mathematics - Inadmissible theorems in research

One of my engineering friends told me how he once had to take a make-up calculus I exam due to being hospitalised and so self-studied a lot of the missed topics. For the make-up exam, he used L'Hôpital's rule, although we weren't taught that until 1 or 2 exams later. My friend told me that the professor wrote

'You are not yet allowed to use L'Hôpital's rule.'

So, I like to say that L'Hôpital's rule was inadmissible in that exam.

Now, it absolutely makes sense that if you're the student that you're not allowed to use propositions, theorems, etc from future topics, all the more for future classes and especially for something as basic as calculus I. It also makes sense to adjust for majors: Certainly maths majors shouldn't be allowed to use topics in discrete mathematics or linear algebra to have an edge over their business, environmental science or engineering (who take linear algebra later than maths majors in my university) classmates in calculus I or II.

But after bachelor's and master's and maths PhD coursework, you're the researcher and not merely the student: Say, you're doing your maths PhD dissertation or even after you've finished the PhD.

Does maths research have anything inadmissible?

I can't imagine you have something to prove and then you find some paper that helps you prove something and then you go to your advisor who would then tell you, 'You are not yet allowed to use Poincaré theorem' or for something proven true more than 12 years ago: 'You are not yet allowed to use Cauchy's differentiation formula'.

Actually what about outside maths, say, physics or computer science?

Answer

The error, such as it is, your friend made was not the use of l'Hôpital, but the lack of proof that it is correct. If he had stated l'Hôpital as a lemma and provided a sufficiently elementary proof, then presumably the lecturer would not have had an issue with the solution.

An analogous phenomenon happens in research mathematics. There are plenty of folklore results, where researchers are pretty sure the result is true, and the techniques for proving the result are known, but nobody happens to have written the proof down or at least published it. These can be found, for example, in the classical regularity theory for partial differential equations.

Should one provide a proof of such a result when using it as a tool? Sometimes people simply refer to the result without being explicit about it. Sometimes they prove it "because we cannot find a proof in the literature", even if the proof is simple or not to the point of a given article. There is no absolutely right solution in these cases.

I think that folklore results are as close to "inadmissible" as one gets in research mathematics; one should be careful about them, sometimes prove them, but sometimes they are also used without proof.