In Our Time - Infinity

Episode Date: October 23, 2003

Melvyn Bragg and guests discuss the nature and existence of mathematical infinity. Jonathan Swift encapsulated the counter-intuitive character of infinity with insouciant style:“So, naturalists observe, a fleaHath smaller fleas on him that preyAnd these hath smaller fleas to bite ‘emAnd so proceed ad infinitum.”Alas, the developing utility mathematicians put to the idea of infinity did not find the English philosopher Thomas Hobbes quite so relaxed. When confronted with a diagram depicting an infinite solid whose volume was finite, he wrote, “To understand this for sense, it is not required that a man should be a geometrician or logician, but that he should be mad”. Yet philosophers and mathematicians have continued to grapple with the unending, and it is a core concept in modern maths.So, what is mathematical infinity? Are some infinities bigger than others? And does infinity exist in nature?With Ian Stewart, Professor of Mathematics at the University of Warwick; Robert Kaplan, co-founder of The Math Circle at Harvard University and author of The Art of the Infinite: Our Lost Language of Numbers; Sarah Rees, Reader in Pure Mathematics at the University of Newcastle.

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or wherever you get your pods. 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, Jonathan Swift encapsulated the counterintuitive character of infinity with insoucant's style. So naturalists observe a flea, he wrote,

has smaller fleas on him that prey. And these has smaller fleas to bite him, and so proceed, add infinity. item. Alas, the developing utility which mathematicians put to the idea of infinity didn't find the English philosopher Thomas Hobbes quite so relaxed. When confronted with the diagram depicting

an infinite solid, whose volume was finite, he wrote, to understand this for sense, it is not required that a man should be a geometrician or logician, but he should be mad. Yet philosophers and mathematicians have continued to grapple with the unending, and it's a core concept

in modern maths. So what is is mathematical infinity, are some infinites bigger than others, and does infinity exist in nature? With me to discuss the mathematics of the infinite, is Ian Stewart, Professor of Mathematics at the University of Warwick, Sarah Rees, reader in pure mathematics at the University of Newcastle, and Robert Kaplan, co-founder and co-director of the Math Circle at Harvard University, and author of a new book, The Art of the Infinite, Our Lost Language of Numbers.

Ian Stewart, let's start this with Zeno, who illustrated the problem by using you. using the infinitesimal. And one of his paradox, the best known, is the tortoise and the hare, or Achilles and the hare, the race between them and how Achilles or the hair can never catch the tortoise according to his mathematics.

That's right. Zeno, what he was really trying to do was to explain and show up certain logical problems with assumptions about the structure of space and time. So just to show that this little story has a serious philosophical point. But the... So Achilles and the tortoise start off, they're having a race, the tortoise gets head start,

but Achilles can run much faster. And so we all know that Achilles will catch the tortoise up and very rapidly be out in front. But Zina says, no, he won't. He says, because think about what happens. The tortoise is out in front, let's say, you know, some specific amount in front. Achilles has to get to the point where the tortoise starts from, but by the time he's got there, the tortoise has moved on a bit.

So then Achilles has to get to where the tortoise is now. But by the time he gets there, tortoise has moved on a bit again. And this just goes on and on forever. And if it goes on and on forever, there's the infinity coming in, then Achilles can't catch the tortoise at all. Now, if you think about that, you say, oh, how old you know, this is complete nonsense, because those times are getting shorter and shorter and shorter and shorter.

And a mathematician nowadays would say, well, what is proved is that there's a whole lot of times at which Achilles has not caught the tortoise. That doesn't mean there isn't some other. time at which he has caught the tortoise, but you do end up concluding that in order to catch the tortoise, he has to do infinitely many different things, get to where it was when it started, get to where it is next, get to where it is next, can you do infinitely many things and then do something else after that? It's all very puzzling.

Well, I thought it was similar until you explained it, you know? I thought it was a clear example of creative mathematics defying common sense. But why is that so important? What you have said? in the way you've said it, you've put it in a very interesting, an important way, and it was important for mathematicians for a long time, and is still something that you dwell on with obviously interest. Yeah, it's really about the logical structure of mathematics, the way the Greeks thought about things like geometry,

the things that we now consider to be very basic mathematics, there are all sorts of little philosophical problems when you try and make the whole thing fit together logically, and Zeno was just, he was being provocative, He was in almost deliberately obscuring things by saying, hang on, you don't understand this stuff as well as you think. And there was an approach to geometry there still is in terms of movement. It's very, very intuitive that if you take a triangle and move it around, it doesn't change shape, and you can fit it on top of another triangle and so forth.

And you can do a lot of geometrical proofs in this way. But Zeno's saying, hold it, you've got to tell me what you mean by movement, and it's not as straightforward as you think. Was the influential? Was that paradox particularly influential? It was influential in the sense that the Greeks actually took this fairly seriously, some of them anyway, and it led... It's a very complicated, indirect sort of story, but it led to some wonderful work by a Greek called Eudoxus, which was all about the mathematics of the irrational, about numbers that are not exact fractions. And suppose you want to compare two numbers of that kind.

You get tangled up in the kind of things that cause Zeno all of this thought. And Udox has sorted it out pretty well. It was then largely ignored by almost everyone who came after. It's the bit of Euclid's geometry book that was never taught to anybody, because it was too difficult. And it resurfaced in about the 1800s in a much more modern guise. And are we talking, Robert Kaplan, how we're talking there about it resurfacing with Leibniz, mutant. Yes, and I think if we put the infinitesimal in the context of use, so that this doesn't

seem like some abstract game or meaningless play, Zeno was the hired gun of the philosopher Permanides. Permanities wanted to show that there was no motion. There was only being. Being is, is the only thing we can think, he said. And Zeno, quick to help him, wanted to show that when you do try to think of motion, you get tangled up, hopelessly tangled up in paradoxes like this. Now turn, as you know, saying to modern concerns about motion. Think of trying to describe the slope of a hill, a one in four hill. Well, easy enough. It rises one unit up for every four units you go over. We all know how much trouble we have with our cars climbing such a hill. But can you talk about the slope of a curve, not of a straight line, but of a curve. Think of the flight of a ball,

a parabola that the ball makes. How can you talk about its slope? Well, it doesn't have a slope. Its slope keeps changing, growing less and less steep as you reach the height, the peak where the ball stops, where the tennis player is supposed to hit his serve, flat at the top, and then gradually increasing more and more steeply as the ball goes down. How can we talk about slopes of curved lines? Ah, using the infinitesimal. By saying, well, we could take shorter and shorter straight lines to approximate this curve. Think of pasting those straight lines onto it,

or making a curve by taking a series of threads, tighter and tighter threads, which approximate. which hug this curve. Now, we really don't want to talk about straight lines at all, not even very short ones, but very, very, very short ones, infinitesimally short ones. We want the length of these tangent lines to go to zero. And so how did the work of Leibniz and Newton on the calculus

help us to discern that? They invented a part or together. It's a great argument who had priority. This wonderful art calculus, which allows us to talk about change in a cogent way, not via paradoxes, but by regulating our thought about the very small, so that we can actually take a sum over very short lengths, or talk about the slope of a curve at a point. How do they do this? By the notion of limit, letting a quantity go toward,

0. Think, for example, of the sum, 1-half plus a fourth, plus an eighth, plus a 16th, I'm going to go on adding half of what I had to this series. What does it add up to? Well, if you stopped at the second term, a half plus a fourth, it adds up to three-fourths. One more term, 7-8s. One more term, 15th, 16th. What does the whole series add up to, as I add smaller and smaller units to it. It adds up to precisely one if you take an infinite number of these terms. Now then,

Newton's idea was challenged, as mathematical ideas, I understand it. I'm born very thin, I see her, okay, so I need a lot of help. Newton's mathematical idea was challenged by Bishop Barclay, the philosopher, who said that if you're doing these smaller and smaller intervals and calling a curve of straight line, a conjunction of many, many, many straight lines,

the intervals of movement actually are reduced at certain points to zero. And so you're talking about lots of zeros, and that refutes the theory, because the zero is nothing, and therefore if it reduces itself to nothing, you're back where you started from. He would be absolutely right in his criticism, were one reducing these intervals to zero, but we're not. We're going toward... I think the great change in Western thought in the Renaissance is the replacing of two by toward.

Process, going toward, not getting there. How well we know at the beginning of the 21st century that we're never getting there, but we're always on the way. So that you would refute Berkeley by saying that Berkeley thought that there was an end, whereas Newton understood that there was always a process. Instead of kicking a stone as Samuel Johnson did to refute Berkeley, I would refute him that way. Right, we got through that. No, Sir Ries, despite Barkner objection, space is infinitely divisible, isn't it? Yes.

Can you explain how Henri Poincaray demonstrated this in the 19th century with the Poincaray disk? I think what you're wanting me to explain is... The Poincarot disc can be used. It's just a circle of radius one. It's quite a very finite object, and it's just a part of the Euclidean plane. It's a finite object. Puancares used it to describe the whole of hyperbolic space, an infinite hyperbolic space. So as a Euclidean disk, it's got radius one.

In Euclidean geometry, we know how to measure lengths. We get our rulers out, and using what Robert just described, we can measure the length of any curve, because we can divide it up into infinitesimal pieces and measure the lengths of each of these pieces and then add them together, and we get the length of a curve. And, well, so if we do that, and we find the shortest curve between any two points in Euclidean geometry is a straight line.

Now, suppose we redefine the way we measure an infinitesimal. Suppose we're within the Poincoré disk, and we take two points that are infinitesimally close, and when we're deciding how far apart they are, we take into account how far they are from the boundary, the circumference of this disk. So we measure the distance between them differently, depending on where they are in the disk.

and now we measure the length of the curve by adding together these infinitesimal distances. Then we'll find that the shortest curve between two points is no longer a straight line. It's, at least to our eyes, it's a curve that is an arc of a circle that meets the circumference of this disk at two right angles where it meets it. And now if we take two points that are close to the edge of the disc, they're actually a long way apart, although they may look to us with our Euclidean eyes very, very close. and now if we measure the length of a diameter of this disk, it's actually infinite. We've got the entire hyperbolic plane within this tiny little circle.

So what do you mean by the hyperbolic plane, and why is this significant? So the hyperbolic plane is an example of, if we take Euclin's axioms of geometry, then we can drop just one of them, the parallel axiom, which says, effectively, if we take a point here and two points here, and a line here, then we can draw a line through the first point which is parallel to the second line and we can draw just one such line. When hyperbolic geometry, you can draw infinitely many such lines.

You drop that condition, you can draw infinitely many such lines. It's the geometry that you see if you look on a saddle. It's a geometry of a space where there's an awful lot of space around a single point, much more than you see in the Euclidean space because you're going up and down and up and down, there's just much more space around a point. It's the geometry where the angles of a triangle, angle at up to less than 180 degrees.

Would only be a lighter comment on that before I move on? Yeah, when... Yeah, when... Well, Ian, and then Robert, you. Yeah, well, when I were a little lad, when I was a kid, I read a book by a wonderful man called Warwick Sawyer, who was a teacher who explained mathematics, and he explained the hyperbolic plane in terms of creatures that lived on a finite world.

It was shaped like a disc, and it got very cold towards the edge, and all their measuring instruments shrank as they got near a... and nearer the edge. And if they tried to measure the length of a line, as they moved out towards the edge, putting their ruler along this line, the ruler shrank. They didn't know this because they shrank too.

Everything shrank. And so they just happily kept going, measuring this. So they've moved, this is not quite the right formula, but halfway out from the edge, the ruler's half as long. And then halfway out again, it's half as long again. And if you start adding all this up, you've got infinitely many steps, each of which is one whole ruler long.

So they're going out, say, one, two, three, four, five, haven't got to the edge yet, six, seven, just goes on forever. And this is a different kind of geometry from our usual Euclidean flat plane where we imagine that our rulers don't shrink as we move them around. But of course the creatures on this disk

don't know that it's shrinking because to them everything looks the same. I think you're seeing here in this other giant. this hyperbolic geometry, the fecund heart of mathematics, invention, imagination, coming to grips with what we know and what we think we know. Sarah's point about dropping the parallel postulate, Euclid says, given a line and a point not on it,

there is one and only one parallel to that line through that point. What a reasonable assumption. Well, reasonable but troublesome for the Greeks. Think of the Greeks, sailors like Odysseus, but coast-hugging sailors in the Mediterranean, not going through the gates of Hercules toward the Infinite beyond. They were puzzled by the Infinite out there. They were troubled by the idea of two lines which never met, which never met here or here or even there, but didn't meet at all.

They tried to prove this parallel postul. it and we're unable to. It turns out it can't be proven from the other axioms. And we're still talking about the infinite here, but can we move, if you don't mind you? Can I move to big infinitism? Can you tell us the difference between what we're now going to talk about and what we have been talking about?

I'm not quite sure what you're leading is on to here, Melwyn. I'm taking, the other side. We've been talking about the infinitesimal. Oh, okay, big infinites is right. Yes. The big infinites. The big infinites. Okay, this friend of mine told me a lovely story.

other day about a friend of his who was talking to their four-year-old or so son. And as happens in these things, you know, kids are natural mathematicians, and the question of what's the biggest number came up. And so this chap asked you, saying, said, well, okay, what is the biggest number? And the kid thinks about it for a moment, says, 380. And his dad looks at him and says, well, what about 381? There's a silence for a moment, and then the boy says, well, I was close. the point of this story

there isn't a biggest number and whatever it is you can always add one and get one bigger now there are two ways to say that and one of them is respectable and one of them leads into all these problems with infinity and the respectable way which is what actually people like Euclid

did is just to say there is no such thing as a biggest number the less respectable but actually much more interesting ways to say that there are infinitely many different numbers and they can be as big as you like. They can become infinitely big. The question is, what do we mean by that? Sarah, what do you mean by that? There's no such thing as a biggest number.

And that they can be infinitely big. It's this word infinite that's a good teasing word. What does it mean to you in the context of what Ian said? Well, I suppose it means that if you give me any number, I can give you a bigger one. And where does that take us in regard to coming back from a mathematical world to the world in which attempting to get through at the moment? It means that I don't have to worry about whether I've got enough memory on my computer because there's infinitely much memory out there that I can go out and buy for it. That's actually a very good point. Yes, you can always add a bit extra on.

So it's better what Robert was saying about the difference between things and processes. If you think of infinity as a thing, you get into awful trouble, although actually mathematicians do think of infinity as a thing, and they do get into awful trouble, but it's great fun. But if you think of it as a process, the process at any stage is only handling finite things. If I have a computer and I start adding extra bits of memory, at any stage I have a finite amount of memory. It's just that if I run out, I can always put a bit more on.

Robert, the 9th century court astronomer of Baghdad, Tabit, introduced a paradox for large infinities, as I have read. You know, that infinity could be split into any number of parts, each of which would be the same size as the whole. Is that right? It is. In that case, what's important about that? Well, he argued, as Galileo did later in 1638,

that if you take the numbers we've been talking about, the counting numbers, which go on forever, and pull out... Counting numbers, 1, 2, 3, 4.5, and so on. And pull out every other one, the even numbers, 2, 4, 6, 8. Well, you clearly have half as many now as you have. the even numbers are half as many as all the counting numbers.

Seems perfectly straightforward. Until you begin to notice that, well, two is the first even number, four is the second, six is the third, I can match up the even numbers with the counting numbers. There are exactly as many. If you think of them as moving through two doors, the counting numbers coming through one door and the even numbers through a second,

they come out holding hands. Well, how can this be? Galileo said, this is too troublesome. I think the best approach is just not to think about it. And it wasn't until the 19th century with an amazing man named Georg contour that we began seriously to think about

the behavior of infinite sets. I haven't got to contour yet, and we've got to try to get through this. Why is it so astonishing to a mind, largely untroupled by mathematics until this last week, I must tell you, that the even numbers and the counting numbers come out of parallel doors holding hands? Why did Galileo find that so troubling?

Because the even numbers should be what we call a subset of the counting numbers. The even numbers belong to the counting numbers, but they've left out all the odds. There are all those odds. numbers left behind. I get it. Ah, yeah.

So how could there be exactly as many? In fact, there are exactly as many odd numbers as there are counting numbers. Whoops. Yeah, I think of it as knives and forks. It doesn't trouble me, but I see the problem. Suppose you had a table laid with knives and forks. And at every place setting there's one knife and one fork. And somebody counts the number of knives, and it's exactly the same as the number of pieces of cutlery.

The number of knives plus the number of forks is the same as the number of knives, and the number of forks is the same as the number of knives, and yet it is not that there's zero of these. There's an awful lot of knives, there's an awful lot of forks, and it doesn't matter whether you look at knives, forks, or the whole lot, you get the same number every time. So, Erie, how did Galileo, then, as Robert has alluded to Galilee, can you tell us a bit more how he built on Tavit's work?

How did he build on Tavis' work, well? I think he had to be very good. careful because the Spanish Inquisition was out there and didn't like people who claimed that the infinity really existed. What else did he do? He also proved there were the same number of square numbers, I think, as whole numbers, which is astonishing, really, because we all know that many numbers aren't perfect squares, you know, I mean, you square 100, you get 10,000, you square 101, you get something which is, well, quite a lot bigger than... 2001? Something like that, yeah, or 121 or something like that. Yeah, I think you're right.

But, I mean, you know, they get very sparsely spread out as you go on and on and on. And yet there as many of those as there are whole numbers. So what else do Galileo do? I'm a bit sunk on Galileo. I think that's such an astonishing insight. Take, well, Melvin, what's your favorite whole number? Seven. Take seven and its multiples. 14, 21, 28, and so on.

We're leaving out a lot of numbers now from the counting numbers. And yet, there are exactly as many multiples of seven as there are counting numbers. Seven holds hands with one, 14 with two, 21 with three, and so. on, something wrong here. And Aristotle, to go back from Galileo, Aristotle said, well, you can't think it's illegitimate to talk about infinity as a completed whole. And to say that there are as many multiples of seven as there are counting numbers is to think about each of those infinities as a completed whole. Perhaps that's where the trouble lies. in terms of infinity and the idea of completed hells and did the Big Bang theory have any influence on the way excuse me this was thought about it influences the way we think about the relation between mathematics

and the real world in as much as the Big Bang theory is some people are not convinced this has a great deal to do with the real world on the other hand cosmogist believed that's where it all came from if you look at the the instant of creation you're looking at an infinitesimal period of time and you're looking at a zero-sized point universe which suddenly explodes into a much bigger. You know, a universe who's actually, maybe it's a football size and then it's very much

bigger than that. It's bigger and bigger and bigger. So this notion of a singularity comes up. And perhaps you think about a black hole, which is another one of these related things. the closest that real physics gets to real infinities is inside a black hole. Now the great thing about inside a black hole is we have no idea what's really there because you can't go in and take a look, at least if you did, you can't come out. That's right.

So let's back off a little bit. Black hole arises when you have a very, very massive star. It's so heavy it starts to collapse under its own gravitational field. And in fact, it's so heavy that even light cannot escape from it. and that's why they call them black. Actually, they're probably red, but that's because light can come very close and it gets slowed down. But inside a black hole,

according to the mathematical theories of these things, what happens is that star keeps collapsing, and all of the mass in the star collapses right down to a single point, at which point its density is infinite, genuinely infinite. And if you believe that the model is a correct description of the physics, if you could go inside a black hole, what you would find is a point singularity of infinite density. Of course, you wouldn't be able to come out of the black hole

to tell anyone about it, and in practice you'd be ripped to shreds by gravitational fields as you went in. So this is, again, a paradox, because no physicist really likes to believe that there are genuine real infinities anywhere in the universe. Did Cantor, to come to Canada, which you wanted to come to earlier, Robert,

can you explain who Cantor was, why is he important? Did he challenge these paradoxes? He did. He was the reincarnation in the 19th century of Alcibiades, that Greek spirit of contrariness. He took on all comers in the intellectual world, Cantor did, by saying, well, Aristotle says we can't talk about completed infinities.

Let's do so. Let's see what happens if we do. Think of a set. The notion set is due to Cantor and his friend Dedic. think of a set of all the counting numbers. Think of a set of all the multiples of seven. Now we can compare the two, and we see that Galileo's problem evaporates.

There are as many of one set as of the other. You can match the elements up in a one-to-one way. This is true of any rhythmic pattern in the counting numbers. What about the fractions? There must be many more fractions than counting numbers. Just between zero and one, you subdivide and subdivide and subdivide forever. No, Cantor comes up with a marvelously ingenious, brilliant way of showing that there are exactly as many fractions as there are counting numbers.

Ah, so all these different infinites turn out to be the same. So far, so good. What about decimals? Every fraction can be turned to a decimal, but not every decimal can be turned into a fraction. The decimals that have no repeating pattern aren't in correspondence with fractions. Like the square root of two. Like the square root of two, that troublesome fellow. What Cantor showed, remarkably enough, was that the infinity of decimals is larger than the infinity of counting numbers and all their friends. And then he gets larger infinities than that, and larger and larger still.

you'd think that this mind-splitting idea would be too difficult for anyone but a genius like Cantor to handle and yet an eight-year-old can come to grips with it because the only idea, the only technique involved is making one-to-one correspondences or showing that they can't be made. So, Rees, Kantor's ideas came under great attack

and he had a breakdown, maybe as a result of that attack, his collaborator David Hilbert declared no one will drive us from the paradise that Cantor has created. Why is his work so important and this use of the word paradise is intriguing? It's a very difficult question, yes. Because it opens up the whole of post-Cantorian mathematics, if you wish, where I think we just have a different view of what the foundations of mathematics. David Hilbert called it a point. paradise. Roberts explained how important it is and how, and Roberts explained, cantoe in the highest possible terms, we obviously challenged authority of Aristotle, but an atrial

understanding, but what is paradisial about it? You can really have fun romping through this stuff if you like this kind of game. This was a rather pivotal period in the development of mathematics where an awful lot of what everyone thought they knew was true was getting torn up, thrown away and we were getting a much more subtle view of the kind of assumptions everyone had been making for a long, long time. And I think Hilbert came face to face with this because he was working on various bits of mathematics to do with, he was actually working on what later turned into the foundations of quantum mechanics. He was working with the mathematics of sequences of numbers, not just a single number.

number, but whole list of numbers, and these were infinitely long lists, but they were infinitely long in the smallest of the cantorian infinites. And there's a big distinction in mathematics between finite and infinite, but there's an equally important distinction when you get into the infinite sets between the smallest ones, the ones that match the counting numbers, what we call countable sets, and all of the others, which are just lumped together and called uncountable. and you can do all sorts of wonderful mathematics with countable sets. You can actually add up a countable set of numbers with some chance of getting a sensible answer.

You just cannot do this with an uncountable set. And Hilbert was very much focused on these analytic operations of adding together infinitely many numbers and this kind of thing. And Cantor sorted out the basics of what you do here and made a big distinction between what is legitimate and what's not. Robert, does this bring us to the question of whether infinites are real or imaginary? Does this bring it anywhere near the real world? Does this bring us anywhere near the discovery of zero and imaginary numbers?

I think it does. I think it takes us from the world of appearances to the real world, and that's the paradise that Hilbert referred to, that wonderful quatrain of Jonathan Swift's that you quoted earlier. We could add to it the following, for at the gates that Cantor flung apart, and Hilbert later, angelic flees, cavort in hosts inordinately greater. What Cantor had done was to reveal that the world of appearances,

the world of our infinite time, stretching backward and forward to infinity, infinite space, that these are small infinities nested within a larger one, that we come on to what mind rather than I is in contact with, that we're seeing with the imagination, with the inner eye, we're seeing those things which cannot but be true, paradoxical as they sound, like imaginary numbers, I, the square root of minus one, which when it first arose, people said, well, this is ridiculous. There can't be a number which times itself is negative one,

and how right they were. There is no real number like that. A positive times a positive is positive. A negative times a negative is positive. So what sort of number times itself could be a negative, like negative 1? Well, in that alcibiades spirit, a mathematician says, let's invent it. Let's imagine it. And Descartes gave it the name imaginary number.

Wallace, the great English mathematician, called them fictional numbers. but these are fictions more real than our facts. In what way? Can you just take that? Or maybe so. Would you like to tell us why these imaginary numbers are so useful in mathematics? Well, they enable us to solve equations, for instance. I mean, so any equation which can be written over the real numbers,

the numbers that we know, can be solved using the imaginary numbers. If we don't have the imaginary numbers, then we have some equations we can solve, and some we can't. Some simple, like quadratic equation. To be only able to solve half of them would be incredibly unsatisfactory. So it makes the universe complete. In what areas are they being useful?

Can you give us an example of, in this particular area of inquiry in mathematics, the imaginary numbers are essential? They're taking us towards this. They're absolutely basic in a whole part of applied mathematics, engineering, physics. This is what really gave them a boost. The mathematicians were interested just because these were nice numbers, and you could do really nice things with them and instead of saying some equations can be solved

and some kind, you say, well, they can all be solved, but some of the answers make sense, and some of them don't. But the engineers discovered that even simple things like alternating current in electricity is very, very convenient. If you want to do calculations with alternating current, if you do them using these imaginary numbers, it all makes sense, it's all very straightforward,

and you get nice answers, which are correct. If you try and do these calculations without using these, imaginary numbers, it can be done, but it's an arcane sort of art. Calculations get very, very complicated. At the end of the day, you get the same answer, but why go all around the houses when these so-called imaginary numbers give you a beautiful shortcut that gets you straight to the answer you want? I think, beside their use, I think there's a very important point here about mathematics

as an art. Compare a folk song to a late Beethoven quartet. One goes a very great distance from one to the other, but it's a gap we can bridge with our emotions and with our sensations. To go from the mathematics of counting objects on a table to these vast infinities and these imaginary numbers, or even zero, is the same sort of gap that there is from a folk song to a late Beethoven quartet. and again it's a gap we can bridge with our finite human minds as long as we're allowed to let our imagination loose. Was there discovery of zero the key to this? I think it was that... Was that about around...

You're the expert on zero. The Sumerian's back? I'm facing the expert on zero, ladies and gentlemen. It's not a bad morning. I know nothing. But it came into use in the... No, it came me to mathematical use in the 13th century in any sort of... It did, but it was invented by...

the Sumerians 5,000 years ago for their bookkeeping for place notation. Instead of having to write different symbols for larger and larger numbers, you could just say a number in this column is a unit, but in that column is a 10, and that column is a hundred, using a mark for nothing in this column to stand for what we now call zero. Not only for notation for placeholding, for making numbers convenient, but zero is the the fulcrum of the number line, the positive numbers going off to the right, the negative to the left, it stands in the midst of all our calculations. By means of zero, we can solve equations that looked otherwise impossible.

You take one complicated expression, set it equal to another, well, why not subtract that one on the right-hand side of the equal sign, set the whole thing equal to zero, and what was a sum can turn into a product, and then you can solve the product in various straightforward ways, which you couldn't have done before. So we're talking about mathematicians existing in fictional worlds here, aren't we, Zara? You can be all the hard questions. Mathematicians existing in fictional worlds. So you're saying, this isn't real stuff, really.

This is just all made up. I'm taking the word from the language of mathematics. I'm taking the word imaginary numbers, numbers which, I mean, I can't see minus one plastic cup here, but I can't see minus one plastic cup over there where it isn't. indeed you can't no so I have to make it up if I have one on the left I'd say imagine ladies there's a plastic cup which isn't there so that's an active

imagination on your part and my part to imagine it is there so you're dealing in a fictional world I think we all have to decide where the fact stops and the fiction starts I mean so it doesn't take a great deal it isn't a very big jump to start believing that things have been taken away

to move from from the positive numbers to the negative numbers I think and you make it Eventually, a lot of mathematics is just about accepting things. A lot of understanding mathematics is just about accepting things. When you become sufficiently familiar with something, you believe it's true, it's natural, it's real. And I think the step from the real numbers to the imaginary numbers

isn't really much more than the step from the positive numbers to the negative ones. It's just an acceptance. People in the north know how real negative numbers are when they think of the temperature. All of us know how real negative numbers are when we think of debt. We had a child in our math circle who we asked, what does five take away six? And he said, it's three. And we said, hmm, interesting answer.

Why do you say that? He said, well, because the answer is negative one, but I'm not allowed to say that. He was five. Does the existence of infinities and mathematics depend on the existence of an infinite universe? I don't think so. I think philosophers may disagree.

and we have a long argument. I think if you really look into the mathematical nature, the way mathematicians think about infinity, it's all a big fraud, actually. It's there not as a thing that you want to exist. It's there to avoid thinking about things going on for a while and then stopping. Why did Euclid have an infinite plane? Well, actually, Euclid didn't have an infinite plane.

What Euclid had was a plane that you always make a bit bigger by extending your lines. And what he says is not that any two lines that aren't parallel meet. He said if you take two lines that aren't parallel, then if you extend them far enough, they will meet. And he's really mostly thinking about finite length lines, but that length can be as big as you like. And mathematicians introduce various notions of infinity

so that you don't have to worry about how far you extend things. You don't have to worry about how far you can count. As Sarah said earlier, so that your computer memory can be as big as you need. If I want a big number, there's going to be a big enough one around. Instead of saying, let's work with numbers up to 100 digits, and then you have this horrible thought that if you add two of those together, it might have more digits than that.

Oh dear, what's happened is falling off the end of the number line. Let's get rid of the end. And when you say endless, on the one hand, you think, oh, it goes on forever, but on the other hand, it's a much simpler thing. You say, I'm not going to talk about the end of this thing. So you're saying we need a concept of infinity. concept of infinity in order to satisfy certain mathematical imperatives, but we don't need a concept of infinity in order to live in the world we live in.

Need. Oh, reason not the need, says Lear, because while I want to pick up Ian's point about mathematics and fraud, just as art surrounds life and gives it meaning, so the infinite surrounds finite mathematics and makes sense of it. It's our one contact with certainty mathematics, and it isn't here now, it's everywhere, and it leads us inevitably to the infinitely small, to the infinitely large, to the infinity of our imagination. So that's why it's been said, you quote it, I think I'm probably quoting you, don't it? Of all the arts, mathematics most puts into question the distinction between creation and

discovery, was that? Yes, yes. That's a big thing to say. It's a big claim of all the arts. infinite. It's like music, mathematics. It's so much like music. We appreciate it through our feeling for structure.

But I still think that people listening to this program will want you to, wouldn't mind the three of you saying very briefly, whether you think what you've been talking about, and I think given the time available, having a really good guard to try to make this understandable, has a relationship towards received traditional religious, let's say religious ideas of infinity, the infinite wisdom, the infinite life, the eternal life.

Whether there's any relationship, whether the mathematics, as it were, shores this up, bears this out, there's nothing to do with it, or where it stands in relation to it. Can I ask you each in time? Yes, first of all, yeah. Very brief. The hyperspace philosophers of the Victorian period. I do very brief.

Okay. I see in the clock. Where does God live? Well, the fourth dimension is a great place for God because he can see every part of the third dimension laid out in front of him. But this is much too limiting, so let's put him in the world. so let's put him in the fifth, sixth, seventh, infinitiath dimension. And this was really done in Victorian times.

This was a point of view. Robert, final point is I started with your quotation from you. We look for meaning. We want the world to make sense to us. It makes sense by reverting to these seeming fictions that turn out to be our facts. And Sarah? I think, I find it very hard to see the infinity out there

in a sort of concrete kind of way. I can't imagine living infinitely often. What I cannot deny is the existence of the infinitesimal. In order to have things happen continuously, we have to have the infinitesimal, and it's just the twin of infinity. Well, I have infinitesimally time left. Thank you very much for being here, and thank you all 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.