On 21–23 June 1993, Andrew Wiles gave three lectures at the Isaac Newton Institute in Cambridge announcing a proof of the Taniyama–Shimura conjecture for semistable elliptic curves, and with it Fermat's Last Theorem. In late summer 1993 the referee, Nick Katz, found a gap in the manuscript: the Euler system used to extend the method of Kolyvagin and Flach was incomplete. Wiles later told NOVA it was “an error in a crucial part of the argument, but it was something so subtle that I'd missed it completely until that point.” For nearly a year he tried to repair it alone, and then, at Peter Sarnak's suggestion, brought in his former student Richard Taylor. Together they still found no way through.

On 19 September 1994 Wiles took one last look at his original route (Iwasawa theory) while studying why the Kolyvagin–Flach method failed. He saw that the failure itself meant the original route could be made to work, once strengthened with what he had learned from the method that had failed. In his words: “Suddenly, totally unexpectedly, I had this incredible revelation,” and it was “so indescribably beautiful; it was so simple and so elegant.” On 24 October 1994 he submitted two manuscripts, the second written with Taylor; both appeared in the Annals of Mathematics in May 1995.

This is the tenth ray, the Beauty of the Self, mainly in its generative face. What a judge cannot not acknowledge here is where the error went. A proof that fails can be blamed on the world, on a method that was never going to work, or on someone else. What happens here is that the wrong seeing lands on the one who saw: this is the proof I announced, this is the place I did not see, and that “I” has an origin (a route he had himself chosen), an edge (a spot he could not see), and a direction not yet his own. Then what lets him go on is that he looks hard at the route that failed: the failed method is not thrown away, it says what the original route was missing. That is the chisel-construct cycle: the chisel is the already constructed judgment that Kolyvagin–Flach could stand in for the original route, the construct is the original route as strengthened by it.

What it is not: not perseverance and not humility, both of which are praise of a person, whereas what is to be seen here is structure, not character; not the relief of “the problem is finally solved”, which belongs to usefulness and correctness; not reverence for a famous result. We can say only what is on record: when the gap appeared, a year of failure, one specific date, and his own description of that moment. Nobody is ranked against anyone; what is pointed to is one act: the error folding back onto its maker did not crush him but became the way this person saw his own first choice clearly.

Its remainder is plain: here there is an “I”, and where that I goes next is not a question for this ray; it belongs to the beauty of purpose, and this text has no standing to answer it for Wiles. It is worth seeing again now because the story is so easily told as a genius's triumph, when what is on record is finer: a gap he had not seen himself, and the one time he looked back.

Han Qin (秦汉) · @hqinjarsy · SAE Judgment & Aesthetics · Ray 10