Andrew Wiles on proving Fermat’s Last Theorem (1995) [video]

TL;DR: Andrew Wiles recalls the final crucial turning point in his proof of Fermat's Last Theorem: after what seemed like an insurmountable obstacle, a sudden flash of insight made the proof emerge whole from the "ashes." ## Seven Years: A Private Battle During the first seven years of working on the proof of Fermat's Last Theorem, Wiles says he loved every minute of it. No matter how difficult the problem, there were always setbacks and often seemingly insurmountable obstacles, but this was a private, deeply personal battle he had committed himself to. ## September: Back to the Original Structure By September, Wiles decided to go back and re-examine the original structure proposed by Flack and Colyvan, trying to pinpoint exactly why it didn't work. He wanted to attempt to formulate it precisely. He says that in mathematics one can never truly do that, but he just wanted to put his mind at ease and confirm that the approach really couldn't be made to work. ## The Incredible Revelation In early September, he was sitting at his desk. All of a sudden, completely unexpectedly, he had an incredible revelation. Wiles described it as the most important moment of his career: > It was so indescribably beautiful. So simple, so elegant. He stared at it in disbelief for 20 minutes. During the day, he walked around the department, constantly returning to his desk to see if it was still there. It was still there. ## The Answer Rising from the Ashes Wiles realized that what almost seemed to block the Flack and Colyvan approach was precisely the key that could make the theory-less horizontal structure work—and his original approach to the problem from three years earlier happened to be exactly what could make it work. And so, from the ashes, the real answer to the problem seemed to rise. ## "I've Got It" That first evening, he went back and mulled it over carefully. The next morning, he checked it again. By 11 o'clock, he was satisfied and went downstairs to tell his wife: > I've got it. I think I've got it. The statement was so unexpected that his wife thought he was talking about some toy for the children or something. She asked: "Got what?" He said: "I've repaired my proof. I've found it." ## The Reward of a Childhood Dream Wiles says that for him, no other problem would carry the same significance. He had a very rare privilege: the ability to pursue his childhood dream in his adult life. > I know it's a rare privilege, but if one can do it, its rewards exceed anything I could imagine. Source: [Andrew Wiles on proving Fermat's Last Theorem (1995) [video]](https://www.youtube.com/watch?v=GS7CxAtV5Ks)