In Our Time - Maths in the Early Islamic World

Episode Date: February 16, 2017

Melvyn Bragg and guests discuss the flourishing of maths in the early Islamic world, as thinkers from across the region developed ideas in places such as Baghdad's House of Wisdom. Among them were the Persians Omar Khayyam, who worked on equations, and Al-Khwarizmi, latinised as Algoritmi and pictured above, who is credited as one of the fathers of algebra, and the Jewish scholar Al-Samawal, who converted to Islam and worked on mathematical induction. As well as the new ideas, there were many advances drawing on Indian, Babylonian and Greek work and, thanks to the recording or reworking by mathematicians in the Islamic world, that broad range of earlier maths was passed on to western Europe for further study.With Colva Roney-Dougal Reader in Pure Mathematics at the University of St AndrewsPeter Pormann Professor of Classics & Graeco-Arabic Studies at the University of ManchesterAndJim Al-Khalili Professor of Physics at the University of SurreyProducer: Simon Tillotson.

This is the BBC. Thanks for downloading this episode of In Our Time. There's a reading list to go with it on our website, and you can get news about our programs if you follow us on Twitter at BBC In Our Time. I hope you enjoy the programmes. Hello, mathematics flourished in the early Islamic world from the 8th century onwards. Astonishingly versatile minds consumed all they could from Indian, Greek and Babylonian traditions, among others, and made extraordinary leaps of their own,

which still affect what children in simple form are to. at school today. One, Choirismi, Al-Qarizmi, May 780 to 850, made his reputation for algebra, a word taken from one of his great books, just as algorithm comes from his name, westernised into Latin. Another Persian, Oma Kiyam, now known for his poetry, solved complex cubic equations. They and many others were inspired by new calculations called for by the Quran, by translations radiating from Baghdad under the Abbasid caliphs, and by their religious duty to seek knowledge. to discuss maths in the early Islamic world are Colver O'Nidougal,

reader in pure mathematics at the University of St Andrews, Peter Paulman, Professor of Classics and Greco-Arabic Studies at the University of Manchester, and Jim Al-Khalili, Professor of Physics at the University of Surrey. Colva, many mathematical traditions were drawn together in this period. Let's start with the Babylonians. What can you tell us about them and what did they offer to this period? So first, in terms of the location,

the Babylonian Empire was centred in modern-day Iraq, and neighbouring countries, so along the rivers Tigris and Euphrates. So it's in exactly the same place as the locations we're going to be talking about later. It's much, much older, though. So Babylonian culture started really flourishing about 3,000 BC until about 200 BC. So it's as far before the death of the prophet as, say, Chaucer is from us now. In terms of what the maths of the Babylonian Empire was, they had an incredibly advanced culture in a very early age.

big cities, maybe as big as 60,000 people by maybe 2000 BC. So there's a lot of maths involved in the storing of grain, in issuing grain for agriculture, so not just the corn for the oxen, who are going to plough the fields, but also the seed corn, and then an expectation of a certain amount of corn to come back again. There was an extensive canal network, because this is a river-based society. So digging of canals, remeasuring of fields after the rivers have flooded,

meaning that you need to work out the correct area of the fields again. And then also, of course, astronomical observations for fixing the calendar to work out when you should plant, when the floods are going to come, when you should harvest, all of that kind of thing. So Babylonian maths can probably be best described as algorithmic. We've got a lot of evidence of them solving... Well, people who don't get that word easily. Yeah. What would you say that meant?

So what I mean is that we've got lots of clay tablets, which is how they wrote, where they're repeatedly solving similar kinds of problems. So a problem you might find in a Babylonian math tablet is there's a rectangular field. The length is 10, let's say meters, so we're not involved in Babylonian units, 10 meters longer than the width. And the area is 300 square meters, 3,000 square meters. What is the width? And the tablet will go on and describe the steps you need to take to solve that problem. To us, that's a quadratic equation.

I'm saying that the breadth times the breadth plus 10 is 3,000 and so I'd write that a little quadratic. But to them it's a series of steps. How's it algorithmic? So the solution would say something like you... But how does that make it armurismic? Because they're not writing an equation, they're not drawing a diagram,

they're telling you what you do with the numbers in the question to reach the number in the answer. Why was there a number based 60? We don't know. It's the very short answer. So the 60 is why we have... That's a long time, isn't it? We're still in 60 at the moment. I'm looking at that ticker going, tick, tick, tick, tick, 60 seconds.

So the 60 and the 360 degrees in a circle is also thanks to the Babylonians. We really don't know. I mean, my preferred belief is that it's coming from the fact that they're using lots of different units of measurements. So much like the imperial system, we've got 12 inches in a foot three feet in a yard and then some number of yards in a mile. The Babylonians had lots and lots of different units of measurement which slowly coalesced into 60s. What about the Indian contribution? The Indian contribution.

And we're talking about 300 BC when the Babylonians stung in, is that right? Kind of that, right? What about the Indian? So the Indian contribution is from say 800 BC through to the period we're going to be talking about. So, I mean, the Indian mathematics continued to be amazing up to about 1,200 and even beyond. the Indian contribution is much more about mathematics in the service of astronomy and religion. So one of the interesting things with Indian mathematics is there's an early obsession with very, very large numbers.

They were interested in finding the point at which all of the planets and the solar system would be realigned in the place that they were in the beginning and this naturally leads to enormous numbers being considered. They were interested in measurement of the heavens so that you could predict eclipses so that you could say when the various plans planets were going to be in the different signs of the zodiac and that kind of thing.

So again, they're solving equations, but they're not really thinking of them as equations per se. And the way in which they come down to us are often in the form of incredibly compressed poetical phrases, because the tradition is mostly oral not written, and this makes a problem for historians. Jim, Jim Alcali, this is called the Golden Age, often a great cross-fertilization of ideas. What made that possible? Well, a lot of people assume that it started because of the birth of this new religion, Islam in Arabia, whereas in fact for a whole century there was an Islamic dynasty, the Umayyads, where scholarship didn't really flourish. The golden age is really thought to have begun with the Abbasid dynasty, which began in the middle of the 8th century.

Abbasid Caliph built his new capital, Baghdad, around about that time. And for me, I think the most important thing is the Abbasids were very much influenced by Persian culture. And in Persia there was this long, long tradition of scholarship and learning. And so they became obsessed with texts and books. So the Golden Age really began with a wonderful flourishing translation movement. This Islamic empire for which the language was Arabic, because that was the language of the Holy Book, the Quran. They realised there was all these great texts from Greece, from India, from Persia, that they wanted to translate into Arabic.

And that kick-started an obsession with learning, an obsession with scholarship. First of all, translating the great texts of the Greeks, Aristotle, Plato, Euclid, Galen on medicine, and then writing their own texts. So we have that, but it is, when you say an obsession with it. Are we talking about Mamun Mahoud? Amman. And what Moon was, I guess, the person. time as Chorismism, isn't it? Yes, so early 9th century.

So they're getting going in a very big way. Absolutely. I mean, they'd already started getting going before then. He was born into a culture and a barsid culture where there was an obsession with learning. He was the son of Harunur Rashid, who famous because he appears as a character in the 2001 Knights, Arabian Knights, and during the golden age of Baghdad. Baghdad was by this time the early 9th century, the most important city in the world, probably the biggest city in the world. Do we have a key?

Is it a key in the Quran that you shall seek knowledge? Is that the key to the pursuit of learning and the devotion to learning of the Caliphs? To a certain extent, yes, but I don't think we should push it too far. Certainly in the Quran it does say you seek knowledge from the cradle to the grave, and certainly in the hadith of the prophets, that you know, seek knowledge,

even if you have to go as far as China. So there was certainly an obsession with wanting to find out about the natural world and how things work but it was also the influence of, as I said, Persian obsession with culture. And when we were talking about Indian influences coming in, that we think that we had a tincture of the Chinese in it as well, so we're bringing in a lot of things.

Can we talk about this House of Wisdom? What did they mean by the House of Wisdom? What sort of house was it? It's supposed to last it for 400 years. It is contested. It is contested, and I'd probably get into hot water with historians. But let's say what I'm. I think of it. There was certainly potentially something called the House of Wisdom,

a bit like the Library of Alexandria many centuries earlier, which was a place where books were stored. It may have also been a translation. This was in Baghdad. This was during the time of Al-Mat-moon. It may have existed in some form or another in his father's palace. It might be a physical building separate. It might be part of a palace. But there's no archaeological evidence of such a real... Well, there's an archaeological evidence of much at that time because it's all clay, wasn't it?

That's true. So that archaeology is probably the wrong way to look, the wrong place to look. But was it a research centre? Was it a place where people went? We can go there and be paid by the Caliphs to get on with the work we want to do in mathematics? I believe it very well could have been.

It may not have been the only such place in Baghdad. Baghdad will have been full of libraries and translation houses and places where the wealthy high society of the Abbasids would provide. the money and the patronage to these scholars. But there probably was a place where people like El Corizmi and other scholars would gravitate to in the same way as the Library of Alexandria was a place where all the greatest thinkers would go and work.

Peter Paulman, do you agree with that? Well, I mean, so there's the myth of the House of Wisdom as this research school academy and so on and so forth. And basically there's very little evidence. So the House of Wisdom, Bait al-Hekma, or Qizanat al-Hikma, as it appears in our sources, is linked to some of the mathematicians we're going to talk about, like Al-Qa Rizmi. But his name is mentioned only in a late 10th century source in connection with this. And certainly the translation movement has no direct link to this House of Wisdom. So the most important translator, Hunayne Ibn Ishaq, wrote a letter explaining how he translated Galen. Galen is a physician of the second century.

He dies in 216 AD. And basically, he accounts of his translation activity and tells us how he translated, and he never mentions the House of Wisdom. We mentioned the Babylonians when we started with that, and then mentioned rather slightly, we must come back to them to the Indians. But a big influence was the Greeks, the great translation movements which preceded the advances and the inventions. They sometimes thought of as just translators the Arabic, not at all.

They built on that and built their own thing. The great translation movement, they translate in almost all the Greek texts, but particularly the mathematics that's what we concentrated on. Can you tell us about that? Especially Euclid. Yes, of course. So basically the two Greek mathematicians

that most people know are Pythagoras and Euclid, and they have become household names here in England, in Europe more generally. But they were also, to a certain extent, household names in the Arab world. And if If you look at Pythagoras, we all remember the Pythagorean theorem, A square plus B square equals

C square, you know, like for a rectangular triangle, that was certainly something that was known through Euclid. So if you look at the Greek tradition, basically all the great philosophers like Plato and Aristotle had a keen interest in mathematics. But we don't have mathematical textbook by them. Of course, they sometimes discuss mathematics in their dialogues for Plato or elsewhere in their writings as in Aristotle. But the first massive textbook for mathematics that we have,

and certainly one of the most successful textbooks of all time, are Euclid's elements. Now, Euclid is a mathematician about whose life we know very little. He probably was active in Alexandria in the 3rd century, BC, and he wrote this massive textbook, the Elements. What impact did his elements have on the Arab scholars when they translated it? Oh, it's absolutely massive. So Euclid is one of the earlier texts that gets translated into Arabic.

And basically the whole discipline of mathematics, as it later develops, is to a certain extent based on the translation of the elements and other Greek mathematical texts such as Diophantus. Is there any way you can explain how the Greek and the Babylonian and the Indian intermingled? Do you have instances of that? So basically, the Greek culture, sometimes people used to talk about the Greek miracle. All of a sudden, in the 5th century BC, we have this great Greek culture, and everything is the Greek invention. But we know, for instance, that, you know, like the degrees of angle of the 360, about which we talked,

these things came from the Babylonians to the Greeks, and the Greeks also took from the Egyptians to a large extent. So basically in the preface of the, or in the frame narrative of the Timeas, the young Socrates talks to the old Solon, and Solon says to him that basically the Greeks are like children, and we have taken all our knowledge from the Egyptians. So Greek culture, like the culture we are going to talk about, was a culture that was able to take all sorts of knowledge from different cultures that integrated. But Euclid, and we don't know, for Euclid, for instance,

we don't know exactly what did he invent himself, what were his own discoveries, and what did he take from his predecessors? Because we simply don't have the mathematical text of his predecessors. We're never going to get at the bottom everything, because influences, in all you bring, influences go back infinitely. But sort of talking about this time. And now, that's our subject this morning. They translated the Greeks.

The Greeks were a massive influence. Their geometry was a massive influence in this extraordinary mix. And Colva, how did algebra come into it? So, Greek mathematics was essentially geometric. But what the Greeks had that the other cultures we've been talking about didn't was the concept of proof. And that you start with a bunch of what they call axioms and common notions. Those are in some sense the rules of the game.

So one of Euclids might be two things which are equal to a third thing are equal to each other. That's one of his common notions. And from those you construct a proof-based edifice. Algebra then was more about solving problems of areas and volumes. It wasn't a provable thing. Why did they think they needed algebra? They needed algebra to solve practical problems. So they needed algebra to solve problems like if I'm wanting to make a field,

have a certain size because all the field markers have been washed away. and I know that it has this much width, how long do I need to make the field. But they wrote their algebra in, the early algebra was written out in prose. So why did they need to translate that prose, which had worked for quite a while, into algebraic symbols? Essentially because the prose is very difficult to understand. So the process of going symbolic takes a very long time. The process of going fully symbolic doesn't happen for, and really until about the 16, 1700s. I mean, that's a long time later.

But what we're talking about is the beginning of the understanding that there is a system called algebra which can solve all problems of this sort rather than merely a collection of instances of special cases and we can solve this one and that one and the other one. So it was necessary to take the next step to have these symbols because words were not enough? Yes. Except words had been enough until then to put forward

how you divided up fields, how you divided up inheritance. The laws of inheritance in the Quran, you left a quarter, the wife died, a quarter to her husband, two to one, to boys over girls. and so and so, became very, very complicated. Words had solved out until then.

I just would love to get hold of fingers and said, yes, but one day they thought, no, won't work, we need to do symbols. Well, no, they're still writing in words. I mean, words is the case for almost all of this period. We're not about to go symbolic. The sense in which algebra is born is the understanding, it's a conceptual step,

that this is now a topic of study in its own right, that problems like this, form a class of problems which, whilst related to geometry, any question about area could get turned, into a geometric diagram is also related to a process of things being on either side of something like an equal sign and adjusting them accordingly. So we're in a cast of mind, Jim, Jamakily. In a cast of mind, very like the Greeks, which one of you described as aristocratic thinking,

of thinking for thinking sake, this is very interesting. We will sit and consider triangles all day because we need that. That's what we do with our lives. Is that what's happening here? No, actually what is happening now in a way that the Greeks didn't do was you, mathematics to solve real practical problems. Issues like dividing up land for irrigation or sorting out your taxes, running a huge empire.

The Greeks were very much at the abstract thinking and a lot of their geometry that they developed. But how did they get to the algebra to do that? They were already doing quite well with words. Well, as Kovas says, we shouldn't say that algebra is all about symbols, X and Y. Symbolic algebra is a way of doing things more efficiently. But algebra itself, all the way through the Islamic period, didn't have symbols. That had to wait for people like Descartes. So they were writing everything rhetorically, everything in prose,

and that didn't slow them down in terms of doing algebra. So when we say Al-Qarizmi is a founder or reinventer of algebra, what is he reinvented that we would consider to be algebra? Well, what he was doing as opposed to someone like Diophantus, the Greek, was that he wasn't looking at specific, or indeed the Babbal. He wasn't looking at specific examples. So when I teach something in my physics lectures, I give a general method, derive an equation, general ideas,

and then I say, okay, let's look at some examples. What if you have this and this, and let's use this method to solve this example? The Greeks and Babylonians were just looking at specific examples. What Al-Huarsmi did for the first time in his text and his book of algebra was give the general recipe, the general algorithm. And only later, in the second half of the book, does he say, let's look at specific examples if, you know, you emancipate a slave and you die and you have three children and so on. So for him, and the reason why we say he is the father of the field of algebra,

is because he described it as a discipline in its own right in general rather than looking at specific problems. So this is a different way of thinking from that employed by the Greeks where they thought in an abstract way. You don't think he's thinking in an abstract way here. No, he's very much thinking about something. His book was very much a book for the wider public. He's saying, here are lots of different problems that you will encounter in daily lives. Taxation, for example. Here's how you go about solving these problems.

I'm going to give you the technique that you need to use for solving any such problem of this type. Peter Porman, could we develop this? He is still called, you emphatically call him a father of algebra and he changed mathematics at that. By doing that, what do you mean by that? Well, I mean, to put it very simply, the Greeks solve problems, the Arabs or Persians in this case

solved equations. So, you know, when you solve these general equations, you have to you know, I mean, you take it to another level of abstraction. Let me give you just one very quick example from Diophantus. So Diophantus was a mathematician who lived roughly in the 3rd century AD and he opens his arithmetic, his book, with a problem where he says, Okay, assume that you want to add two numbers.

The difference between these two numbers is 40, and you add the two numbers together and you get to 100. So how do you solve this? And then he will go through concrete steps. He will say, okay, there's one number. Let's call it X. He doesn't call it X. He calls it just number.

And then there's another number, which is 40 more. So it's X plus 40. So then you have twice X to X plus 40. equals 100, then you take 40 away on each side, then you have 2x equals 60, and then you divide it by two on both sides, and then you have x equals 30, and then you know, aha, the lower number is 30, and then you add 40 to the lower number, and then you have 70, and you have solved this particular problem. I mean, that's the first problem of the easiest, and this is something which she didn't invent

this has long, long, long, long been solved, but, I mean, not to go into very complicated But that's the kind of thing the Greeks do. So you have concrete examples and concrete solutions, whereas in Al-Hawarismi's book on algebra, basically you solve equations, general equations, and not these concrete examples. Why is that better, and why do you think that was the turning point? Well, I mean, it's another level of abstraction.

It opens a whole field of mathematics, basically. What's that field? Well, algebra. I know that. No, I'm sorry. People want to know what we're talking about. And that field therefore includes what, which had not been included before?

Well, you know, solving quadratic, cubic equations. That's what I wanted to know, yeah. Sorry. No, please continue. No, and so basically solving ever more complicated equations that require ever greater levels of abstraction, that's really what happens there. So it opened the door to a whole new area of knowledge is what you're saying.

Yes, absolutely. Or a whole way of dealing with undiscovered knowledge. Indeed, yes. And go far beyond the Greeks. Of course, they could never have done anything without also looking at the heritage, the Greek, the Babylonian, indirectly the Babylonian and the Indian heritage. But they really make progress. And it's not for nothing.

This is why we use these terms still, you know, both an astronomer. and mathematics, we have these Arab words because throughout the Middle Ages, basically, this field of mathematics was always associated with the Arabs. Colman? Yeah, so just as a maybe more concrete example of one of the things that Alchorismi did that changed what algebra was, he was the first time ever in history

to systematically say these are the types of quadratic equation. If you might have a quadratic equation, so he wouldn't have called it a quadratic equation, but an equation in squares, roots and things. So we've got some squares, which is the x squaredy bits. We've got some roots, which is the multiples of X, and we've got some things, which are the numbers. For him, there's six different kinds of quadratic equation

because he's not happy yet with negative numbers or zero. So depending on how they were arranged either side of the equal sign, he gets different sorts. He recognises that the study of things like that is a topic in its own right, and that it's worth classifying those six things and giving methods of solution to those six things so that now any problem involving squares, roots and things, the reader can solve.

How did the modern number system develop in the period in which you're talking in this high Islamic culture? This is a fascinating story, but it's a bit murky in terms of sources. So we've already mentioned that the Babylonians had a base 60 system for their numbers. So, for example, if they wanted to write 61, they'd write it like 11 because it's one times 60 plus 1. The Indians had developed essentially our modern way of writing numbers where we use the symbols 1 through 9 plus the special 1-0, which means there's nothing there. And each symbol, when we write it down,

has more than one piece of information associated with it. So when I write down 31, the 3 is indicating both some kind of threiness and because it's got a 1 after it, that it's 310s rather than just 3-1s. So the 3 there is being 30 because of its position, and the actual symbol written down. That system was invented by the Indians, we don't know exactly when, maybe 200, 300 AD, something around that time period,

and slowly spread west. We know that it had reached as far as Syria by the 600s because there's a lovely quote from a Syrian Christian bishop complaining about the Greeks, thinking they know everything and the Indians are marvellous. So it definitely made it through into the Arabic, Islamic world by that time period. And it's a massive advance on either the Roman numeral system, which is horrendous for division,

or the Babylonian system, which doesn't match how we say numbers. You would go along with that way, did you? Yes, I'd just say that I think the decimal system, which we now call Hindu numerals, or Hindu-Arabic numerals, we haven't really talked about the, we're talking about it without having described it. So could you say what the decimal system is and how? No, that came in. Well, it's a positional numbering system, as Colvo explains. So whereas the Babylonians would have used base 60, so you go 1 to 59, and then you start again.

Binary, the computers use, zeros and ones, that's base 2, because you go 0, 1, and then you start again the next unit. The Indians gave the world the decimal system, 1 to 9, and then you start again in units of 10 and 100 and so on. And although that was transferred to the Middle East and the Middle East and the, This lovely story, this Syrian bishop, Severus Sabacht, or whatever, his name was. It didn't really catch on. The scholars in the Islamic world at that time were using a mixture.

They're using Babylonian, sexagesimal, base 60. They were using the Greek method of letters denoting numbers. And they were very reluctant to use this new decimal system. Even when it transferred to Europe, there were Europeans who came and translated texts from Arabic into Latin, took it back. Europeans were very skeptical. They thought it was evil Islamic number system that they should never use. So there was

this wonderful method that we all use around the world today that took centuries to catch on, both in the Islamic world and later on in Europe. And so the fractions and the decimals took a long time. They weren't used at the time we're talking about in the time of the golden age of Islamic mathematics. They started to be.

They were discovered. Yeah, discovered. So we have to make a distinction between the decimal system, the 1 to 9 plus the 0 that was inherited from India, and then the decimal fractions. Because even after people start to use decimal systems, they were still using fractions instead of, you know, nought point something. The earliest we know of the use of the decimal fractions is a translator of Euclid, called the Euclidean, a luclidacy in a text in the 10th century, Baghdad. He first uses, he says it's much easier actually to do calculations, rather using fractions, dividing one number over another, you can write it in this method. He doesn't

put a decimal point. He puts a prime over the last number before the decimal point. Peter Paulman, while this is going on, most of the ideas, as I understand it, you'll correct me if I'm wrong, are still being expressed in prose. What effect did this have on the transmission of ideas at the time? We're talking about 8th, 9, 10, 11, 12th centuries. Well, there's a continuity in a way, because as we said, the Greeks largely, with the exception of Diophantus used prose to describe mathematics and when this is then translated it is translated as prose into Arabic

and so that's relatively straightforward although of course you have to invent certain terms for certain concepts that are novel but the text that are then written the whole field of algebra as it develops very much in prose as we have said so basically you would describe one plus a number plus a root

plus a square equals that or the other. So it is all written out in prose and the symbols come in again much, much later. And that also facilitates the transition from Arabic into Latin, which is so important for the developments in the European Middle Ages. And we're talking about a time when the spread of knowledge, if you're in this House of Wisdom,

whether it's real or virtual House of Wisdom, which put that to one side at the moment. We're talking about a time when knowledge was being transferred very slowly outside those tight groups. We're talking about manuscripts, a book, Britain, which would take a very long time for a person to do to be passed on. So can you tell us what the consequence of that was, that knowledge spread so slowly? Well, I would slightly disagree with that proposition because what happens in the Islamic Empire is that paper becomes available. So from China, this is invented in China, but spreads to the Arab world. and we have many, many more manuscripts

and much higher production of books in, let's say, 9th century Baghdad than anywhere else in Europe at the time. And a text that is written in one part of the Islamic world could very quickly travel to other parts of the Islamic world. So even Al-Andalus, you know, like Muslim Spain, there are mathematicians there, their contributions, there are scientists, their physicians who kind of partake in the same scientific discourse

as those people who are in the heartlands and the same goes for the East if you think of a philosopher like Avicenna for instance who's also a physician he hails of Persian origin hails from the East spends most of his life in what is nowadays Afghanistan and Uzbekistan

but is very quickly discussed and received and analyzed in the West so these texts do travel and that's very important and so the the development of the language, basically, that Arabic is the scientific language that is used from the shores of the Guadalte Kibir to the shores of the Ganges.

That really made all this possible. Thank you for correcting me. Very good. Holby. Colbert, we know of Oma Kayam. Most people know about him for the Rubayat that he wrote, but he was more, my people like you,

he's considered more highly as a mathematician than a poet. Anyway, so he was a great mathematician. Why? Well, so I'd already mentioned, that Al-Huizmi was the first person to really classify quadratic equations and say this is all the sorts there are. Kayam takes it one step further. He classifies all of the kinds of cubic equations. So this is all of the equations beginning with x cubed, x times x times x and then some squares and then some roots and then some numbers. He breaks them all down into different types and then he's longing to

give what we would call an algebraic solution to these types of cubics, but he can't. The method's just aren't there yet. The maths isn't there yet. But what he does that's completely astonishing is he's able to give a geometric solution, by which I mean, he shows that if you construct a curve of this type and then you intersect it with a curve of that type, then the length of some lengths like this are going to be the roots to your equation. And he comments that he'd love to give an algebraic solution but hasn't been able to. He's also able to give approximate solutions to cubic equations. And so he sets the whole world of maths off on a race to solve cubics that wasn't eventually settled for another four or five hundred years after him.

This took a long time to happen. Jim, Jim, Alkalili, let's talk about the number zero. There are three, you emphasised there are three sorts of zeros. There's the Babylonian zero. There's a Greek zero and there's a later zero. So can you clip through the first two and get to the third one? And the reason why it's fascinating is because for a lot of people, if they say, oh, you know, what did the Arab world give us?

Oh, they invented zero. And of course that's one of the things they didn't give us. And so what you mean by zero is important. If we just mean zero as a symbol, as a placeholder, so let's say how do you tell the difference between the numbers 11 and 101? There's got to be something between the two ones to show this. The Babylonians did that. The Babylonians did that.

They came up with a – first of all, it was just a gap, then they came out of a symbol. So they first came up with that notion of a placeholder around about 300 BC. Later the Mayans developed their own symbol. So a symbol for zero. Babylonians. And the next one? Around about the same time, the concept of zero, and that I think we will attribute to Aristotle. He describes the concept of nothingness.

How do you describe it? It was just nothingness, a void. The void. The vacuum. He wasn't thinking of it as a number. He wasn't doing the mathematics. It was philosophical, rather mathematical. Indeed, indeed. But when we talk about zero as a number that sits halfway between plus one and minus one, that is given credit to Indian mathematicians. People like Ariavata in the early 6th century, Brahma Gupta,

and it was sort of inherited as a package with the decimal systems 1 to 9 by the Islamic world. So this gave you a continuous line of numbers going infinitely back minuses and infinitely forward pluses. Did that give you the first taste of that? The idea of infinity in division by infinity I think comes much later. Well, Aristotle talked about zero. Yes, but that's because he didn't. like the concept of infinity. He didn't think infinity existed.

So, all right. When we got this zero, then, when we got the third zero between the two ones, did that or did that not give us a continuous line in numbering that they hadn't had before? No, no. It took a while. Well, the, even in the Islamic word,

so if people like Khorismi in his algebra still didn't use the notion of zero. So a nice example is what number, if you multiply it by two, gives an answer that's the same as if you square that number. And the answer is two, because two times two is four, and the square of two is four. But there's another solution, zero, because if you double zero at zero, and you square zero at zero. In the Arab mathematicians, people like Chaharismid, I should say Arab, he's a Persian.

Those who wrote in Arabic didn't acknowledge zero as a solution. When they wrote their equations, even in prose, they avoided having something, something, something equals zero. They'd always stick a number on the other side of the equation. So it was very slow in being part of mathematics as a real useful number. That's right. Because they couldn't find a use for it, was that? They didn't see a need for it. I think they could avoid it. That took sort of later in Europe before it really caught on. Peter Pullman, let's say we're getting towards a 12th, 37. What is the interchange with Europe at this time? With what we, with Euro. I mean, there in the, the, the, I don't know, they're

and over Spain and so on, but with the rest of Europe. Yes, so basically, just as a lot of Greek texts got translated in the ninth entry into Arabic, so a lot of Arabic texts got translated into Latin. That's true for philosophy, for medicine, but also in particular for mathematics. For instance, somebody called Robert of Chester translated the book on algebra by Al-Hawarizmi. And that became a fundamental text for Europe. and on the basis of this Latin translation, basically mathematics in Europe in Latin took off.

And translation is again at the core of the spread of knowledge and the development of new mathematics. And so it became rather hithy-missive what got translated and what didn't get translated. Latin took over as the language of learning from Arabic, which had been the language of learning for four or five, it depended high. for at least several hundred years of the golden age.

And what affected this up? Can we talk about Fibonacci and give us a just concrete example of Fibonacci? What did he do that was important for the development of this story? So Fibonacci was born probably in Pisa and his dad was a representative of the Pisan merchant in North Africa. So Europe is now where at the 12th, 13th centuries, Europe is beginning to trade again, properly send out trading missions.

So his dad is living in what's now Algeria. and Fibonacci grows up there and then travels widely through North Africa. He must have just been interested in things. And in particular, he learnt what he calls the Hindu numerals, and he also learned how to solve basic algebra problems. And he went back to Europe, and he wrote one of the first books that really took off. So it's called The Book of the Abacus. And it's describing how to use the Indian numerals for trade and accountancy,

how to use methods with them to convert currency. and how to solve various other simple problems. He was also an amazing number theorist, but that's by the bye. Let's get on. So, Jim, can I just take us towards the end of this story? When Arabic, the great ascendancy of Arabic, it went on, and Peter is very stern about this, that it didn't stop in the 13th century.

He went on, moved to Samakhan and continued. But the European scientists, we have Galileo and are moving in, as it were, and the translation of Latin. What was, you feel that there was a great loss and even a belittling of what the Islamic mathematicians had done, they had discovered a lot of these things, which the Europeans

built on but didn't acknowledge or they got lost, as it were, in translation. I don't think there's anything particularly insidious about trying to hide it. But you do use the word belittle. Yeah. Yeah, I think first of all, it's serendipity, which Arabic text tended to get translated into

Latin. You think, things like star charts in astronomy. Very often it's not the greatest astronomers whose work is then used by the likes of Marco Polo or Christopher Columbus. It's other lesser known as astronomers. But certainly the great scientists in Europe

during the Renaissance, the birth of science in Europe, people like Copernicus, in his text on astronomy, he acknowledges Islamic astronomers and mathematicians and he uses their techniques. But Europe then, you know, with the likes of, of Galileo and Kepler and Newton.

Science developed and mathematics developed so quickly that it's almost inevitable that earlier mathematicians of the Islamic world do get forgotten. Why were they developed so quickly and we still have, as you've all suggested, it's going on in the Irish-speaking world. Why didn't they join in this push forward?

Various reasons. I would argue that it's the fragmentation of the great empire. We're not talking about a huge unified Abbasid Empire as there was in the 9th, 10th century. There are kingdoms. fiefdoms and caliphates and they're arguing and fighting against each other. I also think it's important that in the Islamic world, the Arabic-speaking world, they didn't take on the printing press.

That was seen as very much a no-no. It's part of their culture that they would use scribes and calligraphy was very important. And just as Peter mentioned, that paper was very important to propagating books in the early translation movement in the Islamic world, not adopting the printing press until much later, kept them held back behind Europe. Pretty, I wouldn't it right, just not doing it

that way but doing it the other way, stopped them, sort of, aborted their intellectual engagement. Well, it just meant Europe overtook them far more quickly. Would you agree with that, Peter? Well, I mean, so basically, there's this idea that there's decline, and we had a golden age, and I don't know, after Al-Gazali

who dies in the early 12th century, 11-11, or after another date, everything goes down, downhill. But is that really true? Basically, people have looked at philosophy, have looked at medicine and see that there are flourishing traditions of medicine, of philosophy to less extent also of mathematics that develop in the 13, 14, 15, 16, 17, 18th centuries.

And the interchange with Europe does not start. So basically, if you look at the Ottoman code in the 17th century, the Latin text that are translated into Arabic. There's a Greek text like the philosopher Plethe, for instance, who is very much a Renaissance man is translated into Arabic. So the exchange doesn't stop and of course ultimately the Ottoman Empire declines.

Finally, Jim, what would you say as the chief legacy of this four or five hundred years up? Well, it's certainly more than just looking after the great knowledge of the Greeks and in order to hand it back to Europe when it reawakens. We've been talking about mathematics, but that's just one of many disciplines in knowledge.

I'm going to stick to mathematics. What was the legacy of that? I would say the development beyond what the Greeks did, which was essentially number theory and geometry, they gave us algebra, they gave us trigonometry, two other independent branches of mathematics. I'm afraid. I know you want to come in, Colvo,

Times Up, I'm afraid. Thank you very much, Peter Pover, Peter Pover, Rone Dugle, and Jim Alcalile. Next week we'll be talking about Seneca the Younger, the philosopher and tutor to the Emperor Nero. Thank you for listening. And the In Our Time podcast gets some extra time now with a few minutes of bonus material from Melvin and his guests. Anyway, what I really wanted to say was, I think they defined mathematics.

They defined what we now think of as mathematics. So for the Greeks, maths was an exercise in proof and geometry, and that was it. It was quite inward-looking and quite sterile, essentially, and would probably have fizzled out in some anyway. What the Babylonians did was, what the Islamic golden age did for us was mathematics is about shapes, it's about

numbers, it's about proofs, it's about maths applied to itself for its own sake, it's also about math applied to real world problems like physics or like engineering or all the rest of it. And this notion about the shape of the discipline that is mathematics is I think what the Arab world gave to Europe.

And they applied it to other disciplines. So they people like Ibn Al-Hatham mathematicised optics, mathematicalised astronomy. So whereas these were observational disciplines and ideas that the Greeks had, they applied rigorous mathematics to it. And to philosophy, I mean, we didn't mention Al-Kindy,

but so the translation of Euclid, you have like these proofs which, you know, you basically, it's called a reductue at absurdum, so you basically have a proof you say, A is can't be the case, B can't be the case, C can't be the case, D can't be the case, these are the four possibilities therefore the whole thing can't be the case

and this is a method of proof which you find in Euclid which then is taken up by Alkini and this is how he does his philosophical proofs if you say this is the case this is the case yes of course does Nalcindi prove that infinity

can't exist using some simple mathematics that's very very clever for his time yeah so basically there are proofs and you set up dilemmas or dichotomies and you prove various subsections as impossible and therefore prove the whole proposition as impossible. And that is a friend of mine, Peter Adamson,

who was on the program many times, he called this reductu at absalom at infinitum because how kind he does it so often, you know? My argument about the House of Wisdom against historians who are more skeptical is the... Be hell held back.

He did help. With great dignity. Yes, he did. Exactly. That's right. Yes, yeah. And I would say the absence of evidence

isn't the evidence of absence. But symbolically is such a powerful notion that it's sort of I'm sort of not that interested. Yeah, I don't care if there was a building of a special room inside the Calus Palace. For me, it was

and the whole of Baghdad was a magnet. But that's it. So basically Baghdad is the city of probably half a million, possibly even a million inhabitants from all over the place. It's a melting pot. And, you know, if you could make it there, you could make it everywhere. You know, so this was really what is so fascinating. And it's not like this House of Wisdom.

So there was probably this House of Wisdom as a Palace Library, but it didn't play such a role in the whole translation movement. But that's in a way not important. And if you look at the sponsorship for the translation movement, the sponsors were not the cadets themselves, but you like high-ranking court officials, such as the Banamusa, you know, mathematicians, you know, sons of a highwayman turned plutocrat.

Or the Persian, the Barma kids, you know, the Persian viziers, you know, that obsession of Persian culture with learning. And they were rich. They were, you know, pouring money into translations. One of the things we didn't really get into is the attitudes to education. So, I mean, for the Babylonians, One of the other reasons they thought maths was important was that they believed that the future elite, the future rulers,

needed to go through years of having algebra beaten into them to train the brain of the future directors of things. Whereas I think the Greeks had a more kind of humanities including mathematics notion for what the upper class is should be doing. And then once we get into the Indian world, then it's only certain families that need to learn mathematics. You only need to learn it if you're from an astronomical family and you've inherited the role of astronomer.

In the Islamic Goal Made, we suddenly reached this phase where people are meant to be informed. It is good for them in general to be informed and to learn and to study and to study incredibly broadly, but maths as a core part of what any cultured person should be able to master. The Greek thing is, I was going to ask if you could assess the relative weight of the different influences.

The Greek has a heck of a start with Plato saying, on Bubby's Academy if nobody comes in if they don't know mathematics better words to that effect Yeah Yeah the Greek You've got to give them their due

They're got to give in their idea They're very lucky about this Yeah They are I mean I mean The point is So many of the Greek scholars

Are still household names today Aristotle and Plato and Ukraine But not as many in the Islamic period Are And it's just to show that In so many of these disciplines it's a continuum. It's a baton handing over knowledge from one civilization culture to the other. So it's filling in that gap between the Greeks and the Renaissance in Europe that is important.

But it's not to say that I remember once giving a talk saying there's a football match between the Islamic scholars and the Greek scholars. And you can very easily list all 11, the whole sort of first team of the Greeks are household names. But not as many. You know, you've got Avesenna and Baruni, you know, bossing the midfield. And you've got Tussin somewhere. Kayaan somewhere. as far as me, you know, in defence. The real problem would be finding the referee, wouldn't it? That is true, yes, that would be a different problem.

One thing we didn't mention was a lot of the other, you know, a lot of the later mathematics. So Al-Qaashi is a particular favourite of mine. 15th century. He's regarded as the greatest mathematician in the world in the 15th century. So at the end of the Islamic golden age, but before the great mathematicians of Europe start emerging. And he, you know, he developed. so many ideas. He calculates pi to 16 decimal places.

And far better than Archimedes could do before I'm in agreement. But not only does he do that, he says in advance, I'm going to calculate it to this accuracy. Therefore, I need it to this much. So this is how accurate. There's also a calculation of the circumference of the world, isn't there? That's Beiruni, yes. The Da Vinci of the Islamic world, absolutely.

Yeah, and we had length of the year to within milliseconds. It's just astonishing. Yeah, that was Kayam. Yeah, that was Kayam. I mean, I was busy talking about cubic because I couldn't do the year as well. Probably better come next to. Yeah, yeah, there's a part two. Part two.

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