Bourbaki and foundations

martdowd at martdowd at
Tue May 24 20:03:44 EDT 2022

Robert Black writes:

Today I think everyone would agree with
 I don't agree.  I think universal algebra should be taught first, then groups, rings, and modules, then category theory.  Category theory is an advanced topic, and is best introduced after some exposure to basic categories of mathematics as examples.  My book this,  Basic topology is covered after category theory because the important limits and colimits can be constructed via the forgetful functor to Set.  I also cover linear algebra, model theory (including first order logic), and computability before category theory.  The intention of the book is to provide an introduction to a moderately wide range of mathematical subjects to 4th year undergraduates headed to graduate school.
Note also that naive set theory is required for universal algebra, and students are well exposed to this by 4th year.

Martin Dowd
-----Original Message-----
From: Robert Black <mongre at>
To: Foundations of Mathematics <fom at>
Sent: Tue, May 24, 2022 10:32 am
Subject: Re: Bourbaki and foundations

This whole thread began with the claim that Grothendieck left Bourbaki
because of the unwillingness of its other members to replace set theory
by category theory as a foundation of mathematics. But just as a piece
of history this can't be true, can it? Grothendieck left Bourbaki in
high dudgeon in the late 1950s. He certainly thought that the first
volume should be rewritten so as to include the basic concepts of
category theory from early on, and that chapter IV, 'Structures' should
be completely replaced treating roughly the same material from a
category-theoretic viewpoint. Today I think everyone would agree with
this, whatever they think about category-theoretic foundations. But he
surely can't have been arguing for *replacing* set theory by category
theory, since in the 1950s nobody, not even Grothendieck, knew how to do

Robert Black

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