From mumford@math.harvard.edu Wed Nov 23 09:39:37 1994 Received: from math.harvard.edu (tara.harvard.edu [128.103.28.11]) by antares.mcs.anl.gov (8.6.4/8.6.4) with SMTP id JAA21364 for ; Wed, 23 Nov 1994 09:39:35 -0600 Date: Wed, 23 Nov 94 10:41:22 EST From: mumford@math.harvard.edu (David Mumford) Message-Id: <9411231541.AA03670@math.harvard.edu> To: T.Forster@pmms.cam.ac.uk, owner-qed@mcs.anl.gov Subject: Re: Errors in Mathematics Cc: mumford In reply to: > Lyle says: > > 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. > > Isn't this what people call a selection effect? We don't > remember false proofs of falsehoods! > > Thomas > there are examples of the following kind: schools of mathematics which went slowly astray, starting with quite careful rigorous mathematicians and slowly acquiring bad habits, until patent falsehoods were published. Of course, the rest of the world, although at first they didn't notice it, became after a while more and more skeptical, until finally the mistakes were pointed out in print. The best known case is the Italian school of algebraic geometry, which produced extremely good and deep results for some 50 years, but then went to pieces. There are 3 key names here -- Castelnuovo, Enriques and Severi. C was earliest and was totally rigorous, a splendid mathematician. E came next and, as far as I know, never published anything that was false, though he openly acknowledged that some of his proofs didn't cover every possible case (there were often special highly singular cases which later turned out to be central to understanding a situation). He used to talk about posing "critical doubts". He had his own standards and was happy to reexamine a "proof" and make it more nearly complete. Unfortunately Severi, the last in the line, a fascist with a dictatorial temperament, really killed the whole school because, although he started off with brilliant and correct discoveries, later published books full of garbage (this was in the 30's and 40's). The rest of the world was uncertain what had been proven and what not. He gave a keynote speech at the first Int Congress after the war in 1950, but his mistakes were becoming clearer and clearer. It took the efforts of 2 great men, Zariski and Weil, to clean up the mess in the 40's and 50's although dredging this morass for its correct results continues occasionally to this day. David Mumford