Concerning the debate on how often false theorems are asserted
in real-life mathematics, I showed some of the qed correspondence
to my colleague Barry Mazur and he gave me a very thoughtful response.
I think this may help the ongoing discussion and give some insight, esp.
into the odd ways in which the social nature of mathematics prevents
false things from spreading too widely.
----- Begin Included Message -----
>From mazur Wed Nov 23 13:14:52 1994
Date: Wed, 23 Nov 94 13:14:50 EST
From: mazur (Barry Mazur)
Return-Receipt-To: mazur (Barry Mazur)
To: mumford
Subject: false proofs
Content-Length: 3544
X-Lines: 91
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