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A typo, a spelling error etc., in a published article, is definitely not enough for issuing an erratum.

If a mistake destroys a main result, then an erratum is definitely necessary, and the proof should be rewritten, if it's possible to fix.

What about cases in between? What if a small lemma, remark is not correctly stated? What about a mistake in the proof that is actually very easy to fix? What about a wrong number in the calculation? What if ... (you name the case) ...?

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    $\begingroup$ Should this be moved to academia.stackexchange.com or is it to be expected that answers will be specific for mathematics? $\endgroup$
    – Dirk
    Commented Nov 20, 2015 at 12:15
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    $\begingroup$ @Dirk more specific for mathematics, in the sense that I'm not talking about data, experiment and so on, I'm talking about proofs and statements. $\endgroup$
    – Hao Chen
    Commented Nov 20, 2015 at 12:18
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    $\begingroup$ Mathematicians are generally very shy at admitting mistakes and publishing errata. I also think that there is a very big difference between the publishing culture of mathematicians and the publishing culture in other scientific disciplines. So migrating this question to academia.stackexchange.com is probably not such a great idea. $\endgroup$
    – André Henriques
    Commented Nov 20, 2015 at 12:18
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    $\begingroup$ Jordan Ellenberg wrote a fascinating blog post about an error that was found in one of his preprints: quomodocumque.wordpress.com/2013/11/23/… He was rightly commended by many of his readers for the way he dealt with this error. The post, and the comments that follow, is well worth a read. $\endgroup$
    – Nick Gill
    Commented Nov 20, 2015 at 15:37
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    $\begingroup$ @AndréHenriques Academia has a tag mathematics "On standards or conventions specific to mathematics as an academic discipline, and programs that lead to a degree in this field." with more than 500 questions. So it is not quite clear to me why it would be "not such a great idea" to migrate it there. Anyway it is too late now. But for future occasions it could be relevant. $\endgroup$
    – user9072
    Commented Nov 20, 2015 at 17:58

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The evidence on arXiv does not seem to support the assertion that mathematicians are significantly more reluctant to issue an erratum than, say, physicists:

The factor of two in the number of errata between physics and mathematics correlates well with the total number of arXiv submissions (about three times as many in physics than in mathematics). Incidentally, only a very small percentage of arXiv submissions are errata (the total number math articles is about 150,000).

I do notice that many of the mathematics errata are limited to the arXiv version, and not incorporated in the journal. The idea being, I would think, that issuing an erratum for a relatively minor issue is a service to the reader, who will likely consult the arXiv version and will therefore be alerted to the error. This would obviously be unacceptable for an error that invalidates the entire work, but those happen more rarely.

In this connection I find it interesting to note that mathematicians withdraw more papers from the arXiv than physicists, 128 versus 90 in the last year. I presume this says something about the different consequences of a small mistake in the two disciplines.

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    $\begingroup$ +1 for arXiv. Do not forget to point out that the arXiv version is corrected in the comment field, or else many readers will think that the published version is the more up to date. $\endgroup$
    – darij grinberg
    Commented Nov 20, 2015 at 14:03
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    $\begingroup$ There was an article in the Notices a couple of years ago on this topic. ams.org/notices/201304/rnoti-p418.pdf Among other things, the article said, "There is no reason to think engineers and mathematicians make mistakes less often than other researchers, while there is some evidence that the research cultures of these fields discourage publishing corrections." Neal Koblitz, however, has a different opinion. ams.org/notices/201307/rnoti-p839.pdf $\endgroup$
    – Timothy Chow
    Commented Nov 20, 2015 at 22:40
  • $\begingroup$ In order to compare two fields (such as math and physics), instead of absolute numbers, it might be a better idea to give the percentage of arxiv submissions which are errata. $\endgroup$
    – André Henriques
    Commented Nov 24, 2015 at 9:38
  • $\begingroup$ @AndréHenriques --- good point, I added this info $\endgroup$
    – Carlo Beenakker
    Commented Nov 24, 2015 at 9:58
  • $\begingroup$ "Errata in Mathematics" – sounds like a good title for a new journal.... $\endgroup$
    – Gerry Myerson
    Commented Nov 30, 2015 at 3:19
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It happened to me, too. On my thesis. Published in Annals of Math. I got a note from A. Fröhlich saying that in a seminar in London, they hadn’t been able to verify an involved inequality-computation in a proof. Indeed, my computation was completely wrong. Fortunately, the lemma wasn’t essential to the whole except for producing useful counterexamples. What was most embarrassing was that if I had tried to pull the wool over the reader’s eyes, I couldn’t have chosen a better method of deception. I sent in a one-page correction with a replacement-statement that was rather weaker, but still good for the counterexamples. A few years later, I found a correct proof of the original statement.

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I can't speak for everyone, but since you ask for examples, I've published "errata" at least a couple of times. Once was a paper (in Inventionnes, quite embarrassing) where the statement of the main theorem was false due to a mistake in the proof! Luckily, a small modification of the proof gave a weaker, but still interesting, true result. This comes under your heading of "If a mistake destroys a main result,..." The other situation, which you didn't mention, was a case where one of the main results of a paper was an immediate consequence of an earlier paper that we had not known about. After our paper was in press, someone pointed this out, so we submitted an "addendum".

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Checking MathSciNet in the five-year period from 2010 to 2014, there were 1319 entries with "erratum" or "errata" in the title. Sampling a few dozen of them, most cases involve a misstatement severe enough to affect the statement of a theorem. In a few cases, something was inadvertently left out, such as MATLAB code or an author's name. For perspective, during this period, there were 525795 items added to MathSciNet.

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