hm.. this isn't coming from gearing toward them being a mathematician, but it is coming from gearing toward them being able to "theorize seriously" in some general sense, like be able to see deeper into things than the intuitions we develop naturally. maybe we differ in that goal
... and then not getting bored with trying really hard to answer the questions, even though they're not otherwise interesting)
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all this about math being about "seeing into the core of things" cashes out in many ways, but for example, you have lots of physics insights coming along seemingly necessarily with tough math, eg electromagnetism (field theory), GR (riemannian geometry), QM (hilbert spaces)
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like, only doing math when you're led to it directly by something else you were doing... is actually a cool thing people should try; but i think even better is accounting for the straightforward induction saying that that strategy doesn't shave the yak enough.
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