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I try to go through every exercise in a book when I'm reading it. Of course, there are going to be a range of routine calculations/more trivial results/harder results that are in each book. With the easier exercises there are many times where I read them, think about them for a bit, see how it follows in my head, and simply move on. Similarly, if I'm in the middle of writing a proof that wasn't immediate and I see in my head how to finish it, I'll just stop writing and move on.

This might be a silly question but is there any point to physically writing down each proof from start to finish? Or is it fine to do what I've been doing?

beeclu
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  • I'd say it's fine to do what you're doing. It would take too much time to write out every last detail of every proof. Seeing the big idea is more important in my opinion. – morrowmh Aug 11 '22 at 22:43
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    Whatever works for you is fine. You might consider writing down enough so that should you return to your work later there's enough so that you can remember what you were thinking when you felt you understood it and didn't need to write any more. – Ethan Bolker Aug 11 '22 at 22:46
  • I agree with Ethan. Also, most (all?) proofs have "key/core ideas" that glue them together. Recognizing these and even jotting them down can help. – morrowmh Aug 11 '22 at 22:47
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    With textbook exercises, you can pretty-much stop when you see that the result is in sight. That said, proof-writing itself is something of an art, so there is value in practicing how to structure and streamline an argument. – Blue Aug 11 '22 at 22:51
  • Personally, I use LaTex to keep a private notebook of all of my private work. So, I take every problem to completion. This way, 6 months (or 6 years) later, I can review the pdf to refresh my memory on the topic. – user2661923 Aug 12 '22 at 00:17

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An interesting question.

On one hand, you should do what you feel like doing! Stop doing a thing when it becomes boring or "toooo obvious".

Also, even if your study/work needs to satisfy some external diagnostic (exams, etc.), it is entirely possible that you understand things well (enough) without writing things down to the bitter end.

On the other hand, still, we can inadvertently deceive ourselves, so (as has happened to me many times over the years) what I had thought was a trivial part was actually the crucial point... and maybe didn't even work out at all.

I do tend to recommend "moving forward" more quickly than not, because seeing later stuff usually gives extremely useful perspectives on the earlier stuff. And, all too often in textbooks, we see exercises given which are pretty ugly to do without later methods, but are obvious corollaries of those later methods. An argument in favor of this prank is that it teaches us appreciation of the later methods... and maybe that's a good thing, for some people, in some situations. E.g., when I was much younger, I was skeptical that "fancy ideas" would help address seemingly simple issues. But encounters with some very good mathematicians changed my mind. :)

paul garrett
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  • This all sounds like good advice to me. But what about proof writing for the purpose of demonstrating a fact to someone else? Textbook exercises often give an opportunity to do this -- but if one is self-studying, with nobody to show the proof to, perhaps it is not such a useful exercise. – David K Aug 11 '22 at 23:01
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    True, even apart from the "persuade the grader you know what you're talking about", imagining we're outside of a grades-and-exams situation, and just trying to persuade a curious or mildly skeptical person that something is true... yes, examples are good, and a sketch-of-proof, maybe or maybe not executing all the gory details, depending on the situation. And, yes, to be deprived of this "advocacy exercise" in a self-study situation is one of the limitations of it. – paul garrett Aug 11 '22 at 23:13
  • It occurs to me now that on occasion, when I've had a work-related problem whose solution was an algorithm that I needed to be sure about, I've written up a proof as if I were writing it for an audience where the audience is me at some later time. But that relies on skills of writing and critiquing proofs that (as you mention) are hard to acquire by self-study alone. – David K Aug 11 '22 at 23:20
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The act of writing/LaTeXing helps us to organize our sometimes muddled thoughts. It is in some ways simulating giving a presentation on the proof. Indeed, when we write something down, perhaps we have an audience (whether that audience is ourselves or someone else) in mind.

I’m not sure if this is true but I have heard that Einstein and Feynman (and probably others) would, when stuck on a problem, pretend as though they were presenting the problem to a class.

(I have tried to perform this practice as well but with a twist: adding a bright student in the audience who challenges every nontrivial claim or logical step I make in my argument forcing me to justify them)

Perhaps the act of writing/presenting forces an organizational task on ourselves which in turn gives us a deeper understanding of the topic we are writing about.

To get a bit meta, the act of writing this answer has given me a deeper understanding of why we write down our proofs.

JazzGuitar7
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