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I am in an extremely stressed out situation right now. I am pursuing Mathematical Sciences, as of now. I experience absolutely no difficulty in understanding the motivation or meaning underlying the several concepts and ideas one is taught in undergraduate courses. I am taking a few advanced and graduate courses as well and recently obtained a scholarship to pursue a research project.

I am extremely enthusiastic regarding what I do and I really love and have a passion for mathematics.

But the major problem I am facing right now is that I find myself incompetent in writing proofs in flawless rigour. I figure out the solution to any assigned problem as a rough intuitive idea, which in almost all cases is the underlying idea behind an elegant solution to the given problem. But once I can "see" the solution, I can no longer formulate it into a standard rigorous proof and this incompetence is hurting me severely in every step in "my career of exploration ". As a result, though I find that I can solve almost every problem assigned to me (provided I am given time to ponder over it), however hard they might seem to be, still I feel I lack the skill to formulate it into a perfect solution.

I am passing through an extremely depressing phase of my life and I would be highly indebted to anyone who lends me a helping hand in this regard. You can suggest me any literature to study, any class or lecture to attend or anything else that you feel would be beneficial for me.

closed as off-topic by Alexander Woo, scaaahu, user3209815, Federico Poloni, Massimo Ortolano Jun 5 '18 at 9:01

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  • First you should create your own account... – Solar Mike Jun 5 '18 at 6:38
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    You should ask your professors for help, not a bunch of strangers on the Internet who don't know you or your work. Helping you is, after all, part of their job. – Alexander Woo Jun 5 '18 at 7:08
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    What leads you to say that you are "incompetent in writing proofs in flawless rigour"? Has someone told you this or is it your own judgement? What do other people think of your proofs? Writing proofs is a communication exercise - if other people can read and accept your proofs then you are done. If you want full rigour then perhaps looking at an interactive theorem prover such as Isabelle might help. – Mark Wassell Jun 5 '18 at 7:09
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    I am afraid this is off-topic here. Learning to write rigorous proofs properly is one of the objectives of undergraduate courses in mathematics (often taught using copious amounts of epsilon and deltas). – Federico Poloni Jun 5 '18 at 8:21
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Your experience in dealing with proofs is not unusual for someone at your level (late undergraduate by the sounds of it), and it is likely that the situation will improve with practice and experience. It is quite difficult for many students at your level to obtain any intuitive grasp of mathematical problems, so if you are already able to intuit a rough solution, without proof, you are probably already ahead of the curve. Nevertheless, there is a substantial gap between being able to "see" a result intuitively and being able to prove that result. Here are some suggestions for learning to bridge that gap:

  • If you have an intuitive idea of the solution to a problem, this means that you have some intuitive idea of "why" certain conditions imply a certain result. See if you can break this down into smaller steps, and learn to write your steps out as conjectures (i.e., statements that might be true but you're not sure yet). Try to prove (or disprove) these smaller conjectures. Pretend you are trying to explain it to a small child, who keeps incessantly asking, "but why?".

  • Try to prove a step in your result, and if you keep hitting a wall, try to find a counter-example to show it is false. Work both ends of the problem, and let your failures on one end inform your search on the other end. (For example, if all your attempted counter-examples fail, try to find the reason they are failing, which might give you an approach for positive proof.) If a statement is true you should be able to prove it to be true; if it is false, you should be able to find a counter-example to show its falsity.**

  • If the theorem you are trying to prove is too hard, try taking an applied example and working it out to see the mechanics of what is happening. Do this for a few examples until you see a pattern at work that might be a basis for a broader proof. Try proving a more specific theorem and then generalise this to a broader theorem.

  • Whenever things get too complex, consider re-framing your notation to simplify what you're doing. Always look for avenues to simplify the problem, or simplify your descriptive notation. Bad notation can often obscure a problem, and good notation can get to the heart of a matter. Try to keep your writing clear and explain intuitively what you are doing as you go. Don't let formalities overwhelm the intuitive explanation of what is occurring in the proof.

  • Read lots and lots of proofs and make sure you understand the techniques that other authors have used to prove their theorems. This will expand your "tool-kit" for when you encounter a statement that you want to prove.

Lastly, don't get depressed over the difficulty of acquiring a difficult skill. Mathematicians are rare because learning mathematics to that level is hard. It takes talent and ability, but it also requires practice and experience. Math students usually don't become competent in proving theorems until they are at least part-way through grad-school, and many are not great at it even when they finish. If your grades are okay, that means you are learning at the expected pace.


** Technically there are statements that are true but not provable in mathematics, but you won't encounter these in any courses outside of advanced courses in logic/foundations of math.

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