Re: WHY ARE WE CONCERNED? I

[Colin McLarty <colin.mclarty@case.edu>]

> 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