# Bourbaki and foundations

Wed May 18 21:43:54 EDT 2022

```FOR THE THREAD BOURBAKI AND FOUNDATIONS

In the initial post by Michael Sheard which generated this very rich and interesting thread, there is a certain ambiguity that begs for clarification. Let me explain what a mean, by an example. So, one question is (I quote):

“How widespread was the belief that category theory is an obviously better foundation for mathematics than set theory, among mathematicians in 2006 (when the book was published)?”

Now, “foundation” above can have two meanings. It might mean either (i) the foundational toolkit the mathematician should better work with or (ii) the domain of objects, one can reduce (try to reduce) all mathematical objects to.

These two are obviously not the same. For example. One can reduce (make an effort to reduce) all mathematics to set theory, but one can hardly expect from an expert of, e.g., analysis to employ ZFC as her toolkit, when proving theorems.

There are several similar analogies that make the same point.

Let me make one in the form of two questions.

Some logicist could probably say that all mathematics can be reduced to set theory. Would she also opt for all mathematical theorems she proves to have the shape of theorems of ZFC? On the other hand, someone who feels well at ease in category theory, would she also claim that all mathematical objects can be reduced (in the logicist sense) to categories?

Best,
Doukas

From: FOM <fom-bounces at cs.nyu.edu> on behalf of Michael Sheard <msheard at stlawu.edu>
Date: Monday, 9 May 2022 - 2:58 AM
To: fom at cs.nyu.edu <fom at cs.nyu.edu>
Subject: Bourbaki and foundations

I recently read the book "The Artist and the Mathematician," which is Amir Aczel's "biography" of Nicolas Bourbaki.  In the concluding chapter, we find this passage:

"Bourbaki lost an incredibly important opportunity to remake its oeuvre in the new form of the theory of categories, something that would have better suited the study of structures than did the old theory of sets with its myriad problems and inadequacies. ... Bourbaki had a chance, through the work of Grothendieck and his students, to refound modern mathematics on the theory of categories, but Bourbaki missed that chance.  In part, this missed opportunity led to the demise of Bourbaki.  For mathematics remained based on a flawed system, set theory, rather than something that would have been much more appropriate. ..."

Earlier, Aczel had already said that Bourbaki's "greatest error" was "letting Grothendieck go and disagreeing with his vision for the future of mathematics" -- specifically, its unwillingness to revise its earlier volumes in order to replace set theory, a "discipline full of theoretical holes," with category theory, which "does not suffer from the inherent limitations of set theory," as the foundation of mathematics.

Although much of my work is in set theory, I am open to the possibility that category theory could serve as a better foundation for mathematics in its full scope than set theory.  I just do not feel that I have ever seen much evidence for that proposition.  So, I would be very interested to hear reactions to any or all of the following questions:

(1) How widespread was the belief that category theory is an obviously better foundation for mathematics than set theory, among mathematicians in 2006 (when the book was published)?

(2) How widespread is this belief today?

(3) What is your position on this belief?

Thanks and best wishes,
Mike

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