I was reading over some of my supervisor's old papers and I have found one from 7 years ago in which part of the analysis is fundamentally flawed which effectively undermines half of the conclusions. The issue is in a paper in a subfield of a subfield so it's unlikely that any other people will find this mistake.

My question is what should I do. On one hand, I feel like I have moral obligation to let him know so that the issue can be resolved. On the other hand, I feel uncomfortable bringing it up as I don't want it to have any negative repercussions on our working relationship.

What should I do in this scenario?

Thank you.

Edit: This question is different from Should I warn my professor about some errors that I've found in his paper? as in that case it was a paper in advanced access (e.g. one that was worked on recently), while in this case it is a 7 year old paper.

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    Have you considered that he probably already knows? People are human, mistakes happen. – astronat Mar 30 at 18:29
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    @user1150512, Maybe, you can simply approach him at the "RIGHT" time, and politely present your question/finding to him directly ? It could be that either you or he made a mistake in this case. It should not be big deal to figure this out as both of you are professional educator and student. – Job_September_2020 Mar 30 at 18:33
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    If you can only find one error in my past papers, you aren't looking closely enough. Understanding comes in fits and starts and isn't always correct. – Jon Custer Mar 30 at 23:02

First, I'd double-check your work. Are you certain this is a mistake? Is there something you're missing? Do the best you can to understand what's going on.

Second, I would definitely raise this with your supervisor, but do so from a perspective of questioning and seeking clarification. Rather than appearing in judicial robes with the paper in hand and declaring it flawed by the laws of humankind and the universe, approach it as a question: There's a step in the paper that you don't understand. You think the step should say "XYZ", but the paper says "XZY". You're not sure if you're making a mistake or if there is an error in the paper.

There's nothing disrespectful about this humble approach, and you get a double benefit that if it turns out you are the one that is wrong then you have no face to save, and hopefully learn something along the way.

Mistakes happen, and they don't indicate the person making the mistake is dishonest or flawed in some fundamental way. Next steps to take after this will depend on how crucial the flaw is and what other work it might impact. If you take these steps and there are still some negative impacts on your working relationship, well... probably that person wasn't worth working with anyways, and some rift was going to develop sooner or later anyways. It would not be your fault.

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    I don't think you need to be quite that weak kneed. "One of us has an error. Can we work through it?" or "I can't seem to follow the logic here. Is there an error, perhaps?" But certainly double check your reasoning first. Think of errors in old papers as an opportunity for new work - new thinking. It is a positive when caught. Beware of fragile egos, of course. – Buffy Mar 30 at 19:16
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    @Buffy I would say my advice certainly tilts towards massaging the fragile ego, but in my mind the examples you give are perfectly within my suggestion to approach with a questioning/clarifying approach: both "one of us has an error" and "is there an error" avoid any declaration that someone else is wrong before even starting the discussion, which is something that might provoke unneeded defensiveness. That said, I would have taken a more direct approach with my own advisor, but I also knew his personality and that doing so wasn't going to be an issue. – Bryan Krause Mar 30 at 19:26
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    Yep. Not a good idea to run down the hall in the department yelling "Dr. Buffy screwed it up. Again. Ha". (Even if I did - again.) – Buffy Mar 30 at 19:41
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    When Prof. Nick Katz at Princeton found the mistake in Wiles' original attempted proof of Fermat's Last Theorem, according to the Nova episode on the proof, he went to Wiles and said something like "I don't understand this step here, can you clarify it?" They went back and forth a few times before it became clear that there really was a mistake. Katz is no shrinking violet. Just an example of how Krause's suggestion is actually what people really do. – Mark Foskey Mar 31 at 3:53
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    This is good advice for any similar situation, in academia or out, and regardless of which way the power dynamic goes. Obviously, more specific relationships might have different answers, but this is the right general model. – fectin Mar 31 at 14:03

Regardless of your "moral obligations", any reasonable advisor would be happy to hear this from their PhD student, especially if you make it sound like a (potentially tricky) question, not an accusation of making a simple mistake.

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