# How to correct a fundamental error in a series of papers made by a colleague of your adviser?

Talk to your adviser as soon as possible. I'm a bit surprised that in your previous interactions your adviser did not already tell you whether and how he planned to contact the author(s) of the fallacious papers -- especially given that at least one was your adviser's collaborator.

Here is my advice (it comes from a pure mathematician, if that is relevant to you):

• Spend up to one work day writing up a careful description of the error. If a result is false, give a counterexample. If you can give one specific, easy to verify counterexample, start with that. Then if you have a further sense of the "terrain of counterexamples" -- e.g. if you know the result is never true, or you know that it is true precisely under some additional hypothesis, then include that.

The point of this is that in many cases what we perceive to be errors in mathematical work stem from misunderstandings between the reader and the author, including different use of the same terminology. I would say that approximately half of all the errors I think I see in my colleagues' work turn out to have such innocent explanations. Another big percentage of suspected errors stem from misunderstandings of the part of the reader: it is one thing if you come across a result that contradicts another result you have already written up. However, if you are just reading then the odds are fair that the cognitive dissonance in your mind is not actually caused by a mistake in the paper: for instance when I read other people's work, I try to do so with an eye towards its relationships to my own work and my own problems. As a proud student of the "Richard P. Feynman school of situational genius" I heartily endorse this approach, but sometimes it turns out that what I think is an error is due to an entanglement of "my situation" and the author's. Finally, when an error has actually been made, there is usually (very understandable) psychological resistance on the part of the author. This is why it helps to arrive with individual, crisp counterexamples.

• If the theorem is correct but the proof is faulty, the situation may be more nebulous.

Everyone has different standards of what constitutes an acceptable proof. Moreover, mathematicians are still human beings, and the percentage of inessential slipups we make in the course of our written work is non-negligible. Most people do not appreciate their expository flaws and bone-headed but unimportant mistakes being taken as evidence that their proofs are incorrect. In fact, where possible, when you claim that someone's proof is faulty it is an honorable thing to give them a chance to correct it.

In this case you speak of a "fundamental error", which I guess means that it is worse than the relatively innocuous mistakes described above. Do you mean though that the theorems are still true, and you know this because you know how to prove them? If so:

• Spend up to two or three work days writing up as much as possible of what you feel is a correct proof. If you can only give a sketch, so be it, but try to include all the ideas which convince you that your argument is correct. Your target audience includes people who are deeply committed to and knowledgeable in this particular area: you can write accordingly.

I would meet with your advisor as soon as you have written each document. (If you can write both in one day, great.) Then the question becomes obvious and unavoidable: what should you do with these documents? If he can vouch for their accuracy, I think you will certainly want to contact the author of the flawed paper, but whether to do this yourself or through your adviser is something you should ask him about.

Why do I place time limitations on these tasks? It is because of the following important observation, which in my understanding is somewhat peculiar to mathematics:

• Even if the author's proof is wrong and yours is right, your corrected proof may or may not be publishable.

There are too many nuances here for me to go over them all, but just one quick thing: you say that you fixed the result using material from old works. Because of this, it is possible (perhaps; I don't know the situation) that the authors and/or the editors in question will still regard the mistake as a "slip up" (even if you do not). A math paper which does not contain "new results", "new ideas" or "new techniques" may be very hard to publish...let me readily admit that this practice is not entirely fair or wise, but it most certainly is extant. So you don't want to spend substantial work time on what may end up being a corrigendum written by the author but identifying you as the provider of both the problem and the solution.

Good luck.

• Thank you very much for all the details you provided. The theorem in question is a "To good to be true theorem". The first time I saw it I said "No way". I checked the proof and it was a beautiful proof, I said to myself "Why no one in the past didn't have this idea ?". It was very simple. But I still couldn't believe it. So after detailed examinations, I found that the theorem uses a Lemma but the conditions to use the Lemma are not all satisfied. I think the counterexample is difficult to find. What I did is proving the theorem by a different approach with additional assumptions. Commented Jul 22, 2014 at 1:30
• @user165633: Thanks for the additional details. It sounds like both the counterexample and the partial fix are of real value. I would recommend that you follow my advice (especially the part about talking to your adviser). Honestly, this does sound like a good situation for you to get a publication out of it...but you should still proceed carefully and not spend too much time writing it up until you learn how this will be received. Commented Jul 22, 2014 at 3:28
• I think it is important for you to write it up for your own benefit, as well as the benefit of future researchers. The potential political situation associated with this should be explored as well. mathoverflow.net/questions/31337/… has some advice, one key part of which is to talk to more than one senior person who is familiar with the authors and the situation. It is important to fix things, but it is also important to pay due respect and care to whom and what came before. Talk to more than one person about how to proceed. Commented Jul 22, 2014 at 4:36
• I think it's really important to make sure that your refutation is ironclad. So, it does make sense to write it up carefully and show to the parties concerned. Commented Jul 22, 2014 at 14:04
• I'd also strongly encourage you to try to find a counterexample, even (or especially) if it's difficult. If you can't find one for the full theorem (and if so, that's interesting in itself), at least try to find one where the incorrectly applied lemma fails, even if the final result still holds. Also, since you say you have a new proof with additional assumptions, try to construct an example of what happens if those assumptions don't hold. Does the full theorem fail (great, you have your counterexample) or does it still hold (which means your new proof might not be as general as it could be)? Commented Jul 22, 2014 at 14:17