Sunday, July 26, 2015

The Tao of Mathematics: Playing Chess with the Devil

Terry Tao
Wikipedia

Gareth Cook tells us of "The Singular Mind of Terry Tao: A prodigy grows up to become one of the greatest mathematicians in the world" (New York Times, July 24, 2015), but even for him, the process wasn't easy:
The true work of the mathematician is not experienced until the later parts of graduate school, when the student is challenged to create knowledge in the form of a novel proof. It is common to fill page after page with an attempt, the seasons turning, only to arrive precisely where you began, empty-handed - or to realize that a subtle flaw of logic doomed the whole enterprise from its outset. 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.
Does one ever defeat the Devil in this game? Not really. One might win a game, but that game is simply part of a larger game. One merely gains ground, game by game, in the great game of an infinite number of games, and the Devil has all the time in the world, but you don't. Interestingly, Fefferman takes the chess analogy further, as we see in an interview with Daniela Martínez Nava for Ciencia Nostra, “Math is a chess game against the devil” (June 10, 2012):
Math is like playing chess but you are playing a game against the devil. But you get to take steps back; you get to take back as many moves as you like. So if you play against the devil[,] you get crush[ed] and you think about why you lost and you try to change your move and again you are crush[ed]; you are simply wrong, whatever reason you thought was the reason for you being crush[ed] that's not the reason at all. But sooner or later maybe you get an idea and then you get a little scare[d] to play and get crush[ed] again[,] but you give it a try because the devil will have to make another move, and pretty soon after a few years of playing many games[,] you start to see what is going on[,] and after a while of fighting[,] you win. But while you play this game God is whispering in your ear "move your peon [i.e., pawn] over there, is all you have to do and then he is in real trouble[.]" But you can't hear, you are deaf . . . well not deaf[,] but you have to pay a lot of attention [because] if not[,] you can't know what God is telling you. But that is the spirit of the thing.
Interesting. God plays a role. I gather mathematicians sense that they dealing with a logical mind when they attempt to elucidate proofs . . .

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