1 00:00:17,280 --> 00:00:21,400 Mathematics is about solving problems 2 00:00:21,400 --> 00:00:26,240 and it's the great unsolved problems that make maths really alive. 3 00:00:28,240 --> 00:00:29,440 In the summer of 1900, 4 00:00:29,440 --> 00:00:32,160 the International Congress of Mathematicians 5 00:00:32,160 --> 00:00:34,440 was held here in Paris in the Sorbonne. 6 00:00:34,440 --> 00:00:36,680 It was a pretty shambolic affair, 7 00:00:36,680 --> 00:00:39,280 not helped by the sultry August heat. 8 00:00:39,280 --> 00:00:43,000 But it will be remembered as one of the greatest congresses of all time 9 00:00:43,000 --> 00:00:47,360 thanks to a lecture given by the up-and-coming David Hilbert. 10 00:00:48,880 --> 00:00:51,600 Hilbert, a young German mathematician, 11 00:00:51,600 --> 00:00:56,360 boldly set out what he believed were the 23 most important problems 12 00:00:56,360 --> 00:00:58,560 for mathematicians to crack. 13 00:00:58,560 --> 00:01:04,200 He was trying to set the agenda for 20th-century maths and he succeeded. 14 00:01:04,200 --> 00:01:09,480 These Hilbert problems would define the mathematics of the modern age. 15 00:01:09,480 --> 00:01:15,360 Of those who tried to crack Hilbert's challenges, some would experience immense triumphs, 16 00:01:15,360 --> 00:01:18,920 whilst others would be plunged into infinite despair. 17 00:01:30,120 --> 00:01:33,120 The first problem on Hilbert's list emerged from here, 18 00:01:33,120 --> 00:01:36,320 Halle, in East Germany. 19 00:01:36,320 --> 00:01:41,200 It was where the great mathematician Georg Cantor spent all his adult life. 20 00:01:41,200 --> 00:01:45,280 And where he became the first person to really understand the meaning 21 00:01:45,280 --> 00:01:50,160 of infinity and give it mathematical precision. 22 00:01:50,160 --> 00:01:52,480 The statue in the town square, however, 23 00:01:52,480 --> 00:01:57,360 honours Halle's other famous son, the composer George Handel. 24 00:01:57,360 --> 00:02:03,600 To discover more about Cantor, I had to take a tram way out of town. 25 00:02:03,600 --> 00:02:07,000 For 50 years, Halle was part of Communist East Germany 26 00:02:07,000 --> 00:02:10,320 and the Communists loved celebrating their scientists. 27 00:02:10,320 --> 00:02:15,000 So much so, they put Cantor on the side of a large cube that they commissioned. 28 00:02:15,000 --> 00:02:17,640 But, being communists, they didn't put the cube 29 00:02:17,640 --> 00:02:20,720 in the middle of town. They put it out amongst the people. 30 00:02:24,320 --> 00:02:27,840 When I eventually found the estate, I started to fear 31 00:02:27,840 --> 00:02:31,200 that maybe I had got the location wrong. 32 00:02:34,480 --> 00:02:38,680 This looks the most unlikely venue for a statue to a mathematician. 33 00:02:39,960 --> 00:02:41,840 Excuse me? 34 00:02:42,840 --> 00:02:43,960 Ein Frage. 35 00:02:43,960 --> 00:02:47,560 - Can you help me a minute? - Wie bitte? - Do you speak English? - No! - No? 36 00:02:47,560 --> 00:02:49,600 Ich suche ein Wurfel. 37 00:02:49,600 --> 00:02:51,600 Ein Wurfel, ja? 38 00:02:51,600 --> 00:02:52,880 Is that right? A "Wurfel"? 39 00:02:52,880 --> 00:02:55,160 A cube? Yeah? Like that? 40 00:02:55,160 --> 00:02:58,680 Mit ein Bild der Mathematiker? 41 00:02:58,680 --> 00:03:01,160 Yeah? Go round there? 42 00:03:01,160 --> 00:03:02,440 Die Name ist Cantor. 43 00:03:02,440 --> 00:03:04,560 Somewhere over here. Ah! There it is! 44 00:03:04,560 --> 00:03:06,080 It's much bigger than I thought. 45 00:03:06,080 --> 00:03:09,720 I thought it was going to be something like this sort of size. 46 00:03:09,720 --> 00:03:13,760 Aha, here we are. On the dark side of the cube. 47 00:03:13,760 --> 00:03:16,120 here's the man himself, Cantor. 48 00:03:16,120 --> 00:03:18,880 Cantor's one of my big heroes actually. 49 00:03:18,880 --> 00:03:23,040 I think if I had to choose some theorems that I wish I'd proved, 50 00:03:23,040 --> 00:03:25,040 I think the couple that Cantor proved 51 00:03:25,040 --> 00:03:27,920 would be up there in my top ten. 52 00:03:27,920 --> 00:03:30,200 'This is because before Cantor, 53 00:03:30,200 --> 00:03:33,200 'no-one had really understood infinity.' 54 00:03:33,200 --> 00:03:38,080 It was a tricky, slippery concept that didn't seem to go anywhere. 55 00:03:38,080 --> 00:03:42,680 But Cantor showed that infinity could be perfectly understandable. 56 00:03:42,680 --> 00:03:45,640 Indeed, there wasn't just one infinity, 57 00:03:45,640 --> 00:03:48,120 but infinitely many infinities. 58 00:03:48,120 --> 00:03:54,400 First Cantor took the numbers 1, 2, 3, 4 and so on. 59 00:03:54,400 --> 00:03:58,000 Then he thought about comparing them with a much smaller set... 60 00:03:58,000 --> 00:04:02,720 something like 10, 20, 30, 40... 61 00:04:02,720 --> 00:04:06,320 What he showed is that these two infinite sets of numbers 62 00:04:06,320 --> 00:04:10,640 actually have the same size because we can pair them up - 63 00:04:10,640 --> 00:04:14,520 1 with 10, 2 with 20, 3 with 30 and so on. 64 00:04:14,520 --> 00:04:17,880 So these are the same sizes of infinity. 65 00:04:20,640 --> 00:04:22,680 But what about the fractions? 66 00:04:22,680 --> 00:04:27,520 After all, there are infinitely many fractions between any two whole numbers. 67 00:04:27,520 --> 00:04:30,760 Surely the infinity of fractions is much bigger 68 00:04:30,760 --> 00:04:33,360 than the infinity of whole numbers. 69 00:04:38,360 --> 00:04:41,600 Well, what Cantor did was to find a way to pair up 70 00:04:41,600 --> 00:04:45,400 all of the whole numbers with an infinite load of fractions. 71 00:04:45,400 --> 00:04:47,280 And this is how he did it. 72 00:04:47,280 --> 00:04:52,520 He started by arranging all the fractions in an infinite grid. 73 00:04:52,520 --> 00:04:57,160 The first row contained the whole numbers, fractions with one on the bottom. 74 00:04:57,160 --> 00:05:01,720 In the second row came the halves, fractions with two on the bottom. 75 00:05:01,720 --> 00:05:06,280 And so on. Every fraction appears somewhere in this grid. 76 00:05:06,280 --> 00:05:10,320 Where's two thirds? Second column, third row. 77 00:05:10,320 --> 00:05:15,560 Now imagine a line snaking back and forward diagonally through the fractions. 78 00:05:18,080 --> 00:05:24,920 By pulling this line straight, we can match up every fraction with one of the whole numbers. 79 00:05:24,920 --> 00:05:29,280 This means the fractions are the same sort of infinity 80 00:05:29,280 --> 00:05:31,080 as the whole numbers. 81 00:05:31,080 --> 00:05:34,120 So perhaps all infinities have the same size. 82 00:05:34,120 --> 00:05:36,680 Well, here comes the really exciting bit 83 00:05:36,680 --> 00:05:41,080 because Cantor now considers the set of all infinite decimal numbers. 84 00:05:41,080 --> 00:05:45,320 And here he proves that they give us a bigger infinity because 85 00:05:45,320 --> 00:05:49,320 however you tried to list all the infinite decimals, Cantor produced 86 00:05:49,320 --> 00:05:52,480 a clever argument to show how to construct a new decimal number 87 00:05:52,480 --> 00:05:54,200 that was missing from your list. 88 00:05:54,200 --> 00:05:58,240 Suddenly, the idea of infinity opens up. 89 00:05:58,240 --> 00:06:01,840 There are different infinities, some bigger than others. 90 00:06:01,840 --> 00:06:03,600 It's a really exciting moment. 91 00:06:03,600 --> 00:06:07,880 For me, this is like the first humans understanding how to count. 92 00:06:07,880 --> 00:06:12,120 But now we're counting in a different way. We are counting infinities. 93 00:06:12,120 --> 00:06:18,080 A door has opened and an entirely new mathematics lay before us. 94 00:06:19,320 --> 00:06:21,480 But it never helped Cantor much. 95 00:06:21,480 --> 00:06:25,240 I was in the cemetery in Halle where he is buried 96 00:06:25,240 --> 00:06:28,280 and where I had arranged to meet Professor Joe Dauben. 97 00:06:28,280 --> 00:06:32,720 He was keen to make the connections between Cantor's maths and his life. 98 00:06:33,720 --> 00:06:36,280 He suffered from manic depression. 99 00:06:36,280 --> 00:06:39,680 One of the first big breakdowns he has is in 1884 100 00:06:39,680 --> 00:06:42,160 but then around the turn of the century 101 00:06:42,160 --> 00:06:44,720 these recurrences of the mental illness 102 00:06:44,720 --> 00:06:46,760 become more and more frequent. 103 00:06:46,760 --> 00:06:49,720 A lot of people have tried to say that his mental illness 104 00:06:49,720 --> 00:06:53,120 was triggered by the incredible abstract mathematics he dealt with. 105 00:06:53,120 --> 00:06:57,280 Well, he was certainly struggling, so there may have been a connection. 106 00:06:57,280 --> 00:07:01,920 Yeah, I mean I must say, when you start to contemplate the infinite... 107 00:07:01,920 --> 00:07:05,080 I am pretty happy with the bottom end of the infinite, 108 00:07:05,080 --> 00:07:07,240 but as you build it up more and more, 109 00:07:07,240 --> 00:07:09,920 I must say I start to feel a bit unnerved 110 00:07:09,920 --> 00:07:13,280 about what's going on here and where is it going. 111 00:07:13,280 --> 00:07:17,880 For much of Cantor's life, the only place it was going was here - 112 00:07:17,880 --> 00:07:20,280 the university's sanatorium. 113 00:07:20,280 --> 00:07:24,040 There was no treatment then for manic depression 114 00:07:24,040 --> 00:07:27,920 or indeed for the paranoia that often accompanied Cantor's attacks. 115 00:07:27,920 --> 00:07:30,800 Yet the clinic was a good place to be - 116 00:07:30,800 --> 00:07:33,560 comfortable, quiet and peaceful. 117 00:07:33,560 --> 00:07:37,800 And Cantor often found his time here gave him the mental strength 118 00:07:37,800 --> 00:07:41,120 to resume his exploration of the infinite. 119 00:07:41,120 --> 00:07:44,480 Other mathematicians would be bothered by the paradoxes 120 00:07:44,480 --> 00:07:46,400 that Cantor's work had created. 121 00:07:46,400 --> 00:07:50,480 Curiously, this was one thing Cantor was not worried by. 122 00:07:50,480 --> 00:07:53,800 He was never as upset about the paradox of the infinite 123 00:07:53,800 --> 00:07:56,960 as everybody else was because Cantor believed that 124 00:07:56,960 --> 00:08:00,240 there are certain things that I have been able to show, 125 00:08:00,240 --> 00:08:03,640 we can establish with complete mathematical certainty 126 00:08:03,640 --> 00:08:08,040 and then the absolute infinite which is only in God. 127 00:08:08,040 --> 00:08:12,360 He can understand all of this and there's still that final paradox 128 00:08:12,360 --> 00:08:15,400 that is not given to us to understand, but God does. 129 00:08:18,000 --> 00:08:22,280 But there was one problem that Cantor couldn't leave 130 00:08:22,280 --> 00:08:23,720 in the hands of the Almighty, 131 00:08:23,720 --> 00:08:26,320 a problem he wrestled with for the rest of his life. 132 00:08:26,320 --> 00:08:29,920 It became known as the continuum hypothesis. 133 00:08:29,920 --> 00:08:33,200 Is there an infinity sitting between the smaller infinity 134 00:08:33,200 --> 00:08:37,760 of all the whole numbers and the larger infinity of the decimals? 135 00:08:40,640 --> 00:08:45,080 Cantor's work didn't go down well with many of his contemporaries 136 00:08:45,080 --> 00:08:48,760 but there was one mathematician from France who spoke up for him, 137 00:08:48,760 --> 00:08:51,680 arguing that Cantor's new mathematics of infinity 138 00:08:51,680 --> 00:08:54,840 was "beautiful, if pathological". 139 00:08:54,840 --> 00:09:00,480 Fortunately this mathematician was the most famous and respected mathematician of his day. 140 00:09:00,480 --> 00:09:04,160 When Bertrand Russell was asked by a French politician who he thought 141 00:09:04,160 --> 00:09:08,720 the greatest man France had produced, he replied without hesitation, "Poincare". 142 00:09:08,720 --> 00:09:10,840 The politician was surprised that he'd chosen 143 00:09:10,840 --> 00:09:14,680 the prime minister Raymond Poincare above the likes of Napoleon, Balzac. 144 00:09:14,680 --> 00:09:19,040 Russell replied, "I don't mean Raymond Poincare but his cousin, 145 00:09:19,040 --> 00:09:21,720 "the mathematician, Henri Poincare." 146 00:09:25,480 --> 00:09:28,720 Henri Poincare spent most of his life in Paris, 147 00:09:28,720 --> 00:09:32,520 a city that he loved even with its uncertain climate. 148 00:09:32,520 --> 00:09:36,400 In the last decades of the 19th century, Paris was a centre 149 00:09:36,400 --> 00:09:40,720 for world mathematics and Poincare became its leading light. 150 00:09:40,720 --> 00:09:44,680 Algebra, geometry, analysis, he was good at everything. 151 00:09:44,680 --> 00:09:47,800 His work would lead to all kinds of applications, 152 00:09:47,800 --> 00:09:50,600 from finding your way around on the underground, 153 00:09:50,600 --> 00:09:54,280 to new ways of predicting the weather. 154 00:09:54,280 --> 00:09:57,240 Poincare was very strict about his working day. 155 00:09:57,240 --> 00:09:59,040 Two hours of work in the morning 156 00:09:59,040 --> 00:10:01,080 and two hours in the early evening. 157 00:10:01,080 --> 00:10:02,480 Between these periods, 158 00:10:02,480 --> 00:10:06,160 he would let his subconscious carry on working on the problem. 159 00:10:06,160 --> 00:10:10,240 He records one moment when he had a flash of inspiration which occurred 160 00:10:10,240 --> 00:10:14,760 almost out of nowhere, just as he was getting on a bus. 161 00:10:16,720 --> 00:10:21,480 And one such flash of inspiration led to an early success. 162 00:10:21,480 --> 00:10:25,120 In 1885, King Oscar II of Sweden and Norway 163 00:10:25,120 --> 00:10:32,320 offered a prize of 2,500 crowns for anyone who could establish mathematically once and for all 164 00:10:32,320 --> 00:10:36,680 whether the solar system would continue turning like clockwork, 165 00:10:36,680 --> 00:10:38,640 or might suddenly fly apart. 166 00:10:38,640 --> 00:10:44,720 If the solar system has two planets then Newton had already proved that their orbits would be stable. 167 00:10:44,720 --> 00:10:48,720 The two bodies just travel in ellipsis round each other. 168 00:10:48,720 --> 00:10:53,600 But as soon as soon as you add three bodies like the earth, moon and sun, 169 00:10:53,600 --> 00:10:58,880 the question of whether their orbits were stable or not stumped even the great Newton. 170 00:10:58,880 --> 00:11:03,040 The problem is that now you have some 18 different variables, 171 00:11:03,040 --> 00:11:05,280 like the exact coordinates of each body 172 00:11:05,280 --> 00:11:07,440 and their velocity in each direction. 173 00:11:07,440 --> 00:11:10,720 So the equations become very difficult to solve. 174 00:11:10,720 --> 00:11:15,680 But Poincare made significant headway in sorting them out. 175 00:11:15,680 --> 00:11:21,720 Poincare simplified the problem by making successive approximations to the orbits which he believed 176 00:11:21,720 --> 00:11:24,880 wouldn't affect the final outcome significantly. 177 00:11:24,880 --> 00:11:28,120 Although he couldn't solve the problem in its entirety, 178 00:11:28,120 --> 00:11:33,240 his ideas were sophisticated enough to win him the prize. 179 00:11:33,240 --> 00:11:36,640 He developed this great sort of arsenal of techniques, 180 00:11:36,640 --> 00:11:38,320 mathematical techniques 181 00:11:38,320 --> 00:11:40,880 in order to try and solve it 182 00:11:40,880 --> 00:11:44,040 and in fact, the prize that he won was essentially 183 00:11:44,040 --> 00:11:47,520 more for the techniques than for solving the problem. 184 00:11:47,520 --> 00:11:51,280 But when Poincare's paper was being prepared for publication 185 00:11:51,280 --> 00:11:54,320 by the King's scientific advisor, Mittag-Leffler, 186 00:11:54,320 --> 00:11:56,360 one of the editors found a problem. 187 00:11:58,920 --> 00:12:02,440 Poincare realised he'd made a mistake. 188 00:12:02,440 --> 00:12:06,560 Contrary to what he had originally thought, even a small change in the 189 00:12:06,560 --> 00:12:10,720 initial conditions could end up producing vastly different orbits. 190 00:12:10,720 --> 00:12:13,440 His simplification just didn't work. 191 00:12:13,440 --> 00:12:17,040 But the result was even more important. 192 00:12:17,040 --> 00:12:24,080 The orbits Poincare had discovered indirectly led to what we now know as chaos theory. 193 00:12:24,080 --> 00:12:29,120 Understanding the mathematical rules of chaos explain why a butterfly's wings 194 00:12:29,120 --> 00:12:31,600 could create tiny changes in the atmosphere 195 00:12:31,600 --> 00:12:33,320 that ultimately might cause 196 00:12:33,320 --> 00:12:37,520 a tornado or a hurricane to appear on the other side of the world. 197 00:12:37,520 --> 00:12:40,400 So this big subject of the 20th century, chaos, 198 00:12:40,400 --> 00:12:43,600 actually came out of a mistake that Poincare made 199 00:12:43,600 --> 00:12:45,400 and he spotted at the last minute. 200 00:12:45,400 --> 00:12:49,040 Yes! So the essay had actually been published in its original form, 201 00:12:49,040 --> 00:12:54,240 and was ready to go out and Mittag-Leffler had sent copies out to various people, 202 00:12:54,240 --> 00:12:59,400 and it was to his horror when Poincare wrote to him to say, "Stop!" 203 00:12:59,400 --> 00:13:03,120 Oh, my God. This is every mathematician's worst nightmare. 204 00:13:03,120 --> 00:13:04,640 Absolutely. "Pull it!" 205 00:13:04,640 --> 00:13:06,160 Hold the presses! 206 00:13:07,360 --> 00:13:10,280 Owning up to his mistake, if anything, 207 00:13:10,280 --> 00:13:12,920 enhanced Poincare's reputation. 208 00:13:12,920 --> 00:13:15,800 He continued to produce a wide range of original work 209 00:13:15,800 --> 00:13:16,960 throughout his life. 210 00:13:16,960 --> 00:13:20,000 Not just specialist stuff either. 211 00:13:20,000 --> 00:13:24,480 He also wrote popular books, extolling the importance of maths. 212 00:13:24,480 --> 00:13:28,560 Here we go. Here's a section on the future of mathematics. 213 00:13:30,080 --> 00:13:34,240 It starts, "If we wish to foresee the future of mathematics, 214 00:13:34,240 --> 00:13:39,240 "our proper course is to study the history and present the condition of the science." 215 00:13:39,240 --> 00:13:45,000 So, I think Poincare might have approved of my journey to uncover the story of maths. 216 00:13:45,000 --> 00:13:48,120 He certainly would have approved of the next destination. 217 00:13:48,120 --> 00:13:53,400 Because to discover perhaps Poincare's most important contribution to modern mathematics, 218 00:13:53,400 --> 00:13:56,320 I had to go looking for a bridge. 219 00:13:59,800 --> 00:14:01,480 Seven bridges in fact. 220 00:14:01,480 --> 00:14:04,160 The Seven bridges of Konigsberg. 221 00:14:04,160 --> 00:14:09,280 Today the city is known as Kaliningrad, a little outpost 222 00:14:09,280 --> 00:14:14,240 of Russia on the Baltic Sea surrounded by Poland and Lithuania. 223 00:14:14,240 --> 00:14:18,000 Until 1945, however, when it was ceded to the Soviet Union, 224 00:14:18,000 --> 00:14:21,120 it was the great Prussian City of Konigsberg. 225 00:14:22,640 --> 00:14:25,920 Much of the old town sadly has been demolished. 226 00:14:25,920 --> 00:14:29,800 There is now no sign at all of two of the original seven bridges 227 00:14:29,800 --> 00:14:34,400 and several have changed out of all recognition. 228 00:14:34,400 --> 00:14:38,000 This is one of the original bridges. 229 00:14:38,000 --> 00:14:44,640 It may seem like an unlikely setting for the beginning of a mathematical story, but bear with me. 230 00:14:44,640 --> 00:14:47,920 It started as an 18th-century puzzle. 231 00:14:47,920 --> 00:14:53,160 Is there a route around the city which crosses each of these seven bridges only once? 232 00:14:53,160 --> 00:14:57,240 Finding the solution is much more difficult than it looks. 233 00:15:07,200 --> 00:15:11,040 It was eventually solved by the great mathematician Leonhard Euler, 234 00:15:11,040 --> 00:15:15,400 who in 1735 proved that it wasn't possible. 235 00:15:15,400 --> 00:15:19,680 There could not be a route that didn't cross at least one bridge twice. 236 00:15:19,680 --> 00:15:23,200 He solved the problem by making a conceptual leap. 237 00:15:23,200 --> 00:15:27,440 He realised, you don't really care what the distances are between the bridges. 238 00:15:27,440 --> 00:15:31,480 What really matters is how the bridges are connected together. 239 00:15:31,480 --> 00:15:37,920 This is a problem of a new sort of geometry of position - a problem of topology. 240 00:15:37,920 --> 00:15:40,960 Many of us use topology every day. 241 00:15:40,960 --> 00:15:43,440 Virtually all metro maps the world over 242 00:15:43,440 --> 00:15:46,040 are drawn on topological principles. 243 00:15:46,040 --> 00:15:49,400 You don't care how far the stations are from each other 244 00:15:49,400 --> 00:15:51,200 but how they are connected. 245 00:15:51,200 --> 00:15:53,840 There isn't a metro in Kaliningrad, 246 00:15:53,840 --> 00:15:58,560 but there is in the nearest other Russian city, St Petersburg. 247 00:15:58,560 --> 00:16:00,720 The topology is pretty easy on this map. 248 00:16:00,720 --> 00:16:03,200 It's the Russian I am having difficulty with. 249 00:16:03,200 --> 00:16:06,360 - Can you tell me which...? - What's the problem? 250 00:16:06,360 --> 00:16:09,760 I want to know what station this one was. 251 00:16:09,760 --> 00:16:12,840 I had it the wrong way round even! 252 00:16:14,640 --> 00:16:18,280 Although topology had its origins in the bridges of Konigsberg, 253 00:16:18,280 --> 00:16:22,520 it was in the hands of Poincare that the subject evolved 254 00:16:22,520 --> 00:16:26,120 into a powerful new way of looking at shape. 255 00:16:26,120 --> 00:16:29,840 Some people refer to topology as bendy geometry 256 00:16:29,840 --> 00:16:34,640 because in topology, two shapes are the same if you can bend or morph 257 00:16:34,640 --> 00:16:37,240 one into another without cutting it. 258 00:16:37,240 --> 00:16:42,360 So for example if I take a football or rugby ball, topologically they 259 00:16:42,360 --> 00:16:46,480 are the same because one can be morphed into the other. 260 00:16:46,480 --> 00:16:51,960 Similarly a bagel and a tea-cup are the same because one can be morphed into the other. 261 00:16:51,960 --> 00:16:58,720 Even very complicated shapes can be unwrapped to become much simpler from a topological point of view. 262 00:16:58,720 --> 00:17:02,760 But there is no way to continuously deform a bagel to morph it into a ball. 263 00:17:02,760 --> 00:17:06,560 The hole in the middle makes these shapes topologically different. 264 00:17:06,560 --> 00:17:11,800 Poincare knew all the possible two-dimensional topological surfaces. 265 00:17:11,800 --> 00:17:15,560 But in 1904 he came up with a topological problem 266 00:17:15,560 --> 00:17:17,480 he just couldn't solve. 267 00:17:17,480 --> 00:17:21,320 If you've got a flat two-dimensional universe then Poincare worked out 268 00:17:21,320 --> 00:17:24,520 all the possible shapes he could wrap that universe up into. 269 00:17:24,520 --> 00:17:29,600 It could be a ball or a bagel with one hole, two holes or more holes in. 270 00:17:29,600 --> 00:17:35,200 But we live in a three-dimensional universe so what are the possible shapes that our universe can be? 271 00:17:35,200 --> 00:17:39,240 That question became known as the Poincare Conjecture. 272 00:17:39,240 --> 00:17:43,960 It was finally solved in 2002 here in St Petersburg 273 00:17:43,960 --> 00:17:47,560 by a Russian mathematician called Grisha Perelman. 274 00:17:47,560 --> 00:17:51,240 His proof is very difficult to understand, even for mathematicians. 275 00:17:51,240 --> 00:17:57,960 Perelman solved the problem by linking it to a completely different area of mathematics. 276 00:17:57,960 --> 00:18:03,800 To understand the shapes, he looked instead at the dynamics of the way things can flow over the shape 277 00:18:03,800 --> 00:18:06,880 which led to a description of all the possible ways 278 00:18:06,880 --> 00:18:11,320 that three dimensional space can be wrapped up in higher dimensions. 279 00:18:11,320 --> 00:18:16,000 I wondered if the man himself could help in unravelling the intricacies of his proof, 280 00:18:16,000 --> 00:18:23,400 but I'd been told that finding Perelman is almost as difficult as understanding the solution. 281 00:18:23,400 --> 00:18:26,040 The classic stereotype of a mathematician 282 00:18:26,040 --> 00:18:29,800 is a mad eccentric scientist, but I think that's a little bit unfair. 283 00:18:29,800 --> 00:18:33,040 Most of my colleagues are normal. Well, reasonably. 284 00:18:33,040 --> 00:18:35,120 But when it comes to Perelman, 285 00:18:35,120 --> 00:18:37,920 there is no doubt he is a very strange character. 286 00:18:37,920 --> 00:18:40,880 He's received prizes and offers of professorships 287 00:18:40,880 --> 00:18:43,560 from distinguished universities in the West 288 00:18:43,560 --> 00:18:46,280 but he's turned them all down. 289 00:18:46,280 --> 00:18:49,800 Recently he seems to have given up mathematics completely 290 00:18:49,800 --> 00:18:52,000 and retreated to live as a semi-recluse 291 00:18:52,000 --> 00:18:54,720 in this very modest housing estate with his mum. 292 00:18:54,720 --> 00:19:01,320 He refuses to talk to the media but I thought he might just talk to me as a fellow mathematician. 293 00:19:01,320 --> 00:19:03,480 I was wrong. 294 00:19:03,480 --> 00:19:07,320 Well, it's interesting. I think he's actually turned off his buzzer. 295 00:19:07,320 --> 00:19:09,600 Probably too many media have been buzzing it. 296 00:19:09,600 --> 00:19:12,920 I tried a neighbour and that rang but his doesn't ring at all. 297 00:19:12,920 --> 00:19:18,560 I think his papers, his mathematics speaks for itself in a way. 298 00:19:18,560 --> 00:19:21,080 I don't really need to meet the mathematician 299 00:19:21,080 --> 00:19:23,560 and in this age of Big Brother and Big Money, 300 00:19:23,560 --> 00:19:26,840 I think there's something noble about the fact he gets his kick 301 00:19:26,840 --> 00:19:29,520 out of proving theorems and not winning prizes. 302 00:19:32,960 --> 00:19:36,000 One mathematician would certainly have applauded. 303 00:19:36,000 --> 00:19:40,440 For solving any of his 23 problems, David Hilbert offered no prize 304 00:19:40,440 --> 00:19:45,760 or reward beyond the admiration of other mathematicians. 305 00:19:45,760 --> 00:19:49,360 When he sketched out the problems in Paris in 1900, 306 00:19:49,360 --> 00:19:52,360 Hilbert himself was already a mathematical star. 307 00:19:52,360 --> 00:19:56,320 And it was in Gottingen in northern Germany that he really shone. 308 00:19:59,440 --> 00:20:05,560 He was by far the most charismatic mathematician of his age. 309 00:20:05,560 --> 00:20:09,960 It's clear that everyone who knew him thought he was absolutely wonderful. 310 00:20:12,880 --> 00:20:17,600 He studied number theory and brought everything together that was there 311 00:20:17,600 --> 00:20:20,760 and then within a year or so he left that completely 312 00:20:20,760 --> 00:20:24,320 and revolutionised the theory of integral equation. 313 00:20:24,320 --> 00:20:26,880 It's always change and always something new, 314 00:20:26,880 --> 00:20:29,840 and there's hardly anybody who is... 315 00:20:29,840 --> 00:20:34,800 who was so flexible and so varied in his approach as Hilbert was. 316 00:20:34,800 --> 00:20:41,800 His work is still talked about today and his name has become attached to many mathematical terms. 317 00:20:41,800 --> 00:20:46,160 Mathematicians still use the Hilbert Space, the Hilbert Classification, 318 00:20:46,160 --> 00:20:51,120 the Hilbert Inequality and several Hilbert theorems. 319 00:20:51,120 --> 00:20:54,800 But it was his early work on equations that marked him out 320 00:20:54,800 --> 00:20:57,520 as a mathematician thinking in new ways. 321 00:20:57,520 --> 00:21:01,480 Hilbert showed that although there are infinitely many equations, 322 00:21:01,480 --> 00:21:04,800 there are ways to divide them up so that they are built 323 00:21:04,800 --> 00:21:08,160 out of just a finite set, like a set of building blocks. 324 00:21:08,160 --> 00:21:13,880 The most striking element of Hilbert's proof was that he couldn't actually construct this finite set. 325 00:21:13,880 --> 00:21:17,440 He just proved it must exist. 326 00:21:17,440 --> 00:21:20,760 Somebody criticised this as theology and not mathematics 327 00:21:20,760 --> 00:21:22,400 but they'd missed the point. 328 00:21:22,400 --> 00:21:26,280 What Hilbert was doing here was creating a new style of mathematics, 329 00:21:26,280 --> 00:21:28,840 a more abstract approach to the subject. 330 00:21:28,840 --> 00:21:31,280 You could still prove something existed, 331 00:21:31,280 --> 00:21:34,240 even though you couldn't construct it explicitly. 332 00:21:34,240 --> 00:21:37,960 It's like saying, "I know there has to be a way to get 333 00:21:37,960 --> 00:21:42,360 "from Gottingen to St Petersburg even though I can't tell you 334 00:21:42,360 --> 00:21:44,440 "how to actually get there." 335 00:21:44,440 --> 00:21:49,120 As well as challenging mathematical orthodoxies, Hilbert was also happy 336 00:21:49,120 --> 00:21:54,840 to knock the formal hierarchies that existed in the university system in Germany at the time. 337 00:21:54,840 --> 00:22:01,000 Other professors were quite shocked to see Hilbert out bicycling and drinking with his students. 338 00:22:01,000 --> 00:22:03,440 - He liked very much parties. - Yeah? 339 00:22:03,440 --> 00:22:07,240 - Yes. - Party animal. That's my kind of mathematician. 340 00:22:07,240 --> 00:22:13,360 He liked very much dancing with young women. He liked very much to flirt. 341 00:22:13,360 --> 00:22:17,880 Really? Most mathematicians I know are not the biggest of flirts. 342 00:22:17,880 --> 00:22:22,000 'Yet this lifestyle went hand in hand with an absolute commitment to maths.' 343 00:22:22,000 --> 00:22:26,200 Hilbert was of course somebody who thought 344 00:22:26,200 --> 00:22:30,240 that everybody who has a mathematical skill, 345 00:22:30,240 --> 00:22:36,400 a penguin, a woman, a man, or black, white or yellow, 346 00:22:36,400 --> 00:22:40,280 it doesn't matter, he should do mathematics 347 00:22:40,280 --> 00:22:42,360 and he should be admired for his. 348 00:22:42,360 --> 00:22:46,200 The mathematics speaks for itself in a way. 349 00:22:46,200 --> 00:22:49,720 - It doesn't matter... - When you're a penguin. 350 00:22:49,720 --> 00:22:54,360 Yeah, if you can prove the Riemann hypothesis, we really don't mind. 351 00:22:54,360 --> 00:22:58,280 - Yes, mathematics was for him a universal language. - Yes. 352 00:22:58,280 --> 00:23:02,080 Hilbert believed that this language was powerful enough 353 00:23:02,080 --> 00:23:04,360 to unlock all the truths of mathematics, 354 00:23:04,360 --> 00:23:07,640 a belief he expounded in a radio interview he gave 355 00:23:07,640 --> 00:23:11,400 on the future of mathematics on the 8th September 1930. 356 00:23:16,080 --> 00:23:20,280 In it, he had no doubt that all his 23 problems would soon be solved 357 00:23:20,280 --> 00:23:23,720 and that mathematics would finally be put 358 00:23:23,720 --> 00:23:26,840 on unshakeable logical foundations. 359 00:23:26,840 --> 00:23:30,160 There are absolutely no unsolvable problems, he declared, 360 00:23:30,160 --> 00:23:32,520 a belief that's been held by mathematicians 361 00:23:32,520 --> 00:23:34,480 since the time of the Ancient Greeks. 362 00:23:34,480 --> 00:23:40,040 He ended with this clarion call, "We must know, we will know." 363 00:23:40,040 --> 00:23:44,640 'Wir mussen wissen, wir werden wissen.' 364 00:23:45,960 --> 00:23:48,480 Unfortunately for him, the very day before 365 00:23:48,480 --> 00:23:52,320 in a scientific lecture that was not considered worthy of broadcast, 366 00:23:52,320 --> 00:23:55,520 another mathematician would shatter Hilbert's dream 367 00:23:55,520 --> 00:23:59,480 and put uncertainty at the heart of mathematics. 368 00:23:59,480 --> 00:24:02,400 The man responsible for destroying Hilbert's belief 369 00:24:02,400 --> 00:24:05,520 was an Austrian mathematician, Kurt Godel. 370 00:24:10,400 --> 00:24:12,440 And it all started here - Vienna. 371 00:24:12,440 --> 00:24:15,360 Even his admirers, and there are a great many, 372 00:24:15,360 --> 00:24:19,920 admit that Kurt Godel was a little odd. 373 00:24:19,920 --> 00:24:23,840 As a child, he was bright, sickly and very strange. 374 00:24:23,840 --> 00:24:25,880 He couldn't stop asking questions. 375 00:24:25,880 --> 00:24:30,720 So much so, that his family called him Herr Warum - Mr Why. 376 00:24:30,720 --> 00:24:35,160 Godel lived in Vienna in the 1920s and 1930s, 377 00:24:35,160 --> 00:24:38,000 between the fall of the Austro-Hungarian Empire 378 00:24:38,000 --> 00:24:39,960 and its annexation by the Nazis. 379 00:24:39,960 --> 00:24:45,520 It was a strange, chaotic and exciting time to be in the city. 380 00:24:45,520 --> 00:24:48,160 Godel studied mathematics at Vienna University 381 00:24:48,160 --> 00:24:50,600 but he spent most of his time in the cafes, 382 00:24:50,600 --> 00:24:52,960 the internet chat rooms of their time, 383 00:24:52,960 --> 00:24:55,920 where amongst games of backgammon and billiards, 384 00:24:55,920 --> 00:24:59,040 the real intellectual excitement was taking place. 385 00:24:59,040 --> 00:25:02,320 Particularly amongst a highly influential group 386 00:25:02,320 --> 00:25:05,920 of philosophers and scientists called the Vienna Circle. 387 00:25:05,920 --> 00:25:10,080 In their discussions, Kurt Godel would come up with an idea 388 00:25:10,080 --> 00:25:13,000 that would revolutionise mathematics. 389 00:25:13,000 --> 00:25:15,960 He'd set himself a difficult mathematical test. 390 00:25:15,960 --> 00:25:18,760 He wanted to solve Hilbert's second problem 391 00:25:18,760 --> 00:25:22,000 and find a logical foundation for all mathematics. 392 00:25:22,000 --> 00:25:25,520 But what he came up with surprised even him. 393 00:25:25,520 --> 00:25:28,960 All his efforts in mathematical logic not only couldn't provide 394 00:25:28,960 --> 00:25:33,840 the guarantee Hilbert wanted, instead he proved the opposite. 395 00:25:33,840 --> 00:25:35,440 Got it. 396 00:25:35,440 --> 00:25:38,800 It's called the Incompleteness Theorem. 397 00:25:38,800 --> 00:25:42,360 Godel proved that within any logical system for mathematics 398 00:25:42,360 --> 00:25:46,200 there will be statements about numbers which are true 399 00:25:46,200 --> 00:25:48,200 but which you cannot prove. 400 00:25:48,200 --> 00:25:53,000 He starts with the statement, "This statement cannot be proved." 401 00:25:53,000 --> 00:25:55,480 This is not a mathematical statement yet. 402 00:25:55,480 --> 00:25:58,360 But by using a clever code based on prime numbers, 403 00:25:58,360 --> 00:26:03,480 Godel could change this statement into a pure statement of arithmetic. 404 00:26:03,480 --> 00:26:08,640 Now, such statements must be either true or false. 405 00:26:08,640 --> 00:26:13,320 Hold on to your logical hats as we explore the possibilities. 406 00:26:13,320 --> 00:26:17,960 If the statement is false, that means the statement could be proved, 407 00:26:17,960 --> 00:26:21,320 which means it would be true, and that's a contradiction. 408 00:26:21,320 --> 00:26:23,880 So that means, the statement must be true. 409 00:26:23,880 --> 00:26:28,320 In other words, here is a mathematical statement that is true 410 00:26:28,320 --> 00:26:30,840 but can't be proved. 411 00:26:30,840 --> 00:26:32,440 Blast. 412 00:26:32,440 --> 00:26:35,520 Godel's proof led to a crisis in mathematics. 413 00:26:35,520 --> 00:26:39,320 What if the problem you were working on, the Goldbach conjecture, say, 414 00:26:39,320 --> 00:26:43,600 or the Riemann hypothesis, would turn out to be true but unprovable? 415 00:26:43,600 --> 00:26:46,720 It led to a crisis for Godel too. 416 00:26:46,720 --> 00:26:50,400 In the autumn of 1934, he suffered the first of what became 417 00:26:50,400 --> 00:26:55,520 a series of breakdowns and spent time in a sanatorium. 418 00:26:55,520 --> 00:26:58,960 He was saved by the love of a good woman. 419 00:26:58,960 --> 00:27:02,880 Adele Nimbursky was a dancer in a local night club. 420 00:27:02,880 --> 00:27:06,200 She kept Godel alive. 421 00:27:06,200 --> 00:27:10,040 One day, she and Godel were walking down these very steps. 422 00:27:10,040 --> 00:27:13,120 Suddenly he was attacked by Nazi thugs. 423 00:27:13,120 --> 00:27:17,360 Godel himself wasn't Jewish, but many of his friends in the Vienna Circle were. 424 00:27:17,360 --> 00:27:19,840 Adele came to his rescue. 425 00:27:19,840 --> 00:27:24,400 But it was only a temporary reprieve for Godel and for maths. 426 00:27:24,400 --> 00:27:29,680 All across Austria and Germany, mathematics was about to die. 427 00:27:33,680 --> 00:27:36,240 In the new German empire in the late 1930s 428 00:27:36,240 --> 00:27:39,760 there weren't colourful balloons flying over the universities, 429 00:27:39,760 --> 00:27:41,600 but swastikas. 430 00:27:41,600 --> 00:27:46,280 The Nazis passed a law allowing the removal of any civil servant 431 00:27:46,280 --> 00:27:47,680 who wasn't Aryan. 432 00:27:47,680 --> 00:27:51,200 Academics were civil servants in Germany then and now. 433 00:27:53,520 --> 00:27:56,200 Mathematicians suffered more than most. 434 00:27:56,200 --> 00:27:59,600 144 in Germany would lose their jobs. 435 00:27:59,600 --> 00:28:04,040 14 were driven to suicide or died in concentration camps. 436 00:28:07,680 --> 00:28:10,600 But one brilliant mathematician stayed on. 437 00:28:10,600 --> 00:28:12,400 David Hilbert helped arrange 438 00:28:12,400 --> 00:28:15,000 for some of his brightest students to flee. 439 00:28:15,000 --> 00:28:17,640 And he spoke out for a while about the dismissal 440 00:28:17,640 --> 00:28:19,200 of his Jewish colleagues. 441 00:28:19,200 --> 00:28:23,400 But soon, he too became silent. 442 00:28:26,720 --> 00:28:29,240 It's not clear why he didn't flee himself 443 00:28:29,240 --> 00:28:31,320 or at least protest a little more. 444 00:28:31,320 --> 00:28:33,600 He did fall ill towards the end of his life 445 00:28:33,600 --> 00:28:35,800 so maybe he just didn't have the energy. 446 00:28:35,800 --> 00:28:38,440 All around him, mathematicians and scientists 447 00:28:38,440 --> 00:28:42,160 were fleeing the Nazi regime until it was only Hilbert left 448 00:28:42,160 --> 00:28:47,480 to witness the destruction of one of the greatest mathematical centres of all time. 449 00:28:50,000 --> 00:28:53,640 David Hilbert died in 1943. 450 00:28:53,640 --> 00:28:56,360 Only ten people attended the funeral 451 00:28:56,360 --> 00:28:59,600 of the most famous mathematician of his time. 452 00:28:59,600 --> 00:29:01,880 The dominance of Europe, 453 00:29:01,880 --> 00:29:05,680 the centre for world maths for 500 years, was over. 454 00:29:05,680 --> 00:29:12,000 It was time for the mathematical baton to be handed to the New World. 455 00:29:13,840 --> 00:29:17,120 Time in fact for this place. 456 00:29:17,120 --> 00:29:22,040 The Institute for Advanced Study had been set up in Princeton in 1930. 457 00:29:22,040 --> 00:29:24,880 The idea was to reproduce the collegiate atmosphere 458 00:29:24,880 --> 00:29:28,880 of the old European universities in rural New Jersey. 459 00:29:28,880 --> 00:29:32,200 But to do this, it needed to attract the very best, 460 00:29:32,200 --> 00:29:34,280 and it didn't need to look far. 461 00:29:34,280 --> 00:29:37,480 Many of the brightest European mathematicians 462 00:29:37,480 --> 00:29:39,920 were fleeing the Nazis for America. 463 00:29:39,920 --> 00:29:42,520 People like Hermann Weyl, whose research 464 00:29:42,520 --> 00:29:45,680 would have major significance for theoretical physics. 465 00:29:45,680 --> 00:29:48,280 And John Von Neumann, who developed game theory 466 00:29:48,280 --> 00:29:50,840 and was one of the pioneers of computer science. 467 00:29:50,840 --> 00:29:55,400 The Institute quickly became the perfect place 468 00:29:55,400 --> 00:29:59,440 to create another Gottingen in the woods. 469 00:29:59,440 --> 00:30:04,760 One mathematician in particular made the place a home from home. 470 00:30:04,760 --> 00:30:06,320 Every morning Kurt Godel, 471 00:30:06,320 --> 00:30:09,360 dressed in a white linen suit and wearing a fedora, 472 00:30:09,360 --> 00:30:13,040 would walk from his home along Mercer Street to the Institute. 473 00:30:13,040 --> 00:30:16,520 On his way, he would stop here at number 112, 474 00:30:16,520 --> 00:30:22,640 to pick up his closest friend, another European exile, Albert Einstein. 475 00:30:22,640 --> 00:30:26,960 But not even relaxed, affluent Princeton could help Godel 476 00:30:26,960 --> 00:30:29,040 finally escape his demons. 477 00:30:29,040 --> 00:30:31,640 Einstein was always full of laughter. 478 00:30:31,640 --> 00:30:35,520 He described Princeton as a banishment to paradise. 479 00:30:35,520 --> 00:30:40,080 But the much younger Godel became increasingly solemn and pessimistic. 480 00:30:43,160 --> 00:30:46,400 Slowly this pessimism turned into paranoia. 481 00:30:46,400 --> 00:30:50,520 He spent less and less time with his fellow mathematicians in Princeton. 482 00:30:50,520 --> 00:30:54,200 Instead, he preferred to come here to the beach, next to the ocean, 483 00:30:54,200 --> 00:30:59,240 walk alone, thinking about the works of the great German mathematician, Leibniz. 484 00:31:01,400 --> 00:31:05,320 But as Godel was withdrawing into his own interior world, 485 00:31:05,320 --> 00:31:09,320 his influence on American mathematics paradoxically 486 00:31:09,320 --> 00:31:12,000 was growing stronger and stronger. 487 00:31:12,000 --> 00:31:16,160 And a young mathematician from just along the New Jersey coast 488 00:31:16,160 --> 00:31:19,840 eagerly took on some of the challenges he posed. 489 00:31:19,840 --> 00:31:23,760 # One, two, three, four, five, six, seven, eight, nine, ten 490 00:31:23,760 --> 00:31:25,880 # Times a day I could love you... # 491 00:31:25,880 --> 00:31:27,040 In 1950s America, 492 00:31:27,040 --> 00:31:31,440 the majority of youngsters weren't preoccupied with mathematics. 493 00:31:31,440 --> 00:31:35,160 Most went for a more relaxed, hedonistic lifestyle 494 00:31:35,160 --> 00:31:38,840 in this newly affluent land of ice-cream and doughnuts. 495 00:31:38,840 --> 00:31:42,560 But one teenager didn't indulge in the normal pursuits 496 00:31:42,560 --> 00:31:45,640 of American adolescence but chose instead 497 00:31:45,640 --> 00:31:49,200 to grapple with some of the major problems in mathematics. 498 00:31:49,200 --> 00:31:50,680 From a very early age, 499 00:31:50,680 --> 00:31:55,080 Paul Cohen was winning mathematical competitions and prizes. 500 00:31:55,080 --> 00:31:58,960 But he found it difficult at first to discover a field in mathematics 501 00:31:58,960 --> 00:32:01,280 where he could really make his mark... 502 00:32:01,280 --> 00:32:05,720 Until he read about Cantor's continuum hypothesis. 503 00:32:05,720 --> 00:32:09,280 That's the one problem, as I had learned back in Halle, 504 00:32:09,280 --> 00:32:11,760 that Cantor just couldn't resolve. 505 00:32:11,760 --> 00:32:15,400 It asks whether there is or there isn't an infinite set of numbers 506 00:32:15,400 --> 00:32:18,080 bigger than the set of all whole numbers 507 00:32:18,080 --> 00:32:20,960 but smaller than the set of all decimals. 508 00:32:20,960 --> 00:32:24,280 It sounds straightforward, but it had foiled all attempts 509 00:32:24,280 --> 00:32:29,160 to solve it since Hilbert made it his first problem way back in 1900. 510 00:32:29,160 --> 00:32:31,480 With the arrogance of youth, 511 00:32:31,480 --> 00:32:36,040 the 22-year-old Paul Cohen decided that he could do it. 512 00:32:36,040 --> 00:32:40,720 Cohen came back a year later with the extraordinary discovery 513 00:32:40,720 --> 00:32:43,200 that both answers could be true. 514 00:32:43,200 --> 00:32:47,160 There was one mathematics where the continuum hypothesis 515 00:32:47,160 --> 00:32:49,080 could be assumed to be true. 516 00:32:49,080 --> 00:32:51,800 There wasn't a set between the whole numbers 517 00:32:51,800 --> 00:32:53,440 and the infinite decimals. 518 00:32:55,160 --> 00:32:59,200 But there was an equally consistent mathematics 519 00:32:59,200 --> 00:33:03,440 where the continuum hypothesis could be assumed to be false. 520 00:33:03,440 --> 00:33:08,280 Here, there was a set between the whole numbers and the infinite decimals. 521 00:33:08,280 --> 00:33:11,480 It was an incredibly daring solution. 522 00:33:11,480 --> 00:33:13,840 Cohen's proof seemed true, 523 00:33:13,840 --> 00:33:19,160 but his method was so new that nobody was absolutely sure. 524 00:33:19,160 --> 00:33:22,720 There was only one person whose opinion everybody trusted. 525 00:33:22,720 --> 00:33:26,640 There was a lot of scepticism and he had to come and make a trip here, 526 00:33:26,640 --> 00:33:29,320 to the Institute right here, to visit Godel. 527 00:33:29,320 --> 00:33:32,720 And it was only after Godel gave his stamp of approval 528 00:33:32,720 --> 00:33:34,240 in quite an unusual way, 529 00:33:34,240 --> 00:33:37,880 He said, "Give me your paper", and then on Monday he put it back 530 00:33:37,880 --> 00:33:40,360 in the box and said, "Yes, it's correct." 531 00:33:40,360 --> 00:33:42,040 Then everything changed. 532 00:33:43,240 --> 00:33:46,200 Today mathematicians insert a statement 533 00:33:46,200 --> 00:33:50,840 that says whether the result depends on the continuum hypothesis. 534 00:33:50,840 --> 00:33:54,880 We've built up two different mathematical worlds 535 00:33:54,880 --> 00:33:57,320 in which one answer is yes and the other is no. 536 00:33:57,320 --> 00:34:01,440 Paul Cohen really has rocked the mathematical universe. 537 00:34:01,440 --> 00:34:05,680 It gave him fame, riches, and prizes galore. 538 00:34:07,680 --> 00:34:12,880 He didn't publish much after his early success in the '60s. 539 00:34:12,880 --> 00:34:15,040 But he was absolutely dynamite. 540 00:34:15,040 --> 00:34:18,840 I can't imagine anyone better to learn from, and he was very eager 541 00:34:18,840 --> 00:34:23,840 to learn, to teach you anything he knew or even things he didn't know. 542 00:34:23,840 --> 00:34:27,640 With the confidence that came from solving Hilbert's first problem, 543 00:34:27,640 --> 00:34:30,320 Cohen settled down in the mid 1960s 544 00:34:30,320 --> 00:34:34,440 to have a go at the most important Hilbert problem of them all - 545 00:34:34,440 --> 00:34:36,960 the eighth, the Riemann hypothesis. 546 00:34:36,960 --> 00:34:43,000 When he died in California in 2007, 40 years later, he was still trying. 547 00:34:43,000 --> 00:34:46,200 But like many famous mathematicians before him, 548 00:34:46,200 --> 00:34:48,280 Riemann had defeated even him. 549 00:34:48,280 --> 00:34:52,440 But his approach has inspired others to make progress towards a proof, 550 00:34:52,440 --> 00:34:55,560 including one of his most famous students, Peter Sarnak. 551 00:34:55,560 --> 00:34:59,440 I think, overall, absolutely loved the guy. 552 00:34:59,440 --> 00:35:01,840 He was my inspiration. 553 00:35:01,840 --> 00:35:04,600 I'm really glad I worked with him. 554 00:35:04,600 --> 00:35:06,800 He put me on the right track. 555 00:35:09,960 --> 00:35:14,240 Paul Cohen is a good example of the success of the great American Dream. 556 00:35:14,240 --> 00:35:16,800 The second generation Jewish immigrant 557 00:35:16,800 --> 00:35:18,960 becomes top American professor. 558 00:35:18,960 --> 00:35:23,640 But I wouldn't say that his mathematics was a particularly American product. 559 00:35:23,640 --> 00:35:25,720 Cohen was so fired up by this problem 560 00:35:25,720 --> 00:35:29,680 that he probably would have cracked it whatever the surroundings. 561 00:35:31,200 --> 00:35:33,680 Paul Cohen had it relatively easy. 562 00:35:33,680 --> 00:35:36,640 But another great American mathematician of the 1960s 563 00:35:36,640 --> 00:35:40,320 faced a much tougher struggle to make an impact. 564 00:35:40,320 --> 00:35:43,440 Not least because she was female. 565 00:35:43,440 --> 00:35:48,240 In the story of maths, nearly all the truly great mathematicians have been men. 566 00:35:48,240 --> 00:35:51,560 But there have been a few exceptions. 567 00:35:51,560 --> 00:35:54,000 There was the Russian Sofia Kovalevskaya 568 00:35:54,000 --> 00:35:58,920 who became the first female professor of mathematics in Stockholm in 1889, 569 00:35:58,920 --> 00:36:03,400 and won a very prestigious French mathematical prize. 570 00:36:03,400 --> 00:36:07,080 And then Emmy Noether, a talented algebraist who fled from the Nazis 571 00:36:07,080 --> 00:36:10,600 to America but then died before she fully realised her potential. 572 00:36:10,600 --> 00:36:15,920 Then there is the woman who I am crossing America to find out about. 573 00:36:15,920 --> 00:36:19,680 Julia Robinson, the first woman ever to be elected president 574 00:36:19,680 --> 00:36:22,080 of the American Mathematical Society. 575 00:36:31,440 --> 00:36:34,840 She was born in St Louis in 1919, 576 00:36:34,840 --> 00:36:38,160 but her mother died when she was two. 577 00:36:38,160 --> 00:36:42,360 She and her sister Constance moved to live with their grandmother 578 00:36:42,360 --> 00:36:45,720 in a small community in the desert near Phoenix, Arizona. 579 00:36:47,720 --> 00:36:49,800 Julia Robinson grew up around here. 580 00:36:49,800 --> 00:36:53,440 I've got a photo which shows her cottage in the 1930s, 581 00:36:53,440 --> 00:36:55,480 with nothing much around it. 582 00:36:55,480 --> 00:36:58,160 The mountains pretty much match those over there 583 00:36:58,160 --> 00:37:00,640 so I think she might have lived down there. 584 00:37:01,600 --> 00:37:04,160 Julia grew up a shy, sickly girl, 585 00:37:04,160 --> 00:37:09,440 who, when she was seven, spent a year in bed because of scarlet fever. 586 00:37:09,440 --> 00:37:12,240 Ill-health persisted throughout her childhood. 587 00:37:12,240 --> 00:37:15,120 She was told she wouldn't live past 40. 588 00:37:15,120 --> 00:37:20,400 But right from the start, she had an innate mathematical ability. 589 00:37:20,400 --> 00:37:25,240 Under the shade of the native Arizona cactus, she whiled away the time 590 00:37:25,240 --> 00:37:28,720 playing endless counting games with stone pebbles. 591 00:37:28,720 --> 00:37:31,960 This early searching for patterns would give her a feel 592 00:37:31,960 --> 00:37:35,320 and love of numbers that would last for the rest of her life. 593 00:37:35,320 --> 00:37:39,160 But despite showing an early brilliance, she had to continually 594 00:37:39,160 --> 00:37:44,080 fight at school and college to simply be allowed to keep doing maths. 595 00:37:44,080 --> 00:37:47,920 As a teenager, she was the only girl in the maths class 596 00:37:47,920 --> 00:37:50,600 but had very little encouragement. 597 00:37:50,600 --> 00:37:55,480 The young Julia sought intellectual stimulation elsewhere. 598 00:37:55,480 --> 00:37:59,440 Julia loved listening to a radio show called the University Explorer 599 00:37:59,440 --> 00:38:02,440 and the 53rd episode was all about mathematics. 600 00:38:02,440 --> 00:38:04,960 The broadcaster described how he discovered 601 00:38:04,960 --> 00:38:08,560 despite their esoteric language and their seclusive nature, 602 00:38:08,560 --> 00:38:12,320 mathematicians are the most interesting and inspiring creatures. 603 00:38:12,320 --> 00:38:16,240 For the first time, Julia had found out that there were mathematicians, 604 00:38:16,240 --> 00:38:17,920 not just mathematics teachers. 605 00:38:17,920 --> 00:38:20,440 There was a world of mathematics out there, 606 00:38:20,440 --> 00:38:22,240 and she wanted to be part of it. 607 00:38:26,080 --> 00:38:29,680 The doors to that world opened here at the University of California, 608 00:38:29,680 --> 00:38:31,960 at Berkeley near San Francisco. 609 00:38:33,760 --> 00:38:38,680 She was absolutely obsessed that she wanted to go to Berkeley. 610 00:38:38,680 --> 00:38:44,200 She wanted to go away to some place where there were mathematicians. 611 00:38:44,200 --> 00:38:46,720 Berkeley certainly had mathematicians, 612 00:38:46,720 --> 00:38:50,320 including a number theorist called Raphael Robinson. 613 00:38:50,320 --> 00:38:53,400 In their frequent walks around the campus 614 00:38:53,400 --> 00:38:59,960 they found they had more than just a passion for mathematics. They married in 1952. 615 00:38:59,960 --> 00:39:03,200 Julia got her PhD and settled down 616 00:39:03,200 --> 00:39:05,720 to what would turn into her lifetime's work - 617 00:39:05,720 --> 00:39:07,280 Hilbert's tenth problem. 618 00:39:07,280 --> 00:39:10,000 She thought about it all the time. 619 00:39:10,000 --> 00:39:14,120 She said to me she just wouldn't wanna die without knowing that answer 620 00:39:14,120 --> 00:39:16,240 and it had become an obsession. 621 00:39:17,280 --> 00:39:21,200 Julia's obsession has been shared with many other mathematicians 622 00:39:21,200 --> 00:39:24,560 since Hilbert had first posed it back in 1900. 623 00:39:24,560 --> 00:39:28,400 His tenth problem asked if there was some universal method 624 00:39:28,400 --> 00:39:34,200 that could tell whether any equation had whole number solutions or not. 625 00:39:34,200 --> 00:39:36,520 Nobody had been able to solve it. 626 00:39:36,520 --> 00:39:39,520 In fact, the growing belief was 627 00:39:39,520 --> 00:39:42,440 that no such universal method was possible. 628 00:39:42,440 --> 00:39:44,520 How on earth could you prove that, 629 00:39:44,520 --> 00:39:48,400 however ingenious you were, you'd never come up with a method? 630 00:39:50,080 --> 00:39:51,800 With the help of colleagues, 631 00:39:51,800 --> 00:39:55,640 Julia developed what became known as the Robinson hypothesis. 632 00:39:55,640 --> 00:39:58,920 This argued that to show no such method existed, 633 00:39:58,920 --> 00:40:03,280 all you had to do was to cook up one equation whose solutions 634 00:40:03,280 --> 00:40:06,040 were a very specific set of numbers. 635 00:40:06,040 --> 00:40:09,280 The set of numbers needed to grow exponentially, 636 00:40:09,280 --> 00:40:13,960 like taking powers of two, yet still be captured by the equations 637 00:40:13,960 --> 00:40:16,520 at the heart of Hilbert's problem. 638 00:40:16,520 --> 00:40:21,600 Try as she might, Robinson just couldn't find this set. 639 00:40:21,600 --> 00:40:25,880 For the tenth problem to be finally solved, 640 00:40:25,880 --> 00:40:28,880 there needed to be some fresh inspiration. 641 00:40:28,880 --> 00:40:34,280 That came from 5,000 miles away - St Petersburg in Russia. 642 00:40:34,280 --> 00:40:37,840 Ever since the great Leonhard Euler set up shop here 643 00:40:37,840 --> 00:40:39,040 in the 18th century, 644 00:40:39,040 --> 00:40:42,960 the city has been famous for its mathematics and mathematicians. 645 00:40:42,960 --> 00:40:44,760 Here in the Steklov Institute, 646 00:40:44,760 --> 00:40:47,480 some of the world's brightest mathematicians 647 00:40:47,480 --> 00:40:50,160 have set out their theorems and conjectures. 648 00:40:50,160 --> 00:40:54,320 This morning, one of them is giving a rare seminar. 649 00:40:57,120 --> 00:41:00,040 It's tough going even if you speak Russian, 650 00:41:00,040 --> 00:41:02,080 which unfortunately I don't. 651 00:41:02,080 --> 00:41:06,320 But we do get a break in the middle to recover before the final hour. 652 00:41:06,320 --> 00:41:08,320 There is a kind of rule in seminars. 653 00:41:08,320 --> 00:41:12,880 The first third is for everyone, the second third for the experts 654 00:41:12,880 --> 00:41:16,080 and the last third is just for the lecturer. 655 00:41:16,080 --> 00:41:19,080 I think that's what we're going to get next. 656 00:41:19,080 --> 00:41:22,800 The lecturer is Yuri Matiyasevich and he is explaining 657 00:41:22,800 --> 00:41:26,520 his latest work on - what else? - the Riemann hypothesis. 658 00:41:28,720 --> 00:41:33,160 As a bright young graduate student in 1965, Yuri's tutor 659 00:41:33,160 --> 00:41:36,000 suggested he have a go at another Hilbert problem, 660 00:41:36,000 --> 00:41:39,000 the one that had in fact preoccupied Julia Robinson. 661 00:41:39,000 --> 00:41:40,280 Hilbert's tenth. 662 00:41:43,160 --> 00:41:45,080 It was the height of the Cold War. 663 00:41:45,080 --> 00:41:48,440 Perhaps Matiyasevich could succeed for Russia 664 00:41:48,440 --> 00:41:52,080 where Julia and her fellow American mathematicians had failed. 665 00:41:52,080 --> 00:41:55,000 - At first I did not like their approach. - Oh, right. 666 00:41:55,000 --> 00:41:59,640 The statement looked to me rather strange and artificial 667 00:41:59,640 --> 00:42:03,520 but after some time I understood it was quite natural, 668 00:42:03,520 --> 00:42:07,200 and then I understood that she had a new brilliant idea 669 00:42:07,200 --> 00:42:10,000 and I just started to further develop it. 670 00:42:11,520 --> 00:42:17,000 In January 1970, he found the vital last piece in the jigsaw. 671 00:42:17,000 --> 00:42:21,880 He saw how to capture the famous Fibonacci sequence of numbers 672 00:42:21,880 --> 00:42:26,040 using the equations that were at the heart of Hilbert's problem. 673 00:42:26,040 --> 00:42:28,920 Building on the work of Julia and her colleagues, 674 00:42:28,920 --> 00:42:30,720 he had solved the tenth. 675 00:42:30,720 --> 00:42:34,240 He was just 22 years old. 676 00:42:34,240 --> 00:42:37,920 The first person he wanted to tell was the woman he owed so much to. 677 00:42:39,800 --> 00:42:41,720 I got no answer 678 00:42:41,720 --> 00:42:44,600 and I believed they were lost in the mail. 679 00:42:44,600 --> 00:42:47,720 It was quite natural because it was Soviet time. 680 00:42:47,720 --> 00:42:50,800 But back in California, Julia had heard rumours 681 00:42:50,800 --> 00:42:54,840 through the mathematical grapevine that the problem had been solved. 682 00:42:54,840 --> 00:42:57,120 And she contacted Yuri herself. 683 00:42:58,120 --> 00:43:01,480 She said, I just had to wait for you to grow up 684 00:43:01,480 --> 00:43:06,160 to get the answer, because she had started work in 1948. 685 00:43:06,160 --> 00:43:07,960 When Yuri was just a baby. 686 00:43:07,960 --> 00:43:11,240 Then he responds by thanking her 687 00:43:11,240 --> 00:43:16,160 and saying that the credit is as much hers as it is his. 688 00:43:18,240 --> 00:43:20,520 YURI: I met Julia one year later. 689 00:43:20,520 --> 00:43:25,080 It was in Bucharest. I suggested after the conference in Bucharest 690 00:43:25,080 --> 00:43:30,120 Julia and her husband Raphael came to see me here in Leningrad. 691 00:43:30,120 --> 00:43:35,400 Together, Julia and Yuri worked on several other mathematical problems 692 00:43:35,400 --> 00:43:39,160 until shortly before Julia died in 1985. 693 00:43:39,160 --> 00:43:41,960 She was just 55 years old. 694 00:43:41,960 --> 00:43:45,640 She was able to find the new ways. 695 00:43:45,640 --> 00:43:49,640 Many mathematicians just combine previous known methods 696 00:43:49,640 --> 00:43:55,560 to solve new problems and she had really new ideas. 697 00:43:55,560 --> 00:43:59,160 Although Julia Robinson showed there was no universal method 698 00:43:59,160 --> 00:44:01,560 to solve all equations in whole numbers, 699 00:44:01,560 --> 00:44:05,840 mathematicians were still interested in finding methods 700 00:44:05,840 --> 00:44:08,760 to solve special classes of equations. 701 00:44:08,760 --> 00:44:11,320 It would be in France in the early 19th century, 702 00:44:11,320 --> 00:44:13,560 in one of the most extraordinary stories 703 00:44:13,560 --> 00:44:17,120 in the history of mathematics, that methods were developed 704 00:44:17,120 --> 00:44:20,240 to understand why certain equations could be solved 705 00:44:20,240 --> 00:44:21,760 while others couldn't. 706 00:44:27,840 --> 00:44:32,520 It's early morning in Paris on the 29th May 1832. 707 00:44:32,520 --> 00:44:37,120 Evariste Galois is about to fight for his very life. 708 00:44:37,120 --> 00:44:40,680 It is the reign of the reactionary Bourbon King, Charles X, 709 00:44:40,680 --> 00:44:43,960 and Galois, like many angry young men in Paris then, 710 00:44:43,960 --> 00:44:46,680 is a republican revolutionary. 711 00:44:46,680 --> 00:44:52,000 Unlike the rest of his comrades though, he has another passion - mathematics. 712 00:44:53,560 --> 00:44:56,480 He had just spent four months in jail. 713 00:44:56,480 --> 00:45:00,160 Then, in a mysterious saga of unrequited love, 714 00:45:00,160 --> 00:45:02,280 he is challenged to a duel. 715 00:45:02,280 --> 00:45:04,280 He'd been up the whole previous night 716 00:45:04,280 --> 00:45:07,360 refining a new language for mathematics he'd developed. 717 00:45:07,360 --> 00:45:14,160 Galois believed that mathematics shouldn't be the study of number and shape, but the study of structure. 718 00:45:14,160 --> 00:45:17,240 Perhaps he was still pre-occupied with his maths. 719 00:45:17,240 --> 00:45:18,800 GUNSHOT 720 00:45:18,800 --> 00:45:21,680 There was only one shot fired that morning. 721 00:45:21,680 --> 00:45:27,280 Galois died the next day, just 20 years old. 722 00:45:27,280 --> 00:45:30,320 It was one of mathematics greatest losses. 723 00:45:30,320 --> 00:45:33,080 Only by the beginning of the 20th century 724 00:45:33,080 --> 00:45:37,640 would Galois be fully appreciated and his ideas fully realised. 725 00:45:42,400 --> 00:45:46,520 Galois had discovered new techniques to be able to tell 726 00:45:46,520 --> 00:45:49,920 whether certain equations could have solutions or not. 727 00:45:49,920 --> 00:45:54,000 The symmetry of certain geometric objects seemed to be the key. 728 00:45:54,000 --> 00:45:58,520 His idea of using geometry to analyse equations 729 00:45:58,520 --> 00:46:03,880 would be picked up in the 1920s by another Parisian mathematician, Andre Weil. 730 00:46:03,880 --> 00:46:09,520 I was very much interested and so far as school was concerned 731 00:46:09,520 --> 00:46:13,720 quite successful in all possible branches. 732 00:46:13,720 --> 00:46:17,480 And he was. After studying in Germany as well as France, 733 00:46:17,480 --> 00:46:21,000 Andre settled down at this apartment in Paris 734 00:46:21,000 --> 00:46:25,760 which he shared with his more-famous sister, the writer Simone Weil. 735 00:46:25,760 --> 00:46:31,040 But when the Second World War broke out, he found himself in very different circumstances. 736 00:46:31,040 --> 00:46:37,040 He dodged the draft by fleeing to Finland where he was almost executed for being a Russian spy. 737 00:46:37,040 --> 00:46:42,720 On his return to France he was put in prison in Rouen to await trial for desertion. 738 00:46:42,720 --> 00:46:45,320 At the trial, the judge gave him a choice. 739 00:46:45,320 --> 00:46:49,120 Five more years in prison or serve in a combat unit. 740 00:46:49,120 --> 00:46:52,240 He chose to join the French army, a lucky choice 741 00:46:52,240 --> 00:46:56,120 because just before the Germans invaded a few months later, 742 00:46:56,120 --> 00:46:58,280 all the prisoners were killed. 743 00:46:58,280 --> 00:47:05,400 Weil only spent a few months in prison, but this time was crucial for his mathematics. 744 00:47:05,400 --> 00:47:11,000 Because here he built on the ideas of Galois and first developed algebraic geometry 745 00:47:11,000 --> 00:47:15,720 a whole new language for understanding solutions to equations. 746 00:47:15,720 --> 00:47:18,720 Galois had shown how new mathematical structures 747 00:47:18,720 --> 00:47:22,600 can be used to reveal the secrets behind equations. 748 00:47:22,600 --> 00:47:24,640 Weil's work led him to theorems 749 00:47:24,640 --> 00:47:28,800 that connected number theory, algebra, geometry and topology 750 00:47:28,800 --> 00:47:33,720 and are one of the greatest achievements of modern mathematics. 751 00:47:33,720 --> 00:47:36,760 Without Andre Weil, we would never have heard 752 00:47:36,760 --> 00:47:41,400 of the strangest man in the history of maths, Nicolas Bourbaki. 753 00:47:43,720 --> 00:47:50,400 There are no photos of Bourbaki in existence but we do know he was born in this cafe in the Latin Quarter 754 00:47:50,400 --> 00:47:54,520 in 1934 when it was a proper cafe, the cafe Capoulade, 755 00:47:54,520 --> 00:47:58,000 and not the fast food joint it has now become. 756 00:47:58,000 --> 00:48:03,200 Just down the road, I met up with Bourbaki expert David Aubin. 757 00:48:03,200 --> 00:48:06,400 When I was a graduate student I got quite frightened 758 00:48:06,400 --> 00:48:08,120 when I used to go into the library 759 00:48:08,120 --> 00:48:10,960 because this guy Bourbaki had written so many books. 760 00:48:10,960 --> 00:48:14,400 Something like 30 or 40 altogether. 761 00:48:14,400 --> 00:48:19,680 In analysis, in geometry, in topology, it was all new foundations. 762 00:48:19,680 --> 00:48:23,360 Virtually everyone studying Maths seriously anywhere in the world 763 00:48:23,360 --> 00:48:28,200 in the 1950s, '60s and '70s would have read Nicolas Bourbaki. 764 00:48:28,200 --> 00:48:31,160 He applied for membership of the American Maths Society, I heard. 765 00:48:31,160 --> 00:48:33,360 At which point he was denied membership 766 00:48:33,360 --> 00:48:36,320 - on the grounds that he didn't exist. - Oh! 767 00:48:36,320 --> 00:48:38,160 The Americans were right. 768 00:48:38,160 --> 00:48:41,880 Nicolas Bourbaki does not exist at all. And never has. 769 00:48:41,880 --> 00:48:46,200 Bourbaki is in fact the nom de plume for a group of French mathematicians 770 00:48:46,200 --> 00:48:49,880 led by Andre Weil who decided to write a coherent account 771 00:48:49,880 --> 00:48:52,480 of the mathematics of the 20th century. 772 00:48:52,480 --> 00:48:57,200 Most of the time mathematicians like to have their own names on theorems. 773 00:48:57,200 --> 00:48:59,600 But for the Bourbaki group, 774 00:48:59,600 --> 00:49:03,440 the aims of the project overrode any desire for personal glory. 775 00:49:03,440 --> 00:49:07,120 After the Second World War, the Bourbaki baton was handed down 776 00:49:07,120 --> 00:49:10,080 to the next generation of French mathematicians. 777 00:49:10,080 --> 00:49:15,400 And their most brilliant member was Alexandre Grothendieck. 778 00:49:15,400 --> 00:49:17,000 Here at the IHES in Paris, 779 00:49:17,000 --> 00:49:21,520 the French equivalent of Princeton's Institute for Advanced Study, 780 00:49:21,520 --> 00:49:27,160 Grothendieck held court at his famous seminars in the 1950s and 1960s. 781 00:49:29,920 --> 00:49:33,600 He had this incredible charisma. 782 00:49:33,600 --> 00:49:40,240 He had this amazing ability to see a young person and somehow know 783 00:49:40,240 --> 00:49:46,280 what kind of contribution this person could make to this incredible vision 784 00:49:46,280 --> 00:49:48,920 he had of how mathematics could be. 785 00:49:48,920 --> 00:49:54,520 And this vision enabled him to get across some very difficult ideas indeed. 786 00:49:54,520 --> 00:49:58,240 He says, "Suppose you want to open a walnut. 787 00:49:58,240 --> 00:50:02,200 "So the standard thing is you take a nutcracker and you just break it open." 788 00:50:02,200 --> 00:50:04,800 And he says his approach is more like, 789 00:50:04,800 --> 00:50:08,120 you take this walnut and you put it out in the snow 790 00:50:08,120 --> 00:50:10,160 and you leave it there for a few months 791 00:50:10,160 --> 00:50:13,760 and then when you come back to it, it just opens. 792 00:50:13,760 --> 00:50:15,760 Grothendieck is a Structuralist. 793 00:50:15,760 --> 00:50:19,720 What he's interested in are the hidden structures 794 00:50:19,720 --> 00:50:22,120 underneath all mathematics. 795 00:50:22,120 --> 00:50:27,560 Only when you get down to the very basic architecture and think in very general terms 796 00:50:27,560 --> 00:50:31,160 will the patterns in mathematics become clear. 797 00:50:31,160 --> 00:50:37,120 Grothendieck produced a new powerful language to see structures in a new way. 798 00:50:37,120 --> 00:50:39,720 It was like living in a world of black and white 799 00:50:39,720 --> 00:50:42,960 and suddenly having the language to see the world in colour. 800 00:50:42,960 --> 00:50:46,640 It's a language that mathematicians have been using ever since 801 00:50:46,640 --> 00:50:51,640 to solve problems in number theory, geometry, even fundamental physics. 802 00:50:53,160 --> 00:50:56,440 But in the late 1960s, Grothendieck decided 803 00:50:56,440 --> 00:51:01,640 to turn his back on mathematics after he discovered politics. 804 00:51:01,640 --> 00:51:06,320 He believed that the threat of nuclear war and the questions 805 00:51:06,320 --> 00:51:12,440 of nuclear disarmament were more important than mathematics 806 00:51:12,440 --> 00:51:17,480 and that people who continue to do mathematics 807 00:51:17,480 --> 00:51:21,240 rather than confront this threat of nuclear war 808 00:51:21,240 --> 00:51:22,920 were doing harm in the world. 809 00:51:26,440 --> 00:51:29,040 Grothendieck decided to leave Paris 810 00:51:29,040 --> 00:51:32,040 and move back to the south of France where he grew up. 811 00:51:32,040 --> 00:51:36,680 Bursts of radical politics followed and then a nervous breakdown. 812 00:51:36,680 --> 00:51:40,720 He moved to the Pyrenees and became a recluse. 813 00:51:40,720 --> 00:51:45,600 He's now lost all contact with his old friends and mathematical colleagues. 814 00:51:46,600 --> 00:51:51,040 Nevertheless, the legacy of his achievements means that Grothendieck stands 815 00:51:51,040 --> 00:51:57,440 alongside Cantor, Godel and Hilbert as someone who has transformed the mathematical landscape. 816 00:51:59,200 --> 00:52:03,800 He changed the whole subject in a really fundamental way. It will never go back. 817 00:52:03,800 --> 00:52:08,800 Certainly, he's THE dominant figure of the 20th century. 818 00:52:16,200 --> 00:52:18,280 I've come back to England, though, 819 00:52:18,280 --> 00:52:22,440 thinking again about another seminal figure of the 20th century. 820 00:52:22,440 --> 00:52:26,640 The person that started it all off, David Hilbert. 821 00:52:26,640 --> 00:52:32,400 Of the 23 problems Hilbert set mathematicians in the year 1900, 822 00:52:32,400 --> 00:52:34,880 most have now been solved. 823 00:52:34,880 --> 00:52:37,160 However there is one great exception. 824 00:52:37,160 --> 00:52:40,360 The Riemann hypothesis, the eighth on Hilbert's list. 825 00:52:40,360 --> 00:52:43,160 That is still the holy grail of mathematics. 826 00:52:44,960 --> 00:52:50,200 Hilbert's lecture inspired a generation to pursue their mathematical dreams. 827 00:52:50,200 --> 00:52:55,120 This morning, in the town where I grew up, I hope to inspire another generation. 828 00:52:55,120 --> 00:52:57,280 CHEERING AND APPLAUSE 829 00:53:01,680 --> 00:53:04,120 Thank you. Hello. My name's Marcus du Sautoy 830 00:53:04,120 --> 00:53:05,960 and I'm a Professor of Mathematics 831 00:53:05,960 --> 00:53:08,120 up the road at the University of Oxford. 832 00:53:08,120 --> 00:53:10,320 It was actually in this school here, 833 00:53:10,320 --> 00:53:14,520 in fact this classroom is where I discovered my love for mathematics. 834 00:53:14,520 --> 00:53:17,120 'This love of mathematics that I first acquired 835 00:53:17,120 --> 00:53:20,400 'here in my old comprehensive school still drives me now. 836 00:53:20,400 --> 00:53:22,280 'It's a love of solving problems. 837 00:53:22,280 --> 00:53:25,680 'There are so many problems I could tell them about, 838 00:53:25,680 --> 00:53:27,720 'but I've chosen my favourite.' 839 00:53:27,720 --> 00:53:30,840 I think that a mathematician is a pattern searcher 840 00:53:30,840 --> 00:53:33,960 and that's really what mathematicians try and do. 841 00:53:33,960 --> 00:53:37,080 We try and understand the patterns and the structure 842 00:53:37,080 --> 00:53:40,440 and the logic to explain the way the world around us works. 843 00:53:40,440 --> 00:53:43,480 And this is really at the heart of the Riemann hypothesis. 844 00:53:43,480 --> 00:53:48,360 The task is - is there any pattern in these numbers which can help me say 845 00:53:48,360 --> 00:53:50,440 where the next number will be? 846 00:53:50,440 --> 00:53:52,760 What's the next one after 31? How can I tell? 847 00:53:52,760 --> 00:53:55,760 'These numbers are, of course, prime numbers - 848 00:53:55,760 --> 00:53:58,200 'the building blocks of mathematics.' 849 00:53:58,200 --> 00:54:01,520 'The Riemann hypothesis, a conjecture about the distribution 850 00:54:01,520 --> 00:54:04,720 'of the primes, goes to the very heart of our subject.' 851 00:54:04,720 --> 00:54:07,560 Why on earth is anybody interested in these primes? 852 00:54:07,560 --> 00:54:11,040 Why is the army interested in primes, why are spies interested? 853 00:54:11,040 --> 00:54:14,800 - Isn't it to encrypt stuff? - Exactly. 854 00:54:14,800 --> 00:54:18,280 I study this stuff cos I think it's all really beautiful and elegant 855 00:54:18,280 --> 00:54:20,200 but actually, there's a lot of people 856 00:54:20,200 --> 00:54:24,440 who are interested in these numbers because of their very practical use. 857 00:54:24,440 --> 00:54:28,720 'The bizarre thing is that the more abstract and difficult mathematics becomes, 858 00:54:28,720 --> 00:54:32,480 'the more it seems to have applications in the real world. 859 00:54:32,480 --> 00:54:36,560 'Mathematics now pervades every aspect of our lives. 860 00:54:36,560 --> 00:54:41,560 'Every time we switch on the television, plug in a computer, pay with a credit card. 861 00:54:41,560 --> 00:54:46,160 'There's now a million dollars for anyone who can solve the Riemann hypothesis. 862 00:54:46,160 --> 00:54:48,600 'But there's more at stake than that.' 863 00:54:48,600 --> 00:54:51,800 Anybody who proves this theorem will be remembered forever. 864 00:54:51,800 --> 00:54:55,640 They'll be on that board ahead of any of those other mathematicians. 865 00:54:55,640 --> 00:54:59,600 'That's because the Riemann hypothesis is a corner-stone of maths. 866 00:54:59,600 --> 00:55:02,800 'Thousands of theorems depend on it being true. 867 00:55:02,800 --> 00:55:06,000 'Very few mathematicians think that it isn't true. 868 00:55:06,000 --> 00:55:10,640 'But mathematics is about proof and until we can prove it 869 00:55:10,640 --> 00:55:12,840 'there will still be doubt.' 870 00:55:12,840 --> 00:55:17,160 Maths has grown out of this passion to get rid of doubt. 871 00:55:17,160 --> 00:55:20,760 This is what I have learned in my journey through the history of mathematics. 872 00:55:20,760 --> 00:55:25,080 Mathematicians like Archimedes and al-Khwarizmi, Gauss and Grothendieck 873 00:55:25,080 --> 00:55:30,520 were driven to understand the precise way numbers and space work. 874 00:55:30,520 --> 00:55:33,200 Maths in action, that one. 875 00:55:33,200 --> 00:55:35,440 It's beautiful. Really nice. 876 00:55:35,440 --> 00:55:39,200 Using the language of mathematics, they have told us stories 877 00:55:39,200 --> 00:55:43,760 that remain as true today as they were when they were first told. 878 00:55:43,760 --> 00:55:48,760 In the Mediterranean, I discovered the origins of geometry. 879 00:55:48,760 --> 00:55:51,840 Mathematicians and philosophers flocked to Alexandria 880 00:55:51,840 --> 00:55:55,240 driven by a thirst for knowledge and the pursuit of excellence. 881 00:55:55,240 --> 00:55:59,080 In India, I learned about another discovery 882 00:55:59,080 --> 00:56:02,880 that it would be impossible to imagine modern life without. 883 00:56:02,880 --> 00:56:07,240 So here we are in one of the true holy sites of the mathematical world. 884 00:56:07,240 --> 00:56:10,080 Up here are some numbers, 885 00:56:10,080 --> 00:56:12,680 and here's the new number. 886 00:56:12,680 --> 00:56:14,320 Its zero. 887 00:56:14,320 --> 00:56:19,600 In the Middle East, I was amazed at al-Khwarizmi's invention of algebra. 888 00:56:19,600 --> 00:56:22,480 He developed systematic ways to analyse problems 889 00:56:22,480 --> 00:56:26,160 so that the solutions would work whatever numbers you took. 890 00:56:26,160 --> 00:56:28,080 In the Golden Age of Mathematics, 891 00:56:28,080 --> 00:56:31,600 in Europe in the 18th and 19th centuries, I found how maths 892 00:56:31,600 --> 00:56:35,760 discovered new ways for analysing bodies in motion and new geometries 893 00:56:35,760 --> 00:56:40,520 that helped us understand the very strange shape of space. 894 00:56:40,520 --> 00:56:43,840 It is with Riemann's work that we finally have 895 00:56:43,840 --> 00:56:49,280 the mathematical glasses to be able to explore such worlds of the mind. 896 00:56:49,280 --> 00:56:53,480 And now my journey into the abstract world of 20th-century mathematics 897 00:56:53,480 --> 00:56:56,600 has revealed that maths is the true language 898 00:56:56,600 --> 00:56:58,800 the universe is written in, 899 00:56:58,800 --> 00:57:02,120 the key to understanding the world around us. 900 00:57:02,120 --> 00:57:05,840 Mathematicians aren't motivated by money and material gain 901 00:57:05,840 --> 00:57:09,160 or even by practical applications of their work. 902 00:57:09,160 --> 00:57:13,400 For us, it is the glory of solving one of the great unsolved problems 903 00:57:13,400 --> 00:57:18,560 that have outwitted previous generations of mathematicians. 904 00:57:18,560 --> 00:57:21,920 Hilbert was right. It's the unsolved problems of mathematics 905 00:57:21,920 --> 00:57:23,720 that make it a living subject, 906 00:57:23,720 --> 00:57:27,160 which obsess each new generation of mathematicians. 907 00:57:27,160 --> 00:57:30,960 Despite all the things we've discovered over the last seven millennia, 908 00:57:30,960 --> 00:57:33,600 there are still many things we don't understand. 909 00:57:33,600 --> 00:57:39,960 And its Hilbert's call of, "We must know, we will know", which drives mathematics. 910 00:57:42,240 --> 00:57:45,440 You can learn more about The Story Of Maths 911 00:57:45,440 --> 00:57:48,480 with the Open University at... 912 00:58:00,600 --> 00:58:03,640 Subtitled by Red Bee Media Ltd 913 00:58:03,640 --> 00:58:06,680 E-mail subtitling@bbc.co.uk