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Every child is a gifted child.
I think some intuition leaks out in every step of an induction proof.
MacPherson told me that my theorem can be viewed as blah blah blah Grothendieck blah blah blah, which makes it much more respectable.
What promotes math progress even more than new ideas are new technical tools and habits of thought that encapsulate existing ideas, so that insights of one generation become the instincts of the next.