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?

16 Answers
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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.

Does maths research have anything inadmissible?

No, but trying to prove X without using Y is still a very useful concept even in research, because it can lead to interesting generalizations, or new proof techniques that can be applied to a larger set of problems.

For instance, in some sense the Lebesgue integral is "just" trying to prove the properties of integrals without using the continuity of f, or the theory of matroids is "just" trying to prove the properties of linearly independent vectors without using a lot of properties from the vector space structure.

So this is far from being a pointless exercise, if that's what you had in mind.

In the sense that you are asking, I cannot imagine there ever being a method that is ruled inadmissible because the researcher is "not ready for it." Every intellectual approach is potentially fair game.

If the specific goal of a work is to find an alternate approach to establishing something, however, it could well be the case that one or more prior methods are ruled out of scope, as it would assume the result that you want to establish by another independent path. For example, the constant e has been derived in multiple ways.

Finally, once you step outside of pure theory and into experimental work, one must also consider the ethics of an experimental method. Many potential approaches are considered inadmissible due to the objectionable nature of the experiment. In extreme cases, such Nazi medical experiments, even referencing the prior work may be considered inadmissible.

It is worth pointing out, that theorems are usually inadmissible if they lead to circular theorem proving. If you study math you learn how mathematical theories are built lemma by lemma and theorem by theorem. These theorems and their dependencies form a directed acyclic graph (DAG).

If you are asked to reproduce the proof of a certain theorem and you use a "later" result, this results usually depends on the theorem you are supposed to prove, so using it is not just inadmissible for educational reasons, it actually would lead to an incorrect proof in the context of the DAG.

In that sense there cannot be any inadmissible theorems in research, because research usually consists of proving the "latest" theorems. However, if you publish a shorter, more elegant or more beautiful proof of a known result, you might have to look out for inadmissible theorems again.

While there are indeed no inadmissible theorems in research, there are certain things that one sometimes tries to avoid.

Two examples come to mind:

The first is the classification of finite simple groups. The classification itself is not particularly complicated, but the proof is absurdly so. This makes mathematicians working in group theory prefer to avoid using it when possible. It is in fact quite often explicitly pointed out in a paper if a key result relies on it.

The reason for this preference was probably to some extend originally that the proof was too complicated for people to have full confidence in, but my impression is that this is no longer the case, and the preference is now due to the fact that relying on the classification makes the "real reason" for the truth of a result more opaque and thus less likely to lead to further insights.

The other example is the huge effort that has gone into trying to prove the so-called Kazhdan-Lusztig conjecture using purely algebraic methods.

The result itself is algebraic in nature, but the original proof uses a lot of very deep results from geometry, which made it impossible to use it as a stepping stone to settings not allowing for this geometric structure.

Such an algebraic proof was achieved in 2012 by Elias and Williamson, when they proved Soergel's conjecture, which has the Kazhdan-Lusztig conjecture as one of several consequences.

The techniques used in this proof allowed just the sort of generalizations hoped for, leading first to a disproof of Lusztig's conjecture in 2013 (a characteristic $p$ analogue of the Kazhdan-Lusztig conjecture), and then to a proof of a replacement for Lusztig's conjecture in 2015 (for type $A$) and 2017 (in general), at least under some mild assumptions on the characteristic.

There are cases where the researcher restricts himself not to use certain theorems. Example:

Atle Selberg,"An elementary proof of the prime-number theorem". Ann. of Math. (2) 50 (1949), 305--313.

The author restricts himself to use only "elementary" (in a technical sense) methods.

Other cases may be proofs in geometry using only straightedge and compasses. Gauss showed that the regular 257-gon may be constructed with straightedge and compasses. I would not consider that to be "a new proof of a known result".

It is perhaps worth noting that some results are in a sense inadmissible because they aren't actually theorems. Some conjectures/axioms are so central that they are widely used, even though they haven't yet been established. Proofs relying on these should make that clear in the hypotheses. However, it wouldn't be that hard to have a bad day and forget that something you use frequently hasn't actually been proved yet, or that it is needed for a later result you want to use.

In intuitionistic logic and constructive mathematics we try to prove stuff without the law of excluded middle, which excludes many of the normal tools used in math. And in logic in general we often try to prove stuff using only a defined set of axioms, which often means that we are not allowed to follow our 'normal' intuitions. Especially when proving something in multiple axiomatic systems of different strengt you can get that some tool only become available towards the end(the more powerful systems) , and are as such inadmissible in the weaker systems.

To answer your main question, no. Nothing is disallowed. Any advisor would (or at least should) allow any valid mathematics. There is nothing in mathematics that is disallowed, especially in doctoral research. Of course this implies acceptance (now settled) on Poincaré's theorem. Prior to an accepted proof you couldn't depend on it.

In fact, you can even write a dissertation based on a hypothetical (If Prof Buffy's Large Theorem is true, then it follows that...). You can explore the consequences of things not proven. Sometimes it helps connect them to known results, leading to a proof of the "large theorem" and sometimes it helps to lead to a contradiction showing it false.

However, I have an issue with the background you have given on what is appropriate in teaching and examining students. I question the wisdom of the first professor disallowing anything that the student knows. That seems shortsighted and turns the professor into a gate that allows only some things to trickle through.

Of course, if the professor wants to test the student on a particular technique he can try to find questions that do so, but this also points up the basic stupidity of exams in general. There are other ways to assure that the student learns essential techniques.

A university education isn't about competition with other students and the (horrors) problem of an unfair advantage. it is about learning. If the professor or the system grades students competitively they are doing a poor job.

If you have the 20 absolutely best students in the world and grade purely competitively, then half of them will be below average.

I don't think there are inadmissible theorems in research, although obviously one has to care not to rely on assumptions that has yet to be proven for a particular problem.

However, in terms of PhD or postdoc work, I feel that some approaches may be rather "off-topic" because of not-really-academic reasons. For example, if you secure a PhD funding to study topic X, you should not normally use it to study Y. Similarly, if you secure a postdoc in a team which develops method A, and you want to study your competitor's method B, your PI may want to keep the time you spend on B limited, so it does not exceed the time you spend to develop A. Some PIs are quite notorious in a sense that they won't tolerate you even touching some method C, because of their important reasons, so even though you have full academic freedom to go and explore method C if you like it, it may be "inadmissible" to do so within your current work arrangements.

I'm going to give a related point of view from outside of academia, namely a commercial/government research organisation.

I have come across researchers and managers who are hindered by what I call an exam mentality, whereby they assume that a research question can only be answered with a data set provided, and cannot make reference to other data, results, studies etc.

I've found this exam mentality to be extremely limiting and comes about because the researcher or manager has a misconception about research that has been indoctrinated from their (mostly exam-based) education.

The fact of the matter is that by not using data/techniques/studies on arbitrary grounds stifles research. It leads to missed opportunities for commercial organisations to make profit, or missed consequences when governments introduce new policy, or missed side-effects of new drugs etc.

I will add a small example from Theoretical Computer Science and algorithm design.

It is a very important open problem to find a combinatorial (or even LP based) algorithm that achieves the Goemans-Williamson bound (0.878) for approximating the MaxCut problem in polynomial time.

We know that using Semidefinite Programming techniques, a bound on the approximation factor of alpha = 0.878 can be achieved in poly time. But can we achieve this bound using other techniques? Slightly less ambitiously but probably equally important: Can we find a combinatorial algorithm with approximation guarantee strictle better than 1/2?

Luca Trevisan had made important progress towards that direction using spectral techniques.

In research you would use the most applicable method (that you know) to demonstrate a solution, and would possibly also be in situations where you are asked about or offered alternative approaches to your solution (and then you learn a new method).

In the example where L'Hôpital's rule was "not permitted", it could be that the question could have been worded better as it sounds like a "solve this" question, assuming that only the methods taught in the course are known to students and therefore only the methods taught in the course will be used in the exam.

Well in pure Math research I am sure brute force approximations by computer are disallowed except as a way to introduce the interest in the topic and possibly a way to narrow the area to be explored. Perhaps even to suggest an approach to solution.

Math research requires equations that describe an exact answer and proof that the answer is correct by derivation from established math facts and theorems. Computer approximations may use ever smaller intervals to narrow the range of an answer but they don't actually reach the infinitely small limit of L'hospital style.

The separate area of computerized derivations basically just automates what is already known. I am sure many places leave researchers free to use such computerization to speed documentation of work as far as such software goes. I am sure that plenty of human guidance is still needed to formulate the problem, introduce postulates and choose which available solution steps to try. But the key thing is that all such software derivation would have to verified by hand before any outside review for software error and that techniques stay within allowed boundaries (the IF portion of theorems etc).

And after such hand checks...how many mathematical researchers would credit computer software for assistance?

Well I saw applications mathematicians cite software as a quick check method for colleagues to check the reasonableness of the work way back in the 1980s. In that applications mathematics sometimes has an almost engineering math view of practical results, I suppose that they still do give the computer software approximations as quick demonstration AFTER the formal derivations. And I hear that applications math sometimes solves the nearest approximation to the problem possible when solution to the exact problem still evades them. So again more room for assistance by computer software derivation. Not sure that such operations research type topics fits everyone's definition of mathematical research though.

In shorter terms, yes computer approximation techniques are often used in a shotgun manner to look for areas of potential convergence on solutions. As in "give me a hint". Especially in applied math topics where real world boundaries can be described.

Again a there is the question of whether real world problems other than fundamental physics are true math research or the much looser applied math or even operations research.

But in the actual derivation of theorems from underlying and proven theorems to new theorems...computers are more limited to documentation tools similar to word processors for prose. Still tools becoming more and more important to speed the more common equation checking of documented work as word processors check spelling and grammar for prose. And more areas where human must override or redirect.

The axiom of choice (and its corollaries) are pretty well-accepted these days in the mathematical community, but you might occasionally run across a few old-school mathematicians who think that it's "wrong", and therefore that any corollary that you use the axiom of choice to prove is also "wrong". (Of course, what it even means for the axiom of choice to be "wrong" is a largely philosophical question.)

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