Terence Tao on Nostr: Even within pure mathematics, we have learned that sometimes the best way to solve a ...

Even within pure mathematics, we have learned that sometimes the best way to solve a problem is to abstract away information that one would intuitively think to be highly relevant. Much of the progress in analytic number theory, for instance, has been obtained by adopting the perspective that strongly number-theoretic structures, such as the set of primes, should in fact often be treated as much less structured objects, for instance as a rather arbitrary set of numbers obeying some minimal set of combinatorial properties. Abstract too much, and one no longer has enough information to solve the problem; but with just the right amount of abstraction, the problem can move into sharper focus, suggesting the right set of techniques to attack it, and also exploiting the freedom of the setup to perform additional transformations that would not have made sense in the initial setting.

I sometimes like to joke that applied mathematicians need to know the first two chapters of every pure math textbook, but after that the subsequent chapters may have little (or even negative) value to them. On the other hand, the quest to locate Chapters 3-12 are often what made Chapters 1-2 the perfectly polished and useful gems of mathematics that have such broad utility... (3/3)