Saturday, July 25, 2015

What it's like to do research

The steady state of mathematical research is to be completely stuck. It is a process that Charles Fefferman of Princeton, himself a onetime math prodigy turned Fields medalist, likens to "playing chess with the devil." The rules of the devil’s game are special, though: The devil is vastly superior at chess, but, Fefferman explained, you may take back as many moves as you like, and the devil may not. You play a first game, and, of course, "he crushes you." So you take back moves and try something different, and he crushes you again, "in much the same way." If you are sufficiently wily, you will eventually discover a move that forces the devil to shift strategy; you still lose, but — aha! — you have your first clue.

As a group, the people drawn to mathematics tend to value certainty and logic and a neatness of outcome, so this game becomes a special kind of torture. And yet this is what any ­would-be mathematician must summon the courage to face down: weeks, months, years on a problem that may or may not even be possible to unlock. You find yourself sitting in a room without doors or windows, and you can shout and carry on all you want, but no one is listening. ...

In the game of devil’s chess, players have no real hope if they haven’t studied the winning games of the masters. A proof establishes facts that can be used in subsequent proofs, but it also offers a set of moves and strategies that forced the devil to submit — a devious way to pin one of his pieces or shut down a counterattack, or an endgame move that sacrifices a bishop to gain a winning position. Just as a chess player might examine variations of the Ruy Lopez and King’s Indian Defense, a mathematician might study particularly clever applications of the Chinese remainder theorem or the sieve of Eratosthenes. The wise player has a vast repertoire to draw on, and the crafty player intuits the move that suits the moment. ...

Early encounters with math can be misleading. The subject seems to be about learning rules — how and when to apply ancient tricks to arrive at an answer. Four cookies remain in the cookie jar; the ball moves at 12.5 feet per second. Really, though, to be a mathematician is to experiment. Mathematical research is a fundamentally creative act. Lore has it that when David Hilbert, arguably the most influential mathematician of fin de si├Ęcle Europe, heard that a colleague had left to pursue fiction, he quipped: "He did not have enough imagination for mathematics."