To Infinity and Beyond The Story of Maths


To Infinity and Beyond

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Mathematics is about solving problems

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and it's the great unsolved problems that make maths really alive.

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In the summer of 1900,

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the International Congress of Mathematicians

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was held here in Paris in the Sorbonne.

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It was a pretty shambolic affair,

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not helped by the sultry August heat.

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But it will be remembered as one of the greatest congresses of all time

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thanks to a lecture given by the up-and-coming David Hilbert.

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Hilbert, a young German mathematician,

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boldly set out what he believed were the 23 most important problems

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for mathematicians to crack.

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He was trying to set the agenda for 20th-century maths and he succeeded.

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These Hilbert problems would define the mathematics of the modern age.

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Of those who tried to crack Hilbert's challenges, some would experience immense triumphs,

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whilst others would be plunged into infinite despair.

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The first problem on Hilbert's list emerged from here,

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Halle, in East Germany.

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It was where the great mathematician Georg Cantor spent all his adult life.

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And where he became the first person to really understand the meaning

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of infinity and give it mathematical precision.

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The statue in the town square, however,

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honours Halle's other famous son, the composer George Handel.

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To discover more about Cantor, I had to take a tram way out of town.

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For 50 years, Halle was part of Communist East Germany

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and the Communists loved celebrating their scientists.

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So much so, they put Cantor on the side of a large cube that they commissioned.

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But, being communists, they didn't put the cube

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in the middle of town. They put it out amongst the people.

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When I eventually found the estate, I started to fear

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that maybe I had got the location wrong.

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This looks the most unlikely venue for a statue to a mathematician.

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Excuse me?

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Ein Frage.

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-Can you help me a minute?

-Wie bitte?

-Do you speak English?

-No!

-No?

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Ich suche ein Wurfel.

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Ein Wurfel, ja?

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Is that right? A "Wurfel"?

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A cube? Yeah? Like that?

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Mit ein Bild der Mathematiker?

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Yeah? Go round there?

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Die Name ist Cantor.

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Somewhere over here. Ah! There it is!

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It's much bigger than I thought.

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I thought it was going to be something like this sort of size.

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Aha, here we are. On the dark side of the cube.

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here's the man himself, Cantor.

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Cantor's one of my big heroes actually.

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I think if I had to choose some theorems that I wish I'd proved,

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I think the couple that Cantor proved

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would be up there in my top ten.

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'This is because before Cantor,

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'no-one had really understood infinity.'

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It was a tricky, slippery concept that didn't seem to go anywhere.

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But Cantor showed that infinity could be perfectly understandable.

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Indeed, there wasn't just one infinity,

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but infinitely many infinities.

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First Cantor took the numbers 1, 2, 3, 4 and so on.

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Then he thought about comparing them with a much smaller set...

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something like 10, 20, 30, 40...

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What he showed is that these two infinite sets of numbers

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actually have the same size because we can pair them up -

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1 with 10, 2 with 20, 3 with 30 and so on.

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So these are the same sizes of infinity.

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But what about the fractions?

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After all, there are infinitely many fractions between any two whole numbers.

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Surely the infinity of fractions is much bigger

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than the infinity of whole numbers.

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Well, what Cantor did was to find a way to pair up

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all of the whole numbers with an infinite load of fractions.

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And this is how he did it.

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He started by arranging all the fractions in an infinite grid.

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The first row contained the whole numbers, fractions with one on the bottom.

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In the second row came the halves, fractions with two on the bottom.

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And so on. Every fraction appears somewhere in this grid.

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Where's two thirds? Second column, third row.

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Now imagine a line snaking back and forward diagonally through the fractions.

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By pulling this line straight, we can match up every fraction with one of the whole numbers.

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This means the fractions are the same sort of infinity

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as the whole numbers.

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So perhaps all infinities have the same size.

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Well, here comes the really exciting bit

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because Cantor now considers the set of all infinite decimal numbers.

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And here he proves that they give us a bigger infinity because

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however you tried to list all the infinite decimals, Cantor produced

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a clever argument to show how to construct a new decimal number

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that was missing from your list.

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Suddenly, the idea of infinity opens up.

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There are different infinities, some bigger than others.

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It's a really exciting moment.

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For me, this is like the first humans understanding how to count.

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But now we're counting in a different way. We are counting infinities.

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A door has opened and an entirely new mathematics lay before us.

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But it never helped Cantor much.

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I was in the cemetery in Halle where he is buried

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and where I had arranged to meet Professor Joe Dauben.

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He was keen to make the connections between Cantor's maths and his life.

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He suffered from manic depression.

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One of the first big breakdowns he has is in 1884

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but then around the turn of the century

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these recurrences of the mental illness

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become more and more frequent.

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A lot of people have tried to say that his mental illness

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was triggered by the incredible abstract mathematics he dealt with.

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Well, he was certainly struggling, so there may have been a connection.

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Yeah, I mean I must say, when you start to contemplate the infinite...

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I am pretty happy with the bottom end of the infinite,

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but as you build it up more and more,

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I must say I start to feel a bit unnerved

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about what's going on here and where is it going.

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For much of Cantor's life, the only place it was going was here -

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the university's sanatorium.

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There was no treatment then for manic depression

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or indeed for the paranoia that often accompanied Cantor's attacks.

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Yet the clinic was a good place to be -

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comfortable, quiet and peaceful.

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And Cantor often found his time here gave him the mental strength

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to resume his exploration of the infinite.

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Other mathematicians would be bothered by the paradoxes

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that Cantor's work had created.

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Curiously, this was one thing Cantor was not worried by.

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He was never as upset about the paradox of the infinite

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as everybody else was because Cantor believed that

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there are certain things that I have been able to show,

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we can establish with complete mathematical certainty

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and then the absolute infinite which is only in God.

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He can understand all of this and there's still that final paradox

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that is not given to us to understand, but God does.

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But there was one problem that Cantor couldn't leave

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in the hands of the Almighty,

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a problem he wrestled with for the rest of his life.

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It became known as the continuum hypothesis.

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Is there an infinity sitting between the smaller infinity

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of all the whole numbers and the larger infinity of the decimals?

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Cantor's work didn't go down well with many of his contemporaries

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but there was one mathematician from France who spoke up for him,

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arguing that Cantor's new mathematics of infinity

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was "beautiful, if pathological".

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Fortunately this mathematician was the most famous and respected mathematician of his day.

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When Bertrand Russell was asked by a French politician who he thought

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the greatest man France had produced, he replied without hesitation, "Poincare".

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The politician was surprised that he'd chosen

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the prime minister Raymond Poincare above the likes of Napoleon, Balzac.

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Russell replied, "I don't mean Raymond Poincare but his cousin,

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"the mathematician, Henri Poincare."

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Henri Poincare spent most of his life in Paris,

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a city that he loved even with its uncertain climate.

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In the last decades of the 19th century, Paris was a centre

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for world mathematics and Poincare became its leading light.

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Algebra, geometry, analysis, he was good at everything.

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His work would lead to all kinds of applications,

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from finding your way around on the underground,

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to new ways of predicting the weather.

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Poincare was very strict about his working day.

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Two hours of work in the morning

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and two hours in the early evening.

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Between these periods,

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he would let his subconscious carry on working on the problem.

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He records one moment when he had a flash of inspiration which occurred

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almost out of nowhere, just as he was getting on a bus.

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And one such flash of inspiration led to an early success.

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In 1885, King Oscar II of Sweden and Norway

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offered a prize of 2,500 crowns for anyone who could establish mathematically once and for all

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whether the solar system would continue turning like clockwork,

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or might suddenly fly apart.

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If the solar system has two planets then Newton had already proved that their orbits would be stable.

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The two bodies just travel in ellipsis round each other.

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But as soon as soon as you add three bodies like the earth, moon and sun,

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the question of whether their orbits were stable or not stumped even the great Newton.

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The problem is that now you have some 18 different variables,

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like the exact coordinates of each body

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and their velocity in each direction.

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So the equations become very difficult to solve.

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But Poincare made significant headway in sorting them out.

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Poincare simplified the problem by making successive approximations to the orbits which he believed

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wouldn't affect the final outcome significantly.

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Although he couldn't solve the problem in its entirety,

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his ideas were sophisticated enough to win him the prize.

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He developed this great sort of arsenal of techniques,

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mathematical techniques

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in order to try and solve it

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and in fact, the prize that he won was essentially

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more for the techniques than for solving the problem.

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But when Poincare's paper was being prepared for publication

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by the King's scientific advisor, Mittag-Leffler,

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one of the editors found a problem.

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Poincare realised he'd made a mistake.

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Contrary to what he had originally thought, even a small change in the

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initial conditions could end up producing vastly different orbits.

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His simplification just didn't work.

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But the result was even more important.

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The orbits Poincare had discovered indirectly led to what we now know as chaos theory.

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Understanding the mathematical rules of chaos explain why a butterfly's wings

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could create tiny changes in the atmosphere

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that ultimately might cause

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a tornado or a hurricane to appear on the other side of the world.

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So this big subject of the 20th century, chaos,

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actually came out of a mistake that Poincare made

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and he spotted at the last minute.

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Yes! So the essay had actually been published in its original form,

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and was ready to go out and Mittag-Leffler had sent copies out to various people,

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and it was to his horror when Poincare wrote to him to say, "Stop!"

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Oh, my God. This is every mathematician's worst nightmare.

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Absolutely. "Pull it!"

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Hold the presses!

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Owning up to his mistake, if anything,

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enhanced Poincare's reputation.

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He continued to produce a wide range of original work

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throughout his life.

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Not just specialist stuff either.

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He also wrote popular books, extolling the importance of maths.

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Here we go. Here's a section on the future of mathematics.

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It starts, "If we wish to foresee the future of mathematics,

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"our proper course is to study the history and present the condition of the science."

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So, I think Poincare might have approved of my journey to uncover the story of maths.

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He certainly would have approved of the next destination.

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Because to discover perhaps Poincare's most important contribution to modern mathematics,

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I had to go looking for a bridge.

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Seven bridges in fact.

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The Seven bridges of Konigsberg.

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Today the city is known as Kaliningrad, a little outpost

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of Russia on the Baltic Sea surrounded by Poland and Lithuania.

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Until 1945, however, when it was ceded to the Soviet Union,

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it was the great Prussian City of Konigsberg.

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Much of the old town sadly has been demolished.

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There is now no sign at all of two of the original seven bridges

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and several have changed out of all recognition.

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This is one of the original bridges.

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It may seem like an unlikely setting for the beginning of a mathematical story, but bear with me.

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It started as an 18th-century puzzle.

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Is there a route around the city which crosses each of these seven bridges only once?

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Finding the solution is much more difficult than it looks.

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It was eventually solved by the great mathematician Leonhard Euler,

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who in 1735 proved that it wasn't possible.

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There could not be a route that didn't cross at least one bridge twice.

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He solved the problem by making a conceptual leap.

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He realised, you don't really care what the distances are between the bridges.

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What really matters is how the bridges are connected together.

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This is a problem of a new sort of geometry of position - a problem of topology.

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Many of us use topology every day.

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Virtually all metro maps the world over

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are drawn on topological principles.

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You don't care how far the stations are from each other

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but how they are connected.

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There isn't a metro in Kaliningrad,

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but there is in the nearest other Russian city, St Petersburg.

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The topology is pretty easy on this map.

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It's the Russian I am having difficulty with.

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-Can you tell me which...?

-What's the problem?

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I want to know what station this one was.

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I had it the wrong way round even!

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Although topology had its origins in the bridges of Konigsberg,

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it was in the hands of Poincare that the subject evolved

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into a powerful new way of looking at shape.

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Some people refer to topology as bendy geometry

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because in topology, two shapes are the same if you can bend or morph

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one into another without cutting it.

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So for example if I take a football or rugby ball, topologically they

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are the same because one can be morphed into the other.

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Similarly a bagel and a tea-cup are the same because one can be morphed into the other.

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Even very complicated shapes can be unwrapped to become much simpler from a topological point of view.

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But there is no way to continuously deform a bagel to morph it into a ball.

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The hole in the middle makes these shapes topologically different.

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Poincare knew all the possible two-dimensional topological surfaces.

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But in 1904 he came up with a topological problem

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he just couldn't solve.

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If you've got a flat two-dimensional universe then Poincare worked out

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all the possible shapes he could wrap that universe up into.

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It could be a ball or a bagel with one hole, two holes or more holes in.

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But we live in a three-dimensional universe so what are the possible shapes that our universe can be?

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That question became known as the Poincare Conjecture.

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It was finally solved in 2002 here in St Petersburg

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by a Russian mathematician called Grisha Perelman.

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His proof is very difficult to understand, even for mathematicians.

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Perelman solved the problem by linking it to a completely different area of mathematics.

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To understand the shapes, he looked instead at the dynamics of the way things can flow over the shape

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which led to a description of all the possible ways

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that three dimensional space can be wrapped up in higher dimensions.

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I wondered if the man himself could help in unravelling the intricacies of his proof,

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but I'd been told that finding Perelman is almost as difficult as understanding the solution.

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The classic stereotype of a mathematician

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is a mad eccentric scientist, but I think that's a little bit unfair.

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Most of my colleagues are normal. Well, reasonably.

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But when it comes to Perelman,

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there is no doubt he is a very strange character.

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He's received prizes and offers of professorships

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from distinguished universities in the West

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but he's turned them all down.

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Recently he seems to have given up mathematics completely

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and retreated to live as a semi-recluse

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in this very modest housing estate with his mum.

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He refuses to talk to the media but I thought he might just talk to me as a fellow mathematician.

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I was wrong.

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Well, it's interesting. I think he's actually turned off his buzzer.

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Probably too many media have been buzzing it.

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I tried a neighbour and that rang but his doesn't ring at all.

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I think his papers, his mathematics speaks for itself in a way.

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I don't really need to meet the mathematician

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and in this age of Big Brother and Big Money,

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I think there's something noble about the fact he gets his kick

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out of proving theorems and not winning prizes.

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One mathematician would certainly have applauded.

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For solving any of his 23 problems, David Hilbert offered no prize

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or reward beyond the admiration of other mathematicians.

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When he sketched out the problems in Paris in 1900,

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Hilbert himself was already a mathematical star.

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And it was in Gottingen in northern Germany that he really shone.

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He was by far the most charismatic mathematician of his age.

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It's clear that everyone who knew him thought he was absolutely wonderful.

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He studied number theory and brought everything together that was there

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and then within a year or so he left that completely

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and revolutionised the theory of integral equation.

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It's always change and always something new,

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and there's hardly anybody who is...

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who was so flexible and so varied in his approach as Hilbert was.

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His work is still talked about today and his name has become attached to many mathematical terms.

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Mathematicians still use the Hilbert Space, the Hilbert Classification,

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the Hilbert Inequality and several Hilbert theorems.

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But it was his early work on equations that marked him out

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as a mathematician thinking in new ways.

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Hilbert showed that although there are infinitely many equations,

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there are ways to divide them up so that they are built

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out of just a finite set, like a set of building blocks.

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The most striking element of Hilbert's proof was that he couldn't actually construct this finite set.

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He just proved it must exist.

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Somebody criticised this as theology and not mathematics

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but they'd missed the point.

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What Hilbert was doing here was creating a new style of mathematics,

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a more abstract approach to the subject.

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You could still prove something existed,

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even though you couldn't construct it explicitly.

0:21:310:21:34

It's like saying, "I know there has to be a way to get

0:21:340:21:37

"from Gottingen to St Petersburg even though I can't tell you

0:21:370:21:42

"how to actually get there."

0:21:420:21:44

As well as challenging mathematical orthodoxies, Hilbert was also happy

0:21:440:21:49

to knock the formal hierarchies that existed in the university system in Germany at the time.

0:21:490:21:54

Other professors were quite shocked to see Hilbert out bicycling and drinking with his students.

0:21:540:22:01

-He liked very much parties.

-Yeah?

0:22:010:22:03

-Yes.

-Party animal. That's my kind of mathematician.

0:22:030:22:07

He liked very much dancing with young women. He liked very much to flirt.

0:22:070:22:13

Really? Most mathematicians I know are not the biggest of flirts.

0:22:130:22:17

'Yet this lifestyle went hand in hand with an absolute commitment to maths.'

0:22:170:22:22

Hilbert was of course somebody who thought

0:22:220:22:26

that everybody who has a mathematical skill,

0:22:260:22:30

a penguin, a woman, a man, or black, white or yellow,

0:22:300:22:36

it doesn't matter, he should do mathematics

0:22:360:22:40

and he should be admired for his.

0:22:400:22:42

The mathematics speaks for itself in a way.

0:22:420:22:46

-It doesn't matter...

-When you're a penguin.

0:22:460:22:49

Yeah, if you can prove the Riemann hypothesis, we really don't mind.

0:22:490:22:54

-Yes, mathematics was for him a universal language.

-Yes.

0:22:540:22:58

Hilbert believed that this language was powerful enough

0:22:580:23:02

to unlock all the truths of mathematics,

0:23:020:23:04

a belief he expounded in a radio interview he gave

0:23:040:23:07

on the future of mathematics on the 8th September 1930.

0:23:070:23:11

In it, he had no doubt that all his 23 problems would soon be solved

0:23:160:23:20

and that mathematics would finally be put

0:23:200:23:23

on unshakeable logical foundations.

0:23:230:23:26

There are absolutely no unsolvable problems, he declared,

0:23:260:23:30

a belief that's been held by mathematicians

0:23:300:23:32

since the time of the Ancient Greeks.

0:23:320:23:34

He ended with this clarion call, "We must know, we will know."

0:23:340:23:40

'Wir mussen wissen, wir werden wissen.'

0:23:400:23:44

Unfortunately for him, the very day before

0:23:450:23:48

in a scientific lecture that was not considered worthy of broadcast,

0:23:480:23:52

another mathematician would shatter Hilbert's dream

0:23:520:23:55

and put uncertainty at the heart of mathematics.

0:23:550:23:59

The man responsible for destroying Hilbert's belief

0:23:590:24:02

was an Austrian mathematician, Kurt Godel.

0:24:020:24:05

And it all started here - Vienna.

0:24:100:24:12

Even his admirers, and there are a great many,

0:24:120:24:15

admit that Kurt Godel was a little odd.

0:24:150:24:19

As a child, he was bright, sickly and very strange.

0:24:190:24:23

He couldn't stop asking questions.

0:24:230:24:25

So much so, that his family called him Herr Warum - Mr Why.

0:24:250:24:30

Godel lived in Vienna in the 1920s and 1930s,

0:24:300:24:35

between the fall of the Austro-Hungarian Empire

0:24:350:24:38

and its annexation by the Nazis.

0:24:380:24:39

It was a strange, chaotic and exciting time to be in the city.

0:24:390:24:45

Godel studied mathematics at Vienna University

0:24:450:24:48

but he spent most of his time in the cafes,

0:24:480:24:50

the internet chat rooms of their time,

0:24:500:24:52

where amongst games of backgammon and billiards,

0:24:520:24:55

the real intellectual excitement was taking place.

0:24:550:24:59

Particularly amongst a highly influential group

0:24:590:25:02

of philosophers and scientists called the Vienna Circle.

0:25:020:25:05

In their discussions, Kurt Godel would come up with an idea

0:25:050:25:10

that would revolutionise mathematics.

0:25:100:25:13

He'd set himself a difficult mathematical test.

0:25:130:25:15

He wanted to solve Hilbert's second problem

0:25:150:25:18

and find a logical foundation for all mathematics.

0:25:180:25:22

But what he came up with surprised even him.

0:25:220:25:25

All his efforts in mathematical logic not only couldn't provide

0:25:250:25:28

the guarantee Hilbert wanted, instead he proved the opposite.

0:25:280:25:33

Got it.

0:25:330:25:35

It's called the Incompleteness Theorem.

0:25:350:25:38

Godel proved that within any logical system for mathematics

0:25:380:25:42

there will be statements about numbers which are true

0:25:420:25:46

but which you cannot prove.

0:25:460:25:48

He starts with the statement, "This statement cannot be proved."

0:25:480:25:53

This is not a mathematical statement yet.

0:25:530:25:55

But by using a clever code based on prime numbers,

0:25:550:25:58

Godel could change this statement into a pure statement of arithmetic.

0:25:580:26:03

Now, such statements must be either true or false.

0:26:030:26:08

Hold on to your logical hats as we explore the possibilities.

0:26:080:26:13

If the statement is false, that means the statement could be proved,

0:26:130:26:17

which means it would be true, and that's a contradiction.

0:26:170:26:21

So that means, the statement must be true.

0:26:210:26:23

In other words, here is a mathematical statement that is true

0:26:230:26:28

but can't be proved.

0:26:280:26:30

Blast.

0:26:300:26:32

Godel's proof led to a crisis in mathematics.

0:26:320:26:35

What if the problem you were working on, the Goldbach conjecture, say,

0:26:350:26:39

or the Riemann hypothesis, would turn out to be true but unprovable?

0:26:390:26:43

It led to a crisis for Godel too.

0:26:430:26:46

In the autumn of 1934, he suffered the first of what became

0:26:460:26:50

a series of breakdowns and spent time in a sanatorium.

0:26:500:26:55

He was saved by the love of a good woman.

0:26:550:26:58

Adele Nimbursky was a dancer in a local night club.

0:26:580:27:02

She kept Godel alive.

0:27:020:27:06

One day, she and Godel were walking down these very steps.

0:27:060:27:10

Suddenly he was attacked by Nazi thugs.

0:27:100:27:13

Godel himself wasn't Jewish, but many of his friends in the Vienna Circle were.

0:27:130:27:17

Adele came to his rescue.

0:27:170:27:19

But it was only a temporary reprieve for Godel and for maths.

0:27:190:27:24

All across Austria and Germany, mathematics was about to die.

0:27:240:27:29

In the new German empire in the late 1930s

0:27:330:27:36

there weren't colourful balloons flying over the universities,

0:27:360:27:39

but swastikas.

0:27:390:27:41

The Nazis passed a law allowing the removal of any civil servant

0:27:410:27:46

who wasn't Aryan.

0:27:460:27:47

Academics were civil servants in Germany then and now.

0:27:470:27:51

Mathematicians suffered more than most.

0:27:530:27:56

144 in Germany would lose their jobs.

0:27:560:27:59

14 were driven to suicide or died in concentration camps.

0:27:590:28:04

But one brilliant mathematician stayed on.

0:28:070:28:10

David Hilbert helped arrange

0:28:100:28:12

for some of his brightest students to flee.

0:28:120:28:15

And he spoke out for a while about the dismissal

0:28:150:28:17

of his Jewish colleagues.

0:28:170:28:19

But soon, he too became silent.

0:28:190:28:23

It's not clear why he didn't flee himself

0:28:260:28:29

or at least protest a little more.

0:28:290:28:31

He did fall ill towards the end of his life

0:28:310:28:33

so maybe he just didn't have the energy.

0:28:330:28:35

All around him, mathematicians and scientists

0:28:350:28:38

were fleeing the Nazi regime until it was only Hilbert left

0:28:380:28:42

to witness the destruction of one of the greatest mathematical centres of all time.

0:28:420:28:47

David Hilbert died in 1943.

0:28:500:28:53

Only ten people attended the funeral

0:28:530:28:56

of the most famous mathematician of his time.

0:28:560:28:59

The dominance of Europe,

0:28:590:29:01

the centre for world maths for 500 years, was over.

0:29:010:29:05

It was time for the mathematical baton to be handed to the New World.

0:29:050:29:12

Time in fact for this place.

0:29:130:29:17

The Institute for Advanced Study had been set up in Princeton in 1930.

0:29:170:29:22

The idea was to reproduce the collegiate atmosphere

0:29:220:29:24

of the old European universities in rural New Jersey.

0:29:240:29:28

But to do this, it needed to attract the very best,

0:29:280:29:32

and it didn't need to look far.

0:29:320:29:34

Many of the brightest European mathematicians

0:29:340:29:37

were fleeing the Nazis for America.

0:29:370:29:39

People like Hermann Weyl, whose research

0:29:390:29:42

would have major significance for theoretical physics.

0:29:420:29:45

And John Von Neumann, who developed game theory

0:29:450:29:48

and was one of the pioneers of computer science.

0:29:480:29:50

The Institute quickly became the perfect place

0:29:500:29:55

to create another Gottingen in the woods.

0:29:550:29:59

One mathematician in particular made the place a home from home.

0:29:590:30:04

Every morning Kurt Godel,

0:30:040:30:06

dressed in a white linen suit and wearing a fedora,

0:30:060:30:09

would walk from his home along Mercer Street to the Institute.

0:30:090:30:13

On his way, he would stop here at number 112,

0:30:130:30:16

to pick up his closest friend, another European exile, Albert Einstein.

0:30:160:30:22

But not even relaxed, affluent Princeton could help Godel

0:30:220:30:26

finally escape his demons.

0:30:260:30:29

Einstein was always full of laughter.

0:30:290:30:31

He described Princeton as a banishment to paradise.

0:30:310:30:35

But the much younger Godel became increasingly solemn and pessimistic.

0:30:350:30:40

Slowly this pessimism turned into paranoia.

0:30:430:30:46

He spent less and less time with his fellow mathematicians in Princeton.

0:30:460:30:50

Instead, he preferred to come here to the beach, next to the ocean,

0:30:500:30:54

walk alone, thinking about the works of the great German mathematician, Leibniz.

0:30:540:30:59

But as Godel was withdrawing into his own interior world,

0:31:010:31:05

his influence on American mathematics paradoxically

0:31:050:31:09

was growing stronger and stronger.

0:31:090:31:12

And a young mathematician from just along the New Jersey coast

0:31:120:31:16

eagerly took on some of the challenges he posed.

0:31:160:31:19

# One, two, three, four, five, six, seven, eight, nine, ten

0:31:190:31:23

# Times a day I could love you... #

0:31:230:31:25

In 1950s America,

0:31:250:31:27

the majority of youngsters weren't preoccupied with mathematics.

0:31:270:31:31

Most went for a more relaxed, hedonistic lifestyle

0:31:310:31:35

in this newly affluent land of ice-cream and doughnuts.

0:31:350:31:38

But one teenager didn't indulge in the normal pursuits

0:31:380:31:42

of American adolescence but chose instead

0:31:420:31:45

to grapple with some of the major problems in mathematics.

0:31:450:31:49

From a very early age,

0:31:490:31:50

Paul Cohen was winning mathematical competitions and prizes.

0:31:500:31:55

But he found it difficult at first to discover a field in mathematics

0:31:550:31:58

where he could really make his mark...

0:31:580:32:01

Until he read about Cantor's continuum hypothesis.

0:32:010:32:05

That's the one problem, as I had learned back in Halle,

0:32:050:32:09

that Cantor just couldn't resolve.

0:32:090:32:11

It asks whether there is or there isn't an infinite set of numbers

0:32:110:32:15

bigger than the set of all whole numbers

0:32:150:32:18

but smaller than the set of all decimals.

0:32:180:32:20

It sounds straightforward, but it had foiled all attempts

0:32:200:32:24

to solve it since Hilbert made it his first problem way back in 1900.

0:32:240:32:29

With the arrogance of youth,

0:32:290:32:31

the 22-year-old Paul Cohen decided that he could do it.

0:32:310:32:36

Cohen came back a year later with the extraordinary discovery

0:32:360:32:40

that both answers could be true.

0:32:400:32:43

There was one mathematics where the continuum hypothesis

0:32:430:32:47

could be assumed to be true.

0:32:470:32:49

There wasn't a set between the whole numbers

0:32:490:32:51

and the infinite decimals.

0:32:510:32:53

But there was an equally consistent mathematics

0:32:550:32:59

where the continuum hypothesis could be assumed to be false.

0:32:590:33:03

Here, there was a set between the whole numbers and the infinite decimals.

0:33:030:33:08

It was an incredibly daring solution.

0:33:080:33:11

Cohen's proof seemed true,

0:33:110:33:13

but his method was so new that nobody was absolutely sure.

0:33:130:33:19

There was only one person whose opinion everybody trusted.

0:33:190:33:22

There was a lot of scepticism and he had to come and make a trip here,

0:33:220:33:26

to the Institute right here, to visit Godel.

0:33:260:33:29

And it was only after Godel gave his stamp of approval

0:33:290:33:32

in quite an unusual way,

0:33:320:33:34

He said, "Give me your paper", and then on Monday he put it back

0:33:340:33:37

in the box and said, "Yes, it's correct."

0:33:370:33:40

Then everything changed.

0:33:400:33:42

Today mathematicians insert a statement

0:33:430:33:46

that says whether the result depends on the continuum hypothesis.

0:33:460:33:50

We've built up two different mathematical worlds

0:33:500:33:54

in which one answer is yes and the other is no.

0:33:540:33:57

Paul Cohen really has rocked the mathematical universe.

0:33:570:34:01

It gave him fame, riches, and prizes galore.

0:34:010:34:05

He didn't publish much after his early success in the '60s.

0:34:070:34:12

But he was absolutely dynamite.

0:34:120:34:15

I can't imagine anyone better to learn from, and he was very eager

0:34:150:34:18

to learn, to teach you anything he knew or even things he didn't know.

0:34:180:34:23

With the confidence that came from solving Hilbert's first problem,

0:34:230:34:27

Cohen settled down in the mid 1960s

0:34:270:34:30

to have a go at the most important Hilbert problem of them all -

0:34:300:34:34

the eighth, the Riemann hypothesis.

0:34:340:34:36

When he died in California in 2007, 40 years later, he was still trying.

0:34:360:34:43

But like many famous mathematicians before him,

0:34:430:34:46

Riemann had defeated even him.

0:34:460:34:48

But his approach has inspired others to make progress towards a proof,

0:34:480:34:52

including one of his most famous students, Peter Sarnak.

0:34:520:34:55

I think, overall, absolutely loved the guy.

0:34:550:34:59

He was my inspiration.

0:34:590:35:01

I'm really glad I worked with him.

0:35:010:35:04

He put me on the right track.

0:35:040:35:06

Paul Cohen is a good example of the success of the great American Dream.

0:35:090:35:14

The second generation Jewish immigrant

0:35:140:35:16

becomes top American professor.

0:35:160:35:18

But I wouldn't say that his mathematics was a particularly American product.

0:35:180:35:23

Cohen was so fired up by this problem

0:35:230:35:25

that he probably would have cracked it whatever the surroundings.

0:35:250:35:29

Paul Cohen had it relatively easy.

0:35:310:35:33

But another great American mathematician of the 1960s

0:35:330:35:36

faced a much tougher struggle to make an impact.

0:35:360:35:40

Not least because she was female.

0:35:400:35:43

In the story of maths, nearly all the truly great mathematicians have been men.

0:35:430:35:48

But there have been a few exceptions.

0:35:480:35:51

There was the Russian Sofia Kovalevskaya

0:35:510:35:54

who became the first female professor of mathematics in Stockholm in 1889,

0:35:540:35:58

and won a very prestigious French mathematical prize.

0:35:580:36:03

And then Emmy Noether, a talented algebraist who fled from the Nazis

0:36:030:36:07

to America but then died before she fully realised her potential.

0:36:070:36:10

Then there is the woman who I am crossing America to find out about.

0:36:100:36:15

Julia Robinson, the first woman ever to be elected president

0:36:150:36:19

of the American Mathematical Society.

0:36:190:36:22

She was born in St Louis in 1919,

0:36:310:36:34

but her mother died when she was two.

0:36:340:36:38

She and her sister Constance moved to live with their grandmother

0:36:380:36:42

in a small community in the desert near Phoenix, Arizona.

0:36:420:36:45

Julia Robinson grew up around here.

0:36:470:36:49

I've got a photo which shows her cottage in the 1930s,

0:36:490:36:53

with nothing much around it.

0:36:530:36:55

The mountains pretty much match those over there

0:36:550:36:58

so I think she might have lived down there.

0:36:580:37:00

Julia grew up a shy, sickly girl,

0:37:010:37:04

who, when she was seven, spent a year in bed because of scarlet fever.

0:37:040:37:09

Ill-health persisted throughout her childhood.

0:37:090:37:12

She was told she wouldn't live past 40.

0:37:120:37:15

But right from the start, she had an innate mathematical ability.

0:37:150:37:20

Under the shade of the native Arizona cactus, she whiled away the time

0:37:200:37:25

playing endless counting games with stone pebbles.

0:37:250:37:28

This early searching for patterns would give her a feel

0:37:280:37:31

and love of numbers that would last for the rest of her life.

0:37:310:37:35

But despite showing an early brilliance, she had to continually

0:37:350:37:39

fight at school and college to simply be allowed to keep doing maths.

0:37:390:37:44

As a teenager, she was the only girl in the maths class

0:37:440:37:47

but had very little encouragement.

0:37:470:37:50

The young Julia sought intellectual stimulation elsewhere.

0:37:500:37:55

Julia loved listening to a radio show called the University Explorer

0:37:550:37:59

and the 53rd episode was all about mathematics.

0:37:590:38:02

The broadcaster described how he discovered

0:38:020:38:04

despite their esoteric language and their seclusive nature,

0:38:040:38:08

mathematicians are the most interesting and inspiring creatures.

0:38:080:38:12

For the first time, Julia had found out that there were mathematicians,

0:38:120:38:16

not just mathematics teachers.

0:38:160:38:17

There was a world of mathematics out there,

0:38:170:38:20

and she wanted to be part of it.

0:38:200:38:22

The doors to that world opened here at the University of California,

0:38:260:38:29

at Berkeley near San Francisco.

0:38:290:38:31

She was absolutely obsessed that she wanted to go to Berkeley.

0:38:330:38:38

She wanted to go away to some place where there were mathematicians.

0:38:380:38:44

Berkeley certainly had mathematicians,

0:38:440:38:46

including a number theorist called Raphael Robinson.

0:38:460:38:50

In their frequent walks around the campus

0:38:500:38:53

they found they had more than just a passion for mathematics. They married in 1952.

0:38:530:38:59

Julia got her PhD and settled down

0:38:590:39:03

to what would turn into her lifetime's work -

0:39:030:39:05

Hilbert's tenth problem.

0:39:050:39:07

She thought about it all the time.

0:39:070:39:10

She said to me she just wouldn't wanna die without knowing that answer

0:39:100:39:14

and it had become an obsession.

0:39:140:39:16

Julia's obsession has been shared with many other mathematicians

0:39:170:39:21

since Hilbert had first posed it back in 1900.

0:39:210:39:24

His tenth problem asked if there was some universal method

0:39:240:39:28

that could tell whether any equation had whole number solutions or not.

0:39:280:39:34

Nobody had been able to solve it.

0:39:340:39:36

In fact, the growing belief was

0:39:360:39:39

that no such universal method was possible.

0:39:390:39:42

How on earth could you prove that,

0:39:420:39:44

however ingenious you were, you'd never come up with a method?

0:39:440:39:48

With the help of colleagues,

0:39:500:39:51

Julia developed what became known as the Robinson hypothesis.

0:39:510:39:55

This argued that to show no such method existed,

0:39:550:39:58

all you had to do was to cook up one equation whose solutions

0:39:580:40:03

were a very specific set of numbers.

0:40:030:40:06

The set of numbers needed to grow exponentially,

0:40:060:40:09

like taking powers of two, yet still be captured by the equations

0:40:090:40:13

at the heart of Hilbert's problem.

0:40:130:40:16

Try as she might, Robinson just couldn't find this set.

0:40:160:40:21

For the tenth problem to be finally solved,

0:40:210:40:25

there needed to be some fresh inspiration.

0:40:250:40:28

That came from 5,000 miles away - St Petersburg in Russia.

0:40:280:40:34

Ever since the great Leonhard Euler set up shop here

0:40:340:40:37

in the 18th century,

0:40:370:40:39

the city has been famous for its mathematics and mathematicians.

0:40:390:40:42

Here in the Steklov Institute,

0:40:420:40:44

some of the world's brightest mathematicians

0:40:440:40:47

have set out their theorems and conjectures.

0:40:470:40:50

This morning, one of them is giving a rare seminar.

0:40:500:40:54

It's tough going even if you speak Russian,

0:40:570:41:00

which unfortunately I don't.

0:41:000:41:02

But we do get a break in the middle to recover before the final hour.

0:41:020:41:06

There is a kind of rule in seminars.

0:41:060:41:08

The first third is for everyone, the second third for the experts

0:41:080:41:12

and the last third is just for the lecturer.

0:41:120:41:16

I think that's what we're going to get next.

0:41:160:41:19

The lecturer is Yuri Matiyasevich and he is explaining

0:41:190:41:22

his latest work on - what else? - the Riemann hypothesis.

0:41:220:41:26

As a bright young graduate student in 1965, Yuri's tutor

0:41:280:41:33

suggested he have a go at another Hilbert problem,

0:41:330:41:36

the one that had in fact preoccupied Julia Robinson.

0:41:360:41:39

Hilbert's tenth.

0:41:390:41:40

It was the height of the Cold War.

0:41:430:41:45

Perhaps Matiyasevich could succeed for Russia

0:41:450:41:48

where Julia and her fellow American mathematicians had failed.

0:41:480:41:52

-At first I did not like their approach.

-Oh, right.

0:41:520:41:55

The statement looked to me rather strange and artificial

0:41:550:41:59

but after some time I understood it was quite natural,

0:41:590:42:03

and then I understood that she had a new brilliant idea

0:42:030:42:07

and I just started to further develop it.

0:42:070:42:10

In January 1970, he found the vital last piece in the jigsaw.

0:42:110:42:17

He saw how to capture the famous Fibonacci sequence of numbers

0:42:170:42:21

using the equations that were at the heart of Hilbert's problem.

0:42:210:42:26

Building on the work of Julia and her colleagues,

0:42:260:42:28

he had solved the tenth.

0:42:280:42:30

He was just 22 years old.

0:42:300:42:34

The first person he wanted to tell was the woman he owed so much to.

0:42:340:42:37

I got no answer

0:42:390:42:41

and I believed they were lost in the mail.

0:42:410:42:44

It was quite natural because it was Soviet time.

0:42:440:42:47

But back in California, Julia had heard rumours

0:42:470:42:50

through the mathematical grapevine that the problem had been solved.

0:42:500:42:54

And she contacted Yuri herself.

0:42:540:42:57

She said, I just had to wait for you to grow up

0:42:580:43:01

to get the answer, because she had started work in 1948.

0:43:010:43:06

When Yuri was just a baby.

0:43:060:43:07

Then he responds by thanking her

0:43:070:43:11

and saying that the credit is as much hers as it is his.

0:43:110:43:16

YURI: I met Julia one year later.

0:43:180:43:20

It was in Bucharest. I suggested after the conference in Bucharest

0:43:200:43:25

Julia and her husband Raphael came to see me here in Leningrad.

0:43:250:43:30

Together, Julia and Yuri worked on several other mathematical problems

0:43:300:43:35

until shortly before Julia died in 1985.

0:43:350:43:39

She was just 55 years old.

0:43:390:43:41

She was able to find the new ways.

0:43:410:43:45

Many mathematicians just combine previous known methods

0:43:450:43:49

to solve new problems and she had really new ideas.

0:43:490:43:55

Although Julia Robinson showed there was no universal method

0:43:550:43:59

to solve all equations in whole numbers,

0:43:590:44:01

mathematicians were still interested in finding methods

0:44:010:44:05

to solve special classes of equations.

0:44:050:44:08

It would be in France in the early 19th century,

0:44:080:44:11

in one of the most extraordinary stories

0:44:110:44:13

in the history of mathematics, that methods were developed

0:44:130:44:17

to understand why certain equations could be solved

0:44:170:44:20

while others couldn't.

0:44:200:44:21

It's early morning in Paris on the 29th May 1832.

0:44:270:44:32

Evariste Galois is about to fight for his very life.

0:44:320:44:37

It is the reign of the reactionary Bourbon King, Charles X,

0:44:370:44:40

and Galois, like many angry young men in Paris then,

0:44:400:44:43

is a republican revolutionary.

0:44:430:44:46

Unlike the rest of his comrades though, he has another passion - mathematics.

0:44:460:44:52

He had just spent four months in jail.

0:44:530:44:56

Then, in a mysterious saga of unrequited love,

0:44:560:45:00

he is challenged to a duel.

0:45:000:45:02

He'd been up the whole previous night

0:45:020:45:04

refining a new language for mathematics he'd developed.

0:45:040:45:07

Galois believed that mathematics shouldn't be the study of number and shape, but the study of structure.

0:45:070:45:14

Perhaps he was still pre-occupied with his maths.

0:45:140:45:17

GUNSHOT

0:45:170:45:18

There was only one shot fired that morning.

0:45:180:45:21

Galois died the next day, just 20 years old.

0:45:210:45:27

It was one of mathematics greatest losses.

0:45:270:45:30

Only by the beginning of the 20th century

0:45:300:45:33

would Galois be fully appreciated and his ideas fully realised.

0:45:330:45:37

Galois had discovered new techniques to be able to tell

0:45:420:45:46

whether certain equations could have solutions or not.

0:45:460:45:49

The symmetry of certain geometric objects seemed to be the key.

0:45:490:45:54

His idea of using geometry to analyse equations

0:45:540:45:58

would be picked up in the 1920s by another Parisian mathematician, Andre Weil.

0:45:580:46:03

I was very much interested and so far as school was concerned

0:46:030:46:09

quite successful in all possible branches.

0:46:090:46:13

And he was. After studying in Germany as well as France,

0:46:130:46:17

Andre settled down at this apartment in Paris

0:46:170:46:21

which he shared with his more-famous sister, the writer Simone Weil.

0:46:210:46:25

But when the Second World War broke out, he found himself in very different circumstances.

0:46:250:46:31

He dodged the draft by fleeing to Finland where he was almost executed for being a Russian spy.

0:46:310:46:37

On his return to France he was put in prison in Rouen to await trial for desertion.

0:46:370:46:42

At the trial, the judge gave him a choice.

0:46:420:46:45

Five more years in prison or serve in a combat unit.

0:46:450:46:49

He chose to join the French army, a lucky choice

0:46:490:46:52

because just before the Germans invaded a few months later,

0:46:520:46:56

all the prisoners were killed.

0:46:560:46:58

Weil only spent a few months in prison, but this time was crucial for his mathematics.

0:46:580:47:05

Because here he built on the ideas of Galois and first developed algebraic geometry

0:47:050:47:11

a whole new language for understanding solutions to equations.

0:47:110:47:15

Galois had shown how new mathematical structures

0:47:150:47:18

can be used to reveal the secrets behind equations.

0:47:180:47:22

Weil's work led him to theorems

0:47:220:47:24

that connected number theory, algebra, geometry and topology

0:47:240:47:28

and are one of the greatest achievements of modern mathematics.

0:47:280:47:33

Without Andre Weil, we would never have heard

0:47:330:47:36

of the strangest man in the history of maths, Nicolas Bourbaki.

0:47:360:47:41

There are no photos of Bourbaki in existence but we do know he was born in this cafe in the Latin Quarter

0:47:430:47:50

in 1934 when it was a proper cafe, the cafe Capoulade,

0:47:500:47:54

and not the fast food joint it has now become.

0:47:540:47:58

Just down the road, I met up with Bourbaki expert David Aubin.

0:47:580:48:03

When I was a graduate student I got quite frightened

0:48:030:48:06

when I used to go into the library

0:48:060:48:08

because this guy Bourbaki had written so many books.

0:48:080:48:10

Something like 30 or 40 altogether.

0:48:100:48:14

In analysis, in geometry, in topology, it was all new foundations.

0:48:140:48:19

Virtually everyone studying Maths seriously anywhere in the world

0:48:190:48:23

in the 1950s, '60s and '70s would have read Nicolas Bourbaki.

0:48:230:48:28

He applied for membership of the American Maths Society, I heard.

0:48:280:48:31

At which point he was denied membership

0:48:310:48:33

-on the grounds that he didn't exist.

-Oh!

0:48:330:48:36

The Americans were right.

0:48:360:48:38

Nicolas Bourbaki does not exist at all. And never has.

0:48:380:48:41

Bourbaki is in fact the nom de plume for a group of French mathematicians

0:48:410:48:46

led by Andre Weil who decided to write a coherent account

0:48:460:48:49

of the mathematics of the 20th century.

0:48:490:48:52

Most of the time mathematicians like to have their own names on theorems.

0:48:520:48:57

But for the Bourbaki group,

0:48:570:48:59

the aims of the project overrode any desire for personal glory.

0:48:590:49:03

After the Second World War, the Bourbaki baton was handed down

0:49:030:49:07

to the next generation of French mathematicians.

0:49:070:49:10

And their most brilliant member was Alexandre Grothendieck.

0:49:100:49:15

Here at the IHES in Paris,

0:49:150:49:17

the French equivalent of Princeton's Institute for Advanced Study,

0:49:170:49:21

Grothendieck held court at his famous seminars in the 1950s and 1960s.

0:49:210:49:27

He had this incredible charisma.

0:49:290:49:33

He had this amazing ability to see a young person and somehow know

0:49:330:49:40

what kind of contribution this person could make to this incredible vision

0:49:400:49:46

he had of how mathematics could be.

0:49:460:49:48

And this vision enabled him to get across some very difficult ideas indeed.

0:49:480:49:54

He says, "Suppose you want to open a walnut.

0:49:540:49:58

"So the standard thing is you take a nutcracker and you just break it open."

0:49:580:50:02

And he says his approach is more like,

0:50:020:50:04

you take this walnut and you put it out in the snow

0:50:040:50:08

and you leave it there for a few months

0:50:080:50:10

and then when you come back to it, it just opens.

0:50:100:50:13

Grothendieck is a Structuralist.

0:50:130:50:15

What he's interested in are the hidden structures

0:50:150:50:19

underneath all mathematics.

0:50:190:50:22

Only when you get down to the very basic architecture and think in very general terms

0:50:220:50:27

will the patterns in mathematics become clear.

0:50:270:50:31

Grothendieck produced a new powerful language to see structures in a new way.

0:50:310:50:37

It was like living in a world of black and white

0:50:370:50:39

and suddenly having the language to see the world in colour.

0:50:390:50:42

It's a language that mathematicians have been using ever since

0:50:420:50:46

to solve problems in number theory, geometry, even fundamental physics.

0:50:460:50:51

But in the late 1960s, Grothendieck decided

0:50:530:50:56

to turn his back on mathematics after he discovered politics.

0:50:560:51:01

He believed that the threat of nuclear war and the questions

0:51:010:51:06

of nuclear disarmament were more important than mathematics

0:51:060:51:12

and that people who continue to do mathematics

0:51:120:51:17

rather than confront this threat of nuclear war

0:51:170:51:21

were doing harm in the world.

0:51:210:51:22

Grothendieck decided to leave Paris

0:51:260:51:29

and move back to the south of France where he grew up.

0:51:290:51:32

Bursts of radical politics followed and then a nervous breakdown.

0:51:320:51:36

He moved to the Pyrenees and became a recluse.

0:51:360:51:40

He's now lost all contact with his old friends and mathematical colleagues.

0:51:400:51:45

Nevertheless, the legacy of his achievements means that Grothendieck stands

0:51:460:51:51

alongside Cantor, Godel and Hilbert as someone who has transformed the mathematical landscape.

0:51:510:51:57

He changed the whole subject in a really fundamental way. It will never go back.

0:51:590:52:03

Certainly, he's THE dominant figure of the 20th century.

0:52:030:52:08

I've come back to England, though,

0:52:160:52:18

thinking again about another seminal figure of the 20th century.

0:52:180:52:22

The person that started it all off, David Hilbert.

0:52:220:52:26

Of the 23 problems Hilbert set mathematicians in the year 1900,

0:52:260:52:32

most have now been solved.

0:52:320:52:34

However there is one great exception.

0:52:340:52:37

The Riemann hypothesis, the eighth on Hilbert's list.

0:52:370:52:40

That is still the holy grail of mathematics.

0:52:400:52:43

Hilbert's lecture inspired a generation to pursue their mathematical dreams.

0:52:440:52:50

This morning, in the town where I grew up, I hope to inspire another generation.

0:52:500:52:55

CHEERING AND APPLAUSE

0:52:550:52:57

Thank you. Hello. My name's Marcus du Sautoy

0:53:010:53:04

and I'm a Professor of Mathematics

0:53:040:53:05

up the road at the University of Oxford.

0:53:050:53:08

It was actually in this school here,

0:53:080:53:10

in fact this classroom is where I discovered my love for mathematics.

0:53:100:53:14

'This love of mathematics that I first acquired

0:53:140:53:17

'here in my old comprehensive school still drives me now.

0:53:170:53:20

'It's a love of solving problems.

0:53:200:53:22

'There are so many problems I could tell them about,

0:53:220:53:25

'but I've chosen my favourite.'

0:53:250:53:27

I think that a mathematician is a pattern searcher

0:53:270:53:30

and that's really what mathematicians try and do.

0:53:300:53:33

We try and understand the patterns and the structure

0:53:330:53:37

and the logic to explain the way the world around us works.

0:53:370:53:40

And this is really at the heart of the Riemann hypothesis.

0:53:400:53:43

The task is - is there any pattern in these numbers which can help me say

0:53:430:53:48

where the next number will be?

0:53:480:53:50

What's the next one after 31? How can I tell?

0:53:500:53:52

'These numbers are, of course, prime numbers -

0:53:520:53:55

'the building blocks of mathematics.'

0:53:550:53:58

'The Riemann hypothesis, a conjecture about the distribution

0:53:580:54:01

'of the primes, goes to the very heart of our subject.'

0:54:010:54:04

Why on earth is anybody interested in these primes?

0:54:040:54:07

Why is the army interested in primes, why are spies interested?

0:54:070:54:11

-Isn't it to encrypt stuff?

-Exactly.

0:54:110:54:14

I study this stuff cos I think it's all really beautiful and elegant

0:54:140:54:18

but actually, there's a lot of people

0:54:180:54:20

who are interested in these numbers because of their very practical use.

0:54:200:54:24

'The bizarre thing is that the more abstract and difficult mathematics becomes,

0:54:240:54:28

'the more it seems to have applications in the real world.

0:54:280:54:32

'Mathematics now pervades every aspect of our lives.

0:54:320:54:36

'Every time we switch on the television, plug in a computer, pay with a credit card.

0:54:360:54:41

'There's now a million dollars for anyone who can solve the Riemann hypothesis.

0:54:410:54:46

'But there's more at stake than that.'

0:54:460:54:48

Anybody who proves this theorem will be remembered forever.

0:54:480:54:51

They'll be on that board ahead of any of those other mathematicians.

0:54:510:54:55

'That's because the Riemann hypothesis is a corner-stone of maths.

0:54:550:54:59

'Thousands of theorems depend on it being true.

0:54:590:55:02

'Very few mathematicians think that it isn't true.

0:55:020:55:06

'But mathematics is about proof and until we can prove it

0:55:060:55:10

'there will still be doubt.'

0:55:100:55:12

Maths has grown out of this passion to get rid of doubt.

0:55:120:55:17

This is what I have learned in my journey through the history of mathematics.

0:55:170:55:20

Mathematicians like Archimedes and al-Khwarizmi, Gauss and Grothendieck

0:55:200:55:25

were driven to understand the precise way numbers and space work.

0:55:250:55:30

Maths in action, that one.

0:55:300:55:33

It's beautiful. Really nice.

0:55:330:55:35

Using the language of mathematics, they have told us stories

0:55:350:55:39

that remain as true today as they were when they were first told.

0:55:390:55:43

In the Mediterranean, I discovered the origins of geometry.

0:55:430:55:48

Mathematicians and philosophers flocked to Alexandria

0:55:480:55:51

driven by a thirst for knowledge and the pursuit of excellence.

0:55:510:55:55

In India, I learned about another discovery

0:55:550:55:59

that it would be impossible to imagine modern life without.

0:55:590:56:02

So here we are in one of the true holy sites of the mathematical world.

0:56:020:56:07

Up here are some numbers,

0:56:070:56:10

and here's the new number.

0:56:100:56:12

Its zero.

0:56:120:56:14

In the Middle East, I was amazed at al-Khwarizmi's invention of algebra.

0:56:140:56:19

He developed systematic ways to analyse problems

0:56:190:56:22

so that the solutions would work whatever numbers you took.

0:56:220:56:26

In the Golden Age of Mathematics,

0:56:260:56:28

in Europe in the 18th and 19th centuries, I found how maths

0:56:280:56:31

discovered new ways for analysing bodies in motion and new geometries

0:56:310:56:35

that helped us understand the very strange shape of space.

0:56:350:56:40

It is with Riemann's work that we finally have

0:56:400:56:43

the mathematical glasses to be able to explore such worlds of the mind.

0:56:430:56:49

And now my journey into the abstract world of 20th-century mathematics

0:56:490:56:53

has revealed that maths is the true language

0:56:530:56:56

the universe is written in,

0:56:560:56:58

the key to understanding the world around us.

0:56:580:57:02

Mathematicians aren't motivated by money and material gain

0:57:020:57:05

or even by practical applications of their work.

0:57:050:57:09

For us, it is the glory of solving one of the great unsolved problems

0:57:090:57:13

that have outwitted previous generations of mathematicians.

0:57:130:57:18

Hilbert was right. It's the unsolved problems of mathematics

0:57:180:57:21

that make it a living subject,

0:57:210:57:23

which obsess each new generation of mathematicians.

0:57:230:57:27

Despite all the things we've discovered over the last seven millennia,

0:57:270:57:30

there are still many things we don't understand.

0:57:300:57:33

And its Hilbert's call of, "We must know, we will know", which drives mathematics.

0:57:330:57:39

You can learn more about The Story Of Maths

0:57:420:57:45

with the Open University at...

0:57:450:57:48

Subtitled by Red Bee Media Ltd

0:58:000:58:03

E-mail [email protected]

0:58:030:58:06

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