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There is a story I read about tiling the plane with convex pentagons.

You can read about it in this article on pages 1 and 2.

Summary of the story: A guy showed in his doctorate work all classes of tiling the plane with convex pentagons, and proved that they are indeed all possible cases. Later, riddle was published in popular science magazine to find all these classes. One of the readers found a tiling that did not belong to any of the classes, and so the claim and the proof turned out to be wrong.

Reading this made me think about some questions.

Is it rare when a theorem was proved and the proof was published, and later it turned out that the theorem is wrong?

Can we somehow guess how many theorems out there that we think are right, but actually are wrong? I bet that if in our case, the theorem was about tiling in $R^3$ nobody would ever notice.

What can be the effect of such theorems on mathematics in general? Can it be a serious issue for mathematics even if the wrong theorems were not very important, but still, some stuff was based on them?

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Where in that article does it say that anyone gave an invalid "proof"? Reinhardt only implied that his list was complete, and although Kershner claimed that his list was complete, he gave no proof. – jwodder Dec 21 '11 at 0:08
This is pretty common I think. Also, usually mathematicians are more likely to have the right idea, and just not have the details of a correct proof. – AnonymousCoward Dec 21 '11 at 0:50
@jwodder You are correct. I am deeply sorry for I overlooked the first sentence in the second page: "he did not publish a proof of the assertion". Will be more careful in the future. – Artium Dec 23 '11 at 0:28
up vote 18 down vote accepted

It is not rare, but is uncommon, for false theorems to be published. It is somewhat more common, although still not frequent, for flawed proofs to be published, even though the theorem as stated is correct. One way to find these is to search for "correction", "corrigendum", or "retraction" on MathSciNet. Peer review can catch some of these errors, but mathematics is a human field in the end.

Every specialty has its own anecdotes about flawed proofs that are used to scare some caution into graduate students. For example, I would guess many logicians and analysts have heard how Lebesgue falsely claimed in print that every analytic set is Borel (in 1905).

In principle, the discovery of a flawed proof could mean that people have to re-check many other results. But in practice the effect is usually localized. Many researchers are cautious about using new results "blindly", once they realize that errors are not rare. In particular, being cautious means making sure you understand how to prove the results you use in your own proofs, whenever possible, so that nobody can later say your proof is flawed. By the time you are working on research papers, it becomes extremely unsatisfying to use a result of someone else as a "black box" without understanding it. On the other hand, it would be perfectly possible for a flawed proof to linger for a while if nobody else needs to use the result.

One of the roles of monographs and textbooks is to give another vetting to the theorems. A result that is proved in secondary books is somehow more reliable than a result that can only be cited to one research paper. This is another reason that errors tend to cause only localized problems, because by the time a result becomes standard in its field, a large number of mathematicians will have looked at it.

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Lebesgue falsely claimed in print that every analytic set is Borel ^ sounds interesting, any reference for this – FiniteA Dec 21 '11 at 2:21
@FiniteA: There is some more info at… . – Carl Mummert Dec 21 '11 at 3:00
@FiniteA: See also Souslin's note where the mistake was first pointed out (see what follows after Théorème V on page 91). [Thanks to J. M. for finding the link for me] – t.b. Dec 21 '11 at 3:23
@t.b. Thanks, that's very interesting. It also made me smile. Lebesgue's proof "must, therefore, be modified". – Carl Mummert Dec 21 '11 at 3:32

I think Carl Mummert's answer is spot-on. One other thing worth mentioning is that more often than being wrong, or even having a recognizably flawed proof, are papers which are just really hard to understand. (This is sometimes compounded by language barriers... more than once I've had to deal with papers that were written in Russian, then either never translated or translated incoherently.) Some of these proofs are not complete, others are complete but hard to understand, some are mostly correct but have a few errors, and so on...

So people generally take a cautious approach. One good piece of advice my PhD adviser in grad school told me, never use a result whose proof you don't understand. Some people rely on proofs that they don't understand but that are accepted and many people understand, so they figure it's an acceptable risk. Others (I'm one of them) won't use any proof they don't understand. This may be more doable in some fields than others. So while there are wrong theorems out there, the effect is minimized by people recognizing this fact and proceeding with caution.

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I think that for many graduate students "Never use a result whose proof you don't understand" is dangerously impractical advice. In fact, as a current adviser of PhD students that advice seems to be something of an abdication of advisorial responsibility: one of the important roles of an adviser is to strike a balance between requiring comprehensive mastery of the foundations of one's field and encouraging the student to go on to work on novel problems. Both extremes are poor solutions to this problem (although "take absolutely everything on faith" is worse, of course). – Pete L. Clark Dec 21 '11 at 23:21
Upon further reflection, it seems that we both agree that your adviser's advice is good for some and bad for others. This is undoubtedly true but not so useful to the random passersby. Can you zero in a bit more on why this advice was good for you? – Pete L. Clark Dec 21 '11 at 23:25
@PeteL.Clark I'm not in a field like algebraic number theory where this is nearly impossible. It actually isn't much of a burden for me to understand what I use, and not use the few results that I might want to use but whose proofs I don't understand. There just isn't a huge amount of background needed in my research. – Zarrax Dec 22 '11 at 2:39

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