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<p>I am not at all an authority on the subject, but I am a math PhD candidate at Stanford and I can give you my own perspective. </p>
<p>In my opinion, you don’t really find out if you are good at mathematical research until you do mathematical research, i.e. towards the end of college (for the very accelerated students) or in graduate school itself. Short of doing actual research, a good track record in extra-curricular problem solving seems to be the next best indicator. Especially if you like problems that are not solved by “standard” techniques, but rather require you to be creative and think of a new approach yourself. (Suppose you live in the middle ages and calculus has not been invented yet. Pi isn’t defined yet either. How would you determine the area of a disk?) </p>
<p>I disagree with ucbalumnus that your performance in “rigorous” math courses is a good indicator. Certainly being able to write a proof is a prerequisite for successful graduate study but it’s nowhere near sufficient. (Mastery of spelling and grammar does not make a good playwright either.) Good research actually requires a very different skill set from algebra and analysis problem sets. </p>
<p>Here’s a successful mathematician’s take on the subject: [There?s</a> more to mathematics than rigour and proofs | What’s new](<a href=“http://terrytao.■■■■■■■■■■■■■/career-advice/there’s-more-to-mathematics-than-rigour-and-proofs/]There?s”>http://terrytao.■■■■■■■■■■■■■/career-advice/there’s-more-to-mathematics-than-rigour-and-proofs/)</p>