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But besides the Proof side of mathematics there's the art of defining ideas in a mathematical fashion/rigorous fashion ( like the definition of limit of a sequence ) which feels to me as being doable by someone who is trained in doing mathematical proofs but also requires some experience of taking ideas and attempting to formulate them/define them ( this is an aspect of some people who are researching mathematics would try to gain experience in).

One book that I consider the holy grail of learning to do mathematical proofs ( which helped me tremendously ) is

*How to prove it*by

*Velleman.*( Another helpful book I've read some parts of it is

*Polya's*book )

I learned all the topics in the book, did almost all of the exercises, I've been doing math courses which are very proof-oriented like Real Analysis, Set Theory..., all of which are close hand to the mathematician's framework of proofs, rigour and language. But sometimes when I want to take a concept and formulate it, I feel stumbled ( I did learn about formulation with regards to learning predicate logic. But I'm talking about deeper stuff, how to be able to formulate constructions/ideas either for the sake of themselves or for the sake of solving a problem, such as the one provided in the answer in this question: https://cs.stackexchange.com/questions/54990/filling-bins-with-pairs-of-balls)

I don't know of any book that teaches you how to take abstract ideas and define/formulate them in a rigorous/mathematical fashion, and I'm interested in finding such a book to learn from, do you know of any such book that you can recommend about? or do you have a recommendation as to how to get better at this skill? ( because just doing proofs doesn't help )