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.
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://meta.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