion over at graph.

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeAug 28th 2012

This was also discussed on the Category Theory List not long ago. But I don’t have the pointer right now.

• CommentRowNumber20.
• CommentAuthorTodd_Trimble
• CommentTimeJun 2nd 2013

I have done some further tweaking of Bourbaki. Since (as has already been pointed out) the formal Bourbaki notion of “structure” had little impact, even in the Elements (past the book Theory of Sets, taking the books in their intended logical order), I though it would be a little clearer to change “a central achievement of Bourbaki is the emphasis of mathematics as the study of structure” to “a central point of view of Bourbaki is the emphasis of mathematics as the study of structure”. A “point of view” can be something more nebulous and informal than what the word “achievement” might suggest (concrete results), and indeed it is an informal notion of structure that has taken root in the popular mind as a guiding central idea behind Bourbaki and the Elements.

• CommentRowNumber21.
• CommentAuthorzskoda
• CommentTimeJun 3rd 2013
• (edited Jun 3rd 2013)

I would also like to know to which extent the entry structure in model theory should really be independent of model.

They should stay separated. Structure (in the sense of model theory) is attached to a language and does not care about axioms, and a model is attached to a theory and cares strictly about the axioms. There are also models for different kind of logics which are not selected structures any more (like Kripke models, for example); in additions models can be viewed as functors, what is not a point of view that useful for structures.