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 questions and answers (more coming soon):
- The difference between maximum and maximal. (JDH)
- What is this axiom saying? (Zhen Lin)
- Why we use the word 'compact' for compact spaces? (dtldarek)
- Is "imposing" one function onto another ever used in mathematics? (Steven Fontaine)
- Is every Noetherian module finitely-generated? (Arturo Magidin)