resume his exploration of the infinite. Other mathematicians would be bothered by the paradoxes that Cantor's work had created. Curiously, this was

Transcription

1 The Story of Maths D Mathematics is about solving problems and it's the great unsolved problems that make maths really alive. In the summer of 1900, the International Congress of Mathematicians was held here in Paris in the Sorbonne. It was a pretty shambolic affair, not helped by the sultry August heat. But it will be remembered as one of the greatest congresses of all time thanks to a lecture given by the up-and-coming David Hilbert. Hilbert, a young German mathematician, boldly set out what he believed were the 23 most important problems for mathematicians to crack. He was trying to set the agenda for 20th-century maths and he succeeded. These Hilbert problems would define the mathematics of the modern age. Of those who tried to crack Hilbert's challenges, some would experience immense triumphs, whilst others would be plunged into infinite despair. The first problem on Hilbert's list emerged from here, Halle, in East Germany. It was where the great mathematician Georg Cantor spent all his adult life. And where he became the first person to really understand the meaning of infinity and give it mathematical precision. The statue in the town square, however, honours Halle's other famous son, the composer George Handel. To discover more about Cantor, I had to take a tram way out of town. For 50 years, Halle was part of Communist East Germany and the Communists loved celebrating their scientists. So much so, they put Cantor on the side of a large cube that they commissioned. But, being communists, they didn't put the cube in the middle of town. They put it out amongst the people. When I eventually found the estate, I started to fear that maybe I had got the location wrong. This looks the most unlikely venue for a statue to a mathematician. Excuse me? Ein Frage. Can you help me a minute? Wie bitte? Do you speak English? No! No? Ich suche ein Wurfel. Ein Wurfel, ja? Is that right? A "Wurfel"? A cube? Yeah? Like that? Mit ein Bild der Mathematiker? Yeah? Go round there? Die Name ist Cantor. Somewhere over here. Ah! There it is! It's much bigger than I thought. I thought it was going to be something like this sort of size. Aha, here we are. On the dark side of the cube. here's the man himself, Cantor. Cantor's one of my big heroes actually. I think if I had to choose some theorems that I wish I'd proved, I think the couple that Cantor proved would be up there in my top ten. 'This is because before Cantor, 'no-one had really understood infinity.' It was a tricky, slippery concept that didn't seem to go anywhere. But Cantor showed that infinity could be perfectly understandable. Indeed, there wasn't just one infinity, but infinitely many infinities. First Cantor took the numbers 1, 2, 3, 4 and so on. Then he thought about comparing them with a much smaller set... something like 10, 20, 30, What he showed is that these two infinite sets of numbers actually have the same size because we can pair them up - 1 with 10, 2 with 20, 3 with 30 and so on. So these are the same sizes of infinity. But what about the fractions? After all, there are infinitely many fractions between any two whole numbers. Surely the infinity of fractions is much bigger than the infinity of whole numbers. Well, what Cantor did was to find a way to pair up all of the whole numbers with an infinite load of fractions. And this is how he did it. He started by arranging all the fractions in an infinite grid. The first row contained the whole numbers, fractions with one on the bottom. In the second row came the halves, fractions with two on the bottom. And so on. Every fraction appears somewhere in this grid. Where's two thirds? Second column, third row. Now imagine a line snaking back and forward diagonally through the fractions. By pulling this line straight, we can match up every fraction with one of the whole numbers. This means the fractions are the same sort of infinity as the whole numbers. So perhaps all infinities have the same size. Well, here comes the really exciting bit because Cantor now considers the set of all infinite decimal numbers. And here he proves that they give us a bigger infinity because however you tried to list all the infinite decimals, Cantor produced a clever argument to show how to construct a new decimal number that was missing from your list. Suddenly, the idea of infinity opens up. There are different infinities, some bigger than others. It's a really exciting moment. For me, this is like the first humans understanding how to count. But now we're counting in a different way. We are counting infinities. A door has opened and an entirely new mathematics lay before us. But it never helped Cantor much. I was in the cemetery in Halle where he is buried and where I had arranged to meet Professor Joe Dauben. He was keen to make the connections between Cantor's maths and his life. He suffered from manic depression. One of the first big breakdowns he has is in 1884 but then around the turn of the century these recurrences of the mental illness become more and more frequent. A lot of people have tried to say that his mental illness was triggered by the incredible abstract mathematics he dealt with. Well, he was certainly struggling, so there may have been a connection. Yeah, I mean I must say, when you start to contemplate the infinite... I am pretty happy with the bottom end of the infinite, but as you build it up more and more, I must say I start to feel a bit unnerved about what's going on here and where is it going. For much of Cantor's life, the only place it was going was here - the university's sanatorium. There was no treatment then for manic depression or indeed for the paranoia that often accompanied Cantor's attacks. Yet the clinic was a good place to be - comfortable, quiet and peaceful. And Cantor often found his time here gave him the mental strength to 1

2 resume his exploration of the infinite. Other mathematicians would be bothered by the paradoxes that Cantor's work had created. Curiously, this was one thing Cantor was not worried by. He was never as upset about the paradox of the infinite as everybody else was because Cantor believed that there are certain things that I have been able to show, we can establish with complete mathematical certainty and then the absolute infinite which is only in God. He can understand all of this and there's still that final paradox that is not given to us to understand, but God does. But there was one problem that Cantor couldn't leave in the hands of the Almighty, a problem he wrestled with for the rest of his life. It became known as the continuum hypothesis. Is there an infinity sitting between the smaller infinity of all the whole numbers and the larger infinity of the decimals? Cantor's work didn't go down well with many of his contemporaries but there was one mathematician from France who spoke up for him, arguing that Cantor's new mathematics of infinity was "beautiful, if pathological". Fortunately this mathematician was the most famous and respected mathematician of his day. When Bertrand Russell was asked by a French politician who he thought the greatest man France had produced, he replied without hesitation, "Poincare". The politician was surprised that he'd chosen the prime minister Raymond Poincare above the likes of Napoleon, Balzac. Russell replied, "I don't mean Raymond Poincare but his cousin, "the mathematician, Henri Poincare. Henri Poincare spent most of his life in Paris, a city that he loved even with its uncertain climate. In the last decades of the 19th century, Paris was a centre for world mathematics and Poincare became its leading light. Algebra, geometry, analysis, he was good at everything. His work would lead to all kinds of applications, from finding your way around on the underground, to new ways of predicting the weather. Poincare was very strict about his working day. Two hours of work in the morning and two hours in the early evening. Between these periods, he would let his subconscious carry on working on the problem. He records one moment when he had a flash of inspiration which occurred almost out of nowhere, just as he was getting on a bus. And one such flash of inspiration led to an early success. In 1885, King Oscar II of Sweden and Norway offered a prize of 2,500 crowns for anyone who could establish mathematically once and for all whether the solar system would continue turning like clockwork, or might suddenly fly apart. If the solar system has two planets then Newton had already proved that their orbits would be stable. The two bodies just travel in ellipsis round each other. But as soon as soon as you add three bodies like the earth, moon and sun, the question of whether their orbits were stable or not stumped even the great Newton. The problem is that now you have some 18 different variables, like the exact coordinates of each body and their velocity in each direction. So the equations become very difficult to solve. But Poincare made significant headway in sorting them out. Poincare simplified the problem by making successive approximations to the orbits which he believed wouldn't affect the final outcome significantly. Although he couldn't solve the problem in its entirety, his ideas were sophisticated enough to win him the prize. He developed this great sort of arsenal of techniques, mathematical techniques in order to try and solve it and in fact, the prize that he won was essentially more for the techniques than for solving the problem. But when Poincare's paper was being prepared for publication by the King's scientific advisor, Mittag-Leffler, one of the editors found a problem. Poincare realised he'd made a mistake. Contrary to what he had originally thought, even a small change in the initial conditions could end up producing vastly different orbits. His simplification just didn't work. But the result was even more important. The orbits Poincare had discovered indirectly led to what we now know as chaos theory. Understanding the mathematical rules of chaos explain why a butterfly's wings could create tiny changes in the atmosphere that ultimately might cause a tornado or a hurricane to appear on the other side of the world. So this big subject of the 20th century, chaos, actually came out of a mistake that Poincare made and he spotted at the last minute. Yes! So the essay had actually been published in its original form, and was ready to go out and Mittag- Leffler had sent copies out to various people, and it was to his horror when Poincare wrote to him to say, "Stop!" Oh, my God. This is every mathematician's worst nightmare. Absolutely. "Pull it!" Hold the presses! Owning up to his mistake, if anything, enhanced Poincare's reputation. He continued to produce a wide range of original work throughout his life. Not just specialist stuff either. He also wrote popular books, extolling the importance of maths. Here we go. Here's a section on the future of mathematics. It starts, "If we wish to foresee the future of mathematics, "our proper course is to study the history and present the condition of the science." So, I think Poincare might have approved of my journey to uncover the story of maths. He certainly would have approved of the next destination. Because to discover perhaps Poincare's most important contribution to modern mathematics, I had to go looking for a bridge. Seven bridges in fact. The Seven bridges of Konigsberg. Today the city is known as Kaliningrad, a little outpost of Russia on the Baltic Sea surrounded by Poland and Lithuania. Until 1945, however, when it was ceded to the Soviet Union, it was the great Prussian City of Konigsberg. Much of the old town sadly has been demolished. There is now no sign at all of two of the original seven bridges and several have changed out of all recognition. This is one of the original bridges. It may seem like an unlikely setting for the beginning of a mathematical story, but bear with me. It started as an 18th-century puzzle. Is there a route around the city which crosses each of these seven bridges only once? Finding the solution is much more difficult than it looks. It was eventually solved by the great mathematician Leonhard Euler, who in 1735 proved that it wasn't possible. There could not be a route that didn't cross at least one bridge twice. 2

3 He solved the problem by making a conceptual leap. He realised, you don't really care what the distances are between the bridges. What really matters is how the bridges are connected together. This is a problem of a new sort of geometry of position - a problem of topology. Many of us use topology every day. Virtually all metro maps the world over are drawn on topological principles. You don't care how far the stations are from each other but how they are connected. There isn't a metro in Kaliningrad, but there is in the nearest other Russian city, St Petersburg. The topology is pretty easy on this map. It's the Russian I am having difficulty with. Can you tell me which...? What's the problem? I want to know what station this one was. I had it the wrong way round even! Although topology had its origins in the bridges of Konigsberg, it was in the hands of Poincare that the subject evolved into a powerful new way of looking at shape. Some people refer to topology as bendy geometry because in topology, two shapes are the same if you can bend or morph one into another without cutting it. So for example if I take a football or rugby ball, topologically they are the same because one can be morphed into the other. Similarly a bagel and a teacup are the same because one can be morphed into the other. Even very complicated shapes can be unwrapped to become much simpler from a topological point of view. But there is no way to continuously deform a bagel to morph it into a ball. The hole in the middle makes these shapes topologically different. Poincare knew all the possible twodimensional topological surfaces. But in 1904 he came up with a topological problem he just couldn't solve. If you've got a flat two-dimensional universe then Poincare worked out all the possible shapes he could wrap that universe up into. It could be a ball or a bagel with one hole, two holes or more holes in. But we live in a three-dimensional universe so what are the possible shapes that our universe can be? That question became known as the Poincare Conjecture. It was finally solved in 2002 here in St Petersburg by a Russian mathematician called Grisha Perelman. His proof is very difficult to understand, even for mathematicians. Perelman solved the problem by linking it to a completely different area of mathematics. To understand the shapes, he looked instead at the dynamics of the way things can flow over the shape which led to a description of all the possible ways that three dimensional space can be wrapped up in higher dimensions. I wondered if the man himself could help in unravelling the intricacies of his proof, but I'd been told that finding Perelman is almost as difficult as understanding the solution. The classic stereotype of a mathematician is a mad eccentric scientist, but I think that's a little bit unfair. Most of my colleagues are normal. Well, reasonably. But when it comes to Perelman, there is no doubt he is a very strange character. He's received prizes and offers of professorships from distinguished universities in the West but he's turned them all down. Recently he seems to have given up mathematics completely and retreated to live as a semi-recluse in this very modest housing estate with his mum. He refuses to talk to the media but I thought he might just talk to me as a fellow mathematician. I was wrong. Well, it's interesting. I think he's actually turned off his buzzer. Probably too many media have been buzzing it. I tried a neighbour and that rang but his doesn't ring at all. I think his papers, his mathematics speaks for itself in a way. I don't really need to meet the mathematician and in this age of Big Brother and Big Money, I think there's something noble about the fact he gets his kick out of proving theorems and not winning prizes. One mathematician would certainly have applauded. For solving any of his 23 problems, David Hilbert offered no prize or reward beyond the admiration of other mathematicians. When he sketched out the problems in Paris in 1900, Hilbert himself was already a mathematical star. And it was in Gottingen in northern Germany that he really shone. He was by far the most charismatic mathematician of his age. It's clear that everyone who knew him thought he was absolutely wonderful. He studied number theory and brought everything together that was there and then within a year or so he left that completely and revolutionised the theory of integral equation. It's always change and always something new, and there's hardly anybody who is... who was so flexible and so varied in his approach as Hilbert was. His work is still talked about today and his name has become attached to many mathematical terms. Mathematicians still use the Hilbert Space, the Hilbert Classification, the Hilbert Inequality and several Hilbert theorems. But it was his early work on equations that marked him out as a mathematician thinking in new ways. Hilbert showed that although there are infinitely many equations, there are ways to divide them up so that they are built out of just a finite set, like a set of building blocks. The most striking element of Hilbert's proof was that he couldn't actually construct this finite set. He just proved it must exist. Somebody criticised this as theology and not mathematics but they'd missed the point. What Hilbert was doing here was creating a new style of mathematics, a more abstract approach to the subject. You could still prove something existed, even though you couldn't construct it explicitly. It's like saying, "I know there has to be a way to get "from Gottingen to St Petersburg even though I can't tell you "how to actually get there." As well as challenging mathematical orthodoxies, Hilbert was also happy to knock the formal hierarchies that existed in the university system in Germany at the time. Other professors were quite shocked to see Hilbert out bicycling and drinking with his students. He liked very much parties. Yeah? Yes. Party animal. That's my kind of mathematician. He liked very much dancing with young women. He liked very much to flirt. Really? Most mathematicians I know are not the biggest of flirts. 'Yet this lifestyle went hand in hand with an absolute commitment to maths.' Hilbert was of course somebody who thought that everybody who has a mathematical skill, a penguin, a woman, a man, or black, white or yellow, it doesn't matter, he should do mathematics and he should be admired for his. The mathematics speaks for itself in a way. It doesn't matter... When you're a penguin. Yeah, if you can prove the Riemann hypothesis, we really don't mind. Yes, mathematics was 3

4 for him a universal language. Yes. Hilbert believed that this language was powerful enough to unlock all the truths of mathematics, a belief he expounded in a radio interview he gave on the future of mathematics on the 8th September In it, he had no doubt that all his 23 problems would soon be solved and that mathematics would finally be put on unshakeable logical foundations. There are absolutely no unsolvable problems, he declared, a belief that's been held by mathematicians since the time of the Ancient Greeks. He ended with this clarion call, "We must know, we will know." 'Wir mussen wissen, wir werden wissen.' Unfortunately for him, the very day before in a scientific lecture that was not considered worthy of broadcast, another mathematician would shatter Hilbert's dream and put uncertainty at the heart of mathematics. The man responsible for destroying Hilbert's belief was an Austrian mathematician, Kurt Godel. And it all started here - Vienna. Even his admirers, and there are a great many, admit that Kurt Godel was a little odd. As a child, he was bright, sickly and very strange. He couldn't stop asking questions. So much so, that his family called him Herr Warum - Mr Why. Godel lived in Vienna in the 1920s and 1930s, between the fall of the Austro-Hungarian Empire and its annexation by the Nazis. It was a strange, chaotic and exciting time to be in the city. Godel studied mathematics at Vienna University but he spent most of his time in the cafes, the internet chat rooms of their time, where amongst games of backgammon and billiards, the real intellectual excitement was taking place. Particularly amongst a highly influential group of philosophers and scientists called the Vienna Circle. In their discussions, Kurt Godel would come up with an idea that would revolutionise mathematics. He'd set himself a difficult mathematical test. He wanted to solve Hilbert's second problem and find a logical foundation for all mathematics. But what he came up with surprised even him. All his efforts in mathematical logic not only couldn't provide the guarantee Hilbert wanted, instead he proved the opposite. Got it. It's called the Incompleteness Theorem. Godel proved that within any logical system for mathematics there will be statements about numbers which are true but which you cannot prove. He starts with the statement, "This statement cannot be proved." This is not a mathematical statement yet. But by using a clever code based on prime numbers, Godel could change this statement into a pure statement of arithmetic. Now, such statements must be either true or false. Hold on to your logical hats as we explore the possibilities. If the statement is false, that means the statement could be proved, which means it would be true, and that's a contradiction. So that means, the statement must be true. In other words, here is a mathematical statement that is true but can't be proved. Blast. Godel's proof led to a crisis in mathematics. What if the problem you were working on, the Goldbach conjecture, say, or the Riemann hypothesis, would turn out to be true but unprovable? It led to a crisis for Godel too. In the autumn of 1934, he suffered the first of what became a series of breakdowns and spent time in a sanatorium. He was saved by the love of a good woman. Adele Nimbursky was a dancer in a local night club. She kept Godel alive. One day, she and Godel were walking down these very steps. Suddenly he was attacked by Nazi thugs. Godel himself wasn't Jewish, but many of his friends in the Vienna Circle were. Adele came to his rescue. But it was only a temporary reprieve for Godel and for maths. All across Austria and Germany, mathematics was about to die. In the new German empire in the late 1930s there weren't colourful balloons flying over the universities, but swastikas. The Nazis passed a law allowing the removal of any civil servant who wasn't Aryan. Academics were civil servants in Germany then and now. Mathematicians suffered more than most. 144 in Germany would lose their jobs. 14 were driven to suicide or died in concentration camps. But one brilliant mathematician stayed on. David Hilbert helped arrange for some of his brightest students to flee. And he spoke out for a while about the dismissal of his Jewish colleagues. But soon, he too became silent. It's not clear why he didn't flee himself or at least protest a little more. He did fall ill towards the end of his life so maybe he just didn't have the energy. All around him, mathematicians and scientists were fleeing the Nazi regime until it was only Hilbert left to witness the destruction of one of the greatest mathematical centres of all time. David Hilbert died in Only ten people attended the funeral of the most famous mathematician of his time. The dominance of Europe, the centre for world maths for 500 years, was over. It was time for the mathematical baton to be handed to the New World. Time in fact for this place. The Institute for Advanced Study had been set up in Princeton in The idea was to reproduce the collegiate atmosphere of the old European universities in rural New Jersey. But to do this, it needed to attract the very best, and it didn't need to look far. Many of the brightest European mathematicians were fleeing the Nazis for America. People like Hermann Weyl, whose research would have major significance for theoretical physics. And John Von Neumann, who developed game theory and was one of the pioneers of computer science. The Institute quickly became the perfect place to create another Gottingen in the woods. One mathematician in particular made the place a home from home. Every morning Kurt Godel, dressed in a white linen suit and wearing a fedora, would walk from his home along Mercer Street to the Institute. On his way, he would stop here at number 112, to pick up his closest friend, another European exile, Albert Einstein. But not even relaxed, affluent Princeton could help Godel finally escape his demons. Einstein was always full of laughter. He described Princeton as a banishment to paradise. But the much younger Godel became increasingly solemn and pessimistic. Slowly this pessimism turned into paranoia. He spent less and less time with his fellow mathematicians in Princeton. Instead, he preferred to come here to the beach, next to the ocean, walk alone, thinking about the works of the great German mathematician, Leibniz. But as Godel was withdrawing into his own interior world, his influence on American mathematics paradoxically was growing stronger and stronger. And a young mathematician from just along the New Jersey coast eagerly took on some of the challenges he posed. 4

5 # One, two, three, four, five, six, seven, eight, nine, ten # Times a day I could love you... # In 1950s America, the majority of youngsters weren't preoccupied with mathematics. Most went for a more relaxed, hedonistic lifestyle in this newly affluent land of ice-cream and doughnuts. But one teenager didn't indulge in the normal pursuits of American adolescence but chose instead to grapple with some of the major problems in mathematics. From a very early age, Paul Cohen was winning mathematical competitions and prizes. But he found it difficult at first to discover a field in mathematics where he could really make his mark... Until he read about Cantor's continuum hypothesis. That's the one problem, as I had learned back in Halle, that Cantor just couldn't resolve. It asks whether there is or there isn't an infinite set of numbers bigger than the set of all whole numbers but smaller than the set of all decimals. It sounds straightforward, but it had foiled all attempts to solve it since Hilbert made it his first problem way back in With the arrogance of youth, the 22-year-old Paul Cohen decided that he could do it. Cohen came back a year later with the extraordinary discovery that both answers could be true. There was one mathematics where the continuum hypothesis could be assumed to be true. There wasn't a set between the whole numbers and the infinite decimals. But there was an equally consistent mathematics where the continuum hypothesis could be assumed to be false. Here, there was a set between the whole numbers and the infinite decimals. It was an incredibly daring solution. Cohen's proof seemed true, but his method was so new that nobody was absolutely sure. There was only one person whose opinion everybody trusted. There was a lot of scepticism and he had to come and make a trip here, to the Institute right here, to visit Godel. And it was only after Godel gave his stamp of approval in quite an unusual way, He said, "Give me your paper", and then on Monday he put it back in the box and said, "Yes, it's correct." Then everything changed. Today mathematicians insert a statement that says whether the result depends on the continuum hypothesis. We've built up two different mathematical worlds in which one answer is yes and the other is no. Paul Cohen really has rocked the mathematical universe. It gave him fame, riches, and prizes galore. He didn't publish much after his early success in the '60s. But he was absolutely dynamite. I can't imagine anyone better to learn from, and he was very eager to learn, to teach you anything he knew or even things he didn't know. With the confidence that came from solving Hilbert's first problem, Cohen settled down in the mid 1960s to have a go at the most important Hilbert problem of them all - the eighth, the Riemann hypothesis. When he died in California in 2007, 40 years later, he was still trying. But like many famous mathematicians before him, Riemann had defeated even him. But his approach has inspired others to make progress towards a proof, including one of his most famous students, Peter Sarnak. I think, overall, absolutely loved the guy. He was my inspiration. I'm really glad I worked with him. He put me on the right track. Paul Cohen is a good example of the success of the great American Dream. The second generation Jewish immigrant becomes top American professor. But I wouldn't say that his mathematics was a particularly American product. Cohen was so fired up by this problem that he probably would have cracked it whatever the surroundings. Paul Cohen had it relatively easy. But another great American mathematician of the 1960s faced a much tougher struggle to make an impact. Not least because she was female. In the story of maths, nearly all the truly great mathematicians have been men. But there have been a few exceptions. There was the Russian Sofia Kovalevskaya who became the first female professor of mathematics in Stockholm in 1889, and won a very prestigious French mathematical prize. And then Emmy Noether, a talented algebraist who fled from the Nazis to America but then died before she fully realised her potential. Then there is the woman who I am crossing America to find out about. Julia Robinson, the first woman ever to be elected president of the American Mathematical Society. She was born in St Louis in 1919, but her mother died when she was two. She and her sister Constance moved to live with their grandmother in a small community in the desert near Phoenix, Arizona. Julia Robinson grew up around here. I've got a photo which shows her cottage in the 1930s, with nothing much around it. The mountains pretty much match those over there so I think she might have lived down there. Julia grew up a shy, sickly girl, who, when she was seven, spent a year in bed because of scarlet fever. Ill-health persisted throughout her childhood. She was told she wouldn't live past 40. But right from the start, she had an innate mathematical ability. Under the shade of the native Arizona cactus, she whiled away the time playing endless counting games with stone pebbles. This early searching for patterns would give her a feel and love of numbers that would last for the rest of her life. But despite showing an early brilliance, she had to continually fight at school and college to simply be allowed to keep doing maths. As a teenager, she was the only girl in the maths class but had very little encouragement. The young Julia sought intellectual stimulation elsewhere. Julia loved listening to a radio show called the University Explorer and the 53rd episode was all about mathematics. The broadcaster described how he discovered despite their esoteric language and their seclusive nature, mathematicians are the most interesting and inspiring creatures. For the first time, Julia had found out that there were mathematicians, not just mathematics teachers. There was a world of mathematics out there, and she wanted to be part of it. The doors to that world opened here at the University of California, at Berkeley near San Francisco. She was absolutely obsessed that she wanted to go to Berkeley. She wanted to go away to some place where there were mathematicians. Berkeley certainly had mathematicians, including a number theorist called Raphael Robinson. In their frequent walks around the campus they found they had more than just a passion for mathematics. They married in Julia got her PhD and settled down to what would turn into her lifetime's work - Hilbert's tenth problem. 5

6 She thought about it all the time. She said to me she just wouldn't wanna die without knowing that answer and it had become an obsession. Julia's obsession has been shared with many other mathematicians since Hilbert had first posed it back in His tenth problem asked if there was some universal method that could tell whether any equation had whole number solutions or not. Nobody had been able to solve it. In fact, the growing belief was that no such universal method was possible. How on earth could you prove that, however ingenious you were, you'd never come up with a method? With the help of colleagues, Julia developed what became known as the Robinson hypothesis. This argued that to show no such method existed, all you had to do was to cook up one equation whose solutions were a very specific set of numbers. The set of numbers needed to grow exponentially, like taking powers of two, yet still be captured by the equations at the heart of Hilbert's problem. Try as she might, Robinson just couldn't find this set. For the tenth problem to be finally solved, there needed to be some fresh inspiration. That came from 5,000 miles away - St Petersburg in Russia. Ever since the great Leonhard Euler set up shop here in the 18th century, the city has been famous for its mathematics and mathematicians. Here in the Steklov Institute, some of the world's brightest mathematicians have set out their theorems and conjectures. This morning, one of them is giving a rare seminar. It's tough going even if you speak Russian, which unfortunately I don't. But we do get a break in the middle to recover before the final hour. There is a kind of rule in seminars. The first third is for everyone, the second third for the experts and the last third is just for the lecturer. I think that's what we're going to get next. The lecturer is Yuri Matiyasevich and he is explaining his latest work on - what else? - the Riemann hypothesis. As a bright young graduate student in 1965, Yuri's tutor suggested he have a go at another Hilbert problem, the one that had in fact preoccupied Julia Robinson. Hilbert's tenth. It was the height of the Cold War. Perhaps Matiyasevich could succeed for Russia where Julia and her fellow American mathematicians had failed. At first I did not like their approach. Oh, right. The statement looked to me rather strange and artificial but after some time I understood it was quite natural, and then I understood that she had a new brilliant idea and I just started to further develop it. In January 1970, he found the vital last piece in the jigsaw. He saw how to capture the famous Fibonacci sequence of numbers using the equations that were at the heart of Hilbert's problem. Building on the work of Julia and her colleagues, he had solved the tenth. He was just 22 years old. The first person he wanted to tell was the woman he owed so much to. I got no answer and I believed they were lost in the mail. It was quite natural because it was Soviet time. But back in California, Julia had heard rumours through the mathematical grapevine that the problem had been solved. And she contacted Yuri herself. She said, I just had to wait for you to grow up to get the answer, because she had started work in When Yuri was just a baby. Then he responds by thanking her and saying that the credit is as much hers as it is his. YURI: I met Julia one year later. It was in Bucharest. I suggested after the conference in Bucharest Julia and her husband Raphael came to see me here in Leningrad. Together, Julia and Yuri worked on several other mathematical problems until shortly before Julia died in She was just 55 years old. She was able to find the new ways. Many mathematicians just combine previous known methods to solve new problems and she had really new ideas. Although Julia Robinson showed there was no universal method to solve all equations in whole numbers, mathematicians were still interested in finding methods to solve special classes of equations. It would be in France in the early 19th century, in one of the most extraordinary stories in the history of mathematics, that methods were developed to understand why certain equations could be solved while others couldn't. It's early morning in Paris on the 29th May Evariste Galois is about to fight for his very life. It is the reign of the reactionary Bourbon King, Charles X, and Galois, like many angry young men in Paris then, is a republican revolutionary. Unlike the rest of his comrades though, he has another passion - mathematics. He had just spent four months in jail. Then, in a mysterious saga of unrequited love, he is challenged to a duel. He'd been up the whole previous night refining a new language for mathematics he'd developed. Galois believed that mathematics shouldn't be the study of number and shape, but the study of structure. Perhaps he was still pre-occupied with his maths. GUNSHOT There was only one shot fired that morning. Galois died the next day, just 20 years old. It was one of mathematics greatest losses. Only by the beginning of the 20th century would Galois be fully appreciated and his ideas fully realised. Galois had discovered new techniques to be able to tell whether certain equations could have solutions or not. The symmetry of certain geometric objects seemed to be the key. His idea of using geometry to analyse equations would be picked up in the 1920s by another Parisian mathematician, Andre Weil. I was very much interested and so far as school was concerned quite successful in all possible branches. And he was. After studying in Germany as well as France, Andre settled down at this apartment in Paris which he shared with his more-famous sister, the writer Simone Weil. But when the Second World War broke out, he found himself in very different circumstances. He dodged the draft by fleeing to Finland where he was almost executed for being a Russian spy. On his return to France he was put in prison in Rouen to await trial for desertion. At the trial, the judge gave him a choice. Five more years in prison or serve in a combat unit. He chose to join the French army, a lucky choice because just before the Germans invaded a few months later, all the prisoners were killed. Weil only spent a few months in prison, but this time was crucial for his mathematics. Because here he built on the ideas of Galois and first developed algebraic geometry a whole new language for understanding solutions to equations. Galois had shown how new mathematical structures can be used to reveal the secrets behind equations. Weil's work led him to theorems that connected number theory, algebra, geometry and topology and are one of the greatest achievements of modern 6

7 mathematics. Without Andre Weil, we would never have heard of the strangest man in the history of maths, Nicolas Bourbaki. There are no photos of Bourbaki in existence but we do know he was born in this cafe in the Latin Quarter in 1934 when it was a proper cafe, the cafe Capoulade, and not the fast food joint it has now become. Just down the road, I met up with Bourbaki expert David Aubin. When I was a graduate student I got quite frightened when I used to go into the library because this guy Bourbaki had written so many books. Something like 30 or 40 altogether. In analysis, in geometry, in topology, it was all new foundations. Virtually everyone studying Maths seriously anywhere in the world in the 1950s, '60s and '70s would have read Nicolas Bourbaki. He applied for membership of the American Maths Society, I heard. At which point he was denied membership on the grounds that he didn't exist. Oh! The Americans were right. Nicolas Bourbaki does not exist at all. And never has. Bourbaki is in fact the nom de plume for a group of French mathematicians led by Andre Weil who decided to write a coherent account of the mathematics of the 20th century. Most of the time mathematicians like to have their own names on theorems. But for the Bourbaki group, the aims of the project overrode any desire for personal glory. After the Second World War, the Bourbaki baton was handed down to the next generation of French mathematicians. And their most brilliant member was Alexandre Grothendieck. Here at the IHES in Paris, the French equivalent of Princeton's Institute for Advanced Study, Grothendieck held court at his famous seminars in the 1950s and 1960s. He had this incredible charisma. He had this amazing ability to see a young person and somehow know what kind of contribution this person could make to this incredible vision he had of how mathematics could be. And this vision enabled him to get across some very difficult ideas indeed. He says, "Suppose you want to open a walnut. "So the standard thing is you take a nutcracker and you just break it open." And he says his approach is more like, you take this walnut and you put it out in the snow and you leave it there for a few months and then when you come back to it, it just opens. Grothendieck is a Structuralist. What he's interested in are the hidden structures underneath all mathematics. Only when you get down to the very basic architecture and think in very general terms will the patterns in mathematics become clear. Grothendieck produced a new powerful language to see structures in a new way. It was like living in a world of black and white and suddenly having the language to see the world in colour. It's a language that mathematicians have been using ever since to solve problems in number theory, geometry, even fundamental physics. But in the late 1960s, Grothendieck decided to turn his back on mathematics after he discovered politics. He believed that the threat of nuclear war and the questions of nuclear disarmament were more important than mathematics and that people who continue to do mathematics rather than confront this threat of nuclear war were doing harm in the world. Grothendieck decided to leave Paris and move back to the south of France where he grew up. Bursts of radical politics followed and then a nervous breakdown. He moved to the Pyrenees and became a recluse. He's now lost all contact with his old friends and mathematical colleagues. Nevertheless, the legacy of his achievements means that Grothendieck stands alongside Cantor, Godel and Hilbert as someone who has transformed the mathematical landscape. He changed the whole subject in a really fundamental way. It will never go back. Certainly, he's THE dominant figure of the 20th century. I've come back to England, though, thinking again about another seminal figure of the 20th century. The person that started it all off, David Hilbert. Of the 23 problems Hilbert set mathematicians in the year 1900, most have now been solved. However there is one great exception. The Riemann hypothesis, the eighth on Hilbert's list. That is still the holy grail of mathematics. Hilbert's lecture inspired a generation to pursue their mathematical dreams. This morning, in the town where I grew up, I hope to inspire another generation. CHEERING AND APPLAUSE Thank you. Hello. My name's Marcus du Sautoy and I'm a Professor of Mathematics up the road at the University of Oxford. It was actually in this school here, in fact this classroom is where I discovered my love for mathematics. 'This love of mathematics that I first acquired 'here in my old comprehensive school still drives me now. 'It's a love of solving problems. 'There are so many problems I could tell them about, 'but I've chosen my favourite.' I think that a mathematician is a pattern searcher and that's really what mathematicians try and do. We try and understand the patterns and the structure and the logic to explain the way the world around us works. And this is really at the heart of the Riemann hypothesis. The task is - is there any pattern in these numbers which can help me say where the next number will be? What's the next one after 31? How can I tell? 'These numbers are, of course, prime numbers - 'the building blocks of mathematics.' 'The Riemann hypothesis, a conjecture about the distribution 'of the primes, goes to the very heart of our subject.' Why on earth is anybody interested in these primes? Why is the army interested in primes, why are spies interested? Isn't it to encrypt stuff? Exactly. I study this stuff cos I think it's all really beautiful and elegant but actually, there's a lot of people who are interested in these numbers because of their very practical use. 'The bizarre thing is that the more abstract and difficult mathematics becomes, 'the more it seems to have applications in the real world. 'Mathematics now pervades every aspect of our lives. 'Every time we switch on the television, plug in a computer, pay with a credit card. 'There's now a million dollars for anyone who can solve the Riemann hypothesis. 'But there's more at stake than that.' Anybody who proves this theorem will be remembered forever. They'll be on that board ahead of any of those other mathematicians. 'That's because the Riemann hypothesis is a corner-stone of maths. 'Thousands of theorems depend on it being true. 'Very few mathematicians think that it isn't true. 'But mathematics is 7

8 about proof and until we can prove it 'there will still be doubt.' Maths has grown out of this passion to get rid of doubt. This is what I have learned in my journey through the history of mathematics. Mathematicians like Archimedes and al- Khwarizmi, Gauss and Grothendieck were driven to understand the precise way numbers and space work. Maths in action, that one. It's beautiful. Really nice. Using the language of mathematics, they have told us stories that remain as true today as they were when they were first told. In the Mediterranean, I discovered the origins of geometry. Mathematicians and philosophers flocked to Alexandria driven by a thirst for knowledge and the pursuit of excellence. In India, I learned about another discovery that it would be impossible to imagine modern life without. So here we are in one of the true holy sites of the mathematical world. Up here are some numbers, and here's the new number. Its zero. In the Middle East, I was amazed at al-khwarizmi's invention of algebra. He developed systematic ways to analyse problems so that the solutions would work whatever numbers you took. In the Golden Age of Mathematics, in Europe in the 18th and 19th centuries, I found how maths discovered new ways for analysing bodies in motion and new geometries that helped us understand the very strange shape of space. It is with Riemann's work that we finally have the mathematical glasses to be able to explore such worlds of the mind. And now my journey into the abstract world of 20th-century mathematics has revealed that maths is the true language the universe is written in, the key to understanding the world around us. Mathematicians aren't motivated by money and material gain or even by practical applications of their work. For us, it is the glory of solving one of the great unsolved problems that have outwitted previous generations of mathematicians. Hilbert was right. It's the unsolved problems of mathematics that make it a living subject, which obsess each new generation of mathematicians. Despite all the things we've discovered over the last seven millennia, there are still many things we don't understand. And its Hilbert's call of, "We must know, we will know", which drives mathematics. You can learn more about The Story Of Maths with the Open University at... Subtitled by Red Bee Media Ltd subtitling@bbc.co.uk 8