I'm near the end of Velleman's How to Prove It, self-studying and learning a lot about proofs. This book teaches you how to express ideas rigorously in logic notation, prove the theorem logically, and then "translate" it back to English for the written proof. I've noticed that because of the way it was taught I have a really hard time even approaching a proof without first expressing everything rigorously in logic statements. Is that a problem? I feel like I should be able to manipulate the concepts correctly enough without having to literally encode everything. Is logic a crutch? Or is it normal to have to do that?
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It's perfectly normal. In fact, I think that's how a mathematitian's mind grows.
At least, that's how it's worked for me. Mind you, the intuition in the final step is very different than the original intuition. The original one is the brash "d'ah, how can you ever doubt that?!" kind of thing that is embarasingly bad at doing math.
The more developed intuition in step 3 is much different. It has a lot more thought put into it. It's more "yeah, this is so, and I know approximately how I can prove this using strict logic, but since it would take 3 pages, I won't use strict logic."
Sure, the second intuition can also be wrong. And every once in a while, it is. But its performance is way way way way way better than in the beginning, and it also has a fall-back. If all else fails, your final answer is no longer "well, but, but how could it not be true?", the final answer is "oh alright fine, I'll spell it out rigorously!"
Logic is not a crutch. It is a way to check that you haven't made a mistake. It might not be the best first step to make a proof, but it is definitely a necessary step.
As you get more experienced, you tend to start by sketching the proof in very informal terms. This is like navigating by large landmarks: "I am headed towards that mountain over there, hm, that ridge seems like a good approach. OK, lets go."
However, to actually get to that mountain you still put one foot in front of another, one step at a time for hours. Logic is like walking one step at a time, while looking at your feet and the nearby ground. It is necessary, but sometimes you need to look up to see where you should be going.
Now, if you are going to write the proof in English eventually, you don't have to actually write down all the logic in symbols. I find that I can write English (or Norwegian) while keeping the formal symbols in my mind.
Some published proofs aren't very formal. The reason for this is that formal proofs can be very long. Instead they give arguments that convince the reader that it is possible to work out the proof in all its details, even if neither the writer or the reader have actually done so.
Professional mathematicians become good at telling whether such an informal argument is valid or not, but they occasionally slip up, even peer-reviewed articles can be wrong. So, take care.