When to give up on a problem (for maximal learning of a specific area)?

Question

There are many things to admire about the so-called Moore method, in which all the theorems and aspects of a course (e.g. real analysis) become problems for you to solve on your own.

But sometimes, for time management or out of frustration, you want to just give up on a problem.

With the pragmatic aim of maximising what I understand of the subject and honing techniques I have for problem-solving, when should I give up?

There is a similar question, but from the point of view of enjoyment of Olympiads, which is not the same:

When to give up on a hard math problem?

EDIT: To make things specific, could you name a moment when you yourself gave up on a problem deservedly? Or name a mathematical fact whose proof is not that complicated but is really really hard to find out on your own?

A real-life example from today: I tried to prove the Lebesgue Density Theorem today. I had gone through a brief course on the Lebesgue integral. During today's thoughts, I independently found the concept of a Vitali Cover (without knowing that that was the name of it) but didn't articulate the crucial main theorem on Vitali Covers. In around eight hours, I gave up and looked up the solution.

Answer 1

As mentioned in the comments, there will not be a single or straightforward answer, but I cannot resists making a few observations and comments. I always felt a strong urge to discover everything myself, but this approach is limited by available time, and not very efficient. While it is nice to try this yourself, I am sceptical whether the benefits (better understanding) are significant.

1. Solving a problem on your own will ensure you understand the problem (and its solution, e.g. a Theorem or Proof) well enough - it is much easier to miss the depth of mathematical results when you just read about them. However, even if you have solved a problem, it can happen that you still have not grasped the problem in its full depth, or are still not aware of all of its aspects (I am speaking from personal experience in research). Think about proofs in which every step makes sense, but you don't really understand the underlying mechanics.
2. In most fields it took dozens or even hundreds of mathematicians many decades to arrive at a consistent theory. Even if you "peek behind the curtains" and know which direction your self-study is going, it is hard to replicate all those mathematicians' work in your lifetime. So I think that some peeking/"cheating" is allowed and inevitable.
3. For "simple" problems, solving them by yourself, the benefit is usually great: You make sure you understand all relevant definitions and results, as well as the problem itself. For harder problems, the situation is less clear-cut. You certainly will spend more time on the problem than would be needed to simply understand the relevant theory - you will definitely have a good understanding of the problem at the end of the day. The big issue is that some proofs will not necessarily add much to the intuition of the problem - sometimes the difficulty consists in finding the right "trick", which is not really improving the understanding.
4. Working for a long time on hard problem will help you practice research- and problem-solving-skills, but given you only have limited time, I would ask myself how much time you would want to spend on the problems.

Addendum If you do Maths in your free time, it could be an enjoyable pastime to "re-discover" mathematical concepts, or prove known results. As a researcher who wants to learn about a new topic, the approach would usually be to read about it, and try some simple exercise problems, or simply to sanity checks like "Why does the author claim that X is true?". Especially in graduate topics, there is no time to re-discover a new field all by yourself. This is hard enough for undergraduate topics - how long would it take to discover all the proofs of a Linear Algebra lecture, or the like?

Answer 2

Just a personal opinion, but I think the issue is not so much how much time to spend on a problem you are stuck on, but on diagnosing the obstruction and using the diagnosis to decide how to proceed, depending on your more general learning or research goals. Here are some examples.

1. You are missing a trick. Such problems are hardly worth spending much time on. The solution might come to you in an hour, or in a month, and you won't learn any more in the latter case than in the former.

2. Close to your situation, you are teaching yourself by trying to prove the main theorems, perhaps by a guided series of exercises. Here I find that you will often see the main idea, but struggle to see how to put it together. Then I think it's worth just taking a small look at the solution just get some extra hint, even just to see how the proof is being organised, then going back to the problem.

3. Sometimes it becomes clear that despite being sure that you are supposed to apply such and such a concept or theorem to a problem, and despite knowing in principle how to define the concept or state the theorem, the exercise teaches you that you have not understood them well enough. Then it may be worth going back to some simpler examples or exercises to improve your understanding before returning to the problem. It can also happen that the examples you have already seen are too simple, and are not displaying the full power of the theorem. Then a search like "What is theorem X used for?" may be useful.

4. Sometimes you may find that your problems are much simpler. For example, the obstruction may be that the notation that you have set up is awkward, and e.g. makes it difficult to keep track of how objects are being identified. The literature will have figured out a good solution by trial and error, and it's not worth reinventing that wheel. Alternatively, you may be lacking those little lemmas that state basic facts, and it may be time to rustle up some of those. For example, if you are used to working with rings with identity, you may be unconfident about which basic facts about them carry over to rings without identity, and that may be making your life harder than necessary. Or you may have a good conceptual understanding of a topic, but you are struggling with exercises because you are not adept at the standard calculations that come up again and again.

I'm sure there will be many more, and better, examples than these. But the main idea is to use difficulties with the exercises as tutors that provide hints about what best to do next.