From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3193 Path: news.gmane.org!not-for-mail From: Colin McLarty Newsgroups: gmane.science.mathematics.categories Subject: Re: WHY ARE WE CONCERNED? I Date: Thu, 30 Mar 2006 18:44:19 -0500 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019148 7651 80.91.229.2 (29 Apr 2009 15:32:28 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:32:28 +0000 (UTC) To: Categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Mar 30 20:53:28 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 30 Mar 2006 20:53:28 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FP7sf-00076E-3F for categories-list@mta.ca; Thu, 30 Mar 2006 20:52:49 -0400 Content-Language: en Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 139 Original-Lines: 32 Xref: news.gmane.org gmane.science.mathematics.categories:3193 Archived-At: > 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