Re: Bourbaki and Categories

[David Spivak <dspivak@uoregon.edu>]

I agree with Andre.  Encapsulating a group of mathematicians inside a  
single named entity fosters a kind of collaborative spirit in which  
good ideas are not kept for personal use later but are shared amongst  
the community.  When ideas are shared in real time, good mathematics  
can be produced faster.  Anyone who wants to join the collective can  
do so, and the collective produces highly useful material.  Of course  
such an enterprise is orthogonal to name-recognition, and maybe to  
getting tenure!  But there is certainly something good about it, as  
there is about wikipedia and the open source movement.

I also agree that the internet could be used in a better way to  
transfer knowledge of mathematics.  Math papers are written linearly,  
in the bottom-up (Euclid/Bourbaki) style, to some extent.  Whereas  
words on paper are in this sense one-dimensional, computers offer  
many more dimensions for knowledge transfer.

Even more interesting to me would be a kind of zoom-feature on  
proofs.  Proofs are in the eye of the beholder: for example it has  
been debated as to whether Perelman's 70 pages was a full proof of  
geometrization.  Given a proof with a statement which one does not  
understand, a mathematician may find himself reproving something that  
was obvious to (or wrongly assumed to be obvious by) another  
mathematician.  The community could benefit if a mathematician who  
proves such a statement then uploaded the proof, even in rough form,  
to some kind of math wiki.  If it were well-organized, this math wiki  
could revolutionize how mathematics is done.  In fact, choosing the  
"right way" to organize such a site may itself be a problem which  
could produce interesting mathematics.

Whatever the case may be, I am all for the idea of a new Bourbaki- 
style enterprise in some form or another.  I think it may first  
require interested parties to get together at some physical location.

David


On Sep 13, 2008, at 10:17 AM, Andre Joyal wrote:

> 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
>
>