In Our Time - Indian Mathematics

Episode Date: December 14, 2006

Melvyn Bragg and guests discuss the contribution Indian mathematicians have made to our understanding of the subject. Mathematics from the Indian subcontinent has provided foundations for much of our modern thinking on the subject. They were thought to be the first to use zero as a number. Our modern numerals have their roots there too. And mathematicians in the area that is now India, Pakistan and Bangladesh were grappling with concepts such as infinity centuries before Europe got to grips with it. There’s even a suggestion that Indian mathematicians discovered Pythagoras’ theorem before Pythagoras. Some of these advances have their basis in early religious texts which describe the geometry necessary for building falcon-shaped altars of precise dimensions. Astronomical calculations used to decide the dates of religious festivals also encouraged these mathematical developments. So how were these advances passed on to the rest of the world? And why was the contribution of mathematicians from this area ignored by Europe for centuries?With George Gheverghese Joseph, Honorary Reader in Mathematics Education at Manchester University; Colva Roney-Dougal, Lecturer in Pure Mathematics at the University of St Andrews; Dennis Almeida, Lecturer in Mathematics Education at Exeter University and the Open University.

This BBC podcast is supported by ads outside the UK. 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 4. I hope you enjoy the program. Hello, mathematics from the Indian subcontinent have provided foundations for much of our modern thinking on the subject. They were thought to be the first to use zero as a number.

Our modern numerals have their roots there too. And mathematicians in the area that's now India, Pakistan and Bangladesh, were grappling with concepts such as infinity centuries before Europe got to grips with it. There's even a suggestion that Indian mathematicians discovered Pythagoras' theorem before Pythagoras. Some of these advances have their basis in early religious texts, which describe the geometry necessary for building falcon-shaped altars of precise dimensions. Astronomical calculations used to decide the dates of religious festivals also encourage these mathematical developments.

So how were these advances passed on to the rest of the world? and why was the contribution of mathematics from this area ignored by Europe for centuries? Joining me to discusses is George Gieveguise Joseph, an honorary reader in mathematics education at Manchester University, Dennis Almida, Mathematics Educator at Exeter University and the Open University, and Colvaroni Dougal lecturer in pure mathematics at the University of St. Andrews. George Joseph, the beginnings of Indian maths, I'm told, can be traced back to the Indus civilization,

about two and a half thousand years BC, in the main cities Harappa, which is now in Pakistan. Can you tell us what's been found there, which would confirm the claim that mathematics can be sort of begun on the Indian subcontinent there? Well, I think the Indus Valley civilization is probably more appropriately called the Harappa civilization, because it seemed to stretch over a very large area,

not just the Indus Valley alone, but about 1.2 million square meters. Now, because the script has not been deciphered, the information that we have is from excavated artifacts which include plumb bulbs or sort of units of weighing things, which were quite a complicated set of weights. extremely accurate. They usually took on the sort of binary

or the decimal scale ratio between the sort of weights. There was also a they also found a particular scale on a bone which has very accurate graduations. This is sort of

about nine parallel lines on this particular scale the length of which is about 66.2 millimeters. And this particular scale is very interesting because you can associate a number of other measurements in the ancient world with the Indus measurements. And then finally, there are quite interesting things coming out of the brick technology that was used, which was very advanced in the Harapa civilization.

And the brick technology shows, for instance, there were 15 different types of bricks of sort of great uniformity. And also having a ratio of forest to two is to one, that is the length, the breadth and the thickness being forest to two is to one, considered to be the one that is good for optimal bonding the bricks. And so you're suggesting that serious mathematics was involved in the construction of these artefacts, even though we can't decipher as yet the Harappan script. So there's enough evidence there to say that they were employing mathematics.

Although maybe some... Is it possible that these were extremely skillful artisans who did it until it worked? Or do you think the abstract notions were there Are they implicit in the objects themselves? Well, I don't think we've got any evidence either way, but I would assume that with such accurate measurements, there must have been some type of mathematics,

if necessary, some practical mathematics that were involved in it. Can we come to the Vedas text, which much later, they later formed the basis of Hinduism, but they're written in Sanskrit from about 1,500 BC, so I have some writing here. What evidence of mathematical advances can be? be found in these texts? Well, the first recorded account of mathematics or rather of geometries found in a set of

of texts called the Sulbas Sutras, which basically means rules relating to ropes. Now, this particular text, which was a surveyist manual, which is used for constructing altars. And these altars could be of different shapes, including what you mentioned, like the falcon shape, was the more complex ones, but there were ones which were square altars, as well as circular altars, which are used usually for domestic worship. Where the mathematics came in is when they tried to equate the areas of, for instance, a square with a circular altar. And they immediately realized that this was not possible, so they had to arrive at some form of approximation. And similarly, there's a whole range of other things relating to, for instance, the Pythagorean

theorem, which is stated very interestingly as the... Sorry, I'm not interrupting you. But can I just bring in something from the text there, which shows the way that they regard in mathematics, as a quotation, which I read in the note, you gave me. It says just as the feathers of a peacock and the jewel zone of the snake are placed at the highest point of the body. So similarly, the position of mathematics is the highest of all branches of the bea.

So they were aware of, very much aware, of how important mathematics was. Can I come to you now, Colver, given that awareness, and given George was going to move on, to the Pythagoras theorem. Can you give us some idea of what they were discovering, which you now think would be in advance of the Greeks? Sure. So George just mentioned the Sivasutras, which were the rules for building altars and

constructing things out of bricks. Now, the earliest one of these is from about 800 BC. It's called the Bodhiana Sivasutra. And it contains, so Pythagoras theorem tells you that if you've got a triangle where one of the corners is a right angle, two of the sides meet at that corner that's right angle. If you take the square of each of their lengths and add them together, the number you get is the same as if you take the length of the third side, the one that doesn't come to the right angle, square it. That's going to be the same as the sum of the other two. So what we find in the Bodhiana-Sulversutra, which is definitely before Pythagoras, is a version of Pythagoras theorem stated just for right-angled triangles where the two sides meeting at the right angle are the same length.

So he says, take a square, cut it down the diagonal. Then if you look, at the area that you make by making a square on one of the short sides, that's half of the area that you make by making a square on the long side. In other words, the long side is root two times the length of the short side. There's also... And as George said, they did, they used ropes to... And they used ropes. So that was stated as the length of a rope across the diagonal.

If you make an area with side that rope, you get twice as much. But there's also versions of Pythagoras theorem explicit inside other constructions. For instance, there's constructions where they're trying to merge two squares to make a third square whose area is the same as the sum of the previous two. And although they don't explicitly state a theorem, you can see from the way in which they're saying, lay this square off on this square, draw this thing, make this new square, that they must have known the general version.

Right. And we've seen, so the basis of Hinduism is in their Vedas, and that supported the developed mathematics. In fact, it put it at a very high point, which is extremely. interesting at that stage they're putting that. But we see the rise of Jainism in the 6th century and they're interested in very large numbers which led them to construct ideas and theories about infinity.

Can you fill us in on that? Sure. So Jain texts about mathematics are particularly important in the period from about 300 BC to about 400 AD. After that, Jainism didn't die out. I mean, there's still Jains in India now, but it became less significant. Now, the Jains continued with all the standard maths

that we find in the VEDAs as well, but as you mentioned, they were particularly interested in two things, which are startling to our eyes now. One is the presence of very, very large numbers. For instance, you find in an early Jane text that they define a radju to be the distance travelled by a god in six months if he travels a million kilometres in each blink of an eye. And I think the idea with these huge numbers is

you're just meant to sit back and marvel at the scale of them. Can you drop one second? Sure. It's fascinating. Just to tell us. So the idea of... numbers is very intricately bound up with the religious texts, with the notions of religion. We're talking about religious...

Yes. Them feeding off each other very closely. Very much, so. I mean, the Janes believed that space was infinite and that time was infinite, and so they were marvelling at the scale and the size of the world. And they thought that the more you could understand bigness in some way, the closer to understanding the wonder of the world you were. In terms of infinity, one of the most remarkable things they did is there's some very early texts where, they divide all numbers up into a lot of families, but three broad classes, one of which now corresponds to what we'd call finite numbers.

And again, they were aware that the finite went very, very big. One of which looks a lot like what we'd now call the countable numbers. So this is infinity, but it's infinity like the whole counting numbers, where you can imagine that given an infinite amount of time, you could list it. You could slowly work your way, one, two, three, four, five, six, seven. And you'd need to go on forever, but you could at least list it. and then the next infinity, which is what they associate with space, which is what we now call uncountable infinity,

and that's infinity that's so big that you couldn't even begin to start putting the things into a list. And they were on to that. Can I talk to you, Dennis Amadeh. Can we develop more what they brought in, I mean, in terms of cosmology, in terms of their abstract thought about mathematics? Yes, the chain of religion is quite interesting for many, many reasons,

including the fact that it's strictly adheres to non-violence. But his contribution to mathematics comes from its philosophy and cosmology, where large numbers are conspicuous in every facet of their beliefs. For example, the Jains believe that the religion is founded by groups of 24 teachers. The first one came around 8 million years ago, so you have a very large number occurring from the very beginning. then these groups of 24 teachers will occur in infinite succession. So you have groups of 24 occurring.

If we only have the first 24, now we're going to have 24 occurring with infinite succession. So there's the concept of infinity as well as very large numbers, 8 million. How do you deal with it? You need a system or a structure of thinking whereby you can embody that kind of largeness. Then you have the ideas of infinity in terms of souls. There are an infinite number of souls each an individual. Each individual has an infinite capacity for liberation and goodness. So you have infinity times infinity occurring.

So another concept, how do you deal with infinity times infinity? Then, of course, you have the question of time. Time has no beginning and no end. There is no creator in the gender religion. So time, there's no backwards and there's no forward. It goes on forever and the direction. How do you deal with that? So, amazingly enough, we find that the religious texts

also talk about these mathematical notions. The ideas of countability, as Colva said, do come in. You first have the finite sets, 8 million. That's finite. It's large, but it's finite. And then you have the concept of infinity about individuals existing. They must be counted.

one, two, three, four, and you go on forever. But then you come to the large infinities of time and space. It's around the third century BC, as I understand it. We begin to see the numerals, which are the basis of modern numbers, one, two, three, and so on. How did these emerge? Well, obviously, the question of counting has always kind of grasped the consciousness of thinkers for many, many years, 28,000 years

since the first signs that human beings were counting to the advent of number systems. which could be used for arithmetic. The Brahmi symbols, which were the first kind of conceptualization of symbols being attached to number words, occurred around about 250 BC. They superseded the earlier Karošti system, which had a few number symbols, but usually numbers were enumerated by a series of strokes. Two would be two strokes.

Three would be three strokes. The Brahmi system advanced that by having number symbols for one to ten, except for two and three. So two and three would be two dashes and three dashes. The Brahmi system had the limitation that there was no zero. You need a zero, both as a number and a placeholder. When did that come into place, as it were? Number simple and placeholder came around about the fourth century,

the first evidence of it anyway, in the Bakshali manuscript, which is discovered in Western Pakistan. In that you have the existence of zero both as a number and a placeholder. So you could write numbers such as 203, the two standing for the 100, the 0 for no tens, and 3 for the units. So 200 plus 0 plus 3, 203. The main advantage of the Indian number system which developed was that you could write it either way.

You could write it from right to left so that the smaller units are on the right, so 203, the 3 is the units, or you could write it the other way around. For example, in the Panchasidantica, you have the number 0252. That stands for 2,000, written the other way around. The two, the right end, is the thousands. And this had the versatility that attracted Arabs who wrote their script the other way around. That versatility helped them to incorporate that number system

to the way they wrote and thought. Can I just go back to you briefly, George? Buddhism was implicit and implicated in bringing about the number zeros, I understand it. Yes, I think behind the sort of emergence of zero as a number, there was a sort of very interesting philosophical or metaphysical doctrine which was associated with Buddhism. It was known as Sunyata, which is the sort of spiritual practice of emptying your mind or creating a world.

void. And this was, in fact, suggested not just for religious practices, but suggested for any form of aesthetic, creative work. Writers were supposed to empty their minds before they wrote. Similarly, painters were supposed to sort of start with the void before they sort of drew something. And similarly, with the architects where the emphasis was it was actually the void or the space that was more important than what the object that was placed on the... So there's a direct interrelation now, yes. There seems to be called a... There seems to be a gap now until about the 5th century AD when the classical period gets underway.

Is that simply because the simple lack of evidence? I mean, George has talked about the very beginning. They still can't decipher the scripts then. So are we running into an evidence gap here? There is an evidence gap. We're not quite sure what triggers the evidence gap. One possibility is, obviously, with the rise of Jainism, there's a decline in the need to build these big altars for sacrifices anymore.

The Vedas aren't being studied quite as much as they used to be. There was also an enormous amount of political upheaval going on at the time, which only finally settled down in about the 4th century AD with the rise of the Gupta Empire, and that then heralds the classical period. But another reason may have been the way in which maths was studied at the time. Maths was very much studied in families, so you'd get sort of high-cast Indians who were also interested in astrology

and hence studying the stars and needing to understand maths to predict astrological events. And they would teach their children mathematics and it would get passed on from generation to generation. The family would have a library of mathematical texts, most of which were commentaries on earlier texts, and they would sit there writing copies and copies.

So in a time of both political and religious upheaval, It's very possible for links in that chain to be broken. The next mathematician, the mathematician and astronomer of note, I believe, is Aria Bata working behind the northeast. Why was his work so important? Well, he said that's a good question, Melvin, because most of Arirabata's work in his only known work, the Ariabata contains essentially mathematical rules,

that were already known. His main contribution and the reason for his significance in subsequent developments in the mathematics was his work in infinite series. He didn't actually do infinite series. He actually summed finite series, but he gave kind of cryptographical

techniques whereby one could actually do that. For example, in one of the verses, he says that you can't actually measure curved lines by breaking them down into very small line segments in such a way that you can actually sum them to infinity. I didn't actually do it, but later on in the Kerala school, they were able to take on these hints and methods signified in his text and developed techniques to sum to infinity.

And thereby construct functions and series for certain trigonometric functions such as sign theta and cost theta, and also find the value of pi by summing the circumference of the circle. He got very near the value of pi, didn't he? He did. He very near the circumference of the earth, and he explained eclipses extremely clearly and elegantly.

That's right. We believe, however, that his value of pi was constructed by using the old Archimedean method of inscribing the circle a bit larger and larger hexagons, which is different from the method that he let his followers develop in the later centuries. Are you talking about the same method, or are you talking about an information flow that came from?

There's certainly indications that there was a synthesis of ideas from both Persia, Greece, and from the east as well, China and so on, in India, because the major university in India at the time was in Patna where Ari Bhati himself studied, and that was as large as Cambridge in Oxford at this present time thousands of students, thousands of professors and teachers and there is every evidence that Arirapata himself

was a professor there, so he must, and in some sense has been influenced by teachings from other cultures. It is worth of pause for thought because I think not many of our listeners will know that in the fifth and sixth century idea there were these massive universities there

and working at such an extraordinary high level. Colva, I'm told that the greatest Hindu mathematician was Bhaskara II. I'm afraid we have to fast forward to the 12th century. I'm very sorry. And he's best known for a book he wrote for his daughter, Lilavati. How did he come to write this book and why is it important? Well, so we've actually got several texts from Bascaria.

As you said, we're up to the 12th century now. The Lilavati story is possibly the nicest. So he was also an astronomer as well as a mathematician. Almost all of these early people were. And as such, he was very interested in finding the most allspoken. time for events to happen. So he had calculated the perfect hour at which his daughter should be married. And the day approached and she put on her wedding dress and they, an hour before the ceremony was due to happen, they put what's called an hour cup of water. So that's a little cup that floats in a

bowl of water and it's got a hole in the bottom such that after precisely an hour, the cup will sink down and you'll know the time has arrived. Now, Lillivetti was so excited about her impending wedding that she was looking over at the cup and one of the pearls fell off her dress and blocked the hole in the bottom of this cup. So without anybody noticing the cup stopped filling up, never sank. The auspicious hour for her wedding had passed. So to console his no doubt dejected daughter,

he said he would write a book of mathematics and dedicate it to her so that if nothing else, at least her name would now live forever. So I guess he succeeded there. Lillivati itself is quite a simple math text, but you can imagine teaching children with it. It contains quadratic equations. It contains things like the rule of three, which is problems like, to take a Christmas example,

if six minutes pies cost £1.20, how much will $8 cost? You work out that one cost 20 pence, and so eight will cost you £1.60. But in terms of really pushing mathematics on, some of the most important things that Basquechariah did were further advancing this notion of pre-calculus. So calculus is the mathematics of rates of change. One operation in calculus is differentiation and the opposite is integration. So for instance, if I've got something describing my distance and I look at its rate of change, I differentiate it, I get my speed. And if I look at my speed and I look at its rate of change, I differentiate it,

I get my acceleration or deceleration. So this is the kind of maths we're talking about. he was looking at planetary orbits and he was interested in working out how fast the planets are going at any point in time. Now the problem is if you pick an exact point of time, then nothing's moving. Because you've specified a precise point. It's at a particular place. So what he realized you have to do is you've got to move down towards smaller and smaller and smaller time intervals so that you can start to approximate instantaneous time. And he went all the way down to 133,750th of a century.

second in his hope to get hold of this instantaneous time. There's a lot of what you're saying sort of has distinct resonance for those of us brought up on a Western tradition with what was happening several centuries later. But I hope we'll have time to come to that. George Joseph, just to finish, not to finish, but to continue this story of Indian mathematics before we come to the transmission of Indian mathematics.

And those extraordinary work came about from 1,300 to 1,600, as I'm told, in southern India, in Kerala. Can you tell us what happened there and why it was so important? Yeah, can I just very quickly add that the reason why Kerala became such an important centre? I think partly it was its distance and geography. The geography meant that it was sort of relatively protected from all the political and social instability by the mountains that surrounded it. And of course on the east and on the west there were the oceans.

So it remained relatively protected. Very long tradition, even going before Ayurabata, in mathematical developments in Kerala, indigenous developments in Kerala itself. The emergence of Jains and Buddhists were very important elements as well. Because the Buddhists and the Jains came and set up institutions within Kerala. So there's various reasons why Kerala was so important. I think the significance of it is taking on point that Dennis made was that for the first time,

in the whole history of mathematics, one sort of saw constants or the parameters like pi and others could be expressed in terms of an infinite series. Now, that was a very major development because it, apart from, there was a practical side to it. It sort of helped in sort of arriving at another way of estimating or getting accurate values of what we would call pi by using the infinite series. And similarly with also trigonometric functions like the sine cosine and so on. I think that was a very critical element in doing that.

But more important than that was also that a lot of these ideas may have actually been indigenous to Kerala itself. It is within Kerala that it developed, you know. So you're talking about no outside influences at all? So the influences would have been, apart from the sort of texts of Ariapata and the sort of disciples that came in, a lot of the influences were sort of,

or the creation was something very original within Kerala. I think that is what is very interesting about that. Dennis, can you, Dennis, Almonda, Can you take this up? As I understand, the school was founded by Madhava, a mathematician and an astronomer. We know his work's been lost, but his contribution is known. Briefly, how is it known though? His work has been lost, and what is his contribution?

Well, Madova, who was born in the 14th century, developed quite a lot of results in mathematics that had tremendous significance, both to Indian mathematics and later on in Europe. how do we know that he did these things, given the fact that all his works are lost? Well, he was quoted by his followers in their own works, which are known. So Neil Akanta, Somiaji, Jastadeva, and so on, actually sent the master or the teacher has told us these results. So that's how we know that he developed some, for the time, quite astonishing results.

I mentioned earlier that the Carol mathematicians, and this started with Mardava, developed series which were infinite sums which would give the value of pi. They developed infinite series also that could sum to sign and cosine and inverse ten. Later on, 250 years later, we find exactly those series developed by the Renaissance mathematicians such as Newton and Leibniz. Madawa therefore, by priority, developed these series by prototypical calculus results, and that's his significance, that they were able to do quite sophisticated things without the tools that a pure mathematician in modern times possesses. Can I just stay with you for a moment, Dennis, and to start on this idea of the, I just take the next section as to how these ideas were transmitted.

And you, I'm saying because you touch on it earlier when you talk about Archimedes have an involvement because the people who know about the history of mathematics listening, I think they'll be riveted, but they'll also say, well, what about the Babylonians, well, about the Egyptians, what about the Greeks, where is the interflow here? We are concentrating on Indian mathematics and it's a tremendous story. But if we, can we turn about how these ideas were transmitted

and why they were, where they weren't, and what happened? So, were they, did they go directly to the Arabs and bypass the Greeks? What went on there? Let's talk about the sort of early Middle Ages in our time frame. Well, if you start off with the tenet adopted by a famous historian of mathematics, Van der Wadden, who says that the general rule in the history of mathematics is dependence, rather than independence, then we have to assume that there was an interchange of ideas. Famous historians of mathematics, but Indian mathematics say that,

Indeed, classical Indian mathematics was introduced into India, but Indian mathematics developed its own unique identity. For example, they were much more driven by the search for results than on proof. That's another epistemological difference. But if we then say what happened to Indian mathematics in the classical period, we know that it was transmitted to the Arab lands. There is a famous book, I written in the 11th century, by Saeed Alandaluisi, Science in the Medieval World, and he says, yes, our number system, our trigonometry is owed to the Indians.

So they develop it, they transmit it to other lands. It's how human nature is. When we've come across information, we want to share it. We want to try and understand it and develop it further. The question about the later period of Indian mathematics is rather sketchy, because no research has been done. But it would be outstandingly an outstanding fact if the Carolla mathematics remained isolated in Carolina.

It would be the only incident in the history of post-medieval mathematics, but an important discovery just remained where it was. George Joseph, can I bring us to one man who, the great Hussainer, the great Uzbek philosopher and scientist in the late 10th century, he was brought into contact with Indian numerals. Now, what would he have his dispersal?

Would he have Euclid? Would he have the Greeks? Would Indian be the most important thing he had? Can you give us some idea of what fed his mind in this area? I think the influence of Indian mathematics came into the Arab world first. The Greek text came in later. And the influences from the Indian maths that came in were mainly concerned with our number systems and operations with them.

And there is a sort of a story that says that it may have come through a particular mission that came from Sindh, one of the areas in India, to the court of Khalif al-Manzul. And there was one person called Kanaka who was sort of mentioned. And the influence therefore came through this particular mission who brought in both astronomical texts as well as some of the arithmetical texts from India. I think the Greek influences probably came in much later. And one of the great contributions in my view of the Arab mathematicians or those Arabic mathematicians were that they were able to combine these two influences.

And from about the 10th 11th century, these influences in turn, was developed and synthesized and new creations were made by the Arab mathematicians. Well, that's very clear. Thank you very much. Colva, the Jesuits were very influential. Do you want us to be to say with George, say, you bit your bottom lip. George, the Jesuit were very... It's very... Actually, it would have been...

It would have managed to see it on television. It wasn't a big bite. The Jesuits were influential in spreading Indian numbers, spreading Indian mouths, weren't they? Yes, not so much the Indian numbers, but much later, I think we've been sort of working on a project which sort of is studying the possibility of the Jesuits, bringing some of the results from Kerala mathematics into Europe.

One must remember that the Jesuits were at those times quite mathematically accomplished. Some of the Jesuits scientists who went to Kerala were also some of the very notable astronomers and mathematicians of the day. I mean, I could mention some of the names of people like Matthew Riki and others who were in constant correspondence with clavius, who was a very important figure in Koloiji Romano. So though there is, we haven't found any documentary evidence as such, there is a strong circumstantial evidence which would indicate that a lot of the information from Kerala Mathematical, which was quite advanced mathematics,

did actually were transmitted by the Jesuits to some of the mathematicians in Europe. I come in a moment, Dennis. I just want to bring in Culver here. She's been out of the conversation in a little while. How are these Indian advances, mathematics, must have been revelations, how are they greeted in Europe

when the Jesuits brought them back and others brought them back in Europe? Well, they spread into Europe very, very slowly. The earliest evidence we've got of Indian numerals in Europe comes from a 10th century Spanish. Codex where, I mean, they're used, but they're not particularly discussed. Gerbett of Oriac, who went on to become Pope Sylvester, also encountered Arabic numerals, well, Muslim numerals during his time in Spain, but he doesn't seem to have ever popularised

how to do additional multiplication or anything like that with them. Instead, what happened was people on their counting boards, rather than using five dots in the tens column to mean 50, suddenly had a single piece with a symbol for five on it. So they sort of got the beginnings of the idea, but it didn't really catch on. The person who really introduced Europe finally to Indian numerals was Fibonacci. So now we're all the way up to the 11th, 12th century AD. Now his father was a diplomat stationed in Algiers at the time, organizing trading on part of the North and Italians. And so Fibonacci grew up there and says that he was introduced to the Indian numerals when at quite an early age and suddenly could think of almost nothing else.

So he then came back to Europe and wrote a book called Liber Abakai, the book of the Abacus, where he introduces Indian numerals, shows how to do addition and multiplication and so forth with them, and really helps proselytize that they're very, very useful for bookkeeping. But still it was really slow to take on. As late as the 15th century, you find the mayor of Frankfurt in Germany banning the use of Indian numerals in accounts books because he's so worried they could be fair. And there's a quote from 15th century Germany of a man wanting to know where to send his son to university being told, if you only want him to be able to do addition and subtraction, then any French or German university would do.

But if you want him to be able to understand multiplication and possibly maybe even long division, then you're just going to have to send him to Italy because there the numerals had properly taken on. Dennis, can you develop this? Because I just want to get in, it was, as I understand it, I mean, Culver's been very eloquent there. but the archives in Sicily are full of praise for these mathematics and awe, really. But then over the next few centuries that seems to die away. The Renaissance mathematicians come forward, absorbing all this. And can you just talk about the influence and reputation over the next few centuries of Indian mathematics?

Well, let me just answer the Colvus point about the resistance to the acceptance of information from English. India and the East in particular. The number system was banned in Germany, quite rightly, in the 15th century, but near a home, accounts were not allowed to be kept with the Indian number system until the 19th century. So... Why was that? That's simply because they were not trusted.

Why were they not trusted? There was a question about being able to do the arithmetic, which, as I always said, was not quite introduced into. So there wasn't just because they couldn't do it? It wasn't a question. There was also a question of cipher. The question of zero was very difficult to comprehend. It had a kind of magical connotation.

The word zero is cipher in Italian. So there were these kind of people who kind of did underground mathematics, so to speak. But this led actually, as Charles Dickon said, in one of his lectures, that this led directly to the burning down of the house of loss. It's quite an interesting story because they stored all the accounts on tally sticks, to be burnt at some point in time. They were burnt in the ovens and the houses of Lords, which set fire to the panelling,

would set fire to the house and then set fire at the lower house as well. We've got a nice new building. So, yes, it's a question, it's actually a good kind of significance. If you don't accept technology, you pay a big, big price. Yes, it was very slow. And it probably coincided with the crusades and the kind of animosity

that was building up between cultures. So the Arab numerals had the word Arab. So there was a question of, do we trust a system that came from an alien culture, which is now kind of antagonistic towards our own? It's absolutely fascinating, isn't it? Because we're talking about politics as well as long. In the 19th century, you find kind of explicit vocations of this kind of attitude.

There were people like Sidiya kind of talking in quite reprehensible terms about Brahman's and the filthy habits and the statisticious ideas. And even claiming that the Indian number system was kind of derived from the Roman number. system, a complete kind of an antagonism that could not be explained rationally. So one could also explain the flow in those terms.

And Carolina mathematics occurred exactly at that point in time when imperialism and colonialism started becoming endemic in the East. Sorry, George, first. I think just taking up Dennis' point, I think knowledge,

What he meant, I think, was that the knowledge of Kerala mathematics occurred around that time with a person called Vish who actually did it. What I was just going to go on to add is that I think we've had some problems with the Eurocentrism of understanding, development of mathematics, of that matter science, as basically arise from a dialogue between civilizations. to me that idea has been the main idea that has sort of made me work in this area at all. The whole idea that different cultures and different civilizations have dialogues which could be extremely creative, whether it came to the sort of birth of modern science or the birth of modern mathematics. I think this particular concept was very important. And also further than that was one has to tie it up with a sort of political change.

changes that was taking place at that time and the political reality at that time, which was sort of based on the idea of growing empires all over the place and therefore the need to denigrate the cultures. Would you agree with that, Colonel? Yes, very much so. And the one thing one would also add to that is that I think lots of the early historians of maths, at least partly got into it because of their veneration for the Greeks. And so there really is just a desire to see the Greeks as the source of absolutely everything

because we are intellectually descended from the Greeks. So, I mean, as later as the 1930s, coming back to the question of numerals, you find people insisting that the Pythagoreans invented our current numerals and that they're based around a system of dots that joined together. And that the only reason that there's no evidence for this is that the Pythagorean's realized it was so important that they kept it secret. So there's a desperate belief to have us descended from the originators of all of the important ideas in the world.

Can you put a final word in, Dennis, and then we'll have to move a one and five. There's definitely the case that the Greek mathematics and the Indian mathematics came from different epistemological bases. To understand Indian mathematics within the Greek context would require you to actually get into somebody else's frame set. It was possible for Hardy to understand the great Indian mathematician Ramuddinja, who said at the end, if I were to give a score out of 100 for absolute, invention, I would give myself 25 and I'll give Ramona John 100. Thank you very much. I'm sorry we have to end there. Thanks, George Joseph, Colvo, Ronnie Douglas and Dennis Almeida.

And next week we'll be talking about the history of hell. Thank you for listening. We hope you've enjoyed this Radio 4 podcast. You can find hundreds of other programs about history, science and philosophy at BBC.com.com.uk forward slash radio 4.