7

I have a few results during my math PhD that I used in a black-box or grey-box sense. I have a rough idea of how it is proved but the proof itself is extremely long and detailed. I cannot reproduce it off the top of my head without looking at the paper and going through it line by line again. Some of these proofs also themselves refer to other lemmas and proofs in different papers and it can become quite a deep rabbit hole.

I am at a stage now where I can either spend some weeks studying these proofs in detail or I can focus more new research and take these results as black box results. What is the right/expected thing to do from a scientific mindset? On the one hand, understanding everything from scratch would be nice but on the other hand, the reason we write lemmas is so that others can use them as a springboard to develop new ideas.

TL;DR how deeply do mathematicians understand other people's work before using their results? My practical concern would be my thesis defence but the broader scientific "best practice" would be good to know too.

4
  • 1
  • Most people I know would be okay in using the result in your situation.
    – user111388
    Sep 3, 2020 at 11:29
  • 2
    @GoodDeeds, thank you. I'd say it's related although I'm working in a field where computational implementations aren't involved. Also, the questioner there is a biologist using a math result while my case is that the results I'm building on are from my own field (linear algebra). But it certainly helps!
    – Gomie
    Sep 3, 2020 at 11:49
  • As an analogy, we usually do not take apart the legs of a chair just to see why they provide adequate support. Unless you are interested in building a 'better' chair or have a clear goal in mind, like using them for a table, then I would say you should make better use of your time. In general, I learn only those concepts/tools/ideas that would help me achieve a research aim or solve a problem. Sep 5, 2020 at 20:00

4 Answers 4

1

This depends on the individual case and requires judgment.

On the one hand, understanding the past work completely gives you a better base of understanding going forward.

On the other hand, there are circumstances in which you understand the past work sufficiently to go forward, and you have enough trust in it to know that you won't be embarrassed in future for doing dead end work.

That complete understanding takes time, as you note. It is seldom necessary for a mathematician to recapitulate the entire history of mathematics since, say, Euclid, to do significant work.

For some problems in some (sub) fields, more understanding is needed. In others, less. But there is an old saying "Don't let the perfect be the enemy of the good." (Usually attributed to Voltaire.)

If you are good enough to recognize problems with lines of thought and you have insight into the underlying structure of the field, then trust your judgement. If you smell a skunk, delve deeper, otherwise, move on. It isn't a perfect defense against error, but.... Voltaire, again.

1

I think this does depend a bit on each person's approach, but what you've described to me sounds totally fine and normal. I quite regularly use results where I could give a gauzy explanation of the proof, but nothing good enough for, say, a Ph.D. qualifying exam, and I feel no compunction about using well-established results where I could not even summarize the proof (for example, the classification of finite simple groups). I can't imagine deciding to dedicate even a full work day, let alone weeks, to trying to understand a proof without some clear goal in mind for using that understanding.

0

You don't need to know all the proofs. That wouldn't be practical. If it helps you prove your result or uses a common technique then it could be worth learning. Otherwise you would be spending lots of time reading a proof, but for what purpose? Just knowing why a theorem is true can be satisfying, but that's about it.

It does help to keep in mind whether a theorem you read follows from the definitions in a straightforward way or whether it uses some deep facts.

1
  • 1
    Hmmm, actually proofs are the most important thing in mathematics. They bring insight. This might be a safe move if you already had the needed insight, but a poor way to develop it. Of course, trivial theorems have trivial proofs.
    – Buffy
    Sep 3, 2020 at 12:55
0

Whether people expect you to know the proofs of results that you use will depend on (at least) two factors. First, how complicated is the proof? I wouldn't expect anyone to know the whole proof of the classification of the finite simple groups or the solution of the Kepler problem, etc. Second, how close is the result to your research area? I once used, in a paper about the axiom of choice, the result (which I hope I'm remembering correctly) that, over the group ring of a cyclic group of order 23, not all projective modules are free. I didn't learn the proof of that, and I don't feel guilty about not learning it. I just cited the paper where it was proved. But, depending on the complexity of the proof, I might well feel guilty in the same situation if the result was in set theory.

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .