What do I do if my supervisor insists that the classical CLT is false and wants me to write that in my paper?

Question

We are in a physics department. My supervisor, upon reading a Physical Review Letters (PRL) paper, concludes (and claims that the paper also concludes) that the classical central limit theorem (CLT) is false under the (usual) assumptions stated. He insists on this result and wants me to say/imply that in my manuscript because he finds fault with me making use of the classical CLT in a few statistical tests that I have utilized in the paper.

Question: (1) Would you say that my supervisor is scientifically incompetent? (2) How should I handle such a situation?

The paper is single-authored by me but he says that the university requires him to vet journal submissions (for quality) even though he is not a co-author which I accept.

(EDIT) More info regarding the context:

Just to clarify further, copied from my comment below: In my paper, I was simply stating the assumptions of classical CLT so that I could make use of the resulting normal distributions for some statistical tests. It was not a particular usage/application of classical CLT but simply a statement of the assumptions that would need to be held to recover normal distributions for the application of statistical tests I was using.

Answer 1

(1) Would you say that my supervisor is scientifically incompetent?

If he says the CLT as stated and proved in textbooks is false, he is obviously wrong. "Scientifically incompetent" sounds like a very sweeping statement that I would avoid endorsing given the information currently available. He could simply be misunderstanding something in the language of the paper, or his knowledge of rigorous math could have an embarrassing gap (wouldn't be the first time), or it could be a failure of communication between you where you have a subtle misunderstanding of his position.

In any case, textbook CLT is correct, no question about it.

(2) How should I handle such a situation?

An honest intellectual dialogue with him to clarify the issue would be ideal, if he is the kind of person who can have such a dialogue when he is in the wrong and not back himself into an emotional corner and become upset and unable to reason. If you think that's risky, try to find another professor you trust and can consult about the issue, and if they agree with you, ask them to participate in a discussion with the advisor to make him understand your point. And try approaching any discussions with an open mind, in case it's you who are in error about what his position is exactly.

Finally, it's possible that your advisor really is scientifically incompetent, so if you are unable to make him come around to your view and conclude that he is intellectually unsuited to be your advisor, you should probably switch advisors. In any case, under no circumstance should you concede to his demand to deny the truth of a standard formulation of CLT in your paper. Good luck!

Answer 2

Be absolutely certain about what you both mean here. CLT has a very well established proof. It is almost inconceivable to think that a disproof of such a fundamental mathematical theorem would ever be published in PRL over a pure mathematics journal.

This leads us to infer, quite justifiably, I think, that the PRL paper must be addressing a popular application of CLT in some field of Physics. The conclusion here would then be that CLT is an inappropriate tool to analyze the system under study - not that CLT itself is invalid. It's not uncommon at all for Physics to change its mathematical tools from time to time as more is learned about the systems those tools are meant to model.

I would suggest that you take the time to sit down and discuss this with your supervisor until you both have a clear understanding of each others' ideas. Confrontation is not going to be productive here. If you can't clearly express your ideas to each other then the likelihood of producing a quality paper is likewise quite low. Sorting out this misunderstanding will likely do both of you some good.

Answer 3

I think that some supervisors may try to "teach" you something, which they perceive as passing on some sort of "knowledge" of theirs.

However, it may turn out (as it probably did in your case) that this "knowledge" was simply a hunch, a misunderstanding, or simply an uncompleted thought. Maybe they (half-heartedly) intended for this to start a discussion that would result in a new, revolutionary way of thinking about the CLT. I say half-heartedly, because they didn't even explain their idea.

I think there are many more and better ways to assess the scientific competence of your supervisor, like scientific output, quality/impact of lectures, etc. If he is indeed not suited to be advising students on how to do science, it should be apparent elsewhere (or even everywhere else).

Answer 4

Another approach to the one given by other excellent answers is to avoid the problem altogether! Do you really need to use the central limit theorem? Maybe not, maybe you could do away with approximations altogether, or switch to some other approximation tool, like some simulations (maybe the bootstrap).

If you want advice about alternative methods, you could ask that as a new question at https://stats.stackexchange.com/

Answer 5

As other commentators have pointed out, it is important to establish exactly what your supervisor is asserting. It would be somewhat unusual to claim that a well-known and well-vetted mathematical theorem is wrong, without some specific argument constituting a counter-proof. Before you make any big decisions, make absolutely sure that your supervisor is objecting to the actual theorem (when correctly stated) and does not have an objection merely to a misstatement or misapplication of the theorem.

Now, supposing that he really is objecting to the CLT, as correctly stated, I think the next step is this. If he is of the view that a well-known mathematical theorem is false, then the onus is on him to clearly and rigorously show his reasoning and establish the correctness of his own view with the mathematical community. If he is right (he is not) and he can prove it (he can't) then his discovery would be one of the most impressive discoveries in the history of probability theory, and it would win him permanent acclaim in the field. It is up to him to publish his argument if he has one. In the meantime, there is a wide literature asserting proofs of the theorem that have been accepted by the mathematical community, with thousands of trained eyes scrutinising it for centuries. He has no legitimate standing to block your work on the basis that he does not accept the theorem.

I would suggest that you firstly bring the matter up with him in a polite but forceful manner, and insist that you regard the classical CLT to be correct, and you will not say anything to the contrary in your work. If he is going to use his vetting prerogative to seek to pressure you to say something that you (along with the rest of the mathematical community) regard as untrue, then you should bypass him completely and publish your work without his input. If necessary you could ask another academic to take over the vetting role, or simply bypass it altogether.

Lastly, I disagree a bit with some other commentators regarding the issue of competence. Even if your supervisor is wrong on this issue, I wouldn't assume he is incompetent in any broader sense (at least not on this evidence alone). Mathematical theorems can be hard, and the proof of the CLT is not trivial. It is entirely possible that your supervisor is a competent physicist in other respects and just has a "blind spot" in this aspect of probability theory (which is not even within his direct field). If it is remains feasible to work effectively with this supervisor despite your disagreement on this issue, and if that is the only thing you notice where he is in error, there is probably scope to learn plenty of other things from him.