I was this guy to certain professors: at the end of the day it's less about formality and more about someone's ability to understand you at a high level and assess you. To me, I was trying to convey my ideas as precisely as possible. To my professor, I was being so verbose that it would have taken him at least an hour to thoroughly assess the rigor with which I had approach my proof, which simply doesn't scale to all students.
Your friend in math is definitions. IME clever definitions minimize the sheer amount of rigor you need to get from point A to point B through their abstractions. The more "natural" or easily understandable a definition is, the easier it is to use that definition as a ground truth in your theorems.
For example, I have an unpublished proof of a graph/game-theory conjecture. Proving the theorem's correctness is extremely convoluted if you rely on atomic definitions of graphs, valid actions, etc. However, as you define new relationships precisely, it becomes much easier. The more abstractly you approach the problem, the simpler the problem becomes, given the correct abstractions.
Your friend in math is definitions. IME clever definitions minimize the sheer amount of rigor you need to get from point A to point B through their abstractions. The more "natural" or easily understandable a definition is, the easier it is to use that definition as a ground truth in your theorems.
For example, I have an unpublished proof of a graph/game-theory conjecture. Proving the theorem's correctness is extremely convoluted if you rely on atomic definitions of graphs, valid actions, etc. However, as you define new relationships precisely, it becomes much easier. The more abstractly you approach the problem, the simpler the problem becomes, given the correct abstractions.