[Update June 28, 2014: This post originally included stories about Poincare and Hilbert making some mistakes. At some point after posting this I realised how unfair it is to talk about somebody else’s mistake (even if it is Hilbert and Poincare) without giving precise references. Instead of deleting the stories, I’ll insert some comments where I think I am unfair. Sorry!]
I was recently forced to reflect on mistakes in mathematics. The reason was that my collaborators and I discovered a mistake in an old paper (16 years old), which forced us to make a significant revision to two of our papers.
A young student of mathematics may consider a paper which contains a mistake to be a complete disaster. (By “mistake” I don’t mean a gap – some step that is not sufficiently well justified (where “sufficiently well” can be a source of great controversy). By “mistake” I mean a false claim). But it turns out that mistakes are inevitable. A paper that contains a mistake is a terrible headache, indeed, but not a disaster.
Arveson once told me: “Everybody makes mistakes. And I mean EVERYBODY”. And he was right. There are two well known stories about Hilbert and Poincare which I’d like to repeat for the reader’s entertainment, and also to make myself feel better before telling you about the mistake my collaborators and I overlooked.
First story: [I think I first read the story about Hilbert in Rota’s “Ten Lesson’s I wish I had been Taught” (lesson 6)]: When a new set of Hilbert’s collected papers was prepared (for his birthday, the story tells), it was discovered that the papers were full of mistakes and could not be published as they were. A young and promising mathematician (Olga Taussky-Todd) worked for three years to correct (almost) all the mistakes. Finally, when the new volume of collected (and corrected) papers was presented to Hilbert, he did not notice any change. What is the moral here? One moral, I suppose, is that even Hilbert made mistakes (hence we are all allowed to). The second is that many mistakes — say, the type of mistakes Hilbert would make — are not fatal: if the mistakes are planted in healthy garden, they can be weeded out and replaced by true alternatives, often-times leaving the important corollaries intact.
[Update June 28: A reference to Rota’s “10 Lessons” is not good enough, and neither is reference to the Wikipedia article on Olga Taussky-Todd, which in turn references Rota’s “Indiscrete Thoughts”, where “10 Lessons” appear.]
Second story: actually two stories, about Poincare. Poincare made two very important mistakes! First mistake: in 1888 Poincare submitted a manuscript to Acta Mathematica – as part of a competition in honour of the King of Norway and Sweden – in which, among other things (for example inventing the field of dynamical systems), he claimed that the solutions of the 3-body problem (restricted to the plane) are stable (meaning roughly that the inhabitants of a solar system with a sun and two planets can rest assured that the planets in their solar system will continue orbiting more or less as they do forever, without collapsing to the sun or diverging to infinity). After winning the competition, and after the paper was published (and probably in part due to the assistant editor of Acta, Edvard Phragmen, asking Poincare for numerous clarifications during the editorial process), Poincare discovered that his manuscript had a serious error in it. Poincare corrected his mistake, inventing Chaos while he was at it.
[Update June 28: This story is well documented. I learned it from Donal Oshea’s book “The Poincare Conjecture: In Search of the Shape of the Universe” , but it is easy to find online references, too].
Second mistake: In 1900 Poincare claimed that if the homology of a compact 3 manifold is trivial, then it is homeomorphic to a sphere. He himself found out his mistake, and provided a counterexample. In order to show that his example is indeed a counter example he had to invent a new topological invariant: the fundamental group. He computed the fundamental group of his example and saw that it is different from the one of the sphere. But this led him to ask: if a closed manifold has a trivial fundamental group, must it be homeomorphic to the 3-sphere? This is known as the Poincare conjecture, of course, and the rest is history.
[Update June 28: Here I should have given a reference of where exactly Poincare claimed that trivial homology implies a space is a sphere. I don’t know it (it probably also appears in Oshea’s book)]
The moral here? I don’t know. But it is nice to add that after making his first mistake, Poincare and Mittag-Leffler (the editor) set a good example by recalling all published editions and replacing them with a new and correct version.
So that’s what I’ll try to imitate now.