One of the things I enjoy reading about is the intuitions of experts in cold, hard, logical fields like mathematics and physics. Years ago I assumed that a breakthrough in such a field must require a similar thought process – rigid, methodological, with no room for intuition or guesswork. But it turns out this isn't true. Einstein's thought experiments appear to have been incredibly creative and heavily driven by intuition. Dirac wrote a paper on some vague intuitions about where the next breakthroughs towards a theory of everything might be.
Similarly, mathematicians also seem to embrace intuition:
But some problems look entirely unlike any solved problems. For example, two of the biggest open problems in the field of number theory are the twin primes conjecture and the Goldbach conjecture. They look a lot like each other, but they’re also distinct from anything else mathematicians have managed to prove.
Maynard thinks of them as a pair of islands — a remote archipelago. Their distance from the shores of mathematical knowledge implies that it’s going to take a big discovery to get there.