The Real Story of Calculus: The Kerala School's Contributions

[00:00] So uh a wonderful physicist and mathematician,  professor PP Divakaran, who was uh retired from   Tata Institute of Fundamental Research, it's a  premier institute in India for physics and maths,  

[00:13] passed away at the age of 89 on 23rd of August  and he was a close uh friend of mine. He has done   incredible amount of work in particle physics and  more recently in bringing Indian Calculus that was  

[00:28] done in 14th century and 15th century into the  world. He has published a lot of papers on this   topic and he has this wonderful book published in  Springer and since he passed away I thought it's   a great opportunity for us to talk about Indian  Calculus in the 14th and 15th century. Before you  

[00:45] get there you say he's a particle physicist  but he's also talking about 14th century or   something mathematics. How come this man who's  passed away was doing both of those things? Yes,   great question. So actually he was mostly a  particle physicist all of his life. Um he was  

[01:00] picked up by a Indian national talent search uh  in the 1950s and after that he went to Chicago   for his PhD. He also came to Oxford and worked  with professor Dalitz. Then he returned back to  

[01:13] India and became a professor in particle physics  and did a lot of work on quantum field theory and   particle physics. He retired in late 90s and  after retiring he got a fellowship to visit   other institutions in India. So he was visiting  Chennai Mathematical Institute, that's where he  

[01:28] came in contact with historians of mathematics  doing research on Indian Calculus and since he   was a mathematical physicist he could read these  books and appreciate it much better than purely a  

[01:42] historian of mathematics. It's a fortunate thing  that I  met such a eminent person in my life.   Um I knew about him from my friend circles, from  my professors, a lot, because he was well known.  

[01:54] Um I met him twice in my life. Uh the first time I  met when I was a first year PhD student and he was   visiting my institute to give a lot of talks on  physics that time, not on mathematics. The second  

[02:06] time I met him was last year and I spent about  two months with him, every day interacting all   throughout the day. So in some in some sense  I was kind of taking care of him while he was  

[02:18] there. It's an institute called Inter-University  Centre for Astronomy and Astrophysics in India.   And after I came back to UK we kept in touch,  we exchange emails regularly, mostly on physics  

[02:31] and mathematics. So so we we knew each other  very well, and it's really sad that he passed   away. Tell me about Indian calculus then because  when I think of calculus I think of Leibniz and   Isaac Newton. I've seen all the documents at the  Royal Society. There was famously a dispute as to  

[02:46] which one of them was first and things like that.  You're going to tell me a different story via your   friend. Yes. First of all, all credit to Newton  and Leibniz. Newton is one of my greatest heroes   in physics because of physics reasons, but also  he discovered calculus, as well as Leibniz. And so  

[03:01] there's no connection between Newton's discovery  and the discovery in India that I'm gonna to talk   about. Coming back to Kerala school. Kerala in the  modern India is a state, it's the southern state.   It's a very intellectual state. It's well known  for mathematics and culture and physics. In those  

[03:18] days it was under a kingdom called Vijayanagaram  kingdom. So 14th century, second half. And there   was a particular mathematician called Madhava  who kind of immigrated from 200 kilometers up  

[03:30] in north towards a place where he settled finally  and built his school of mathematics. We know a lot   about his mathematics because of his disciples.  His students wrote a lot about all the theorems  

[03:42] and and proofs but we don't know much about his  life. We kind of know which village he was born   in. So at least that much is clear. And we also  know his birth year,  which I think is 1340. And  

[03:54] very soon he he established this school which  today we call Kerala school of Mathematics. In   in the western world it is known as that. Uh but I  think professor Divakaran will call it Nila School  

[04:06] of Mathematics. Now this word 'Nila' is a name of  a river, and the mathematicians sort of lived in   villages around this river, and it went on, this  school of mathematics, for about 300 years. How  

[04:18] kind of formalized would this, what would this  have looked like if I was there? Yes, that's a   great question. So, I think this was a Gurukula  culture in the sense the word 'Gurukula' means   uh teacherhood in some sense. So the disciples or  students would go to a grand teacher or a master  

[04:34] and they would stay around that place, and  learn. But it was not a big campus basically,   it may be just the house of your teacher, and you  live around it. But again, I'm not very sure in   this particular question. It obviously became the  place to go did it and people who had an affinity  

[04:48] with mathematics would travel there. Yes, and I  should say that this was also a place of cultural   revolution in Kerala, in the state. The language  is called Malayalam, it's Kerala's language,   and a lot of cultural and linguistic and even  Ayurveda revolution happened around that place.  

[05:03] So definitely that was an important place. Most of  the evidence that that we know in the modern day,   especially in the western world, about the Kerala  School comes from a particular book. It's called   Yuktibhasa. It simply means the language of logic  and proofs, basically formal mathematics, and it  

[05:20] talks about the calculus that was done for the  past 150 years. This book was written in 1520s and   Madhava arrived in 1350s, so you see the gap. And  it talks about several important things that we  

[05:34] know in modern calculus. So this book that you're  talking about which is talking about something   that had happened hundreds of years before is  already 100 years before Newton and Leibniz. Yes,   200, more than 200 years before. And it is well  known now on Wikipedia on mathematical sources.  

[05:50] All - most mathematicians would know this by now.  So it's an important uh thing. But Newton and   Leibniz wouldn't have had access to this book. So,  of course I cannot deny, but there's no evidence   and I don't want to push that angle because you  know, um I have talked to professor Divakaran  

[06:05] who has read seven volumes of Newton's work and  he says he did not find any evidence. The ideas   were so different - the Indian cultural thinking  was so different from the Renaissance thinking uh  

[06:17] that uh it sounds unlikely. But who knows! One of  the fundamental things about calculus is measuring   the length of a curve or an arc. The Indian  mathematicians were concerned with measuring the  

[06:30] length of an arc. It might have come originally  from Astronomy because you know planets rotate   around uh sun and you need to cover their arc. The  way Newton came to it was through motion, which   is very similar. You have a curve which is not  linear. It's not a straight line. Your velocity  

[06:46] is changing as a function of time and he wanted  to capture it. So it's something which is changing   as a function of time. The Indian mathematicians  were concerned with circles. So they're looking   at something which is changing as a function of  theta, angle. Okay. So how do you measure the  

[07:01] length? If if something is linear, you know the  length. If something is curving, to calculate that   length, you have to discover calculus. So what you  do is take a curved object like a like an arc and  

[07:14] split it into small small small small segment and  treat each segment as a straight line in the limit   when the length goes to zero and then add them  up and then you get the arc. So they use this  

[07:28] uh to find many important series. So they found  the series expansion of sin x, the so-called   MacLaurin series. They found the series expansion  of cos x and they found the series expansion of   arc tangent, tan inverse x, which is very famous  Gregory series. Okay, how different were the  

[07:46] techniques they were using to what Newton and  Leibniz ended up using? If you map it all back,   is it all the same stuff fundamentally, or is it  really different? Fundamentally, technique-wise,   it was exactly the same. The same techniques was  being used. splitting a object into small small  

[08:02] parts and taking a limit. So, so this was the  first time in the history of mathematics where   limits were taken carefully. Um, and um, uh,  so it's same in that sense. It's different in  

[08:15] the sense that Newton and Leibniz talked about all  sorts of functions. Any general function you give   me, I can give you its derivative. I can integrate  it. Uh, in Indian mathematical school in Kerala,  

[08:27] it was only trigonometric functions because they  were talking about circle. So they are sine,   cosine and tangent. That's the main difference.  Things like exponential function, logarithm  

[08:39] function, never appeared in uh in the Indian  school. I won't be surprised if they they they   would have been able to do it but they were they  got so concerned with the foundational issues that  

[08:52] they they stuck to few classes of functions but  went into foundational issues of how to expand a   function, add corrections carefully and carefully  and carefully. So they spent all of those years   focusing on that and circles were all they were  thinking about. It's a sort of Indian thinking  

[09:07] at that time. How did these ideas discovered in  the Kerala school not spread around the world   and become famous before Newton and Leibniz  were even born. There are multiple reasons   why this could have happened. First of all, this  happened in Kerala and this grand textbook that  

[09:24] I'm talking about called Yuktibhasa was,  I should not say the word unfortunately,   but it was distinctively written in the local  Kerala language Malayalam and most scholars   knew Sanskrit and they could read and understand  Sanskrit. So the later scholars, including western  

[09:39] scholars could not get hold of a Malayalam book  where detailed of complicated advanced proofs were   written in that language. So that is one major  reason. The second reason is Kerala was under   Portuguese colony. So there were a lot of fights  going on at that time. That's one of the reason  

[09:55] why the School sort of declined and died and with  that the mathematical culture sort of died off,   until it was picked up only 50-60 years ago. In  fact, a book that was written before this book,  

[10:07] by the previous scholar called Nilakantha few  years before he wrote his book Aryabhatiya   Bhasya. It's about the older Indian mathematician  Aryabhata and what he did in Sanskrit and that was   very popularly known. We do not know exactly why  this book was written in the Malayalam language.  

[10:26] Uh but once it is written in Malayalam language it  was difficult to break through, especially because   the proofs that we are discussing are extremely  formal - series expansion, power series expansion   proofs which is which was not even conceived in  the northern part of India at that time. What did  

[10:43] professor Divakaran have to say about this on like  a personal level? Like did he feel pride? Did he   feel an injustice or a sadness that it never got  recognized sooner or what was his opinion about   all this? Great question and I know this answer  from him directly, but also he has talked about  

[10:59] it in many of the lectures. So before I answer  let me give reference to some of the lectures.   So he gave a set of three famous lectures  at the International Centre for Theoretical   Sciences, it's in Bangalore in India, in 2020. And  those lectures are recorded in a very nice high  

[11:16] quality and one can actually watch and see all the  proofs he's giving. So it'd be nice to watch those   lectures but I've also had this conversation with  him in person. He wasn't sad uh that it wasn't   well known. He understood the reason why! uh it's  sort of nobody's fault, nobody was maliciously  

[11:33] trying to hide it. So he understood the complexity  of reasons why, but also he worked hard and   uh got it out. uh so he was very happy with it  and he went deep into it - not just Kerala school,  

[11:45] he went back and back. In fact, this book that I'm  holding is divided into three sections. The first   section goes to 1500 BC Indian number theory, the  second one goes to 500 AD Aryabhata, so astronomy  

[11:57] and mathematics, and then finally this Kerala  School of Mathematics, which is I would call   as a very pinnacle of mathematics. Madhava was  definitely the most prominent mathematician in the   world in the medieval ages, without a doubt. So he  would have been known outside of India. It's only  

[12:13] in retrospect. If somehow you know this had broken  out of India and got into the western world at the   time it was happening. Is there any speculation  as to where mathematics and physics would be now?   Would we be 200 years ahead? I think it would  have accelerated a lot, but it also depends on  

[12:28] western world understanding decimal system. You  might know that Newton was intrigued when he had   this polonom  binomial expansions, and he was  thinking of decimal systems and he's thinking   this is a new paradigm. But decimal system has  been used in India for thousands of years. So if  

[12:44] people understood decimal system very well, then  it would have easily broken out. How do you feel   about it? Do you feel a pride in this? Do you feel  an injustice? What's your opinion of it having you   know read the book and spoken to the man and  understanding it more? Uh first of all I'm not  

[12:59] surprised because I'm born, I I'm an Indian, so  I know the cultural heritage of mathematics. So   something like mathematics happened in India  - I'm not surprised.  If something like law   of gravitation was known I might be surprised, and  be intrigued. Mathematics was was part of culture.  

[13:17] So I'm proud, I also feel proud, but not  necessarily as an Indian but as a scholar,   to know that this modern thinking of mathematics  and calculus existed so long back and these  

[13:30] people are thinking very very modern way of  expanding functions and approximating them.