Re: Bourbaki and Categories

["R Brown" <ronnie.profbrown@btinternet.com>]

Dear All,

The importance of Bourbaki should be stessed, as it was started when, so we 
are told, texts were very bad. There are many beautiful things in the books: 
I developed part of an undergraduate course from the account of the 
classification of closed subgroups of R^n. This relates to old questions on 
orbits of the planets, and also gives some nice exercises and even exam 
questions of a calculation type. It is good to present students with a 
classification theorem.

The difficulties for Bourbaki seem to arise from the presentation (a) as a 
final and definitive view in toto, and (b) without enough context, as Andre 
points out.

On (a), there is the old childish joke: what happens if you put worms in a 
straight line from Marble Arch to Picadilly Circus? One of them would be 
bound to wriggle and spoil it all! So some mathematical worms have not only 
wriggled but grown large and marched off in a different direction.

On (b), there is the old debating society tag:
text without context is merely pretext.
See more questions in Tim and my article on `Mathematics in Context'.

What is wrong is to present, or take, the whole account as totally 
authoritative, and  will last indefinitely.

What Bourbaki also shows is the value for at least the writers of taking a 
viewpoint and following it through as far as it will go: if it seems in the 
end to go too far, or to be inadequate, then that is valuable information 
for them and others. See my Dirac quote in `Out of Line'.

Ronnie








----- Original Message ----- 
From: "Andre Joyal" <joyal.andre@uqam.ca>
To: <categories@mta.ca>
Sent: Saturday, September 13, 2008 6:17 PM
Subject: categories: Re: Bourbaki and Categories


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