Is there much benefit in memorising proofs outside of an exam setting?

Question

This is a question I have been thinking about for a while, which seems especially important as I hope to transition from undergraduate to graduate study in the next year, where studying for an exam becomes less important and studying for actual learning becomes more important.

I have always been required as part of my degree to learn proofs. Seemingly, basically, just memorising them line-by-line - there might be other methods for answering "prove [random proposition in the lecture notes]" questions, but memorisation is most effective and effectively forced by a tight time limit.

I always thought there had to be a good reason these kind of questions were put - in fact, apparently required by guidelines to be put - in the examinations. Therefore, even when teaching myself maths, I put in the effort to memorise as many proofs of theorems and propositions as I could. This especially seemed useful for maintaining understanding over long periods of time.

Now, I wonder if this was all just a bit of a waste of time. I have read many people commenting here that doing problems is far more useful - "you learn maths by doing it".

Am I wrong? Is there value in memorising proofs (after having read and understood them, not as opposed to skipping them entirely)? Or is this just a waste of time? What about when reading research papers?

Answer 1

You do not study proofs because you need to memorize them, you study them because you want to understand them, and when you understand them you should be able to more or less easily reproduce them to someone, if needed.

When you understand proofs of some statement and when you also understand that statement then you can try to find some other proofs of that same statement, and, it could happen that with other proof-approach, that that statement could be generalized in a way that those proofs other than yours were not able to accomplish.

It is natural that some proofs can be called too technical, or too long, or tedious, or even ugly (did I really wrote this?) and then there exists a need to give as simple and as beautiful proof as possible of some statement and that is possible only if you know techniques and constructs that appear in rather different approaches to the same problem.

For me, the key feature of understanding proofs is the understanding of techniques that are "beyond" proof, for example, if you want to prove some fact about some class of functions you could approximate that class with some other classes that has some feature and then pass to a limit to obtain conclusion to a class that is the limit of those classes.

Here, a notion of approximation and of the limit are important, and they arise almost everywhere inside analytical studies, whether you approximate irrationals with rationals, or some enough times differentiable function with its Taylor polynomial.

When you learn laws of powers in arithmetic then you start to observe that those laws do not depend on the actual nature of the elements but just on associativity and commutativity of an operation that operates on them, but that could not be easily understood if you did not understand that on some concrete structure first.

So, if you were to ask me, more important is to understand general techniques that appear in wide variety of proofs along different branches of mathematics than just concrete proofs, but general techniques can hardly be understood if you did not first understand proofs in concrete settings.

So, if you do mathematics and like it, surely it is good to understand proofs as best as you can, to that point that you can explain them to your fellow colleagues if they stumble upon something they do not understand. The key is to co-operate and discuss those topics that you study and do.

Answer 2

I fell into the trap of memorisation. Work through the proof, understand what is being done. You will not need to memorise. You will remember key plot points simply because you've spent time on the proof and relevant definitions (learn math by doing it). You can carry on from there independently.

Also, use the whole semester to study. Don't jumpstart your engines 2 weeks before the session.

Answer 3

Understanding a proof means, you need to understand the full idea as a whole, getting every line of a proof but not getting the whole picture is not actual understanding. So, if you understand the proof, no need to memorize it. It will not harm to understand proofs outside your course.

Answer 4

One way in which it can be helpful to memorize a proof is to learn the techniques. Diagonalization comes up in both Cantor’s proof of the countability of the rationals and Turing’s proof of the undecidability of the Halting Problem, among others. I’ve had the chance to apply it to several other problems since I saw it. The original proof of the Fourier transform was incorrect as published, but I was asked to memorize a cleaned-up version specifically because both the result and the integration technique were useful. At a more basic level, every high-school student is given examples of proof by induction less for the specific formulas (although I’ve often used closed-form formulas for series) than to teach the structure of a proof by induction.

Usually, I would memorize at least a sketch of a proof in order to keep track of the preconditions for the theorem to hold. If you remember that the geometric proof of the Pythagorean Theorem involves rearranging triangles into rectangles, with the angle opposite the hypotenuse at the corners, you’ll remember that it only holds for right triangles.

Most of the proofs I was asked to memorize in school were a matter of remembering, “Here’s where the proof uses this property, here’s where it uses that property, and that’s why I needed all those assumptions and no others.” If you know all the conditions, along with some idea of how they were used and the theorems and lemmas you needed to prove first, you probably can reconstruct any proof a professor will expect undergrads to memorize. In fact, they’ll be happier if you write an equivalent proof that you really understand than if you regurgitate one word-for-word (which I wouldn’t have been able to do anyway). On an exam, you’ll usually be reminded what the conditions of the theorem are and you can double-check that you used each one and no others. When you’re using the theorem in your own proofs, remembering the proof is helpful to catch misuses of the theorem. And you’ll often find yourself thinking, “Doesn’t this sort of look like ...?”

Answer 5

In learning any subject, even sports, the student starts by memorizing what they do not yet understand. This can be identification with the teacher and copying what they do. The understanding is meant to catch up. When the exam approaches and the proof is still not internalized then memorization is relied upon. As stated this starts to look silly when the proof comes out wrong having been nearly memorized but not understood. At the end of the course you will likely have only the memory of memorizing proofs rather than the tool kit those proofs are meant to demonstrate. I'm sure most posters here have been there, whether at the deep end of P-Chem or the first week of quantum mechanics. It takes a balance of approaches to both master the material and ace the exams. Don't forget to go back and learn the material.