1 00:00:16,580 --> 00:00:20,700 Mathematics is about solving problems 2 00:00:20,700 --> 00:00:25,540 and it's the great unsolved problems that make maths really alive. 3 00:00:27,540 --> 00:00:28,740 In the summer of 1900, 4 00:00:28,740 --> 00:00:31,460 the International Congress of Mathematicians 5 00:00:31,460 --> 00:00:33,740 was held here in Paris in the Sorbonne. 6 00:00:33,740 --> 00:00:35,980 It was a pretty shambolic affair, 7 00:00:35,980 --> 00:00:38,580 not helped by the sultry August heat. 8 00:00:38,580 --> 00:00:42,300 But it will be remembered as one of the greatest congresses of all time 9 00:00:42,300 --> 00:00:46,660 thanks to a lecture given by the up-and-coming David Hilbert. 10 00:00:48,180 --> 00:00:50,900 Hilbert, a young German mathematician, 11 00:00:50,900 --> 00:00:55,660 boldly set out what he believed were the 23 most important problems 12 00:00:55,660 --> 00:00:57,860 for mathematicians to crack. 13 00:00:57,860 --> 00:01:03,500 He was trying to set the agenda for 20th-century maths and he succeeded. 14 00:01:03,500 --> 00:01:08,780 These Hilbert problems would define the mathematics of the modern age. 15 00:01:08,780 --> 00:01:14,660 Of those who tried to crack Hilbert's challenges, some would experience immense triumphs, 16 00:01:14,660 --> 00:01:18,220 whilst others would be plunged into infinite despair. 17 00:01:29,420 --> 00:01:32,420 The first problem on Hilbert's list emerged from here, 18 00:01:32,420 --> 00:01:35,620 Halle, in East Germany. 19 00:01:35,620 --> 00:01:40,500 It was where the great mathematician Georg Cantor spent all his adult life. 20 00:01:40,500 --> 00:01:44,580 And where he became the first person to really understand the meaning 21 00:01:44,580 --> 00:01:49,460 of infinity and give it mathematical precision. 22 00:01:49,460 --> 00:01:51,780 The statue in the town square, however, 23 00:01:51,780 --> 00:01:56,660 honours Halle's other famous son, the composer George Handel. 24 00:01:56,660 --> 00:02:02,900 To discover more about Cantor, I had to take a tram way out of town. 25 00:02:02,900 --> 00:02:06,300 For 50 years, Halle was part of Communist East Germany 26 00:02:06,300 --> 00:02:09,620 and the Communists loved celebrating their scientists. 27 00:02:09,620 --> 00:02:14,300 So much so, they put Cantor on the side of a large cube that they commissioned. 28 00:02:14,300 --> 00:02:16,940 But, being communists, they didn't put the cube 29 00:02:16,940 --> 00:02:20,020 in the middle of town. They put it out amongst the people. 30 00:02:23,620 --> 00:02:27,140 When I eventually found the estate, I started to fear 31 00:02:27,140 --> 00:02:30,500 that maybe I had got the location wrong. 32 00:02:33,780 --> 00:02:37,980 This looks the most unlikely venue for a statue to a mathematician. 33 00:02:39,260 --> 00:02:41,140 Excuse me? 34 00:02:42,140 --> 00:02:43,260 Ein Frage. 35 00:02:43,260 --> 00:02:46,860 Can you help me a minute? Wie bitte? Do you speak English? No! No? 36 00:02:46,860 --> 00:02:48,900 Ich suche ein Wurfel. 37 00:02:48,900 --> 00:02:50,900 Ein Wurfel, ja? 38 00:02:50,900 --> 00:02:52,180 Is that right? A "Wurfel"? 39 00:02:52,180 --> 00:02:54,460 A cube? Yeah? Like that? 40 00:02:54,460 --> 00:02:57,980 Mit ein Bild der Mathematiker? 41 00:02:57,980 --> 00:03:00,460 Yeah? Go round there? 42 00:03:00,460 --> 00:03:01,740 Die Name ist Cantor. 43 00:03:01,740 --> 00:03:03,860 Somewhere over here. Ah! There it is! 44 00:03:03,860 --> 00:03:05,380 It's much bigger than I thought. 45 00:03:05,380 --> 00:03:09,020 I thought it was going to be something like this sort of size. 46 00:03:09,020 --> 00:03:13,060 Aha, here we are. On the dark side of the cube. 47 00:03:13,060 --> 00:03:15,420 here's the man himself, Cantor. 48 00:03:15,420 --> 00:03:18,180 Cantor's one of my big heroes actually. 49 00:03:18,180 --> 00:03:22,340 I think if I had to choose some theorems that I wish I'd proved, 50 00:03:22,340 --> 00:03:24,340 I think the couple that Cantor proved 51 00:03:24,340 --> 00:03:27,220 would be up there in my top ten. 52 00:03:27,220 --> 00:03:29,500 'This is because before Cantor, 53 00:03:29,500 --> 00:03:32,500 'no-one had really understood infinity.' 54 00:03:32,500 --> 00:03:37,380 It was a tricky, slippery concept that didn't seem to go anywhere. 55 00:03:37,380 --> 00:03:41,980 But Cantor showed that infinity could be perfectly understandable. 56 00:03:41,980 --> 00:03:44,940 Indeed, there wasn't just one infinity, 57 00:03:44,940 --> 00:03:47,420 but infinitely many infinities. 58 00:03:47,420 --> 00:03:53,700 First Cantor took the numbers 1, 2, 3, 4 and so on. 59 00:03:53,700 --> 00:03:57,300 Then he thought about comparing them with a much smaller set... 60 00:03:57,300 --> 00:04:02,020 something like 10, 20, 30, 40... 61 00:04:02,020 --> 00:04:05,620 What he showed is that these two infinite sets of numbers 62 00:04:05,620 --> 00:04:09,940 actually have the same size because we can pair them up - 63 00:04:09,940 --> 00:04:13,820 1 with 10, 2 with 20, 3 with 30 and so on. 64 00:04:13,820 --> 00:04:17,180 So these are the same sizes of infinity. 65 00:04:19,940 --> 00:04:21,980 But what about the fractions? 66 00:04:21,980 --> 00:04:26,820 After all, there are infinitely many fractions between any two whole numbers. 67 00:04:26,820 --> 00:04:30,060 Surely the infinity of fractions is much bigger 68 00:04:30,060 --> 00:04:32,660 than the infinity of whole numbers. 69 00:04:37,660 --> 00:04:40,900 Well, what Cantor did was to find a way to pair up 70 00:04:40,900 --> 00:04:44,700 all of the whole numbers with an infinite load of fractions. 71 00:04:44,700 --> 00:04:46,580 And this is how he did it. 72 00:04:46,580 --> 00:04:51,820 He started by arranging all the fractions in an infinite grid. 73 00:04:51,820 --> 00:04:56,460 The first row contained the whole numbers, fractions with one on the bottom. 74 00:04:56,460 --> 00:05:01,020 In the second row came the halves, fractions with two on the bottom. 75 00:05:01,020 --> 00:05:05,580 And so on. Every fraction appears somewhere in this grid. 76 00:05:05,580 --> 00:05:09,620 Where's two thirds? Second column, third row. 77 00:05:09,620 --> 00:05:14,860 Now imagine a line snaking back and forward diagonally through the fractions. 78 00:05:17,380 --> 00:05:24,220 By pulling this line straight, we can match up every fraction with one of the whole numbers. 79 00:05:24,220 --> 00:05:28,580 This means the fractions are the same sort of infinity 80 00:05:28,580 --> 00:05:30,380 as the whole numbers. 81 00:05:30,380 --> 00:05:33,420 So perhaps all infinities have the same size. 82 00:05:33,420 --> 00:05:35,980 Well, here comes the really exciting bit 83 00:05:35,980 --> 00:05:40,380 because Cantor now considers the set of all infinite decimal numbers. 84 00:05:40,380 --> 00:05:44,620 And here he proves that they give us a bigger infinity because 85 00:05:44,620 --> 00:05:48,620 however you tried to list all the infinite decimals, Cantor produced 86 00:05:48,620 --> 00:05:51,780 a clever argument to show how to construct a new decimal number 87 00:05:51,780 --> 00:05:53,500 that was missing from your list. 88 00:05:53,500 --> 00:05:57,540 Suddenly, the idea of infinity opens up. 89 00:05:57,540 --> 00:06:01,140 There are different infinities, some bigger than others. 90 00:06:01,140 --> 00:06:02,900 It's a really exciting moment. 91 00:06:02,900 --> 00:06:07,180 For me, this is like the first humans understanding how to count. 92 00:06:07,180 --> 00:06:11,420 But now we're counting in a different way. We are counting infinities. 93 00:06:11,420 --> 00:06:17,380 A door has opened and an entirely new mathematics lay before us. 94 00:06:18,620 --> 00:06:20,780 But it never helped Cantor much. 95 00:06:20,780 --> 00:06:24,540 I was in the cemetery in Halle where he is buried 96 00:06:24,540 --> 00:06:27,580 and where I had arranged to meet Professor Joe Dauben. 97 00:06:27,580 --> 00:06:32,020 He was keen to make the connections between Cantor's maths and his life. 98 00:06:33,020 --> 00:06:35,580 He suffered from manic depression. 99 00:06:35,580 --> 00:06:38,980 One of the first big breakdowns he has is in 1884 100 00:06:38,980 --> 00:06:41,460 but then around the turn of the century 101 00:06:41,460 --> 00:06:44,020 these recurrences of the mental illness 102 00:06:44,020 --> 00:06:46,060 become more and more frequent. 103 00:06:46,060 --> 00:06:49,020 A lot of people have tried to say that his mental illness 104 00:06:49,020 --> 00:06:52,420 was triggered by the incredible abstract mathematics he dealt with. 105 00:06:52,420 --> 00:06:56,580 Well, he was certainly struggling, so there may have been a connection. 106 00:06:56,580 --> 00:07:01,220 Yeah, I mean I must say, when you start to contemplate the infinite... 107 00:07:01,220 --> 00:07:04,380 I am pretty happy with the bottom end of the infinite, 108 00:07:04,380 --> 00:07:06,540 but as you build it up more and more, 109 00:07:06,540 --> 00:07:09,220 I must say I start to feel a bit unnerved 110 00:07:09,220 --> 00:07:12,580 about what's going on here and where is it going. 111 00:07:12,580 --> 00:07:17,180 For much of Cantor's life, the only place it was going was here - 112 00:07:17,180 --> 00:07:19,580 the university's sanatorium. 113 00:07:19,580 --> 00:07:23,340 There was no treatment then for manic depression 114 00:07:23,340 --> 00:07:27,220 or indeed for the paranoia that often accompanied Cantor's attacks. 115 00:07:27,220 --> 00:07:30,100 Yet the clinic was a good place to be - 116 00:07:30,100 --> 00:07:32,860 comfortable, quiet and peaceful. 117 00:07:32,860 --> 00:07:37,100 And Cantor often found his time here gave him the mental strength 118 00:07:37,100 --> 00:07:40,420 to resume his exploration of the infinite. 119 00:07:40,420 --> 00:07:43,780 Other mathematicians would be bothered by the paradoxes 120 00:07:43,780 --> 00:07:45,700 that Cantor's work had created. 121 00:07:45,700 --> 00:07:49,780 Curiously, this was one thing Cantor was not worried by. 122 00:07:49,780 --> 00:07:53,100 He was never as upset about the paradox of the infinite 123 00:07:53,100 --> 00:07:56,260 as everybody else was because Cantor believed that 124 00:07:56,260 --> 00:07:59,540 there are certain things that I have been able to show, 125 00:07:59,540 --> 00:08:02,940 we can establish with complete mathematical certainty 126 00:08:02,940 --> 00:08:07,340 and then the absolute infinite which is only in God. 127 00:08:07,340 --> 00:08:11,660 He can understand all of this and there's still that final paradox 128 00:08:11,660 --> 00:08:14,700 that is not given to us to understand, but God does. 129 00:08:17,300 --> 00:08:21,580 But there was one problem that Cantor couldn't leave 130 00:08:21,580 --> 00:08:23,020 in the hands of the Almighty, 131 00:08:23,020 --> 00:08:25,620 a problem he wrestled with for the rest of his life. 132 00:08:25,620 --> 00:08:29,220 It became known as the continuum hypothesis. 133 00:08:29,220 --> 00:08:32,500 Is there an infinity sitting between the smaller infinity 134 00:08:32,500 --> 00:08:37,060 of all the whole numbers and the larger infinity of the decimals? 135 00:08:39,940 --> 00:08:44,380 Cantor's work didn't go down well with many of his contemporaries 136 00:08:44,380 --> 00:08:48,060 but there was one mathematician from France who spoke up for him, 137 00:08:48,060 --> 00:08:50,980 arguing that Cantor's new mathematics of infinity 138 00:08:50,980 --> 00:08:54,140 was "beautiful, if pathological". 139 00:08:54,140 --> 00:08:59,780 Fortunately this mathematician was the most famous and respected mathematician of his day. 140 00:08:59,780 --> 00:09:03,460 When Bertrand Russell was asked by a French politician who he thought 141 00:09:03,460 --> 00:09:08,020 the greatest man France had produced, he replied without hesitation, "Poincare". 142 00:09:08,020 --> 00:09:10,140 The politician was surprised that he'd chosen 143 00:09:10,140 --> 00:09:13,980 the prime minister Raymond Poincare above the likes of Napoleon, Balzac. 144 00:09:13,980 --> 00:09:18,340 Russell replied, "I don't mean Raymond Poincare but his cousin, 145 00:09:18,340 --> 00:09:21,020 "the mathematician, Henri Poincare." 146 00:09:24,780 --> 00:09:28,020 Henri Poincare spent most of his life in Paris, 147 00:09:28,020 --> 00:09:31,820 a city that he loved even with its uncertain climate. 148 00:09:31,820 --> 00:09:35,700 In the last decades of the 19th century, Paris was a centre 149 00:09:35,700 --> 00:09:40,020 for world mathematics and Poincare became its leading light. 150 00:09:40,020 --> 00:09:43,980 Algebra, geometry, analysis, he was good at everything. 151 00:09:43,980 --> 00:09:47,100 His work would lead to all kinds of applications, 152 00:09:47,100 --> 00:09:49,900 from finding your way around on the underground, 153 00:09:49,900 --> 00:09:53,580 to new ways of predicting the weather. 154 00:09:53,580 --> 00:09:56,540 Poincare was very strict about his working day. 155 00:09:56,540 --> 00:09:58,340 Two hours of work in the morning 156 00:09:58,340 --> 00:10:00,380 and two hours in the early evening. 157 00:10:00,380 --> 00:10:01,780 Between these periods, 158 00:10:01,780 --> 00:10:05,460 he would let his subconscious carry on working on the problem. 159 00:10:05,460 --> 00:10:09,540 He records one moment when he had a flash of inspiration which occurred 160 00:10:09,540 --> 00:10:14,060 almost out of nowhere, just as he was getting on a bus. 161 00:10:16,020 --> 00:10:20,780 And one such flash of inspiration led to an early success. 162 00:10:20,780 --> 00:10:24,420 In 1885, King Oscar II of Sweden and Norway 163 00:10:24,420 --> 00:10:31,620 offered a prize of 2,500 crowns for anyone who could establish mathematically once and for all 164 00:10:31,620 --> 00:10:35,980 whether the solar system would continue turning like clockwork, 165 00:10:35,980 --> 00:10:37,940 or might suddenly fly apart. 166 00:10:37,940 --> 00:10:44,020 If the solar system has two planets then Newton had already proved that their orbits would be stable. 167 00:10:44,020 --> 00:10:48,020 The two bodies just travel in ellipsis round each other. 168 00:10:48,020 --> 00:10:52,900 But as soon as soon as you add three bodies like the earth, moon and sun, 169 00:10:52,900 --> 00:10:58,180 the question of whether their orbits were stable or not stumped even the great Newton. 170 00:10:58,180 --> 00:11:02,340 The problem is that now you have some 18 different variables, 171 00:11:02,340 --> 00:11:04,580 like the exact coordinates of each body 172 00:11:04,580 --> 00:11:06,740 and their velocity in each direction. 173 00:11:06,740 --> 00:11:10,020 So the equations become very difficult to solve. 174 00:11:10,020 --> 00:11:14,980 But Poincare made significant headway in sorting them out. 175 00:11:14,980 --> 00:11:21,020 Poincare simplified the problem by making successive approximations to the orbits which he believed 176 00:11:21,020 --> 00:11:24,180 wouldn't affect the final outcome significantly. 177 00:11:24,180 --> 00:11:27,420 Although he couldn't solve the problem in its entirety, 178 00:11:27,420 --> 00:11:32,540 his ideas were sophisticated enough to win him the prize. 179 00:11:32,540 --> 00:11:35,940 He developed this great sort of arsenal of techniques, 180 00:11:35,940 --> 00:11:37,620 mathematical techniques 181 00:11:37,620 --> 00:11:40,180 in order to try and solve it 182 00:11:40,180 --> 00:11:43,340 and in fact, the prize that he won was essentially 183 00:11:43,340 --> 00:11:46,820 more for the techniques than for solving the problem. 184 00:11:46,820 --> 00:11:50,580 But when Poincare's paper was being prepared for publication 185 00:11:50,580 --> 00:11:53,620 by the King's scientific advisor, Mittag-Leffler, 186 00:11:53,620 --> 00:11:55,660 one of the editors found a problem. 187 00:11:58,220 --> 00:12:01,740 Poincare realised he'd made a mistake. 188 00:12:01,740 --> 00:12:05,860 Contrary to what he had originally thought, even a small change in the 189 00:12:05,860 --> 00:12:10,020 initial conditions could end up producing vastly different orbits. 190 00:12:10,020 --> 00:12:12,740 His simplification just didn't work. 191 00:12:12,740 --> 00:12:16,340 But the result was even more important. 192 00:12:16,340 --> 00:12:23,380 The orbits Poincare had discovered indirectly led to what we now know as chaos theory. 193 00:12:23,380 --> 00:12:28,420 Understanding the mathematical rules of chaos explain why a butterfly's wings 194 00:12:28,420 --> 00:12:30,900 could create tiny changes in the atmosphere 195 00:12:30,900 --> 00:12:32,620 that ultimately might cause 196 00:12:32,620 --> 00:12:36,820 a tornado or a hurricane to appear on the other side of the world. 197 00:12:36,820 --> 00:12:39,700 So this big subject of the 20th century, chaos, 198 00:12:39,700 --> 00:12:42,900 actually came out of a mistake that Poincare made 199 00:12:42,900 --> 00:12:44,700 and he spotted at the last minute. 200 00:12:44,700 --> 00:12:48,340 Yes! So the essay had actually been published in its original form, 201 00:12:48,340 --> 00:12:53,540 and was ready to go out and Mittag-Leffler had sent copies out to various people, 202 00:12:53,540 --> 00:12:58,700 and it was to his horror when Poincare wrote to him to say, "Stop!" 203 00:12:58,700 --> 00:13:02,420 Oh, my God. This is every mathematician's worst nightmare. 204 00:13:02,420 --> 00:13:03,940 Absolutely. "Pull it!" 205 00:13:03,940 --> 00:13:05,460 Hold the presses! 206 00:13:06,660 --> 00:13:09,580 Owning up to his mistake, if anything, 207 00:13:09,580 --> 00:13:12,220 enhanced Poincare's reputation. 208 00:13:12,220 --> 00:13:15,100 He continued to produce a wide range of original work 209 00:13:15,100 --> 00:13:16,260 throughout his life. 210 00:13:16,260 --> 00:13:19,300 Not just specialist stuff either. 211 00:13:19,300 --> 00:13:23,780 He also wrote popular books, extolling the importance of maths. 212 00:13:23,780 --> 00:13:27,860 Here we go. Here's a section on the future of mathematics. 213 00:13:29,380 --> 00:13:33,540 It starts, "If we wish to foresee the future of mathematics, 214 00:13:33,540 --> 00:13:38,540 "our proper course is to study the history and present the condition of the science." 215 00:13:38,540 --> 00:13:44,300 So, I think Poincare might have approved of my journey to uncover the story of maths. 216 00:13:44,300 --> 00:13:47,420 He certainly would have approved of the next destination. 217 00:13:47,420 --> 00:13:52,700 Because to discover perhaps Poincare's most important contribution to modern mathematics, 218 00:13:52,700 --> 00:13:55,620 I had to go looking for a bridge. 219 00:13:59,100 --> 00:14:00,780 Seven bridges in fact. 220 00:14:00,780 --> 00:14:03,460 The Seven bridges of Konigsberg. 221 00:14:03,460 --> 00:14:08,580 Today the city is known as Kaliningrad, a little outpost 222 00:14:08,580 --> 00:14:13,540 of Russia on the Baltic Sea surrounded by Poland and Lithuania. 223 00:14:13,540 --> 00:14:17,300 Until 1945, however, when it was ceded to the Soviet Union, 224 00:14:17,300 --> 00:14:20,420 it was the great Prussian City of Konigsberg. 225 00:14:21,940 --> 00:14:25,220 Much of the old town sadly has been demolished. 226 00:14:25,220 --> 00:14:29,100 There is now no sign at all of two of the original seven bridges 227 00:14:29,100 --> 00:14:33,700 and several have changed out of all recognition. 228 00:14:33,700 --> 00:14:37,300 This is one of the original bridges. 229 00:14:37,300 --> 00:14:43,940 It may seem like an unlikely setting for the beginning of a mathematical story, but bear with me. 230 00:14:43,940 --> 00:14:47,220 It started as an 18th-century puzzle. 231 00:14:47,220 --> 00:14:52,460 Is there a route around the city which crosses each of these seven bridges only once? 232 00:14:52,460 --> 00:14:56,540 Finding the solution is much more difficult than it looks. 233 00:15:06,500 --> 00:15:10,340 It was eventually solved by the great mathematician Leonhard Euler, 234 00:15:10,340 --> 00:15:14,700 who in 1735 proved that it wasn't possible. 235 00:15:14,700 --> 00:15:18,980 There could not be a route that didn't cross at least one bridge twice. 236 00:15:18,980 --> 00:15:22,500 He solved the problem by making a conceptual leap. 237 00:15:22,500 --> 00:15:26,740 He realised, you don't really care what the distances are between the bridges. 238 00:15:26,740 --> 00:15:30,780 What really matters is how the bridges are connected together. 239 00:15:30,780 --> 00:15:37,220 This is a problem of a new sort of geometry of position - a problem of topology. 240 00:15:37,220 --> 00:15:40,260 Many of us use topology every day. 241 00:15:40,260 --> 00:15:42,740 Virtually all metro maps the world over 242 00:15:42,740 --> 00:15:45,340 are drawn on topological principles. 243 00:15:45,340 --> 00:15:48,700 You don't care how far the stations are from each other 244 00:15:48,700 --> 00:15:50,500 but how they are connected. 245 00:15:50,500 --> 00:15:53,140 There isn't a metro in Kaliningrad, 246 00:15:53,140 --> 00:15:57,860 but there is in the nearest other Russian city, St Petersburg. 247 00:15:57,860 --> 00:16:00,020 The topology is pretty easy on this map. 248 00:16:00,020 --> 00:16:02,500 It's the Russian I am having difficulty with. 249 00:16:02,500 --> 00:16:05,660 Can you tell me which...? What's the problem? 250 00:16:05,660 --> 00:16:09,060 I want to know what station this one was. 251 00:16:09,060 --> 00:16:12,140 I had it the wrong way round even! 252 00:16:13,940 --> 00:16:17,580 Although topology had its origins in the bridges of Konigsberg, 253 00:16:17,580 --> 00:16:21,820 it was in the hands of Poincare that the subject evolved 254 00:16:21,820 --> 00:16:25,420 into a powerful new way of looking at shape. 255 00:16:25,420 --> 00:16:29,140 Some people refer to topology as bendy geometry 256 00:16:29,140 --> 00:16:33,940 because in topology, two shapes are the same if you can bend or morph 257 00:16:33,940 --> 00:16:36,540 one into another without cutting it. 258 00:16:36,540 --> 00:16:41,660 So for example if I take a football or rugby ball, topologically they 259 00:16:41,660 --> 00:16:45,780 are the same because one can be morphed into the other. 260 00:16:45,780 --> 00:16:51,260 Similarly a bagel and a tea-cup are the same because one can be morphed into the other. 261 00:16:51,260 --> 00:16:58,020 Even very complicated shapes can be unwrapped to become much simpler from a topological point of view. 262 00:16:58,020 --> 00:17:02,060 But there is no way to continuously deform a bagel to morph it into a ball. 263 00:17:02,060 --> 00:17:05,860 The hole in the middle makes these shapes topologically different. 264 00:17:05,860 --> 00:17:11,100 Poincare knew all the possible two-dimensional topological surfaces. 265 00:17:11,100 --> 00:17:14,860 But in 1904 he came up with a topological problem 266 00:17:14,860 --> 00:17:16,780 he just couldn't solve. 267 00:17:16,780 --> 00:17:20,620 If you've got a flat two-dimensional universe then Poincare worked out 268 00:17:20,620 --> 00:17:23,820 all the possible shapes he could wrap that universe up into. 269 00:17:23,820 --> 00:17:28,900 It could be a ball or a bagel with one hole, two holes or more holes in. 270 00:17:28,900 --> 00:17:34,500 But we live in a three-dimensional universe so what are the possible shapes that our universe can be? 271 00:17:34,500 --> 00:17:38,540 That question became known as the Poincare Conjecture. 272 00:17:38,540 --> 00:17:43,260 It was finally solved in 2002 here in St Petersburg 273 00:17:43,260 --> 00:17:46,860 by a Russian mathematician called Grisha Perelman. 274 00:17:46,860 --> 00:17:50,540 His proof is very difficult to understand, even for mathematicians. 275 00:17:50,540 --> 00:17:57,260 Perelman solved the problem by linking it to a completely different area of mathematics. 276 00:17:57,260 --> 00:18:03,100 To understand the shapes, he looked instead at the dynamics of the way things can flow over the shape 277 00:18:03,100 --> 00:18:06,180 which led to a description of all the possible ways 278 00:18:06,180 --> 00:18:10,620 that three dimensional space can be wrapped up in higher dimensions. 279 00:18:10,620 --> 00:18:15,300 I wondered if the man himself could help in unravelling the intricacies of his proof, 280 00:18:15,300 --> 00:18:22,700 but I'd been told that finding Perelman is almost as difficult as understanding the solution. 281 00:18:22,700 --> 00:18:25,340 The classic stereotype of a mathematician 282 00:18:25,340 --> 00:18:29,100 is a mad eccentric scientist, but I think that's a little bit unfair. 283 00:18:29,100 --> 00:18:32,340 Most of my colleagues are normal. Well, reasonably. 284 00:18:32,340 --> 00:18:34,420 But when it comes to Perelman, 285 00:18:34,420 --> 00:18:37,220 there is no doubt he is a very strange character. 286 00:18:37,220 --> 00:18:40,180 He's received prizes and offers of professorships 287 00:18:40,180 --> 00:18:42,860 from distinguished universities in the West 288 00:18:42,860 --> 00:18:45,580 but he's turned them all down. 289 00:18:45,580 --> 00:18:49,100 Recently he seems to have given up mathematics completely 290 00:18:49,100 --> 00:18:51,300 and retreated to live as a semi-recluse 291 00:18:51,300 --> 00:18:54,020 in this very modest housing estate with his mum. 292 00:18:54,020 --> 00:19:00,620 He refuses to talk to the media but I thought he might just talk to me as a fellow mathematician. 293 00:19:00,620 --> 00:19:02,780 I was wrong. 294 00:19:02,780 --> 00:19:06,620 Well, it's interesting. I think he's actually turned off his buzzer. 295 00:19:06,620 --> 00:19:08,900 Probably too many media have been buzzing it. 296 00:19:08,900 --> 00:19:12,220 I tried a neighbour and that rang but his doesn't ring at all. 297 00:19:12,220 --> 00:19:17,860 I think his papers, his mathematics speaks for itself in a way. 298 00:19:17,860 --> 00:19:20,380 I don't really need to meet the mathematician 299 00:19:20,380 --> 00:19:22,860 and in this age of Big Brother and Big Money, 300 00:19:22,860 --> 00:19:26,140 I think there's something noble about the fact he gets his kick 301 00:19:26,140 --> 00:19:28,820 out of proving theorems and not winning prizes. 302 00:19:32,260 --> 00:19:35,300 One mathematician would certainly have applauded. 303 00:19:35,300 --> 00:19:39,740 For solving any of his 23 problems, David Hilbert offered no prize 304 00:19:39,740 --> 00:19:45,060 or reward beyond the admiration of other mathematicians. 305 00:19:45,060 --> 00:19:48,660 When he sketched out the problems in Paris in 1900, 306 00:19:48,660 --> 00:19:51,660 Hilbert himself was already a mathematical star. 307 00:19:51,660 --> 00:19:55,620 And it was in Gottingen in northern Germany that he really shone. 308 00:19:58,740 --> 00:20:04,860 He was by far the most charismatic mathematician of his age. 309 00:20:04,860 --> 00:20:09,260 It's clear that everyone who knew him thought he was absolutely wonderful. 310 00:20:12,180 --> 00:20:16,900 He studied number theory and brought everything together that was there 311 00:20:16,900 --> 00:20:20,060 and then within a year or so he left that completely 312 00:20:20,060 --> 00:20:23,620 and revolutionised the theory of integral equation. 313 00:20:23,620 --> 00:20:26,180 It's always change and always something new, 314 00:20:26,180 --> 00:20:29,140 and there's hardly anybody who is... 315 00:20:29,140 --> 00:20:34,100 who was so flexible and so varied in his approach as Hilbert was. 316 00:20:34,100 --> 00:20:41,100 His work is still talked about today and his name has become attached to many mathematical terms. 317 00:20:41,100 --> 00:20:45,460 Mathematicians still use the Hilbert Space, the Hilbert Classification, 318 00:20:45,460 --> 00:20:50,420 the Hilbert Inequality and several Hilbert theorems. 319 00:20:50,420 --> 00:20:54,100 But it was his early work on equations that marked him out 320 00:20:54,100 --> 00:20:56,820 as a mathematician thinking in new ways. 321 00:20:56,820 --> 00:21:00,780 Hilbert showed that although there are infinitely many equations, 322 00:21:00,780 --> 00:21:04,100 there are ways to divide them up so that they are built 323 00:21:04,100 --> 00:21:07,460 out of just a finite set, like a set of building blocks. 324 00:21:07,460 --> 00:21:13,180 The most striking element of Hilbert's proof was that he couldn't actually construct this finite set. 325 00:21:13,180 --> 00:21:16,740 He just proved it must exist. 326 00:21:16,740 --> 00:21:20,060 Somebody criticised this as theology and not mathematics 327 00:21:20,060 --> 00:21:21,700 but they'd missed the point. 328 00:21:21,700 --> 00:21:25,580 What Hilbert was doing here was creating a new style of mathematics, 329 00:21:25,580 --> 00:21:28,140 a more abstract approach to the subject. 330 00:21:28,140 --> 00:21:30,580 You could still prove something existed, 331 00:21:30,580 --> 00:21:33,540 even though you couldn't construct it explicitly. 332 00:21:33,540 --> 00:21:37,260 It's like saying, "I know there has to be a way to get 333 00:21:37,260 --> 00:21:41,660 "from Gottingen to St Petersburg even though I can't tell you 334 00:21:41,660 --> 00:21:43,740 "how to actually get there." 335 00:21:43,740 --> 00:21:48,420 As well as challenging mathematical orthodoxies, Hilbert was also happy 336 00:21:48,420 --> 00:21:54,140 to knock the formal hierarchies that existed in the university system in Germany at the time. 337 00:21:54,140 --> 00:22:00,300 Other professors were quite shocked to see Hilbert out bicycling and drinking with his students. 338 00:22:00,300 --> 00:22:02,740 He liked very much parties. Yeah? 339 00:22:02,740 --> 00:22:06,540 Yes. Party animal. That's my kind of mathematician. 340 00:22:06,540 --> 00:22:12,660 He liked very much dancing with young women. He liked very much to flirt. 341 00:22:12,660 --> 00:22:17,180 Really? Most mathematicians I know are not the biggest of flirts. 342 00:22:17,180 --> 00:22:21,300 'Yet this lifestyle went hand in hand with an absolute commitment to maths.' 343 00:22:21,300 --> 00:22:25,500 Hilbert was of course somebody who thought 344 00:22:25,500 --> 00:22:29,540 that everybody who has a mathematical skill, 345 00:22:29,540 --> 00:22:35,700 a penguin, a woman, a man, or black, white or yellow, 346 00:22:35,700 --> 00:22:39,580 it doesn't matter, he should do mathematics 347 00:22:39,580 --> 00:22:41,660 and he should be admired for his. 348 00:22:41,660 --> 00:22:45,500 The mathematics speaks for itself in a way. 349 00:22:45,500 --> 00:22:49,020 It doesn't matter... When you're a penguin. 350 00:22:49,020 --> 00:22:53,660 Yeah, if you can prove the Riemann hypothesis, we really don't mind. 351 00:22:53,660 --> 00:22:57,580 Yes, mathematics was for him a universal language. Yes. 352 00:22:57,580 --> 00:23:01,380 Hilbert believed that this language was powerful enough 353 00:23:01,380 --> 00:23:03,660 to unlock all the truths of mathematics, 354 00:23:03,660 --> 00:23:06,940 a belief he expounded in a radio interview he gave 355 00:23:06,940 --> 00:23:10,700 on the future of mathematics on the 8th September 1930. 356 00:23:15,380 --> 00:23:19,580 In it, he had no doubt that all his 23 problems would soon be solved 357 00:23:19,580 --> 00:23:23,020 and that mathematics would finally be put 358 00:23:23,020 --> 00:23:26,140 on unshakeable logical foundations. 359 00:23:26,140 --> 00:23:29,460 There are absolutely no unsolvable problems, he declared, 360 00:23:29,460 --> 00:23:31,820 a belief that's been held by mathematicians 361 00:23:31,820 --> 00:23:33,780 since the time of the Ancient Greeks. 362 00:23:33,780 --> 00:23:39,340 He ended with this clarion call, "We must know, we will know." 363 00:23:39,340 --> 00:23:43,940 'Wir mussen wissen, wir werden wissen.' 364 00:23:45,260 --> 00:23:47,780 Unfortunately for him, the very day before 365 00:23:47,780 --> 00:23:51,620 in a scientific lecture that was not considered worthy of broadcast, 366 00:23:51,620 --> 00:23:54,820 another mathematician would shatter Hilbert's dream 367 00:23:54,820 --> 00:23:58,780 and put uncertainty at the heart of mathematics. 368 00:23:58,780 --> 00:24:01,700 The man responsible for destroying Hilbert's belief 369 00:24:01,700 --> 00:24:04,820 was an Austrian mathematician, Kurt Godel. 370 00:24:09,700 --> 00:24:11,740 And it all started here - Vienna. 371 00:24:11,740 --> 00:24:14,660 Even his admirers, and there are a great many, 372 00:24:14,660 --> 00:24:19,220 admit that Kurt Godel was a little odd. 373 00:24:19,220 --> 00:24:23,140 As a child, he was bright, sickly and very strange. 374 00:24:23,140 --> 00:24:25,180 He couldn't stop asking questions. 375 00:24:25,180 --> 00:24:30,020 So much so, that his family called him Herr Warum - Mr Why. 376 00:24:30,020 --> 00:24:34,460 Godel lived in Vienna in the 1920s and 1930s, 377 00:24:34,460 --> 00:24:37,300 between the fall of the Austro-Hungarian Empire 378 00:24:37,300 --> 00:24:39,260 and its annexation by the Nazis. 379 00:24:39,260 --> 00:24:44,820 It was a strange, chaotic and exciting time to be in the city. 380 00:24:44,820 --> 00:24:47,460 Godel studied mathematics at Vienna University 381 00:24:47,460 --> 00:24:49,900 but he spent most of his time in the cafes, 382 00:24:49,900 --> 00:24:52,260 the internet chat rooms of their time, 383 00:24:52,260 --> 00:24:55,220 where amongst games of backgammon and billiards, 384 00:24:55,220 --> 00:24:58,340 the real intellectual excitement was taking place. 385 00:24:58,340 --> 00:25:01,620 Particularly amongst a highly influential group 386 00:25:01,620 --> 00:25:05,220 of philosophers and scientists called the Vienna Circle. 387 00:25:05,220 --> 00:25:09,380 In their discussions, Kurt Godel would come up with an idea 388 00:25:09,380 --> 00:25:12,300 that would revolutionise mathematics. 389 00:25:12,300 --> 00:25:15,260 He'd set himself a difficult mathematical test. 390 00:25:15,260 --> 00:25:18,060 He wanted to solve Hilbert's second problem 391 00:25:18,060 --> 00:25:21,300 and find a logical foundation for all mathematics. 392 00:25:21,300 --> 00:25:24,820 But what he came up with surprised even him. 393 00:25:24,820 --> 00:25:28,260 All his efforts in mathematical logic not only couldn't provide 394 00:25:28,260 --> 00:25:33,140 the guarantee Hilbert wanted, instead he proved the opposite. 395 00:25:33,140 --> 00:25:34,740 Got it. 396 00:25:34,740 --> 00:25:38,100 It's called the Incompleteness Theorem. 397 00:25:38,100 --> 00:25:41,660 Godel proved that within any logical system for mathematics 398 00:25:41,660 --> 00:25:45,500 there will be statements about numbers which are true 399 00:25:45,500 --> 00:25:47,500 but which you cannot prove. 400 00:25:47,500 --> 00:25:52,300 He starts with the statement, "This statement cannot be proved." 401 00:25:52,300 --> 00:25:54,780 This is not a mathematical statement yet. 402 00:25:54,780 --> 00:25:57,660 But by using a clever code based on prime numbers, 403 00:25:57,660 --> 00:26:02,780 Godel could change this statement into a pure statement of arithmetic. 404 00:26:02,780 --> 00:26:07,940 Now, such statements must be either true or false. 405 00:26:07,940 --> 00:26:12,620 Hold on to your logical hats as we explore the possibilities. 406 00:26:12,620 --> 00:26:17,260 If the statement is false, that means the statement could be proved, 407 00:26:17,260 --> 00:26:20,620 which means it would be true, and that's a contradiction. 408 00:26:20,620 --> 00:26:23,180 So that means, the statement must be true. 409 00:26:23,180 --> 00:26:27,620 In other words, here is a mathematical statement that is true 410 00:26:27,620 --> 00:26:30,140 but can't be proved. 411 00:26:30,140 --> 00:26:31,740 Blast. 412 00:26:31,740 --> 00:26:34,820 Godel's proof led to a crisis in mathematics. 413 00:26:34,820 --> 00:26:38,620 What if the problem you were working on, the Goldbach conjecture, say, 414 00:26:38,620 --> 00:26:42,900 or the Riemann hypothesis, would turn out to be true but unprovable? 415 00:26:42,900 --> 00:26:46,020 It led to a crisis for Godel too. 416 00:26:46,020 --> 00:26:49,700 In the autumn of 1934, he suffered the first of what became 417 00:26:49,700 --> 00:26:54,820 a series of breakdowns and spent time in a sanatorium. 418 00:26:54,820 --> 00:26:58,260 He was saved by the love of a good woman. 419 00:26:58,260 --> 00:27:02,180 Adele Nimbursky was a dancer in a local night club. 420 00:27:02,180 --> 00:27:05,500 She kept Godel alive. 421 00:27:05,500 --> 00:27:09,340 One day, she and Godel were walking down these very steps. 422 00:27:09,340 --> 00:27:12,420 Suddenly he was attacked by Nazi thugs. 423 00:27:12,420 --> 00:27:16,660 Godel himself wasn't Jewish, but many of his friends in the Vienna Circle were. 424 00:27:16,660 --> 00:27:19,140 Adele came to his rescue. 425 00:27:19,140 --> 00:27:23,700 But it was only a temporary reprieve for Godel and for maths. 426 00:27:23,700 --> 00:27:28,980 All across Austria and Germany, mathematics was about to die. 427 00:27:32,980 --> 00:27:35,540 In the new German empire in the late 1930s 428 00:27:35,540 --> 00:27:39,060 there weren't colourful balloons flying over the universities, 429 00:27:39,060 --> 00:27:40,900 but swastikas. 430 00:27:40,900 --> 00:27:45,580 The Nazis passed a law allowing the removal of any civil servant 431 00:27:45,580 --> 00:27:46,980 who wasn't Aryan. 432 00:27:46,980 --> 00:27:50,500 Academics were civil servants in Germany then and now. 433 00:27:52,820 --> 00:27:55,500 Mathematicians suffered more than most. 434 00:27:55,500 --> 00:27:58,900 144 in Germany would lose their jobs. 435 00:27:58,900 --> 00:28:03,340 14 were driven to suicide or died in concentration camps. 436 00:28:06,980 --> 00:28:09,900 But one brilliant mathematician stayed on. 437 00:28:09,900 --> 00:28:11,700 David Hilbert helped arrange 438 00:28:11,700 --> 00:28:14,300 for some of his brightest students to flee. 439 00:28:14,300 --> 00:28:16,940 And he spoke out for a while about the dismissal 440 00:28:16,940 --> 00:28:18,500 of his Jewish colleagues. 441 00:28:18,500 --> 00:28:22,700 But soon, he too became silent. 442 00:28:26,020 --> 00:28:28,540 It's not clear why he didn't flee himself 443 00:28:28,540 --> 00:28:30,620 or at least protest a little more. 444 00:28:30,620 --> 00:28:32,900 He did fall ill towards the end of his life 445 00:28:32,900 --> 00:28:35,100 so maybe he just didn't have the energy. 446 00:28:35,100 --> 00:28:37,740 All around him, mathematicians and scientists 447 00:28:37,740 --> 00:28:41,460 were fleeing the Nazi regime until it was only Hilbert left 448 00:28:41,460 --> 00:28:46,780 to witness the destruction of one of the greatest mathematical centres of all time. 449 00:28:49,300 --> 00:28:52,940 David Hilbert died in 1943. 450 00:28:52,940 --> 00:28:55,660 Only ten people attended the funeral 451 00:28:55,660 --> 00:28:58,900 of the most famous mathematician of his time. 452 00:28:58,900 --> 00:29:01,180 The dominance of Europe, 453 00:29:01,180 --> 00:29:04,980 the centre for world maths for 500 years, was over. 454 00:29:04,980 --> 00:29:11,300 It was time for the mathematical baton to be handed to the New World. 455 00:29:13,140 --> 00:29:16,420 Time in fact for this place. 456 00:29:16,420 --> 00:29:21,340 The Institute for Advanced Study had been set up in Princeton in 1930. 457 00:29:21,340 --> 00:29:24,180 The idea was to reproduce the collegiate atmosphere 458 00:29:24,180 --> 00:29:28,180 of the old European universities in rural New Jersey. 459 00:29:28,180 --> 00:29:31,500 But to do this, it needed to attract the very best, 460 00:29:31,500 --> 00:29:33,580 and it didn't need to look far. 461 00:29:33,580 --> 00:29:36,780 Many of the brightest European mathematicians 462 00:29:36,780 --> 00:29:39,220 were fleeing the Nazis for America. 463 00:29:39,220 --> 00:29:41,820 People like Hermann Weyl, whose research 464 00:29:41,820 --> 00:29:44,980 would have major significance for theoretical physics. 465 00:29:44,980 --> 00:29:47,580 And John Von Neumann, who developed game theory 466 00:29:47,580 --> 00:29:50,140 and was one of the pioneers of computer science. 467 00:29:50,140 --> 00:29:54,700 The Institute quickly became the perfect place 468 00:29:54,700 --> 00:29:58,740 to create another Gottingen in the woods. 469 00:29:58,740 --> 00:30:04,060 One mathematician in particular made the place a home from home. 470 00:30:04,060 --> 00:30:05,620 Every morning Kurt Godel, 471 00:30:05,620 --> 00:30:08,660 dressed in a white linen suit and wearing a fedora, 472 00:30:08,660 --> 00:30:12,340 would walk from his home along Mercer Street to the Institute. 473 00:30:12,340 --> 00:30:15,820 On his way, he would stop here at number 112, 474 00:30:15,820 --> 00:30:21,940 to pick up his closest friend, another European exile, Albert Einstein. 475 00:30:21,940 --> 00:30:26,260 But not even relaxed, affluent Princeton could help Godel 476 00:30:26,260 --> 00:30:28,340 finally escape his demons. 477 00:30:28,340 --> 00:30:30,940 Einstein was always full of laughter. 478 00:30:30,940 --> 00:30:34,820 He described Princeton as a banishment to paradise. 479 00:30:34,820 --> 00:30:39,380 But the much younger Godel became increasingly solemn and pessimistic. 480 00:30:42,460 --> 00:30:45,700 Slowly this pessimism turned into paranoia. 481 00:30:45,700 --> 00:30:49,820 He spent less and less time with his fellow mathematicians in Princeton. 482 00:30:49,820 --> 00:30:53,500 Instead, he preferred to come here to the beach, next to the ocean, 483 00:30:53,500 --> 00:30:58,540 walk alone, thinking about the works of the great German mathematician, Leibniz. 484 00:31:00,700 --> 00:31:04,620 But as Godel was withdrawing into his own interior world, 485 00:31:04,620 --> 00:31:08,620 his influence on American mathematics paradoxically 486 00:31:08,620 --> 00:31:11,300 was growing stronger and stronger. 487 00:31:11,300 --> 00:31:15,460 And a young mathematician from just along the New Jersey coast 488 00:31:15,460 --> 00:31:19,140 eagerly took on some of the challenges he posed. 489 00:31:19,140 --> 00:31:23,060 # One, two, three, four, five, six, seven, eight, nine, ten 490 00:31:23,060 --> 00:31:25,180 # Times a day I could love you... # 491 00:31:25,180 --> 00:31:26,340 In 1950s America, 492 00:31:26,340 --> 00:31:30,740 the majority of youngsters weren't preoccupied with mathematics. 493 00:31:30,740 --> 00:31:34,460 Most went for a more relaxed, hedonistic lifestyle 494 00:31:34,460 --> 00:31:38,140 in this newly affluent land of ice-cream and doughnuts. 495 00:31:38,140 --> 00:31:41,860 But one teenager didn't indulge in the normal pursuits 496 00:31:41,860 --> 00:31:44,940 of American adolescence but chose instead 497 00:31:44,940 --> 00:31:48,500 to grapple with some of the major problems in mathematics. 498 00:31:48,500 --> 00:31:49,980 From a very early age, 499 00:31:49,980 --> 00:31:54,380 Paul Cohen was winning mathematical competitions and prizes. 500 00:31:54,380 --> 00:31:58,260 But he found it difficult at first to discover a field in mathematics 501 00:31:58,260 --> 00:32:00,580 where he could really make his mark... 502 00:32:00,580 --> 00:32:05,020 Until he read about Cantor's continuum hypothesis. 503 00:32:05,020 --> 00:32:08,580 That's the one problem, as I had learned back in Halle, 504 00:32:08,580 --> 00:32:11,060 that Cantor just couldn't resolve. 505 00:32:11,060 --> 00:32:14,700 It asks whether there is or there isn't an infinite set of numbers 506 00:32:14,700 --> 00:32:17,380 bigger than the set of all whole numbers 507 00:32:17,380 --> 00:32:20,260 but smaller than the set of all decimals. 508 00:32:20,260 --> 00:32:23,580 It sounds straightforward, but it had foiled all attempts 509 00:32:23,580 --> 00:32:28,460 to solve it since Hilbert made it his first problem way back in 1900. 510 00:32:28,460 --> 00:32:30,780 With the arrogance of youth, 511 00:32:30,780 --> 00:32:35,340 the 22-year-old Paul Cohen decided that he could do it. 512 00:32:35,340 --> 00:32:40,020 Cohen came back a year later with the extraordinary discovery 513 00:32:40,020 --> 00:32:42,500 that both answers could be true. 514 00:32:42,500 --> 00:32:46,460 There was one mathematics where the continuum hypothesis 515 00:32:46,460 --> 00:32:48,380 could be assumed to be true. 516 00:32:48,380 --> 00:32:51,100 There wasn't a set between the whole numbers 517 00:32:51,100 --> 00:32:52,740 and the infinite decimals. 518 00:32:54,460 --> 00:32:58,500 But there was an equally consistent mathematics 519 00:32:58,500 --> 00:33:02,740 where the continuum hypothesis could be assumed to be false. 520 00:33:02,740 --> 00:33:07,580 Here, there was a set between the whole numbers and the infinite decimals. 521 00:33:07,580 --> 00:33:10,780 It was an incredibly daring solution. 522 00:33:10,780 --> 00:33:13,140 Cohen's proof seemed true, 523 00:33:13,140 --> 00:33:18,460 but his method was so new that nobody was absolutely sure. 524 00:33:18,460 --> 00:33:22,020 There was only one person whose opinion everybody trusted. 525 00:33:22,020 --> 00:33:25,940 There was a lot of scepticism and he had to come and make a trip here, 526 00:33:25,940 --> 00:33:28,620 to the Institute right here, to visit Godel. 527 00:33:28,620 --> 00:33:32,020 And it was only after Godel gave his stamp of approval 528 00:33:32,020 --> 00:33:33,540 in quite an unusual way, 529 00:33:33,540 --> 00:33:37,180 He said, "Give me your paper", and then on Monday he put it back 530 00:33:37,180 --> 00:33:39,660 in the box and said, "Yes, it's correct." 531 00:33:39,660 --> 00:33:41,340 Then everything changed. 532 00:33:42,540 --> 00:33:45,500 Today mathematicians insert a statement 533 00:33:45,500 --> 00:33:50,140 that says whether the result depends on the continuum hypothesis. 534 00:33:50,140 --> 00:33:54,180 We've built up two different mathematical worlds 535 00:33:54,180 --> 00:33:56,620 in which one answer is yes and the other is no. 536 00:33:56,620 --> 00:34:00,740 Paul Cohen really has rocked the mathematical universe. 537 00:34:00,740 --> 00:34:04,980 It gave him fame, riches, and prizes galore. 538 00:34:06,980 --> 00:34:12,180 He didn't publish much after his early success in the '60s. 539 00:34:12,180 --> 00:34:14,340 But he was absolutely dynamite. 540 00:34:14,340 --> 00:34:18,140 I can't imagine anyone better to learn from, and he was very eager 541 00:34:18,140 --> 00:34:23,140 to learn, to teach you anything he knew or even things he didn't know. 542 00:34:23,140 --> 00:34:26,940 With the confidence that came from solving Hilbert's first problem, 543 00:34:26,940 --> 00:34:29,620 Cohen settled down in the mid 1960s 544 00:34:29,620 --> 00:34:33,740 to have a go at the most important Hilbert problem of them all - 545 00:34:33,740 --> 00:34:36,260 the eighth, the Riemann hypothesis. 546 00:34:36,260 --> 00:34:42,300 When he died in California in 2007, 40 years later, he was still trying. 547 00:34:42,300 --> 00:34:45,500 But like many famous mathematicians before him, 548 00:34:45,500 --> 00:34:47,580 Riemann had defeated even him. 549 00:34:47,580 --> 00:34:51,740 But his approach has inspired others to make progress towards a proof, 550 00:34:51,740 --> 00:34:54,860 including one of his most famous students, Peter Sarnak. 551 00:34:54,860 --> 00:34:58,740 I think, overall, absolutely loved the guy. 552 00:34:58,740 --> 00:35:01,140 He was my inspiration. 553 00:35:01,140 --> 00:35:03,900 I'm really glad I worked with him. 554 00:35:03,900 --> 00:35:06,100 He put me on the right track. 555 00:35:09,260 --> 00:35:13,540 Paul Cohen is a good example of the success of the great American Dream. 556 00:35:13,540 --> 00:35:16,100 The second generation Jewish immigrant 557 00:35:16,100 --> 00:35:18,260 becomes top American professor. 558 00:35:18,260 --> 00:35:22,940 But I wouldn't say that his mathematics was a particularly American product. 559 00:35:22,940 --> 00:35:25,020 Cohen was so fired up by this problem 560 00:35:25,020 --> 00:35:28,980 that he probably would have cracked it whatever the surroundings. 561 00:35:30,500 --> 00:35:32,980 Paul Cohen had it relatively easy. 562 00:35:32,980 --> 00:35:35,940 But another great American mathematician of the 1960s 563 00:35:35,940 --> 00:35:39,620 faced a much tougher struggle to make an impact. 564 00:35:39,620 --> 00:35:42,740 Not least because she was female. 565 00:35:42,740 --> 00:35:47,540 In the story of maths, nearly all the truly great mathematicians have been men. 566 00:35:47,540 --> 00:35:50,860 But there have been a few exceptions. 567 00:35:50,860 --> 00:35:53,300 There was the Russian Sofia Kovalevskaya 568 00:35:53,300 --> 00:35:58,220 who became the first female professor of mathematics in Stockholm in 1889, 569 00:35:58,220 --> 00:36:02,700 and won a very prestigious French mathematical prize. 570 00:36:02,700 --> 00:36:06,380 And then Emmy Noether, a talented algebraist who fled from the Nazis 571 00:36:06,380 --> 00:36:09,900 to America but then died before she fully realised her potential. 572 00:36:09,900 --> 00:36:15,220 Then there is the woman who I am crossing America to find out about. 573 00:36:15,220 --> 00:36:18,980 Julia Robinson, the first woman ever to be elected president 574 00:36:18,980 --> 00:36:21,380 of the American Mathematical Society. 575 00:36:30,740 --> 00:36:34,140 She was born in St Louis in 1919, 576 00:36:34,140 --> 00:36:37,460 but her mother died when she was two. 577 00:36:37,460 --> 00:36:41,660 She and her sister Constance moved to live with their grandmother 578 00:36:41,660 --> 00:36:45,020 in a small community in the desert near Phoenix, Arizona. 579 00:36:47,020 --> 00:36:49,100 Julia Robinson grew up around here. 580 00:36:49,100 --> 00:36:52,740 I've got a photo which shows her cottage in the 1930s, 581 00:36:52,740 --> 00:36:54,780 with nothing much around it. 582 00:36:54,780 --> 00:36:57,460 The mountains pretty much match those over there 583 00:36:57,460 --> 00:36:59,940 so I think she might have lived down there. 584 00:37:00,900 --> 00:37:03,460 Julia grew up a shy, sickly girl, 585 00:37:03,460 --> 00:37:08,740 who, when she was seven, spent a year in bed because of scarlet fever. 586 00:37:08,740 --> 00:37:11,540 Ill-health persisted throughout her childhood. 587 00:37:11,540 --> 00:37:14,420 She was told she wouldn't live past 40. 588 00:37:14,420 --> 00:37:19,700 But right from the start, she had an innate mathematical ability. 589 00:37:19,700 --> 00:37:24,540 Under the shade of the native Arizona cactus, she whiled away the time 590 00:37:24,540 --> 00:37:28,020 playing endless counting games with stone pebbles. 591 00:37:28,020 --> 00:37:31,260 This early searching for patterns would give her a feel 592 00:37:31,260 --> 00:37:34,620 and love of numbers that would last for the rest of her life. 593 00:37:34,620 --> 00:37:38,460 But despite showing an early brilliance, she had to continually 594 00:37:38,460 --> 00:37:43,380 fight at school and college to simply be allowed to keep doing maths. 595 00:37:43,380 --> 00:37:47,220 As a teenager, she was the only girl in the maths class 596 00:37:47,220 --> 00:37:49,900 but had very little encouragement. 597 00:37:49,900 --> 00:37:54,780 The young Julia sought intellectual stimulation elsewhere. 598 00:37:54,780 --> 00:37:58,740 Julia loved listening to a radio show called the University Explorer 599 00:37:58,740 --> 00:38:01,740 and the 53rd episode was all about mathematics. 600 00:38:01,740 --> 00:38:04,260 The broadcaster described how he discovered 601 00:38:04,260 --> 00:38:07,860 despite their esoteric language and their seclusive nature, 602 00:38:07,860 --> 00:38:11,620 mathematicians are the most interesting and inspiring creatures. 603 00:38:11,620 --> 00:38:15,540 For the first time, Julia had found out that there were mathematicians, 604 00:38:15,540 --> 00:38:17,220 not just mathematics teachers. 605 00:38:17,220 --> 00:38:19,740 There was a world of mathematics out there, 606 00:38:19,740 --> 00:38:21,540 and she wanted to be part of it. 607 00:38:25,380 --> 00:38:28,980 The doors to that world opened here at the University of California, 608 00:38:28,980 --> 00:38:31,260 at Berkeley near San Francisco. 609 00:38:33,060 --> 00:38:37,980 She was absolutely obsessed that she wanted to go to Berkeley. 610 00:38:37,980 --> 00:38:43,500 She wanted to go away to some place where there were mathematicians. 611 00:38:43,500 --> 00:38:46,020 Berkeley certainly had mathematicians, 612 00:38:46,020 --> 00:38:49,620 including a number theorist called Raphael Robinson. 613 00:38:49,620 --> 00:38:52,700 In their frequent walks around the campus 614 00:38:52,700 --> 00:38:59,260 they found they had more than just a passion for mathematics. They married in 1952. 615 00:38:59,260 --> 00:39:02,500 Julia got her PhD and settled down 616 00:39:02,500 --> 00:39:05,020 to what would turn into her lifetime's work - 617 00:39:05,020 --> 00:39:06,580 Hilbert's tenth problem. 618 00:39:06,580 --> 00:39:09,300 She thought about it all the time. 619 00:39:09,300 --> 00:39:13,420 She said to me she just wouldn't wanna die without knowing that answer 620 00:39:13,420 --> 00:39:15,540 and it had become an obsession. 621 00:39:16,580 --> 00:39:20,500 Julia's obsession has been shared with many other mathematicians 622 00:39:20,500 --> 00:39:23,860 since Hilbert had first posed it back in 1900. 623 00:39:23,860 --> 00:39:27,700 His tenth problem asked if there was some universal method 624 00:39:27,700 --> 00:39:33,500 that could tell whether any equation had whole number solutions or not. 625 00:39:33,500 --> 00:39:35,820 Nobody had been able to solve it. 626 00:39:35,820 --> 00:39:38,820 In fact, the growing belief was 627 00:39:38,820 --> 00:39:41,740 that no such universal method was possible. 628 00:39:41,740 --> 00:39:43,820 How on earth could you prove that, 629 00:39:43,820 --> 00:39:47,700 however ingenious you were, you'd never come up with a method? 630 00:39:49,380 --> 00:39:51,100 With the help of colleagues, 631 00:39:51,100 --> 00:39:54,940 Julia developed what became known as the Robinson hypothesis. 632 00:39:54,940 --> 00:39:58,220 This argued that to show no such method existed, 633 00:39:58,220 --> 00:40:02,580 all you had to do was to cook up one equation whose solutions 634 00:40:02,580 --> 00:40:05,340 were a very specific set of numbers. 635 00:40:05,340 --> 00:40:08,580 The set of numbers needed to grow exponentially, 636 00:40:08,580 --> 00:40:13,260 like taking powers of two, yet still be captured by the equations 637 00:40:13,260 --> 00:40:15,820 at the heart of Hilbert's problem. 638 00:40:15,820 --> 00:40:20,900 Try as she might, Robinson just couldn't find this set. 639 00:40:20,900 --> 00:40:25,180 For the tenth problem to be finally solved, 640 00:40:25,180 --> 00:40:28,180 there needed to be some fresh inspiration. 641 00:40:28,180 --> 00:40:33,580 That came from 5,000 miles away - St Petersburg in Russia. 642 00:40:33,580 --> 00:40:37,140 Ever since the great Leonhard Euler set up shop here 643 00:40:37,140 --> 00:40:38,340 in the 18th century, 644 00:40:38,340 --> 00:40:42,260 the city has been famous for its mathematics and mathematicians. 645 00:40:42,260 --> 00:40:44,060 Here in the Steklov Institute, 646 00:40:44,060 --> 00:40:46,780 some of the world's brightest mathematicians 647 00:40:46,780 --> 00:40:49,460 have set out their theorems and conjectures. 648 00:40:49,460 --> 00:40:53,620 This morning, one of them is giving a rare seminar. 649 00:40:56,420 --> 00:40:59,340 It's tough going even if you speak Russian, 650 00:40:59,340 --> 00:41:01,380 which unfortunately I don't. 651 00:41:01,380 --> 00:41:05,620 But we do get a break in the middle to recover before the final hour. 652 00:41:05,620 --> 00:41:07,620 There is a kind of rule in seminars. 653 00:41:07,620 --> 00:41:12,180 The first third is for everyone, the second third for the experts 654 00:41:12,180 --> 00:41:15,380 and the last third is just for the lecturer. 655 00:41:15,380 --> 00:41:18,380 I think that's what we're going to get next. 656 00:41:18,380 --> 00:41:22,100 The lecturer is Yuri Matiyasevich and he is explaining 657 00:41:22,100 --> 00:41:25,820 his latest work on - what else? - the Riemann hypothesis. 658 00:41:28,020 --> 00:41:32,460 As a bright young graduate student in 1965, Yuri's tutor 659 00:41:32,460 --> 00:41:35,300 suggested he have a go at another Hilbert problem, 660 00:41:35,300 --> 00:41:38,300 the one that had in fact preoccupied Julia Robinson. 661 00:41:38,300 --> 00:41:39,580 Hilbert's tenth. 662 00:41:42,460 --> 00:41:44,380 It was the height of the Cold War. 663 00:41:44,380 --> 00:41:47,740 Perhaps Matiyasevich could succeed for Russia 664 00:41:47,740 --> 00:41:51,380 where Julia and her fellow American mathematicians had failed. 665 00:41:51,380 --> 00:41:54,300 At first I did not like their approach. Oh, right. 666 00:41:54,300 --> 00:41:58,940 The statement looked to me rather strange and artificial 667 00:41:58,940 --> 00:42:02,820 but after some time I understood it was quite natural, 668 00:42:02,820 --> 00:42:06,500 and then I understood that she had a new brilliant idea 669 00:42:06,500 --> 00:42:09,300 and I just started to further develop it. 670 00:42:10,820 --> 00:42:16,300 In January 1970, he found the vital last piece in the jigsaw. 671 00:42:16,300 --> 00:42:21,180 He saw how to capture the famous Fibonacci sequence of numbers 672 00:42:21,180 --> 00:42:25,340 using the equations that were at the heart of Hilbert's problem. 673 00:42:25,340 --> 00:42:28,220 Building on the work of Julia and her colleagues, 674 00:42:28,220 --> 00:42:30,020 he had solved the tenth. 675 00:42:30,020 --> 00:42:33,540 He was just 22 years old. 676 00:42:33,540 --> 00:42:37,220 The first person he wanted to tell was the woman he owed so much to. 677 00:42:39,100 --> 00:42:41,020 I got no answer 678 00:42:41,020 --> 00:42:43,900 and I believed they were lost in the mail. 679 00:42:43,900 --> 00:42:47,020 It was quite natural because it was Soviet time. 680 00:42:47,020 --> 00:42:50,100 But back in California, Julia had heard rumours 681 00:42:50,100 --> 00:42:54,140 through the mathematical grapevine that the problem had been solved. 682 00:42:54,140 --> 00:42:56,420 And she contacted Yuri herself. 683 00:42:57,420 --> 00:43:00,780 She said, I just had to wait for you to grow up 684 00:43:00,780 --> 00:43:05,460 to get the answer, because she had started work in 1948. 685 00:43:05,460 --> 00:43:07,260 When Yuri was just a baby. 686 00:43:07,260 --> 00:43:10,540 Then he responds by thanking her 687 00:43:10,540 --> 00:43:15,460 and saying that the credit is as much hers as it is his. 688 00:43:17,540 --> 00:43:19,820 YURI: I met Julia one year later. 689 00:43:19,820 --> 00:43:24,380 It was in Bucharest. I suggested after the conference in Bucharest 690 00:43:24,380 --> 00:43:29,420 Julia and her husband Raphael came to see me here in Leningrad. 691 00:43:29,420 --> 00:43:34,700 Together, Julia and Yuri worked on several other mathematical problems 692 00:43:34,700 --> 00:43:38,460 until shortly before Julia died in 1985. 693 00:43:38,460 --> 00:43:41,260 She was just 55 years old. 694 00:43:41,260 --> 00:43:44,940 She was able to find the new ways. 695 00:43:44,940 --> 00:43:48,940 Many mathematicians just combine previous known methods 696 00:43:48,940 --> 00:43:54,860 to solve new problems and she had really new ideas. 697 00:43:54,860 --> 00:43:58,460 Although Julia Robinson showed there was no universal method 698 00:43:58,460 --> 00:44:00,860 to solve all equations in whole numbers, 699 00:44:00,860 --> 00:44:05,140 mathematicians were still interested in finding methods 700 00:44:05,140 --> 00:44:08,060 to solve special classes of equations. 701 00:44:08,060 --> 00:44:10,620 It would be in France in the early 19th century, 702 00:44:10,620 --> 00:44:12,860 in one of the most extraordinary stories 703 00:44:12,860 --> 00:44:16,420 in the history of mathematics, that methods were developed 704 00:44:16,420 --> 00:44:19,540 to understand why certain equations could be solved 705 00:44:19,540 --> 00:44:21,060 while others couldn't. 706 00:44:27,140 --> 00:44:31,820 It's early morning in Paris on the 29th May 1832. 707 00:44:31,820 --> 00:44:36,420 Evariste Galois is about to fight for his very life. 708 00:44:36,420 --> 00:44:39,980 It is the reign of the reactionary Bourbon King, Charles X, 709 00:44:39,980 --> 00:44:43,260 and Galois, like many angry young men in Paris then, 710 00:44:43,260 --> 00:44:45,980 is a republican revolutionary. 711 00:44:45,980 --> 00:44:51,300 Unlike the rest of his comrades though, he has another passion - mathematics. 712 00:44:52,860 --> 00:44:55,780 He had just spent four months in jail. 713 00:44:55,780 --> 00:44:59,460 Then, in a mysterious saga of unrequited love, 714 00:44:59,460 --> 00:45:01,580 he is challenged to a duel. 715 00:45:01,580 --> 00:45:03,580 He'd been up the whole previous night 716 00:45:03,580 --> 00:45:06,660 refining a new language for mathematics he'd developed. 717 00:45:06,660 --> 00:45:13,460 Galois believed that mathematics shouldn't be the study of number and shape, but the study of structure. 718 00:45:13,460 --> 00:45:16,540 Perhaps he was still pre-occupied with his maths. 719 00:45:16,540 --> 00:45:18,100 GUNSHOT 720 00:45:18,100 --> 00:45:20,980 There was only one shot fired that morning. 721 00:45:20,980 --> 00:45:26,580 Galois died the next day, just 20 years old. 722 00:45:26,580 --> 00:45:29,620 It was one of mathematics greatest losses. 723 00:45:29,620 --> 00:45:32,380 Only by the beginning of the 20th century 724 00:45:32,380 --> 00:45:36,940 would Galois be fully appreciated and his ideas fully realised. 725 00:45:41,700 --> 00:45:45,820 Galois had discovered new techniques to be able to tell 726 00:45:45,820 --> 00:45:49,220 whether certain equations could have solutions or not. 727 00:45:49,220 --> 00:45:53,300 The symmetry of certain geometric objects seemed to be the key. 728 00:45:53,300 --> 00:45:57,820 His idea of using geometry to analyse equations 729 00:45:57,820 --> 00:46:03,180 would be picked up in the 1920s by another Parisian mathematician, Andre Weil. 730 00:46:03,180 --> 00:46:08,820 I was very much interested and so far as school was concerned 731 00:46:08,820 --> 00:46:13,020 quite successful in all possible branches. 732 00:46:13,020 --> 00:46:16,780 And he was. After studying in Germany as well as France, 733 00:46:16,780 --> 00:46:20,300 Andre settled down at this apartment in Paris 734 00:46:20,300 --> 00:46:25,060 which he shared with his more-famous sister, the writer Simone Weil. 735 00:46:25,060 --> 00:46:30,340 But when the Second World War broke out, he found himself in very different circumstances. 736 00:46:30,340 --> 00:46:36,340 He dodged the draft by fleeing to Finland where he was almost executed for being a Russian spy. 737 00:46:36,340 --> 00:46:42,020 On his return to France he was put in prison in Rouen to await trial for desertion. 738 00:46:42,020 --> 00:46:44,620 At the trial, the judge gave him a choice. 739 00:46:44,620 --> 00:46:48,420 Five more years in prison or serve in a combat unit. 740 00:46:48,420 --> 00:46:51,540 He chose to join the French army, a lucky choice 741 00:46:51,540 --> 00:46:55,420 because just before the Germans invaded a few months later, 742 00:46:55,420 --> 00:46:57,580 all the prisoners were killed. 743 00:46:57,580 --> 00:47:04,700 Weil only spent a few months in prison, but this time was crucial for his mathematics. 744 00:47:04,700 --> 00:47:10,300 Because here he built on the ideas of Galois and first developed algebraic geometry 745 00:47:10,300 --> 00:47:15,020 a whole new language for understanding solutions to equations. 746 00:47:15,020 --> 00:47:18,020 Galois had shown how new mathematical structures 747 00:47:18,020 --> 00:47:21,900 can be used to reveal the secrets behind equations. 748 00:47:21,900 --> 00:47:23,940 Weil's work led him to theorems 749 00:47:23,940 --> 00:47:28,100 that connected number theory, algebra, geometry and topology 750 00:47:28,100 --> 00:47:33,020 and are one of the greatest achievements of modern mathematics. 751 00:47:33,020 --> 00:47:36,060 Without Andre Weil, we would never have heard 752 00:47:36,060 --> 00:47:40,700 of the strangest man in the history of maths, Nicolas Bourbaki. 753 00:47:43,020 --> 00:47:49,700 There are no photos of Bourbaki in existence but we do know he was born in this cafe in the Latin Quarter 754 00:47:49,700 --> 00:47:53,820 in 1934 when it was a proper cafe, the cafe Capoulade, 755 00:47:53,820 --> 00:47:57,300 and not the fast food joint it has now become. 756 00:47:57,300 --> 00:48:02,500 Just down the road, I met up with Bourbaki expert David Aubin. 757 00:48:02,500 --> 00:48:05,700 When I was a graduate student I got quite frightened 758 00:48:05,700 --> 00:48:07,420 when I used to go into the library 759 00:48:07,420 --> 00:48:10,260 because this guy Bourbaki had written so many books. 760 00:48:10,260 --> 00:48:13,700 Something like 30 or 40 altogether. 761 00:48:13,700 --> 00:48:18,980 In analysis, in geometry, in topology, it was all new foundations. 762 00:48:18,980 --> 00:48:22,660 Virtually everyone studying Maths seriously anywhere in the world 763 00:48:22,660 --> 00:48:27,500 in the 1950s, '60s and '70s would have read Nicolas Bourbaki. 764 00:48:27,500 --> 00:48:30,460 He applied for membership of the American Maths Society, I heard. 765 00:48:30,460 --> 00:48:32,660 At which point he was denied membership 766 00:48:32,660 --> 00:48:35,620 on the grounds that he didn't exist. Oh! 767 00:48:35,620 --> 00:48:37,460 The Americans were right. 768 00:48:37,460 --> 00:48:41,180 Nicolas Bourbaki does not exist at all. And never has. 769 00:48:41,180 --> 00:48:45,500 Bourbaki is in fact the nom de plume for a group of French mathematicians 770 00:48:45,500 --> 00:48:49,180 led by Andre Weil who decided to write a coherent account 771 00:48:49,180 --> 00:48:51,780 of the mathematics of the 20th century. 772 00:48:51,780 --> 00:48:56,500 Most of the time mathematicians like to have their own names on theorems. 773 00:48:56,500 --> 00:48:58,900 But for the Bourbaki group, 774 00:48:58,900 --> 00:49:02,740 the aims of the project overrode any desire for personal glory. 775 00:49:02,740 --> 00:49:06,420 After the Second World War, the Bourbaki baton was handed down 776 00:49:06,420 --> 00:49:09,380 to the next generation of French mathematicians. 777 00:49:09,380 --> 00:49:14,700 And their most brilliant member was Alexandre Grothendieck. 778 00:49:14,700 --> 00:49:16,300 Here at the IHES in Paris, 779 00:49:16,300 --> 00:49:20,820 the French equivalent of Princeton's Institute for Advanced Study, 780 00:49:20,820 --> 00:49:26,460 Grothendieck held court at his famous seminars in the 1950s and 1960s. 781 00:49:29,220 --> 00:49:32,900 He had this incredible charisma. 782 00:49:32,900 --> 00:49:39,540 He had this amazing ability to see a young person and somehow know 783 00:49:39,540 --> 00:49:45,580 what kind of contribution this person could make to this incredible vision 784 00:49:45,580 --> 00:49:48,220 he had of how mathematics could be. 785 00:49:48,220 --> 00:49:53,820 And this vision enabled him to get across some very difficult ideas indeed. 786 00:49:53,820 --> 00:49:57,540 He says, "Suppose you want to open a walnut. 787 00:49:57,540 --> 00:50:01,500 "So the standard thing is you take a nutcracker and you just break it open." 788 00:50:01,500 --> 00:50:04,100 And he says his approach is more like, 789 00:50:04,100 --> 00:50:07,420 you take this walnut and you put it out in the snow 790 00:50:07,420 --> 00:50:09,460 and you leave it there for a few months 791 00:50:09,460 --> 00:50:13,060 and then when you come back to it, it just opens. 792 00:50:13,060 --> 00:50:15,060 Grothendieck is a Structuralist. 793 00:50:15,060 --> 00:50:19,020 What he's interested in are the hidden structures 794 00:50:19,020 --> 00:50:21,420 underneath all mathematics. 795 00:50:21,420 --> 00:50:26,860 Only when you get down to the very basic architecture and think in very general terms 796 00:50:26,860 --> 00:50:30,460 will the patterns in mathematics become clear. 797 00:50:30,460 --> 00:50:36,420 Grothendieck produced a new powerful language to see structures in a new way. 798 00:50:36,420 --> 00:50:39,020 It was like living in a world of black and white 799 00:50:39,020 --> 00:50:42,260 and suddenly having the language to see the world in colour. 800 00:50:42,260 --> 00:50:45,940 It's a language that mathematicians have been using ever since 801 00:50:45,940 --> 00:50:50,940 to solve problems in number theory, geometry, even fundamental physics. 802 00:50:52,460 --> 00:50:55,740 But in the late 1960s, Grothendieck decided 803 00:50:55,740 --> 00:51:00,940 to turn his back on mathematics after he discovered politics. 804 00:51:00,940 --> 00:51:05,620 He believed that the threat of nuclear war and the questions 805 00:51:05,620 --> 00:51:11,740 of nuclear disarmament were more important than mathematics 806 00:51:11,740 --> 00:51:16,780 and that people who continue to do mathematics 807 00:51:16,780 --> 00:51:20,540 rather than confront this threat of nuclear war 808 00:51:20,540 --> 00:51:22,220 were doing harm in the world. 809 00:51:25,740 --> 00:51:28,340 Grothendieck decided to leave Paris 810 00:51:28,340 --> 00:51:31,340 and move back to the south of France where he grew up. 811 00:51:31,340 --> 00:51:35,980 Bursts of radical politics followed and then a nervous breakdown. 812 00:51:35,980 --> 00:51:40,020 He moved to the Pyrenees and became a recluse. 813 00:51:40,020 --> 00:51:44,900 He's now lost all contact with his old friends and mathematical colleagues. 814 00:51:45,900 --> 00:51:50,340 Nevertheless, the legacy of his achievements means that Grothendieck stands 815 00:51:50,340 --> 00:51:56,740 alongside Cantor, Godel and Hilbert as someone who has transformed the mathematical landscape. 816 00:51:58,500 --> 00:52:03,100 He changed the whole subject in a really fundamental way. It will never go back. 817 00:52:03,100 --> 00:52:08,100 Certainly, he's THE dominant figure of the 20th century. 818 00:52:15,500 --> 00:52:17,580 I've come back to England, though, 819 00:52:17,580 --> 00:52:21,740 thinking again about another seminal figure of the 20th century. 820 00:52:21,740 --> 00:52:25,940 The person that started it all off, David Hilbert. 821 00:52:25,940 --> 00:52:31,700 Of the 23 problems Hilbert set mathematicians in the year 1900, 822 00:52:31,700 --> 00:52:34,180 most have now been solved. 823 00:52:34,180 --> 00:52:36,460 However there is one great exception. 824 00:52:36,460 --> 00:52:39,660 The Riemann hypothesis, the eighth on Hilbert's list. 825 00:52:39,660 --> 00:52:42,460 That is still the holy grail of mathematics. 826 00:52:44,260 --> 00:52:49,500 Hilbert's lecture inspired a generation to pursue their mathematical dreams. 827 00:52:49,500 --> 00:52:54,420 This morning, in the town where I grew up, I hope to inspire another generation. 828 00:52:54,420 --> 00:52:56,580 CHEERING AND APPLAUSE 829 00:53:00,980 --> 00:53:03,420 Thank you. Hello. My name's Marcus du Sautoy 830 00:53:03,420 --> 00:53:05,260 and I'm a Professor of Mathematics 831 00:53:05,260 --> 00:53:07,420 up the road at the University of Oxford. 832 00:53:07,420 --> 00:53:09,620 It was actually in this school here, 833 00:53:09,620 --> 00:53:13,820 in fact this classroom is where I discovered my love for mathematics. 834 00:53:13,820 --> 00:53:16,420 'This love of mathematics that I first acquired 835 00:53:16,420 --> 00:53:19,700 'here in my old comprehensive school still drives me now. 836 00:53:19,700 --> 00:53:21,580 'It's a love of solving problems. 837 00:53:21,580 --> 00:53:24,980 'There are so many problems I could tell them about, 838 00:53:24,980 --> 00:53:27,020 'but I've chosen my favourite.' 839 00:53:27,020 --> 00:53:30,140 I think that a mathematician is a pattern searcher 840 00:53:30,140 --> 00:53:33,260 and that's really what mathematicians try and do. 841 00:53:33,260 --> 00:53:36,380 We try and understand the patterns and the structure 842 00:53:36,380 --> 00:53:39,740 and the logic to explain the way the world around us works. 843 00:53:39,740 --> 00:53:42,780 And this is really at the heart of the Riemann hypothesis. 844 00:53:42,780 --> 00:53:47,660 The task is - is there any pattern in these numbers which can help me say 845 00:53:47,660 --> 00:53:49,740 where the next number will be? 846 00:53:49,740 --> 00:53:52,060 What's the next one after 31? How can I tell? 847 00:53:52,060 --> 00:53:55,060 'These numbers are, of course, prime numbers - 848 00:53:55,060 --> 00:53:57,500 'the building blocks of mathematics.' 849 00:53:57,500 --> 00:54:00,820 'The Riemann hypothesis, a conjecture about the distribution 850 00:54:00,820 --> 00:54:04,020 'of the primes, goes to the very heart of our subject.' 851 00:54:04,020 --> 00:54:06,860 Why on earth is anybody interested in these primes? 852 00:54:06,860 --> 00:54:10,340 Why is the army interested in primes, why are spies interested? 853 00:54:10,340 --> 00:54:14,100 Isn't it to encrypt stuff? Exactly. 854 00:54:14,100 --> 00:54:17,580 I study this stuff cos I think it's all really beautiful and elegant 855 00:54:17,580 --> 00:54:19,500 but actually, there's a lot of people 856 00:54:19,500 --> 00:54:23,740 who are interested in these numbers because of their very practical use. 857 00:54:23,740 --> 00:54:28,020 'The bizarre thing is that the more abstract and difficult mathematics becomes, 858 00:54:28,020 --> 00:54:31,780 'the more it seems to have applications in the real world. 859 00:54:31,780 --> 00:54:35,860 'Mathematics now pervades every aspect of our lives. 860 00:54:35,860 --> 00:54:40,860 'Every time we switch on the television, plug in a computer, pay with a credit card. 861 00:54:40,860 --> 00:54:45,460 'There's now a million dollars for anyone who can solve the Riemann hypothesis. 862 00:54:45,460 --> 00:54:47,900 'But there's more at stake than that.' 863 00:54:47,900 --> 00:54:51,100 Anybody who proves this theorem will be remembered forever. 864 00:54:51,100 --> 00:54:54,940 They'll be on that board ahead of any of those other mathematicians. 865 00:54:54,940 --> 00:54:58,900 'That's because the Riemann hypothesis is a corner-stone of maths. 866 00:54:58,900 --> 00:55:02,100 'Thousands of theorems depend on it being true. 867 00:55:02,100 --> 00:55:05,300 'Very few mathematicians think that it isn't true. 868 00:55:05,300 --> 00:55:09,940 'But mathematics is about proof and until we can prove it 869 00:55:09,940 --> 00:55:12,140 'there will still be doubt.' 870 00:55:12,140 --> 00:55:16,460 Maths has grown out of this passion to get rid of doubt. 871 00:55:16,460 --> 00:55:20,060 This is what I have learned in my journey through the history of mathematics. 872 00:55:20,060 --> 00:55:24,380 Mathematicians like Archimedes and al-Khwarizmi, Gauss and Grothendieck 873 00:55:24,380 --> 00:55:29,820 were driven to understand the precise way numbers and space work. 874 00:55:29,820 --> 00:55:32,500 Maths in action, that one. 875 00:55:32,500 --> 00:55:34,740 It's beautiful. Really nice. 876 00:55:34,740 --> 00:55:38,500 Using the language of mathematics, they have told us stories 877 00:55:38,500 --> 00:55:43,060 that remain as true today as they were when they were first told. 878 00:55:43,060 --> 00:55:48,060 In the Mediterranean, I discovered the origins of geometry. 879 00:55:48,060 --> 00:55:51,140 Mathematicians and philosophers flocked to Alexandria 880 00:55:51,140 --> 00:55:54,540 driven by a thirst for knowledge and the pursuit of excellence. 881 00:55:54,540 --> 00:55:58,380 In India, I learned about another discovery 882 00:55:58,380 --> 00:56:02,180 that it would be impossible to imagine modern life without. 883 00:56:02,180 --> 00:56:06,540 So here we are in one of the true holy sites of the mathematical world. 884 00:56:06,540 --> 00:56:09,380 Up here are some numbers, 885 00:56:09,380 --> 00:56:11,980 and here's the new number. 886 00:56:11,980 --> 00:56:13,620 Its zero. 887 00:56:13,620 --> 00:56:18,900 In the Middle East, I was amazed at al-Khwarizmi's invention of algebra. 888 00:56:18,900 --> 00:56:21,780 He developed systematic ways to analyse problems 889 00:56:21,780 --> 00:56:25,460 so that the solutions would work whatever numbers you took. 890 00:56:25,460 --> 00:56:27,380 In the Golden Age of Mathematics, 891 00:56:27,380 --> 00:56:30,900 in Europe in the 18th and 19th centuries, I found how maths 892 00:56:30,900 --> 00:56:35,060 discovered new ways for analysing bodies in motion and new geometries 893 00:56:35,060 --> 00:56:39,820 that helped us understand the very strange shape of space. 894 00:56:39,820 --> 00:56:43,140 It is with Riemann's work that we finally have 895 00:56:43,140 --> 00:56:48,580 the mathematical glasses to be able to explore such worlds of the mind. 896 00:56:48,580 --> 00:56:52,780 And now my journey into the abstract world of 20th-century mathematics 897 00:56:52,780 --> 00:56:55,900 has revealed that maths is the true language 898 00:56:55,900 --> 00:56:58,100 the universe is written in, 899 00:56:58,100 --> 00:57:01,420 the key to understanding the world around us. 900 00:57:01,420 --> 00:57:05,140 Mathematicians aren't motivated by money and material gain 901 00:57:05,140 --> 00:57:08,460 or even by practical applications of their work. 902 00:57:08,460 --> 00:57:12,700 For us, it is the glory of solving one of the great unsolved problems 903 00:57:12,700 --> 00:57:17,860 that have outwitted previous generations of mathematicians. 904 00:57:17,860 --> 00:57:21,220 Hilbert was right. It's the unsolved problems of mathematics 905 00:57:21,220 --> 00:57:23,020 that make it a living subject, 906 00:57:23,020 --> 00:57:26,460 which obsess each new generation of mathematicians. 907 00:57:26,460 --> 00:57:30,260 Despite all the things we've discovered over the last seven millennia, 908 00:57:30,260 --> 00:57:32,900 there are still many things we don't understand. 909 00:57:32,900 --> 00:57:39,260 And its Hilbert's call of, "We must know, we will know", which drives mathematics. 910 00:57:41,540 --> 00:57:44,740 You can learn more about The Story Of Maths 911 00:57:44,740 --> 00:57:47,780 with the Open University at... 912 00:57:59,900 --> 00:58:02,940 Subtitled by Red Bee Media Ltd 913 00:58:02,940 --> 00:58:05,980 E-mail subtitling@bbc.co.uk