From owner-qed Mon Nov 21 11:56:30 1994 Received: from localhost (listserv@localhost) by antares.mcs.anl.gov (8.6.4/8.6.4) id LAA21961 for qed-out; Mon, 21 Nov 1994 11:53:37 -0600 Received: from cs.albany.edu (root@cs.albany.edu [128.204.2.7]) by antares.mcs.anl.gov (8.6.4/8.6.4) with ESMTP id LAA21956 for ; Mon, 21 Nov 1994 11:53:29 -0600 Received: from ramanujan.cs.albany.edu (mosh@ramanujan.cs.albany.edu [128.204.2.57]) by cs.albany.edu (8.6.9/HUB01) with ESMTP id MAA09771; Mon, 21 Nov 1994 12:52:40 -0500 From: Mohsin Ahmed Received: (mosh@localhost) by ramanujan.cs.albany.edu (8.6.9/CLI) id MAA16442; Mon, 21 Nov 1994 12:52:35 -0500 Date: Mon, 21 Nov 1994 12:52:35 -0500 Message-Id: <199411211752.MAA16442@ramanujan.cs.albany.edu> To: LYBRHED@delphi.com CC: qed@mcs.anl.gov In-reply-to: <01HJQ1HO7NH89GY4ZQ@delphi.com> (message from Lyle Burkhead on Mon, 21 Nov 1994 01:18:04 -0500 (EST)) Subject: Re: Errors in Mathematics Reply-to: mosh@cs.albany.edu Sender: owner-qed@mcs.anl.gov Precedence: bulk From: Lyle Burkhead Subject: Errors in Mathematics About false theorems: we still don't have any. The examples given were all correct results with incorrect or incomplete proofs. The Four Color Theorem, the Hard Lefschetz Theorem, and Dehn's Lemma all turned out to be true. Not only that, the erroneous proofs were noticed by human mathematicians, not by automated reasoning systems. I have never encountered a false theorem that was used as the foundation for other theorems, with disastrous results. And I don't think anybody else has, either. There are many imperfections in the mathematical literature, and some incomplete proofs, but I don't think there are any substantive errors that affect the integrity of mathematics. To repeat, Euclid's parallel postulate CANNOT be proved, yet we had many (false) proofs of it for 2000 years. The whole of classical mechanics is based on the assumption our space is euclidean (as opposed to hyperbolic), because no other consistent geometry was imaginable at that time. I agree with you that the results were not disastrous, except in the theory of relativity. Of course, this doesn't affect the integrity of mathematics, it only effects the integrity of our ideas of space. You are also right in saying most of these errors were noticed by humans, and not machines. This is because logical errors or missing cases in proofs easy to spot, and belong to domain of computer checking. What is difficult to spot is wrong use of definitions and concepts - spotting these proofs leads to advancement of mathematics. Such errors become apparent as soon as you try to formalize a proof for computer verification. Considering this, it seems computer aided verification of chips and algorithms would be a immediate use of proof systems, rather than verifying pure mathematics. -- - Mohsin.