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Prompted by this bit of news, http://www.wired.co.uk/article/fmri-bug-brain-scans-results where a bug in MRI software has the potential to nullify up to 40,000 published papers. Has anything analogous happened in mathematics? Obviously not on this scale, but was that ever a bug which, once discovered, caused a large number of follow-up results to become falsified as well?

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    $\begingroup$ Probably relevant: mathoverflow.net/questions/35468/… $\endgroup$ – Wojowu Jul 6 '16 at 17:29
  • $\begingroup$ Indeed relevant. I guess my question can be specialized further: (1) which proofs collapsed because they relied on other wrong proofs [but the claim wasn't necessarily wrong] (2) which results turned out to be wrong because they relied on false claims. I guess I had (2) in mind, mostly. $\endgroup$ – Aryeh Kontorovich Jul 6 '16 at 17:52
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    $\begingroup$ Wikipedia contains a long list of flawed math results, but actual retractions (recalls) of math papers are very rare, for reasons analysed in Errors and Corrections in Mathematics Literature $\endgroup$ – Carlo Beenakker Jul 6 '16 at 22:55
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    $\begingroup$ I've heard that there were quite a number of papers using properties of certain class of groups (whose name I've forgotten) -- folklore knowledge one could say -- but it turned out that nobody had an actual proof. It supposedly traced back to some Artin's students and their "proof" by authority: "My advisor said so." It didn't lead to retractions as it was unclear whether these properties hold or not. Rather, people sought alternative proofs without using these "Artin's lemmas". $\endgroup$ – Vít Tuček Jul 7 '16 at 17:56
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    $\begingroup$ I see several answers appearing, but has there even been a_single_instance of a math paper that was retracted because it relied on a wrong proof in the previous literature? As you can see here there have been various retracted papers because of an error in the paper itself, but I have not found a single case of a retraction because of an error by someone else. In particular, the errors in the "Italian school" literature mentioned by Gerald Edgar do not seem to have produced any retraction. $\endgroup$ – Carlo Beenakker Jul 8 '16 at 10:12
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One I have heard of is the Italian school of algebraic geometry, 1885 to 1935.

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I would risk to state that in mathematics such things (on large scale) do not happen:-) There are indeed some published and "accepted" false proofs, and false statements. The link given in the comment of Wojowu gives plenty of examples. However, I am sure that the list of results which depend on these incorrect results and thus are false will be very small.

The reason is that most mathematicians (at least most of those who prove significant results:-) usually take care to check what their proofs rely on. This is in the nature of mathematics: if you use a fact that you do not really understand, you do not understand what you are doing yourself.

I would state this as a bit ambitious principle: "You have to know the proofs of all results that you use". Of course this is not always so in practice, but this is an ideal, a goal. The result is that more some theorem is used, more times it is checked by various people. And the theorems which are really used much are checked so well, that one can be reasonably sure that they are correct.

I mean that none of the examples listed in the answers to Wojowu's question produced a chain of false results.

By the way, this is one reason why "computer-assisted proofs" are considered less satisfactory by many mathematicians than real "human" proofs.

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    $\begingroup$ We mathematicians have a huge advantage over other scientists in that our "laboratories" are largely our own heads: rechecking a result doesn't carry costs in time and use of equipment as it does other scientists. I think Feynman noticed that in psychology say, there doesn't seem to be much reward or incentive for going back and rechecking or duplicating old experiments; one instead has to publish a new study and just cite the old ones in support. $\endgroup$ – Todd Trimble Jul 7 '16 at 19:26
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    $\begingroup$ Interestingly, advocates of machine-checkable proofs will often regard real human proofs as less satisfactory than machine-checked proofs for the same reason---i.e., humans are more prone to error, at least when long complicated calculations are involved. $\endgroup$ – Timothy Chow Jul 7 '16 at 21:24

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