What's your approach to digesting proofs?

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What's your approach to digesting proofs?

Postby theodds » Tue Jun 21, 2011 7:45 pm UTC

This is mainly aimed how to deal with proofs that are technical. My reason for asking is that I've been slogging through about 300 pages of notes of the form Theorem-Proof-Theorem-Proof...with very little in the way of context, motivation, or examples. The proofs seem to just wander aimlessly until everything comes together so that the result is there, but not so much intuition (even though there often IS intuition to be had). I can go for awhile doing what I have done in the past (try to prove the result, if I get stuck peek at the proof for help, if the situation is hopeless then read proof line by line slowly), but I burn out very quickly given the way the notes are structured.

My method so far has been to first try and put the result in context (figure out why it's useful and try to figure out if the result is intuitive), then read the proof and try to get an idea of the skeleton; without the technical details, what is the overall goal? The proof becomes easier to chop up into pieces that way, and makes the technical details of each piece easier to deal with.

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Re: What's your approach to digesting proofs?

Postby OverBored » Tue Jun 21, 2011 8:05 pm UTC

What sort of area are the proofs in? I tend to try to read some notes and then a little later (say an hour or so) try to reprove the result, remembering that if my proof isn't the same then it could still be valid, but I need to be careful and check my proof fully.

In certain areas, you get a lot of very similar proofs, so it might be worth grouping them together and learning/understanding a more general form of the proof. Proofs are often a lot easier to understand if there is some motivating example or at least an example of application of the proof, so if one isn't provided, you could always try to come up with your own example.

I guess the final thing is that sometimes, the proof really doesn't matter much, just the result. In this case having an example is even more important, since reading the proof will shed little or no light on the theorem itself.

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Re: What's your approach to digesting proofs?

Postby Talith » Tue Jun 21, 2011 8:10 pm UTC

The context differs if you're reading from a technical paper aimed at concisely demonstrating a proof, or if you're reading material aimed at teaching a topic such as text books and lecture notes. Ideally, you'd want a published proof to give some kind of motivation and even examples for it to be accessible to as many people as possible but that's probably not true in the majority of cases. If you're aiming this more towards learning proofs for an exam, it's my experience that learning proofs is much easier if you recognise the proof from a lecture setting and not as just words in a textbook or set of lecture notes - trying to learn a proof by just bashing it in from rote memorisation is nothing compared to being slowly and confidently taught by someone that fully understands the proof (preferably with no time constraints and with a chance for you to ask questions but we can't have everything).

Some of my friends tend to disagree with me on this: they seem to have an ability to spend hours writing carbon copies of proofs until they can write it from pure memory, whereas I just wouldn't be able to do this so find it easier (as well as a bit more fulfilling) to understand the basic concepts of a proofs, try and get some kind of geometric interpretation if possible, and then if I'm still struggling (maybe it's a proof where some terms seem to just be plucked from thin air) then I write a short list of objects to memorise and then apply my usual technique to the rest of the proof.

It's also often really useful to try and justify a proof to yourself by looking for counterexamples and seeing where they fail to hold true. If nothing else, doing this will at least help you to gain an intuition for how concepts of the overall topic act together.

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Re: What's your approach to digesting proofs?

Postby achan1058 » Tue Jun 21, 2011 10:41 pm UTC

I really agree with the "trying to look for counterexample" part of looking at a theorem/proof. Often it gives you the reasons of why the conditions of the theorem is there, and what direction is the proof going. I also agree on memorizing line by line = BAD, BAD, BAD. You will probably ended up missing something on the exam. If you remember the key steps, you can just deduce the rest from logic anyways, and it saves you plenty of studying time. If you are doing something such as set theory or topology or analysis on some "strange" set of objects, usually they are a generalization of something well known, such as the reals of the complex. Try doing the proof on that space instead. It will give you some good insight on why the proof works.

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Re: What's your approach to digesting proofs?

Postby theodds » Tue Jun 21, 2011 11:55 pm UTC

Thanks for the replies. I don't anticipate ever having to reproduce these proofs for the most part - the ones I would be expected to reproduce are usually pretty easy once you have some intuition about the topic (measure-theoretic probability, mostly various kinds of convergence results, if anyone cares). Studying the proofs just helps one understand the material better. I would never memorize something by rote memorization. Usually if I end up knowing a proof by heart it is because I've studied the mechanics of the proof so much that the steps just stick in my head.

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Re: What's your approach to digesting proofs?

Postby Yakk » Wed Jun 22, 2011 8:24 pm UTC

I map up and down.

Map down to some "real" examples, or less abstract. Relate an infinite dimensional case down to a finite dimensional one, or a finite to a fixed dimensional one. Take a proof in the less abstract space and work out why it doesn't directly apply to the more abstract one.

Map up to some "general" versions. This rabbit hole is deep.

Break the assumptions down, and figure out why each is needed (the counterexample route -- or, rather, highlight the part of the proof that requires a certain assumption).

Clump and deconstruct the proof. Clump the proof into broad "goals" and "steps". Deconstruct the proof, associating one part with adjacent parts.

Understand the proof by necessary instead of sufficient -- understand the proof of the contra-positive as if it was a different problem entirely.

Build a lattice, instead of a narrative, so if a piece of the lattice is missing you can reconstruct it from the hole it leaves.

Steal a page from Feynman, and do the "concrete example" trick. Turn the proof into a description of a concrete example, and reason what is true there (or if it could be false). Figure out why your concrete example differs from the abstract one you are working on.
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Re: What's your approach to digesting proofs?

Postby coffeesneeze » Thu Jun 23, 2011 6:50 pm UTC

I have a friend that swears by latex. He studies by writing all his proofs in latex language. Something about that act helps solidify the concepts in his mind.
I always "draw a picture" of the parts of the proof I don't understand. Meaning I look for a geometric interpretation, as was already suggested in this thread.

I've noticed it is very difficult for me to predict what part of a proof will become clear to me next. Thanks for starting this discussion.

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Re: What's your approach to digesting proofs?

Postby mdyrud » Thu Jun 23, 2011 6:57 pm UTC

Definitely TeX a proof that is important and you don't quite get. It takes more concentration to write down the symbols in TeX than it does if you're just writing on paper so it gives you a chance to think and digest what is going on in the proof. Plus you can make a .pdf of the important proofs written in your own words that you can go back to and have as a backup in case something happens to your notes.

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