Terence Tao on Nostr: Pure mathematics is largely a field of curiosity-driven research, and as such has ...

Pure mathematics is largely a field of curiosity-driven research, and as such has significantly more flexibility in problem selection and solution technique selection than in other disciplines. A civil engineer, tasked to build a bridge across a river, cannot simply declare that one should first work on the toy problem of building a shorter bridge over a different creek; or a surgeon who is tasked to remove a tumor on a patient, cannot simply declare that there is an experimental alternative procedure that they would like to perform instead to eliminate the tumor. But a mathematician can, and often does, start working on a challenging problem by working out simpler model problems first; and the exercise of applying an experimental technique different from the traditional ones, or by varying the inputs or parameters of a method to be far from what textbooks would recommend, is a very fruitful one.

This flexibility can be hard to adapt to for some students of mathematics. I have had undergraduates in my graduate complex analysis class ask me in worried tones if it is permitted to solve a homework problem using a mathematical method they learned in a different class, or if they should work on an unassigned homework problem because they suspect that its solution may be relevant to an assigned one.
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