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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.
295
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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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00:19:12,920 –> 00:19:18,560
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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00:19:32,960 –> 00:19:36,000
One mathematician would certainly have applauded.
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00:19:36,000 –> 00:19:40,440
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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00:19:59,440 –> 00:20:05,560
He was by far the most charismatic mathematician of his age.
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00:20:05,560 –> 00:20:09,960
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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00:20:29,840 –> 00:20:34,800
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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00:20:54,800 –> 00:20:57,520
as a mathematician thinking in new ways.
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00:20:57,520 –> 00:21:01,480
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.
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It’s like saying, “I know there has to be a way to get
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“from Gottingen to St Petersburg even though I can’t tell you
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“how to actually get there.”
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As well as challenging mathematical orthodoxies, Hilbert was also happy
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to knock the formal hierarchies that existed in the university system in Germany at the time.
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Other professors were quite shocked to see Hilbert out bicycling and drinking with his students.
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- He liked very much parties.
- Yeah?
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00:22:03,440 –> 00:22:07,240
- Yes.
- Party animal. That’s my kind of mathematician.
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00:22:07,240 –> 00:22:13,360
He liked very much dancing with young women. He liked very much to flirt.
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00:22:13,360 –> 00:22:17,880
Really? Most mathematicians I know are not the biggest of flirts.
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00:22:17,880 –> 00:22:22,000
‘Yet this lifestyle went hand in hand with an absolute commitment to maths.’
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00:22:22,000 –> 00:22:26,200
Hilbert was of course somebody who thought
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00:22:26,200 –> 00:22:30,240
that everybody who has a mathematical skill,
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00:22:30,240 –> 00:22:36,400
a penguin, a woman, a man, or black, white or yellow,
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it doesn’t matter, he should do mathematics
347
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and he should be admired for his.
348
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The mathematics speaks for itself in a way.
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- It doesn’t matter…
- When you’re a penguin.
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Yeah, if you can prove the Riemann hypothesis, we really don’t mind.
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00:22:54,360 –> 00:22:58,280
- Yes, mathematics was for him a universal language.
- Yes.
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Hilbert believed that this language was powerful enough
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to unlock all the truths of mathematics,
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a belief he expounded in a radio interview he gave
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on the future of mathematics on the 8th September 1930.
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In it, he had no doubt that all his 23 problems would soon be solved
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and that mathematics would finally be put
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on unshakeable logical foundations.
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There are absolutely no unsolvable problems, he declared,
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a belief that’s been held by mathematicians
361
00:23:32,520 –> 00:23:34,480
since the time of the Ancient Greeks.
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00:23:34,480 –> 00:23:40,040
He ended with this clarion call, “We must know, we will know.”
363
00:23:40,040 –> 00:23:44,640
‘Wir mussen wissen, wir werden wissen.’
364
00:23:45,960 –> 00:23:48,480
Unfortunately for him, the very day before
365
00:23:48,480 –> 00:23:52,320
in a scientific lecture that was not considered worthy of broadcast,
366
00:23:52,320 –> 00:23:55,520
another mathematician would shatter Hilbert’s dream
367
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and put uncertainty at the heart of mathematics.
368
00:23:59,480 –> 00:24:02,400
The man responsible for destroying Hilbert’s belief
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00:24:02,400 –> 00:24:05,520
was an Austrian mathematician, Kurt Godel.
370
00:24:10,400 –> 00:24:12,440
And it all started here - Vienna.
371
00:24:12,440 –> 00:24:15,360
Even his admirers, and there are a great many,
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00:24:15,360 –> 00:24:19,920
admit that Kurt Godel was a little odd.
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00:24:19,920 –> 00:24:23,840
As a child, he was bright, sickly and very strange.
374
00:24:23,840 –> 00:24:25,880
He couldn’t stop asking questions.
375
00:24:25,880 –> 00:24:30,720
So much so, that his family called him Herr Warum - Mr Why.
376
00:24:30,720 –> 00:24:35,160
Godel lived in Vienna in the 1920s and 1930s,
377
00:24:35,160 –> 00:24:38,000
between the fall of the Austro-Hungarian Empire
378
00:24:38,000 –> 00:24:39,960
and its annexation by the Nazis.
379
00:24:39,960 –> 00:24:45,520
It was a strange, chaotic and exciting time to be in the city.
380
00:24:45,520 –> 00:24:48,160
Godel studied mathematics at Vienna University
381
00:24:48,160 –> 00:24:50,600
but he spent most of his time in the cafes,
382
00:24:50,600 –> 00:24:52,960
the internet chat rooms of their time,
383
00:24:52,960 –> 00:24:55,920
where amongst games of backgammon and billiards,
384
00:24:55,920 –> 00:24:59,040
the real intellectual excitement was taking place.
385
00:24:59,040 –> 00:25:02,320
Particularly amongst a highly influential group
386
00:25:02,320 –> 00:25:05,920
of philosophers and scientists called the Vienna Circle.
387
00:25:05,920 –> 00:25:10,080
In their discussions, Kurt Godel would come up with an idea
388
00:25:10,080 –> 00:25:13,000
that would revolutionise mathematics.
389
00:25:13,000 –> 00:25:15,960
He’d set himself a difficult mathematical test.
390
00:25:15,960 –> 00:25:18,760
He wanted to solve Hilbert’s second problem
391
00:25:18,760 –> 00:25:22,000
and find a logical foundation for all mathematics.
392
00:25:22,000 –> 00:25:25,520
But what he came up with surprised even him.
393
00:25:25,520 –> 00:25:28,960
All his efforts in mathematical logic not only couldn’t provide
394
00:25:28,960 –> 00:25:33,840
the guarantee Hilbert wanted, instead he proved the opposite.
395
00:25:33,840 –> 00:25:35,440
Got it.
396
00:25:35,440 –> 00:25:38,800
It’s called the Incompleteness Theorem.
397
00:25:38,800 –> 00:25:42,360
Godel proved that within any logical system for mathematics
398
00:25:42,360 –> 00:25:46,200
there will be statements about numbers which are true
399
00:25:46,200 –> 00:25:48,200
but which you cannot prove.
400
00:25:48,200 –> 00:25:53,000
He starts with the statement, “This statement cannot be proved.”
401
00:25:53,000 –> 00:25:55,480
This is not a mathematical statement yet.
402
00:25:55,480 –> 00:25:58,360
But by using a clever code based on prime numbers,
403
00:25:58,360 –> 00:26:03,480
Godel could change this statement into a pure statement of arithmetic.
404
00:26:03,480 –> 00:26:08,640
Now, such statements must be either true or false.
405
00:26:08,640 –> 00:26:13,320
Hold on to your logical hats as we explore the possibilities.
406
00:26:13,320 –> 00:26:17,960
If the statement is false, that means the statement could be proved,
407
00:26:17,960 –> 00:26:21,320
which means it would be true, and that’s a contradiction.
408
00:26:21,320 –> 00:26:23,880
So that means, the statement must be true.
409
00:26:23,880 –> 00:26:28,320
In other words, here is a mathematical statement that is true
410
00:26:28,320 –> 00:26:30,840
but can’t be proved.
411
00:26:30,840 –> 00:26:32,440
Blast.
412
00:26:32,440 –> 00:26:35,520
Godel’s proof led to a crisis in mathematics.
413
00:26:35,520 –> 00:26:39,320
What if the problem you were working on, the Goldbach conjecture, say,
414
00:26:39,320 –> 00:26:43,600
or the Riemann hypothesis, would turn out to be true but unprovable?
415
00:26:43,600 –> 00:26:46,720
It led to a crisis for Godel too.
416
00:26:46,720 –> 00:26:50,400
In the autumn of 1934, he suffered the first of what became
417
00:26:50,400 –> 00:26:55,520
a series of breakdowns and spent time in a sanatorium.
418
00:26:55,520 –> 00:26:58,960
He was saved by the love of a good woman.
419
00:26:58,960 –> 00:27:02,880
Adele Nimbursky was a dancer in a local night club.
420
00:27:02,880 –> 00:27:06,200
She kept Godel alive.
421
00:27:06,200 –> 00:27:10,040
One day, she and Godel were walking down these very steps.
422
00:27:10,040 –> 00:27:13,120
Suddenly he was attacked by Nazi thugs.
423
00:27:13,120 –> 00:27:17,360
Godel himself wasn’t Jewish, but many of his friends in the Vienna Circle were.
424
00:27:17,360 –> 00:27:19,840
Adele came to his rescue.
425
00:27:19,840 –> 00:27:24,400
But it was only a temporary reprieve for Godel and for maths.
426
00:27:24,400 –> 00:27:29,680
All across Austria and Germany, mathematics was about to die.
427
00:27:33,680 –> 00:27:36,240
In the new German empire in the late 1930s
428
00:27:36,240 –> 00:27:39,760
there weren’t colourful balloons flying over the universities,
429
00:27:39,760 –> 00:27:41,600
but swastikas.
430
00:27:41,600 –> 00:27:46,280
The Nazis passed a law allowing the removal of any civil servant
431
00:27:46,280 –> 00:27:47,680
who wasn’t Aryan.
432
00:27:47,680 –> 00:27:51,200
Academics were civil servants in Germany then and now.
433
00:27:53,520 –> 00:27:56,200
Mathematicians suffered more than most.
434
00:27:56,200 –> 00:27:59,600
144 in Germany would lose their jobs.
435
00:27:59,600 –> 00:28:04,040
14 were driven to suicide or died in concentration camps.
436
00:28:07,680 –> 00:28:10,600
But one brilliant mathematician stayed on.
437
00:28:10,600 –> 00:28:12,400
David Hilbert helped arrange
438
00:28:12,400 –> 00:28:15,000
for some of his brightest students to flee.
439
00:28:15,000 –> 00:28:17,640
And he spoke out for a while about the dismissal
440
00:28:17,640 –> 00:28:19,200
of his Jewish colleagues.
441
00:28:19,200 –> 00:28:23,400
But soon, he too became silent.
442
00:28:26,720 –> 00:28:29,240
It’s not clear why he didn’t flee himself
443
00:28:29,240 –> 00:28:31,320
or at least protest a little more.
444
00:28:31,320 –> 00:28:33,600
He did fall ill towards the end of his life
445
00:28:33,600 –> 00:28:35,800
so maybe he just didn’t have the energy.
446
00:28:35,800 –> 00:28:38,440
All around him, mathematicians and scientists
447
00:28:38,440 –> 00:28:42,160
were fleeing the Nazi regime until it was only Hilbert left
448
00:28:42,160 –> 00:28:47,480
to witness the destruction of one of the greatest mathematical centres of all time.
449
00:28:50,000 –> 00:28:53,640
David Hilbert died in 1943.
450
00:28:53,640 –> 00:28:56,360
Only ten people attended the funeral
451
00:28:56,360 –> 00:28:59,600
of the most famous mathematician of his time.
452
00:28:59,600 –> 00:29:01,880
The dominance of Europe,
453
00:29:01,880 –> 00:29:05,680
the centre for world maths for 500 years, was over.
454
00:29:05,680 –> 00:29:12,000
It was time for the mathematical baton to be handed to the New World.
455
00:29:13,840 –> 00:29:17,120
Time in fact for this place.
456
00:29:17,120 –> 00:29:22,040
The Institute for Advanced Study had been set up in Princeton in 1930.
457
00:29:22,040 –> 00:29:24,880
The idea was to reproduce the collegiate atmosphere
458
00:29:24,880 –> 00:29:28,880
of the old European universities in rural New Jersey.
459
00:29:28,880 –> 00:29:32,200
But to do this, it needed to attract the very best,
460
00:29:32,200 –> 00:29:34,280
and it didn’t need to look far.
461
00:29:34,280 –> 00:29:37,480
Many of the brightest European mathematicians
462
00:29:37,480 –> 00:29:39,920
were fleeing the Nazis for America.
463
00:29:39,920 –> 00:29:42,520
People like Hermann Weyl, whose research
464
00:29:42,520 –> 00:29:45,680
would have major significance for theoretical physics.
465
00:29:45,680 –> 00:29:48,280
And John Von Neumann, who developed game theory
466
00:29:48,280 –> 00:29:50,840
and was one of the pioneers of computer science.
467
00:29:50,840 –> 00:29:55,400
The Institute quickly became the perfect place
468
00:29:55,400 –> 00:29:59,440
to create another Gottingen in the woods.
469
00:29:59,440 –> 00:30:04,760
One mathematician in particular made the place a home from home.
470
00:30:04,760 –> 00:30:06,320
Every morning Kurt Godel,
471
00:30:06,320 –> 00:30:09,360
dressed in a white linen suit and wearing a fedora,
472
00:30:09,360 –> 00:30:13,040
would walk from his home along Mercer Street to the Institute.
473
00:30:13,040 –> 00:30:16,520
On his way, he would stop here at number 112,
474
00:30:16,520 –> 00:30:22,640
to pick up his closest friend, another European exile, Albert Einstein.
475
00:30:22,640 –> 00:30:26,960
But not even relaxed, affluent Princeton could help Godel
476
00:30:26,960 –> 00:30:29,040
finally escape his demons.
477
00:30:29,040 –> 00:30:31,640
Einstein was always full of laughter.
478
00:30:31,640 –> 00:30:35,520
He described Princeton as a banishment to paradise.
479
00:30:35,520 –> 00:30:40,080
But the much younger Godel became increasingly solemn and pessimistic.
480
00:30:43,160 –> 00:30:46,400
Slowly this pessimism turned into paranoia.
481
00:30:46,400 –> 00:30:50,520
He spent less and less time with his fellow mathematicians in Princeton.
482
00:30:50,520 –> 00:30:54,200
Instead, he preferred to come here to the beach, next to the ocean,
483
00:30:54,200 –> 00:30:59,240
walk alone, thinking about the works of the great German mathematician, Leibniz.
484
00:31:01,400 –> 00:31:05,320
But as Godel was withdrawing into his own interior world,
485
00:31:05,320 –> 00:31:09,320
his influence on American mathematics paradoxically
486
00:31:09,320 –> 00:31:12,000
was growing stronger and stronger.
487
00:31:12,000 –> 00:31:16,160
And a young mathematician from just along the New Jersey coast
488
00:31:16,160 –> 00:31:19,840
eagerly took on some of the challenges he posed.
489
00:31:19,840 –> 00:31:23,760
One, two, three, four, five, six, seven, eight, nine, ten
490
00:31:23,760 –> 00:31:25,880
Times a day I could love you…
491
00:31:25,880 –> 00:31:27,040
In 1950s America,
492
00:31:27,040 –> 00:31:31,440
the majority of youngsters weren’t preoccupied with mathematics.
493
00:31:31,440 –> 00:31:35,160
Most went for a more relaxed, hedonistic lifestyle
494
00:31:35,160 –> 00:31:38,840
in this newly affluent land of ice-cream and doughnuts.
495
00:31:38,840 –> 00:31:42,560
But one teenager didn’t indulge in the normal pursuits
496
00:31:42,560 –> 00:31:45,640
of American adolescence but chose instead
497
00:31:45,640 –> 00:31:49,200
to grapple with some of the major problems in mathematics.
498
00:31:49,200 –> 00:31:50,680
From a very early age,
499
00:31:50,680 –> 00:31:55,080
Paul Cohen was winning mathematical competitions and prizes.
500
00:31:55,080 –> 00:31:58,960
But he found it difficult at first to discover a field in mathematics
501
00:31:58,960 –> 00:32:01,280
where he could really make his mark…
502
00:32:01,280 –> 00:32:05,720
Until he read about Cantor‘s continuum hypothesis.
503
00:32:05,720 –> 00:32:09,280
That’s the one problem, as I had learned back in Halle,
504
00:32:09,280 –> 00:32:11,760
that Cantor just couldn’t resolve.
505
00:32:11,760 –> 00:32:15,400
It asks whether there is or there isn’t an infinite set of numbers
506
00:32:15,400 –> 00:32:18,080
bigger than the set of all whole numbers
507
00:32:18,080 –> 00:32:20,960
but smaller than the set of all decimals.
508
00:32:20,960 –> 00:32:24,280
It sounds straightforward, but it had foiled all attempts
509
00:32:24,280 –> 00:32:29,160
to solve it since Hilbert made it his first problem way back in 1900.
510
00:32:29,160 –> 00:32:31,480
With the arrogance of youth,
511
00:32:31,480 –> 00:32:36,040
the 22-year-old Paul Cohen decided that he could do it.
512
00:32:36,040 –> 00:32:40,720
Cohen came back a year later with the extraordinary discovery
513
00:32:40,720 –> 00:32:43,200
that both answers could be true.
514
00:32:43,200 –> 00:32:47,160
There was one mathematics where the continuum hypothesis
515
00:32:47,160 –> 00:32:49,080
could be assumed to be true.
516
00:32:49,080 –> 00:32:51,800
There wasn’t a set between the whole numbers
517
00:32:51,800 –> 00:32:53,440
and the infinite decimals.
518
00:32:55,160 –> 00:32:59,200
But there was an equally consistent mathematics
519
00:32:59,200 –> 00:33:03,440
where the continuum hypothesis could be assumed to be false.
520
00:33:03,440 –> 00:33:08,280
Here, there was a set between the whole numbers and the infinite decimals.
521
00:33:08,280 –> 00:33:11,480
It was an incredibly daring solution.
522
00:33:11,480 –> 00:33:13,840
Cohen’s proof seemed true,
523
00:33:13,840 –> 00:33:19,160
but his method was so new that nobody was absolutely sure.
524
00:33:19,160 –> 00:33:22,720
There was only one person whose opinion everybody trusted.
525
00:33:22,720 –> 00:33:26,640
There was a lot of scepticism and he had to come and make a trip here,
526
00:33:26,640 –> 00:33:29,320
to the Institute right here, to visit Godel.
527
00:33:29,320 –> 00:33:32,720
And it was only after Godel gave his stamp of approval
528
00:33:32,720 –> 00:33:34,240
in quite an unusual way,
529
00:33:34,240 –> 00:33:37,880
He said, “Give me your paper”, and then on Monday he put it back
530
00:33:37,880 –> 00:33:40,360
in the box and said, “Yes, it’s correct.”
531
00:33:40,360 –> 00:33:42,040
Then everything changed.
532
00:33:43,240 –> 00:33:46,200
Today mathematicians insert a statement
533
00:33:46,200 –> 00:33:50,840
that says whether the result depends on the continuum hypothesis.
534
00:33:50,840 –> 00:33:54,880
We’ve built up two different mathematical worlds
535
00:33:54,880 –> 00:33:57,320
in which one answer is yes and the other is no.
536
00:33:57,320 –> 00:34:01,440
Paul Cohen really has rocked the mathematical universe.
537
00:34:01,440 –> 00:34:05,680
It gave him fame, riches, and prizes galore.
538
00:34:07,680 –> 00:34:12,880
He didn’t publish much after his early success in the ‘60s.
539
00:34:12,880 –> 00:34:15,040
But he was absolutely dynamite.
540
00:34:15,040 –> 00:34:18,840
I can’t imagine anyone better to learn from, and he was very eager
541
00:34:18,840 –> 00:34:23,840
to learn, to teach you anything he knew or even things he didn’t know.
542
00:34:23,840 –> 00:34:27,640
With the confidence that came from solving Hilbert’s first problem,
543
00:34:27,640 –> 00:34:30,320
Cohen settled down in the mid 1960s
544
00:34:30,320 –> 00:34:34,440
to have a go at the most important Hilbert problem of them all -
545
00:34:34,440 –> 00:34:36,960
the eighth, the Riemann hypothesis.
546
00:34:36,960 –> 00:34:43,000
When he died in California in 2007, 40 years later, he was still trying.
547
00:34:43,000 –> 00:34:46,200
But like many famous mathematicians before him,
548
00:34:46,200 –> 00:34:48,280
Riemann had defeated even him.
549
00:34:48,280 –> 00:34:52,440
But his approach has inspired others to make progress towards a proof,
550
00:34:52,440 –> 00:34:55,560
including one of his most famous students, Peter Sarnak.
551
00:34:55,560 –> 00:34:59,440
I think, overall, absolutely loved the guy.
552
00:34:59,440 –> 00:35:01,840
He was my inspiration.
553
00:35:01,840 –> 00:35:04,600
I’m really glad I worked with him.
554
00:35:04,600 –> 00:35:06,800
He put me on the right track.
555
00:35:09,960 –> 00:35:14,240
Paul Cohen is a good example of the success of the great American Dream.
556
00:35:14,240 –> 00:35:16,800
The second generation Jewish immigrant
557
00:35:16,800 –> 00:35:18,960
becomes top American professor.
558
00:35:18,960 –> 00:35:23,640
But I wouldn’t say that his mathematics was a particularly American product.
559
00:35:23,640 –> 00:35:25,720
Cohen was so fired up by this problem
560
00:35:25,720 –> 00:35:29,680
that he probably would have cracked it whatever the surroundings.
561
00:35:31,200 –> 00:35:33,680
Paul Cohen had it relatively easy.
562
00:35:33,680 –> 00:35:36,640
But another great American mathematician of the 1960s
563
00:35:36,640 –> 00:35:40,320
faced a much tougher struggle to make an impact.
564
00:35:40,320 –> 00:35:43,440
Not least because she was female.
565
00:35:43,440 –> 00:35:48,240
In the story of maths, nearly all the truly great mathematicians have been men.
566
00:35:48,240 –> 00:35:51,560
But there have been a few exceptions.
567
00:35:51,560 –> 00:35:54,000
There was the Russian Sofia Kovalevskaya
568
00:35:54,000 –> 00:35:58,920
who became the first female professor of mathematics in Stockholm in 1889,
569
00:35:58,920 –> 00:36:03,400
and won a very prestigious French mathematical prize.
570
00:36:03,400 –> 00:36:07,080
And then Emmy Noether, a talented algebraist who fled from the Nazis
571
00:36:07,080 –> 00:36:10,600
to America but then died before she fully realised her potential.
572
00:36:10,600 –> 00:36:15,920
Then there is the woman who I am crossing America to find out about.
573
00:36:15,920 –> 00:36:19,680
Julia Robinson, the first woman ever to be elected president
574
00:36:19,680 –> 00:36:22,080
of the American Mathematical Society.
575
00:36:31,440 –> 00:36:34,840
She was born in St Louis in 1919,
576
00:36:34,840 –> 00:36:38,160
but her mother died when she was two.
577
00:36:38,160 –> 00:36:42,360
She and her sister Constance moved to live with their grandmother
578
00:36:42,360 –> 00:36:45,720
in a small community in the desert near Phoenix, Arizona.
579
00:36:47,720 –> 00:36:49,800
Julia Robinson grew up around here.
580
00:36:49,800 –> 00:36:53,440
I’ve got a photo which shows her cottage in the 1930s,
581
00:36:53,440 –> 00:36:55,480
with nothing much around it.
582
00:36:55,480 –> 00:36:58,160
The mountains pretty much match those over there
583
00:36:58,160 –> 00:37:00,640
so I think she might have lived down there.
584
00:37:01,600 –> 00:37:04,160
Julia grew up a shy, sickly girl,
585
00:37:04,160 –> 00:37:09,440
who, when she was seven, spent a year in bed because of scarlet fever.
586
00:37:09,440 –> 00:37:12,240
Ill-health persisted throughout her childhood.
587
00:37:12,240 –> 00:37:15,120
She was told she wouldn’t live past 40.
588
00:37:15,120 –> 00:37:20,400
But right from the start, she had an innate mathematical ability.
589
00:37:20,400 –> 00:37:25,240
Under the shade of the native Arizona cactus, she whiled away the time
590
00:37:25,240 –> 00:37:28,720
playing endless counting games with stone pebbles.
591
00:37:28,720 –> 00:37:31,960
This early searching for patterns would give her a feel
592
00:37:31,960 –> 00:37:35,320
and love of numbers that would last for the rest of her life.
593
00:37:35,320 –> 00:37:39,160
But despite showing an early brilliance, she had to continually
594
00:37:39,160 –> 00:37:44,080
fight at school and college to simply be allowed to keep doing maths.
595
00:37:44,080 –> 00:37:47,920
As a teenager, she was the only girl in the maths class
596
00:37:47,920 –> 00:37:50,600
but had very little encouragement.
597
00:37:50,600 –> 00:37:55,480
The young Julia sought intellectual stimulation elsewhere.
598
00:37:55,480 –> 00:37:59,440
Julia loved listening to a radio show called the University Explorer
599
00:37:59,440 –> 00:38:02,440
and the 53rd episode was all about mathematics.
600
00:38:02,440 –> 00:38:04,960
The broadcaster described how he discovered
601
00:38:04,960 –> 00:38:08,560
despite their esoteric language and their seclusive nature,
602
00:38:08,560 –> 00:38:12,320
mathematicians are the most interesting and inspiring creatures.
603
00:38:12,320 –> 00:38:16,240
For the first time, Julia had found out that there were mathematicians,
604
00:38:16,240 –> 00:38:17,920
not just mathematics teachers.
605
00:38:17,920 –> 00:38:20,440
There was a world of mathematics out there,
606
00:38:20,440 –> 00:38:22,240
and she wanted to be part of it.
607
00:38:26,080 –> 00:38:29,680
The doors to that world opened here at the University of California,
608
00:38:29,680 –> 00:38:31,960
at Berkeley near San Francisco.
609
00:38:33,760 –> 00:38:38,680
She was absolutely obsessed that she wanted to go to Berkeley.
610
00:38:38,680 –> 00:38:44,200
She wanted to go away to some place where there were mathematicians.
611
00:38:44,200 –> 00:38:46,720
Berkeley certainly had mathematicians,
612
00:38:46,720 –> 00:38:50,320
including a number theorist called Raphael Robinson.
613
00:38:50,320 –> 00:38:53,400
In their frequent walks around the campus
614
00:38:53,400 –> 00:38:59,960
they found they had more than just a passion for mathematics. They married in 1952.
615
00:38:59,960 –> 00:39:03,200
Julia got her PhD and settled down
616
00:39:03,200 –> 00:39:05,720
to what would turn into her lifetime’s work -
617
00:39:05,720 –> 00:39:07,280
Hilbert’s tenth problem.
618
00:39:07,280 –> 00:39:10,000
She thought about it all the time.
619
00:39:10,000 –> 00:39:14,120
She said to me she just wouldn’t wanna die without knowing that answer
620
00:39:14,120 –> 00:39:16,240
and it had become an obsession.
621
00:39:17,280 –> 00:39:21,200
Julia’s obsession has been shared with many other mathematicians
622
00:39:21,200 –> 00:39:24,560
since Hilbert had first posed it back in 1900.
623
00:39:24,560 –> 00:39:28,400
His tenth problem asked if there was some universal method
624
00:39:28,400 –> 00:39:34,200
that could tell whether any equation had whole number solutions or not.
625
00:39:34,200 –> 00:39:36,520
Nobody had been able to solve it.
626
00:39:36,520 –> 00:39:39,520
In fact, the growing belief was
627
00:39:39,520 –> 00:39:42,440
that no such universal method was possible.
628
00:39:42,440 –> 00:39:44,520
How on earth could you prove that,
629
00:39:44,520 –> 00:39:48,400
however ingenious you were, you’d never come up with a method?
630
00:39:50,080 –> 00:39:51,800
With the help of colleagues,
631
00:39:51,800 –> 00:39:55,640
Julia developed what became known as the Robinson hypothesis.
632
00:39:55,640 –> 00:39:58,920
This argued that to show no such method existed,
633
00:39:58,920 –> 00:40:03,280
all you had to do was to cook up one equation whose solutions
634
00:40:03,280 –> 00:40:06,040
were a very specific set of numbers.
635
00:40:06,040 –> 00:40:09,280
The set of numbers needed to grow exponentially,
636
00:40:09,280 –> 00:40:13,960
like taking powers of two, yet still be captured by the equations
637
00:40:13,960 –> 00:40:16,520
at the heart of Hilbert’s problem.
638
00:40:16,520 –> 00:40:21,600
Try as she might, Robinson just couldn’t find this set.
639
00:40:21,600 –> 00:40:25,880
For the tenth problem to be finally solved,
640
00:40:25,880 –> 00:40:28,880
there needed to be some fresh inspiration.
641
00:40:28,880 –> 00:40:34,280
That came from 5,000 miles away - St Petersburg in Russia.
642
00:40:34,280 –> 00:40:37,840
Ever since the great Leonhard Euler set up shop here
643
00:40:37,840 –> 00:40:39,040
in the 18th century,
644
00:40:39,040 –> 00:40:42,960
the city has been famous for its mathematics and mathematicians.
645
00:40:42,960 –> 00:40:44,760
Here in the Steklov Institute,
646
00:40:44,760 –> 00:40:47,480
some of the world’s brightest mathematicians
647
00:40:47,480 –> 00:40:50,160
have set out their theorems and conjectures.
648
00:40:50,160 –> 00:40:54,320
This morning, one of them is giving a rare seminar.
649
00:40:57,120 –> 00:41:00,040
It’s tough going even if you speak Russian,
650
00:41:00,040 –> 00:41:02,080
which unfortunately I don’t.
651
00:41:02,080 –> 00:41:06,320
But we do get a break in the middle to recover before the final hour.
652
00:41:06,320 –> 00:41:08,320
There is a kind of rule in seminars.
653
00:41:08,320 –> 00:41:12,880
The first third is for everyone, the second third for the experts
654
00:41:12,880 –> 00:41:16,080
and the last third is just for the lecturer.
655
00:41:16,080 –> 00:41:19,080
I think that’s what we’re going to get next.
656
00:41:19,080 –> 00:41:22,800
The lecturer is Yuri Matiyasevich and he is explaining
657
00:41:22,800 –> 00:41:26,520
his latest work on - what else? - the Riemann hypothesis.
658
00:41:28,720 –> 00:41:33,160
As a bright young graduate student in 1965, Yuri’s tutor
659
00:41:33,160 –> 00:41:36,000
suggested he have a go at another Hilbert problem,
660
00:41:36,000 –> 00:41:39,000
the one that had in fact preoccupied Julia Robinson.
661
00:41:39,000 –> 00:41:40,280
Hilbert’s tenth.
662
00:41:43,160 –> 00:41:45,080
It was the height of the Cold War.
663
00:41:45,080 –> 00:41:48,440
Perhaps Matiyasevich could succeed for Russia
664
00:41:48,440 –> 00:41:52,080
where Julia and her fellow American mathematicians had failed.
665
00:41:52,080 –> 00:41:55,000
- At first I did not like their approach.
- Oh, right.
666
00:41:55,000 –> 00:41:59,640
The statement looked to me rather strange and artificial
667
00:41:59,640 –> 00:42:03,520
but after some time I understood it was quite natural,
668
00:42:03,520 –> 00:42:07,200
and then I understood that she had a new brilliant idea
669
00:42:07,200 –> 00:42:10,000
and I just started to further develop it.
670
00:42:11,520 –> 00:42:17,000
In January 1970, he found the vital last piece in the jigsaw.
671
00:42:17,000 –> 00:42:21,880
He saw how to capture the famous Fibonacci sequence of numbers
672
00:42:21,880 –> 00:42:26,040
using the equations that were at the heart of Hilbert’s problem.
673
00:42:26,040 –> 00:42:28,920
Building on the work of Julia and her colleagues,
674
00:42:28,920 –> 00:42:30,720
he had solved the tenth.
675
00:42:30,720 –> 00:42:34,240
He was just 22 years old.
676
00:42:34,240 –> 00:42:37,920
The first person he wanted to tell was the woman he owed so much to.
677
00:42:39,800 –> 00:42:41,720
I got no answer
678
00:42:41,720 –> 00:42:44,600
and I believed they were lost in the mail.
679
00:42:44,600 –> 00:42:47,720
It was quite natural because it was Soviet time.
680
00:42:47,720 –> 00:42:50,800
But back in California, Julia had heard rumours
681
00:42:50,800 –> 00:42:54,840
through the mathematical grapevine that the problem had been solved.
682
00:42:54,840 –> 00:42:57,120
And she contacted Yuri herself.
683
00:42:58,120 –> 00:43:01,480
She said, I just had to wait for you to grow up
684
00:43:01,480 –> 00:43:06,160
to get the answer, because she had started work in 1948.
685
00:43:06,160 –> 00:43:07,960
When Yuri was just a baby.
686
00:43:07,960 –> 00:43:11,240
Then he responds by thanking her
687
00:43:11,240 –> 00:43:16,160
and saying that the credit is as much hers as it is his.
688
00:43:18,240 –> 00:43:20,520
YURI: I met Julia one year later.
689
00:43:20,520 –> 00:43:25,080
It was in Bucharest. I suggested after the conference in Bucharest
690
00:43:25,080 –> 00:43:30,120
Julia and her husband Raphael came to see me here in Leningrad.
691
00:43:30,120 –> 00:43:35,400
Together, Julia and Yuri worked on several other mathematical problems
692
00:43:35,400 –> 00:43:39,160
until shortly before Julia died in 1985.
693
00:43:39,160 –> 00:43:41,960
She was just 55 years old.
694
00:43:41,960 –> 00:43:45,640
She was able to find the new ways.
695
00:43:45,640 –> 00:43:49,640
Many mathematicians just combine previous known methods
696
00:43:49,640 –> 00:43:55,560
to solve new problems and she had really new ideas.
697
00:43:55,560 –> 00:43:59,160
Although Julia Robinson showed there was no universal method
698
00:43:59,160 –> 00:44:01,560
to solve all equations in whole numbers,
699
00:44:01,560 –> 00:44:05,840
mathematicians were still interested in finding methods
700
00:44:05,840 –> 00:44:08,760
to solve special classes of equations.
701
00:44:08,760 –> 00:44:11,320
It would be in France in the early 19th century,
702
00:44:11,320 –> 00:44:13,560
in one of the most extraordinary stories
703
00:44:13,560 –> 00:44:17,120
in the history of mathematics, that methods were developed
704
00:44:17,120 –> 00:44:20,240
to understand why certain equations could be solved
705
00:44:20,240 –> 00:44:21,760
while others couldn’t.
706
00:44:27,840 –> 00:44:32,520
It’s early morning in Paris on the 29th May 1832.
707
00:44:32,520 –> 00:44:37,120
Evariste Galois is about to fight for his very life.
708
00:44:37,120 –> 00:44:40,680
It is the reign of the reactionary Bourbon King, Charles X,
709
00:44:40,680 –> 00:44:43,960
and Galois, like many angry young men in Paris then,
710
00:44:43,960 –> 00:44:46,680
is a republican revolutionary.
711
00:44:46,680 –> 00:44:52,000
Unlike the rest of his comrades though, he has another passion - mathematics.
712
00:44:53,560 –> 00:44:56,480
He had just spent four months in jail.
713
00:44:56,480 –> 00:45:00,160
Then, in a mysterious saga of unrequited love,
714
00:45:00,160 –> 00:45:02,280
he is challenged to a duel.
715
00:45:02,280 –> 00:45:04,280
He’d been up the whole previous night
716
00:45:04,280 –> 00:45:07,360
refining a new language for mathematics he’d developed.
717
00:45:07,360 –> 00:45:14,160
Galois believed that mathematics shouldn’t be the study of number and shape, but the study of structure.
718
00:45:14,160 –> 00:45:17,240
Perhaps he was still pre-occupied with his maths.
719
00:45:17,240 –> 00:45:18,800
GUNSHOT
720
00:45:18,800 –> 00:45:21,680
There was only one shot fired that morning.
721
00:45:21,680 –> 00:45:27,280
Galois died the next day, just 20 years old.
722
00:45:27,280 –> 00:45:30,320
It was one of mathematics greatest losses.
723
00:45:30,320 –> 00:45:33,080
Only by the beginning of the 20th century
724
00:45:33,080 –> 00:45:37,640
would Galois be fully appreciated and his ideas fully realised.
725
00:45:42,400 –> 00:45:46,520
Galois had discovered new techniques to be able to tell
726
00:45:46,520 –> 00:45:49,920
whether certain equations could have solutions or not.
727
00:45:49,920 –> 00:45:54,000
The symmetry of certain geometric objects seemed to be the key.
728
00:45:54,000 –> 00:45:58,520
His idea of using geometry to analyse equations
729
00:45:58,520 –> 00:46:03,880
would be picked up in the 1920s by another Parisian mathematician, Andre Weil.
730
00:46:03,880 –> 00:46:09,520
I was very much interested and so far as school was concerned
731
00:46:09,520 –> 00:46:13,720
quite successful in all possible branches.
732
00:46:13,720 –> 00:46:17,480
And he was. After studying in Germany as well as France,
733
00:46:17,480 –> 00:46:21,000
Andre settled down at this apartment in Paris
734
00:46:21,000 –> 00:46:25,760
which he shared with his more-famous sister, the writer Simone Weil.
735
00:46:25,760 –> 00:46:31,040
But when the Second World War broke out, he found himself in very different circumstances.
736
00:46:31,040 –> 00:46:37,040
He dodged the draft by fleeing to Finland where he was almost executed for being a Russian spy.
737
00:46:37,040 –> 00:46:42,720
On his return to France he was put in prison in Rouen to await trial for desertion.
738
00:46:42,720 –> 00:46:45,320
At the trial, the judge gave him a choice.
739
00:46:45,320 –> 00:46:49,120
Five more years in prison or serve in a combat unit.
740
00:46:49,120 –> 00:46:52,240
He chose to join the French army, a lucky choice
741
00:46:52,240 –> 00:46:56,120
because just before the Germans invaded a few months later,
742
00:46:56,120 –> 00:46:58,280
all the prisoners were killed.
743
00:46:58,280 –> 00:47:05,400
Weil only spent a few months in prison, but this time was crucial for his mathematics.
744
00:47:05,400 –> 00:47:11,000
Because here he built on the ideas of Galois and first developed algebraic geometry
745
00:47:11,000 –> 00:47:15,720
a whole new language for understanding solutions to equations.
746
00:47:15,720 –> 00:47:18,720
Galois had shown how new mathematical structures
747
00:47:18,720 –> 00:47:22,600
can be used to reveal the secrets behind equations.
748
00:47:22,600 –> 00:47:24,640
Weil’s work led him to theorems
749
00:47:24,640 –> 00:47:28,800
that connected number theory, algebra, geometry and topology
750
00:47:28,800 –> 00:47:33,720
and are one of the greatest achievements of modern mathematics.
751
00:47:33,720 –> 00:47:36,760
Without Andre Weil, we would never have heard
752
00:47:36,760 –> 00:47:41,400
of the strangest man in the history of maths, Nicolas Bourbaki.
753
00:47:43,720 –> 00:47:50,400
There are no photos of Bourbaki in existence but we do know he was born in this cafe in the Latin Quarter
754
00:47:50,400 –> 00:47:54,520
in 1934 when it was a proper cafe, the cafe Capoulade,
755
00:47:54,520 –> 00:47:58,000
and not the fast food joint it has now become.
756
00:47:58,000 –> 00:48:03,200
Just down the road, I met up with Bourbaki expert David Aubin.
757
00:48:03,200 –> 00:48:06,400
When I was a graduate student I got quite frightened
758
00:48:06,400 –> 00:48:08,120
when I used to go into the library
759
00:48:08,120 –> 00:48:10,960
because this guy Bourbaki had written so many books.
760
00:48:10,960 –> 00:48:14,400
Something like 30 or 40 altogether.
761
00:48:14,400 –> 00:48:19,680
In analysis, in geometry, in topology, it was all new foundations.
762
00:48:19,680 –> 00:48:23,360
Virtually everyone studying Maths seriously anywhere in the world
763
00:48:23,360 –> 00:48:28,200
in the 1950s, ‘60s and ‘70s would have read Nicolas Bourbaki.
764
00:48:28,200 –> 00:48:31,160
He applied for membership of the American Maths Society, I heard.
765
00:48:31,160 –> 00:48:33,360
At which point he was denied membership
766
00:48:33,360 –> 00:48:36,320
- on the grounds that he didn’t exist.
- Oh!
767
00:48:36,320 –> 00:48:38,160
The Americans were right.
768
00:48:38,160 –> 00:48:41,880
Nicolas Bourbaki does not exist at all. And never has.
769
00:48:41,880 –> 00:48:46,200
Bourbaki is in fact the nom de plume for a group of French mathematicians
770
00:48:46,200 –> 00:48:49,880
led by Andre Weil who decided to write a coherent account
771
00:48:49,880 –> 00:48:52,480
of the mathematics of the 20th century.
772
00:48:52,480 –> 00:48:57,200
Most of the time mathematicians like to have their own names on theorems.
773
00:48:57,200 –> 00:48:59,600
But for the Bourbaki group,
774
00:48:59,600 –> 00:49:03,440
the aims of the project overrode any desire for personal glory.
775
00:49:03,440 –> 00:49:07,120
After the Second World War, the Bourbaki baton was handed down
776
00:49:07,120 –> 00:49:10,080
to the next generation of French mathematicians.
777
00:49:10,080 –> 00:49:15,400
And their most brilliant member was Alexandre Grothendieck.
778
00:49:15,400 –> 00:49:17,000
Here at the IHES in Paris,
779
00:49:17,000 –> 00:49:21,520
the French equivalent of Princeton’s Institute for Advanced Study,
780
00:49:21,520 –> 00:49:27,160
Grothendieck held court at his famous seminars in the 1950s and 1960s.
781
00:49:29,920 –> 00:49:33,600
He had this incredible charisma.
782
00:49:33,600 –> 00:49:40,240
He had this amazing ability to see a young person and somehow know
783
00:49:40,240 –> 00:49:46,280
what kind of contribution this person could make to this incredible vision
784
00:49:46,280 –> 00:49:48,920
he had of how mathematics could be.
785
00:49:48,920 –> 00:49:54,520
And this vision enabled him to get across some very difficult ideas indeed.
786
00:49:54,520 –> 00:49:58,240
He says, “Suppose you want to open a walnut.
787
00:49:58,240 –> 00:50:02,200
“So the standard thing is you take a nutcracker and you just break it open.”
788
00:50:02,200 –> 00:50:04,800
And he says his approach is more like,
789
00:50:04,800 –> 00:50:08,120
you take this walnut and you put it out in the snow
790
00:50:08,120 –> 00:50:10,160
and you leave it there for a few months
791
00:50:10,160 –> 00:50:13,760
and then when you come back to it, it just opens.
792
00:50:13,760 –> 00:50:15,760
Grothendieck is a Structuralist.
793
00:50:15,760 –> 00:50:19,720
What he’s interested in are the hidden structures
794
00:50:19,720 –> 00:50:22,120
underneath all mathematics.
795
00:50:22,120 –> 00:50:27,560
Only when you get down to the very basic architecture and think in very general terms
796
00:50:27,560 –> 00:50:31,160
will the patterns in mathematics become clear.
797
00:50:31,160 –> 00:50:37,120
Grothendieck produced a new powerful language to see structures in a new way.
798
00:50:37,120 –> 00:50:39,720
It was like living in a world of black and white
799
00:50:39,720 –> 00:50:42,960
and suddenly having the language to see the world in colour.
800
00:50:42,960 –> 00:50:46,640
It’s a language that mathematicians have been using ever since
801
00:50:46,640 –> 00:50:51,640
to solve problems in number theory, geometry, even fundamental physics.
802
00:50:53,160 –> 00:50:56,440
But in the late 1960s, Grothendieck decided
803
00:50:56,440 –> 00:51:01,640
to turn his back on mathematics after he discovered politics.
804
00:51:01,640 –> 00:51:06,320
He believed that the threat of nuclear war and the questions
805
00:51:06,320 –> 00:51:12,440
of nuclear disarmament were more important than mathematics
806
00:51:12,440 –> 00:51:17,480
and that people who continue to do mathematics
807
00:51:17,480 –> 00:51:21,240
rather than confront this threat of nuclear war
808
00:51:21,240 –> 00:51:22,920
were doing harm in the world.
809
00:51:26,440 –> 00:51:29,040
Grothendieck decided to leave Paris
810
00:51:29,040 –> 00:51:32,040
and move back to the south of France where he grew up.
811
00:51:32,040 –> 00:51:36,680
Bursts of radical politics followed and then a nervous breakdown.
812
00:51:36,680 –> 00:51:40,720
He moved to the Pyrenees and became a recluse.
813
00:51:40,720 –> 00:51:45,600
He’s now lost all contact with his old friends and mathematical colleagues.
814
00:51:46,600 –> 00:51:51,040
Nevertheless, the legacy of his achievements means that Grothendieck stands
815
00:51:51,040 –> 00:51:57,440
alongside Cantor, Godel and Hilbert as someone who has transformed the mathematical landscape.
816
00:51:59,200 –> 00:52:03,800
He changed the whole subject in a really fundamental way. It will never go back.
817
00:52:03,800 –> 00:52:08,800
Certainly, he’s THE dominant figure of the 20th century.
818
00:52:16,200 –> 00:52:18,280
I’ve come back to England, though,
819
00:52:18,280 –> 00:52:22,440
thinking again about another seminal figure of the 20th century.
820
00:52:22,440 –> 00:52:26,640
The person that started it all off, David Hilbert.
821
00:52:26,640 –> 00:52:32,400
Of the 23 problems Hilbert set mathematicians in the year 1900,
822
00:52:32,400 –> 00:52:34,880
most have now been solved.
823
00:52:34,880 –> 00:52:37,160
However there is one great exception.
824
00:52:37,160 –> 00:52:40,360
The Riemann hypothesis, the eighth on Hilbert’s list.
825
00:52:40,360 –> 00:52:43,160
That is still the holy grail of mathematics.
826
00:52:44,960 –> 00:52:50,200
Hilbert’s lecture inspired a generation to pursue their mathematical dreams.
827
00:52:50,200 –> 00:52:55,120
This morning, in the town where I grew up, I hope to inspire another generation.
828
00:52:55,120 –> 00:52:57,280
CHEERING AND APPLAUSE
829
00:53:01,680 –> 00:53:04,120
Thank you. Hello. My name’s Marcus du Sautoy
830
00:53:04,120 –> 00:53:05,960
and I’m a Professor of Mathematics
831
00:53:05,960 –> 00:53:08,120
up the road at the University of Oxford.
832
00:53:08,120 –> 00:53:10,320
It was actually in this school here,
833
00:53:10,320 –> 00:53:14,520
in fact this classroom is where I discovered my love for mathematics.
834
00:53:14,520 –> 00:53:17,120
‘This love of mathematics that I first acquired
835
00:53:17,120 –> 00:53:20,400
‘here in my old comprehensive school still drives me now.
836
00:53:20,400 –> 00:53:22,280
‘It’s a love of solving problems.
837
00:53:22,280 –> 00:53:25,680
‘There are so many problems I could tell them about,
838
00:53:25,680 –> 00:53:27,720
‘but I’ve chosen my favourite.’
839
00:53:27,720 –> 00:53:30,840
I think that a mathematician is a pattern searcher
840
00:53:30,840 –> 00:53:33,960
and that’s really what mathematicians try and do.
841
00:53:33,960 –> 00:53:37,080
We try and understand the patterns and the structure
842
00:53:37,080 –> 00:53:40,440
and the logic to explain the way the world around us works.
843
00:53:40,440 –> 00:53:43,480
And this is really at the heart of the Riemann hypothesis.
844
00:53:43,480 –> 00:53:48,360
The task is - is there any pattern in these numbers which can help me say
845
00:53:48,360 –> 00:53:50,440
where the next number will be?
846
00:53:50,440 –> 00:53:52,760
What’s the next one after 31? How can I tell?
847
00:53:52,760 –> 00:53:55,760
‘These numbers are, of course, prime numbers -
848
00:53:55,760 –> 00:53:58,200
‘the building blocks of mathematics.’
849
00:53:58,200 –> 00:54:01,520
‘The Riemann hypothesis, a conjecture about the distribution
850
00:54:01,520 –> 00:54:04,720
‘of the primes, goes to the very heart of our subject.’
851
00:54:04,720 –> 00:54:07,560
Why on earth is anybody interested in these primes?
852
00:54:07,560 –> 00:54:11,040
Why is the army interested in primes, why are spies interested?
853
00:54:11,040 –> 00:54:14,800
- Isn’t it to encrypt stuff?
- Exactly.
854
00:54:14,800 –> 00:54:18,280
I study this stuff cos I think it’s all really beautiful and elegant
855
00:54:18,280 –> 00:54:20,200
but actually, there’s a lot of people
856
00:54:20,200 –> 00:54:24,440
who are interested in these numbers because of their very practical use.
857
00:54:24,440 –> 00:54:28,720
‘The bizarre thing is that the more abstract and difficult mathematics becomes,
858
00:54:28,720 –> 00:54:32,480
‘the more it seems to have applications in the real world.
859
00:54:32,480 –> 00:54:36,560
‘Mathematics now pervades every aspect of our lives.
860
00:54:36,560 –> 00:54:41,560
‘Every time we switch on the television, plug in a computer, pay with a credit card.
861
00:54:41,560 –> 00:54:46,160
‘There’s now a million dollars for anyone who can solve the Riemann hypothesis.
862
00:54:46,160 –> 00:54:48,600
‘But there’s more at stake than that.’
863
00:54:48,600 –> 00:54:51,800
Anybody who proves this theorem will be remembered forever.
864
00:54:51,800 –> 00:54:55,640
They’ll be on that board ahead of any of those other mathematicians.
865
00:54:55,640 –> 00:54:59,600
‘That’s because the Riemann hypothesis is a corner-stone of maths.
866
00:54:59,600 –> 00:55:02,800
‘Thousands of theorems depend on it being true.
867
00:55:02,800 –> 00:55:06,000
‘Very few mathematicians think that it isn’t true.
868
00:55:06,000 –> 00:55:10,640
‘But mathematics is about proof and until we can prove it
869
00:55:10,640 –> 00:55:12,840
‘there will still be doubt.’
870
00:55:12,840 –> 00:55:17,160
Maths has grown out of this passion to get rid of doubt.
871
00:55:17,160 –> 00:55:20,760
This is what I have learned in my journey through the history of mathematics.
872
00:55:20,760 –> 00:55:25,080
Mathematicians like Archimedes and al-Khwarizmi, Gauss and Grothendieck
873
00:55:25,080 –> 00:55:30,520
were driven to understand the precise way numbers and space work.
874
00:55:30,520 –> 00:55:33,200
Maths in action, that one.
875
00:55:33,200 –> 00:55:35,440
It’s beautiful. Really nice.
876
00:55:35,440 –> 00:55:39,200
Using the language of mathematics, they have told us stories
877
00:55:39,200 –> 00:55:43,760
that remain as true today as they were when they were first told.
878
00:55:43,760 –> 00:55:48,760
In the Mediterranean, I discovered the origins of geometry.
879
00:55:48,760 –> 00:55:51,840
Mathematicians and philosophers flocked to Alexandria
880
00:55:51,840 –> 00:55:55,240
driven by a thirst for knowledge and the pursuit of excellence.
881
00:55:55,240 –> 00:55:59,080
In India, I learned about another discovery
882
00:55:59,080 –> 00:56:02,880
that it would be impossible to imagine modern life without.
883
00:56:02,880 –> 00:56:07,240
So here we are in one of the true holy sites of the mathematical world.
884
00:56:07,240 –> 00:56:10,080
Up here are some numbers,
885
00:56:10,080 –> 00:56:12,680
and here’s the new number.
886
00:56:12,680 –> 00:56:14,320
Its zero.
887
00:56:14,320 –> 00:56:19,600
In the Middle East, I was amazed at al-Khwarizmi‘s invention of algebra.
888
00:56:19,600 –> 00:56:22,480
He developed systematic ways to analyse problems
889
00:56:22,480 –> 00:56:26,160
so that the solutions would work whatever numbers you took.
890
00:56:26,160 –> 00:56:28,080
In the Golden Age of Mathematics,
891
00:56:28,080 –> 00:56:31,600
in Europe in the 18th and 19th centuries, I found how maths
892
00:56:31,600 –> 00:56:35,760
discovered new ways for analysing bodies in motion and new geometries
893
00:56:35,760 –> 00:56:40,520
that helped us understand the very strange shape of space.
894
00:56:40,520 –> 00:56:43,840
It is with Riemann’s work that we finally have
895
00:56:43,840 –> 00:56:49,280
the mathematical glasses to be able to explore such worlds of the mind.
896
00:56:49,280 –> 00:56:53,480
And now my journey into the abstract world of 20th-century mathematics
897
00:56:53,480 –> 00:56:56,600
has revealed that maths is the true language
898
00:56:56,600 –> 00:56:58,800
the universe is written in,
899
00:56:58,800 –> 00:57:02,120
the key to understanding the world around us.
900
00:57:02,120 –> 00:57:05,840
Mathematicians aren’t motivated by money and material gain
901
00:57:05,840 –> 00:57:09,160
or even by practical applications of their work.
902
00:57:09,160 –> 00:57:13,400
For us, it is the glory of solving one of the great unsolved problems
903
00:57:13,400 –> 00:57:18,560
that have outwitted previous generations of mathematicians.
904
00:57:18,560 –> 00:57:21,920
Hilbert was right. It’s the unsolved problems of mathematics
905
00:57:21,920 –> 00:57:23,720
that make it a living subject,
906
00:57:23,720 –> 00:57:27,160
which obsess each new generation of mathematicians.
907
00:57:27,160 –> 00:57:30,960
Despite all the things we’ve discovered over the last seven millennia,
908
00:57:30,960 –> 00:57:33,600
there are still many things we don’t understand.
909
00:57:33,600 –> 00:57:39,960
And its Hilbert’s call of, “We must know, we will know”, which drives mathematics.
910
00:57:42,240 –> 00:57:45,440
You can learn more about The Story Of Maths
911
00:57:45,440 –> 00:57:48,480
with the Open University at…
912
00:58:00,600 –> 00:58:03,640
Subtitled by Red Bee Media Ltd
913
00:58:03,640 –> 00:58:06,680
E-mail subtitling@bbc.co.uk
Subtitles by © Red Bee Media Ltd