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I've very convinced that an author made a serious error in the main theorem in one of her 2012 papers in Statistics and Computing, which is a reputed journal in machine learning. I contacted the author, and we did have a respectful conversation, in which she neither agreed nor disagreed with me, but didn't seem to know a mathematically fundamental step mistakenly used in the proof of the theorem. Now, since I'm convinced that this is indeed a serious error, what should be my next step?

Along this line, I followed this question, and ideally I wanted to collaborate with the author, who did very kindly give me half an hour to discuss in real time, but I understand that there's no hope for writing a corrected version of the paper with her, since she neither agreed not disagreed with me.

Now since I'm in the beginning phase of my research career, I'm thinking of leveraging my discovery into writing a paper that'd be along the line of "Notes on the paper X", which'll mostly focus on discussing why the theorem and the proof is wrong. I do of course want to acknowledge the discussion with the author. How can I make this possible?

See the thing is: since the main theorem of the paper is wrong, it makes the subsequent experiments wrong as well, so I can't really do much to improve the results into something to make a brand new paper. However, I do want my contribution to the community (that catching the mistakes and explaining them why it's wrong with potential counterexamples) noted. This is the reason I want to publish my finding. Also I'm wondering even if I write to the editor, they acknowledge that it's a mistake, will they necessarily publish my notes in the same journal? Intuitive it seems no because it's just a correction (albeit to the main theorem), but yes, as this should be the appropriate choice, as this is the very journal the paper was published in. How should I go about it? Thank you!

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    The first thing I'd do is try to talk to another expert (an advisor maybe?) who can help make sure you're indeed correct in your analysis. – Kimball Jul 10 at 13:47
  • Sure I'm all up for that, but finding such an expert interested in another person's work seems a difficult task: I'm not part of a university at this moment but I did finish my PhD in a different area. So speaking with my PhD advisor or department people might not seem feasible. – Science Man Jul 10 at 13:50
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    Regarding your last paragraph: journals regularly publish corrigenda/errata, which are often short notes. Normally these are written by the original authors, but they don't have to be. In your situation, you could still try to make one written by both of you. My suggestion is to write a polite draft, send it to the author and invite her to work with you in making a corrigendum. – Kimball Jul 10 at 16:07
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    @Kimball Thanks for the suggestion, and I'm glad to know that they do so. But do they publish the errata as separate paper(s)? I've already sent her a polite draft detailing the errors. But I'm not sure she went through that in great detail, as was apparent in our real-time meeting. – Science Man Jul 10 at 16:26
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    At least in math, errata are considered separate articles, and indexed as such in publication databases like MathSciNet. – Kimball Jul 10 at 17:25
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If you are sure of your work, there is little reason not to write the paper and submit it. Cite the other work properly, of course, and make your case clear. If the editor of a journal doesn't reject it out of hand it will be passed to a couple of referees who will give their judgement on the work.

But you don't jeopardize your career by doing this. Your paper isn't public until it is published. But asking the editor first is probably a mistake, since you can't develop your argument without writing the paper.

And, it isn't impossible that in writing it up formally you will find an error in your own work. Make the case. Submit it for publication if it is strong enough.

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  • Thanks, that sounds like a good idea - will do so! – Science Man Jul 10 at 14:19
  • The problem is that though, the paper will say, "this statement is wrong as this is a counterexample". But what will it really establish? I mean, the real contribution will be (if I'm correct) that: people who didn't read the math thoroughly before, now will find a significant mistake. The paper will do just that - and not establish a new result. My concern is: is that enough to form a paper alone? – Science Man Jul 10 at 14:22
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    It is hard to say from a distance. But if the paper is widely cited and subsequent results are also wrong, it is important to put it out there. But a statement of what can be proven would be stronger, of course. – Buffy Jul 10 at 14:26
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    @Science Man indeed is a comment on paper x rather than a normal paper. Is the error in paper x a seed for future mistakes and propagating troubles? Or it is just cited but at the end no one uses that specific erroneous point? This would be the main criterion for taking a decision and proceed. – Alchimista Jul 10 at 14:50
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    @ScienceMan That will depend on the particulars. A paper that says "here's a counterexample to Theorem X" but fails to specify what the logical fault in the proof of Theorem X is may be a difficult pill to swallow, especially if no one can find a logical fault in the counterexample. It could be years before a consensus develops and someone presents a solid enough argument for which of the theorem or counterexample is valid. It wouldn't be unprecedented. There's a somewhat famous story with Voevodsky and how he came to favor computerized proof assistants with exactly this situation. – zibadawa timmy Jul 11 at 5:04

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