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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?

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closed as primarily opinion-based by Namaste, Matthew Towers, Paul Frost, Chris Custer, José Carlos Santos Aug 28 '18 at 14:57

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Surely if you understand a proof, that is you grasp the main ideas underlying it, then you don't need to memorise it line-by-line, since you can then just reconstruct it at will? $\endgroup$ – Lord Shark the Unknown Aug 26 '18 at 17:55
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    $\begingroup$ It would be nice if that were true, but when it comes down to the wire and the exam pressure is on you, you want to get the unthinking parts out of the way quickly. It’s simply more effective to memorise. Moreover, I have seen several quite technical proofs come up of quite niche theorems - not just headline theorems, but lemmas and propositions. Things are graded on a curve, and if you’re taking longer to answer a question whose answer can be memorised you lose out. $\endgroup$ – Nethesis Aug 26 '18 at 17:57
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    $\begingroup$ It should be noted that my definition of line by line is not literal, but still more than just “remember the key idea” (often, with technical proofs of propositions, there isn’t really one). $\endgroup$ – Nethesis Aug 26 '18 at 18:03
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    $\begingroup$ @LordSharktheUnknown, Not at least 50% of the theorems. Some must be re-read for many times to be remembered for a long time. $\endgroup$ – user231343 Aug 27 '18 at 1:24
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    $\begingroup$ @Nethesis: It's sad that your exams are set to be so biased towards blind memorizers. Well, do what you need to get sufficient grades, but separate exams from mathematics. Actual mathematics is rarely done under time pressure. =) $\endgroup$ – user21820 Aug 27 '18 at 17:16
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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.

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  • $\begingroup$ 'when you understand them you should be able to more or less easily reproduce them to someone, if needed.' --> This! We're learning logic, not history, law or medicine. If you understand a proof, then you know how to manipulate assumptions to arrive at conclusions. $\endgroup$ – BCLC Aug 27 '18 at 10:40
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    $\begingroup$ when you understand them you should be able to more or less easily reproduce them to someone, if needed. I don't agree this is always the case. There are lots of proofs that invoke weird constructions that are not exactly "logic". A simple example of this is the proof of the cauchy-schwarz inequality. Plus a very long sequence of logical steps, no matter how well they flow, can turn you into memorisation. $\endgroup$ – Aiman Al-Eryani Aug 27 '18 at 16:52
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    $\begingroup$ When I studied chess, I was always amazed that my instructors could memorize 40-move games. Then I realized that their experience allowed them to compartmentalize the moves into high-level concepts from which the moves could be derived. Only a handful of these concepts had to really be memorized for each game. $\endgroup$ – BallpointBen Aug 27 '18 at 20:13
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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.

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    $\begingroup$ + about a million. I have graded countless exams with absurd non sequiturs as a result of mis-memorized proofs. $\endgroup$ – John Brevik Aug 26 '18 at 18:44
  • $\begingroup$ A proof is like designing and building a staircase, or watching someone else do it. You don'r need to recall every nail. You want to understand the concepts and the uses of the tools, $\endgroup$ – DanielWainfleet Aug 27 '18 at 2:31
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    $\begingroup$ @JohnBrevik. The abdomen contains the stomach and the bowels, which are A,E,I,O,U and sometimes Why. $\endgroup$ – DanielWainfleet Aug 27 '18 at 2:33
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    $\begingroup$ @JohnBrevik I remember my linear algebra exam for which I regurgitated things from memory, I thought I'd done good and then the professor invited us to a post exam conversation, where he asks further questions or clarifications. He then asked me about the simplest definitions (what is a polynomial, basis of a vector space, how does one represent a linear operator as a matrix) and I was stumped. Naturally, I failed the exam. $\endgroup$ – Alvin Lepik Aug 27 '18 at 12:12
  • $\begingroup$ @AlvinLepik Exactly! Your professor was actually kinder than I am in these circumstances! $\endgroup$ – John Brevik Aug 27 '18 at 18:31
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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.

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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 ...?”

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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.

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