In Our Time - Zero

Episode Date: May 13, 2004

Melvyn Bragg and guests discuss the history of the number between 1 and -1, which has strange and uniquely beguiling qualities. Shakespeare’s King Lear warned, “Nothing will come of nothing”. The poet and priest John Donne said from the pulpit, “The less anything is, the less we know it: how invisible, unintelligible a thing is nothing”, and the English monk and historian William of Malmesbury called them “dangerous Saracen magic”. They were all talking about zero, the number or symbol that had been part of the mathematics in the East for centuries but was finally taking hold in Europe.What was it about zero that so repulsed their intellects? How was zero invented? And what role does zero play in mathematics today?With Robert Kaplan, co-founder of the Maths Circle at Harvard University and author of The Nothing That Is: A Natural History of Zero; Ian Stewart, Professor of Mathematics at the University of Warwick; Lisa Jardine, Professor of Renaissance Studies at Queen Mary, University of London.

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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, Shakespeare's King Lear warned nothing will come of nothing. It's a thought that had been around for some time and probably borrowed from the Roman poet Lucretius,

Nile Posse Criari de Nilo. The poet and priest John Dunn also warned from the pulpit. The less anything is, the less. we know it, how invisible, unintelligible a thing is nothing. And the English monk and historian Wilhelm Moresbury called the idea dangerous Saracen magic. They were all talking about zero, the worrying, disturbing number or symbol that had been part of the mathematics in the east for centuries, but was in the Renaissance finally taking hold on Europe. What was it about zero that so troubled them? How was zero invented and developed? And what role does Zero play in mathematics

today. With me to discuss the history of zero is Robert Kaplan, co-founder of the math circle at Harvard University, and author of The Nothing That Is, a Natural History of Zero, Lisa Jardin, Professor of Renaissance Studies at Queen Mary University of London, and Ian Stewart, Professor of Mathematics at the University of Warwick. Robert Kaplan, can you tell us how important zero is? Crucially important. From its very beginnings, 5,000 years ago in Samaria,

it allowed us to calculate, to count easily by positional notation, instead of those Roman numerals where how are you going to multiply XXV by LVI, to have a kind of notation where nine different symbols for the nine digits one through nine occur in one column and by moving to the left are valued ten times more, or to the left of that a hundred times more, marvelous. 109, 1,019. But in order to have that positional notation, you have to have a symbol that says nothing in this column.

And some clever Sumerian scribe put two diagonal wedges in his clay tablet to stand for nothing here. And that evolved into our zero. And once it appeared, just as a place marker, just as a comma or a semicolon, it took on a life of its own. And now, in mathematics, it's the fulcrum between the negative and the positive numbers. It allows us to see the balances in the universe. It allows us to calculate what the maximum or the minimum of a function is.

And as the great Swiss mathematician Leonard Euler once said, there is nothing in the world that occurs whose meaning is not that of some maximum or minimum. Zero unlocks the secrets of the universe, and Newton, Leibniz, mathematicians now, are able to calculate and discover meaning through it. Right. Well, we need to stop the program there, although it's a fairly good summary,

to go back to the beginning. That was terrific. The Sumerian culture, 5,000 years ago, how did they come across it? And we just developed that a little. This was a very rich culture in that land, Mesopotamia, between the two rivers, the Euphrates and the Tigris, with contributions to temples coming in daily sheep and cows, and for all I know, loaves of bread, they had to keep track of all these contributions.

So they had to develop a kind of bookkeeping that allowed them to tote up and add. And some scribe, whose name we don't know, thought of this way of, notating in columns, positional notation, which is now so comfortable, so easy for us, and yet is such a wonderfully deep invention. One has to be so proud to be human. That's one of our species, came up with that. Can you just be, now, just one or two people,

listening, just one or two, but I want them to understand precisely and as simply as you can what positional notation absolutely clearly is. If you have 13 beings, and you would like to count them, and you were a Roman, you would write down X I-I-I. That's 13.

X for 10, 1-1-1-1-1-2. Yeah, that's right. Right. If you're us, and if you're an inheritor of this great Sumerian invention, you write down a three in the digits column and a 1 in the tens column because you don't have to think up a new symbol for 13, for 207. You use your old, you cycle back, your old symbol. 1 through 9 by putting them in different columns.

And if you went to 130 and you go move to another column, 130, 130. 1,000, 130. And you just keep moving to the left and you add by tens, hundreds and something. Yes. The Babylonians did it by power 60. Some guy cracked it. Some guy cracked.

Was it like the, do you think it's a similar eureka moment, although Don may have been quietly listed in order to cart more sheep as the Canaanite who invented the alphabet in the 18th. century BC to get on with business. Yes, yes, I think so. It's glorifying businesses in a way which might be slightly uncomfortable for some intellectuals. Well, this is again a way that Zero has become so important

in double entry bookkeeping invented in early Renaissance Italy. We're ahead of ourselves. Stick to Samaria. Back to the Samarians. So they've got positional notation in Samaria and they use that, but it's a way, we're talking about county. We're talking about three

camels and five sheep and so. What happens to it then? Where does it go? Does it spread? Is it a big system? You would think that a wonderful insight like that would spread instantly around at least the European or the Middle Eastern world. But no, it's captured by Alexander when he comes to Babylon in 331 BC

and taken by him along with slaves, women and gold back to Greece. And there it disappears. Why is that answer? Why do you think it disappears in Greece? I think the, certainly the conventional picture, of Greek culture that's come down to us, it's too practical. I mean, you were saying it's business. It's arithmetic which is nowhere near as interesting to the Greeks

as high-flying logic and geometry and more esoteric things. Are we talking about what we would call, use the word snobbish in English, that it's trade, therefore it's not a thing of the mind. It's done by blokes who haul in the camels and the sheep and not by chaps who sit and think. And they didn't leave written records.

of what they were doing, or if they did, they all got lost, and the intellectuals who picked up on the ancient Greek culture wouldn't be interested in them anyway. I mean, there's remains of an ancient Greek calculating device which shows they had a level of technology, which is completely different from what we normally think. But all this got lost. Yeah, this is the artisans, this is the practical people.

So what was important to them in terms of, when you were the word, was geometry, and which did not involve zero. The Greeks got themselves into a frame of mind where instead of using numbers they used lengths. They had a, instead of 3.1 or 3, you had a line of length 3 units. And if you've got the line, you don't really need the number. And you can do all sorts of, and if you want to add two numbers together,

you just stick the two lines together and measure the total length. So everything we would do with arithmetic, they found they could do in geometry. And some things that they didn't know how to do arithmetically, they could also do in geometry, irrational numbers, fancy things like that. And so the geometry was so much superior. in these terms to the arithmetic for the intellectual purposes of mathematics that they got on this geometry kick

and they never really got off it. But meanwhile brooding away in the Middle East is still the zero, which is helpful for humble traders not knowing that they're about to revolutionise the world several thousand years in the distance. In the future, now the Romans were very fine traders. Why didn't they pick up, and they moved to these,

why didn't they pick up zero? The Greeks didn't. Why didn't the Romans? You'd have to ask an ancient Roman, I guess, but they... You can stand in. The Romans are to the Greeks like the Japanese are to the Chinese. They are...

They copy, they improve, they develop. You know, the Romans were great engineers. And they borrowed the Greek way of counting and then turned it into their own version of that system. And that was essentially to use letters of the alphabet for numbers and a different symbol for 10 compared to the symbol for one.

And so on. So the Romans salient. They battled themselves with their system. And then they were also fairly conservative, and they were just stuck with it, I think. It's still strange, before I move across the visa. It's still strange that Alexander brought this back from Babylon, brought zero back, and his influence was tremendous to panic and so forth.

And yet even so, he didn't plant itself. I think the Babylonian zero, as Robert was saying, in the context of Samaria, which is much the same area, fairly similar period, it was more a kind of spacer between other symbols than a symbol in its own right. It had a different role from the other numerical symbols. It was special.

I mean, a good analogy nowadays, if we write a decimal number, we write a lot of digits, then we write a dot, and then we write a lot more digits. And so 5.1, the dot is a different, it's not a number, it's a totally different symbol. And they thought of their zero, more like that, I think. Lisa Janine, as it were, leaping of it, zero reappears in India. We have evidences there, I think, in the 5th and 6th century,

but let's say it reappears in the 9th century. Why India and why is it even so taken so long? We're 4,000 years on from Robert Samaria with the chap hitting on this marvellous invention. First of all, why did it get to India? Well, of course the answer is trade again. I mean, we're back with, we are back with business. And one of the things, of course, as a historian, I would say is, beware of the single items of evidence.

865, maybe the first AD, maybe the first time we have a zero that we, as it were, that is survived for us. I don't believe for one minute that people hadn't been using zeros in trade for an extremely long time before that. Now, when you spoke to Robert and to Ian, the way in which we all decided that trade was sort of lowbrow. And, of course, that also means that the...

relics of trading that uses the zero, which is absolutely needed for anything to do with calculation from lengths of cloth to double-entry bookkeeping. They're ephemeral because this is not the elite part of culture. So it's done on scraps of bark and then on bits of paper, because of course we're in the east where they have paper very early. It's destroyed. So age 65, the trading we now can see trade has brought the zero to India, where all the same. We're Also, the numerals that we recognize have emerged from their script, so that the signs for the integers, for the numbers, one through nine and then a zero,

are all there together in Indian reckoning from that date. And we see the growth of the development, the emergence of Indian mathematicians then, don't we, who seize on this and began to work it? Because, and Robert, I think, would want to say, and I think I would too, that there's an element of the mysticism of the east in this, because if you have six apples on the table

and you allocate the number six to that, that is a marker for the number of apples that are on the table. However, if you start looking at an array of numbers, you begin to see patterns in them. If you are of the patterning kind, that is to say you have a slightly more mystical, I think, you know, as it were, Indian mysticism or Kabbalistic mysticism,

and then you begin to play games with those markers and realize that they have a reality in their own right, that number is something that you can manipulate, you don't need objects to manipulate it, and the zero emerges at that point and Indian mathematicians, as it were, discover what we understand as mathematics at that point. And then it trades through to the Arab mathematicians.

Yes, wonderfully, you see, trade is exactly right. It trades through... It was merely casual. I don't get that. It trades. through to Islam. And I once asked a Muslim friend of mine

if she could tell me why bookkeeping, bills of exchange and mathematics as we know it came through Islam and she just looked at me and she said Muhammad was a merchant. The difference between that culture and the particular sort of spiritualism of Christianity or Judaism

it's all in there. And the Arabs then begin robust mathematics, what we would recognise as robust mathematics. Robert Kaplan, one of the big moves is the philosophical move because it's interesting in many ways, isn't, how children repeat the history of humankind, and children begin numbers with things, three donkeys, two apples, not two and three.

And then there's a stage in their development and in the development of humankind, when it becomes three and two. You don't need the apples. You don't need the donkeys. Away you go. Now, when did that happen and how did it happen? And is what Lisa referred to in India, part of that? Very much part of it. I think your point about children, about ontogeny recapitulating phylogeny, about children recapitulating history, is such a good one. The boy who comes home and says to his mother, we learned at school today that five apples plus two apples is seven apples.

And the mother says, well, that's wonderful. and what's five bananas plus seven bananas? Oh, we haven't done bananas yet. That leap to the abstract, which is so convenient, so comfortable to the Indian mind, if one can speak about such a thing, I think their notion of Shunya, nothingness, as a potential for form,

as a kind of receptacle that can take on shapes, a concept within Hinduism and within Buddhism. I think forms a background to that kind of... But you mean a concept of contaminant shapes? Can you just dig around with that a little bit? Yes. If you think of an empty receptacle, make it out of something hard like

an iron pot, you pour water into it and the water takes the form of the pot. But if you make the receptacle pliant, if its sides are supple, then it takes on the shape of what's put into it. And this is a concept which I think lies behind zero, not only among the Indians,

that you have a potential that can become actual, that can become actualized. When you're thinking in those sort of terms, numbers become your friends, and zero becomes a naturalized citizen of the Republic of Numbers. Ian's point that zero was just a comma, like a decimal point,

a marker different from numbers, it becomes a number among the Indians. Do we know what, among the English, that's when it becomes a number. So it can stand up and be counted. Yes. It isn't just the gap, the space. Indeed. So it becomes a symbol there for, I, and it becomes a symbol.

And it becomes a symbol that's endowed with the same kind of mental images and overtones. There's all the other one, two, three, four, five, six, seven, eight, nine symbols. It's slightly different from them because it represents, don't put anything here, whereas the others represent put something here. But after a while you get so used to it. It's simply on the same level. But you do get used to it. You get used to it, yeah. And yet it isn't like the other numbers.

No, I mean. And I said to somebody when I was coming on to do this program, I said, oh, I'm doing a program on zero. And he said, I hate zero. And I mean, nobody could hate one or two or three. The fact that zero is so tricksy, it isn't quite like the other numbers. There are lots of things you're not allowed to do with zero

that's completely legal with everything else. Yeah, you can't divide zero by zero. That's right. You can't put zero. You can't put zero. And if you multiply anything by zero, it means they're equal, but they're not. Three times zero is zero, four times zero is zero, but three doesn't equal four. I read that in your notes.

That's right. Yeah. You see, I mean, it really is. Yes. No, it doesn't annoy me. It annoys mathematicians. Just part of the bafflement of the universe is right. It's all perfectly logical.

But let's come to, it comes to Europe. Sorry about these jumps. We know, we have an idea, a sort of. when he practically comes to Europe because of Finna Batchi's book. And he meets resistance. One of the ways he meets resistance is in the mechanics of the match, if I can put it that way, Elis Jardium, because what is operating in terms of counting of the sheep and the goats and so forth.

Europe is the abacus. And so you've got a lot of abacus people in Europe, calling it Europe, that's their investment. And these Saracen magic coming in from the Arabian, world, Hindu-Arabic world, is a threat to that. Now, can you say why the abacus was so strong and are there other reasons for the European resistance to zero? After all, it's a 4,000 years old now it's been tinkering about and we're still, we in Europe, they in Europe, them, that lot, are still resisting it. Well, there are almost as many stories of why this happens, I suspect,

as there are people for you to ask. Fibonacci's Liberabaki, which just means calculating book, Abaki, as in Abacus, a calculator. is a really truly wonderful textbook in how to use numbers for lasting records of calculations. Now, the bottom line of the problem with the abacus is it's very good for reckoning, for doing sums, but if your toddler comes into the room and picks up the abacus, you've lost it. All the beads have slid along the wires, and you've lost your sum. Writing down the reckoning was always necessary if you were. going to keep records. Write them down in Roman numerals and you've got a much worse way of keeping

tally than the abacus. Write them down in Roman numerals, what we call, in Arabic numerals, in what we, in the numbers we recognise. For the reasons that Robert explained, you can use the numbers in columns, keep absolute tally of what you have calculated. And it does catch on quite quickly. The problem is that there's a, I think it's a practical gap between the abacus, which you hold in your hand, you do the reckoning, it's for now, right now, still used in lots of eastern countries. Well, you know more than I do, a lot more on this, but does it carry on that quickly? Well, Mr. Robert, discuss between, it catches on quite quickly among the advisors to the new merchant rich, and they begin to use it. But that isn't catching on. There's still, the quotations

I read in the beginning of this programme, John Dunn, Shakespeare, making fun of Vager, That's not catching on quickly. You want to one more word than Rob wants to cover. Okay, because I think that, now you bring in those quotations, that's something slightly different. I think that the new, the zero collides with another old tradition, which is in metaphysical poetry, which is in Shakespeare, which is in the translations of the Bible, about nothing. Hold your finger up. Hold your finger up in an O about nothing where the ring is eternity and the whole.

is nothing inside it. Again, a scary concept. It collides with zero as a number and people get very worried and it doesn't therefore catch on. Yes, and therefore it doesn't catch on. You said it in, it does, well, you don't do it, but it doesn't. But it does in a little way. It does to make the rich people richer and the merchant bank, a person. Because he's awfully anti-commerce. I'm not anti-commerce. I'm not in the slightest. I just, I actually, to the contrary. I think it's absolutely fascinating. I really do that ideas like this can come from. anywhere, because most people think business is just about making loot, and loot is quite interesting. But the fact that Zero wriggled its way through, a 5,000 years, through trade, I think is thrilling. But I just want to keep emphasizing it because I think it's so interesting.

Robert Kaplan, just a second, hold on. Robert Kaplan, you think the resistance it was also to do with Christianity here? That we have an idea. Infinity was, well, easy as a silly word, but infinity was something that Christians were dealing with, the infinite God. nothing was something that Christians found difficult to take on was that part of the

I just want to get this business of resistance over with Annihilation Annihilation has something to do with the devil, it's devil's work This is why William of Malzbury calls Zero Dangerous Saracin magic One of the reasons he does Because here you're trying to invoke

To bring as if by magic Nothingness into reality With the somethingness around you and that's dangerous. So there's certainly that fear of invoking the devil. Why does one stand inside a pentagram when calling up the devil, as Faust does, to protect oneself from those annihilating powers? One thing I'd like to say for merchants and for the abacus,

yes, it's true that with the abacus you reckon quickly and you can lose your sum. But what if you were to dust lightly with sand, your reckoning board? then when you remove a pebble, instead of it being on an abacus, being plunked down on a board in a column, when you remove a pebble, it leaves a trace, it leaves a hollow trace, which may well be the origin of the shape of our zero, that hollow. Can we talk about, for just a moment or two,

before we move on to Newton and get cracking with what zero did to modern mathematics, the battle that was going on between the abacus, people are using the abacus, and the algorithms, people who are using the Arabic numerals, and there were battles, but people were saying, look, I can do six amps faster if I use this, and people said, well, no, but we're stuck with this. So what sort of tension was there there? There's a long-standing subculture in the history of mathematics

of public contests between mathematicians or between advocates of certain methods, and they demonstrate their prowess by trying to solve problems that the opponent has set to, you know, I can solve your problem, but you can't solve mine kind of things. And the battle between the algorithms and the abacists seems to be one of these early examples.

So you could actually pit the people who are using the abacus against the people who are using just written numbers on paper, which are a kind of picture of the abacus. And can you do it faster without an actual abacus? It reminds me of a... There's a recipe I rather love in one of our cookbooks which starts, take 20 chickens, boil them for six hours, then throw away the chickens.

Take an abacus, play with the stones, then throw away the abacus. but you've still got these wonderful symbols on paper that behave like an abacus. And basically in these contests what happens is it's really much more a matter of how practiced are you. And the abacists tend to win. There was a case in about the 1920s when a Japanese using an abacus calculated faster than an American military man using one of these hand-calculator things with cogs that you crank the handle. He was an expert, but the abacus actually was faster even then.

So I think it was touch and go as to what was fastest. Well, I was wrong saying the algorithm that it ticks up more quickly. Well, I don't know. No, I think they probably didn't. I think they're probably didn't. I think it's really, really quick. They still are really quick. If you've got dexterous fingers, and the Japanese cranked up,

the Japanese abacus has, instead of, it doesn't actually have 10. It has two lots of five, and it just has one extra. It has beads worth one, beads worth five in a different part of the column, which speeds it all up, amazing. Anyway, I'm getting off the point. Robert Kaplan, tell us about zero began to come through with double bookkeeping, didn't it? We're still in trade, which is very interesting, but then it began to come through intellectually as well as practically. Can you, in the Renaissance, and then we'll get cracking as to how it sort of took off,

but can you just describe why double bookkeeping was so important to it? This wonderful invention of seeing whether you were in debt or in profit by double entry bookkeeping, by making a column of your debits, a column of your credits, and seeing if they balance that key role of zero. If the two columns are the same, if one minus the other gives you zero, you know that either you've cooked the books or you're doing well.

Now, this is an invention in early Italy. Luca Pacioli, who was the teacher of Piero della Francesca, writes a book on this in which he shows how you can keep records in the end simply, although it takes a while, to learn how to do this accounting,

and know where you are instead of having, as Lisa said earlier, these little scraps of paper running around and falling off the table or bits of bark, now you know where you are. And finally,

before we move on to, as it were, the beginning of modern mathematics, how did that percolate through to the intellectual community? When did they embrace it and say, yes, these abstractions, these Arabic numerals, this way of doing things is the way we're going to do things in the future

and zero is a number. It's not an empty space. It's not a gap. It's a number. Well, it is 15th century. The problem I'm having is just how we get from the practicality. I mean, I think we all still have the problem. Double entry bookkeeping fine. Actually, I can't figure it out even now. It's difficult in itself. Moving over into the realm of using zero

and manipulating it in algebra as if it were identical with the other numbers, but with recognition of its peculiar properties, to get from double entry bookkeeping to that is a huge leap. And I don't think, I'm sorry, but I've read all the books on zero, I think it's one of those things that happens by serendipity. And I would say it happens because all that understanding that Robert is so good at telling us about in his book,

about the nought, as it were, nothing, naught the cultural significance of the mystical quality of nothingness comes into contact with that calculating tradition and intellectuals at some point recognize that the power of this idea in what we would call the mathematical world. Is that fair? Is it any advance on sirenipede? I mean, do you think that Lisa's right with a serendipitous discovery?

Well, let's take an example of that serendipitous discovery. The great Scotsman, John Napier, who up there in the wild north... Didn't all that was. No, I spend my summers there. It's balmy. The weather is beautiful. There he is fighting off his neighbors and inventing logarithms along with other things. He comes up with what he calls his equations to nothing. He sets equations equal...

Which is what data we don't know of? This is about 1588. Right. He sets an equation equal to zero. Now, that doesn't seem like such a big deal, does it? Well, if you have, let's say, 3x squared plus 15x equals minus 32, what on earth are you to do with that? How are you to figure out what X is? I don't know.

I certainly don't know. No, we're together in this. What he does is to move that minus 13, I think I said, from the right-hand. side of the equation to the left, he puts all his X's in one basket and says, let this equal zero. Well, so what? So this. If you multiply two numbers together and their product is zero, at least one of them must itself be zero. So if you could now rewrite the left-hand side of your equation, not as a sum, but as a product of two factors, and those two factors equals zero, one of them at least must be zero, and now you can solve for X.

You can find out what that hidden X was, all because of this equation to nothing. So at that time, 16, 17th century, Ian, what does that enable people to do? Roberts described that very, very clearly. Now, what does that enable people to do? Mathematicians to do. We're now going straight to mathematicians. We're going bang into mathematics. After serendipity, we're in mathematicians.

You get this great flowering of algebra. You get all this stuff with X, which Robert has given us a quick primer in. You get the Renaissance mathematicians understanding how to take equations where what you're given is information about an unknown quantity, the square of that quantity, what you get when you multiply by itself, the cube, various things of this kind. Some combination of those equals 42 watt X. And this goes in parallel with trigonometry. triangles, angles, how you work out the shapes and sizes of triangles, which has a practical connotation. It's to do with surveying.

It's to do with navigation. So the practical side of this is accompanying the theoretical. I think what we're seeing, there are three strands here, and we've talked to two of them. There's the commercial strand on the one hand. There's the pure mathematical, intellectual strand on the other. And in the middle is the burgeoning science of the motion of planets in the heavens, which is one of the things they're used.

these logarithms for. They produced enormous tables of logarithms, which are just digits and digits and digits and digits all over the page, which you can use to convert multiplication, which is difficult, into addition, which is easy. And you can calculate where the planets go and you can do scientific calculations. And all this stuff... So that's beginning to help navigation, that's right. But all this stuff starting to come together into, although we can see the components in the way I've just described, really it's all starting to be part of one game. It's all the same thing. And this is all enabled by zero. All enabled by zero because zero is the key to the notational system that allows you to actually print these things, use them and do them. Where does Newton take this then, Robert Kappa? He does something more than this. In a word. Please, please, we have plenty of time.

Pressure Kaplan can take as long as he wants. Where does this Newton take this? Newton, on the one hand, Leibniz on the other, come up with this idea that if you want to understand motion, what have we been doing all this time in mathematics, understanding the static, the triangles that Ian was talking about? Up to Newton.

Up to Newton. Geometry, which looks at shapes staying fixed. But now, in the Renaissance, we want to understand change. We want to understand how things become and change. If you're a tennis player, you know that you want to hit the ball when you're serving at the top of its art. when it's as if it were standing still there for a moment. If you're a skier and you trudge up the slope to the top and then ski down,

there's that moment at the top when you're neither trudging up nor skiing down, that moment of rest of pause at the top of the hill. If only we could understand when functions that represent change reach their tops or bottoms, their maxima or minima. Well, what Newton discovers, or Leibniz, or both, is that if you draw lines, straight lines, tangent lines to these curves, you'll notice that at their tops or bottoms, those tangent lines are horizontal. Their slopes are zero. When the slope is zero, the function has reached a maximum or a minimum.

Now we can understand change. And now I think you should ask Ian to do that again, so we have two versions of it, and then we'll be able to tease out here. Shall we talk places? I'll take turns. Here, no, will you... Well, follow my good friend of his journey. Are you right? Well, Lisa.

Okay, suppose you want to know how fast your car is travelling. Then you can do a little sum that says, well, in one hour I went 60 miles. I'll do this for the people who still know about miles. So the car was going at 60 miles per hour. Well, if it's going exactly the same speed the whole time, that's true, but in practice it's going at different speeds at different times. So, well, you know, how can I get a more accurate version of the speed it's doing right at this moment?

So, well, suppose it travels one mile in one minute, then it is, in fact, over that minute it's averaging 60 miles an hour. Yeah, but it's still over that minute. I might go fast at the beginning, slow at the end. narrow it down, take the distance you go divided by the time it takes over smaller and smaller and smaller intervals of time. Now what Newton does, and he's clever enough to do it right, but he can't quite explain how he does it, is to say suppose that interval of time gets as small as you like,

which to Newton means it just gets smaller and smaller and smaller to a lot of other people meant, oh, there's a definite size it gets, which is as small as you like. So Newton sees it as dynamic, other people saw it as static. In effect, you're saying take the distance, it goes in time zero and divide it by time zero. And you get a sum that's zero over zero, which we are taught for very good reasons, make no sense whatsoever.

But when Newton did this, miraculously, what should have been zero over zero turns into something like 60 miles an hour. The 60 is still sitting there. It doesn't get divided out. The zeros miraculously vanish, and you get sensible answers. And that's his fluxion. This is his fluxions. This is the calculus.

This is what Bishop Berkeley said was. a discourse addressed to an infidel mathematician that this belief in these strange zeros is much less logical than religious faith. Bishop Berkeley was philosophically right, but Newton turned out to be mathematically right. Well, Newton was a physicist at this point. Think of Newton as a physicist.

Okay, I'm glad to do this bit, because he had a specific problem that he was trying to solve, that in the old system where everybody believed that the Ptolemaic system, that planets traveled around the Earth in circles at uniform speed and that you could calculate where planets were, which is what everybody still needs to know for getting around the world, you could calculate that by just the knowledge of circles, which the Greeks were quite good at. But by the time that in the 17th century, astronomers had realized that that wouldn't do, that planets moved around the sun and that they did so at variable.

speeds, then the problem, and it was a problem that actually Wren set to Newton, went and set it, Hallie and Wren went and said it to Newton was, if a body is moving around the sun with a variable speed, and it isn't a circle, can you tell us what that is? And Newton, and set down, that was when he sat down, and made, and literally by drawing it on a piece of paper, and making tiny increments around the calculated orbit, figured out that it was an ellipse and came back and said it was an ellipse

and developed his mathematics of tiny pieces. Now let's get back to, so we've had three looks at that, three bites of that, three bites of that, cherry, and that's up to, let's get back to this, though, because although Newton, Ian Stewart, intimated that Newton intuited what contemporary mathematicians can now, as it were, in a different way and more, respectively, as we're, prove. He was on to Zero being an enabler, a massive enabler, which takes us back to what you were saying

at the beginning of the Birmingham, Robert, is this just open, not just the gates, but floodgates. You call it one of the great paradigm shifts in the history of ideas. I don't know whether that's your phrase, which is my summary of your phrase. Now, why was it so important? Why was it so important?

Newton comes through with this. What is enabled to happen because of this intellectual? of that problem? I think we can see the immense power of zero if we look not only at Newton and the calculus, but at our computing machines, at our telephones, at any of our electronic devices, which all work with just two numbers, zero and one,

switches that are off or on. All numbers can be written with just zero and one, a binary system. This enables us to send information to calculate rapidly enabled, I like that word very much that zero is the great enabler, enabled by zero and its friend one to produce all of this practical,

logical, logical, logistic work, and at the same time to give us an insight into the nature of the universe, we can derive all numbers, amazingly enough, from zero. Was Lear right that nothing will come of nothing? Perhaps not. Perhaps we can really get everything from nothing. How would you respond to that end, Stuart?

I think this is certainly right. And of course, but that's a notational thing again. That's a technological thing. There is a, this, you referred to the top of the hill where you start to go down again. I think people may not realize just how important and how widespread the use of that kind of idea is. So let's get really technological. You want to put a space habitat up there and collect the sun's rays and being. them down to Earth and have a million people inside a giant cylinder floating around

the Earth somewhere. Where do you put it? And the favourite place to put it is at things called Lagrange points. There are five Lagrange points for the Earth, Moon, Sun system. These are the places where the net gravitational effect of the Earth, Moon and Sun combined is zero. These are the places where there's no gravity. All the textbooks say gravity goes on forever. Well, the gravity of any one planet goes on forever, but they can cancel each other out. Is that a singularity? It's a singularity. It's a critical point. It's a place where the energy surface

is at the top of the hill. How does this relate to what Newton did? We only discovered that singularity because of what Newton did. It's just because the idea of zero became a profound idea inside mathematical and physical thought and therefore it pursued its own course and began the law of unexpected consequences. And it's two

different uses of zero at the same time. It's zero as the place where the hill levels out, where the gravitational energy is level, is flat, and how do you calculate where that is? You use the Newtonian calculus method, which has incorporated zero as an important ingredient in a different way. So it's everywhere.

And it's zero, which is just what, as Ian has explained that, it's zero as the point of stasis in a world in change. If we throw our mind all the way back to the early struggle with zero, that was all about the world in stasis, the fixed world, in triangles or numbers or numbers or, objects. Now we are constantly preoccupied with the world in flux, hence fluxions and calculus that came out of it are the mathematics

that we need on a daily basis. And the singularity or the black hole or the zero point between gravity is the point at which you get stasis again and they are hugely important to our world. But you have to understand that that's because the world around us we understand to be constantly moving. Yes, it's Zero as the great counselor, again, the fear of the Western world when Zero entered it, that this was the annihilator. But here, those points of cancellation, think of two waves approaching one another, and where they meet, they cancel one another out. Those points of annihilation, those points of balance stasis explain to us change as if we had draped bunting the changing curves over these pegs.

of stasis, these pegs of zero. And so do you agree that it is a massively important development for mankind, that it's enabled to do things we simply couldn't do without it? Has it ceased in any real sense to have an association with nothing, with emptiness, with fall? Oh, I think if you talk to any French existentialist, you find it's nothing but nothingness. People are still very troubled by nothing, by naught, by zero, by cipher. and zero mathematically has not come to the end of its troubling, puzzling existence. There's still so much we don't know about how it behaves, what it stands for.

One of the great problems these days is to find the zeros of a certain function, the Riemann, the Zeta function. If we knew that, we'd know so much. Well, we'll have to wait. Thank you all very much for that. That was terrific. Ian Stewart, Robert Cappellee, is a jardine. Thank you very much.

Thank you for listening. Next week we're talking about toleration. 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.