Re: Bourbaki and Categories

[Andre Joyal <joyal.andre@uqam.ca>]

Dear Colin, Zoran, Robert, Eduardo and All,

I find the present discussion on Bourbaki and category theory very important.  
I recall asking the question to Samuel Eilenberg 25 years ago and more recently to Pierre Cartier.  
If my recollection is right, Bourbaki had essentially two options: rewrite the whole treaty using categories, 
or just introduce them in the book on homological algebra, 
The second option won, essentially because of the enormity of the task of rewriting everything. 
Other factors may have contributed on a smaller scale, like some unresolved foundational questions. 
In any cases, it was the beginning of end for Bourbaki.

Bourbaki was a great humanistic and scientific enterprise.
Advanced mathematics was made available to a large number
of students, possibly over the head of their bad teachers. 
It defended the unity and rationality of science in an age
of growing irrationalism (it was conceived in the mid thirties).

I have personally learned a lot of mathematics by reading Bourbaki.  
Everything was proved, and the proofs were logically very clear.
 It was a like a continuation of Euclid Elements two thousand years later!
But after a while, I stopped reading it.
I had realised that something important was missing: the motivation. 
The historical notes were very sketchy and not integrated to the text.
I remember my feeling of frustration in reading the books of functional analysis,
because the applications to partial differential equations were not described.
Everything was presented in a deductive order, from top to down.
We all know that learning is very much an inductive process, from
the particular to the general. This is true also of mathematical research. 

Bourbaki is dead but I hope that the humanistic philosophy behind the enterprise is not.  
Unfortunately, we presently live in an era of growing irrationalism.
Science still needs to be defended against religion.
Civilisation maybe at a turning point with the problem of climate change. 
Millions of people need and want to learn science and mathematics. 

Should we not try to give Bourbaki a second life? 
It will have to be different this time.
Possibly with a new name.
Obviously, internet is the medium of choice.
What do you think?

Andre




-------- Message d'origine--------
De: cat-dist@mta.ca de la part de Colin McLarty
Date: ven. 12/09/2008 14:46
À: categories@mta.ca
Objet : categories: Re:  Bourbaki and Categories
 
From: zoran skoda <zskoda@gmail.com>
Date: Friday, September 12, 2008 2:06 pm

wrote, among other things

> main points of departure. The remark that as a proponent of 
> "structures" Bourbaki
> had to include categories is anyway a bit lacking an argument. 
> First of all, because
> of the size problems one can not take big categories on equal 
> footing with, say groups,
> and considering only small categories would be strange and lacking 
> most interesting examples.

The claim is not that Bourbaki should have studied categories as
structures.  It is that Bourbaki was doomed to fail in trying to use
their structure theory.  Leo Corry shows in his book "Modern Algebra and
the Rise of Mathematical Structures" (Birkhäuser 1996) that they did fail.  

And they should have seen this coming, because their theory had been 

"superseded by that of category and functor, which includes it under a
more general
and convenient form" (Dieudonné "The Work of Nicholas Bourbaki" 1970).

best, Colin