From mumford@math.harvard.edu Wed Nov 23 12:44:12 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 MAA24501 for ; Wed, 23 Nov 1994 12:44:10 -0600 Date: Wed, 23 Nov 94 13:47:39 EST From: mumford@math.harvard.edu (David Mumford) Message-Id: <9411231847.AA03777@math.harvard.edu> To: owner-qed@mcs.anl.gov Subject: False theorems in print? I asked Barry Mazur whether he knew any false theorems on which further work was built (sending him some of the correspondence between Forster and Burkhead) and he sent me a long discussion which may be of interest concerning what happens to false results in the math. community: ----- Begin Included Message ----- Dear David, First a comment on the statement at the end of the e-mail that you forwarded ("Isn't this what people call a selection effect? We don't remember false proofs of false theorems"). My impulse is to say that this comment isn't relevant here because of two things:-- 1. What is asked for is not the isolated false proof or false theorem. We have any number of examples of that. What is asked for, if I understand it, is a false theorem that has been made some use of (i.e., a body of work that must be either rejected or put on hold once the false step is discovered). Now you would think that it would be particularly difficult to not remember a whole theory that went under, by the discovery of a false step in its foundations. 2. We hardly remember anything, false or true. What is taught as important mathematics is such a fine distillation of the vast totality of mathematics that has been done, and we feel utterly justified in ignoring so much (good!) mathematics, that of course our ignorance of the bad, or incorrect, mathematics of our predecessors goes unnoticed. Next, let me try my hand at a bunch of different types of false things-- 1.Yes: All the true theorems that had incorrect, or gappy proofs (which include the Zariski theorem on the fundamental group of the complement of plane curves with nodal singularities, based on Severi, Dehn's lemma, Strong Lefschetz) but which were eventually legitimized. 2. Big incorrect theorems which were vastly publicized but for which one would have to be something of a historian to determine whether or not they were seriously used in any way, so that later results depended upon them. This includes a number of theorems of Poincare (Danny Goroff has just written a long article on the homoclinic point story) and, to my mind would also include Hilbert's "proof" of (was it?) the continuum hypothesis. At least I think that is what is occurring in the second part of one of his essays ("On the infinite") but I am very vague on the story there (e.g., was it ever taken seriously?) 3. Straight errors in articles. These abound of course! There seems to be a self-rectifying principle here, though. Often errors occur exactly where (and because) the text of the article is turgid. Insofar as it IS turgid, the text often is not read, ignored. Here is a curious instance: Joe Harris and I are interested in the condition for a smooth cubic fourfold to be rational. There is a paper published by someone named Morin claiming that they all are. Joe and I realized (immediately) that the paper must be wrong, and a student of Joe's actually found the error. Now for the curious thing: there is an article published by Fano,which appeared after Morin's article appeared. Fano's article is completely correct (as far as we have figured out) and yet Fano quotes often from Morin's article-- BUT Fano only quotes from the correct portion of Morin's article (1) Fano seems to have thoroughly avoided any contact with exactly that part of Morin's article that has the error. I should also say that errors sometimes occur when the author feels that he must say something about a specific mathematical point, but is, in fact, uninterested in the point he himself is making; so he is less than diligent in checking his assertion, and thereby hurls himself, unnecessarily, into an error. But these errors don't mess up too much theory, they just serve to embarrass the author... I guess this doesn't help too much, Barry ----- End Included Message -----