How can we distinguish between crackpots and the intellectually eccentric? Why do crackpots flood the inboxes of mathematics professors everywhere with purported proofs of the Riemann Hypothesis, while the merely eccentric toil endlessly despite the social and career obstacles their interests might pose? To put the question more precisely: why do crackpots almost invariably purport to solve the same problems, while the eccentric only reticently claims success but insists she asked many more interesting questions?

Every student learns that a function is continuous if its graph can be drawn without lifting her pencil from the paper. She also learns that this definition of continuity is not mathematically rigorous and was corrected some centuries after the development of calculus. However, crackpots almost never focus on obvious issues like the continuity of functions. Instead, they purport to have solved the ABC conjecture, undermined the foundations of set theory, or discovered some obscure proof that P = NP. Why?