Is there a point to physically writing down proofs? 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?
 A: 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. :)
A: 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.
