The best mathematics produced in any given century neither completely ignores the dominant paradigm(s) of that century, nor allows itself to be trapped by them. It works at the edges of established mathematics, moving freely between civilization and unfettered jungle. We should create more mathematics that is moderately subversive, that makes use of existing ideas when they're appropriate but ventures deep into the jungle when they're not.
A list of my favorite answers (more coming soon):
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