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fresh_42

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russ_watters

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To expand on the previous answer: the place to go to learn how to do math research is college, to get a PhD.

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mathwonk

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as mentioned above it helps to know a lot of math first, and to learn that math from reading papers written by the best mathematicians you can understand. the best researchers use the most original and insightful ideas, and it helps to learn those. but even when learning from books and courses it helps to take an inquisitive approach to your learning. don't be content just to memorize an argument for a theorem, ask yourself why it works, how someone might think of it, and whether there is a simpler way to do it. when you read a theorem, stop before reading the proof and try to make up your own proof. you will often fail, but when you succeed even partially, either you will have learned the motivation behind the proof in your book, or better, you will have found a new idea of your own. when you learn a new theorem, try to come up with counterexamples to various tweaks of that theorem that relax hypotheses, thus learning why the hypotheses have been chosen as they were. sometimes you may find the theorem is actually true more generally than stated.

for general problem solving tools, take a look at "How to solve it" by the great George Polya. One of his great pieces of advice I remember is: "when you solve a problem, look around for another that is solved by the same method, problems are like grapes, they come in bunches". Also inspiring may be the book "On the psychology of mathematical invention" by Jacques Hadamard. And I second the advice attributed to the genius Niels Abel to read, at least whenever possible, " the masters rather than the pupils". And remember whenever you solve any problem on your own, no matter how small, you have made a start at finding new mathematical insights. if you keep it up long enough you may find one no one else has solved, or may find a solution different from that of others. best of all if you love this endeavor, you will spend many happy hours, even if many are also frustrating. this is a pursuit for years to come, so you must enjoy it.

for general problem solving tools, take a look at "How to solve it" by the great George Polya. One of his great pieces of advice I remember is: "when you solve a problem, look around for another that is solved by the same method, problems are like grapes, they come in bunches". Also inspiring may be the book "On the psychology of mathematical invention" by Jacques Hadamard. And I second the advice attributed to the genius Niels Abel to read, at least whenever possible, " the masters rather than the pupils". And remember whenever you solve any problem on your own, no matter how small, you have made a start at finding new mathematical insights. if you keep it up long enough you may find one no one else has solved, or may find a solution different from that of others. best of all if you love this endeavor, you will spend many happy hours, even if many are also frustrating. this is a pursuit for years to come, so you must enjoy it.

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