WEBVTT 00:00:00.000 --> 00:00:03.379 Imagine locking yourself in an attic for six 00:00:03.379 --> 00:00:06.919 years. Right. Just keeping a massive career -defining 00:00:06.919 --> 00:00:09.720 secret from your colleagues, your friends, I 00:00:09.720 --> 00:00:12.240 mean, the entire world. Which is incredibly hard 00:00:12.240 --> 00:00:14.800 to do in academia. Exactly. And you're doing... 00:00:14.599 --> 00:00:18.379 all of this to solve a 300 year old riddle that 00:00:18.379 --> 00:00:21.320 the absolute greatest minds in history have already 00:00:21.320 --> 00:00:23.719 deemed basically physically impossible. Yeah, 00:00:23.879 --> 00:00:26.679 a true lost cause. So today we are taking a deep 00:00:26.679 --> 00:00:29.399 dive into a stack of Wikipedia excerpts about 00:00:29.399 --> 00:00:33.020 Sir Andrew Wiles. Now you hear British mathematician 00:00:33.020 --> 00:00:35.380 and you might immediately brace yourself for 00:00:35.380 --> 00:00:38.759 a dusty biography. Or an hour of chalk squeaking 00:00:38.759 --> 00:00:40.700 on a blackboard. Right, exactly. But this is 00:00:40.700 --> 00:00:43.100 not that kind of story. This is a story of childhood 00:00:43.100 --> 00:00:46.000 obsession, a devastating, highly public failure, 00:00:46.140 --> 00:00:49.219 and a triumphant comeback that literally unified 00:00:49.219 --> 00:00:51.479 entirely different branches of human knowledge. 00:00:51.560 --> 00:00:53.979 It really is a massive saga. It is. Our mission 00:00:53.979 --> 00:00:56.619 today is to figure out how one man cracked the 00:00:56.619 --> 00:01:00.219 most stubborn problem in mathematics, Fermat's 00:01:00.219 --> 00:01:03.829 Last Theorem. OK, let's unpack this. Whoa. To 00:01:03.829 --> 00:01:06.049 really understand Wiles, we have to start with 00:01:06.049 --> 00:01:10.430 the detail from his early life that kind of sets 00:01:10.430 --> 00:01:12.969 the stage for his entire personality. Okay, where 00:01:12.969 --> 00:01:14.870 are we starting? He was born in Cambridge in 00:01:14.870 --> 00:01:17.989 1953, but he spent a significant chunk of his 00:01:17.989 --> 00:01:20.549 early childhood living in Nigeria with his parents. 00:01:20.650 --> 00:01:23.129 Right. And when it came time for him to start 00:01:23.129 --> 00:01:25.469 formal schooling there, according to his parents' 00:01:25.590 --> 00:01:28.430 letters, he just flat out refused to go. Wait, 00:01:28.430 --> 00:01:31.689 really? He just refused? For months. Just completely 00:01:31.689 --> 00:01:34.810 refused. That is incredibly ironic. I mean, the 00:01:34.810 --> 00:01:37.049 man who goes on to become one of the greatest 00:01:37.049 --> 00:01:40.090 academic minds of our century basically started 00:01:40.090 --> 00:01:42.969 out staging a toddler protest against the very 00:01:42.969 --> 00:01:45.909 concept of school. Exactly. He hated the classroom. 00:01:46.629 --> 00:01:48.920 But the sources make a point to highlight. that 00:01:48.920 --> 00:01:51.519 he already loved solving math problems on his 00:01:51.519 --> 00:01:53.680 own terms. So it wasn't the learning, it was 00:01:53.680 --> 00:01:55.840 the environment. That distinction is everything. 00:01:55.959 --> 00:01:58.920 He was avoiding intellectual work, he was avoiding 00:01:58.920 --> 00:02:01.400 institutional structure, he was drawn to the 00:02:01.400 --> 00:02:03.439 purity of the puzzles themselves, where he could 00:02:03.439 --> 00:02:05.950 just sit and think. Which perfectly brings us 00:02:05.950 --> 00:02:08.169 to a fateful walk home from school when he's 00:02:08.169 --> 00:02:10.169 10 years old back living in Cambridge. Yeah, 00:02:10.169 --> 00:02:12.550 this is the inciting incident of his whole life. 00:02:12.650 --> 00:02:15.250 He stops at his local library and he picks up 00:02:15.250 --> 00:02:18.129 a book by Eric Temple Bell called The Last Problem. 00:02:18.509 --> 00:02:21.569 And inside he discovers Fermat's Last Theorem. 00:02:21.810 --> 00:02:24.289 The ultimate mathematical white whale. Truly. 00:02:24.849 --> 00:02:27.770 Now, for anyone a little rusty on their 17th 00:02:27.770 --> 00:02:30.469 century math puzzles, the theorem is deceptively 00:02:30.469 --> 00:02:33.389 simple. It really is. It states that no three 00:02:33.389 --> 00:02:36.750 positive integers a, b, and c can satisfy the 00:02:36.750 --> 00:02:39.810 equation a to the power of n plus b to the power 00:02:39.810 --> 00:02:42.849 of n equals c to the power of n. Right, for any 00:02:42.849 --> 00:02:45.490 integer value of n that is greater than 2. Exactly. 00:02:45.530 --> 00:02:48.050 And if you think back to basic geometry, if n 00:02:48.050 --> 00:02:51.090 is 2, that's just the Pythagorean theorem. A 00:02:51.090 --> 00:02:53.750 squared plus b squared equals c squared. Which 00:02:53.750 --> 00:02:55.830 we all learned in school. Right, and there are 00:02:55.830 --> 00:02:57.650 endless combinations of whole numbers that make 00:02:57.650 --> 00:03:00.560 that work. 3, 4, and 5, for example. Sure. But 00:03:00.560 --> 00:03:02.819 in the 1630s, period of Fermat claimed that if 00:03:02.819 --> 00:03:05.039 you raise those numbers to the power of 3 or 00:03:05.039 --> 00:03:07.419 4 or 5 or anything higher, the equation just 00:03:07.419 --> 00:03:09.840 breaks. You can never find whole numbers to make 00:03:09.840 --> 00:03:12.340 it balance perfectly. Exactly. And the crazy 00:03:12.340 --> 00:03:15.159 part is, Fermat wrote this claim in the margin 00:03:15.159 --> 00:03:18.240 of a book, adding that he had a truly marvelous 00:03:18.240 --> 00:03:20.780 proof for it, but the margin was too small to 00:03:20.780 --> 00:03:23.580 contain it. The augacity of that guy. I know. 00:03:23.580 --> 00:03:25.979 It's basically the ultimate mathematical mic 00:03:25.979 --> 00:03:27.919 drop. He just leaves it there for history. to 00:03:27.919 --> 00:03:30.939 deal with, but here's what really blows my mind. 00:03:30.960 --> 00:03:34.969 What's up? Wiles is 10 years old. He's looking 00:03:34.969 --> 00:03:38.050 at a problem that has baffled humanity for over 00:03:38.050 --> 00:03:41.090 three centuries. It's like finding a puzzle box 00:03:41.090 --> 00:03:43.370 with instructions so simple a child can read 00:03:43.370 --> 00:03:46.689 them, but the box itself is made of impenetrable 00:03:46.689 --> 00:03:49.310 titanium. That's a great way to put it. Why would 00:03:49.310 --> 00:03:51.789 a 10 -year -old look at that centuries -old titanium 00:03:51.789 --> 00:03:54.509 box and confidently think, yeah, I'm going to 00:03:54.509 --> 00:03:56.610 be the first person to crack this? What's fascinating 00:03:56.610 --> 00:03:59.289 here is the psychological appeal of accessible 00:03:59.289 --> 00:04:02.090 mysteries. OK, tell me more about that. For a 00:04:02.090 --> 00:04:04.780 mathematically inclined kid, The fact that the 00:04:04.780 --> 00:04:07.719 problem is so easy to state is exactly the trap. 00:04:08.400 --> 00:04:10.780 It doesn't require advanced calculus to understand 00:04:10.780 --> 00:04:14.039 the premise. It just requires basic arithmetic. 00:04:14.479 --> 00:04:17.899 Right, so it feels solvable. Exactly. Wiles saw 00:04:17.899 --> 00:04:21.500 a universal truth sitting right there, unproven, 00:04:21.720 --> 00:04:24.980 just mocking everyone. But as the sources note, 00:04:25.399 --> 00:04:27.720 reality eventually set in. Because he was still 00:04:27.720 --> 00:04:30.480 just a kid. Right. He realized that his childhood 00:04:30.480 --> 00:04:33.970 toolbox was just too limited. Passion without 00:04:33.970 --> 00:04:36.709 the proper tools simply isn't enough. That makes 00:04:36.709 --> 00:04:38.709 sense. He had to put the puzzle box back on the 00:04:38.709 --> 00:04:40.850 shelf and wait for the rest of mathematics to 00:04:40.850 --> 00:04:43.329 catch up. And when he did, I mean, he gets his 00:04:43.329 --> 00:04:45.810 bachelor's degree at Oxford, gets his PhD at 00:04:45.810 --> 00:04:48.370 Cambridge, working with his advisor John Coats, 00:04:48.850 --> 00:04:50.970 becomes a professor at Princeton. He's building 00:04:50.970 --> 00:04:53.689 this massive, highly successful career in number 00:04:53.689 --> 00:04:56.470 theory. But the dream is still there, just on 00:04:56.470 --> 00:05:00.240 the shelf until 1986. Yes. Decades later, a specific 00:05:00.240 --> 00:05:02.399 mathematical breakthrough drags him right back 00:05:02.399 --> 00:05:06.459 in. Wiles is 33 at this point. Two mathematicians, 00:05:06.720 --> 00:05:09.000 Yourhard Frey and Ken Rabette, managed to build 00:05:09.000 --> 00:05:11.300 a theoretical bridge between Furman's ancient 00:05:11.300 --> 00:05:14.420 equation and a completely different, highly modern 00:05:14.420 --> 00:05:16.800 area of mathematics. Rabette proved something 00:05:16.800 --> 00:05:19.519 called the epsilon conjecture, which cemented 00:05:19.519 --> 00:05:21.899 a link to the Taniyama -Shimura -Wile conjecture. 00:05:22.029 --> 00:05:24.910 OK, let's not just drop those massive terms without 00:05:24.910 --> 00:05:26.509 breaking them down a little bit for everyone 00:05:26.509 --> 00:05:29.310 listening. Sure enough. We've got this 300 -year 00:05:29.310 --> 00:05:32.110 -old equation on one side and this new Taniyama 00:05:32.110 --> 00:05:34.750 -Shimura -Wild conjecture on the other. What 00:05:34.750 --> 00:05:37.769 is the actual bridge connecting them? Think of 00:05:37.769 --> 00:05:40.529 the Taniyama -Shimura -Wild conjecture as a sort 00:05:40.529 --> 00:05:43.290 of mathematical Rosetta Stone. A Rosetta Stone, 00:05:43.370 --> 00:05:46.689 OK. It proposed that every single elliptic curve 00:05:46.839 --> 00:05:50.120 which our sources describe as equations that 00:05:50.120 --> 00:05:52.500 look like the surface of a torus, or basically 00:05:52.500 --> 00:05:55.480 a donut shape. Oh, I like a donut analogy. Right. 00:05:55.699 --> 00:05:57.959 It proposed that these curves could be perfectly 00:05:57.959 --> 00:05:59.800 translated into something called the modular 00:05:59.800 --> 00:06:02.480 form, which is a highly symmetrical object in 00:06:02.480 --> 00:06:04.800 a totally different branch of math. Okay, so 00:06:04.800 --> 00:06:07.579 it translates donut math into symmetrical math. 00:06:07.839 --> 00:06:10.420 Basically, yes. Fray and Rabette proved that 00:06:10.420 --> 00:06:13.399 if Fermat's equation did have a solution, meaning 00:06:13.399 --> 00:06:15.800 if those whole numbers for powers greater than 00:06:15.800 --> 00:06:18.300 two actually existed, you could use those numbers 00:06:18.300 --> 00:06:21.019 to create a very peculiar elliptic curve. A curve 00:06:21.019 --> 00:06:23.360 that shouldn't exist. A mathematical monster. 00:06:24.160 --> 00:06:27.240 It would be an elliptic curve so bizarre that 00:06:27.240 --> 00:06:29.300 it fundamentally could not be translated into 00:06:29.300 --> 00:06:32.160 a modular form. It would break the Rosetta Stone 00:06:32.160 --> 00:06:35.079 entirely. Ah, I see the logic. So if you can 00:06:35.079 --> 00:06:37.579 prove the Rosetta Stone is real, that the Taniyama 00:06:37.579 --> 00:06:40.959 -Shimura -Weil conjecture is true, and all curves 00:06:40.959 --> 00:06:44.000 must translate by strict rules, then you prove 00:06:44.000 --> 00:06:47.160 the monster curve cannot exist. Precisely. And 00:06:47.160 --> 00:06:49.620 if the monster can't exist, Fermat's equation 00:06:49.620 --> 00:06:52.920 has no solutions. You prove Fermat by proving 00:06:52.920 --> 00:06:55.199 the impossibility of the alternative. Exactly. 00:06:55.370 --> 00:06:58.129 Proof by contradiction. You don't actually attack 00:06:58.129 --> 00:07:00.610 Fermat directly at all. You attack the Taniyama 00:07:00.610 --> 00:07:02.529 -Shimura -Wile conjecture. Here's where it gets 00:07:02.529 --> 00:07:05.670 really interesting. Wiles sees this connection, 00:07:05.910 --> 00:07:08.290 and he decides he is going to be the one to prove 00:07:08.290 --> 00:07:10.350 the Taniyama -Shimura -Wile conjecture for these 00:07:10.350 --> 00:07:12.569 specific curves. Which is a monumental task. 00:07:12.709 --> 00:07:15.089 Huge. But he doesn't start publishing papers, 00:07:15.170 --> 00:07:16.769 and he doesn't collaborate with his brilliant 00:07:16.769 --> 00:07:19.029 colleagues at Princeton. No, he doesn't. He vanishes 00:07:19.029 --> 00:07:21.740 into his attic. He dedicates all of his research 00:07:21.740 --> 00:07:25.079 time to this one problem for over six years in 00:07:25.079 --> 00:07:28.259 near total secrecy. Six. He only tells his wife. 00:07:28.699 --> 00:07:31.120 He even covers his tracks by releasing older, 00:07:31.360 --> 00:07:33.420 unrelated work in small segments just to throw 00:07:33.420 --> 00:07:35.939 people off the set. It's extreme. I really have 00:07:35.939 --> 00:07:38.319 to push back on this approach. was this pure 00:07:38.319 --> 00:07:41.639 hubris. It's like a chef secretly trying to invent 00:07:41.639 --> 00:07:44.579 a completely new flavor of food for six years, 00:07:45.160 --> 00:07:47.379 occasionally tossing out fake recipes just to 00:07:47.379 --> 00:07:49.839 confuse rival chefs. That's a funny way to look 00:07:49.839 --> 00:07:53.100 at it. Why the intense secrecy? Was it arrogance 00:07:53.100 --> 00:07:56.379 or absolute necessity? It's a really vital question. 00:07:56.740 --> 00:07:58.699 To understand it, you have to look at the reality 00:07:58.699 --> 00:08:01.040 of the mathematical community at that time. Okay. 00:08:01.420 --> 00:08:04.519 The Taniyama -Shimura -Wile conjecture was viewed 00:08:04.519 --> 00:08:07.259 by contemporaries as extraordinarily difficult, 00:08:07.660 --> 00:08:09.839 perhaps completely impossible to prove. Like 00:08:09.839 --> 00:08:12.959 a career killer. Yeah. John Coates, his former 00:08:12.959 --> 00:08:15.600 supervisor, stated flatly that it seemed impossible. 00:08:16.540 --> 00:08:18.379 Ken Rabette considered himself one of the vast 00:08:18.379 --> 00:08:20.939 majority who believed it was completely inaccessible. 00:08:21.079 --> 00:08:23.300 Wow. Rabette noted that Wiles was one of the 00:08:23.300 --> 00:08:26.839 few people on earth who had the audacity to even 00:08:26.839 --> 00:08:30.019 dream of trying. Audacity being the polite academic 00:08:30.019 --> 00:08:33.440 way of saying he was crazy for trying. Precisely. 00:08:34.120 --> 00:08:36.659 If Wiles had announced his intentions, every 00:08:36.659 --> 00:08:39.159 mathematician, journalist, and, you know, crank 00:08:39.159 --> 00:08:40.799 with a calculator would have been calling his 00:08:40.799 --> 00:08:43.740 office. Oh, the pressure would be insane. Pointing 00:08:43.740 --> 00:08:46.139 out potential pitfalls, demanding updates, the 00:08:46.139 --> 00:08:48.399 sheer noise would have been deafening. Working 00:08:48.399 --> 00:08:52.259 in a silo allowed him unbroken, obsessive concentration. 00:08:52.480 --> 00:08:55.740 He needed the quiet. He needed to hold a massive 00:08:55.740 --> 00:08:58.580 architecture of abstract concepts in his mind 00:08:58.580 --> 00:09:01.460 without a single interruption. But working in 00:09:01.460 --> 00:09:03.779 a silo comes with a terrifying risk. It does. 00:09:03.820 --> 00:09:05.519 When you don't have anyone checking your math 00:09:05.519 --> 00:09:07.899 for six years, you are basically walking a tightrope 00:09:07.899 --> 00:09:10.879 without a net. And it almost ended in disaster. 00:09:11.379 --> 00:09:15.360 Which brings us to June of 1993. Wiles finally 00:09:15.360 --> 00:09:17.649 steps into the light. He's at a conference in 00:09:17.649 --> 00:09:19.870 his hometown of Cambridge giving a three -day 00:09:19.870 --> 00:09:22.830 lecture series. Right. And Gina Collada, reporting 00:09:22.830 --> 00:09:24.870 for the New York Times, noted that he titled 00:09:24.870 --> 00:09:28.350 his lectures, Modular Forms, Elliptic Curves, 00:09:28.470 --> 00:09:31.490 and Gell -Oll Representations. Notice what is 00:09:31.490 --> 00:09:33.889 conspicuously missing from that title. The word 00:09:33.889 --> 00:09:36.889 firmat. There was absolutely no hint he was going 00:09:36.889 --> 00:09:39.350 to talk about the most famous puzzle in history. 00:09:39.570 --> 00:09:42.620 None. He just grinds through his incredibly dense 00:09:42.620 --> 00:09:45.919 proof for three days. And then at the very end 00:09:45.919 --> 00:09:48.399 of the third lecture, he casually concludes that 00:09:48.399 --> 00:09:50.940 he has proved a general case of the conjecture. 00:09:51.100 --> 00:09:53.940 Almost as an afterthought. Yes. He notes that 00:09:53.940 --> 00:09:56.960 this means Fermat's last theorem is true. He 00:09:56.960 --> 00:10:00.519 writes, QED. Thus, it has been demonstrated on 00:10:00.519 --> 00:10:03.860 the board. And just stop speaking. The 300 year 00:10:03.860 --> 00:10:06.340 mystery is supposedly solved. And the mathematical 00:10:06.340 --> 00:10:08.950 community was electrified. It was front -page 00:10:08.950 --> 00:10:11.149 news around the entire world. But the triumph 00:10:11.149 --> 00:10:13.789 was shockingly short -lived. Very short -lived. 00:10:14.210 --> 00:10:17.629 Just two months later, in August of 1993, while 00:10:17.629 --> 00:10:19.850 the proof is undergoing rigorous peer review, 00:10:20.289 --> 00:10:22.370 a critical flaw is discovered. We're talking 00:10:22.370 --> 00:10:24.629 about a proof that was hundreds of pages long, 00:10:24.809 --> 00:10:27.470 and the reviewers find an issue. The sources 00:10:27.470 --> 00:10:29.809 say the problem involved the properties of something 00:10:29.809 --> 00:10:32.610 called the Selmer Group and Wiles's use of a 00:10:32.610 --> 00:10:35.029 tool called an Euler system. That's right. Wait, 00:10:35.029 --> 00:10:36.950 before we talk about the psychological fallout 00:10:36.950 --> 00:10:39.970 of this, What actually broke? What do those terms 00:10:39.970 --> 00:10:42.210 mean in the context of our architectural metaphor? 00:10:42.450 --> 00:10:45.250 Okay, let's imagine Wiles' proof as a magnificent 00:10:45.250 --> 00:10:48.450 100 -story skyscraper. Okay, I can picture that. 00:10:48.970 --> 00:10:51.570 To prove the whole structure holds, Wiles had 00:10:51.570 --> 00:10:53.889 to prove that a specific foundational pillar, 00:10:54.289 --> 00:10:57.090 which we'll call the Selmer group, had a precise 00:10:57.090 --> 00:10:59.669 mathematical limit for how much stress it could 00:10:59.669 --> 00:11:02.649 take. Because if the stress goes over that limit... 00:11:02.649 --> 00:11:05.460 The whole building falls. The Euler system was 00:11:05.460 --> 00:11:08.100 essentially the highly specialized theoretical 00:11:08.100 --> 00:11:11.480 measuring tape Wiles built to calculate that 00:11:11.480 --> 00:11:14.019 exact stress limit. Okay, so he's measuring the 00:11:14.019 --> 00:11:16.779 load bearing capacity of the pillar. Right. But 00:11:16.779 --> 00:11:19.259 during peer review, they realized his measuring 00:11:19.259 --> 00:11:22.600 tape was flawed under one specific set of extreme 00:11:22.600 --> 00:11:25.299 mathematical conditions. The tape effectively 00:11:25.299 --> 00:11:28.159 stretched. Oh no. Yeah, I couldn't guarantee 00:11:28.159 --> 00:11:29.700 the pillar would hold the weight of the logic 00:11:29.700 --> 00:11:32.220 above it. Imagine the sheer stomach -dropping 00:11:32.220 --> 00:11:35.179 horror of this. You've just told the world you 00:11:35.179 --> 00:11:37.539 solved the unsolvable. The confetti has fallen. 00:11:37.620 --> 00:11:40.139 Right. And then an inspector whispers in your 00:11:40.139 --> 00:11:42.600 ear that there's a microscopic, fatal crack in 00:11:42.600 --> 00:11:45.659 the foundation. The 100 -story building is swaying 00:11:45.659 --> 00:11:48.500 in the wind, and the entire academic world is 00:11:48.500 --> 00:11:51.059 watching you. The psychological resilience required 00:11:51.059 --> 00:11:53.860 to survive that moment is honestly hard to fathom. 00:11:53.980 --> 00:11:56.919 Wiles tried to repair that broken oiler system 00:11:56.919 --> 00:12:00.159 for over a year. A whole year of just staring 00:12:00.159 --> 00:12:02.600 at the crack in the foundation. And he failed 00:12:02.600 --> 00:12:06.399 over and over. According to the sources, He was 00:12:06.399 --> 00:12:08.779 on the verge of giving up completely. Wait, so 00:12:08.779 --> 00:12:11.240 after six years of guarding this secret with 00:12:11.240 --> 00:12:14.200 his life, being utterly paranoid about someone 00:12:14.200 --> 00:12:16.759 stealing his thunder, he just throws in the towel 00:12:16.759 --> 00:12:19.320 and decides to ask for help. What wasn't throwing 00:12:19.320 --> 00:12:22.419 in the towel, it was accepting reality. He realized 00:12:22.419 --> 00:12:25.159 he couldn't fix the crack in the foundation entirely 00:12:25.159 --> 00:12:27.620 isolated in his own mind. He had to let someone 00:12:27.620 --> 00:12:30.460 in. He had to break his six -year habit of secrecy. 00:12:31.039 --> 00:12:33.720 He brought in his former doctoral student Richard 00:12:33.720 --> 00:12:36.220 Taylor to help him interrogate the flaw. And 00:12:36.220 --> 00:12:38.039 together they just patch up the measuring tape. 00:12:38.360 --> 00:12:40.120 That's the most fascinating mechanical shift 00:12:40.120 --> 00:12:42.100 of the entire story. They didn't patch it? They 00:12:42.100 --> 00:12:47.679 didn't. No. On September 19th, 1994, Wiles had 00:12:47.679 --> 00:12:50.379 a sudden insight. He realized that trying to 00:12:50.379 --> 00:12:52.740 fix the broken Euler system was the wrong approach 00:12:52.740 --> 00:12:55.419 entirely. He could circumvent the flawed area 00:12:55.419 --> 00:12:58.019 completely by switching out the core mathematical 00:12:58.019 --> 00:13:01.110 framework. The sources mention he used Galois 00:13:01.110 --> 00:13:03.610 representations to replace the elliptic curves 00:13:03.610 --> 00:13:05.870 in that part of the proof. How did that fix the 00:13:05.870 --> 00:13:07.870 problem with the pillar? It didn't fix the pillar. 00:13:07.909 --> 00:13:11.190 It bypassed it. Oh. They used Galois representations, 00:13:11.309 --> 00:13:13.309 combining them with ideas from a mathematician 00:13:13.309 --> 00:13:16.470 named Victor Kalavajian, to salvage a totally 00:13:16.470 --> 00:13:18.149 different approach that was originally developed 00:13:18.149 --> 00:13:21.049 by Mathias Flack. That sounds incredibly complex. 00:13:21.169 --> 00:13:23.049 Basically, instead of trying to repair the flawed 00:13:23.049 --> 00:13:26.190 measuring tape, they build a massive steel bypass 00:13:26.190 --> 00:13:28.690 around the broken pillar using a completely completely 00:13:28.690 --> 00:13:31.429 different material. It was a brilliant synthesis 00:13:31.429 --> 00:13:33.929 that reduced the problem to a class number formula, 00:13:34.470 --> 00:13:36.590 sidestepping the flaw entirely. And it worked. 00:13:36.750 --> 00:13:39.490 It did. Wiles and Taylor published a second paper 00:13:39.490 --> 00:13:41.970 containing the circumvention alongside the main 00:13:41.970 --> 00:13:45.970 proof. Both are published in May 1995 in a dedicated 00:13:45.970 --> 00:13:48.789 issue of the Annals of Mathematics. And with 00:13:48.789 --> 00:13:52.429 that, the 300 -year -old mystery was unequivocally 00:13:52.429 --> 00:13:55.629 put to rest. The proof held up. It did indeed. 00:13:55.899 --> 00:13:58.360 But that brings us to the ultimate question of 00:13:58.360 --> 00:14:01.360 value. Exactly. So... What does this all mean? 00:14:01.399 --> 00:14:03.480 For anyone listening to this who isn't a number 00:14:03.480 --> 00:14:05.960 theorist, why should you care that a guy proved 00:14:05.960 --> 00:14:08.840 a 300 -year -old equation about numbers raised 00:14:08.840 --> 00:14:11.639 to a power? Fair question. I think about it like 00:14:11.639 --> 00:14:14.639 this. It's like someone spending six years inventing 00:14:14.639 --> 00:14:17.299 a revolutionary new type of combustion engine 00:14:17.299 --> 00:14:20.419 just so they can win a single, very specific, 00:14:20.440 --> 00:14:23.139 historic street race. They win the race, which 00:14:23.139 --> 00:14:25.039 is great for them, but then the rest of the world 00:14:25.039 --> 00:14:27.500 looks at the engine and realizes... Wait a minute, 00:14:27.639 --> 00:14:30.039 this technology is the key to commercial space 00:14:30.039 --> 00:14:32.279 travel. If we connect this to the bigger picture, 00:14:32.799 --> 00:14:36.460 your engine analogy is absolutely spot on. The 00:14:36.460 --> 00:14:39.419 real treasure of Wiles' proof wasn't the solution 00:14:39.419 --> 00:14:42.240 to Fermat's equation itself. It was. No, I mean, 00:14:42.460 --> 00:14:45.419 Fermat's last theorem, in isolation, doesn't 00:14:45.419 --> 00:14:47.919 actually do much for the modern world. It's just 00:14:47.919 --> 00:14:50.539 a rigid statement about numbers. The profound 00:14:50.539 --> 00:14:53.539 value was in the entirely new mathematical tools 00:14:53.539 --> 00:14:56.159 Wiles had to forge in the fires of his obsession 00:14:56.159 --> 00:14:58.340 to get there. The bridge between the islands. 00:14:58.779 --> 00:15:01.679 Exactly. The methods he developed began unifying 00:15:01.679 --> 00:15:04.100 completely disparate ideas and theorems, and 00:15:04.100 --> 00:15:07.659 the impact was immediate. How immediate. By 1999, 00:15:08.220 --> 00:15:11.139 just a few years later, his former students Richard 00:15:11.139 --> 00:15:13.919 Taylor, Brian Conrad, and Fred Diamond working 00:15:13.919 --> 00:15:16.730 with Christophe Bruhl took the tools Wiles had 00:15:16.730 --> 00:15:19.809 built and used them to prove the full modularity 00:15:19.809 --> 00:15:22.649 theorem. Wow. Wiles had only proven the limited 00:15:22.649 --> 00:15:25.169 case needed for Fermat. Right. But his students 00:15:25.169 --> 00:15:26.990 took his engine and drove it across the entire 00:15:26.990 --> 00:15:28.850 mathematical landscape. And the mathematical 00:15:28.850 --> 00:15:31.429 world recognized this massive paradigm shift. 00:15:31.629 --> 00:15:34.029 The list of accolades Wiles received is just 00:15:34.029 --> 00:15:37.169 staggering. It really is. Just in 1995 and 1996, 00:15:37.389 --> 00:15:39.509 he wins the Shock Prize, the Fermat Prize, the 00:15:39.509 --> 00:15:41.909 Wolf Prize, the Royal Medal, and the NASA Award 00:15:41.909 --> 00:15:44.299 in Mathematics. And it kept going. He gets a 00:15:44.299 --> 00:15:46.679 MacArthur Fellowship, you know, the genius grant 00:15:46.679 --> 00:15:50.039 in 1997. He even gets a silver plaque from the 00:15:50.039 --> 00:15:51.860 International Mathematical Union because he was 00:15:51.860 --> 00:15:54.360 technically 41 when he completed the proof. Making 00:15:54.360 --> 00:15:56.960 him just one year too old to receive the famous 00:15:56.960 --> 00:15:59.879 Fields Medal, which has a strict cutoff at age 00:15:59.879 --> 00:16:03.100 40. Which is an amusingly rigid quirk of academic 00:16:03.100 --> 00:16:05.500 bureaucracy, honestly. Very much so. You solved 00:16:05.500 --> 00:16:08.639 the biggest problem in history, but you're 41. 00:16:08.919 --> 00:16:11.639 So here's a lovely silver plaque instead. Right. 00:16:11.980 --> 00:16:14.720 But the honors didn't stop there. No, they didn't. 00:16:14.879 --> 00:16:17.559 He was knighted in 2000. He was appointed the 00:16:17.559 --> 00:16:19.840 first Regis Professor of Mathematics at Oxford 00:16:19.840 --> 00:16:23.659 in 2018. There is literally an asteroid flying 00:16:23.659 --> 00:16:27.500 around out there in space named 900 -999 Wiles. 00:16:27.720 --> 00:16:29.960 It is a testament to the magnitude of his achievement. 00:16:30.139 --> 00:16:32.279 But what I find most compelling is how Wiles 00:16:32.279 --> 00:16:35.440 himself views his legacy. When he won the incredibly 00:16:35.440 --> 00:16:38.340 prestigious ABLE Prize in 2016, he said something 00:16:38.340 --> 00:16:40.659 really revealing. He noted that the methods that 00:16:40.659 --> 00:16:43.100 solved Fermat opened up a new way to attack one 00:16:43.100 --> 00:16:45.240 of the big webs of conjectures in contemporary 00:16:45.240 --> 00:16:48.279 math called the Langlands program. Our sources 00:16:48.279 --> 00:16:50.580 mentioned the Langlands program is a grand vision 00:16:50.580 --> 00:16:53.379 that aims to unify different branches of mathematics. 00:16:53.639 --> 00:16:56.399 It is considered by many to be the grand unified 00:16:56.399 --> 00:16:59.720 theory of mathematics. Wiles said his work gave 00:16:59.720 --> 00:17:02.299 mathematicians a new way to look at that unification. 00:17:02.580 --> 00:17:05.680 So it's all about the process. Yes. This journey 00:17:05.680 --> 00:17:08.700 challenges us to recognize that the process of 00:17:08.700 --> 00:17:11.940 wrestling with a stubborn problem The dead ends, 00:17:12.579 --> 00:17:14.559 the tools you have to invent when the old ones 00:17:14.559 --> 00:17:17.960 break, often yields innovations far more valuable 00:17:17.960 --> 00:17:20.240 than the solution you originally set out to find. 00:17:20.500 --> 00:17:22.539 The struggle wasn't just the price he paid to 00:17:22.539 --> 00:17:25.160 solve the puzzle. The struggle was the prize. 00:17:25.519 --> 00:17:28.329 It generated the new math. Perfectly said. Well, 00:17:28.329 --> 00:17:30.650 we've covered a lot of ground today, from a little 00:17:30.650 --> 00:17:33.470 boy in Nigeria refusing to go to school, to a 00:17:33.470 --> 00:17:35.930 10 -year -old staring down a 300 -year -old mystery 00:17:35.930 --> 00:17:38.690 in a Cambridge library, through six years of 00:17:38.690 --> 00:17:41.349 silent obsession, a heart -stopping public failure, 00:17:41.450 --> 00:17:44.349 and finally, a collaborative triumph that reshaped 00:17:44.349 --> 00:17:46.849 modern mathematics. Thank you for joining us 00:17:46.849 --> 00:17:49.730 on this deep dive into the extraordinary, obsessive, 00:17:49.910 --> 00:17:52.630 and ultimately unifying life of Sir Andrew Wiles. 00:17:53.089 --> 00:17:55.589 But before we let you go, there is one final 00:17:55.589 --> 00:17:57.910 detail from this story to mull over. Oh, there 00:17:57.910 --> 00:18:01.089 is. We talked about how Wiles' crowning achievement 00:18:01.089 --> 00:18:03.809 almost collapsed under the weight of that single 00:18:03.809 --> 00:18:07.210 flaw in 1993, an error that existed, at least 00:18:07.210 --> 00:18:09.710 in part, because he had spent over six years 00:18:09.710 --> 00:18:12.299 working in a near total vacuum. Right, completely 00:18:12.299 --> 00:18:14.420 disconnected from the rigorous peer review of 00:18:14.420 --> 00:18:17.660 his colleagues. Exactly. It took swallowing his 00:18:17.660 --> 00:18:20.160 pride and bringing in Richard Taylor to finally 00:18:20.160 --> 00:18:22.660 discover the circumvention that crossed the finish 00:18:22.660 --> 00:18:25.720 line, which leaves you with a lingering question. 00:18:26.160 --> 00:18:27.839 I think I know where you're going with this. 00:18:28.019 --> 00:18:31.519 Think back to that titanium puzzle box. Is the 00:18:31.519 --> 00:18:35.119 extreme, isolated obsession that Wiles utilized 00:18:35.119 --> 00:18:38.440 the only way the human mind can hold enough complex, 00:18:38.619 --> 00:18:41.240 disparate pieces together to solve a seemingly 00:18:41.240 --> 00:18:45.380 impossible problem? Or did his intense secrecy 00:18:45.380 --> 00:18:48.180 actually needlessly delay the greatest mathematical 00:18:48.180 --> 00:18:50.740 breakthrough of the 20th century? It's something 00:18:50.740 --> 00:18:52.700 to think about the next time you decide to tackle 00:18:52.700 --> 00:18:55.220 a massive project entirely on your own.