In his book, ‘The man of numbers,’ Dr. Keith Devlin, executive director, H-Star Institute, Stanford, writes that while the work of writers and other creative people is readily acknowledged, this is not the case with mathematicians While mathematicians “admire those who make discoveries, their interest is generally in what is discovered, not in who gets there first.”
This is unfortunate, because mathematics too requires as much imagination as do composing music or writing a story. In fact, mathematician Hilbert said of a mathematician who gave up mathematics to become a novelist: “He did not have enough imagination for mathematics, but he had enough for novels.”
Devlin points out that the Hindu numeral system was acknowledged as early as the 10th century, by the Spanish monk Vigila, who wrote of the “subtle talent of the Indians” and that “all other races yield to them in arithmetic and geometry.” In the 13th century, Leonardo of Pisa expressed his admiration for the Hindu numeral system. But we know nothing much about the individuals who developed the system, although we do know that Brahmagupta in his Brahmasphutasiddhanta (628 A.D.) gave some basic properties of zero.
So, given the general lack of accreditation in most cases, is it possible, that other mathematical contributions of Indians, that had a seminal influence on mathematical developments elsewhere, might remain unacknowledged? More specifically, has the work of the Kerala school of mathematicians been denied its due? What part did their work play in the calculus of Newton and Leibniz?
Between 1665 and 1666, Newton came up with his work on infinite series. But did he get the concept of infinity from Greek mathematics? Let us digress a bit here. Greek philosopher Zeno (5th century B.C.E), came up with some problems. Let us examine one of them here. In the Achilles and tortoise problem, it is proposed that Achilles, a fast runner, can never overtake the tortoise. The tortoise gets a head start. By the time Achilles gets to where the tortoise was originally, the tortoise has moved further. By the time Achilles gets to the new location of the tortoise, the tortoise has moved even further. And so it goes on. So technically, Achilles can never overtake the tortoise, because Achilles is always trying to get to where the tortoise was earlier. But we know that in real terms, Achilles will get to a point where he meets the tortoise, and once he gets there, he will overtake the tortoise. But what is the point at which Achilles meets the tortoise? To answer that, one should be able to understand the concept of dividing anything infinitely many times. In other words, one must understand infinitesimals, and infinity. But the Greeks thought in terms of the real world, and were unable to think in terms of abstractions with regard to this problem.
So the Greeks just kept infinity away from their mathematical work. And Zeno’s problems became paradoxes to them. So to cut a long story short, it is unlikely that European mathematicians could have got the concept of infinity from the Greeks. So what could have possibly influenced the work of Newton and Leibniz? Scholar and author Dr. George Gheverghese Joseph, believes that the work of the Kerala school of mathematics, could have influenced their work.
The work of the Kerala school was first brought to light by Charles Whish in 1832. The Kerala school was founded by Madhava (1340- 1425 C.E.) who belonged to a Brahmin caste that had migrated to Kerala from coastal Karnataka. The village in which he lived was Sangamagrama, present day Irinjalakkuda. His only surviving works are astronomical, but he seems to have done a lot more, as is evident from references to him by Kerala mathematicians who came after him. “Thus said Madhava,” they say, when they refer to some mathematical results. Joseph writes that Madhava discovered “infinite series for circular and trigonometric functions, commonly known as the Gregory series for arctangent, Leibniz series for pi, and the Newton power series expansions for sine and cosine, correct to 1/3600 of a degree.”
Madhava’s successors came up with brilliant work too, and the work of the Kerala school, anticipated the work of European mathematicians such as Wallis, Gregory, Taylor, Newton and Leibniz by 200 years.
Joseph believes that the mathematics of the Kerala school might have been transmitted by Jesuit missionaries to Europe.
Dr. Paul Ernest, University of Exeter, feels that one reason for non-acknowledgment of the contributions of Indian mathematicians by the “traditional histories of mathematics,” could be due to “the racial prejudice of Eurocentrism.” Joseph adds that inaccuracies in translation also might have obscured the significance of the Kerala school’s achievements. For instance, Nilakantha in his Aryabhatiyabhashya talks about the unattainability of an exact value for the circumference of a circle. But incorrect translations gave the idea that he was talking of an approximate value. As a result, historians rejected the idea that Indians knew of the incommensurability of pi.
Joseph searched European archives for documents to support his theory of a transmission of Kerala mathematics to Europe, but he drew a blank. Arun Bala, Professor, University of Toronto, feels that the Kerala discoveries might have reached Europe as a set of computing skills, with Indian craftsmen communicating the skills to European craftsmen, rather than as a theoretical exchange from mathematician to mathematician. between mathematicians. Indian craftsmen, would have needed only a truncated version of the infinite series for their practical purposes, which they might have communicated to European craftsmen. European mathematicians might then have reconstructed the original infinite series discovered earlier by the Hindu mathematicians. Since the European mathematicians received only the finite series from India, through the craftsmen, acknowledgment would not have been given to Indians for infinite series.
Dr. Joseph is the author of ‘The crest of the peacock: non-European roots of mathematics’ and ‘A passage to infinity.’ The books are fascinating, and read like the brief of a lawyer, who has a lot of evidence in favour of his client, but for that one crucial clincher. They are like stories, whose end is left tantalisingly unfinished. Maybe future searches might throw up some evidence for the transmission theory, and Dr. Joseph might then write a happy sequel to his account of Kerala mathematics.
(Dr. Joseph has held university appointments in East and Central Africa, Papua New Guinea and New Zealand).
* Mathematics in Kerala acquired a fillip, when the Kulasekhara dynasty (the dynasty of Kulasekara Azhvar) came to power. Sthanu Ravi Varma (844 to 885 C.E.) of this dynasty is said to have established an observatory in his capital Mahodayapuram (present day Kodungallur). His court astronomer- Sankaranarayanan was the student of Govindaswamin (800- 850 C.E.). Govindaswamin did some interesting mathematical work, including a special case of the much later general Newton-Gauss interpolation formula.
* Nilakantha Somayaji’s (1444- 1545 ) work anticipated the work that of Kepler (1571- 1630).
* Nilakantha’s Aryabhatiyabhashya and Jyesthadeva’s Yuktibhasa contain special cases of expansions known today as Taylor series. And in both we find the work is attributed to Madhava.
* Nilakantha developed a computational scheme for planetary motion which is superior to that developed later by Tycho Brahe.
Mathematicians of the Kerala school
: Madhava, Parameswara, Damodara, Nilakantha Somayajin, Jyesthadeva and Citrabhanu of the Kerala School were Brahmins; Sankara Variyar and Achyuta Pisaroti belonged to were non-Brahmin but belonged to castes that had important posts in temple administration, the Variyars being temple accountants.
A short bio sketch
Dr. Joseph has held university appointments in East and Central Africa, Papua New Guinea and New Zealand. He has lectured in many Universities in India, under a Royal Society Visiting Fellowship (twice). In 1992, he addressed a special session of the American Association for the Advancement of Science at Boston. He has lectured in UK, Australia, USA, Singapore, South Africa, Portugal, Spain, Italy, Netherlands, Germany and Norway. He has been on BBC Radio 4’s Programme on “Indian Mathematics.”
Why is it that mathematical contributions are not well documented by historians? Could it be because higher mathematics is beyond the reach of most historians? In India would not the problem be compounded by the fact that the mathematics is in Sanskrit and in verse form?
All that, together with lack of training, especially in historiography. A pursuit in many cases by retired mathematicians, who would not have taken up work in this area earlier, since a full-time career in this area was unavailable
You write that Govinda Bhattathiri, who lived around 1175 A.D. migrated to Tamil Nadu to study under Kaanchanoor Azhvan. Is there any information about Kanchanoor Azhvan? Was he a mathematician or astronomer?
Govinda is a legendary figure of whom there are a lot of unverified stories. I should think it was the practice those days to make no distinction between mathematicians and astronomers since the subjects overlapped in many cases. Some mathematical innovations have been attributed to Govinda without any firm evidence. Contacts mostly known are with a couple of Tamil scholars, although it should be remembered that geographical boundaries and the linguistic overlaps make present divisions somewhat problematic. Also note that the sources of inspiration of Kerala mathematicians were overwhelmingly from the North: Aryabhata and his School, Varahmihira, Bhaskara II etc.
Nilakantha Somayaji of the Kerala school, was in touch with Sundararaja a mathematician of Tamil Nadu. Sundararaja posed some astronomical problems, which Nilakantha answered through his work Sundararajaprasnottara. The two seemed to have had mutual respect for each other, as is evident from the respectful way in which they refer to each other. Did the Kerala mathematicians correspond with mathematicians in other parts of India too? If they did, then could the point of transmission of Kerala mathematics to Europe have been from some other place, say Tamil Nadu, for instance?
Sundararaja is one of the few Tamil mathematician/astronomers figuring in accounts of the time. The relationship between the two clearly was essentially a teacher-student relationship. No evidence exists as yet of contacts with the North. There may be new material to be excavated in the future. Unless that comes about, the links with the North will remain only a conjecture.
The Kerala school shows an awareness of the ideas of integration and differentiation. So they must have come very close to developing the calculus. But why did they stop short of doing so? Why do you think they shied away?
No operational concept of limits existed and the members of the Kerala School were working with a different epistemology and concept of proof. One could say that even without calculus as we understand today, Indian mathematics tackled problems of maximization and minimization as Ramachandra did in the 19th century.
Regarding the transmission of Kerala mathematics through Jesuit missionaries serving in Kerala- how many of them would have known both Sanskrit and mathematics? Ugo Baldini, for example, writes that the standard of mathematics in Jesuit schools was elementary, and that most Jesuit missionaries would not have been able to appreciate the highly refined mathematics of the Kerala school. So except for a few like Ricci and Rubino, others would not have been able to fathom the mathematics of Kerala. And Ricci spent more years in China than in India.
Yes, but the standard of mathematics in the Jesuit College (Collegio Romano) was very high and those Jesuits you mention went to that institution and similar institutions before setting out on their travels. Ricci was believed to have stayed in Cochin for two years
The missionary Rubino, who had trained under French mathematician Viete, found that the Malabar Brahmins could predict “the hour and minute of eclipses of the Sun and the Moon”, and wanted to learn the secret of how they did it. But he says the Brahmins were secretive about their knowledge and wouldn’t share it with him. For the Hindu numeral system to have travelled out of India, the Hindus must have taught the Arab traders the system, for it was the Arabs who spread it far and wide. So why would therebe an unwillingness later to share mathematical knowledge with the Jesuits? Could one reason be the attitude of colonial superiority of the Europeans?
Rubino was a college mate rather than a pupil of Viete. I think this argument about the Brahmins being secretive seems to be specious since we have cases of Europeans having Brahmin informants, De Nobili being a good example. And, in the earlier phase of the European encounter, before colonialism was firmly established, there was a lot of respect for Indian science.
Can you elaborate on the bias against Indian mathematics and science during British rule?
The initial reaction of the British to Indian science was one of awe. But later, as they tightened their grip over the country, they poured scorn on Indian science. And when Charles Whish presented his paper on Kerala Mathematics in 1832, it was met with indifference. We see the British attitude to Indian science changing in accordance with their imperialist goals. One way to control a colonised population is to give them the idea that nothing worthwhile ever originated in their country. Anything native to the country is discarded, and this is what the British did. Attempts were made, as for example by Ramachandra, in the 19th century, to blend modern science and traditional science, but they did not succeed, because of the attitude of the British towards Indian science.
Are opinions in the West becoming less Eurocentric and if so, what is the prevailing opinion on Indian mathematics in the West?
Yes, definitely; and without being immodest, I think my book Crest of the Peacock has had an impact. The next step is to incorporate Kerala mathematics (or rather Medieval Indian mathematics) into the history of world mathematics. I am working towards that objective and hope my next book titled “A History of Indian Mathematics: Engaging with the World from Ancient to Modern Times” will help.
Why did Jyesthadeva write his Yuktibhasa in Malayalam, when all the other works of the Kerala school were in Sanskrit?
A combination of reasons I think: aiming at a wider audience, stimulated by the development of Malayalam as a language of scholarship and dissemination among a wider group of scholars, given the number of manuscripts found.
Since your quest for transmission through Jesuit missionaries was not fruitful, what alternate routes, if any, are you looking at now?
There are still some more archival materials to be explored which would require research funding and research assistants with the necessary linguistic skills. Some of the Jesuit papers are in private collections; some have been destroyed. Meanwhile, there is need to shore up the circumstantial evidence which has been identified in earlier work.
(Dr. Joseph’s book ‘The Crest of the Peacock’, has been translated into Italian, Japanese, Spanish, Farsi and Malayalam. ‘A Passage to Infinity’ has been translated to Malayalam. In 2005, Dr. Joseph organised an International Workshop at Kovalam, on the possibility of transmission of Kerala mathematics to Europe. Presiding over the plenary session, Dr. Romila Thapar spoke on the significance of the project, and suggested that the historical aspects of Kerala mathematics, in relation to other parts of the sub continent also be studied.)