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I am aware that similar questions have been asked here and elsewhere about how to learn from proofs. Some common advice is:

  1. Most proofs are written in a polished form, not how they were first discovered. Look at the polished proof and try to figure out how it was first discovered.

  2. Don't just try to understand a proof line-by-line. Instead, try to capture the main ideas and retain them, instead of retaining the details.

  3. Try to figure out the proof on your own, and use the book proof as a hint.

  4. Try removing one hypothesis at a time, and finding counterexamples.

This is all very good advice, and I have used all of it when studying pure math. However, I have recently changed to studying applied math, and I am unable to apply these strategies sucesfully most of the time. I will try to explain why:

Pure math seems a lot more clean. Take the Sylow Theorems, or the Heine-Borel Theorem as an example. Their proofs may be very tricky to come up with from complete scratch; but you can summarize the proofs in 2-3 key steps, and if you remember these, it's not difficult to reproduce the entire proof. These theorems also have relatively few hypotheses, and its not too difficult to come up with counterexamples if you remove certain hypotheses.

The proofs in applied math are very different. First, they often have many more technical hypotheses; "this constand it less than $1/2$, this variable is bounded by this complicated function", etc. Therefore, it very difficult (and to me, unenlightening) to try to come up with counterexamples which show the neccesity of these very specific hypotheses.

Secondly, the proofs often consist of a lot of heavy manipulations that are very difficult to remember. At each step, you may have 2-6 manipulations that you can consider: Taylor expand this to first order, Taylor expand that to second order, use Triangle Inequality here, make this substitution there, etc. If the proof is 4-5 steps, there may have been 20-50 wrong routes that you could take. This makes the proof both very difficult to remember, and very difficult to come up with.

To illustrate my point visually, here is a proof from pure math that I am used to, and here is a typical proof I encounter in applied math:

Example of pure math proof:

enter image description here

Example of applied math proof:

enter image description here enter image description here

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    $\begingroup$ With your two examples you are, in some sense, comparing apples to oranges. Part of the reason why proofs of metric space theorems are "cleaner" than proofs of applied math theorems is that the statements of metric space theorems are themselves "cleaner", more abstract, simpler. To emphasize the point, in the metric space theorem one can actually understand the statement, perhaps after looking up some rather standard definitions. Whereas in the applied math theorem, I have no idea what "Algorithm 14.1" means, and I could not find out by looking anything up whatsoever. $\endgroup$ – Lee Mosher Aug 4 at 17:47
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    $\begingroup$ If you want some pure math theorems which compare more realistically with applied math theorems, take a look instead at theories of differential topology, differential geometry, or ordinary differential equations. $\endgroup$ – Lee Mosher Aug 4 at 17:48
  • $\begingroup$ @LeeMosher I see. I did just look up a differential geometry book and the theorems are indeed more messy. Do you have any advice on how a student should learn from these messy proofs? $\endgroup$ – Blue Aug 4 at 21:25
  • $\begingroup$ Both of those proofs you linked are about estimates (although the metric space theorem has particularly clean). Learning how to do estimates is very important. Look at lots of theorems that use estimates, try to do the estimates on you own, look for hints from the proof when you get stuck, and so on. Even the theorems with the cleanest estimates, such as that metric space theorem, will help you hone your skills. $\endgroup$ – Lee Mosher Aug 5 at 14:01
  • $\begingroup$ @LeeMosher Thank you $\endgroup$ – Blue Aug 5 at 17:14
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If you were to advance in Pure Mathematics, you would also find that the theorems get more technical, with all kinds of messy hypotheses. What you are seeing in Pure Mathematics are results from a century or so ago. Many books have been written about them, and the has been a lot of time to clean up the results and proofs.

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  • $\begingroup$ That is a good point which I hadn't considered. I guess I just have to deal with it. Do you have any advice on how a student should learn from these messy proofs? $\endgroup$ – Blue Aug 4 at 21:29
  • $\begingroup$ You have to slog through them. After thinking about them for a while, you begin to see the big picture. When you can rewrite the proof so that it is really clean, then you really understand it. But it is a lot of hard work. If you want to go into research mathematics, it will be very helpful acquire this skill of reading long and messy proofs. It will get easier in time. I will admit that it is my least favorite part of being a professional mathematician. But I would also say that my career has suffered because I didn't work hard enough at this. $\endgroup$ – Stephen Montgomery-Smith Aug 4 at 22:11
  • $\begingroup$ Thank you very much. Your advice is very encouraging - I was worried that there was something wrong with me for finding these things so difficult. $\endgroup$ – Blue Aug 5 at 0:43
  • $\begingroup$ If I may, I would like to ask you one more question: is it a good idea to memorize the messy hypotheses, or should I just remember "for nice functions $f$, we have this conclusion"? $\endgroup$ – Blue Aug 6 at 19:35
  • $\begingroup$ I would go with "nice functions." With a lot of these theorems, they create the hypotheses as they try to create a proof. $\endgroup$ – Stephen Montgomery-Smith Aug 6 at 20:17
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It may also be worth mentioning that in the kind of result you're seeing in "Applied Math" (which, as Stephen remarked, you might also see in "Pure Math"), the statement of the theorem is developed at the same time as the proof.
The process might go something like this.

We want to prove some conclusion, say $\lim\inf_{k \to \infty} \|g_k\| = 0$, under some conditions.
What should those conditions be? It's usually not realistic to expect an "if and only if" condition, but other things being equal it's better for our theorem to be as widely applicable as possible, and maybe we have some examples in mind that we want to be covered. We develop an outline of how we might expect to prove the conclusion for something like our examples, and along the way we see what conditions need to be true in order for this to work. Now look at each of these conditions. Might it follow from something else? If so, deriving that condition will become part of the proof. If not, the condition becomes one of the hypotheses.

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  • $\begingroup$ Thanks, that's a nice explanation. Do you have any advice on how a student should learn from these messy proofs? $\endgroup$ – Blue Aug 4 at 21:24

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