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Originally Posted by Urinal Mint
Thanks for your input. It'll be exciting and somewhat intimidating when I ask a question like that.
You think so, but mainly this happens as a byproduct of learning. You observe things, and your natural curiosity leads you to ask a lot of questions, not all of which have readily available answers. So you start thinking harder, and most of the time, you'll come to a pretty uninteresting answer -- uninteresting because when you think about it, the answer will seem obvious in hindsight.
Then there will be the other questions, those which you answer, but it doesn't seem trivial in hindsight. And then there will be those you can't answer.
And you'll have conversations with colleagues, and you'll develop some sort of taste -- i.e., what types of problems people care about and what people don't. You work on and publish what people care about, because that's how the game is played if you want a job.
Learning all of this is the transition from student to mathematician.
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Basically thesis is that students aren't doing any math but are just being told steps to follow and doing them over "exercises." In this system students good at following directions, not necessarily thinking about / proving mathematical principles and solving problems, succeed.
Well, I think you develop a lot of intuition by solving problems. And I think, at least for myself, watching others' methods, techniques, and routines -- learning and understanding their toolboxes -- has made me a better mathematician. In this sense I think "intuition" can actually be learned.
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What struck me is the definition of a mathematician. Does learning and understanding mathematical concepts, even if you aren't coming up with new ones of your own, make you a good mathematician? This question can be further generalized to learning and understanding in all fields as opposed to creating and thinking independently.
I think we can philosophize about this all day, but I am certain that there are folks who have extremely successful careers in mathematics that do nothing more than learn fields that others have developed and do hard calculations or extend others' theorems to new settings (even in relatively uninteresting ways) or whatever. These kind of things won't win you a Fields Medal, but they certainly can get you tenure at a mid-level school. Whether this is acceptable to you all depends on your objective function.