Hofstadter, Dennett - The Mind's I

Prelude . . . Ant Fugue 153certain Diophantine equations. Now Mr. T has been for years trying to find a way ofreconstructing the sounds of Bach playing his harpsichord, which took place over twohundred years ago, from calculations involving the motions of all the molecules in theatmosphere at the present time.ANTEATER: Surely that is impossible! They are irretrievably gone, gone forever!ACHILLES: Thus think the naive ... But Mr. T has devoted many years to this problem, andcame to the realization that the whole thing hinged on the number of solutions to the equationin positive integers, with n > 2.a n +b n = c nTORTOISE: I could explain, of course, just how this equation arises, but I'm sure it wouldbore you.ACHILLES: It turned out that acoustico-retrieval theory predicts that the Bach sounds can beretrieved from the motion of all the molecules in the atmosphere, provided that there existseither at least one solution to the equationCRAB: Amazing!ANTEATER: Fantastic!TORTOISE: Who would have thought!ACHILLES: I was about to say, "provided that there exists either such a solution or a proofthat there are no solutions!" And therefore, Mr, T, in careful fashion, set about working atboth ends of the problem simultaneously. As it turns out, the discovery of the counterexamplewas the key ingredient to finding the proof, so the one led directl) to the other.CRAB: How could that be?TORTOISE: Well, you see, I had shown that the structural layout of an) proof of Fermat'sLast Theorem-if one existed-could be described by an elegant formula, which, it sohappened, depended on the values of a solution to a certain equation. When I found thissecond equation, to my surprise it turned out to be the Fermat equation. An amusingaccidental relationship between form and content. So when I found the counterexample, all Ineeded to do was to use those ( numbers as, a blueprint for constructing my proof that therewere no solutions to the equation. Remarkably simple, when you think