In Our Time - Mathematics' Unintended Consequences

Episode Date: February 11, 2010

Melvyn Bragg and guests John Barrow, Colva Roney-Dougal and Marcus du Sautoy explore the unintended consequences of mathematical discoveries, from the computer to online encryption, to alternating current and predicting the path of asteroids.In his book The Mathematician's Apology (1941), the Cambridge mathematician GH Hardy expressed his reverence for pure maths, and celebrated its uselessness in the real world. Yet one of the branches of pure mathematics in which Hardy excelled was number theory, and it was this field which played a major role in the work of his younger colleague, Alan Turing, as he worked first to crack Nazi codes at Bletchley Park and then on one of the first computers.Melvyn Bragg and guests explore the many surprising and completely unintended uses to which mathematical discoveries have been put. These include:The cubic equations which led, after 400 years, to the development of alternating current - and the electric chair.The centuries-old work on games of chance which eventually contributed to the birth of population statistics.The discovery of non-Euclidean geometry, which crucially provided an 'off-the-shelf' solution which helped Albert Einstein forge his theory of relativity.The 17th-century theorem which became the basis for credit card encryption.In the light of these stories, Melvyn and his guests discuss how and why pure mathematics has had such a range of unintended consequences.John Barrow is Professor of Mathematical Sciences at the University of Cambridge and Professor of Geometry at Gresham College, London; Colva Roney-Dougal is Lecturer in Pure Mathematics at the University of St Andrews; Marcus du Sautoy is Charles Simonyi Professor for the Public Understanding of Science and Professor of Mathematics at the University of Oxford.

This BBC podcast is supported by ads outside the UK. Every Sunday, we talk about the week's tech news on this week in tech. Hi, this is Leo Leport. inviting you to join me this week with Lisa Schmeiser, Dan Patterson, and Yanko Rekkers. We're going to talk about the new 49 megabyte web page. It's the standard, you know. We'll also talk about Elon Musk. You've got some spleenin to do and the Yassify filter, new from Nvidia.

That's this week on this week in tech. you'll find it at twit.tv or wherever you get your podcasts. Thanks for downloading the In Our Time podcast. For more details about In Our Time and for our terms of use, please go to BBC.co.com.uk forward slash radio four. I hope you enjoy the program. Hello, I'm interested in mathematics, wrote the Cambridge mathematician G.H. Hardy only as a creative art.

Hardy revered pure maths and scorned his application to the real world. In great mathematics, he wrote in the same book published in 1941, there is a very high degree of unexpectedness. Yet what Hardy himself never expected was that in a decade since he wrote those words, pure mathematics would yield such an array of unintended consequences, not least through the wartime work of his Cambridge colleague, Alan Turing, whose experiments with prime numbers

led first to his crucial role in beating Hitler's codes and then to the birth of the computer. Yet Hardy shouldn't have been surprised. Pure maths was making itself useful long before the struggle against the Nazis. In 1801, centuries old work on probability help predict the path of an asteroid, then led on to the birth of population statistics, and the discovery of non-Euclidean geometry meanwhile was vital to Einstein's theory of relativity. To explore mathematics how it generates all these unintended consequences, I'm

joined by Colber Roney-Dougall, Lecturer in Pure Mathematics at the University of St Andrews, John Barrow, Professor of Mathematical Sciences at the University of Cambridge and Professor of Geometry at Gresham College London, and Marcus Usoitoy, Charles Cimoni, Professor for the Public Understanding of Science and Professor of Mathematics at the University of Oxford. Colvaroni Dougal, can you start off by giving us an idea of the impact of these unintended consequences? Let's take cubic equations. The work on cubic equations,

how did they lead 400 years later, rather dramatically, to the electric chair? Okay, so we're going to start in 16th century, Italy, and I'd better tell you what a cubic equation is. So a cubic equation is something like, I'm thinking of a number, I'm going to call it X, and I want so that x times x times x is the same thing as x plus six. Now I've picked that as an example because I can do it in my head and the answer there would be two, but in general these things are much harder to solve.

Now, this had been a longstanding problem for thousands of years. People had known how to solve quadratic equations x times x, but 16th century easily they finally discovered a formula for solving cubic equations, a bit like the quadratic formula that you might learn at school, but more complicated. Now, Duralama Cardano published this formula and ran into difficulties. When he was trying to solve certain cubic equations,

he knew that the answer was something straightforward like four. But partway through his calculations, he was getting strange numbers, such as the square root of minus 121. Now, a positive number times a positive number is positive and a negative number times a negative number is positive, so there's no such thing as the square root of minus 121. But he took a deep breath and said, well, let's just pretend that there is.

And this was the birth of what's called complex numbers. If he pretended that there is, then when he finally finished his calculation, all of those funny numbers, the square roots of negative numbers, the complex numbers, cancelled out, and he was left with a perfectly valid number. Now, let's jump forward to Victorian times, and electricity. So the first kind of electricity, which was widely in use is direct current, DC. That's the sort we get out of batteries now. and Thomas Edison was going around electrifying America

and building power stations and connecting it to homes and factories and people were bringing electricity in. There was a later kind of electricity discovered called alternating current where the direction of the electricity changes direction all the time, hence the alternating. Now the mathematics of alternating current is far more complicated because of this change of direction, but given Victorian technology it was possible to transmit alternating current. over far greater distances than direct current without wasting it through resistance.

So Tesla and Westinghouse were strongly arguing that America ought to adopt alternating current and so should the rest of the world. Edison was already wedded to direct current. He'd built a lot of the things. And so there came a period called the current wars when Edison basically was running a PR campaign in favour of direct current and was trying to convince the world that alternating current was far more dangerous, which on some level it is. Now, Edison wasn't particularly pro the death penalty, but he so wanted to win these current wars that he got his employees to design, build,

and later get the state of New York to operate the world's first electric chair running on alternating current to convince everybody it was dangerous. Now, ironically, the very same employee, Thomas Kennelly, who built the electric chair, three years later realized that this horrendous maths of alternating current could be massively simplified by the introduction of complex numbers to describe these waves and surges of electricity changing direction all the time and so then the maths of alternating current became far simpler

and that's why when you plough something in to the wall of our house today it's alternating current that comes down the wire. So 400 centuries he went back to those equations? Yeah, I mean they'd become school maths by that point and he was able to pull it off the shelf. And do you think just briefly, do you think this is, what we're going to discuss is fairly common in mathematics, that unintended consequences follow time and again?

Yes, it is. And the further back we look, the more common it is, because in some sense the bits of the early maths that we remember are the core of maths, and so they have applications. John Barrow, why do you think this is? Is it something about us, human instinct, to spot patterns? Why do you think these equations pop up later to have such an impact? Some people find it almost spooky

that you can be challenged by some problem in the real world that you want to solve or some observations in astronomy or physics that you want to explain, and you discover that a piece of mathematics devised apparently just for the amusement of pure mathematicians who thought it was rather pretty long ago turns out to be just what you need. My view of this is that it's not too surprising in one sense, because a good way of thinking about mathematics is that it's just the collection of all the possible patterns. that there could be. It's a great catalogue of every possible pattern.

Some of those patterns were interesting, some are not, some are useful, some are not. But can you just, sorry, it's an easy word to say, but what do you mean by patterns? Well, it could be patterns like we have on the carpet. They're symmetries if you look at moving one step to the right on your carpet design or your wall design. if it looks the same every time you move one step to the right, or if you stand on your head and look at it and it still looks the same, you would say it's got a symmetry, it's got a pattern, there's something that you can do to it that leaves it unchanged.

And our universe and the world around us has to contain patterns. We couldn't exist otherwise. There would be nothing for natural selection to act upon. So the world must contain patterns. So it's no mystery that what we're, we call mathematics describes them. But what's mysterious is that... Do we have pattern seekers because the universe is full of patterns?

Yes, so we have evolved within the universe in response to what the world is like around us, and it pays to be able to pick patterns and respond to them. And if you have a sensitivity for looking in the woods, as it were, in a rather vague way and sensing that something might be alive, then you have an advantage over someone who does, doesn't have that because you could tell whether that's something is perhaps contemplating having you for lunch, or you might be able to have it for lunch, or it might be a potential mate. So being able to pick patterns out certainly pays. Can you give us an example of how pure maths finds these patterns and then they're picked up by scientists later. Just another almost like a codman did a headline event.

Let's take the astronomer Kepler and his reference to some ancient Greek geometry. If you're an astronomer like Kepler was, you're interested in studying, in his case, the orbits of the planets in the solar system, or if like Galileo you're interested in firing cannonballs in the air and figuring out what the trajectory is followed before it comes back to Earth, then what you require to describe those paths in space are a group of curves that mathematicians call the conics or the conic sections. and they were studied exhaustively and rather beautifully by Apollonius in about 250 BC. And Apollonius invented the names that we now use for them, the parabola, the hyperbola or the ellipse. And if you have a body moving under the influence of a force like gravity, that falls off like the square of the distance the planet is away from the sun that's pulling it,

then its trajectory will be to follow an ellipse if it's bound in orbit around the sun. If it's just flying past, it will follow a different trajectory, which is a hyperbola. If like Galileo we fire a heavy cannonball in the air and make a trace of its trajectory, then that will be following a parabola. So the geometry of these curves, which Apollonia studied by considering an ordinary cone, like an upside-down ice-cream cone and taking slices through it vertically, horizontally

and then at an angle. And the shapes you get left on the surface of the sliced cone gives you either the hyperbola, the ellipse, or the parabola. And that's why they were called conic sections. They're sections or slices through a cone. And did Kepler read about those?

Do he know those? Yes, this type of geometry. So we're talking about 1600 1600. Yeah. So at this period, the staple diet of mathematicians and astronomers like Kepler would have been the work of Euclid and then the work of Apollonius, which follows it. This was the basis for mathematical education. Marcus Isoto, can you, I started with referring to George Hardy and his disdain for applied mathematics in any way.

He wanted them to be kept pure. Can you just give us the... distinction between pure mathematics and applied mathematics? Yes, I would say that for me, an applied mathematician is somebody who's motivated by a problem in the real world that they're trying to unlock solve so that they can do a sort of a physical problem. A pure mathematician, which is, I would count myself among that camp, is one that isn't so much interested in the applications of the real world, but is celebrating the mathematics for its own beauty. and it's just motivated in some sense by a problem for its own sake. And I think that it's interesting, if you go right back to the Babylonians, for example,

the Babylonians are solving sort of math problems that look very practical. They're about measuring areas of fields and things like that, and that's where we get the quadratic equation, and they come up with, as Culver said, an equation, essentially, for a quadratic equation, how to solve it. But when you look at these problems on these tablets, these problems, although they sound very practical, an area of field is 60 units squared,

one side is five units longer than the other, how long is the short side? You'd never know the area of a field without knowing the sides of the actual dimensions of the field. So this sort of problem is, although it sounds very practical, somebody's, one of these scribes is actually doing it just for the pure love of undoing this little puzzle, and I think the motivation there is already gone,

although it sounds practical, a mathematician is just enjoying the satisfaction of undoing one of these little quadratic equations. Do you find out? I do, absolutely. I mean, the wonderful thing is we're going to explore in this programme is the unintended consequences. So although you will be motivated by the pure beauty and the surprise, the thing that Hardy loved, that surprise of finding a new connection between things, time and again, you find that those things probably because they started with problems in the real world,

then go into this wonderful abstract world of mathematics, but quite often they kind of land back down again as the thing that somebody needs. What do you think draws you in? I'd ask you then go right. draws mathematics into exploring mathematics. Is it very much the same as poets wanting to keep right? Is it a...

Because Hardy made that analogy. He did. He talked about poetry and that we're like a pattern searcher. I mean, John's already brought out this idea of pattern and structure. And I think we are drawn to similar structures that creative artists are as well. But actually, I think it's interesting, this split between pure and applied mathematics

actually happened quite late. And if you look, or still in Revolutionary France, for example, Revolutionary France was still very much motivated by practical applications. And when mathematicians in the academies were starting to explore mathematics just for its own sake, you get people complaining about this. Here's somebody complaining about the myth.

mathematician Koshy in France who was doing pure mathematics. He said, it is the opinion of many people that instruction in pure mathematics is being carried too far at the Ecole, and that such an uncalled-for extravagance is prejudicial to the other branches. And so for Revolutionary France, mathematics was empowering the state. It was used actually to do practical things. But then you get a change, and in Germany, their sort of near-humanist movement opens up with Humboldt, who's celebrating intellectual endeavour for its own sake. Why was it so fruitful in Germany then? To move there...

Well, I think what they really saw... In the 19th century we're talking. Yeah, we're talking 19th century. And I think they really saw that education, the university should be taken out of sort of civil service and just celebrated for its own sake. And it really caused an explosion of celebrating mathematics for its own sake. And you suddenly get people the freedom

to think about imaginary worlds that surely couldn't exist in our real world. And actually that allowed mathematics to breathe. which it didn't do in France. And you see a turn in the tide because France used to be very dominant and suddenly during the 19th century, 20th century is German mathematics,

which takes over. And here's Jacobi talking about pure mathematics and complaining about the French. He says the sole object of mathematics is the honour of the human spirit. And that on this view, a problem in the theory of numbers is worth as much as a problem

of the system of the world. So what you're saying is that in mathematics are the solutions to everything, that mathematics is the underpinning. It's back to the Galilei. It's discovered the book of the universe. I've discovered the book of the universe.

It's written in the language of mathematics. Yes, and it's... And you hold to that. I do. But what I think we're going to explore in this program is how you have to free up pure mathematicians to think about things without consequences to actually get the interesting consequences in the long run.

All right, well, let's take a few examples for the rest of the program. Well, let's start with this chain of unintended consequences. Let's talk about the 16th century mathematician, who is also a gambler, tried to work out how to win all the time, and read a book about it, and then what? So, Cardano, who's the man I just mentioned in terms of cubic equations as well, Duralama Cardano, he's a mathematician, he's a doctor,

and he's an extremely compulsive gambler, right, to the extent of having to pawn his wife's jewelry and go into the poor house at some point. He was the first person to calculate the probability of throwing a dice and getting a six of throwing two dices and getting two sixes. And he wrote this book about games of chance in this very, very precise context, particularly of dice games. That was really the beginning of probability.

Nobody had studied this until then. And there was little bits of nibbling away, mostly in the context of dice games. The next person I want to talk about is a Frenchman, Abraham de Mouvre, we're now 17th century. He was a Protestant, and so came over to England to avoid religious persecution in France. became friends with Newton, he started looking at this science of probabilities,

was analysing, again, coin tossing, dice throwing, that kind of thing. And he was the first person in some sense to discover a curve, which we now know as the normal curve or the bell curve. So that's the curve that you may have seen in various discussions of the human population. It looks the same if you reflect it down the middle, it's highest in the middle, and it sort of slopes down gently to both sides, roughly in the shape of the book. bell. And he noticed that if you toss a coin repeatedly, say 10 times, the most common thing to occur would be five heads and five tails. That's the peak of my bell. And then as I get less and less symmetric, so four heads, six tails or six heads, four tails, the probabilities

slowly curve out the way. Now, having got this curve, he started thinking about other ways. So he just get to further lists as, don't we talking about? He picked up on Cardano's idea. So we have a direct connection here. We started with the dice and now we're looking at probabilities, And can you just take us in the direction where he's heading this to Marfrey? Yes. So what does he do with this knowledge? He works, he advances it, and then what does he apply it to? What he applies it to is pensions and annuities.

So he's the first person to consider life expectancy. They'd just started collecting mortality data, and he's the first person to really consider the question of, if I take a 50-year-old man, how long is he likely to live? Until then, William of Orange had been paying out annuities at a rate of 14% a year, irrespective of how old the person was. Hence was losing a lot of money to some rich 20-year-olds and invested with him.

And de Merv was the first person to put on a scientific basis, how long is someone likely to live? John Barrow, given that human beings had played games of chance for thousands of years before Cardano, why did this chain of discoveries only begin to happen with him? It's rather mysterious. Arithmetic goes back to the beginning of recorded history in algebra,

geometry almost to the, again, a similar ancient period of time. But people, as you say, have been playing games of chance since recorded history. But the theory of probability is really very, very late. Why might that be? I can think of two reasons. One, you know, you only got to look at stories like Jonah and the big fish in the Bible where the seaman drew lots in order to decide what to do with Jonah and then cast him overboard. So it was a way of divine. the will of the God.

So chance was the way the God spoke to you. And so it's a rather risky business to start studying that. So there might have been a turning away from a study. Risky in the sense of you're beating God at his goal. Yes, it's rather blasphemous. You're sort of inviting punishment. But even so, you would have think there are people who are willing to take that chance. The other problem I think is more practical.

If you look at what was going on in the marketplace thousands of years, years ago, some dodgy individual would have some chicken bone which he would throw on the floor and you would make a bet as to which piece of it was going to point towards you. Every chicken bone's different and the fellow who owns it understands its biases. But there's no general theory of all the chicken bones. It's only when you have symmetrical dye, like we're used to tossing, where there's an equal chance of any face coming down, that it makes sense. to start having a theory about that.

So I think this very simple idea of equally likely outcomes was what you need to really start developing a large arithmetic and mathematical theory of chance in the long run. Marcus, can we take this story on a bit then? It goes from Cardano, sorry, to Des Moire, who, as Colvincied, became a friend of Newton's, and despite his great advances he made, died in poverty. And does that bring us, is the next stage, in this particular story,

because this is one of two or three stories we're going to tell, do we go to Gauss, to Germany, to Gauss after that? Well, we can do. Would you like somebody? Well, I would... Gauss is one of my favourite mathematicians. In a way, I suppose this is almost an unintended consequence for pure mathematics, you could say,

because Gauss, I think, is probably the mathematician who encapsulates both the pure and applied in one mathematician. I mean, he was somebody who loved data. He became famous, actually, for rediscovering an asteroid that had been lost, they thought it's a new planet. This is the late 18th, early 19th. Yeah, in fact, it was found on the 1st of January 1801,

the first day of the new century, and then they lost this planet behind the sun. They couldn't find it. And here's Gauss the pattern searcher, who was able to take the data and rediscover a pattern in it, and predict where it was going to be. Again, just to keep us, we're still talking about something that kicked off with Cardano with the dice? Well, yes in some way because it's about probabilities and

errors of data and where things going to be. Yeah, but I was going to actually come to prime numbers, which are my favourite numbers, because here again is Scouse who's great at understanding data, and he takes lots of data about prime numbers, and he brings his intuition about probability actually to discover a pattern inside

these numbers. These numbers don't seem to have any pattern to them at all, but by Gauss taking a sort of probabilistic view and saying, well, what is the probability that a number will be prime? And as he looks at the data, he realizes here is a pattern. And he's able to apply the probabilistic model to uncover something very deep about pure mathematics, which we now call the prime number theorem.

So I think here's an interesting example of something which looks rather, you know, probability does look very practical. And of course, it's very relevant to today's, you know, we're all very bad at probability. That's why we need mathematics to help us through it, because their intuition is rubbish when it comes to probability. If you say one in a million people in London might have done this crime, well, there are 10 million people in London.

That means 10 people would do it. For most people, one in a million means it's impossible. So the mathematics of probability has been incredibly important for us in our modern age, in understanding risk. But I think the nice thing about Gauss is that he was able to go backwards and forwards between practical things trying to study an asteroid and pure things, trying to understand the patterns in prime numbers. And even he would spend his time, I mean, I understand this as a pure mathematician,

but he really enjoyed doing the surveying of the German lands. I mean, he'd spend a lot of his time out measuring things. But of course, this leads also to an interesting bit of geometry, which I think John was going to mention. I just want Culver to nail this asteroid first. Sure. 18-0-1 asteroid. Yes.

So, I mean, as Marcus, said it had been tracked for not very long by an Italian mathematician called Piazzi. He found it on the 1st of January. He got sick on the 11th of February, so there was a very, very small data set. Ironically, they were looking for it there because they thought they'd spotted

a numerical pattern in the orbits as predicted by Kepler's laws of planetary orbits, so they had a good idea where they were looking. They found this object. They thought it was a small planet. There's a Hungarian astronomer had taken the data from this Italian, published this very small data set and said, can anyone

tell me where to look for this thing when it appears behind the sun. So this thing, which they thought was a plant, had disappeared behind the sun. And they wanted to know where it would, when it would come back and where it would come and where. That was the... You're going to lose it for a while behind the sun because the sun's quite big in the sky. And so it's going to then re-emerge when you'll be able to look again without being blinded by the sun. And the question is, where is it going to reappear? Where should you look? So Zach published this data and basically challenged the mathematicians of Europe,

including his young friend Gauss, to predict where the thing was going to reappear. and Gauss's prediction was way out of line with everybody else's and he also said, this thing is not a planet, this thing is an asteroid and it turned out that the 24-year-old Gauss was right on both counts and what he'd done was he'd thought about this curve that DeMova had found with the heads and tails and he'd said, I think my errors of observation are going to follow the same shape.

What's going to be most likely and what the average is going to be is no error at all and what's going to become successively less and less likely as bigger and bigger arrows, but an error in one direction is as likely as an error in the other. So by analysing the shape of the errors and seeing that that's the same as the heads and tails curve, he was able to predict where the asteroid was going to reappear. John, John Barry, can you take us onto the next stage of this probability business here? We've got the asteroid and then we're going into the revolution in statistics, aren't we here?

Yes. I mean, this, ironically, there's part of this story where you can see that a branch of mathematics is actually being created from the applications. So those early statisticians like DeMovo were interested in predicting life expectancy. They were part of the actuarial profession in a sense. But then social scientists started to think about systematically investigating trends of behaviour. How many criminals do you expect there to be in a million people in a big city? how many people will die prematurely of TB, how many people will be disabled, and so forth.

And it was argued that you could have a sort of what was called a social physics by Cetale, Adolf Cetale, one of the pioneers of mathematical sociology and criminology. So he wanted to try and predict how the course of large numbers of people would develop in society. one of his great disciples was Florence Nightingale. She was one of the pioneering founding fellows of the Royal Statistical Society. And she was effective as a nurse,

not simply because of applying tender loving care and common sense, but analysing what was the result of moving beds further apart so infection wouldn't spread and using statistics to evaluate the effectiveness of things in medical practice. Conlon, would you like to do? to take that on a bit there. What Catella did. I don't get that.

Yeah, I mean, so Catele was the first person to realize that, again, I'm coming back to this bell curve, our heads and tails curve, which then turned into our eras curve, he realized was applicable to just a huge range of human life. They discovered, for instance, that the circumferences, chest circumferences of Scottish soldiers also follows this same curve. So there was a real astonishment at discovering that you could apply it to physical features of the human form

and also to human behaviour. We've been talking about crime rates already. There was quite an uproar. Coming back to John's earlier point about probability being trying to guess the mind of God, how could it be that crime rates were so constant when humans are endowed with free will? What were we saying about the propensity towards sin? But Ketle really nailed it, basically. He was able to show that one has a certain probability of marrying at a particular age.

That's not going to tell you anything about one individual, but it will certainly tell you about individuals en masse. And to start saying, we can look at how criminal justice systems, for instance, affect crime rates in a way which nowadays we're well aware of being discussed in the news all the time. Do you know I do that before we move on to another example? No, I'm happy to move on. You're happy to move on. It was controversial, as you had hinted at.

In Russia there were mathematicians who regarded statistics really as a thing. threat to religious ideas of free will. And in Victorian England, there were strong opposition to statistics by people like Charles Dickens, who regarded it as a great evil that this concept of the average man, the government could say, well, the lot of the average man is always getting better. And Dickens said, you'll just use this as an excuse not to do anything about the lot of the most underprivilege. So it was a controversial issue. I want to move on a bit now. Can we go back to Gavis, back to a...

Marcus is a hero. The world for a long time believed that Euclid's geometry was the word of truth that parallel lines never met for instance and so on. And his postulates held and, well, goodness knows what,

over 2,000 years. When that was challenged, what did that lead to? You had to go back to the 1700s, beginning of 1800 and try to change your mindset a bit about how people thought about bits of mathematics like Euclid geometry. Today we're used to thinking of it just as a tool that we used to figure out how to build the extension on our house or something.

But for the medieval's and later it was part of the absolute truth of the universe. This is how things were. It wasn't just a description. And so the idea that this might not be the only geometry was something that was really resisted, almost anathema. And so at the beginning of the 1800s, and people started to think about whether you could have geometries

which were not euclids. This is revolutionary in another sense. It's like a type of non-realism about the world. And you saw between beginning of about 1830 and 1832 several different people coming up with

a rigorous way of describing how you would derive something like Pythagoras' theorem on a crumpled surface or a curved surface. So Gauss, again, we just heard about, was one of the players here, claimed to have done it much earlier when he was younger, but Bollier, a Hungarian and Lobacheschi, a Russian, both involved in developing this type of description. And then one of Gauss' students, Bernard Riemann, really completed the story. I think this is a good example of the neo-humanist movement in Germany, allowing you to think about sort of imaginary worlds. I mean, this is I think what Boliath called his geometry,

because he just didn't think it had any application at all. The freedom to think about these other sorts of geometry, which probably don't describe our physical world, is what was important in bringing these alive. Can we just tell Leverett, we've got this geometry of Eukle-Dlaid down. Parallel lines is a good example. parallel lines will never meet, said Euclid, and everybody thought... Although sailors knew that things weren't quite like that.

But people went on believing it intellectually. So when they said, no, we won't, we'll take another course. Let's forget the risk. What did they find and how did they find it? Can you just give us elevation? Well, I think that... Yeah, so the surface of the earth is a good example. What is a straight line on the surface of the earth?

Well, you have to curve, you can't dig in through the earth. And there you find there are no parallel lines, because if you take lines, the straight lines, the shortest path between two points, are the lines of longitude. And you find that those meet at the poles. And so that was a geometry that Riemann discovered said, okay, well, if you restrict yourself to being on a curved surface, you have no parallel lines. So here's a geometry which isn't one of Euclids. And Bollyne, Lobbachevsky and Gauss came up with others where there are many parallel lines through one point compared to another. So here were three different sorts of geometry.

And then the question is, well, actually, which one is our universe? And that was what Gowse was doing when he was surveying Hanover, was he was interested, well, maybe light when it travels, perhaps it does bend and somehow produce triangles with angles less than 180. Now, of course, surveying, he was putting lights up in Hanover, trying to look at the distances between things. But, of course, that was too small a scale to actually see the curvature of space. Now the amazing thing is that these imaginary worlds, these imaginary geometries,

are precisely the geometries that we need to understand the theory of relativity some 50 years later. So that now brings us to Einstein, John Burr. Yes, Einstein always claimed that he wasn't a very good mathematician at all. He was a very good physicist, but he had a lot of help in his mathematics, fortunately for him. He had old college friends who were very strong pure mathematicians and took a lot of advice and tutorials, in effect, on trying to learn about some of this very fancy and to most physicists' minds,

very abstruse pure mathematics that have come along in the 19th century. And until 1900, physicists wouldn't use anything more exotic other than Euclid's geometry, some calculus, and a bit of algebra. But Einstein had a conception of how the universe was fashioned,

where he wanted to give up the idea that space was just like a flat Euclidean tabletop. and planets and stars moved around like balls rolling on this tabletop, that he wanted to give up this idea that space was fixed and unchanging. And his vision was that space was more like a rubber sheet, a little trampoline, and you dropped your planets and stars on it like heavy weights, and as they moved around, they changed the curvature and the shape of the space as they moved. And in order to understand their motion,

you just use this principle that Marcus mentioned that things move so that they get from one point to another in the shortest possible time. And on a flat surface, that means they go on a straight line. But on a curve surface, they don't. They follow the path that, you know, the aircraft I'm travelling in from London to San Francisco follows over the surface of the earth.

But on a very wrinkled wiggly surface, it's a more complicated path. Like the one of mountain stream follows as it wiggles, down from the mountaintop down to ground level. So Einstein's picture of what gravity is and how it acts was bound up with curved geometry for space. That things move around, they change the curvature of space, and that curvature then tells them how to move.

So we have a direct line. Now let's take the last example, Marcus. We've got time for me. We've got plenty of time to talk about it. Back to G.H. Hardy, his number theory, he was rather proud of the fact that it was pure and useful. absolutely there's a mathematician's apology i mean it's a book i think most a lot of mathematicians read when they were younger and brought up on it's sort of um instilled this ethos of celebrating mathematics

for its own sake and it's wonderful he you know he does um celebrate number theory will never be useful for anything at all and um he says the real mathematicians of the real the real mathematics of the real mathematicians the mathematics of firma and oiler and gals and arbal and reman is wholly useless and he's celebrates this. But at the end of the 20th century... Not like Oscar Wilde, isn't it? He's a character you love and hate because of this. No, no, not like Oscar Wilde himself. The phrase, all art is useless.

Oh, right. Well, but the end of the 20th century Hardy has proved to be completely wrong because one of the things he was obsessed with with prime numbers like Gauss. He was trying to understand sort of some structure in these primes. Reamer had come up with some

ideas about the way the primes are distributed. And I certainly thing he thought that primes were never going to be useful at all. They were a lovely play thing. Now, the same time as Hardy was around in Cambridge, you have Turing. Touring was somebody who, you know, we all know perhaps because he cracked the Enigma machine during the Second World War. But what is unknown is that his ideas for using machines to crack codes actually was sparked by his interest in this thing called the Riemann hypothesis about primes. And he applied for a grant from the Royal Society to build this machine

that would try and help to find where, how the primes were distributed and he'd started to build this thing in his room made out of cogs. Did they give him a grant? They'd give him a grant, I think of about 60 pounds or something. Yeah, and the blueprint still exists for this thing. It was never fully built because the Second World War broke out.

Touring is then taken to Bletchley Park where all of these codes are coming in. Again, this is like a little puzzle for him. So it's like a crossword puzzle. You know, here are all these codes. How do you undo them to see actually what the actual settings to the Enigma machine were at. And again, he thinks a machine is what

should be useful for this. And he builds the bomb which actually cracks these things. The bomb? The bomb is B-O-M-B-E. It's the name for... You can go and see this at Bletchley Park, actually. They've got a reconstruction of it. It's like the dessert. Yes, yeah. But the idea, again,

now, of... Originally, Turing's motivation was to try and understand prime numbers. He was able to crack these codes, shortens the wall, by two years as far as Winston Churchill is concerned. After the war, he comes back to this idea of a machine. And he realizes the machines he was building were a bit useless because they could only do one thing.

And he comes up with this idea of a universal machine, which is the forerunner of the computer. And suddenly he's going back, and he comes back to prime numbers. And the first thing that he tests on this computer is to try and beat the current record for the biggest prime number, which unfortunately his machine breaks down. Hardie would be turning in his grave. Not only that.

But then we come to the Internet Cryptoids, which is used on current day computers, which uses a beautiful theorem about prime numbers discovered by firma, 350 years before the internet was ever thought of. And this beautiful little fact about prime numbers, which Hardy loved, we all love as pure mathematicians, is suddenly at the heart of the cryptography

which is used by everyone when they send their credit cards across the internet. This is the most amazing unintended consequence of pure mathematics than I can think of. I mean, okay, relativity is about describing the world, and probability. This is number theory, prime numbers, creating the computers and the internet world that we use today. Right. Colver. Yes.

Do you want to interpolate an opinion or an observation on the, after that area, Macrosaded you so twice. I actually wanted to come back to something which Turing had done, which was far more, so the stuff we've been talking about with the computers he was building sounds quite applied, but another of the ways in which Turing had got into computers

was really thinking about the axioms of mathematics and the limits of mathematics. And he was interested in the question of, can there exist a machine to mechanically verify whether or not a statement is true or false? And as part of proving this, he invented, as Marcus mentioned, the universal machine. Now, the key difference with that compared to previous computers

was that it stores its own program internally. The program is just the same as any other bit of data. This is what means that one machine can mimic any other, because you can feed in the program for any other. It's also, sadly, what's given us internet viruses. So we receive something that we think is a piece of data, just a word attachment, say, or some other attachment. What it's actually got is a program that your computer's going to run, hidden in it.

The pure nanmedities aren't responsible for viruses, so it must be. It's an unintended constant. John, John, to come in on this. Yes, this idea of using prime numbers in this way. It goes back to Leibnitz, you know, Newton's great rival, an adversary. So Leibniz had the idea of building a machine which was going to mechanically solve all human disputes because it performed logic. And the key about the prime numbers is another nice theorem of mathematics that if you take any number, say 10, you can always

split it up into a product of prime numbers, so two times five, in one and only one way. and what this means is that if you take an ingredient of logic, you know, a name or a concept, and you attribute it a prime number like two, and then you have another concept and you attribute five, if you combine them, then there's a unique result, 10. And so Leibniz hoped that by programming his machine with all sorts of political and religious statements, linking them to prime numbers, he could turn the handle and the machine would tell him whether, you know, the doctrine of predestination was correct or whether liberals or Republicans

were the correct party to run the country. It's horribly idealistic, but it was based upon recognising that you can use prime numbers to uniquely characterize the way statements or concepts are joined together and there's just one result. And in some ways this machine of Leibniz

was a precursor to the enormous argument between Newton and Leibniz, which you can. covered not so long ago as to who invented calculus because Leibniz shows up in London with the prototype for his machine, goes to the Royal Society, promises to build them one and never delivers. Right.

Thank you very much. We've got to get out now. Thank you, but Col Maroni-Dougall, Markets Sotoy and John Barrow. Thank you very much indeed. Next week, Indian Rebellion or Indian Mutiny, 1857. Thank you for listening. We hope you've enjoyed this Radio 4 podcast. You can find hundreds of other programmes about history, science and philosophy at BBC.com.com.uk

forward slash radio 4.