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To be clear, I'm $\bf{\text{not}}$ talking about rote memorization.

If I can follow a proof, follow each step on its own, but still have trouble believing that the result is true. I usually try to work through it an additional 1 or more times, trying to shorten the proof into a half dozen key steps, which I memorize so that I can reproduce the proof.

In general, is this a good way to gain insight on why the conclusion ought to be true? or is this just a way of slowing myself down, hindering progress?

roo
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    I would say it's very helpful to get to the point where you can reproduce the proof. – Alex Becker Apr 16 '14 at 21:18
  • It's usually much better to look at different proofs if you feel you understand one and still don't "click" with it. Often, there is something that isn't fully clear about why the proof works, and getting a different perspective can really help. Sometimes, looking at a different proof can help you understand why the other proof made the decisions it did. Even if you don't have different proofs (maybe there is only one known proof), if you can find different expositions of the proof, that can help too. If going over each line of the same proof has failed, elsewhere may be the best option. – ex0du5 Apr 16 '14 at 21:24
  • OK to make the question more concrete, lets work under the assumption that you follow the proof fully from assumption to conclusion, you find the proof completely acceptable, but aren't clueing into the idea behind it. – roo Apr 16 '14 at 21:27
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    I have come across texts that give almost "magical" proofs that leave me dumbfounded. These are usually presented without any intuition and memorizing them will do no good. It usually helps me to find other proofs online or on other sources even if these proofs are not comparatively as nice. – Sandeep Silwal Apr 16 '14 at 21:36

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I'll try and answer the question as I've understood it. Memorizing is never a good idea. You should try working out the proof on your own. If you don't manage the proof even after trying several times, take a quick look and see how it's done and then come back to it. Try explaining the proof to a friend and ask him to challenge you. This is very important, since in your own mind everything seems put together and coherent but when somebody challenges you and asks even the most basics of questions, you realize that you haven't really understood the concept. This may sound cheesy but in my opinion, you are not doing mathematics when you are memorizing something. One part of it is about playing with different ideas that pop up in your head when trying to solve a problem. Some ideas and approaches, will end up being a futile effort and others will end making you feel all warm and fuzzy inside. Develop your own way of doing things. You will end up also understanding the other alternative proofs better. In a book, when you see a proof it all seems logically put together. It's almost like, "Wow, this is brilliant, I woulld have never thought of that." It looks so clean, because it doesn't contain any failed attempts, only those ideas that worked were presented. It's like looking at a Fashion magazine. You think to yourself, "Wow, european girls are hot!". Well, of course they are ,they don't show you the average or simple-looking ones. The failed efforts , I think , are more important. They help you realize what works and what doesn't work. You'll eventually internalize the concepts. All the arguments presented above are just some of them reasons, why you shouldn't memorize the proof even though you understand it. You'll be missing out on a lot!

The very fluffy Panda
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  • I'm not trying to drag this out. I should have been more specific with the context behind my question. The type of proofs I am referring to usually appear in articles, and are quite often, the only place in the literature where the proof is given. All the same, your answer is insightful. Thanks! – roo Apr 20 '14 at 01:15