0:00:17 > 0:00:21Mathematics is about solving problems
0:00:21 > 0:00:26and it's the great unsolved problems that make maths really alive.
0:00:28 > 0:00:29In the summer of 1900,
0:00:29 > 0:00:32the International Congress of Mathematicians
0:00:32 > 0:00:34was held here in Paris in the Sorbonne.
0:00:34 > 0:00:36It was a pretty shambolic affair,
0:00:36 > 0:00:39not helped by the sultry August heat.
0:00:39 > 0:00:43But it will be remembered as one of the greatest congresses of all time
0:00:43 > 0:00:47thanks to a lecture given by the up-and-coming David Hilbert.
0:00:48 > 0:00:51Hilbert, a young German mathematician,
0:00:51 > 0:00:56boldly set out what he believed were the 23 most important problems
0:00:56 > 0:00:58for mathematicians to crack.
0:00:58 > 0:01:04He was trying to set the agenda for 20th-century maths and he succeeded.
0:01:04 > 0:01:09These Hilbert problems would define the mathematics of the modern age.
0:01:09 > 0:01:15Of those who tried to crack Hilbert's challenges, some would experience immense triumphs,
0:01:15 > 0:01:18whilst others would be plunged into infinite despair.
0:01:30 > 0:01:33The first problem on Hilbert's list emerged from here,
0:01:33 > 0:01:36Halle, in East Germany.
0:01:36 > 0:01:41It was where the great mathematician Georg Cantor spent all his adult life.
0:01:41 > 0:01:45And where he became the first person to really understand the meaning
0:01:45 > 0:01:50of infinity and give it mathematical precision.
0:01:50 > 0:01:52The statue in the town square, however,
0:01:52 > 0:01:57honours Halle's other famous son, the composer George Handel.
0:01:57 > 0:02:03To discover more about Cantor, I had to take a tram way out of town.
0:02:03 > 0:02:07For 50 years, Halle was part of Communist East Germany
0:02:07 > 0:02:10and the Communists loved celebrating their scientists.
0:02:10 > 0:02:15So much so, they put Cantor on the side of a large cube that they commissioned.
0:02:15 > 0:02:17But, being communists, they didn't put the cube
0:02:17 > 0:02:20in the middle of town. They put it out amongst the people.
0:02:24 > 0:02:27When I eventually found the estate, I started to fear
0:02:27 > 0:02:31that maybe I had got the location wrong.
0:02:34 > 0:02:38This looks the most unlikely venue for a statue to a mathematician.
0:02:39 > 0:02:41Excuse me?
0:02:42 > 0:02:43Ein Frage.
0:02:43 > 0:02:47- Can you help me a minute?- Wie bitte? - Do you speak English?- No!- No?
0:02:47 > 0:02:49Ich suche ein Wurfel.
0:02:49 > 0:02:51Ein Wurfel, ja?
0:02:51 > 0:02:52Is that right? A "Wurfel"?
0:02:52 > 0:02:55A cube? Yeah? Like that?
0:02:55 > 0:02:58Mit ein Bild der Mathematiker?
0:02:58 > 0:03:01Yeah? Go round there?
0:03:01 > 0:03:02Die Name ist Cantor.
0:03:02 > 0:03:04Somewhere over here. Ah! There it is!
0:03:04 > 0:03:06It's much bigger than I thought.
0:03:06 > 0:03:09I thought it was going to be something like this sort of size.
0:03:09 > 0:03:13Aha, here we are. On the dark side of the cube.
0:03:13 > 0:03:16here's the man himself, Cantor.
0:03:16 > 0:03:18Cantor's one of my big heroes actually.
0:03:18 > 0:03:23I think if I had to choose some theorems that I wish I'd proved,
0:03:23 > 0:03:25I think the couple that Cantor proved
0:03:25 > 0:03:27would be up there in my top ten.
0:03:27 > 0:03:30'This is because before Cantor,
0:03:30 > 0:03:33'no-one had really understood infinity.'
0:03:33 > 0:03:38It was a tricky, slippery concept that didn't seem to go anywhere.
0:03:38 > 0:03:42But Cantor showed that infinity could be perfectly understandable.
0:03:42 > 0:03:45Indeed, there wasn't just one infinity,
0:03:45 > 0:03:48but infinitely many infinities.
0:03:48 > 0:03:54First Cantor took the numbers 1, 2, 3, 4 and so on.
0:03:54 > 0:03:58Then he thought about comparing them with a much smaller set...
0:03:58 > 0:04:02something like 10, 20, 30, 40...
0:04:02 > 0:04:06What he showed is that these two infinite sets of numbers
0:04:06 > 0:04:10actually have the same size because we can pair them up -
0:04:10 > 0:04:141 with 10, 2 with 20, 3 with 30 and so on.
0:04:14 > 0:04:17So these are the same sizes of infinity.
0:04:20 > 0:04:22But what about the fractions?
0:04:22 > 0:04:27After all, there are infinitely many fractions between any two whole numbers.
0:04:27 > 0:04:30Surely the infinity of fractions is much bigger
0:04:30 > 0:04:33than the infinity of whole numbers.
0:04:38 > 0:04:41Well, what Cantor did was to find a way to pair up
0:04:41 > 0:04:45all of the whole numbers with an infinite load of fractions.
0:04:45 > 0:04:47And this is how he did it.
0:04:47 > 0:04:52He started by arranging all the fractions in an infinite grid.
0:04:52 > 0:04:57The first row contained the whole numbers, fractions with one on the bottom.
0:04:57 > 0:05:01In the second row came the halves, fractions with two on the bottom.
0:05:01 > 0:05:06And so on. Every fraction appears somewhere in this grid.
0:05:06 > 0:05:10Where's two thirds? Second column, third row.
0:05:10 > 0:05:15Now imagine a line snaking back and forward diagonally through the fractions.
0:05:18 > 0:05:24By pulling this line straight, we can match up every fraction with one of the whole numbers.
0:05:24 > 0:05:29This means the fractions are the same sort of infinity
0:05:29 > 0:05:31as the whole numbers.
0:05:31 > 0:05:34So perhaps all infinities have the same size.
0:05:34 > 0:05:36Well, here comes the really exciting bit
0:05:36 > 0:05:41because Cantor now considers the set of all infinite decimal numbers.
0:05:41 > 0:05:45And here he proves that they give us a bigger infinity because
0:05:45 > 0:05:49however you tried to list all the infinite decimals, Cantor produced
0:05:49 > 0:05:52a clever argument to show how to construct a new decimal number
0:05:52 > 0:05:54that was missing from your list.
0:05:54 > 0:05:58Suddenly, the idea of infinity opens up.
0:05:58 > 0:06:01There are different infinities, some bigger than others.
0:06:01 > 0:06:03It's a really exciting moment.
0:06:03 > 0:06:07For me, this is like the first humans understanding how to count.
0:06:07 > 0:06:12But now we're counting in a different way. We are counting infinities.
0:06:12 > 0:06:18A door has opened and an entirely new mathematics lay before us.
0:06:19 > 0:06:21But it never helped Cantor much.
0:06:21 > 0:06:25I was in the cemetery in Halle where he is buried
0:06:25 > 0:06:28and where I had arranged to meet Professor Joe Dauben.
0:06:28 > 0:06:32He was keen to make the connections between Cantor's maths and his life.
0:06:33 > 0:06:36He suffered from manic depression.
0:06:36 > 0:06:39One of the first big breakdowns he has is in 1884
0:06:39 > 0:06:42but then around the turn of the century
0:06:42 > 0:06:44these recurrences of the mental illness
0:06:44 > 0:06:46become more and more frequent.
0:06:46 > 0:06:49A lot of people have tried to say that his mental illness
0:06:49 > 0:06:53was triggered by the incredible abstract mathematics he dealt with.
0:06:53 > 0:06:57Well, he was certainly struggling, so there may have been a connection.
0:06:57 > 0:07:01Yeah, I mean I must say, when you start to contemplate the infinite...
0:07:01 > 0:07:05I am pretty happy with the bottom end of the infinite,
0:07:05 > 0:07:07but as you build it up more and more,
0:07:07 > 0:07:09I must say I start to feel a bit unnerved
0:07:09 > 0:07:13about what's going on here and where is it going.
0:07:13 > 0:07:17For much of Cantor's life, the only place it was going was here -
0:07:17 > 0:07:20the university's sanatorium.
0:07:20 > 0:07:24There was no treatment then for manic depression
0:07:24 > 0:07:27or indeed for the paranoia that often accompanied Cantor's attacks.
0:07:27 > 0:07:30Yet the clinic was a good place to be -
0:07:30 > 0:07:33comfortable, quiet and peaceful.
0:07:33 > 0:07:37And Cantor often found his time here gave him the mental strength
0:07:37 > 0:07:41to resume his exploration of the infinite.
0:07:41 > 0:07:44Other mathematicians would be bothered by the paradoxes
0:07:44 > 0:07:46that Cantor's work had created.
0:07:46 > 0:07:50Curiously, this was one thing Cantor was not worried by.
0:07:50 > 0:07:53He was never as upset about the paradox of the infinite
0:07:53 > 0:07:56as everybody else was because Cantor believed that
0:07:56 > 0:08:00there are certain things that I have been able to show,
0:08:00 > 0:08:03we can establish with complete mathematical certainty
0:08:03 > 0:08:08and then the absolute infinite which is only in God.
0:08:08 > 0:08:12He can understand all of this and there's still that final paradox
0:08:12 > 0:08:15that is not given to us to understand, but God does.
0:08:18 > 0:08:22But there was one problem that Cantor couldn't leave
0:08:22 > 0:08:23in the hands of the Almighty,
0:08:23 > 0:08:26a problem he wrestled with for the rest of his life.
0:08:26 > 0:08:29It became known as the continuum hypothesis.
0:08:29 > 0:08:33Is there an infinity sitting between the smaller infinity
0:08:33 > 0:08:37of all the whole numbers and the larger infinity of the decimals?
0:08:40 > 0:08:45Cantor's work didn't go down well with many of his contemporaries
0:08:45 > 0:08:48but there was one mathematician from France who spoke up for him,
0:08:48 > 0:08:51arguing that Cantor's new mathematics of infinity
0:08:51 > 0:08:54was "beautiful, if pathological".
0:08:54 > 0:09:00Fortunately this mathematician was the most famous and respected mathematician of his day.
0:09:00 > 0:09:04When Bertrand Russell was asked by a French politician who he thought
0:09:04 > 0:09:08the greatest man France had produced, he replied without hesitation, "Poincare".
0:09:08 > 0:09:10The politician was surprised that he'd chosen
0:09:10 > 0:09:14the prime minister Raymond Poincare above the likes of Napoleon, Balzac.
0:09:14 > 0:09:19Russell replied, "I don't mean Raymond Poincare but his cousin,
0:09:19 > 0:09:21"the mathematician, Henri Poincare."
0:09:25 > 0:09:28Henri Poincare spent most of his life in Paris,
0:09:28 > 0:09:32a city that he loved even with its uncertain climate.
0:09:32 > 0:09:36In the last decades of the 19th century, Paris was a centre
0:09:36 > 0:09:40for world mathematics and Poincare became its leading light.
0:09:40 > 0:09:44Algebra, geometry, analysis, he was good at everything.
0:09:44 > 0:09:47His work would lead to all kinds of applications,
0:09:47 > 0:09:50from finding your way around on the underground,
0:09:50 > 0:09:54to new ways of predicting the weather.
0:09:54 > 0:09:57Poincare was very strict about his working day.
0:09:57 > 0:09:59Two hours of work in the morning
0:09:59 > 0:10:01and two hours in the early evening.
0:10:01 > 0:10:02Between these periods,
0:10:02 > 0:10:06he would let his subconscious carry on working on the problem.
0:10:06 > 0:10:10He records one moment when he had a flash of inspiration which occurred
0:10:10 > 0:10:14almost out of nowhere, just as he was getting on a bus.
0:10:16 > 0:10:21And one such flash of inspiration led to an early success.
0:10:21 > 0:10:25In 1885, King Oscar II of Sweden and Norway
0:10:25 > 0:10:32offered a prize of 2,500 crowns for anyone who could establish mathematically once and for all
0:10:32 > 0:10:36whether the solar system would continue turning like clockwork,
0:10:36 > 0:10:38or might suddenly fly apart.
0:10:38 > 0:10:44If the solar system has two planets then Newton had already proved that their orbits would be stable.
0:10:44 > 0:10:48The two bodies just travel in ellipsis round each other.
0:10:48 > 0:10:53But as soon as soon as you add three bodies like the earth, moon and sun,
0:10:53 > 0:10:58the question of whether their orbits were stable or not stumped even the great Newton.
0:10:58 > 0:11:03The problem is that now you have some 18 different variables,
0:11:03 > 0:11:05like the exact coordinates of each body
0:11:05 > 0:11:07and their velocity in each direction.
0:11:07 > 0:11:10So the equations become very difficult to solve.
0:11:10 > 0:11:15But Poincare made significant headway in sorting them out.
0:11:15 > 0:11:21Poincare simplified the problem by making successive approximations to the orbits which he believed
0:11:21 > 0:11:24wouldn't affect the final outcome significantly.
0:11:24 > 0:11:28Although he couldn't solve the problem in its entirety,
0:11:28 > 0:11:33his ideas were sophisticated enough to win him the prize.
0:11:33 > 0:11:36He developed this great sort of arsenal of techniques,
0:11:36 > 0:11:38mathematical techniques
0:11:38 > 0:11:40in order to try and solve it
0:11:40 > 0:11:44and in fact, the prize that he won was essentially
0:11:44 > 0:11:47more for the techniques than for solving the problem.
0:11:47 > 0:11:51But when Poincare's paper was being prepared for publication
0:11:51 > 0:11:54by the King's scientific advisor, Mittag-Leffler,
0:11:54 > 0:11:56one of the editors found a problem.
0:11:58 > 0:12:02Poincare realised he'd made a mistake.
0:12:02 > 0:12:06Contrary to what he had originally thought, even a small change in the
0:12:06 > 0:12:10initial conditions could end up producing vastly different orbits.
0:12:10 > 0:12:13His simplification just didn't work.
0:12:13 > 0:12:17But the result was even more important.
0:12:17 > 0:12:24The orbits Poincare had discovered indirectly led to what we now know as chaos theory.
0:12:24 > 0:12:29Understanding the mathematical rules of chaos explain why a butterfly's wings
0:12:29 > 0:12:31could create tiny changes in the atmosphere
0:12:31 > 0:12:33that ultimately might cause
0:12:33 > 0:12:37a tornado or a hurricane to appear on the other side of the world.
0:12:37 > 0:12:40So this big subject of the 20th century, chaos,
0:12:40 > 0:12:43actually came out of a mistake that Poincare made
0:12:43 > 0:12:45and he spotted at the last minute.
0:12:45 > 0:12:49Yes! So the essay had actually been published in its original form,
0:12:49 > 0:12:54and was ready to go out and Mittag-Leffler had sent copies out to various people,
0:12:54 > 0:12:59and it was to his horror when Poincare wrote to him to say, "Stop!"
0:12:59 > 0:13:03Oh, my God. This is every mathematician's worst nightmare.
0:13:03 > 0:13:04Absolutely. "Pull it!"
0:13:04 > 0:13:06Hold the presses!
0:13:07 > 0:13:10Owning up to his mistake, if anything,
0:13:10 > 0:13:12enhanced Poincare's reputation.
0:13:12 > 0:13:15He continued to produce a wide range of original work
0:13:15 > 0:13:16throughout his life.
0:13:16 > 0:13:20Not just specialist stuff either.
0:13:20 > 0:13:24He also wrote popular books, extolling the importance of maths.
0:13:24 > 0:13:28Here we go. Here's a section on the future of mathematics.
0:13:30 > 0:13:34It starts, "If we wish to foresee the future of mathematics,
0:13:34 > 0:13:39"our proper course is to study the history and present the condition of the science."
0:13:39 > 0:13:45So, I think Poincare might have approved of my journey to uncover the story of maths.
0:13:45 > 0:13:48He certainly would have approved of the next destination.
0:13:48 > 0:13:53Because to discover perhaps Poincare's most important contribution to modern mathematics,
0:13:53 > 0:13:56I had to go looking for a bridge.
0:13:59 > 0:14:01Seven bridges in fact.
0:14:01 > 0:14:04The Seven bridges of Konigsberg.
0:14:04 > 0:14:09Today the city is known as Kaliningrad, a little outpost
0:14:09 > 0:14:14of Russia on the Baltic Sea surrounded by Poland and Lithuania.
0:14:14 > 0:14:18Until 1945, however, when it was ceded to the Soviet Union,
0:14:18 > 0:14:21it was the great Prussian City of Konigsberg.
0:14:22 > 0:14:25Much of the old town sadly has been demolished.
0:14:25 > 0:14:29There is now no sign at all of two of the original seven bridges
0:14:29 > 0:14:34and several have changed out of all recognition.
0:14:34 > 0:14:38This is one of the original bridges.
0:14:38 > 0:14:44It may seem like an unlikely setting for the beginning of a mathematical story, but bear with me.
0:14:44 > 0:14:47It started as an 18th-century puzzle.
0:14:47 > 0:14:53Is there a route around the city which crosses each of these seven bridges only once?
0:14:53 > 0:14:57Finding the solution is much more difficult than it looks.
0:15:07 > 0:15:11It was eventually solved by the great mathematician Leonhard Euler,
0:15:11 > 0:15:15who in 1735 proved that it wasn't possible.
0:15:15 > 0:15:19There could not be a route that didn't cross at least one bridge twice.
0:15:19 > 0:15:23He solved the problem by making a conceptual leap.
0:15:23 > 0:15:27He realised, you don't really care what the distances are between the bridges.
0:15:27 > 0:15:31What really matters is how the bridges are connected together.
0:15:31 > 0:15:37This is a problem of a new sort of geometry of position - a problem of topology.
0:15:37 > 0:15:40Many of us use topology every day.
0:15:40 > 0:15:43Virtually all metro maps the world over
0:15:43 > 0:15:46are drawn on topological principles.
0:15:46 > 0:15:49You don't care how far the stations are from each other
0:15:49 > 0:15:51but how they are connected.
0:15:51 > 0:15:53There isn't a metro in Kaliningrad,
0:15:53 > 0:15:58but there is in the nearest other Russian city, St Petersburg.
0:15:58 > 0:16:00The topology is pretty easy on this map.
0:16:00 > 0:16:03It's the Russian I am having difficulty with.
0:16:03 > 0:16:06- Can you tell me which...? - What's the problem?
0:16:06 > 0:16:09I want to know what station this one was.
0:16:09 > 0:16:12I had it the wrong way round even!
0:16:14 > 0:16:18Although topology had its origins in the bridges of Konigsberg,
0:16:18 > 0:16:22it was in the hands of Poincare that the subject evolved
0:16:22 > 0:16:26into a powerful new way of looking at shape.
0:16:26 > 0:16:29Some people refer to topology as bendy geometry
0:16:29 > 0:16:34because in topology, two shapes are the same if you can bend or morph
0:16:34 > 0:16:37one into another without cutting it.
0:16:37 > 0:16:42So for example if I take a football or rugby ball, topologically they
0:16:42 > 0:16:46are the same because one can be morphed into the other.
0:16:46 > 0:16:51Similarly a bagel and a tea-cup are the same because one can be morphed into the other.
0:16:51 > 0:16:58Even very complicated shapes can be unwrapped to become much simpler from a topological point of view.
0:16:58 > 0:17:02But there is no way to continuously deform a bagel to morph it into a ball.
0:17:02 > 0:17:06The hole in the middle makes these shapes topologically different.
0:17:06 > 0:17:11Poincare knew all the possible two-dimensional topological surfaces.
0:17:11 > 0:17:15But in 1904 he came up with a topological problem
0:17:15 > 0:17:17he just couldn't solve.
0:17:17 > 0:17:21If you've got a flat two-dimensional universe then Poincare worked out
0:17:21 > 0:17:24all the possible shapes he could wrap that universe up into.
0:17:24 > 0:17:29It could be a ball or a bagel with one hole, two holes or more holes in.
0:17:29 > 0:17:35But we live in a three-dimensional universe so what are the possible shapes that our universe can be?
0:17:35 > 0:17:39That question became known as the Poincare Conjecture.
0:17:39 > 0:17:43It was finally solved in 2002 here in St Petersburg
0:17:43 > 0:17:47by a Russian mathematician called Grisha Perelman.
0:17:47 > 0:17:51His proof is very difficult to understand, even for mathematicians.
0:17:51 > 0:17:57Perelman solved the problem by linking it to a completely different area of mathematics.
0:17:57 > 0:18:03To understand the shapes, he looked instead at the dynamics of the way things can flow over the shape
0:18:03 > 0:18:06which led to a description of all the possible ways
0:18:06 > 0:18:11that three dimensional space can be wrapped up in higher dimensions.
0:18:11 > 0:18:16I wondered if the man himself could help in unravelling the intricacies of his proof,
0:18:16 > 0:18:23but I'd been told that finding Perelman is almost as difficult as understanding the solution.
0:18:23 > 0:18:26The classic stereotype of a mathematician
0:18:26 > 0:18:29is a mad eccentric scientist, but I think that's a little bit unfair.
0:18:29 > 0:18:33Most of my colleagues are normal. Well, reasonably.
0:18:33 > 0:18:35But when it comes to Perelman,
0:18:35 > 0:18:37there is no doubt he is a very strange character.
0:18:37 > 0:18:40He's received prizes and offers of professorships
0:18:40 > 0:18:43from distinguished universities in the West
0:18:43 > 0:18:46but he's turned them all down.
0:18:46 > 0:18:49Recently he seems to have given up mathematics completely
0:18:49 > 0:18:52and retreated to live as a semi-recluse
0:18:52 > 0:18:54in this very modest housing estate with his mum.
0:18:54 > 0:19:01He refuses to talk to the media but I thought he might just talk to me as a fellow mathematician.
0:19:01 > 0:19:03I was wrong.
0:19:03 > 0:19:07Well, it's interesting. I think he's actually turned off his buzzer.
0:19:07 > 0:19:09Probably too many media have been buzzing it.
0:19:09 > 0:19:12I tried a neighbour and that rang but his doesn't ring at all.
0:19:12 > 0:19:18I think his papers, his mathematics speaks for itself in a way.
0:19:18 > 0:19:21I don't really need to meet the mathematician
0:19:21 > 0:19:23and in this age of Big Brother and Big Money,
0:19:23 > 0:19:26I think there's something noble about the fact he gets his kick
0:19:26 > 0:19:29out of proving theorems and not winning prizes.
0:19:32 > 0:19:36One mathematician would certainly have applauded.
0:19:36 > 0:19:40For solving any of his 23 problems, David Hilbert offered no prize
0:19:40 > 0:19:45or reward beyond the admiration of other mathematicians.
0:19:45 > 0:19:49When he sketched out the problems in Paris in 1900,
0:19:49 > 0:19:52Hilbert himself was already a mathematical star.
0:19:52 > 0:19:56And it was in Gottingen in northern Germany that he really shone.
0:19:59 > 0:20:05He was by far the most charismatic mathematician of his age.
0:20:05 > 0:20:09It's clear that everyone who knew him thought he was absolutely wonderful.
0:20:12 > 0:20:17He studied number theory and brought everything together that was there
0:20:17 > 0:20:20and then within a year or so he left that completely
0:20:20 > 0:20:24and revolutionised the theory of integral equation.
0:20:24 > 0:20:26It's always change and always something new,
0:20:26 > 0:20:29and there's hardly anybody who is...
0:20:29 > 0:20:34who was so flexible and so varied in his approach as Hilbert was.
0:20:34 > 0:20:41His work is still talked about today and his name has become attached to many mathematical terms.
0:20:41 > 0:20:46Mathematicians still use the Hilbert Space, the Hilbert Classification,
0:20:46 > 0:20:51the Hilbert Inequality and several Hilbert theorems.
0:20:51 > 0:20:54But it was his early work on equations that marked him out
0:20:54 > 0:20:57as a mathematician thinking in new ways.
0:20:57 > 0:21:01Hilbert showed that although there are infinitely many equations,
0:21:01 > 0:21:04there are ways to divide them up so that they are built
0:21:04 > 0:21:08out of just a finite set, like a set of building blocks.
0:21:08 > 0:21:13The most striking element of Hilbert's proof was that he couldn't actually construct this finite set.
0:21:13 > 0:21:17He just proved it must exist.
0:21:17 > 0:21:20Somebody criticised this as theology and not mathematics
0:21:20 > 0:21:22but they'd missed the point.
0:21:22 > 0:21:26What Hilbert was doing here was creating a new style of mathematics,
0:21:26 > 0:21:28a more abstract approach to the subject.
0:21:28 > 0:21:31You could still prove something existed,
0:21:31 > 0:21:34even though you couldn't construct it explicitly.
0:21:34 > 0:21:37It's like saying, "I know there has to be a way to get
0:21:37 > 0:21:42"from Gottingen to St Petersburg even though I can't tell you
0:21:42 > 0:21:44"how to actually get there."
0:21:44 > 0:21:49As well as challenging mathematical orthodoxies, Hilbert was also happy
0:21:49 > 0:21:54to knock the formal hierarchies that existed in the university system in Germany at the time.
0:21:54 > 0:22:01Other professors were quite shocked to see Hilbert out bicycling and drinking with his students.
0:22:01 > 0:22:03- He liked very much parties.- Yeah?
0:22:03 > 0:22:07- Yes.- Party animal. That's my kind of mathematician.
0:22:07 > 0:22:13He liked very much dancing with young women. He liked very much to flirt.
0:22:13 > 0:22:17Really? Most mathematicians I know are not the biggest of flirts.
0:22:17 > 0:22:22'Yet this lifestyle went hand in hand with an absolute commitment to maths.'
0:22:22 > 0:22:26Hilbert was of course somebody who thought
0:22:26 > 0:22:30that everybody who has a mathematical skill,
0:22:30 > 0:22:36a penguin, a woman, a man, or black, white or yellow,
0:22:36 > 0:22:40it doesn't matter, he should do mathematics
0:22:40 > 0:22:42and he should be admired for his.
0:22:42 > 0:22:46The mathematics speaks for itself in a way.
0:22:46 > 0:22:49- It doesn't matter... - When you're a penguin.
0:22:49 > 0:22:54Yeah, if you can prove the Riemann hypothesis, we really don't mind.
0:22:54 > 0:22:58- Yes, mathematics was for him a universal language.- Yes.
0:22:58 > 0:23:02Hilbert believed that this language was powerful enough
0:23:02 > 0:23:04to unlock all the truths of mathematics,
0:23:04 > 0:23:07a belief he expounded in a radio interview he gave
0:23:07 > 0:23:11on the future of mathematics on the 8th September 1930.
0:23:16 > 0:23:20In it, he had no doubt that all his 23 problems would soon be solved
0:23:20 > 0:23:23and that mathematics would finally be put
0:23:23 > 0:23:26on unshakeable logical foundations.
0:23:26 > 0:23:30There are absolutely no unsolvable problems, he declared,
0:23:30 > 0:23:32a belief that's been held by mathematicians
0:23:32 > 0:23:34since the time of the Ancient Greeks.
0:23:34 > 0:23:40He ended with this clarion call, "We must know, we will know."
0:23:40 > 0:23:44'Wir mussen wissen, wir werden wissen.'
0:23:45 > 0:23:48Unfortunately for him, the very day before
0:23:48 > 0:23:52in a scientific lecture that was not considered worthy of broadcast,
0:23:52 > 0:23:55another mathematician would shatter Hilbert's dream
0:23:55 > 0:23:59and put uncertainty at the heart of mathematics.
0:23:59 > 0:24:02The man responsible for destroying Hilbert's belief
0:24:02 > 0:24:05was an Austrian mathematician, Kurt Godel.
0:24:10 > 0:24:12And it all started here - Vienna.
0:24:12 > 0:24:15Even his admirers, and there are a great many,
0:24:15 > 0:24:19admit that Kurt Godel was a little odd.
0:24:19 > 0:24:23As a child, he was bright, sickly and very strange.
0:24:23 > 0:24:25He couldn't stop asking questions.
0:24:25 > 0:24:30So much so, that his family called him Herr Warum - Mr Why.
0:24:30 > 0:24:35Godel lived in Vienna in the 1920s and 1930s,
0:24:35 > 0:24:38between the fall of the Austro-Hungarian Empire
0:24:38 > 0:24:39and its annexation by the Nazis.
0:24:39 > 0:24:45It was a strange, chaotic and exciting time to be in the city.
0:24:45 > 0:24:48Godel studied mathematics at Vienna University
0:24:48 > 0:24:50but he spent most of his time in the cafes,
0:24:50 > 0:24:52the internet chat rooms of their time,
0:24:52 > 0:24:55where amongst games of backgammon and billiards,
0:24:55 > 0:24:59the real intellectual excitement was taking place.
0:24:59 > 0:25:02Particularly amongst a highly influential group
0:25:02 > 0:25:05of philosophers and scientists called the Vienna Circle.
0:25:05 > 0:25:10In their discussions, Kurt Godel would come up with an idea
0:25:10 > 0:25:13that would revolutionise mathematics.
0:25:13 > 0:25:15He'd set himself a difficult mathematical test.
0:25:15 > 0:25:18He wanted to solve Hilbert's second problem
0:25:18 > 0:25:22and find a logical foundation for all mathematics.
0:25:22 > 0:25:25But what he came up with surprised even him.
0:25:25 > 0:25:28All his efforts in mathematical logic not only couldn't provide
0:25:28 > 0:25:33the guarantee Hilbert wanted, instead he proved the opposite.
0:25:33 > 0:25:35Got it.
0:25:35 > 0:25:38It's called the Incompleteness Theorem.
0:25:38 > 0:25:42Godel proved that within any logical system for mathematics
0:25:42 > 0:25:46there will be statements about numbers which are true
0:25:46 > 0:25:48but which you cannot prove.
0:25:48 > 0:25:53He starts with the statement, "This statement cannot be proved."
0:25:53 > 0:25:55This is not a mathematical statement yet.
0:25:55 > 0:25:58But by using a clever code based on prime numbers,
0:25:58 > 0:26:03Godel could change this statement into a pure statement of arithmetic.
0:26:03 > 0:26:08Now, such statements must be either true or false.
0:26:08 > 0:26:13Hold on to your logical hats as we explore the possibilities.
0:26:13 > 0:26:17If the statement is false, that means the statement could be proved,
0:26:17 > 0:26:21which means it would be true, and that's a contradiction.
0:26:21 > 0:26:23So that means, the statement must be true.
0:26:23 > 0:26:28In other words, here is a mathematical statement that is true
0:26:28 > 0:26:30but can't be proved.
0:26:30 > 0:26:32Blast.
0:26:32 > 0:26:35Godel's proof led to a crisis in mathematics.
0:26:35 > 0:26:39What if the problem you were working on, the Goldbach conjecture, say,
0:26:39 > 0:26:43or the Riemann hypothesis, would turn out to be true but unprovable?
0:26:43 > 0:26:46It led to a crisis for Godel too.
0:26:46 > 0:26:50In the autumn of 1934, he suffered the first of what became
0:26:50 > 0:26:55a series of breakdowns and spent time in a sanatorium.
0:26:55 > 0:26:58He was saved by the love of a good woman.
0:26:58 > 0:27:02Adele Nimbursky was a dancer in a local night club.
0:27:02 > 0:27:06She kept Godel alive.
0:27:06 > 0:27:10One day, she and Godel were walking down these very steps.
0:27:10 > 0:27:13Suddenly he was attacked by Nazi thugs.
0:27:13 > 0:27:17Godel himself wasn't Jewish, but many of his friends in the Vienna Circle were.
0:27:17 > 0:27:19Adele came to his rescue.
0:27:19 > 0:27:24But it was only a temporary reprieve for Godel and for maths.
0:27:24 > 0:27:29All across Austria and Germany, mathematics was about to die.
0:27:33 > 0:27:36In the new German empire in the late 1930s
0:27:36 > 0:27:39there weren't colourful balloons flying over the universities,
0:27:39 > 0:27:41but swastikas.
0:27:41 > 0:27:46The Nazis passed a law allowing the removal of any civil servant
0:27:46 > 0:27:47who wasn't Aryan.
0:27:47 > 0:27:51Academics were civil servants in Germany then and now.
0:27:53 > 0:27:56Mathematicians suffered more than most.
0:27:56 > 0:27:59144 in Germany would lose their jobs.
0:27:59 > 0:28:0414 were driven to suicide or died in concentration camps.
0:28:07 > 0:28:10But one brilliant mathematician stayed on.
0:28:10 > 0:28:12David Hilbert helped arrange
0:28:12 > 0:28:15for some of his brightest students to flee.
0:28:15 > 0:28:17And he spoke out for a while about the dismissal
0:28:17 > 0:28:19of his Jewish colleagues.
0:28:19 > 0:28:23But soon, he too became silent.
0:28:26 > 0:28:29It's not clear why he didn't flee himself
0:28:29 > 0:28:31or at least protest a little more.
0:28:31 > 0:28:33He did fall ill towards the end of his life
0:28:33 > 0:28:35so maybe he just didn't have the energy.
0:28:35 > 0:28:38All around him, mathematicians and scientists
0:28:38 > 0:28:42were fleeing the Nazi regime until it was only Hilbert left
0:28:42 > 0:28:47to witness the destruction of one of the greatest mathematical centres of all time.
0:28:50 > 0:28:53David Hilbert died in 1943.
0:28:53 > 0:28:56Only ten people attended the funeral
0:28:56 > 0:28:59of the most famous mathematician of his time.
0:28:59 > 0:29:01The dominance of Europe,
0:29:01 > 0:29:05the centre for world maths for 500 years, was over.
0:29:05 > 0:29:12It was time for the mathematical baton to be handed to the New World.
0:29:13 > 0:29:17Time in fact for this place.
0:29:17 > 0:29:22The Institute for Advanced Study had been set up in Princeton in 1930.
0:29:22 > 0:29:24The idea was to reproduce the collegiate atmosphere
0:29:24 > 0:29:28of the old European universities in rural New Jersey.
0:29:28 > 0:29:32But to do this, it needed to attract the very best,
0:29:32 > 0:29:34and it didn't need to look far.
0:29:34 > 0:29:37Many of the brightest European mathematicians
0:29:37 > 0:29:39were fleeing the Nazis for America.
0:29:39 > 0:29:42People like Hermann Weyl, whose research
0:29:42 > 0:29:45would have major significance for theoretical physics.
0:29:45 > 0:29:48And John Von Neumann, who developed game theory
0:29:48 > 0:29:50and was one of the pioneers of computer science.
0:29:50 > 0:29:55The Institute quickly became the perfect place
0:29:55 > 0:29:59to create another Gottingen in the woods.
0:29:59 > 0:30:04One mathematician in particular made the place a home from home.
0:30:04 > 0:30:06Every morning Kurt Godel,
0:30:06 > 0:30:09dressed in a white linen suit and wearing a fedora,
0:30:09 > 0:30:13would walk from his home along Mercer Street to the Institute.
0:30:13 > 0:30:16On his way, he would stop here at number 112,
0:30:16 > 0:30:22to pick up his closest friend, another European exile, Albert Einstein.
0:30:22 > 0:30:26But not even relaxed, affluent Princeton could help Godel
0:30:26 > 0:30:29finally escape his demons.
0:30:29 > 0:30:31Einstein was always full of laughter.
0:30:31 > 0:30:35He described Princeton as a banishment to paradise.
0:30:35 > 0:30:40But the much younger Godel became increasingly solemn and pessimistic.
0:30:43 > 0:30:46Slowly this pessimism turned into paranoia.
0:30:46 > 0:30:50He spent less and less time with his fellow mathematicians in Princeton.
0:30:50 > 0:30:54Instead, he preferred to come here to the beach, next to the ocean,
0:30:54 > 0:30:59walk alone, thinking about the works of the great German mathematician, Leibniz.
0:31:01 > 0:31:05But as Godel was withdrawing into his own interior world,
0:31:05 > 0:31:09his influence on American mathematics paradoxically
0:31:09 > 0:31:12was growing stronger and stronger.
0:31:12 > 0:31:16And a young mathematician from just along the New Jersey coast
0:31:16 > 0:31:19eagerly took on some of the challenges he posed.
0:31:19 > 0:31:23# One, two, three, four, five, six, seven, eight, nine, ten
0:31:23 > 0:31:25# Times a day I could love you... #
0:31:25 > 0:31:27In 1950s America,
0:31:27 > 0:31:31the majority of youngsters weren't preoccupied with mathematics.
0:31:31 > 0:31:35Most went for a more relaxed, hedonistic lifestyle
0:31:35 > 0:31:38in this newly affluent land of ice-cream and doughnuts.
0:31:38 > 0:31:42But one teenager didn't indulge in the normal pursuits
0:31:42 > 0:31:45of American adolescence but chose instead
0:31:45 > 0:31:49to grapple with some of the major problems in mathematics.
0:31:49 > 0:31:50From a very early age,
0:31:50 > 0:31:55Paul Cohen was winning mathematical competitions and prizes.
0:31:55 > 0:31:58But he found it difficult at first to discover a field in mathematics
0:31:58 > 0:32:01where he could really make his mark...
0:32:01 > 0:32:05Until he read about Cantor's continuum hypothesis.
0:32:05 > 0:32:09That's the one problem, as I had learned back in Halle,
0:32:09 > 0:32:11that Cantor just couldn't resolve.
0:32:11 > 0:32:15It asks whether there is or there isn't an infinite set of numbers
0:32:15 > 0:32:18bigger than the set of all whole numbers
0:32:18 > 0:32:20but smaller than the set of all decimals.
0:32:20 > 0:32:24It sounds straightforward, but it had foiled all attempts
0:32:24 > 0:32:29to solve it since Hilbert made it his first problem way back in 1900.
0:32:29 > 0:32:31With the arrogance of youth,
0:32:31 > 0:32:36the 22-year-old Paul Cohen decided that he could do it.
0:32:36 > 0:32:40Cohen came back a year later with the extraordinary discovery
0:32:40 > 0:32:43that both answers could be true.
0:32:43 > 0:32:47There was one mathematics where the continuum hypothesis
0:32:47 > 0:32:49could be assumed to be true.
0:32:49 > 0:32:51There wasn't a set between the whole numbers
0:32:51 > 0:32:53and the infinite decimals.
0:32:55 > 0:32:59But there was an equally consistent mathematics
0:32:59 > 0:33:03where the continuum hypothesis could be assumed to be false.
0:33:03 > 0:33:08Here, there was a set between the whole numbers and the infinite decimals.
0:33:08 > 0:33:11It was an incredibly daring solution.
0:33:11 > 0:33:13Cohen's proof seemed true,
0:33:13 > 0:33:19but his method was so new that nobody was absolutely sure.
0:33:19 > 0:33:22There was only one person whose opinion everybody trusted.
0:33:22 > 0:33:26There was a lot of scepticism and he had to come and make a trip here,
0:33:26 > 0:33:29to the Institute right here, to visit Godel.
0:33:29 > 0:33:32And it was only after Godel gave his stamp of approval
0:33:32 > 0:33:34in quite an unusual way,
0:33:34 > 0:33:37He said, "Give me your paper", and then on Monday he put it back
0:33:37 > 0:33:40in the box and said, "Yes, it's correct."
0:33:40 > 0:33:42Then everything changed.
0:33:43 > 0:33:46Today mathematicians insert a statement
0:33:46 > 0:33:50that says whether the result depends on the continuum hypothesis.
0:33:50 > 0:33:54We've built up two different mathematical worlds
0:33:54 > 0:33:57in which one answer is yes and the other is no.
0:33:57 > 0:34:01Paul Cohen really has rocked the mathematical universe.
0:34:01 > 0:34:05It gave him fame, riches, and prizes galore.
0:34:07 > 0:34:12He didn't publish much after his early success in the '60s.
0:34:12 > 0:34:15But he was absolutely dynamite.
0:34:15 > 0:34:18I can't imagine anyone better to learn from, and he was very eager
0:34:18 > 0:34:23to learn, to teach you anything he knew or even things he didn't know.
0:34:23 > 0:34:27With the confidence that came from solving Hilbert's first problem,
0:34:27 > 0:34:30Cohen settled down in the mid 1960s
0:34:30 > 0:34:34to have a go at the most important Hilbert problem of them all -
0:34:34 > 0:34:36the eighth, the Riemann hypothesis.
0:34:36 > 0:34:43When he died in California in 2007, 40 years later, he was still trying.
0:34:43 > 0:34:46But like many famous mathematicians before him,
0:34:46 > 0:34:48Riemann had defeated even him.
0:34:48 > 0:34:52But his approach has inspired others to make progress towards a proof,
0:34:52 > 0:34:55including one of his most famous students, Peter Sarnak.
0:34:55 > 0:34:59I think, overall, absolutely loved the guy.
0:34:59 > 0:35:01He was my inspiration.
0:35:01 > 0:35:04I'm really glad I worked with him.
0:35:04 > 0:35:06He put me on the right track.
0:35:09 > 0:35:14Paul Cohen is a good example of the success of the great American Dream.
0:35:14 > 0:35:16The second generation Jewish immigrant
0:35:16 > 0:35:18becomes top American professor.
0:35:18 > 0:35:23But I wouldn't say that his mathematics was a particularly American product.
0:35:23 > 0:35:25Cohen was so fired up by this problem
0:35:25 > 0:35:29that he probably would have cracked it whatever the surroundings.
0:35:31 > 0:35:33Paul Cohen had it relatively easy.
0:35:33 > 0:35:36But another great American mathematician of the 1960s
0:35:36 > 0:35:40faced a much tougher struggle to make an impact.
0:35:40 > 0:35:43Not least because she was female.
0:35:43 > 0:35:48In the story of maths, nearly all the truly great mathematicians have been men.
0:35:48 > 0:35:51But there have been a few exceptions.
0:35:51 > 0:35:54There was the Russian Sofia Kovalevskaya
0:35:54 > 0:35:58who became the first female professor of mathematics in Stockholm in 1889,
0:35:58 > 0:36:03and won a very prestigious French mathematical prize.
0:36:03 > 0:36:07And then Emmy Noether, a talented algebraist who fled from the Nazis
0:36:07 > 0:36:10to America but then died before she fully realised her potential.
0:36:10 > 0:36:15Then there is the woman who I am crossing America to find out about.
0:36:15 > 0:36:19Julia Robinson, the first woman ever to be elected president
0:36:19 > 0:36:22of the American Mathematical Society.
0:36:31 > 0:36:34She was born in St Louis in 1919,
0:36:34 > 0:36:38but her mother died when she was two.
0:36:38 > 0:36:42She and her sister Constance moved to live with their grandmother
0:36:42 > 0:36:45in a small community in the desert near Phoenix, Arizona.
0:36:47 > 0:36:49Julia Robinson grew up around here.
0:36:49 > 0:36:53I've got a photo which shows her cottage in the 1930s,
0:36:53 > 0:36:55with nothing much around it.
0:36:55 > 0:36:58The mountains pretty much match those over there
0:36:58 > 0:37:00so I think she might have lived down there.
0:37:01 > 0:37:04Julia grew up a shy, sickly girl,
0:37:04 > 0:37:09who, when she was seven, spent a year in bed because of scarlet fever.
0:37:09 > 0:37:12Ill-health persisted throughout her childhood.
0:37:12 > 0:37:15She was told she wouldn't live past 40.
0:37:15 > 0:37:20But right from the start, she had an innate mathematical ability.
0:37:20 > 0:37:25Under the shade of the native Arizona cactus, she whiled away the time
0:37:25 > 0:37:28playing endless counting games with stone pebbles.
0:37:28 > 0:37:31This early searching for patterns would give her a feel
0:37:31 > 0:37:35and love of numbers that would last for the rest of her life.
0:37:35 > 0:37:39But despite showing an early brilliance, she had to continually
0:37:39 > 0:37:44fight at school and college to simply be allowed to keep doing maths.
0:37:44 > 0:37:47As a teenager, she was the only girl in the maths class
0:37:47 > 0:37:50but had very little encouragement.
0:37:50 > 0:37:55The young Julia sought intellectual stimulation elsewhere.
0:37:55 > 0:37:59Julia loved listening to a radio show called the University Explorer
0:37:59 > 0:38:02and the 53rd episode was all about mathematics.
0:38:02 > 0:38:04The broadcaster described how he discovered
0:38:04 > 0:38:08despite their esoteric language and their seclusive nature,
0:38:08 > 0:38:12mathematicians are the most interesting and inspiring creatures.
0:38:12 > 0:38:16For the first time, Julia had found out that there were mathematicians,
0:38:16 > 0:38:17not just mathematics teachers.
0:38:17 > 0:38:20There was a world of mathematics out there,
0:38:20 > 0:38:22and she wanted to be part of it.
0:38:26 > 0:38:29The doors to that world opened here at the University of California,
0:38:29 > 0:38:31at Berkeley near San Francisco.
0:38:33 > 0:38:38She was absolutely obsessed that she wanted to go to Berkeley.
0:38:38 > 0:38:44She wanted to go away to some place where there were mathematicians.
0:38:44 > 0:38:46Berkeley certainly had mathematicians,
0:38:46 > 0:38:50including a number theorist called Raphael Robinson.
0:38:50 > 0:38:53In their frequent walks around the campus
0:38:53 > 0:38:59they found they had more than just a passion for mathematics. They married in 1952.
0:38:59 > 0:39:03Julia got her PhD and settled down
0:39:03 > 0:39:05to what would turn into her lifetime's work -
0:39:05 > 0:39:07Hilbert's tenth problem.
0:39:07 > 0:39:10She thought about it all the time.
0:39:10 > 0:39:14She said to me she just wouldn't wanna die without knowing that answer
0:39:14 > 0:39:16and it had become an obsession.
0:39:17 > 0:39:21Julia's obsession has been shared with many other mathematicians
0:39:21 > 0:39:24since Hilbert had first posed it back in 1900.
0:39:24 > 0:39:28His tenth problem asked if there was some universal method
0:39:28 > 0:39:34that could tell whether any equation had whole number solutions or not.
0:39:34 > 0:39:36Nobody had been able to solve it.
0:39:36 > 0:39:39In fact, the growing belief was
0:39:39 > 0:39:42that no such universal method was possible.
0:39:42 > 0:39:44How on earth could you prove that,
0:39:44 > 0:39:48however ingenious you were, you'd never come up with a method?
0:39:50 > 0:39:51With the help of colleagues,
0:39:51 > 0:39:55Julia developed what became known as the Robinson hypothesis.
0:39:55 > 0:39:58This argued that to show no such method existed,
0:39:58 > 0:40:03all you had to do was to cook up one equation whose solutions
0:40:03 > 0:40:06were a very specific set of numbers.
0:40:06 > 0:40:09The set of numbers needed to grow exponentially,
0:40:09 > 0:40:13like taking powers of two, yet still be captured by the equations
0:40:13 > 0:40:16at the heart of Hilbert's problem.
0:40:16 > 0:40:21Try as she might, Robinson just couldn't find this set.
0:40:21 > 0:40:25For the tenth problem to be finally solved,
0:40:25 > 0:40:28there needed to be some fresh inspiration.
0:40:28 > 0:40:34That came from 5,000 miles away - St Petersburg in Russia.
0:40:34 > 0:40:37Ever since the great Leonhard Euler set up shop here
0:40:37 > 0:40:39in the 18th century,
0:40:39 > 0:40:42the city has been famous for its mathematics and mathematicians.
0:40:42 > 0:40:44Here in the Steklov Institute,
0:40:44 > 0:40:47some of the world's brightest mathematicians
0:40:47 > 0:40:50have set out their theorems and conjectures.
0:40:50 > 0:40:54This morning, one of them is giving a rare seminar.
0:40:57 > 0:41:00It's tough going even if you speak Russian,
0:41:00 > 0:41:02which unfortunately I don't.
0:41:02 > 0:41:06But we do get a break in the middle to recover before the final hour.
0:41:06 > 0:41:08There is a kind of rule in seminars.
0:41:08 > 0:41:12The first third is for everyone, the second third for the experts
0:41:12 > 0:41:16and the last third is just for the lecturer.
0:41:16 > 0:41:19I think that's what we're going to get next.
0:41:19 > 0:41:22The lecturer is Yuri Matiyasevich and he is explaining
0:41:22 > 0:41:26his latest work on - what else? - the Riemann hypothesis.
0:41:28 > 0:41:33As a bright young graduate student in 1965, Yuri's tutor
0:41:33 > 0:41:36suggested he have a go at another Hilbert problem,
0:41:36 > 0:41:39the one that had in fact preoccupied Julia Robinson.
0:41:39 > 0:41:40Hilbert's tenth.
0:41:43 > 0:41:45It was the height of the Cold War.
0:41:45 > 0:41:48Perhaps Matiyasevich could succeed for Russia
0:41:48 > 0:41:52where Julia and her fellow American mathematicians had failed.
0:41:52 > 0:41:55- At first I did not like their approach.- Oh, right.
0:41:55 > 0:41:59The statement looked to me rather strange and artificial
0:41:59 > 0:42:03but after some time I understood it was quite natural,
0:42:03 > 0:42:07and then I understood that she had a new brilliant idea
0:42:07 > 0:42:10and I just started to further develop it.
0:42:11 > 0:42:17In January 1970, he found the vital last piece in the jigsaw.
0:42:17 > 0:42:21He saw how to capture the famous Fibonacci sequence of numbers
0:42:21 > 0:42:26using the equations that were at the heart of Hilbert's problem.
0:42:26 > 0:42:28Building on the work of Julia and her colleagues,
0:42:28 > 0:42:30he had solved the tenth.
0:42:30 > 0:42:34He was just 22 years old.
0:42:34 > 0:42:37The first person he wanted to tell was the woman he owed so much to.
0:42:39 > 0:42:41I got no answer
0:42:41 > 0:42:44and I believed they were lost in the mail.
0:42:44 > 0:42:47It was quite natural because it was Soviet time.
0:42:47 > 0:42:50But back in California, Julia had heard rumours
0:42:50 > 0:42:54through the mathematical grapevine that the problem had been solved.
0:42:54 > 0:42:57And she contacted Yuri herself.
0:42:58 > 0:43:01She said, I just had to wait for you to grow up
0:43:01 > 0:43:06to get the answer, because she had started work in 1948.
0:43:06 > 0:43:07When Yuri was just a baby.
0:43:07 > 0:43:11Then he responds by thanking her
0:43:11 > 0:43:16and saying that the credit is as much hers as it is his.
0:43:18 > 0:43:20YURI: I met Julia one year later.
0:43:20 > 0:43:25It was in Bucharest. I suggested after the conference in Bucharest
0:43:25 > 0:43:30Julia and her husband Raphael came to see me here in Leningrad.
0:43:30 > 0:43:35Together, Julia and Yuri worked on several other mathematical problems
0:43:35 > 0:43:39until shortly before Julia died in 1985.
0:43:39 > 0:43:41She was just 55 years old.
0:43:41 > 0:43:45She was able to find the new ways.
0:43:45 > 0:43:49Many mathematicians just combine previous known methods
0:43:49 > 0:43:55to solve new problems and she had really new ideas.
0:43:55 > 0:43:59Although Julia Robinson showed there was no universal method
0:43:59 > 0:44:01to solve all equations in whole numbers,
0:44:01 > 0:44:05mathematicians were still interested in finding methods
0:44:05 > 0:44:08to solve special classes of equations.
0:44:08 > 0:44:11It would be in France in the early 19th century,
0:44:11 > 0:44:13in one of the most extraordinary stories
0:44:13 > 0:44:17in the history of mathematics, that methods were developed
0:44:17 > 0:44:20to understand why certain equations could be solved
0:44:20 > 0:44:21while others couldn't.
0:44:27 > 0:44:32It's early morning in Paris on the 29th May 1832.
0:44:32 > 0:44:37Evariste Galois is about to fight for his very life.
0:44:37 > 0:44:40It is the reign of the reactionary Bourbon King, Charles X,
0:44:40 > 0:44:43and Galois, like many angry young men in Paris then,
0:44:43 > 0:44:46is a republican revolutionary.
0:44:46 > 0:44:52Unlike the rest of his comrades though, he has another passion - mathematics.
0:44:53 > 0:44:56He had just spent four months in jail.
0:44:56 > 0:45:00Then, in a mysterious saga of unrequited love,
0:45:00 > 0:45:02he is challenged to a duel.
0:45:02 > 0:45:04He'd been up the whole previous night
0:45:04 > 0:45:07refining a new language for mathematics he'd developed.
0:45:07 > 0:45:14Galois believed that mathematics shouldn't be the study of number and shape, but the study of structure.
0:45:14 > 0:45:17Perhaps he was still pre-occupied with his maths.
0:45:17 > 0:45:18GUNSHOT
0:45:18 > 0:45:21There was only one shot fired that morning.
0:45:21 > 0:45:27Galois died the next day, just 20 years old.
0:45:27 > 0:45:30It was one of mathematics greatest losses.
0:45:30 > 0:45:33Only by the beginning of the 20th century
0:45:33 > 0:45:37would Galois be fully appreciated and his ideas fully realised.
0:45:42 > 0:45:46Galois had discovered new techniques to be able to tell
0:45:46 > 0:45:49whether certain equations could have solutions or not.
0:45:49 > 0:45:54The symmetry of certain geometric objects seemed to be the key.
0:45:54 > 0:45:58His idea of using geometry to analyse equations
0:45:58 > 0:46:03would be picked up in the 1920s by another Parisian mathematician, Andre Weil.
0:46:03 > 0:46:09I was very much interested and so far as school was concerned
0:46:09 > 0:46:13quite successful in all possible branches.
0:46:13 > 0:46:17And he was. After studying in Germany as well as France,
0:46:17 > 0:46:21Andre settled down at this apartment in Paris
0:46:21 > 0:46:25which he shared with his more-famous sister, the writer Simone Weil.
0:46:25 > 0:46:31But when the Second World War broke out, he found himself in very different circumstances.
0:46:31 > 0:46:37He dodged the draft by fleeing to Finland where he was almost executed for being a Russian spy.
0:46:37 > 0:46:42On his return to France he was put in prison in Rouen to await trial for desertion.
0:46:42 > 0:46:45At the trial, the judge gave him a choice.
0:46:45 > 0:46:49Five more years in prison or serve in a combat unit.
0:46:49 > 0:46:52He chose to join the French army, a lucky choice
0:46:52 > 0:46:56because just before the Germans invaded a few months later,
0:46:56 > 0:46:58all the prisoners were killed.
0:46:58 > 0:47:05Weil only spent a few months in prison, but this time was crucial for his mathematics.
0:47:05 > 0:47:11Because here he built on the ideas of Galois and first developed algebraic geometry
0:47:11 > 0:47:15a whole new language for understanding solutions to equations.
0:47:15 > 0:47:18Galois had shown how new mathematical structures
0:47:18 > 0:47:22can be used to reveal the secrets behind equations.
0:47:22 > 0:47:24Weil's work led him to theorems
0:47:24 > 0:47:28that connected number theory, algebra, geometry and topology
0:47:28 > 0:47:33and are one of the greatest achievements of modern mathematics.
0:47:33 > 0:47:36Without Andre Weil, we would never have heard
0:47:36 > 0:47:41of the strangest man in the history of maths, Nicolas Bourbaki.
0:47:43 > 0:47:50There are no photos of Bourbaki in existence but we do know he was born in this cafe in the Latin Quarter
0:47:50 > 0:47:54in 1934 when it was a proper cafe, the cafe Capoulade,
0:47:54 > 0:47:58and not the fast food joint it has now become.
0:47:58 > 0:48:03Just down the road, I met up with Bourbaki expert David Aubin.
0:48:03 > 0:48:06When I was a graduate student I got quite frightened
0:48:06 > 0:48:08when I used to go into the library
0:48:08 > 0:48:10because this guy Bourbaki had written so many books.
0:48:10 > 0:48:14Something like 30 or 40 altogether.
0:48:14 > 0:48:19In analysis, in geometry, in topology, it was all new foundations.
0:48:19 > 0:48:23Virtually everyone studying Maths seriously anywhere in the world
0:48:23 > 0:48:28in the 1950s, '60s and '70s would have read Nicolas Bourbaki.
0:48:28 > 0:48:31He applied for membership of the American Maths Society, I heard.
0:48:31 > 0:48:33At which point he was denied membership
0:48:33 > 0:48:36- on the grounds that he didn't exist. - Oh!
0:48:36 > 0:48:38The Americans were right.
0:48:38 > 0:48:41Nicolas Bourbaki does not exist at all. And never has.
0:48:41 > 0:48:46Bourbaki is in fact the nom de plume for a group of French mathematicians
0:48:46 > 0:48:49led by Andre Weil who decided to write a coherent account
0:48:49 > 0:48:52of the mathematics of the 20th century.
0:48:52 > 0:48:57Most of the time mathematicians like to have their own names on theorems.
0:48:57 > 0:48:59But for the Bourbaki group,
0:48:59 > 0:49:03the aims of the project overrode any desire for personal glory.
0:49:03 > 0:49:07After the Second World War, the Bourbaki baton was handed down
0:49:07 > 0:49:10to the next generation of French mathematicians.
0:49:10 > 0:49:15And their most brilliant member was Alexandre Grothendieck.
0:49:15 > 0:49:17Here at the IHES in Paris,
0:49:17 > 0:49:21the French equivalent of Princeton's Institute for Advanced Study,
0:49:21 > 0:49:27Grothendieck held court at his famous seminars in the 1950s and 1960s.
0:49:29 > 0:49:33He had this incredible charisma.
0:49:33 > 0:49:40He had this amazing ability to see a young person and somehow know
0:49:40 > 0:49:46what kind of contribution this person could make to this incredible vision
0:49:46 > 0:49:48he had of how mathematics could be.
0:49:48 > 0:49:54And this vision enabled him to get across some very difficult ideas indeed.
0:49:54 > 0:49:58He says, "Suppose you want to open a walnut.
0:49:58 > 0:50:02"So the standard thing is you take a nutcracker and you just break it open."
0:50:02 > 0:50:04And he says his approach is more like,
0:50:04 > 0:50:08you take this walnut and you put it out in the snow
0:50:08 > 0:50:10and you leave it there for a few months
0:50:10 > 0:50:13and then when you come back to it, it just opens.
0:50:13 > 0:50:15Grothendieck is a Structuralist.
0:50:15 > 0:50:19What he's interested in are the hidden structures
0:50:19 > 0:50:22underneath all mathematics.
0:50:22 > 0:50:27Only when you get down to the very basic architecture and think in very general terms
0:50:27 > 0:50:31will the patterns in mathematics become clear.
0:50:31 > 0:50:37Grothendieck produced a new powerful language to see structures in a new way.
0:50:37 > 0:50:39It was like living in a world of black and white
0:50:39 > 0:50:42and suddenly having the language to see the world in colour.
0:50:42 > 0:50:46It's a language that mathematicians have been using ever since
0:50:46 > 0:50:51to solve problems in number theory, geometry, even fundamental physics.
0:50:53 > 0:50:56But in the late 1960s, Grothendieck decided
0:50:56 > 0:51:01to turn his back on mathematics after he discovered politics.
0:51:01 > 0:51:06He believed that the threat of nuclear war and the questions
0:51:06 > 0:51:12of nuclear disarmament were more important than mathematics
0:51:12 > 0:51:17and that people who continue to do mathematics
0:51:17 > 0:51:21rather than confront this threat of nuclear war
0:51:21 > 0:51:22were doing harm in the world.
0:51:26 > 0:51:29Grothendieck decided to leave Paris
0:51:29 > 0:51:32and move back to the south of France where he grew up.
0:51:32 > 0:51:36Bursts of radical politics followed and then a nervous breakdown.
0:51:36 > 0:51:40He moved to the Pyrenees and became a recluse.
0:51:40 > 0:51:45He's now lost all contact with his old friends and mathematical colleagues.
0:51:46 > 0:51:51Nevertheless, the legacy of his achievements means that Grothendieck stands
0:51:51 > 0:51:57alongside Cantor, Godel and Hilbert as someone who has transformed the mathematical landscape.
0:51:59 > 0:52:03He changed the whole subject in a really fundamental way. It will never go back.
0:52:03 > 0:52:08Certainly, he's THE dominant figure of the 20th century.
0:52:16 > 0:52:18I've come back to England, though,
0:52:18 > 0:52:22thinking again about another seminal figure of the 20th century.
0:52:22 > 0:52:26The person that started it all off, David Hilbert.
0:52:26 > 0:52:32Of the 23 problems Hilbert set mathematicians in the year 1900,
0:52:32 > 0:52:34most have now been solved.
0:52:34 > 0:52:37However there is one great exception.
0:52:37 > 0:52:40The Riemann hypothesis, the eighth on Hilbert's list.
0:52:40 > 0:52:43That is still the holy grail of mathematics.
0:52:44 > 0:52:50Hilbert's lecture inspired a generation to pursue their mathematical dreams.
0:52:50 > 0:52:55This morning, in the town where I grew up, I hope to inspire another generation.
0:52:55 > 0:52:57CHEERING AND APPLAUSE
0:53:01 > 0:53:04Thank you. Hello. My name's Marcus du Sautoy
0:53:04 > 0:53:05and I'm a Professor of Mathematics
0:53:05 > 0:53:08up the road at the University of Oxford.
0:53:08 > 0:53:10It was actually in this school here,
0:53:10 > 0:53:14in fact this classroom is where I discovered my love for mathematics.
0:53:14 > 0:53:17'This love of mathematics that I first acquired
0:53:17 > 0:53:20'here in my old comprehensive school still drives me now.
0:53:20 > 0:53:22'It's a love of solving problems.
0:53:22 > 0:53:25'There are so many problems I could tell them about,
0:53:25 > 0:53:27'but I've chosen my favourite.'
0:53:27 > 0:53:30I think that a mathematician is a pattern searcher
0:53:30 > 0:53:33and that's really what mathematicians try and do.
0:53:33 > 0:53:37We try and understand the patterns and the structure
0:53:37 > 0:53:40and the logic to explain the way the world around us works.
0:53:40 > 0:53:43And this is really at the heart of the Riemann hypothesis.
0:53:43 > 0:53:48The task is - is there any pattern in these numbers which can help me say
0:53:48 > 0:53:50where the next number will be?
0:53:50 > 0:53:52What's the next one after 31? How can I tell?
0:53:52 > 0:53:55'These numbers are, of course, prime numbers -
0:53:55 > 0:53:58'the building blocks of mathematics.'
0:53:58 > 0:54:01'The Riemann hypothesis, a conjecture about the distribution
0:54:01 > 0:54:04'of the primes, goes to the very heart of our subject.'
0:54:04 > 0:54:07Why on earth is anybody interested in these primes?
0:54:07 > 0:54:11Why is the army interested in primes, why are spies interested?
0:54:11 > 0:54:14- Isn't it to encrypt stuff?- Exactly.
0:54:14 > 0:54:18I study this stuff cos I think it's all really beautiful and elegant
0:54:18 > 0:54:20but actually, there's a lot of people
0:54:20 > 0:54:24who are interested in these numbers because of their very practical use.
0:54:24 > 0:54:28'The bizarre thing is that the more abstract and difficult mathematics becomes,
0:54:28 > 0:54:32'the more it seems to have applications in the real world.
0:54:32 > 0:54:36'Mathematics now pervades every aspect of our lives.
0:54:36 > 0:54:41'Every time we switch on the television, plug in a computer, pay with a credit card.
0:54:41 > 0:54:46'There's now a million dollars for anyone who can solve the Riemann hypothesis.
0:54:46 > 0:54:48'But there's more at stake than that.'
0:54:48 > 0:54:51Anybody who proves this theorem will be remembered forever.
0:54:51 > 0:54:55They'll be on that board ahead of any of those other mathematicians.
0:54:55 > 0:54:59'That's because the Riemann hypothesis is a corner-stone of maths.
0:54:59 > 0:55:02'Thousands of theorems depend on it being true.
0:55:02 > 0:55:06'Very few mathematicians think that it isn't true.
0:55:06 > 0:55:10'But mathematics is about proof and until we can prove it
0:55:10 > 0:55:12'there will still be doubt.'
0:55:12 > 0:55:17Maths has grown out of this passion to get rid of doubt.
0:55:17 > 0:55:20This is what I have learned in my journey through the history of mathematics.
0:55:20 > 0:55:25Mathematicians like Archimedes and al-Khwarizmi, Gauss and Grothendieck
0:55:25 > 0:55:30were driven to understand the precise way numbers and space work.
0:55:30 > 0:55:33Maths in action, that one.
0:55:33 > 0:55:35It's beautiful. Really nice.
0:55:35 > 0:55:39Using the language of mathematics, they have told us stories
0:55:39 > 0:55:43that remain as true today as they were when they were first told.
0:55:43 > 0:55:48In the Mediterranean, I discovered the origins of geometry.
0:55:48 > 0:55:51Mathematicians and philosophers flocked to Alexandria
0:55:51 > 0:55:55driven by a thirst for knowledge and the pursuit of excellence.
0:55:55 > 0:55:59In India, I learned about another discovery
0:55:59 > 0:56:02that it would be impossible to imagine modern life without.
0:56:02 > 0:56:07So here we are in one of the true holy sites of the mathematical world.
0:56:07 > 0:56:10Up here are some numbers,
0:56:10 > 0:56:12and here's the new number.
0:56:12 > 0:56:14Its zero.
0:56:14 > 0:56:19In the Middle East, I was amazed at al-Khwarizmi's invention of algebra.
0:56:19 > 0:56:22He developed systematic ways to analyse problems
0:56:22 > 0:56:26so that the solutions would work whatever numbers you took.
0:56:26 > 0:56:28In the Golden Age of Mathematics,
0:56:28 > 0:56:31in Europe in the 18th and 19th centuries, I found how maths
0:56:31 > 0:56:35discovered new ways for analysing bodies in motion and new geometries
0:56:35 > 0:56:40that helped us understand the very strange shape of space.
0:56:40 > 0:56:43It is with Riemann's work that we finally have
0:56:43 > 0:56:49the mathematical glasses to be able to explore such worlds of the mind.
0:56:49 > 0:56:53And now my journey into the abstract world of 20th-century mathematics
0:56:53 > 0:56:56has revealed that maths is the true language
0:56:56 > 0:56:58the universe is written in,
0:56:58 > 0:57:02the key to understanding the world around us.
0:57:02 > 0:57:05Mathematicians aren't motivated by money and material gain
0:57:05 > 0:57:09or even by practical applications of their work.
0:57:09 > 0:57:13For us, it is the glory of solving one of the great unsolved problems
0:57:13 > 0:57:18that have outwitted previous generations of mathematicians.
0:57:18 > 0:57:21Hilbert was right. It's the unsolved problems of mathematics
0:57:21 > 0:57:23that make it a living subject,
0:57:23 > 0:57:27which obsess each new generation of mathematicians.
0:57:27 > 0:57:30Despite all the things we've discovered over the last seven millennia,
0:57:30 > 0:57:33there are still many things we don't understand.
0:57:33 > 0:57:39And its Hilbert's call of, "We must know, we will know", which drives mathematics.
0:57:42 > 0:57:45You can learn more about The Story Of Maths
0:57:45 > 0:57:48with the Open University at...
0:58:00 > 0:58:03Subtitled by Red Bee Media Ltd
0:58:03 > 0:58:06E-mail subtitling@bbc.co.uk