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Paper A is in the literature, and has been for more than a decade. An error is discovered in paper A and is substantial in that many details are affected, although certain fundamental properties claimed by the theorems are not. (As a poor analogue, it would be like showing that certain solutions to the Navier-Stokes equations had different local properties than what were claimed, but that the global properties were not affected. The error is not of the same caliber as Russell's correction of Frege's work in logic.) The author is notified, who kindly acknowledges the error.

Now what?

Should the remaining action lie fully on the author, or should the discoverer of the error do more, such as contact the journal, or publish his own correction to paper? How long should one wait before suitable action is taken? And what would be suitable action if not done by the author?

Based on remarks from those who previewed this question on meta.mathoverflow, I propose the following

Taxonomy: There are various kinds of error that could be considered.

  • typographical - An error where a change of a character or a word would render the portion of the paper correct. In some cases, the context will provide enough redundancy that the error can be easily fixed by the reader. Addressing these errors by errata lists and other means have their importance, but handling those properly is meant for another question.

  • slip - (This version is slightly different from the source; cf the discussion on meta for the source http://tea.mathoverflow.net/discussion/493/how-do-i-fix-someones-published-error/ ) This is an error in a proof which may be corrected, although not obviously so. In a slip, the claimed main theorem is either true or can be rescued with little cost. In my opinion, the degree of response is proportional to the amount of effort needed to fix it (and is often minor), but there may be slips major enough to warrant the questions above.

  • miscalculation - Often a sign or quantity error. In some cases the results are minor, and lead to better or worse results depending on the calculation. I've included some miscalculations in some of my work to see if anyone would catch them. I've also prepared a response which shows the right calculation and still supports the main claims of the work. (See below on impact as a factor.)

  • oversight or omission - This is stating a fact as true without sufficient folklore to back up that fact. In some cases the author doesn't include the backup to ease (the reading of) the paper and because the author thinks the audience can provide such backup. More seriously, the omission occurs because the author thought the fact was true and that there was an easy proof, when actually the fact may or may not be a fact and the author actually had a faulty argument leading him to think it true.

  • major blunder - This is claiming a result which is true, and turns out not to be true in a socially accepted proof system. Proofs of Euclid's fifth postulate from the other four fall into this type.

The above taxonomy is suggested to help determine the type of response to be made by the discoverer. Also, degree of severity is probably not capable of objective measure, but that doesn't stop one from trying. However, there are at least two other considerations:

  • Degree to which other theorems (even from other papers) depend on the error in the result. I call this impact.

  • Degree to which the error is known in the community.

The case that inspired this question falls, in my mind, into the category of a miscalculation that invalidates a proposition and several results in paper A following from the proposition. However, as I alluded to above in the Navier-Stokes analogy, the corrected results have the same character as the erroneous results. I would walk on a bridge that was built using the general characteristics of the results, and not walk on a bridge that needed the specific results. In this case, I do not know to what degree impact the miscalculation has on other papers, nor how well known this miscalculation is in the community.

If someone thinks they know what area of mathematics my case lies (and are sufficiently experienced in the area), and they are willing to keep information confidential, I am willing to provide more detail in private. Otherwise, in your responses, I ask that no confidentiality be broken, and that no names be used unless to cite instances that are already well-enough known that revealing the names here will do no harm. Also, please include some idea of the three factors listed above (error type, impact on other results, community awareness), as well as other contributing factors.

This feels like a community-wiki question. Please, one response/case per answer. And do no harm.

Motivation: Why do I care about fixing someone else's error? Partly, it adds to my sense of self-worth that I made a contribution, even if the contribution has no originality.
Partly, I want to make sure that no one suffers from the mistake. Partly, I want to bring attention to that area of mathematics and encourage others to contribute. Mostly though, it just makes an empty feeling when one reaches the "Now What?" stage mentioned above. Feel free to include emotional impact, muted sufficiently for civil discourse.

Gerhard "Ask Me About System Design" Paseman, 2010.07.10

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Since some of the discussion has already happened in meta, it would be nice if those who posted relevant interesting stuff there re-post it in here. –  danseetea Jul 10 '10 at 23:31
I've been convinced for quite some time that the mathematical community needs to set up some sort of wiki-style error database to keep track of everything from typos to major errors. But that aside, I'm somewhat concerned about the professional ethics of the practice mentioned here of intentionally leaving miscalculations in published papers just "to see if anyone would catch them". –  Greg Friedman Jul 11 '10 at 6:09
To Greg Friedman: I apologize if I misled you; all the examples of the miscalculations I have are on posted personal web pages. Also, the miscalculations have little impact on the results. I might encourage the practice for the review/referee process, but not for published work. If I were to publish them in a peer-reviewed journal, my stance would be different. In fact, I have written to ask that others who have quoted my work get the calculations correct in print, so far without success, but fortunately with little impact. Gerhard "Ask Me About System Design" Paseman, 2010.07.10 –  Gerhard Paseman Jul 11 '10 at 6:27
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14 Answers

Some advice explicitly directed at less senior people. I would very much advise some who does not yet have tenure to NOT take the nuclear option (e.g. posting a paper on the arXiv accusing someone of being wrong, or writing irate letters to the editors of a journal). In the extremely rare cases in which this has to be done, it is best done by someone who is both pretty senior and very politically skilled. This leads me to my other piece of advice. Namely, talk to other, more senior people in your research area. First, they might be able to convince you that it isn't really as serious an error as you think. Second, they will probably know the personalities involved better, and be more effective at convincing an author to do the right thing if something has to be done.

The two times something like has happened to me, I had ended up proving stronger results than the erroneous papers by pretty different techniques. I buried remarks at the ends of the introductions of my papers mentioning the wrong papers and explaining where they went wrong. On one of those occasions the author had left math and I didn't know how to contact him, so I didn't correspond with him first (after I posted the paper the arXiv, one of his friends contacted him and we exchanged some friendly emails). The other time, I explicitly cleared the language I used with the original author.

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Talking to more senior people is a very good advice, in fact, regardless of the kind of ethical problem one have. The "nuclear option" takes quite a bit of chutzpah and is never a good way to make friends. But sometimes it can be a way to get noticed and have people pay attention to this person's work. Again, I am not advocating it at all, but I do know of a couple of careers which got started that way. In math, being honest and correct is sometimes more important than being politically astute... –  Igor Pak Jul 11 '10 at 4:01
Some people do boneheaded things and have them work out, but still, to any young people contemplating the nuclear option I must say : step away from that ledge! In 99% of cases, it ends badly. And even if it ends "well", you've still probably made an enemy. –  Andy Putman Jul 11 '10 at 4:11
+1: the advice to consult senior people is absolutely key. (This was part of the eventual resolution of the story whose beginning and middle I told in my answer.) If you are actually in the right, it shouldn't be hard to enlist senior allies. From a hard-nosed perspective, the point is that the editor-in-chief of the publishing journal is going to have a harder time ignoring your very senior colleague than s/he is in ignoring you. –  Pete L. Clark Jul 11 '10 at 4:15
While the nuclear option is not the best way (particularly not the "write to the editor" version, with its totalitarian "everyone must hear my opinion" undertone), I can well imagine people being yet more unsatisfied at somebody taking the "consult senior people" path. These "senior people" would usually mean colleagues of the flawed paper's author, and in the author's position, I would definitely prefer direct and honest confrontation to people discussing me and my mistakes in private with my colleagues - particularly in mathematics where people are sure everyone is scheming against them. –  darij grinberg Jul 12 '10 at 9:22
#darji : I'm certainly not suggesting that you should not correspond with the authors of a paper you're having trouble understanding (though I think it is a good idea to be extra polite -- for instance, instead of saying "Theorem X is wrong", say something like "I'm having some trouble understanding Theorem X. Isn't it true that Y"). However, if you feel that they are acting unethical in some way, then it is unwise to try to deal with it on your own. That's what mentors (for instance, PhD and postdoc advisors; I still consult my PhD advisor on a regular basis) are for. –  Andy Putman Jul 12 '10 at 15:31
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I think the question is a bit too detailed. A short version is this: what do people do when they discover an error in other people's papers? Obviously, like the question explains, there is no universal rule - this all depends on the type of an error, relative importance of the results in the paper, relationship between a person who made an error (let's call her/him X) and who discovered it (Y), etc. Let me simply list some relatively standard options.

1) Y tells the error to X. X finds a way to fix it, publishes an "erratum" in the journal, on the arXiv and/or on his/her own webpage. Gives profuse thanks to Y (but only if Y is gives a permission to do so). Occasionally this a joint (X,Y) paper. Either way, this is the most desirable outcome.

1)' Even if the result is false in full generality, X should still publish an "erratum" saying "such-and-such weaker version survives", or even "every hope to prove such-and-such is lost forever"...

2) Y wants to remain anonymous, or X can't be bothered. Then Y writes a letter to editor in chief of the journal which published X's paper. It is their responsibility as much as X's. Let the editor(s) deal with the mess. This is the easiest way out (for Y).

2)' A slightly better way to remain anonymous is for Y (by agreement with the editors) publish a short erratum under an assumed name. I have seen this happen, but in a long run this does not work - eventually people find out who was the author (and in a couple of cases I know, MathSciNet rather counter productively links the pen name to Y). On the other hand, if you really want to remain anonymous, e.g. use an assumed name linked to a fake email account, your erratum submission will not be taken seriously (journals get quite a few crackpot submissions).

3) Y is heavily involved in the field and is writing an article/book (B) on the subject. Y doesn't know how to fix the error. Then sometimes it is a good idea to include this piece of math in the final remarks or an appendix. Y might want to be nice and inform X first, before making the error public. This is a good solid option which allows others to say "error in A was pointed out in B".

4) The error is fundamental, kills paper A, but Y knows how to fix it. Y should publish a new paper explaining the error in full, right in the introduction or the first section. Y should write the paper in such a way as if assuming that X will be refereeing the paper... On rare occasions, it can happen that later Z publishes a paper acknowledging an error in Y's paper, and claiming to have "finally" found "a definite proof", etc. Sometimes an unavoidable chaos ensues, but the good faith decision by Y to publish was still a good one.

5) Y can prove (by different means) a result which follows easily (or even a special case) of that by X. Y should still write a paper. Lots of delicacy is required in trying to explain the whole story. This is the hardest thing to do. Consult a senior expert before making the paper available.

6) In extreme cases, Y can just post a note on the arXiv (this happens occasionally, see the meta discussion), but let me strongly discourage this practice. It should only be used when no other recourse is available. When this kind of thing happens, the allegedly erroneous paper A is refereed, but the erratum is not, so the ousiders no longer know what to think. This can undermine the credibility of the field and turn people away from the working on the problem.

UPDATE: After reading other answers, I realized that I am answering a slightly different question. This is only meant to catalog the possibilities, not to endorse them or explain "how to get there". The latter is often really delicate and difficult, so don't try it if you are not sure! Although some of these outcomes are preferable to others, this is also on case-by-case. Finally, the order is somewhat arbitrary.

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@Igor: 2)' sounds very weird to me, and I have not (knowingly) seen an instance of this. Would you be willing to provide an example? –  Pete L. Clark Jul 11 '10 at 0:43
Thank you for the options. Can you give a relative time scale for some of them? I'd especially like to know about time to wait on X before Y does one of 4,5, or 6. Gerhard "Ask Me About System Design" Paseman, 2010.07.10 –  Gerhard Paseman Jul 11 '10 at 0:45
@Pete: 2)' is both bad and rare, I would like to discourage that and rather not name names (defeating the purpose). As for the MathSciNet linking - this is routine. Try "A. Nilli" for a well known example (of linking not an erratum). –  Igor Pak Jul 11 '10 at 1:02
@Igor: I don't quite understand 5, but I am assuming you are ordering them in terms of what is more socially acceptable. In that case, is there really no middle ground between 3 and 4? 4 sounds already quite extreme to me, and is bound to ruffle a few feathers. –  Willie Wong Jul 11 '10 at 1:02
@Gerhard: there are no rules on this. I recommend being proactive and giving a reasonable deadline to the author (X), something comparable to the amount of time it takes to referee a paper on this subject (this can vary between fields). If the deadline is missed, it's not your fault. But be nice - as Pete writes below, X is already very upset... –  Igor Pak Jul 11 '10 at 1:06
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up vote 15 down vote accepted

UPDATE 07.24 : The set of answers for this question seem to have stabilized. I encourage all who visit this question to review all of the answers and comments posted here and posted behind the meta.mathoverflow link in the question. This answer has an incomplete summary; you might find what you need in one of the other posts.

Thanks to all who have contributed thus far.

    I liked Igor Pak's notion of giving the author the same amount of time as a referee to have the author do the fix on his own. I also liked his list of potential responses, including alternatives to avoid or use as a last resort.
    I appreciated Pete Clark who pointed out that the emotional impact on the author may be considerable.
    I thought algori's notion of substantial error (one that could not be fixed with the methods used in the paper) was a good benchmark regarding severity of error.
    I and others liked Andy Putman's advice to seek out, shall we say, more experienced counsel before acting.
    I thank Timothy Chow for offering an alternative (doing the work for the author of publishing the error) that may well fit my situation.
    I also thank Mike Shulman for his notion of wikifying the correction; perhaps authors who have been so corrected could weigh in on this so that we could determine what social/emotional/academic impact this method might have.
    I thank Daniel Moskovich for his inspiration to move this society toward a perfect world, at least with respect to correcting errors in papers.
    Also, I want to acknowledge the common sense in unknown(yahoo)'s suggestion to continue discussion with the author.

Based on the input so far, I am going to suggest the following as an answer template, to be modified at the dictates of common sense, decency, and situational factors. Recall the assumption that the author has been contacted already and acknowledges the error.

  1. Consult with one or more colleagues in the field who can evaluate the error and suggest a course of action. If they suggest dropping the matter, then stop.

  2. See if the journal involved has already published a correction. If so, then stop.

  3. Contact the author again after a period of time (3 to 6 months) and ask what the author thinks is an appropriate action to take. Offer to assist in writing up a correction, at little or no cost to the author. If the author suggests a reasonable course of action, follow it. Then stop.

  4. Prepare your own version of the correction. If the author has not acted in good faith, and if the colleagues encourage the idea, mail the author a copy of the correction as well as a stated intention to post the correction in 3 to 6 months if the author has problems following up with providing his own correction. Keep the correction for your files.

  5. If a year has passed since the acknowledgment, and several months have passed since you announced your intention of posting the correction, then (given that it is a good thing to do) post an announcement saying what is being corrected, and provide a link to the details.

In the above, do no harm. In particular, approach the situation with the attitude that, regardless of how poorly the author might respond, the goal is to provide a correction to the academic audience, with as much or more sensitivity and respect due to the author as you would expect for yourself.

It is possible a better answer exists out there. If someone can provide it or a link to it, I will acknowledge it. If this answer gets a substantial number of votes from the community, then I will accept it, with the understanding that the other posters contributed to this answer. In any case, I believe this question and all the answers will serve as a helpful resource to those who find themselves close to this situation.

Gerhard "Don't Need No Stinkin' Badges" Paseman, 2010.07.12

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Although there are many different cases to consider, in all of them I think Step 1 is the same: write to the author of Paper A. Your message should convey the sentiment that you believe you have found the following specific mistake(s) in Paper A. Does the author agree?

Among all choices of phrasing which unambiguously convey this sentiment, you should strive to find the one which is maximally polite, respectful and non-confrontational. The tone of your first message will play a large role in determining whether the author responds at all and, if so, the nature of your subsequent correspondence. A past experience of mine amounts to conducting an experiment in this regard: a colleague of mine had pointed out a (fatal) error in Paper A, but the author felt attacked and responded but didn't really engage my colleague mathematically. This went on for a while -- frustratingly, to my colleague -- and culminated in an amazing "J'accuse!" moment at a big math conference -- unfortunately I was in "the wrong" special session at the time so missed out on seeing it with my own eyes by about 50 meters, but I met the author at the same conference, read Paper A, and eventually came to the same conclusions as my colleague. I wrote to the author as nicely as I possibly could, and the response was markedly better than the one my colleague had gotten. That's not the end of the story by any means, but it illustrates my point.

Put yourself in the author's shoes: under any circumstances, it sucks to receive a "your paper is wrong" message. I think that at least 90% of the time the author will not believe it at first, so that some collegial back-and-forth will be required. (When these kind of messages get sent to me, my first response is almost invariably an explanation of why I am correct, whether or not that's the final verdict.)

I think the biggest branching point in the tree of all possible responses is: does the author privately agree with you that there is a mistake? If you can't get to that point, the whole affair becomes much tougher and more unpleasant.

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Sorry, I see now that I missed the point that in your case you are beginning with the assumption that the author has privately acknowledged the error. This makes my answer off-point. Nevertheless others will be in this situation starting at the beginning, so I will leave my answer up in the hope that it may be useful to them. –  Pete L. Clark Jul 11 '10 at 0:50
Pete: I think most of us are more curious about what to do if and after "the whole affair becomes much tougher and more unpleasant". Do you have any words of wisdom? –  Willie Wong Jul 11 '10 at 0:58
Even if you don't answer the question as asked, you may have other useful information to provide. Although I did not emphasize it in my question, I am especially interested in how long to wait on the author before taking additional action. Care to comment on this aspect? Gerhard "Ask Me About System Design" Paseman, 2010.07.10 –  Gerhard Paseman Jul 11 '10 at 1:05
@WW: Not off the top of my head, I'm afraid. I'll let you know if I come up with any... –  Pete L. Clark Jul 11 '10 at 5:15
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No-one's made a similar comment yet, so I'll add my 2 cents. I think that

  • Degree to which the error is known in the community.

Should, ideally, have no effect on the original author's actions.

I say this as someone who quite likes to move between different (sub-)subject areas, and who's a bit anti-social, and who picks up old problems etc. etc. In other words, I'm often trying to learn things directly from papers, without much help from a "community". It would be incredibly frustrating to find an error, to work at correcting it and/or understanding a work-around, only to find out that it was "well-known" to those in the know ten years ago. Part of the beauty of mathematics is that it is, to some extent, eternal, and an article can remain accessible to others for many, many years. So, please, think of this when weighing up whether to make corrections. (My own principle is to keep a list on my webpage of minor corrections: I've also, sadly, had to issue some proper errata...)

(I should say that actually the above scenario has, thankfully, not happened to me. But an analogous issue-- that of preprints circulating in the community and being referenced etc. without ever being published-- is a bugbear of mine).

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I apologize for posting before reading the long discussion; I did a cursory check however and I noticed that Ted Hill has not yet been mentioned. His text How to Publish Counterexamples in 1 2 3 Easy Steps is a first-hand account of dealing with submitting corrections to authors and editors in a rather high-profile case. The facts of the case are intricate enough that one wants to remain circumspect in one's conclusions, but one thing is overwhelmingly clear: the process is not for the fainthearted.

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1+. Very interesting story. And Ted Hill's patience is impressive. –  Martin Brandenburg Dec 16 '10 at 11:41
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I was once in a situation similar to yours. The author acknowledged the error and in principle agreed to publish an erratum, but it gradually became clear that the author would probably never get around to doing so. I eventually resolved the issue by offering to act as the author's secretary, TeXing up the erratum myself but putting only the author's name on it. The author accepted my offer and everything went according to plan. The published erratum bears no sign of my involvement and it's conceivable to me that the journal to this day thinks that I really was just the author's secretary and not another mathematician. In my mind this was nearly a perfect solution. Of course I got no credit but I didn't want any.

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Thank you. Can you tell me how long you waited between the initial contact regarding the error and when you made the offer to act as secretary? Gerhard "Ask Me About System Design" Paseman, 2010.07.10 –  Gerhard Paseman Jul 11 '10 at 3:02
The answer will not be very helpful to you because in my case there were special circumstances that I prefer not to explain. You'll just have to use your judgment I'm afraid. –  Timothy Chow Jul 11 '10 at 4:47
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First of all one should agree on what a substantial error is. Of course, if some or all theorems in a paper turn out to be wrong, this is pretty substantial. But it can also happen that the main results are true as stated, but the proofs are incorrect or incomplete, and then one is confronted with the problem: where to draw the line between substantial and not. I am not proposing a solution to his problem. In fact, I don't think there can be any simple solution: this should be considered on a case by case basis. However, in my opinion the guideline should be: an error or a gap in a proof is substantial when it can't be corrected with the methods used in the paper. This may sound vague but in some cases I'm aware of it was clear what this means.

Now supposing an error has been found, it is for the author to take steps such as publish an erratum or to update the paper on the arxiv or to put a note on the arxiv explaining what is wrong. Now the really difficult case is when the author acknowledges the error but flatly refuses to anything about it. (I'm aware of a couple of such cases; no names will be named.) In this case again there is no universal recipe, I believe, but continue negotiating with the author, editor etc. As a last resort, if everything else should fail, I guess one should put a note on the arxiv explaining that the results and/or proofs of the paper are incorrect, especially when it comes to results other people are likely to use.

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It also helps to be politic about what you put on arxiv, if such measure is to be resorted to. I hope my meaning comes across clearly. :) –  Willie Wong Jul 11 '10 at 0:55
If it were you who was the discoverer in the situation of Paper A in the question, how long would you wait after the author's acknowledgment before doing anything else, like contacting the author again? Gerhard "Ask Me About System Design" Paseman, 2010.07.10 –  Gerhard Paseman Jul 11 '10 at 1:07
Willie -- did you mean "polite" or "political"? –  algori Jul 11 '10 at 1:51
Gerhard -- I think each case has to be considered separately. In my experience (not at all vast) it was the authors who suggested updating their paper and I just agreed that this would be a good idea. I never had to deal with really problematic cases. But I guess if it is likely that other people will try and use an incorrect result I would write to the author once we've both agreed that a serious problem was there politely asking if they would consider updating the arxiv version of the paper and explaining the rationale. –  algori Jul 11 '10 at 2:00
@algori - neither. I meant what I wrote: politic. But my choice of word that brings to mind both the two words you mentioned was not an accident. –  Willie Wong Jul 11 '10 at 2:36
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Something I have done in a similar situation is to write about the content of the paper on the nLab wiki, including the correction. I think this is a great solution, because putting the material on a wiki is itself a service to the community and to the original author (increasing the potential audience for his/her work and helping to disseminate it), and I think including the correction when you do so is unobjectionable, since the culture of a wiki (anyone and everyone is expected to add to it, improve it, and fix mistakes) is different from the culture of publishing refereed papers, or even arXiv preprints.

Of course, this may not be an option when working in a field where there doesn't yet exist any wiki comparable to the nLab -- but then clearly the solution is to start one! (-:O

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Thank you for your suggestion. To be clear, do you put up the page and then notify the author? Or vice-versa? One point on which I want to get some community reaction is how long after notifying the author do I wait before doing anything afterward. What is your take on this point? Gerhard "Ask Me About System Design" Paseman, 2010.07.10 –  Gerhard Paseman Jul 11 '10 at 6:09
I think one should really notify the author first. The hard part then is if the author (a) doesn't do anything reasonable and (b) even after some prodding, doesn't agree to let you do anything reasonable. –  Peter Shor Jul 11 '10 at 13:45
Certainly, it's polite to notify the author. The point I was trying to make is that unlike publishing (or arXiv-posting) a correction, writing something on a wiki yourself is not taking away a privilege normally reserved to the author, so s/he shouldn't object to it. It's common and expected to write on a wiki about mathematics that other people have done, and that may include organizing it in a different way or making incremental improvements -- why shouldn't that include correcting errors? –  Mike Shulman Jul 12 '10 at 5:59
I think it is bad form to post a public claim on the internet (even on a wiki) that a paper has a mistake without first corresponding with the author. The only counterexamples to this would be posting errata for old, famous papers. –  Andy Putman Jul 12 '10 at 19:03
The original question made it clear that the author had already been corresponded with and had acknowledged the error, so I was assuming that as a given in my response. I agree that it is bad form to proclaim an error publically before contacting the author about it; I thought the question was more about how to get the word out about a correction after the author has already agreed that it is needed, but seems disinclined to write it up him/herself. –  Mike Shulman Jul 13 '10 at 4:32
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I've been in various analagous situations, and the following is a dream-list for how I wish things had worked.
In an ideal world, one should be able to make non-essential arxiv updates (typos etc.) on published papers without it generating a new version. This would encourage people to correct minor errors, which can actually seriously confuse a reader. I know that I would not put a new version of a paper on arxiv after catching typo-level errors, although I also know that this is bad for readers.
For more major errors, I wish people would keep errata for their papers on their webpage. For fatal errors, published errata may be necessary.
Politics gets in the way of this working properly, because the incentive to "be friends" is stronger than the inventive to plug minor holes in correct results. This would not be the case if we were doctors working towards a medicine to save the lives of sick loved-ones, for example... then we would care about catching the minor errors too. And so, I wish people aimed for perfect papers more seriously.

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Why is it bad to make typo-level changes on the arXiv? –  Harry Gindi Jul 11 '10 at 16:43
The more result-minded behavior of physicians that you cite is true for some but not all of the medical dramas I have watched on TV, so I don't know whether to believe you or not. –  Pete L. Clark Jul 11 '10 at 16:51
I also agree with Harry about the arXiv. It's probably best not to make a new arXiv version for every single typo, but after the preprint has been up a while and you've found a certain number of typos, I think it's good to post a corrected version. One of the great advantages of the Internet is that it's much easier to change bits in memory than it is to change ink on paper; we're shooting ourselves in the foot if we don't take advantage of it. –  Mike Shulman Jul 12 '10 at 6:02
@Harry : You can make as many edits as you want; however, every time you update your arXiv posting it generates a new version and is included in the daily arXiv mailing. The general etiquette is to make as few updates as possible (I try to make 2 updates : the first a couple of weeks after the initial posting to correct anything that people email me after seeing it, and the second after the paper is accepted to make any corrections the referee requests). –  Andy Putman Jul 12 '10 at 16:27
Actually, I think Noah and Andy are slightly wrong: after about the 5th revision or so, the arXiv stops sending out notification in the daily mailing. (At least, this is what it told me the last time I uploaded a paper.) –  JBL Jul 12 '10 at 17:44
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Sometimes reviews in Math Reviews (i.e. MathSciNet) contain a discussion of errors in the paper under review. This can be very helpful: it is a quick way to communicate the nature of the error to the mathematical community, and even to give a brief explanation of how the error can be fixed (when a fix is known). So in situations where a fix can be described succinctly, updating the review seems like a good way to fix the error, especially since (I think) most readers of a paper see this review. Of course this requires some work on the part of the reviewer, if there is one. Does anyone know if this approach is ever used (after the initial posting of the review)?

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If the author posts an erratum, then the review is updated to link to it. I'm pretty sure that this is the only situation in which the editors of MR allow changes to a review. Remember, MR started life as a printed journal, so its editorial traditions assume that reviews are static. I don't think they are likely to change this. –  Andy Putman Jul 12 '10 at 18:17
MathSciNet does allow changes to reviews. BUT they keep a record of the original review, with a link, so it looks pretty ugly. I've seen a number of instances of this; I think in all cases the changes have actually been pretty trivial (I have not seen, say, a review been changed because of an error, which is your scenario). –  Matthew Daws Jul 12 '10 at 18:42
MathSciNet has rather stringent rules on this, so describing the error is doable but not a good idea as it is not refereed by anyone. If the alleged error is resolved elsewhere (say, on a the arXiv or author's webpage), they don't allow to link. The author of the erroneous paper is not allowed to respond either. What they do recommend is describing some critical typos or adding a link to a review of a published erratum, that's all. (I learned this all from Tim Chow some years ago). –  Igor Pak Jul 12 '10 at 22:39
The examples I can think of all referred to a discussion between the reviewer and the author regarding the error. This seems to fit with what others are saying, I guess. –  Dan Ramras Jul 12 '10 at 23:28
I just re-read my answer, and it is a bit unclear. By "post an erratum", I meant "publish an erratum in a journal". –  Andy Putman Jul 13 '10 at 0:53
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If I published a mistake, then I would be grateful to hear about it. Firstly it shows that someone has actually read my work. Far worse would be to discover your own mistake years later and realise that either nobody read it or nobody cared enough to correct it.

Of course you may be mistaken that there is a mistake, so you shouldn't just say flat out: "there's a mistake", but rather "I believe there's a mistake, and here's what I think it should be. What do you think?"

What Now? Discuss the "what now" with the person involved. If you both agree with what should happen then great. If you disagree about there being a mistake then talk privately with others to see if they agree with you or not. If all else fails then publish the corrections yourself if you think it is worthwhile to do so, but firstly tell the person that you are going to publish, as that gives them one last oppotunity to do it themselves.

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I can tell you a mistake I made - publishing a correction of my own work with an error in print without my coauthor(s) from the original paper. It was a "slip" that stayed in print for over 2 years and had been presented in multiple departments and conferences. Fortunately, no one can tell from my links, so I mention it.

Had it to do over, I would have contacted the coauthor(s) but I didn't think about it. Please don't judge. :(

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If you know that the person reads mathoverflow, then you could post a question asking what to do in these circumstance and maybe the person will take the hint.

Or if you are not the person pointing out the mistake, but the pointee, and wish the pointer had gone about things differently then again you could post a question on MO and hope the person bears this in mind the next time.

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I would rather MO not be cluttered with copies of the same question, with subtle variations. –  Daniel Moskovich Jul 13 '10 at 13:51
I would consider "posting on MO" to be the moral equivalent of publishing, or putting it on one's blog. Since MathOverflow is not for corrections but instead for questions, it might be appropriate to post a question like: "I tried to follow blah in published paper B, but it looks like a typo or other error. Can someone tell me what I am doing wrong?" However, MathOverflow may not be appropriate for this either, so go check out the discussion on meta mentioned under "slip" in the taxonomy above. Gerhard "This has been a PSA" Paseman, 2010.07.13 –  Gerhard Paseman Jul 14 '10 at 0:33
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