Re: Errors in Mathematics

David Mumford (
Wed, 23 Nov 94 10:41:22 EST

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