How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

As F. Scott Fitzgerald said, “The test of a first-rate intelligence is the ability to hold two opposed ideas in the mind at the same time, and still retain the ability to function.” Mathematicians use this ability as a basic tool of thought. It’s essential for the reductio ad absurdum, which requires you to hold in your mind a proposition you believe to be false and reason as if you think it’s true: suppose the square root of 2 is a rational number, even though I’m trying to prove it’s not. . . . It is lucid dreaming of a very systematic kind. And we can do it without short-circuiting ourselves. In fact, it’s a common piece of folk advice—I know I heard it from my PhD adviser, and presumably he from his, etc.—that when you’re working hard on a theorem you should try to prove it by day and disprove it by night. (The precise frequency of the toggle isn’t critical; it’s said of the topologist R. H. Bing that his habit was to split each month between two weeks trying to prove the Poincaré Conjecture and two weeks trying to find a counterexample.*)