WEBVTT 00:00:00.000 --> 00:00:01.740 Welcome back to The Deep Dive, where we take 00:00:01.740 --> 00:00:05.360 these monumental, seemingly impossible stories 00:00:05.360 --> 00:00:07.879 and we really try to break them down into something 00:00:07.879 --> 00:00:10.900 you can, you know, grasp and take with you. Today 00:00:10.900 --> 00:00:13.439 we are going into, well, one of the greatest 00:00:13.439 --> 00:00:16.339 intellectual adventures in human history, I think. 00:00:16.699 --> 00:00:19.660 We're focusing on a figure who achieved something. 00:00:20.359 --> 00:00:22.620 It's like the mathematical equivalent of climbing 00:00:22.620 --> 00:00:25.440 Everest. Right, but alone. In secret and without 00:00:25.440 --> 00:00:27.399 any oxygen. We're talking about Sir Andrew John 00:00:27.399 --> 00:00:29.699 Wiles. And this isn't just going to be a deep 00:00:29.699 --> 00:00:31.640 dive into the math. It's really a story about 00:00:31.640 --> 00:00:36.259 dedication, about obsession, and about catastrophic 00:00:36.259 --> 00:00:38.600 failure, followed by this incredible triumph. 00:00:38.840 --> 00:00:41.219 We're unpacking the quest for Fermat's last theorem. 00:00:41.460 --> 00:00:43.320 And you have to remember, this was a problem 00:00:43.320 --> 00:00:46.479 that just, it mocked the greatest minds for over 00:00:46.479 --> 00:00:48.539 three and a half centuries. Yeah, so our mission 00:00:48.539 --> 00:00:50.439 here is to move past the headline, you know. 00:00:50.729 --> 00:00:53.289 man -solve old math problem. We want you to really 00:00:53.289 --> 00:00:55.590 understand the journey. Exactly. We need to grasp 00:00:55.590 --> 00:00:59.030 the sheer architectural scale of the framework 00:00:59.030 --> 00:01:01.469 he had to basically invent. And that means getting 00:01:01.469 --> 00:01:04.370 into the Taniyama -Shimura vile conjecture. Which 00:01:04.370 --> 00:01:06.549 is now the modularity theorem. Right. And we 00:01:06.549 --> 00:01:08.730 need to track that whole journey, the years of 00:01:08.730 --> 00:01:12.260 sort of... monastic isolation the devastating 00:01:12.260 --> 00:01:14.939 discovery of a flaw in his proof which we'll 00:01:14.939 --> 00:01:16.719 definitely get to and then the ultimate legacy 00:01:16.719 --> 00:01:19.159 which and this is the key part it completely 00:01:19.159 --> 00:01:22.120 redefined modern number theory before we even 00:01:22.120 --> 00:01:24.219 start though we have to talk about why else's 00:01:24.219 --> 00:01:26.980 stature because this wasn't some you know amateur 00:01:26.980 --> 00:01:29.719 genius having a flash of insight this was an 00:01:29.719 --> 00:01:32.840 achievement at the absolute highest levels. Oh, 00:01:32.900 --> 00:01:35.340 absolutely. I mean, right now, he's a Royal Society 00:01:35.340 --> 00:01:38.780 research professor. And since 2018, he's held 00:01:38.780 --> 00:01:42.200 the title of the first Regis Professor of Mathematics 00:01:42.200 --> 00:01:44.739 at Oxford. That's a huge deal. And the awards. 00:01:44.920 --> 00:01:47.819 Yeah. The Abel Prize in 2016, the Copley Medal 00:01:47.819 --> 00:01:50.569 in 2017. And you have to understand, these awards 00:01:50.569 --> 00:01:53.049 weren't just for solving a puzzle. They were 00:01:53.049 --> 00:01:56.230 for creating the tools needed to solve it. Tools 00:01:56.230 --> 00:01:58.670 that then opened up entire new fields of research 00:01:58.670 --> 00:02:00.730 for other mathematicians. It's the difference 00:02:00.730 --> 00:02:04.049 between, say, being a chess grandmaster and being 00:02:04.049 --> 00:02:07.390 the person who invents a new, vastly more complex 00:02:07.390 --> 00:02:09.849 and interesting version of chess. That's a great 00:02:09.849 --> 00:02:12.469 way to put it. His work had these profound ripple 00:02:12.469 --> 00:02:15.030 effects. So to really understand those ripples, 00:02:15.210 --> 00:02:17.030 you have to go all the way back to the beginning. 00:02:17.389 --> 00:02:20.050 to the single moment that kind of set the course 00:02:20.050 --> 00:02:23.270 for his entire life. Okay, let's get into this 00:02:23.270 --> 00:02:25.370 origin story, because it honestly sounds like 00:02:25.370 --> 00:02:27.870 something out of a novel. It all starts with 00:02:27.870 --> 00:02:30.250 a 10 -year -old kid. That's right. Wiles was 00:02:30.250 --> 00:02:34.229 born in Cambridge in 1953. His father was a theologian, 00:02:34.229 --> 00:02:37.009 which is an interesting detail. The family spent 00:02:37.009 --> 00:02:40.150 some time in Nigeria. But the real spark, the 00:02:40.150 --> 00:02:42.490 defining moment, happened back in Cambridge. 00:02:43.100 --> 00:02:45.319 He talks about walking home from school one day 00:02:45.319 --> 00:02:48.659 and, just on a whim, stopping by his local public 00:02:48.659 --> 00:02:50.960 library. And he pulls a book off the shelf, The 00:02:50.960 --> 00:02:53.719 Last Problem by Eric Templebell. And then this 00:02:53.719 --> 00:02:56.219 wasn't some dense academic textbook, right? It 00:02:56.219 --> 00:02:58.520 was a popular science book about math's greatest 00:02:58.520 --> 00:03:01.500 unsolved mysteries. Exactly. And in that book, 00:03:01.520 --> 00:03:03.800 he's introduced to Fermat's Last Theorem, which 00:03:03.800 --> 00:03:05.580 is, you know, it's almost unique in mathematics 00:03:05.580 --> 00:03:08.020 because you can explain it to anyone. A 10 -year 00:03:08.020 --> 00:03:09.639 -old can understand the statement of the problem. 00:03:09.800 --> 00:03:11.719 But no one could prove it. For centuries, no 00:03:11.719 --> 00:03:14.159 one. So just for anyone who needs a quick refresher, 00:03:14.280 --> 00:03:16.960 the theorem is pretty simple to state. It says 00:03:16.960 --> 00:03:19.520 that the equation a to the n plus b to the n 00:03:19.520 --> 00:03:22.300 equals c to the n has no solutions if you're 00:03:22.300 --> 00:03:24.500 using positive whole numbers. Right, for a, b, 00:03:24.580 --> 00:03:27.639 and c. And if that exponent n is any integer 00:03:27.639 --> 00:03:30.139 greater than 2. We all know the case for n equals 00:03:30.139 --> 00:03:33.199 2. That's the Pythagorean theorem. 3 squared 00:03:33.199 --> 00:03:36.120 plus 4 squared equals 5 squared. There are infinite 00:03:36.120 --> 00:03:38.240 solutions. Yeah, that's familiar. But just try 00:03:38.240 --> 00:03:42.530 it for n equals 3. Or four or five. It doesn't 00:03:42.530 --> 00:03:45.030 work. And this all comes from Piero de Fermat 00:03:45.030 --> 00:03:47.969 in the 17th century, who famously scribbled in 00:03:47.969 --> 00:03:50.169 the margin of a book that he had a truly marvelous 00:03:50.169 --> 00:03:53.469 proof. Which this margin is too narrow to contain. 00:03:53.669 --> 00:03:56.490 Yes. The most infamous note in the history of 00:03:56.490 --> 00:04:00.330 science. It became this dare to the entire world. 00:04:00.569 --> 00:04:04.229 And Andrew Wiles, at 10 years old, reads this. 00:04:04.289 --> 00:04:07.280 Yeah. He decides right there and then, I'm going 00:04:07.280 --> 00:04:09.139 to be the one to solve this. I mean, the sheer 00:04:09.139 --> 00:04:11.740 confidence of a child. It's incredible. But that 00:04:11.740 --> 00:04:13.719 confidence obviously runs into reality pretty 00:04:13.719 --> 00:04:16.000 quick. It does. He realized he didn't have the 00:04:16.000 --> 00:04:19.199 tools. He didn't know the math. So he says he, 00:04:19.220 --> 00:04:20.860 you know, abandoned the immediate dream. But 00:04:20.860 --> 00:04:24.120 that idea, that seed, it was planted. It was 00:04:24.120 --> 00:04:26.319 just waiting. So let's talk about how he built 00:04:26.319 --> 00:04:29.540 the foundation. He goes on this very rigorous 00:04:29.540 --> 00:04:32.759 academic path, a B .A. from Merton College, Oxford 00:04:32.759 --> 00:04:37.069 in 74. Then a PhD from Cambridge in 1980. And 00:04:37.069 --> 00:04:39.269 his graduate work is really the key here. He 00:04:39.269 --> 00:04:41.389 didn't just study general mathematics. He dove 00:04:41.389 --> 00:04:43.910 straight into the deepest, most technical parts 00:04:43.910 --> 00:04:46.069 of number theory. The very parts you would need 00:04:46.069 --> 00:04:48.870 decades later. Ironically, yes. He worked under 00:04:48.870 --> 00:04:51.410 a supervisor named John Coates. And his focus 00:04:51.410 --> 00:04:53.509 was on the arithmetic of elliptic curves using 00:04:53.509 --> 00:04:55.769 something called Iwasawa theory. Okay, that sounds 00:04:55.769 --> 00:04:57.740 incredibly specialized. Let's break that down 00:04:57.740 --> 00:04:59.879 for a second. Elliptic curves and Iwasawa theory. 00:05:00.100 --> 00:05:02.379 Right. So very basically, Iwasawa theory is about 00:05:02.379 --> 00:05:04.540 connecting different mathematical worlds. It's 00:05:04.540 --> 00:05:07.600 a way of understanding these really complex algebraic 00:05:07.600 --> 00:05:10.040 structures, like number fields, by looking at 00:05:10.040 --> 00:05:12.079 them through a different lens, the lens of analysis. 00:05:12.560 --> 00:05:15.420 Think of it like a sophisticated system for organizing 00:05:15.420 --> 00:05:18.240 what would otherwise be impossibly messy. So 00:05:18.240 --> 00:05:20.720 he was learning how to take these really thorny 00:05:20.720 --> 00:05:23.639 algebra problems and apply these high -level 00:05:23.639 --> 00:05:26.259 analytical tools to them. He's building a kind 00:05:26.259 --> 00:05:29.019 of... a dictionary between two very different 00:05:29.019 --> 00:05:31.500 languages. That's a perfect way to put it. His 00:05:31.500 --> 00:05:33.639 early work with another mathematician, Barry 00:05:33.639 --> 00:05:36.259 Mazur, was on the main conjecture of Iwasawa 00:05:36.259 --> 00:05:38.899 theory. He was establishing himself as a master 00:05:38.899 --> 00:05:41.420 builder of these conceptual bridges. And his 00:05:41.420 --> 00:05:44.980 PhD thesis, just the title, Reciprocity Laws 00:05:44.980 --> 00:05:46.939 and the Conjecture of Birch and Swinerton Dyer, 00:05:47.040 --> 00:05:49.519 tells you everything. That conjecture is another 00:05:49.519 --> 00:05:51.600 one of these huge foundational problems that 00:05:51.600 --> 00:05:55.439 connects the algebra of elliptic curves to analytic 00:05:55.439 --> 00:05:58.139 functions. So Wiles was fundamentally trained 00:05:58.139 --> 00:06:00.560 to see the connections between geometry, algebra, 00:06:00.740 --> 00:06:03.579 and analysis. That was his unique skill set. 00:06:03.740 --> 00:06:05.579 And he moved around. He was at the Institute 00:06:05.579 --> 00:06:08.139 for Advanced Study, became a professor at Princeton, 00:06:08.360 --> 00:06:11.220 a Guggenheim Fellow in Paris. He was right at 00:06:11.220 --> 00:06:13.220 the center of the number theory world. Which 00:06:13.220 --> 00:06:14.839 really highlights a crucial point, doesn't it? 00:06:14.899 --> 00:06:17.279 For anyone trying to tackle a massive project. 00:06:17.850 --> 00:06:20.389 Wiles didn't just stumble upon an answer. He 00:06:20.389 --> 00:06:23.689 spent 20 years mastering the hardest, most niche 00:06:23.689 --> 00:06:25.850 tools available. So that when the opportunity 00:06:25.850 --> 00:06:28.870 finally came, he was ready. He had the perfect 00:06:28.870 --> 00:06:31.269 toolkit waiting. It's a lesson that you can't 00:06:31.269 --> 00:06:33.810 solve impossible problems without first building 00:06:33.810 --> 00:06:35.910 a new foundation. And that opportunity, as you 00:06:35.910 --> 00:06:39.149 said, arrived in 1986. And it arrived dramatically. 00:06:39.470 --> 00:06:41.529 Yeah. Okay, so let's make that jump. We have 00:06:41.529 --> 00:06:45.449 the 17th century dare. about whole numbers, how 00:06:45.449 --> 00:06:47.569 on earth did that suddenly get connected to these 00:06:47.569 --> 00:06:50.529 incredibly complex modern objects, elliptic curves, 00:06:50.670 --> 00:06:52.910 and modular forms? Well, the connection is just 00:06:52.910 --> 00:06:55.529 a spectacular piece of modern mathematical logic. 00:06:55.829 --> 00:06:57.790 It's a proof by contradiction, but on a grand 00:06:57.790 --> 00:07:00.910 scale. So we have Fermat's last theorem, a to 00:07:00.910 --> 00:07:04.850 the n plus b to the n equals c to the n. No solutions 00:07:04.850 --> 00:07:07.529 for n greater than 2. For centuries, people were 00:07:07.529 --> 00:07:09.370 trying to prove that directly. Right, using number 00:07:09.370 --> 00:07:12.490 theory. The breakthrough came from a completely 00:07:12.490 --> 00:07:14.589 different direction. It started with a mathematician 00:07:14.589 --> 00:07:17.629 named Gerhard Frey in the mid -1980s. And what 00:07:17.629 --> 00:07:20.410 was Frey's idea? Frey suggested that if a solution 00:07:20.410 --> 00:07:23.089 to Fermat's equation actually existed, let's 00:07:23.089 --> 00:07:25.550 just pretend for a second that it does, say a 00:07:25.550 --> 00:07:28.810 to the p plus b to the p equals c to the p. You 00:07:28.810 --> 00:07:32.089 could use those three numbers, A, B, and C, to 00:07:32.089 --> 00:07:34.910 build a very specific elliptic curve. A hypothetical 00:07:34.910 --> 00:07:37.769 curve, based on a hypothetical solution. Exactly. 00:07:38.110 --> 00:07:40.889 And this curve, now known as the Frey curve, 00:07:41.350 --> 00:07:44.850 would be, well, it'd be a monster. Mathematically 00:07:44.850 --> 00:07:46.970 speaking, it would have such strange, irregular 00:07:46.970 --> 00:07:49.170 properties that it just seemed like it couldn't 00:07:49.170 --> 00:07:51.949 possibly fit into the known mathematical universe. 00:07:52.230 --> 00:07:54.189 This is too weird to exist. That was the suspicion. 00:07:54.839 --> 00:07:57.180 Frey conjectured that this curve was so peculiar 00:07:57.180 --> 00:07:59.319 it couldn't be modular. It couldn't have a modular 00:07:59.319 --> 00:08:01.379 form associated with it. This was called the 00:08:01.379 --> 00:08:03.920 epsilon conjecture. Okay, and this is where Kenribbit 00:08:03.920 --> 00:08:05.980 comes in and where the light bulb goes off for 00:08:05.980 --> 00:08:09.500 Wiles in 86. Yes, Kenribbit proved the epsilon 00:08:09.500 --> 00:08:12.600 conjecture. He showed definitively that if a 00:08:12.600 --> 00:08:15.339 solution to Fermat's equation existed, the Frey 00:08:15.339 --> 00:08:17.660 curve it generated would not be modular. This 00:08:17.660 --> 00:08:20.399 was a bombshell. So the logic shifted. Now the 00:08:20.399 --> 00:08:23.240 argument is, if Fermat's last theorem is false, 00:08:23.759 --> 00:08:26.699 then this weird non -modular curve must exist. 00:08:27.139 --> 00:08:30.120 Precisely. And this is when Wiles saw the path. 00:08:30.379 --> 00:08:33.159 He realized, wait a minute, what if I can prove 00:08:33.159 --> 00:08:35.879 that all curves of this type must be modular? 00:08:36.200 --> 00:08:38.940 Because if he could do that, it would create 00:08:38.940 --> 00:08:41.899 a paradox. The perfect contradiction. It would 00:08:41.899 --> 00:08:44.440 mean the Frey curve couldn't exist, which would 00:08:44.440 --> 00:08:46.419 mean the solution that generated it couldn't 00:08:46.419 --> 00:08:49.000 exist, which would mean Fermat's last theorem 00:08:49.000 --> 00:08:52.299 must be true. So his goal became proving a part. 00:08:52.779 --> 00:08:55.259 of what was then called the Taniyama -Shimura 00:08:55.259 --> 00:08:58.259 -Weill conjecture, or TSW. Which is now, because 00:08:58.259 --> 00:09:00.480 of this work, known as the modularity theorem. 00:09:00.659 --> 00:09:02.460 This theorem is the heart of the whole thing. 00:09:02.659 --> 00:09:04.879 So let's slow down and really define the two 00:09:04.879 --> 00:09:07.320 sides of this bridge Weill's had to build. On 00:09:07.320 --> 00:09:09.799 one side, we have elliptic curves. Right. These 00:09:09.799 --> 00:09:12.399 are objects from algebraic geometry defined by 00:09:12.399 --> 00:09:15.059 cubic equations. They are these, you know, smooth, 00:09:15.059 --> 00:09:18.019 often very complex shapes. But their algebra 00:09:18.019 --> 00:09:20.379 is deeply connected to number theory. And on 00:09:20.379 --> 00:09:23.769 the other side... Modular forms. Modular forms 00:09:23.769 --> 00:09:26.889 live in a totally different universe. The world 00:09:26.889 --> 00:09:29.750 of complex analysis. They are functions that 00:09:29.750 --> 00:09:33.070 have these incredible, almost supernatural levels 00:09:33.070 --> 00:09:35.629 of symmetry. They basically stay the same even 00:09:35.629 --> 00:09:37.529 when you twist and transform them in all sorts 00:09:37.529 --> 00:09:40.009 of complicated ways. So one side is algebra and 00:09:40.009 --> 00:09:42.389 geometry. The other is analysis and symmetry. 00:09:43.360 --> 00:09:45.779 totally unrelated. And for most of history, they 00:09:45.779 --> 00:09:48.980 were. The modularity theorem was this incredibly 00:09:48.980 --> 00:09:51.879 audacious claim that these two universes were 00:09:51.879 --> 00:09:54.480 in fact one in the same. It said that for every 00:09:54.480 --> 00:09:56.720 elliptic curve, there's a unique modular form 00:09:56.720 --> 00:09:59.940 that is its perfect twin and vice versa. It was 00:09:59.940 --> 00:10:02.850 like discovering a perfect Rosetta Stone. And 00:10:02.850 --> 00:10:04.929 Wiles' challenge was to prove that this Rosetta 00:10:04.929 --> 00:10:07.070 Stone worked, at least for the specific type 00:10:07.070 --> 00:10:09.389 of curves that the Frey curve belonged to, the 00:10:09.389 --> 00:10:11.570 semi -stable elliptic curve. That's right. The 00:10:11.570 --> 00:10:14.649 full logical chain was now laid out. One, assume 00:10:14.649 --> 00:10:17.649 Fermat is false. So a solution exists. Two, that 00:10:17.649 --> 00:10:20.309 solution creates Frey's peculiar curve. Three, 00:10:20.970 --> 00:10:23.269 Ribbit proved that curve cannot be modular. And 00:10:23.269 --> 00:10:25.370 four, Wiles just had to prove that it must be 00:10:25.370 --> 00:10:28.220 modular. Creating the contradiction. Therefore, 00:10:28.500 --> 00:10:31.940 the initial assumption must be wrong. And Fermat's 00:10:31.940 --> 00:10:34.659 last theorem has to be true. That's just, it's 00:10:34.659 --> 00:10:37.419 beautiful. It's so elegant. It turns this ancient 00:10:37.419 --> 00:10:40.840 problem about numbers into this grand mission 00:10:40.840 --> 00:10:45.100 to unify two huge branches of math. But how hard 00:10:45.100 --> 00:10:46.820 was this? I mean, you mentioned people thought 00:10:46.820 --> 00:10:49.440 it was impossible. The audacity of it cannot 00:10:49.440 --> 00:10:52.879 be overstated. Wiles's own former supervisor, 00:10:53.179 --> 00:10:55.840 John Coates, admitted he told students not to 00:10:55.840 --> 00:10:57.639 work on the conjecture because he considered 00:10:57.639 --> 00:11:00.500 it completely inaccessible. Wow. The consensus 00:11:00.500 --> 00:11:02.740 was that the tools to prove it simply didn't 00:11:02.740 --> 00:11:06.120 exist. This was seen as a multigenerational project, 00:11:06.460 --> 00:11:09.480 something for the 22nd century, not for one person 00:11:09.480 --> 00:11:12.059 to do. And yet Wiles knew this was the only way. 00:11:12.480 --> 00:11:14.700 His specific training, his whole career, had 00:11:14.700 --> 00:11:16.659 led him to this point. He was one of the few 00:11:16.659 --> 00:11:18.820 people on Earth who had deep expertise in both 00:11:18.820 --> 00:11:21.100 sides of the bridge. The elliptic curves and 00:11:21.100 --> 00:11:23.460 the analytic methods like Iwasawa theory needed 00:11:23.460 --> 00:11:26.240 to connect them. He saw the path even if everyone 00:11:26.240 --> 00:11:28.120 else thought it led off a cliff. The stakes were 00:11:28.120 --> 00:11:30.899 astronomical. And the chance of failure was, 00:11:31.100 --> 00:11:34.899 you know, close to 100%. Knowing all that, Wiles 00:11:34.899 --> 00:11:37.980 chose to go it alone. He chose total secrecy. 00:11:38.139 --> 00:11:40.100 Okay, this brings us to part three, which is 00:11:40.100 --> 00:11:44.620 just pure human drama. It's 1986. Wiles decides 00:11:44.620 --> 00:11:47.279 to take this on, but he does it in complete isolation. 00:11:47.980 --> 00:11:51.440 Why? Why not work with someone or at least tell 00:11:51.440 --> 00:11:53.299 people what he was doing? Well, there are a couple 00:11:53.299 --> 00:11:56.240 of big reasons. First, the commitment was total. 00:11:56.440 --> 00:11:59.159 This wasn't a weekend project. He devoted basically 00:11:59.159 --> 00:12:01.340 all of his professional research time to this 00:12:01.340 --> 00:12:03.539 for seven straight years. And the second reason 00:12:03.539 --> 00:12:06.580 was the pressure. The pressure, exactly. If the 00:12:06.580 --> 00:12:08.700 world knew that Andrew Wiles, one of the leading 00:12:08.700 --> 00:12:10.580 experts in the field, was trying to prove the 00:12:10.580 --> 00:12:13.259 impossible, the scrutiny would have been unbearable. 00:12:13.299 --> 00:12:15.340 The media attention, the constant questions, 00:12:15.539 --> 00:12:18.059 it would have been paralyzing. By working in 00:12:18.059 --> 00:12:20.639 secret, he could fail over and over again without 00:12:20.639 --> 00:12:23.000 anyone knowing. He controlled his own timeline. 00:12:23.259 --> 00:12:26.299 He told only his wife. I mean, think about that. 00:12:26.759 --> 00:12:28.700 You're teaching calculus at Princeton. You're 00:12:28.700 --> 00:12:30.600 going to faculty meetings, living a normal academic 00:12:30.600 --> 00:12:33.740 life. And then you go home and secretly try to 00:12:33.740 --> 00:12:36.460 solve the most famous problem in history. The 00:12:36.460 --> 00:12:38.899 mental and emotional burden must have been immense. 00:12:39.690 --> 00:12:41.830 So how did he keep everyone off his back for 00:12:41.830 --> 00:12:44.029 seven years? I mean, academics are expected to 00:12:44.029 --> 00:12:46.490 publish. He used a very clever strategy. He had 00:12:46.490 --> 00:12:49.149 a backlog of previous work, research he had completed 00:12:49.149 --> 00:12:52.809 before 1986. He started releasing that work in 00:12:52.809 --> 00:12:55.690 small segmented papers over the years. A smokescreen. 00:12:55.769 --> 00:12:57.929 A perfect smokescreen. It gave the impression 00:12:57.929 --> 00:13:00.029 that he was just continuing his normal research 00:13:00.029 --> 00:13:02.570 program when in reality he was in his attic deep 00:13:02.570 --> 00:13:05.169 in the wilderness trying to build this mathematical 00:13:05.169 --> 00:13:08.039 cathedral from scratch. That's an incredible 00:13:08.039 --> 00:13:10.480 level of focus. Just knowing that any single 00:13:10.480 --> 00:13:12.759 unfixable flaw could mean that seven years of 00:13:12.759 --> 00:13:15.379 your life were gone. He talked about it, about 00:13:15.379 --> 00:13:17.860 trying one approach for a year or two, realizing 00:13:17.860 --> 00:13:19.919 it was a dead end and having to just throw it 00:13:19.919 --> 00:13:22.440 all away and start over. That iterative failure 00:13:22.440 --> 00:13:25.659 went on for years. And then comes June of 1993. 00:13:26.360 --> 00:13:29.240 The moment he decides to reveal it all at a conference 00:13:29.240 --> 00:13:31.779 in Cambridge. And he did it with such spectacular 00:13:31.779 --> 00:13:34.620 understatement. He was scheduled to give a series 00:13:34.620 --> 00:13:37.960 of three lectures. The title was Modular Forms, 00:13:37.960 --> 00:13:40.440 Elliptic Curves, and Galois Representations. 00:13:40.440 --> 00:13:43.480 No mention of Fermat at all. None. It was billed 00:13:43.480 --> 00:13:46.120 as a highly technical talk for specialists. But, 00:13:46.179 --> 00:13:48.559 you know, word started to spread. People in the 00:13:48.559 --> 00:13:50.480 field knew something big was coming. Because 00:13:50.480 --> 00:13:52.639 of his stature and the subject matter. Right. 00:13:52.799 --> 00:13:55.019 So the lecture hall got more and more crowded 00:13:55.019 --> 00:13:57.950 each day. The first day, it was the experts. 00:13:58.330 --> 00:14:01.230 By the third and final lecture, it was standing 00:14:01.230 --> 00:14:03.470 room only. People were crammed in the aisles, 00:14:03.490 --> 00:14:05.909 and he walks them through it. He builds the argument 00:14:05.909 --> 00:14:09.110 step by step. And then the final moment. How 00:14:09.110 --> 00:14:11.350 did he deliver the punchline? Every account says 00:14:11.350 --> 00:14:14.289 it was just this quiet, profound moment. He finished 00:14:14.289 --> 00:14:16.190 laying out the proof of the modularity theorem 00:14:16.190 --> 00:14:19.090 for semi -stable curves, wrote the conclusion 00:14:19.090 --> 00:14:21.929 on the blackboard, and then he paused. And then, 00:14:22.009 --> 00:14:24.269 as the New York Times famously put it, seemingly 00:14:24.269 --> 00:14:27.429 as an afterthought, he just added that this conclusion 00:14:27.429 --> 00:14:30.970 meant that Fermat's last theorem was true. And 00:14:30.970 --> 00:14:35.110 he wrote QED on the board. Wow. The reaction 00:14:35.110 --> 00:14:37.590 in that room must have been electric. And ammonium. 00:14:37.590 --> 00:14:40.169 People were crying. There were flashbulbs going 00:14:40.169 --> 00:14:43.980 off. It was the resolution to a 350 -year -old 00:14:43.980 --> 00:14:47.340 story achieved by one person in secret. It was 00:14:47.340 --> 00:14:50.419 the ultimate intellectual triumph. A triumph 00:14:50.419 --> 00:14:53.000 that lasted for about two months. The euphoria 00:14:53.000 --> 00:14:56.220 was, yes, catastrophically short -lived. In August 00:14:56.220 --> 00:14:58.740 of 1993, the proof was going through peer review, 00:14:58.919 --> 00:15:01.960 as all major proofs do, and a critical flaw was 00:15:01.960 --> 00:15:04.059 found. Okay, so let's get into the weeds here. 00:15:04.379 --> 00:15:06.379 What was the flaw? Was it just a small calculation 00:15:06.379 --> 00:15:08.779 error? No, it was much deeper than that. It was 00:15:08.779 --> 00:15:10.820 foundational. The problem was with the core piece 00:15:10.820 --> 00:15:13.539 of machinery he was using called an Euler system. 00:15:13.840 --> 00:15:16.240 An Euler system. Right. The flaw was specifically 00:15:16.240 --> 00:15:18.059 related to something called the Selmer group. 00:15:18.240 --> 00:15:20.799 In number theory, Selmer groups are tools that, 00:15:20.820 --> 00:15:23.220 in a way, measure how hard it is to find solutions 00:15:23.220 --> 00:15:25.980 on an elliptic curve. Wiles needed to prove that 00:15:25.980 --> 00:15:28.120 a particular Selmer group was small enough for 00:15:28.120 --> 00:15:30.539 his argument to work. Okay, so the Selmer group 00:15:30.539 --> 00:15:32.840 is like a critical support beam in his proof. 00:15:33.019 --> 00:15:35.899 A perfect analogy. And the Euler system was the 00:15:35.899 --> 00:15:38.340 high -tech crane he was using to put that beam 00:15:38.340 --> 00:15:40.120 in place and prove it was strong enough. And 00:15:40.120 --> 00:15:42.940 the crane broke. The crane had a hidden defect. 00:15:43.629 --> 00:15:46.429 The specific method he was using, an adaptation 00:15:46.429 --> 00:15:48.809 of something called the Cullivag and Flack method, 00:15:48.990 --> 00:15:52.529 just. It didn't work in the exact context he 00:15:52.529 --> 00:15:54.809 needed it to. The machinery wasn't reliable. 00:15:55.350 --> 00:15:57.990 So the triumph just evaporates. He goes from 00:15:57.990 --> 00:16:00.870 being the hero of mathematics to the person who 00:16:00.870 --> 00:16:03.309 has to fix the most famous mistake in the world 00:16:03.309 --> 00:16:05.970 under a global microscope. It was devastating. 00:16:06.149 --> 00:16:08.730 He spent the next year, more than a year, trying 00:16:08.730 --> 00:16:11.269 to fix the original argument. He brought in his 00:16:11.269 --> 00:16:13.379 former student, Richard Taylor, to help. But 00:16:13.379 --> 00:16:16.679 every attempt to patch the hole failed. The original 00:16:16.679 --> 00:16:19.480 path was completely blocked. You have to imagine 00:16:19.480 --> 00:16:21.919 the psychological state he was in. He's on record 00:16:21.919 --> 00:16:24.200 saying he was on the verge of giving up, of admitting 00:16:24.200 --> 00:16:26.779 defeat. He was in despair. He had exhausted every 00:16:26.779 --> 00:16:29.000 possible fix. And then comes the breakthrough, 00:16:29.240 --> 00:16:31.940 the real moment of genius. It was September 19, 00:16:32.320 --> 00:16:35.399 1994. What happened? He was at his desk, just 00:16:35.399 --> 00:16:38.039 about to give up, looking over his failed attempts 00:16:38.039 --> 00:16:41.080 one last time. And he had this sudden realization. 00:16:41.700 --> 00:16:44.500 The solution wasn't to fix the broken machinery. 00:16:44.940 --> 00:16:47.200 The solution was that the reason the machinery 00:16:47.200 --> 00:16:50.559 failed was itself the key to a completely different 00:16:50.559 --> 00:16:53.100 approach. Wait, say that again? The error in 00:16:53.100 --> 00:16:55.200 the Euler system, the very thing that broke his 00:16:55.200 --> 00:16:58.080 proof, could be precisely controlled by a different 00:16:58.080 --> 00:17:01.919 technique. A technique from his own past. Iwasawa 00:17:01.919 --> 00:17:04.640 theory. The very first thing he specialized in. 00:17:04.880 --> 00:17:07.660 20 years earlier. The foundation he built in 00:17:07.660 --> 00:17:10.140 his PhD was the one thing that could save him. 00:17:10.299 --> 00:17:12.539 He realized that the flaw wasn't just a flaw. 00:17:12.640 --> 00:17:15.519 It was a signpost pointing to a new path. He 00:17:15.519 --> 00:17:17.259 didn't have to fix the bridge. He could use his 00:17:17.259 --> 00:17:19.839 old tools to build a new, stronger one right 00:17:19.839 --> 00:17:22.259 next to it. That is the ultimate aha moment, 00:17:22.380 --> 00:17:24.369 the kind of thing you see in movies. It was the 00:17:24.369 --> 00:17:26.589 final piece of the puzzle. He quickly worked 00:17:26.589 --> 00:17:28.910 out the details, brought Taylor back in to help 00:17:28.910 --> 00:17:32.150 formalize one part, and in May of 1995, they 00:17:32.150 --> 00:17:34.230 published two papers in the Annals of Mathematics, 00:17:34.390 --> 00:17:37.430 one by Wiles and a shorter joint paper with Taylor 00:17:37.430 --> 00:17:39.430 that nailed down the technical fix. And that 00:17:39.430 --> 00:17:43.009 was it. The 350 -year quest was for real this 00:17:43.009 --> 00:17:46.599 time. Over. So the story of the proof itself 00:17:46.599 --> 00:17:49.059 is just this incredible drama. But as we move 00:17:49.059 --> 00:17:52.059 into part four, the real legacy, the real weight 00:17:52.059 --> 00:17:54.619 of what Wiles did isn't just solving that one 00:17:54.619 --> 00:17:57.059 equation. It's the mathematical world he left 00:17:57.059 --> 00:17:59.599 behind. Absolutely. The analogy I like is that 00:17:59.599 --> 00:18:01.420 everyone remembers the Eiffel Tower, but the 00:18:01.420 --> 00:18:03.980 real legacy was the invention of structural steel 00:18:03.980 --> 00:18:07.180 and the methods used to build it. Wiles invented 00:18:07.180 --> 00:18:09.539 new tools that mathematicians immediately started 00:18:09.539 --> 00:18:12.069 using everywhere. And how quickly did people 00:18:12.069 --> 00:18:14.430 start picking up on these new methods? Instantly. 00:18:14.650 --> 00:18:16.809 The techniques he developed, especially around 00:18:16.809 --> 00:18:18.849 something called the deformation theory of Galois 00:18:18.849 --> 00:18:21.309 representations, were just fundamental. They 00:18:21.309 --> 00:18:23.269 provided the blueprint for his former student, 00:18:23.470 --> 00:18:25.910 Richard Taylor, and three other mathematicians. 00:18:25.990 --> 00:18:28.809 Brian Conrad, Fred Diamond, and Christoph Briel. 00:18:28.910 --> 00:18:31.289 Right. They took Wiles' platform and used it 00:18:31.289 --> 00:18:33.509 to finish the job. And the job was proving the 00:18:33.509 --> 00:18:35.809 full modularity theorem for all elliptic curves. 00:18:36.109 --> 00:18:38.869 Exactly. Wiles had only proven the semi -stable 00:18:38.869 --> 00:18:41.630 case, which was all he needed for Fermat. By 00:18:41.630 --> 00:18:44.569 the year 2000, thanks to his methods, the entire 00:18:44.569 --> 00:18:47.230 Taniyama -Shomur -Weill conjecture was no longer 00:18:47.230 --> 00:18:50.750 a conjecture. It was a proven foundational theorem 00:18:50.750 --> 00:18:53.930 of mathematics. Which is this massive act of 00:18:53.930 --> 00:18:56.950 unification. And this, you've said, points to 00:18:56.950 --> 00:18:59.690 his even bigger legacy. His contribution to something 00:18:59.690 --> 00:19:02.069 called the Langlands program. Yes. Okay. So we 00:19:02.069 --> 00:19:03.789 need to spend some time on this because the Langlands 00:19:03.789 --> 00:19:06.910 program is, without exaggeration, probably the 00:19:06.910 --> 00:19:09.950 most ambitious and far -reaching vision in all 00:19:09.950 --> 00:19:12.390 of modern mathematics. If the modularity theorem 00:19:12.390 --> 00:19:14.869 was one giant bridge between two mathematical 00:19:14.869 --> 00:19:17.630 continents, the Langlands program is trying to 00:19:17.630 --> 00:19:19.990 build a transportation network connecting every 00:19:19.990 --> 00:19:21.930 continent. That is the perfect way to visualize 00:19:21.930 --> 00:19:24.410 it. It was proposed by Robert Langlands in the 00:19:24.410 --> 00:19:27.410 1960s, and it's this vast web of conjectures, 00:19:27.410 --> 00:19:30.410 this grand roadmap that seeks to unify number 00:19:30.410 --> 00:19:33.309 theory, algebraic geometry, and another field 00:19:33.309 --> 00:19:35.609 called harmonic analysis. And what's the core 00:19:35.609 --> 00:19:38.309 idea? Why is this unification so important? The 00:19:38.309 --> 00:19:41.869 idea is to establish a kind of deep correspondence, 00:19:42.049 --> 00:19:45.109 a reciprocity between objects from these different 00:19:45.109 --> 00:19:48.130 fields. Specifically, it tries to link objects 00:19:48.130 --> 00:19:50.670 from number theory called Galois representations 00:19:50.670 --> 00:19:54.369 with... objects from analysis called automorphic 00:19:54.369 --> 00:19:56.910 forms. Which are basically a generalization of 00:19:56.910 --> 00:19:59.490 modular forms. Exactly. And if these links are 00:19:59.490 --> 00:20:01.829 real, it means you can take a hard problem in 00:20:01.829 --> 00:20:04.210 one field, translate it into the language of 00:20:04.210 --> 00:20:06.509 another field, and potentially solve it there 00:20:06.509 --> 00:20:08.670 much more easily. It's the ultimate mathematical 00:20:08.670 --> 00:20:12.289 translation dictionary. And Wiles' work provided 00:20:12.289 --> 00:20:14.730 a new way to actually build parts of that dictionary. 00:20:15.160 --> 00:20:17.400 It did more than that. When he got the Abel Prize 00:20:17.400 --> 00:20:20.220 in 2016, he said that his methods opened up a 00:20:20.220 --> 00:20:23.039 new way of attacking the Langlands program. And 00:20:23.039 --> 00:20:24.539 what was that new way? What was the specific 00:20:24.539 --> 00:20:27.019 tool? It goes back to how he fixed the flaw. 00:20:27.319 --> 00:20:29.940 It was his work on the deformation theory of 00:20:29.940 --> 00:20:32.380 Galois representations. He showed how you could 00:20:32.380 --> 00:20:35.119 take one of these representations, bend it or 00:20:35.119 --> 00:20:37.940 deform it just slightly, and use that to prove 00:20:37.940 --> 00:20:40.680 it was modular. This became a powerful general 00:20:40.680 --> 00:20:43.000 purpose technique for proving instances of the 00:20:43.000 --> 00:20:45.390 Langlands correspondence. So he didn't just solve 00:20:45.390 --> 00:20:48.250 an old problem. He created a new piece of technology 00:20:48.250 --> 00:20:51.529 that allowed future mathematicians to start building 00:20:51.529 --> 00:20:55.029 the grand unified theory. He provided the first 00:20:55.029 --> 00:20:58.390 major concrete success story for that architecture. 00:20:58.650 --> 00:21:00.910 He proved that this grand vision wasn't just 00:21:00.910 --> 00:21:04.049 a fantasy. It was achievable. And that spurred 00:21:04.049 --> 00:21:07.099 a tidal wave of new research. And this professional 00:21:07.099 --> 00:21:09.779 success also created a kind of intellectual dynasty, 00:21:10.000 --> 00:21:12.019 right? His students have gone on to do incredible 00:21:12.019 --> 00:21:14.480 things. He has an incredible academic tree. You 00:21:14.480 --> 00:21:16.519 have Richard Taylor, of course, who helped with 00:21:16.519 --> 00:21:18.460 the proof and became a world leader in his own 00:21:18.460 --> 00:21:21.079 right. But you also have people like Manjul Bhargava, 00:21:21.119 --> 00:21:23.799 who won the Fields Medal in 2014. The highest 00:21:23.799 --> 00:21:25.900 prize in math. The highest prize. And another 00:21:25.900 --> 00:21:29.579 student, Carl Rubin, who is a giant in algebraic 00:21:29.579 --> 00:21:32.180 number theory. Wiles wasn't just working alone. 00:21:32.319 --> 00:21:34.299 He was training the next generation who would 00:21:34.299 --> 00:21:36.079 build on his foundations. Which explains the 00:21:36.079 --> 00:21:39.480 incredible wave of honors that came after 1995. 00:21:40.420 --> 00:21:43.299 The community knew instantly how big this was. 00:21:43.500 --> 00:21:45.420 It was a flood of awards. In the first few years, 00:21:45.460 --> 00:21:47.380 he got the Shock Prize, the Ostrovsky Prize, 00:21:47.599 --> 00:21:50.160 the Fermat Prize, the Wolf Prize, just one after 00:21:50.160 --> 00:21:52.259 another. But the most interesting story is about 00:21:52.259 --> 00:21:54.390 the Fields Medal, isn't it? The one he didn't 00:21:54.390 --> 00:21:56.230 get. Right. So the Fields Medal, as you said, 00:21:56.269 --> 00:21:58.490 is like the Nobel for math, but it has a strict 00:21:58.490 --> 00:22:01.490 age limit of 40. When Wiles published the final 00:22:01.490 --> 00:22:04.809 correct proof, he was 42. He solved the problem 00:22:04.809 --> 00:22:07.029 of the millennium, but he missed the age cutoff 00:22:07.029 --> 00:22:09.640 by two years. So the International Mathematical 00:22:09.640 --> 00:22:12.299 Union, the IMU, had a problem. They couldn't 00:22:12.299 --> 00:22:14.279 just ignore the biggest achievement in a century 00:22:14.279 --> 00:22:17.619 because of a role. So in 1998, they did something 00:22:17.619 --> 00:22:21.259 unprecedented. They created a special one -time 00:22:21.259 --> 00:22:24.619 award just for him, the IMU silver plaque. They 00:22:24.619 --> 00:22:26.980 literally bent their own rules to honor him. 00:22:27.099 --> 00:22:29.259 Which tells you everything you need to know about 00:22:29.259 --> 00:22:31.619 the magnitude of the work. And the honors just 00:22:31.619 --> 00:22:34.359 kept coming over the decades. A knighthood in 00:22:34.359 --> 00:22:37.559 2000, the Shaw Prize, and finally the Abel Prize 00:22:37.559 --> 00:22:40.799 in 2016, which has no age limit and is on part 00:22:40.799 --> 00:22:43.279 with a Nobel. And there's even a permanent physical 00:22:43.279 --> 00:22:46.940 honor at Oxford. There is. In 2013, the new building 00:22:46.940 --> 00:22:49.279 for the Mathematical Institute at Oxford was 00:22:49.279 --> 00:22:52.299 named the Wiles Building. To have a major academic 00:22:52.299 --> 00:22:54.279 building named after you while you're still alive 00:22:54.279 --> 00:22:56.599 and working there, that's a rare honor indeed. 00:22:57.019 --> 00:22:59.400 So we've tracked this journey, from a 10 -year 00:22:59.400 --> 00:23:01.440 -old with a library book to the man who stared 00:23:01.440 --> 00:23:04.400 into a catastrophic flaw in his life's work and 00:23:04.400 --> 00:23:06.539 found a way to turn it into the foundation for 00:23:06.539 --> 00:23:10.099 a unified mathematical future. Hashtag Outro. 00:23:10.440 --> 00:23:12.779 So to just summarize this incredible journey 00:23:12.779 --> 00:23:15.559 for you, Andrew Wiles didn't solve Fermat's last 00:23:15.559 --> 00:23:18.400 theorem by attacking it head on. He did it by 00:23:18.400 --> 00:23:21.039 first mastering these incredibly deep, separate 00:23:21.039 --> 00:23:24.700 fields, Iwasawa theory, elliptic curves, modular 00:23:24.700 --> 00:23:27.140 forms, and then weaving them together with the 00:23:27.140 --> 00:23:30.059 modularity theorem as his grand unifying bridge. 00:23:30.400 --> 00:23:32.619 He took the huge personal risk of working in 00:23:32.619 --> 00:23:35.740 total secrecy for seven years. He survived the 00:23:35.740 --> 00:23:37.960 public collapse of his first proof because of 00:23:37.960 --> 00:23:40.190 that flaw in the Euler system. And then he found 00:23:40.190 --> 00:23:42.809 redemption by going back to his very first area 00:23:42.809 --> 00:23:45.109 of expertise, to Iowa Solid Theory, to build 00:23:45.109 --> 00:23:47.609 a completely new path around the obstacle. And 00:23:47.609 --> 00:23:49.609 the result wasn't just a proof of format. It 00:23:49.609 --> 00:23:51.829 was the foundation for the full modularity theorem. 00:23:51.950 --> 00:23:54.730 It was a set of brand new tools that gave mathematicians 00:23:54.730 --> 00:23:57.309 a real concrete way to start tackling the grand 00:23:57.309 --> 00:24:00.140 vision of the Langlands program. At the end of 00:24:00.140 --> 00:24:03.380 the day, his success was about synthesis. It 00:24:03.380 --> 00:24:05.720 wasn't about brute force. It was about finding 00:24:05.720 --> 00:24:08.700 this unexpected, profound connection between 00:24:08.700 --> 00:24:10.880 two mathematical worlds that everyone thought 00:24:10.880 --> 00:24:14.000 were strangers. He forced algebraic geometry 00:24:14.000 --> 00:24:17.259 and harmonic analysis to speak the same language. 00:24:17.849 --> 00:24:19.869 And this really is the essential takeaway, isn't 00:24:19.869 --> 00:24:21.730 it? For anyone listening, no matter what your 00:24:21.730 --> 00:24:24.210 field is, the solution to your hardest problems 00:24:24.210 --> 00:24:27.289 might not lie inside that field. It might be 00:24:27.289 --> 00:24:29.230 waiting at the intersection of two things you 00:24:29.230 --> 00:24:31.440 never thought to connect. Exactly. And that's 00:24:31.440 --> 00:24:33.079 the powerful question for you to think about, 00:24:33.140 --> 00:24:35.680 whether you're in science or business or art. 00:24:35.920 --> 00:24:39.039 If Wiles' breakthrough came from forcing a link 00:24:39.039 --> 00:24:41.579 between disparate worlds to make progress on 00:24:41.579 --> 00:24:45.480 this huge, unifying program, what seemingly impossible 00:24:45.480 --> 00:24:48.460 problem in your own life could be unlocked by 00:24:48.460 --> 00:24:50.500 forcing a connection between two disciplines 00:24:50.500 --> 00:24:52.940 or two skills that no one has ever put together 00:24:52.940 --> 00:24:55.519 before? What if unsolvable is just another word 00:24:55.519 --> 00:24:57.779 for not yet connected? The next big breakthrough 00:24:57.779 --> 00:24:59.819 you have might just be sitting there at the junction 00:24:59.819 --> 00:25:01.960 of two things you already know. If you want to 00:25:01.960 --> 00:25:04.279 go deeper into the human story behind all this, 00:25:04.339 --> 00:25:06.880 which is just as fascinating as the math, there 00:25:06.880 --> 00:25:10.000 are two fantastic resources. The BBC documentary, 00:25:10.259 --> 00:25:13.079 which aired in the U .S. on Nova as The Proof. 00:25:13.099 --> 00:25:15.539 A brilliant documentary. And the book by Simon 00:25:15.539 --> 00:25:17.859 Singh, Fermat's Last Theorem. It tells the story 00:25:17.859 --> 00:25:20.000 beautifully and accessibly. We highly recommend 00:25:20.000 --> 00:25:22.400 both. And that's it for this depth dive into 00:25:22.400 --> 00:25:26.740 a 350 year quest and the quiet, persistent dedication 00:25:26.740 --> 00:25:29.099 it took to finally end it. We'll see you next 00:25:29.099 --> 00:25:31.680 time. Welcome to the debate. Today, we're looking 00:25:31.680 --> 00:25:34.900 at the career of Sir Andrew Wiles, the British 00:25:34.900 --> 00:25:37.259 mathematician whose name is, well, it's pretty 00:25:37.259 --> 00:25:39.259 much synonymous with one of the greatest intellectual 00:25:39.259 --> 00:25:41.900 achievements of the modern era. The proof of 00:25:41.900 --> 00:25:46.210 Fermat's last theorem. Right, or FLT, an achievement 00:25:46.210 --> 00:25:48.849 that required him to prove the Taniyama -Shimura 00:25:48.849 --> 00:25:51.289 conjecture for a critical class of equations 00:25:51.289 --> 00:25:55.529 and, in doing so, brought this decisive closure 00:25:55.529 --> 00:25:58.529 to a mathematical riddle that had stumped the 00:25:58.529 --> 00:26:02.049 best minds for over 300 years. So our central 00:26:02.049 --> 00:26:04.349 question today is really about his enduring legacy. 00:26:04.650 --> 00:26:07.609 Is it that monumental, singular historical achievement 00:26:07.609 --> 00:26:11.680 of proving FLT itself? that moment of final conclusive 00:26:11.680 --> 00:26:15.700 victory? Or is it the foundational unifying mathematical 00:26:15.700 --> 00:26:19.200 methods he developed along the way? the new tools 00:26:19.200 --> 00:26:21.700 and frameworks that really pushed number theory 00:26:21.700 --> 00:26:25.259 forward and paved the way for the broader Langlands 00:26:25.259 --> 00:26:29.579 program. And I think the proof of FLT was spectacularly 00:26:29.579 --> 00:26:31.960 important, of course, but I'll be arguing that 00:26:31.960 --> 00:26:34.880 Wiles' creation of these robust new mathematical 00:26:34.880 --> 00:26:37.980 frameworks and the deep structural unification 00:26:37.980 --> 00:26:40.480 that followed, that's what holds the greater 00:26:40.480 --> 00:26:43.420 long -term significance. And I'll hold that the 00:26:43.420 --> 00:26:45.859 historical completion of Fermat's Last Theorem 00:26:46.269 --> 00:26:48.910 is the defining permanent aspect of his career. 00:26:49.170 --> 00:26:53.329 For me, the proof of FLT absolutely must be Wiles' 00:26:53.450 --> 00:26:56.009 defining legacy. And that's precisely because 00:26:56.009 --> 00:27:00.029 of the sheer historical weight and the difficulty 00:27:00.029 --> 00:27:02.869 of that specific problem. I mean, this was a 00:27:02.869 --> 00:27:05.490 personal challenge for him, right? It goes all 00:27:05.490 --> 00:27:07.829 the way back to when he was 10 years old and 00:27:07.829 --> 00:27:11.109 he found a copy of The Last Problem. It was a 00:27:11.109 --> 00:27:14.609 theorem, as he said, So easy to state that he, 00:27:14.690 --> 00:27:16.750 a 10 -year -old, could understand it, yet it 00:27:16.750 --> 00:27:19.849 had resisted proof for centuries. He dedicated 00:27:19.849 --> 00:27:23.269 himself to this one goal with just this exceptional 00:27:23.269 --> 00:27:27.569 discipline, famously working in near total secrecy 00:27:27.569 --> 00:27:30.910 for over six years, starting around mid -1986. 00:27:31.410 --> 00:27:35.750 Right. And the audacity of it is key. His peers, 00:27:35.970 --> 00:27:38.970 people like his former supervisor, John Coates, 00:27:39.049 --> 00:27:42.730 they saw the necessary... prerequisite, the modularity 00:27:42.730 --> 00:27:45.509 theorem, as, and this is a quote, impossible 00:27:45.509 --> 00:27:49.210 to actually prove. Ken Ribbitt called it completely 00:27:49.210 --> 00:27:52.329 inaccessible, while succeeded where centuries 00:27:52.329 --> 00:27:55.690 of mathematical titans had failed. And this led 00:27:55.690 --> 00:27:59.029 to immediate widespread public acclaim and unique 00:27:59.029 --> 00:28:02.230 honors tied specifically to this one achievement. 00:28:03.069 --> 00:28:06.549 like the Wolfsgold Prize in 1997 and the IMU 00:28:06.549 --> 00:28:09.609 Silver Plaque in 98, which was granted specifically 00:28:09.609 --> 00:28:12.269 because he was over the age limit for the Fields 00:28:12.269 --> 00:28:15.630 Medal. This whole context just elevates the achievement 00:28:15.630 --> 00:28:18.890 from a technical step to a defining historical 00:28:18.890 --> 00:28:21.849 monument. I see that, but I come at it from a 00:28:21.849 --> 00:28:24.829 different angle. The solution to FLT was, in 00:28:24.829 --> 00:28:28.369 my view, merely a spectacular consequence, the 00:28:28.369 --> 00:28:30.750 lightning strike, if you will, of the deeper, 00:28:30.829 --> 00:28:33.829 more profound mathematical tools Wiles was developing. 00:28:34.250 --> 00:28:37.210 His work was already deeply rooted in the structural 00:28:37.210 --> 00:28:39.609 side of number theory long before he turned his 00:28:39.609 --> 00:28:42.769 full attention to FLT. I mean, his true expertise 00:28:42.769 --> 00:28:46.529 lay in unifying these disparate fields. Galois 00:28:46.529 --> 00:28:49.309 representations, elliptic curves, and modular 00:28:49.309 --> 00:28:52.849 forms. Okay. This complex framework, it really 00:28:52.849 --> 00:28:55.490 began with his work on Iowa -Sava theory in the 00:28:55.490 --> 00:28:58.390 late 70s, where he was already proving the main 00:28:58.390 --> 00:29:01.069 conjecture for certain extensions of totally 00:29:01.069 --> 00:29:03.369 real fields. And for anyone who's not familiar, 00:29:03.869 --> 00:29:05.910 the Langlands program is essentially a kind of 00:29:05.910 --> 00:29:09.089 grand unification theory for mathematics. It 00:29:09.089 --> 00:29:11.430 predicts these deep, hidden connections between 00:29:11.430 --> 00:29:14.640 fields that seem totally separate. And Wiles' 00:29:14.799 --> 00:29:16.859 breakthrough was establishing a key part of that 00:29:16.859 --> 00:29:20.319 vision, the Taniyama -Shimura conjecture. To 00:29:20.319 --> 00:29:23.140 say his goal was merely FLT I think undervalues 00:29:23.140 --> 00:29:25.579 his entire career trajectory toward this idea 00:29:25.579 --> 00:29:30.059 of mathematical unification. And crucially, Wiles 00:29:30.059 --> 00:29:32.980 himself said as much. When he got the Abel Prize 00:29:32.980 --> 00:29:36.539 in 2016, he stated very clearly that he didn't 00:29:36.539 --> 00:29:40.299 just prove FLT, but he pushed the whole of mathematics 00:29:40.299 --> 00:29:44.430 as a field towards the Langlands program. And 00:29:44.430 --> 00:29:46.569 you see the power of his methods in what happened 00:29:46.569 --> 00:29:49.809 next. His students, Taylor, Conrad, Diamond, 00:29:50.009 --> 00:29:52.750 Bruille, they went on to prove the full modularity 00:29:52.750 --> 00:29:55.769 theorem by 2000, building directly on his work. 00:29:56.210 --> 00:29:59.390 That kind of expansion, that speaks to a foundational 00:29:59.390 --> 00:30:02.690 legacy, not just a conclusive one. I see why 00:30:02.690 --> 00:30:05.509 you think that, and I definitely concede the 00:30:05.509 --> 00:30:07.769 immense conceptual importance of the Langlands 00:30:07.769 --> 00:30:10.670 program, but let's look at the driving force 00:30:10.670 --> 00:30:13.849 from a different perspective. It was the specific 00:30:13.849 --> 00:30:17.569 challenge of FLT, that equation that fascinated 00:30:17.569 --> 00:30:20.569 him as a child, that provided the critical motivation 00:30:20.569 --> 00:30:23.849 to actually create the tools. I mean, if the 00:30:23.849 --> 00:30:26.609 goal had just been some generalized theory, would 00:30:26.609 --> 00:30:28.710 the methods have been pushed to such absolute 00:30:28.710 --> 00:30:31.289 limits? Would they have been subjected to that 00:30:31.289 --> 00:30:34.390 sustained, almost impossible intellectual pressure? 00:30:34.750 --> 00:30:37.950 I don't think so. But is that a meaningful distinction, 00:30:38.289 --> 00:30:40.970 the target versus the method? I think it is. 00:30:41.390 --> 00:30:44.109 The legacy is tied to the successful conclusion 00:30:44.109 --> 00:30:48.250 of that 300 -year quest. The fame, the historical 00:30:48.250 --> 00:30:52.009 context, that's what defines it. This very specific 00:30:52.009 --> 00:30:55.390 hard -won victory led directly to honors that 00:30:55.390 --> 00:30:58.150 cemented his public identity, with the problem, 00:30:58.289 --> 00:30:59.950 you know, being appointed a knight commander 00:30:59.950 --> 00:31:02.410 of the Order of the British Empire, the naming 00:31:02.410 --> 00:31:04.569 of the Mathematical Institute building at Oxford 00:31:04.569 --> 00:31:07.289 after him. The public and academic communities 00:31:07.289 --> 00:31:09.869 recognized that moment of historical closure, 00:31:10.150 --> 00:31:13.250 not the abstract program it supports. That argument 00:31:13.250 --> 00:31:16.049 rests on a separation I don't really accept, 00:31:16.269 --> 00:31:19.430 the separation between the target and the methodological 00:31:19.430 --> 00:31:22.279 innovation. I mean, the sources show Wiles was 00:31:22.279 --> 00:31:25.539 already a respected expert generalizing Iwasawa 00:31:25.539 --> 00:31:27.980 theory, working with these sophisticated objects 00:31:27.980 --> 00:31:30.519 like representations attached to Hilbert modular 00:31:30.519 --> 00:31:34.640 forms well before 1986. His application of these 00:31:34.640 --> 00:31:37.480 complex structural concepts to the Tanyama -Shimura 00:31:37.480 --> 00:31:40.400 conjecture. That is the true innovation. But 00:31:40.400 --> 00:31:43.440 that link was only made because of FLT. Exactly. 00:31:43.559 --> 00:31:47.130 But let's be precise about that link. The modularity 00:31:47.130 --> 00:31:49.890 conjecture says every elliptic curve can be associated 00:31:49.890 --> 00:31:53.289 with a modular form. Wiles was only brought into 00:31:53.289 --> 00:31:55.990 the FLT discussion because of the work of Gerhard 00:31:55.990 --> 00:31:58.910 Frey and then Ken Ribbitt. Ribbitt proved that 00:31:58.910 --> 00:32:00.930 if a counterexample to Fermat's last theorem 00:32:00.930 --> 00:32:03.509 existed, you could construct this very peculiar 00:32:03.509 --> 00:32:06.789 elliptic curve, the Frey curve, with such strange 00:32:06.789 --> 00:32:09.950 properties that it could not be modular. So Wiles' 00:32:10.049 --> 00:32:13.109 task wasn't to prove FLT directly. It was to 00:32:13.109 --> 00:32:15.230 prove that the Taniyama -Tamora conjecture held 00:32:15.230 --> 00:32:17.349 for this class of semi -stable elliptic curves. 00:32:17.809 --> 00:32:20.190 Proving that connection is a far more fundamental 00:32:20.190 --> 00:32:22.130 statement about the deep structure of mathematics 00:32:22.130 --> 00:32:25.630 and the theorem itself. FLT was the target that 00:32:25.630 --> 00:32:27.690 allowed the theoretical weaponry to be deployed, 00:32:27.849 --> 00:32:30.509 yes, but the invention of the weapon is the lasting 00:32:30.509 --> 00:32:33.539 contribution. Okay. I see that connection, but 00:32:33.539 --> 00:32:36.079 I still believe the goal dictated the rigor required. 00:32:36.339 --> 00:32:38.599 And to delve deeper into that, let's talk about 00:32:38.599 --> 00:32:41.319 the period of the flaw and the fix. I think the 00:32:41.319 --> 00:32:43.920 necessity of overcoming that flaw is what fundamentally 00:32:43.920 --> 00:32:47.519 shows that the singular historical goal of FLT 00:32:47.519 --> 00:32:50.640 pushed Wiles to an intellectual resolution that 00:32:50.640 --> 00:32:54.019 just pursuing a generalized theory might not 00:32:54.019 --> 00:32:56.410 have demanded. I agree that period is central, 00:32:56.529 --> 00:32:58.470 but I see it as the strongest evidence for the 00:32:58.470 --> 00:33:01.170 lasting methodological contribution. I'm sorry, 00:33:01.210 --> 00:33:03.589 but I just don't buy that the final proof prioritizes 00:33:03.589 --> 00:33:06.329 the FLT goal over the creation of new methods. 00:33:06.509 --> 00:33:09.569 When that flaw was discovered in August 93, related 00:33:09.569 --> 00:33:11.750 to the Selmer group and Euler systems, he had 00:33:11.750 --> 00:33:13.710 to stop the celebration and completely rethink 00:33:13.710 --> 00:33:17.309 his approach. Right. A huge setback. A huge one. 00:33:17.529 --> 00:33:20.589 And the Selmer group is this key object for studying 00:33:20.589 --> 00:33:23.130 rational points on an elliptic curve. it was 00:33:23.130 --> 00:33:26.069 crucial for his initial approach. When that failed, 00:33:26.230 --> 00:33:28.670 he and Richard Taylor were forced to synthesize 00:33:28.670 --> 00:33:31.769 and build genuinely new methods, combining ideas 00:33:31.769 --> 00:33:34.950 from Kalavagin and Flatch's approach with Awosawa 00:33:34.950 --> 00:33:39.069 theory. This process, building robust new methods 00:33:39.069 --> 00:33:41.230 over the course of a year to overcome a profound 00:33:41.230 --> 00:33:44.150 hurdle, that is the actual lasting technical 00:33:44.150 --> 00:33:46.859 contribution. It wasn't just applying existing 00:33:46.859 --> 00:33:49.440 theory. It was profound theoretical invention 00:33:49.440 --> 00:33:52.240 under extreme duress. That's an interesting point 00:33:52.240 --> 00:33:54.740 on the synthesis, though. I would frame that 00:33:54.740 --> 00:33:57.400 period very differently. The immense intellectual 00:33:57.400 --> 00:34:00.680 focus he maintained on this single problem for 00:34:00.680 --> 00:34:03.460 nearly a decade, even through the, you know, 00:34:03.460 --> 00:34:06.859 the public humiliation and the intellectual agony 00:34:06.859 --> 00:34:10.019 of that flaw, it demonstrates that the goal was 00:34:10.019 --> 00:34:12.920 dictating the methodology. Think about the pressure 00:34:12.920 --> 00:34:15.039 he was under after announcing a result that turned 00:34:15.039 --> 00:34:17.920 out to be incomplete. The ultimate fix wasn't 00:34:17.920 --> 00:34:20.599 some philosophical detour toward a broader program. 00:34:20.800 --> 00:34:23.639 It was an immediate, decisive solution required 00:34:23.639 --> 00:34:27.079 to secure the FLT proof. But it was a new method. 00:34:27.380 --> 00:34:32.579 It was. The insight he had on September 19, 1994, 00:34:32.940 --> 00:34:36.500 which connected his methods to what we call horizontal 00:34:36.500 --> 00:34:39.679 Iwasawa theory, was the final critical piece. 00:34:40.230 --> 00:34:42.750 but it was driven by the urgent need to conclude 00:34:42.750 --> 00:34:46.130 the proof of Fermat's equation. The drive for 00:34:46.130 --> 00:34:49.369 FLT wasn't secondary. It was the pressure vessel 00:34:49.369 --> 00:34:51.789 that ensured the methods were rigorously tested 00:34:51.789 --> 00:34:55.369 and pushed to their absolute limits. If the end 00:34:55.369 --> 00:34:58.030 goal had been less ambitious or less historically 00:34:58.030 --> 00:35:01.070 charged, the framework might never have achieved 00:35:01.070 --> 00:35:03.650 the robustness it needed to survive that level 00:35:03.650 --> 00:35:06.349 of scrutiny. While I do concede the historical 00:35:06.349 --> 00:35:09.289 magnetism of that pursuit, the question remains. 00:35:09.880 --> 00:35:13.599 What holds greater value for mathematics? A conclusive 00:35:13.599 --> 00:35:16.719 historical moment? Or a system that allows for 00:35:16.719 --> 00:35:19.460 exponential future growth? The goal of proving 00:35:19.460 --> 00:35:22.559 LFT was finite. The utility of the framework 00:35:22.559 --> 00:35:26.039 Wiles created is, well, it's effectively infinite. 00:35:26.280 --> 00:35:28.960 Wiles' own approach meant he wasn't looking for 00:35:28.960 --> 00:35:31.960 some numerical trick specific to Fermat's equation, 00:35:32.219 --> 00:35:35.159 which is what so many others had tried. He was 00:35:35.159 --> 00:35:38.110 creating a unified system. And that system, once 00:35:38.110 --> 00:35:40.510 it was complete, it immediately became foundational. 00:35:41.170 --> 00:35:44.070 I mean, just look again at the proof of the full 00:35:44.070 --> 00:35:46.889 modularity theorem. Wiles only needed the semi 00:35:46.889 --> 00:35:50.130 -stable case for FLT. But the fact that his framework 00:35:50.130 --> 00:35:53.349 was so scalable and so comprehensive that it 00:35:53.349 --> 00:35:55.750 allowed others to prove the general case, that 00:35:55.750 --> 00:35:58.329 shows that the output of his method was structurally 00:35:58.329 --> 00:36:01.340 superior to the initial input goal. His work 00:36:01.340 --> 00:36:03.860 provides the enduring structure for future exploration. 00:36:04.159 --> 00:36:06.519 It confirms his own statement that he pushed 00:36:06.519 --> 00:36:09.360 the field toward Langlands. That structure is 00:36:09.360 --> 00:36:11.400 what's going to dictate research trajectories 00:36:11.400 --> 00:36:14.280 50 or 100 years from now, long after the singular 00:36:14.280 --> 00:36:16.599 event of the FLT proof is just absorbed into 00:36:16.599 --> 00:36:19.239 textbook history. But defining a legacy based 00:36:19.239 --> 00:36:22.880 solely on technical utility risks ignoring the 00:36:22.880 --> 00:36:26.059 human element, the inspiration, the galvanizing 00:36:26.059 --> 00:36:29.179 effect of a historical victory. The focus on 00:36:29.179 --> 00:36:31.780 this 300 -year -old problem, it galvanized public 00:36:31.780 --> 00:36:34.679 and academic interest in number theory in a way 00:36:34.679 --> 00:36:37.260 that generalized theorems frankly rarely do. 00:36:37.420 --> 00:36:40.360 The media attention, the documentaries, the popular 00:36:40.360 --> 00:36:43.559 books, that is all a legacy derived from the 00:36:43.559 --> 00:36:47.139 singular historical victory. Wiles' name will 00:36:47.139 --> 00:36:49.400 forever be associated with solving one of the 00:36:49.400 --> 00:36:52.880 most famous problems in history. That clarity, 00:36:52.980 --> 00:36:56.000 that singular victory, is a more powerful and 00:36:56.000 --> 00:36:59.179 defining legacy than a potentially endless program 00:36:59.179 --> 00:37:02.260 of unification, which is, by its very definition, 00:37:02.380 --> 00:37:05.820 incomplete and ongoing. The legacy of Wiles is 00:37:05.820 --> 00:37:08.659 that he's the man who finished the quest. I'm 00:37:08.659 --> 00:37:11.079 just not convinced by that line of reasoning. 00:37:11.599 --> 00:37:14.239 Defining a legacy based on public acclaim, I 00:37:14.239 --> 00:37:17.239 think, risks undervaluing the true mathematical 00:37:17.239 --> 00:37:20.159 depth. While the publicity was great for the 00:37:20.159 --> 00:37:23.639 field, the true measure of Wiles' career is the 00:37:23.639 --> 00:37:27.099 system he left behind. His contribution was creating 00:37:27.099 --> 00:37:29.880 this powerful conceptual bridge, a systematic 00:37:29.880 --> 00:37:32.840 way to move between algebraic geometry and representation 00:37:32.840 --> 00:37:36.980 theory. This unification, the foundational framework 00:37:36.980 --> 00:37:40.059 that supports the Langlands program, is far more 00:37:40.059 --> 00:37:42.400 significant than the solution to one specific 00:37:42.400 --> 00:37:45.079 Diophantine equation, however famous it might 00:37:45.079 --> 00:37:48.300 be. The framework is generative, it's dynamic, 00:37:48.639 --> 00:37:51.219 it's applicable across vast parts of modern math. 00:37:51.460 --> 00:37:54.989 The FLT proof is conclusive. and in a way static. 00:37:55.309 --> 00:37:57.849 The structure he built ensures his work will 00:37:57.849 --> 00:38:00.110 continue to influence breakthroughs long after 00:38:00.110 --> 00:38:03.030 the historical fanfare has faded. Well, while 00:38:03.030 --> 00:38:06.530 acknowledging the brilliance and, yes, the generative 00:38:06.530 --> 00:38:09.230 power of the techniques Wiles developed and the 00:38:09.230 --> 00:38:12.489 conceptual bridges he built, the sheer magnitude 00:38:12.489 --> 00:38:15.670 of the problem he set out to solve, a challenge 00:38:15.670 --> 00:38:18.409 that defeated centuries of brilliant mathematicians, 00:38:18.510 --> 00:38:21.650 that for me remains the pinnacle of his achievement. 00:38:22.539 --> 00:38:25.320 The proof of Fermat's last theorem is the defining 00:38:25.320 --> 00:38:28.519 historical monument of his career. It provided 00:38:28.519 --> 00:38:30.960 unique closure to a question that had defined 00:38:30.960 --> 00:38:33.800 mathematical frustration for generations. And 00:38:33.800 --> 00:38:36.599 I maintain that the intellectual bravery lay 00:38:36.599 --> 00:38:39.639 not just in tackling FLT, but in developing the 00:38:39.639 --> 00:38:42.460 complex machinery required to bridge those distinct 00:38:42.460 --> 00:38:45.440 fields, Galois representations, elliptic curves, 00:38:45.639 --> 00:38:48.460 modular forms, and in doing so, fundamentally 00:38:48.460 --> 00:38:51.429 advancing the Langlands program. This systematic 00:38:51.429 --> 00:38:53.989 approach, confirmed by the ongoing success of 00:38:53.989 --> 00:38:56.369 his methods, offers a legacy that is fundamentally 00:38:56.369 --> 00:38:59.550 generative rather than merely conclusive. It's 00:38:59.550 --> 00:39:02.550 clear we agree on the tremendous importance of 00:39:02.550 --> 00:39:05.849 his career. His work really stands as a testament 00:39:05.849 --> 00:39:08.869 to how the relentless pursuit of a specific, 00:39:09.130 --> 00:39:12.969 deeply held, historical problem can drive the 00:39:12.969 --> 00:39:16.130 most profound and general theoretical advances. 00:39:16.920 --> 00:39:19.619 It leaves us to weigh whether the goal or the 00:39:19.619 --> 00:39:22.940 sophisticated tools created to reach it represents 00:39:22.940 --> 00:39:25.260 the greater and most enduring contribution.