I still feel confused after I prove something.

I don't know about you, but this used to happen to me in the past after prolonged periods (i.e. many months) of no breaks from studying.

I feel like your experience level is needed here. More information is needed.

But I feel like this sometimes if I am trying to over-prove something. I often go too far in the details and sometimes I find I just don’t know those details to completion so, for me, it feels like I don’t prove it because I don’t also single handedly prove all the known theorems I referenced. But that’s when mentors just tell me, “nah- it’s enough!” And it is but for me personally it signals I need to study something else to understand, in that deep spiritual way, why it was working within tje proof.

It could be you’ve just been staring at the problem too long. But also, yeah it can happen you find a solution but don’t really know how or why it works. Or even worse it’s a proof that works but is complicated and not very illuminating.

That does happen a lot, even in research. And only years later after spending a lot of time thinking you find a more elegant or illuminating proof. Is that what you might mean?

Maybe there’s a few levels at work here.

For me to have proved something, I need to at least convince myself that my proof is correct. I’m definitely not satisfied with just someone else saying my writeup/idea is correct. Maybe the words you’ve said or written are enough to someone more proficient, but I think it’s important to go at least as far as getting to a proof that you are confident is correct.

But, being confident in correctness is very different, imo, from understanding/not being confused. This is maybe the next level you can go to with your proof—first correctness, second understanding.

After I’ve proved something, I’m often still very confused. To try to get past this, I’ll stare at my argument and try to extract the “moral”. I’ll ask if maybe my argument could prove something more general. I’ll also ask why can’t my argument prove a contradiction (ie something too general)—this is often very helpful, maybe this step shows you where your argument uses your hypotheses in important ways. This gets me closer to understanding the “why” of my proof, rather than just seeing that every step is logically correct.

Unfortunately, sometimes the morals only come to me later—maybe once I’ve studied some related thing, I’ll realize my argument is adjacent to or even a special case of something grander and better motivated.

Confused about what? Whether you’ve correctly proved the result? If so, then your proof probably isn’t that great if you can’t even convince yourself with it, even if it’s correct. But it’s always a good idea just to go over the main ideas again and make sure they make sense, and check that all your algebra is correct. If anything expand on a few shaky or hand-wavy areas and hopefully your own doubts of your proof should be relieved

I am with u

Think about it as you're going to sleep, - it might 'click' in your brain there or when you wake up. I've followed and used proofs i've not fully understood, and it's frustrating, but it's a good feeling once they do 'click' in your brain

It probably means that you are executing proofs methods and tricks without having the time and/or mental energy to reflect on what it means. You are sequencing logical steps like an automaton, but if someone gave you an "obviously false theorem" (or more likely, if you seriously misinterpret an existing theorem) you might use it without ever realising that the conclusion is absurd.

Depending on the kind of thing you prove (how constructive the proof is), you can sometimes trace an example from the beginning to the end of the proof, and following the path of that example can really help you understand what is the practical meaning of each of the steps you took. Tracing a counter-example and looking at where the proof breaks also does wonders.

Reprove it from scratch a few more times. Sometimes repeating the same method, sometimes with other methods or variations. Do that until you can reproduce the proof from scratch without checking any references. Be able to explain it to an appropriate audience, again without referencing anything. By then your confusion will be minimal.

Being able to prove something doesn't mean all confusion is gone. It just means you can understand each step (with appropriate effort).

I relate. One solution is working with many different examples. When I went into university for maths, I had a naive idea that the heart of mathematics is made up of detached, terse "proofs from the book". The reality is that professional mathematicians work with concrete examples all the time: for motivation, intuition and hints at conjecture. I believe even Richard Borcherds talked in an interview about the importance of working with examples. I have certainly found, in group theory, say, that proofs and examples go hand in hand. Take the proof of Sylow's theorem using group actions - drawing a few Cayley graphs and coset graphs will not only suggest how to come up with the proof, but help you remember it, as if it is second nature.

Is there another way to look at parts of the proof that you understand?

The confusion might arise from your perspective being unexpressed?

There is a difference between proving something within the framework of the definitions and theorems you know in the formal sense, e.g you have chained together enough implications to arrive at the result, and proving something within that framework and motivated by a deeper insight about why the thing you are proving ought to be true, which then governs the sequence of definitions and theorems you use to arrive at the result.

Being able to do the former is a good sign and being able to do the latter is even better but not always necessary for time reasons. Over time it gets slightly easier to do the latter the more you engage with whatever field you are studying.

Just go through every proof of every theorem you used and you should be satisfied.

To avoid that, I always try to understand all the theorems that I use, and I go over their proof.

It just feels wrong for me to use a theorem whose proof I haven't understood

To orove something means that it's correct and you understand it. You have a proof but i wouldn't say that you proved it. Idk maybe it's just ne