Subject: categories: Re: WHY ARE WE CONCERNED? I

From: jim stasheff <[email-masked]>
Date: Fri, 31 Mar 2006 09:30:58 -0500

Proofs may be of ultimate importance
but a lot can be accomplished at the penulitmate level
or even sooner

jim

Colin McLarty wrote:
>> There's a saying about Lefschetz that he "never wrote a valid
>> proof, and never made a false conjecture". Now it's not an attitude
>> that want to encourage, but if you have great mathematicians who
>> are like that (and Lefschetz was not just a good mathematician, but
>> a great mathematician, without whom a good deal of modern algebraic
>> geometry would be unimaginable), then this ought to tell us something.
>
>
> This, and much else about Lefschetz has to tell us a lot.  As to proof,
> Lefschetz also never published a theorem without a purported proof, and
> he often came to feel very strongly that his proofs were not good
> enough.  He wrote two long books on topology in the attempt to repair
> the bad proofs in his influential booklet on cohomology in algebraic
> topology, L'Analysis situs et la Topologie Algebrique.  It was so
> important to him that he enlisted many others.  Notably for us, he
> asked Eilenberg and Mac Lane to contribute an appendix to his 1942
> TOPOLOGY.  This was their first published collaboration "On homology
> groups of infinite complexes and compacta" and pursued the questions
> that quickly led to category theory.
>
> Lefschetz had encouraged work on solving specific problems just over
> the edge of what well-understood foundations for homology could
> handle.  Apparently he believed such solutions would lead to
> significantly deeper understanding.  He had encouraged Steenrod to work
> on p-adic solenoids because existing methods did not seem adequate to
> it.  But whatever his motive, he was determined to see rigorous
> solutions to quite specific problems.
>
> Colin
>
>