Bourbaki and Categories

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

On Fri, Sep 19, 2008, John Baez wrote:

>Do we imagine this new Bourbaki as just systematizing and
>presenting what we know already, or struggling to create
>brand new mathematics?

This is  an important question.
We need to have a clear view of the goal of such an enterprise. 
>From my point of view, the goal should be "educational":
to help students and researchers to learn mathematics
and cross the boundary between fields. 
Mathematics is vast, and every mathematician is a permanent student. 
The traditional way to learn is to read the litterature and to discuss with a master.
I was told that Grothendieck had learned algebraic geometry by discussing with Serre.
But few peoples have this chance.
Obviously, Internet is opening new avenues for learning. 
Many peoples (and myself) have learned a lot by reading your bulletin 
"This Week's Finds in Mathematical Physics". 
You have a real talent to explain a subject by exposing the heuristic!
A discussion forum like the "Categories list" is also very helpful.
Wikipedia is a useful place to gather informations about a subject.
But the Bourbaki Tractate was offering something more:
a unified presentation of mathematics, including the proofs.

The Bourbaki Tractate was the result of a sustained collaboration 
of many generations of mathematicians from different fields. 
Conflicts are inevitable and mathematics evolve quickly.
A unified, final presentation seems impossible.

On Thur, Sep 18, 2008, Ronnie Brown wrote: 

> Something initially less ambitious like this
>might actually get done. Being electronic, it would be  seen as a `current',
>rather than `final account', and so would better reflect the way mathematics
>develops, in which a slight shift of emphasis, or notation (like --> for a
>function), can have profound consequences.

A partially unified evolving presentation of mathematics seems possible.
Today's littérature is already offering something like that!
But I find it cahotic and complicated.
But mathematics is naturally organised and simple!
The complexity of the litterature is often artificial.
Many proofs are complicated, simply because the author
ignores the abstract argument that could simplify everything.
Some statements are left unproved, and peremptory declared obvious
when they are not. The reader who cant see the obvious thing is terrorised.
He may as well quit mathematics. 

What can we do?

Let me submit a few ideas for disccusion.

Mathematics is naturally self-organised. 
The proof of most theorems can be broken in small steps of the form 
A_1,..,A_n --->B, where A_1,..,A_n is the list
of hypothesis and B is the conclusion. 
Each step may have a simple proof.
A complete proof maybe obtained by working
backward from the statement of the theorem to 
the axioms ot to known theorems. 

Everyone who knows a nice proof of a meaningful 
implication A_1,..,A_n --->B should write
a paper about it and put it in a special section of the arXiv (the NB section ?).

On Mon, Sep 19, 2008, Michael Spivack wrote:

>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.

On Tues, Sep 16, 2008, Andrej Bauer wrote:

>My opinion is that we have not yet found the right way to do
>"hive-science", but when we do, it will be a revolution. (A good start
>would be to get out of the hold that the evil publishers have on us.)

A special database for mathematics should be created
(but I dont know how).
Papers in the NB section of the arXves could be selected,
modified and organised with a system of references,
to give a partially unified presentation of mathematics. 
The same database could support different
presentations realised by different competing teams.  
Each team could work like a mathematical journal,
with an editor in chief and an editorial board.

What do you think?

Andre