Re: Explanations

[Ronnie Brown <ronnie.profbrown@btinternet.com>]

I put it in an email to Jean-Pierre Marquis that a proof is conducted in
a conceptual landscape: an analogy is that to describe the way to the
station we may say: go right, then left and turn right again at the
first traffic light. We do not describe the cracks in the pavement; but
we might say: beware of the road works.

Constructing (finding?) landscapes was Grothendieck's amazing
contribution, as suggested by Bill's comments.

I am told that computer assisted proofs have had some success, for
example in Boolean algebra (every Robbins algebra is Boolean.): we can
see that a computer can find its way through a maze, and analogous
things should be useful in mathematics generally. The problem seems to
be that the computer needs a way of prescribing the appropriate
conceptual landscape, analogously to the way we do mathematics, and
there is here a need for programming languages  which can manage the
construction of the variety of hedges in the appropriate high level
maze!  (see Bill's comments).  Even at a given level, it is easy to give
problems which are too hard to do by hand: I gave a course at Bangor in
which one of the exercises was to find a polynomial in x,y over the
reals which had at least 5 critical points, to classify them, and to
produce an illustration of the surface (using Grobner basis methods in
Maple).

There is also an aesthetic element: what do we mean by a good proof? My
own route to groupoids (and then higher groupoids), came about because I
was trying to write a book on topology in the 1960s including the
fundamental group and got fed up with having to get the fundamental
group of the circle by an entirely different method, which really needed
a development of its own.

Raoul Bott said that Grothendieck was amazing in that he was prepared to
work very hard to make things tautological: this was also by playing
with concepts and producing remarkable things. (I overheard this at the
1958 ICM in Edinburgh.)

But Grothendieck wrote to me saying that
"Throughout my whole life as a mathematician, the possibility of making
explicit, elegant computations has always come out by itself, as a
byproduct of a thorough conceptual understanding of what was going on. "

Ronnie

On 26/04/2011 14:45, William Messing wrote:
> I agree with what Andre wrote concerning proofs.
> Ronnie, you will certainly recall Grothendieck's letter to you in
> which he recalled that at the first Seminaire Cartan he was initially
> quite perplexed as to how the singular chain complex of a topological
> space, gigantic in size, could possibly lead to concrete computations
> and applications.  As he said, he soon realized that it is not the
> size that matters, but understanding things properly, that is, in the
> correct order or manner.  In the same letter he recalled that
> initially he was mystified as to how one would ever be able to make
> concrete calculations in etale cohomology, until, after, as he1 put
> it, several days of intense thought, he saw that understanding the
> cohomology of curves, with perhaps arbitrary constuctible torsion
> sheaves (torsion prime to the characteristic of the field over which
> one is working) was the key.
>
> Concerning proofs constructed by people as opposed to computer
> assisted proofs, many years ago Deligne remarked that while he did not
> believe in computer assisted proofs, he was not going to look for a
> counterexample to the proof of the four color theorem.
>
> Best regards,
>
> Bill Messing
>

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