Op-Ed in Pioneer, 8 Oct., 2007

<!--QuoteBegin-->QUOTE<!--QuoteEBegin-->Kerala School absent in annals

Second opinion: KL Jhingan

This has reference to the article, <b>"From Kerala to infinity", the interview with George G Joseph "Knowledge Travels" (August 20) and the editorial, "This is to certify" (August 14).</b>

On going through AL Basham's books, The Wonder that was India and A Cultural History of India, and the chapter on mathematics in ancient India in Jawaharlal Nehru's Discovery of India, one gets a clear indication as to who were the leading lights of Indian mathematics in the past: Aryabhatt (5th century) Brahmagupta (7th Century) Mahavir (9th century) and Bhaskar (12th Century). There is no reference, even obliquely, to the so-called 'Kerala School'.

Basham says that some progress happened in trigonometry, spherical geometry and calculus chiefly in connection with astronomy. He only mentions the above-mentioned doyens of ancient Indian mathematics. And Will Durant states, "Bhaskar crudely anticipated the differential calculus."

The book, A Cultural History of India, says, "KS Shukla lists a minimum 28 commentaries on it (the reference is to Surya Sidhant) by known authors mostly in Sanskrit but two in Telugu reaching to the early 18th century together with at least 17 works based essentially upon its theory, his recent edition includes the commentary of Parameswar (AD 432) written in Kerala in south India."

Discovery of India talks of Narayan (1150 AD), Ganesh (1545 (AD), and mentions a book, History of Hindu Mathematics, by B Dutta and AN Singh 1935. There were earlier books also Baudhayan (eighth century BC) Apastamb and Katyayan (both fifth century BC).

From the above, it can be inferred that the roots of ancient Indian mathematics go back to Vedic times with the flowering of the genius. <b>Such mathematics reached its zenith from fifth to 12th centuries AD.</b>

The observations in The Wonder that was India and Our Oriental Heritage regarding ancient Indian mathematicians' acquaintance with calculus during the above period was decidedly prior to the claim of so-called Kerala School. Thereafter, very little original work on mathematics was done in India after the 12th century, as opined by Nehru; other works being mere repetitions.

There is no reference to Madhava and Nilakantha of the 'Kerala School', to whom the knowledge of infinite series - one of the basic components of calculus in attributed in the said article.

<!--QuoteEnd--><!--QuoteEEnd-->

Will try to get the Aug 20th article and post here.

<!--QuoteBegin-->QUOTE<!--QuoteEBegin--><b>From Kerala to infinity </b>

In their stunning new research, Dennis Francis Almeida and George Gheverghese Joseph show how mathematicians in Kerala developed the infinite series more than 250 years before Isaac Newton is credited to have done so. It was Jesuit missionaries who carried Kerala's knowledge to Europe

According to literature the general methods of the calculus were invented independently by Newton and Leibniz in the late 17th century after exploiting the works of European pioneers such as Fermat, Roberval, Taylor, Gregory, Pascal, and Bernoulli in the preceding half century.Â

However, what appears to be less well known is that the fundamental elements of the calculus including numerical integration methods and infinite series derivations for 'pi' and for trigonometric functions such as sin x, cos x and tan-1 x (the so-called Gregory series) had already been discovered over 250 years earlier in Kerala.

These developments first occurred in the works of the Kerala mathematician Madhava and were subsequently elaborated on by his followers Nilakantha Somayaji, Jyesthadeva, Sankara Variyar and others between the 14th and 16th centuries. In the latter half of the 20th century there has been some acknowledgement of these facts outside India.

There are several modern European histories of mathematics which acknowledge the work of the Kerala school. However it needs to be pointed out that this acknowledgement is not necessarily universal. For example, in the recent past a paper by Fiegenbaum on the history of the calculus makes no acknowledgement of the work of the Kerala school.

However, prior to the publication of Fiegenbaum's paper, several renowned publications detailing the Keralese calculus had already appeared in the West. Such a viewpoint may have its origins in the Eurocentrism that was formulated during the period colonisation by some European nations.

In the early part of the second millennium evaluations of Indian mathematics or, to be precise, astronomy were generally from Arab commentators. They tended to indicate that Indian science and mathematics was independently derived.

Some, like Said Al-Andalusi, claimed it to be of a high order: "(The Indians) have acquired immense information and reached the zenith in their knowledge of the movements of the stars (astronomy) and the secrets of the skies (astrology) as well as other mathematical studies. After all that, they have surpassed all the other peoples in their knowledge of medical science and the strengths of various drugs, the characteristics of compounds, and the peculiarities of substances."

Others like Al-Biruni were more critical. He asserted that Indian mathematics and astronomy was much like the vast mathematical literature of the 21st century - uneven with a few good quality research papers and a majority of error strewn publications.

Nevertheless a common element in these early evaluations is the uniqueness of the development of Indian mathematics. However by the 19th century and contemporaneous with the establishment of European colonies in the East, the views of European scholars about the supposed superiority of European knowledge was developing racist overtones.

This inclination for ignoring advances in and priority of discovery by non-European mathematicians persisted until even very recent times. For example there is no mention of the work of the Kerala School in Edwards' text on the history of the calculus nor in articles on the history of infinite series by historians of mathematics such as Abeles and Fiegenbaum. A possible reason for such puzzling standards in scholarship may have been the rising Eurocentrism that accompanied European colonisation. With this phenomenon, the assumption of White superiority became dominant over a wide range of activities, including the writing of the history of mathematics.

The rise of nationalism in 19th century Europe and the consequent search for the roots of European civilisation, led to an obsession with Greece and the myth of Greek culture as the cradle of all knowledge and values and Europe becoming heir to Greek learning and values.

While we understand the strength of nationalist pride in the evaluation of the achievements of scientists, we do find difficulty in the qualitative comparison between two developments founded on different epistemological bases. It is worthwhile stating here that the initial development of the calculus in 17th century Europe followed the paradigm of Euclidean geometry in which generalisation was important and in which the infinite was a difficult issue.

On the other hand, from the 15th century onwards the Kerala mathematicians employed computational mathematics with floating point numbers to understand the notion of the infinitesimal and derive infinite series for certain targeted functions.

-- Excerpted from 'Kerala Mathematics and its Possible Transmission to Europe' by Dennis Francis Almeida, University of Exeter & George Gheverghese Joseph, University of Manchester. This was originally published in Philosophy of Mathematics Education Journal.

<!--QuoteEnd--><!--QuoteEEnd-->

and

<!--QuoteBegin-->QUOTE<!--QuoteEBegin--><b>Knowledge travels</b>

In the light of recent research at the University of Manchester which shows that the "infinite series" -- and the "Pi series" therein -- were determined by Kerala scholars 250 years before Isaac Newton is credited to have done so, Nandini Jawli spoke to head researcher Dr George Gheverghese Joseph. Excerpts from the interview:

<b>Q. Did you have an inkling of the fact that the "infinite series" and the "Pi series" must have originally been discovered outside Europe? </b>

A. Certainly there was information about it. While researching on my first edition, I did find out that there were quite a number of papers published in India and elsewhere that discussed this.

Here I would like to specify that when you talk of Infinite series, the Pi series is part of it. They are not separate series. Infinite series involve trigonometric functions - sines, cosines and circular functions.

I don't think Indians then knew something called 'pi', that has come later. They were interested in finding out some sort of a way to calculate the circumference of a circle for a given diameter, which virtually comes to the same thing as pi.

<b>Q. How do you think the knowledge of the Infinite series must have finally reached Sir Isaac Newton and Gottfried Leibnitz? </b>

A. This is a conjecture. <b>We do not have any direct evidence. It is unlikely that Newton or Leibnitz knew anything about Kerala mathematics. But they might have obtained their mathematics from a number of European mathematicians, who lived before them. like Fermat, Wallis, James Gregory, who were at some point in touch with the Jesuits for professional reasons. Some of the prominent Jesuit mathematicians and astronomers - like Matteo Ricci, Antonio Rubino and Johann Shreck - had gone to Kerala between middle of 16th to early 17th century. They were sent on a project to know how Indians constructed calendar and did stellar navigation.</b>

Newton and Leibnitz did not directly know about Kerala mathematics. <b>But they were dependent on some of the mathematical ideas of Fermat and others, who could actually have very well been informed by Jesuits. Probably, Jesuits had got the original idea from Kerala.</b>

Q. While the Jesuit missionaries were studying calendars used in different parts of the world, how come they chanced upon scholarly works on calculus, a completely different field of study? Was it a chance finding? An accident?

A. Certainly, it was! <b>The Jesuits were sent to India on an information-gathering mission to find out about Indian calendar. It was the time when Pope had formed a committee to reform Julian calendar. Clavius, the renowned teacher in Rome, had instructed his students to look out for information on Indian calendars.</b>

So if there was any connection to the infinite series, it was not directly the calendar but stellar navigation, which involves accurate values of sines and cosines.

<b>Q. You have told the media: "There were many reasons why the contribution of the Kerala school has not been acknowledged - a legacy of European colonialism and beyond." What is the solution to this prejudice? </b>

A. This is one reason but there are other reasons as well. <b>The activity of the Kerala school was highly localised, just confined to a small area north of Cochin. Then there was the linguistic problem. Very few people who studied, knew the old Malayalam. Many Western scholars knew Sanskrit but not many knew Malayalam.</b>

Another reason was its being a former colony. It's not been part of any colonial power, not just British, to acknowledge that they owe a debt to anybody, particularly someone from a colony.

The solution: We have to more and more excavate the knowledge, make sure we have strong evidence to support any claims. The solution is to break down this prejudice whether from India , Europe or Islamic countries. We would be making same mistakes like Europeans if we try to monopolise mathematical knowledge.

<b>Q. Would it not have been ethical of Newton to acknowledge the contribution of received wisdom to his works on analytical dynamics? </b>

A. Newton was a great scientist. He will be remembered for his genius for a long time. There is nothing unethical about it. His greatness lies in bringing together a number of strands, putting them in a coherent framework and creating something new. His calculus was the starting point of modern mathematics.

I am simply saying that one particular aspect of something attributed to Newton, may have come from Kerala or India.

<!--QuoteEnd--><!--QuoteEEnd-->

Maybe the info about the Kerala school of mathematics was no known outside Kerala?

<!--QuoteBegin-->QUOTE<!--QuoteEBegin-->Kerala School absent in annals

Second opinion: KL Jhingan

This has reference to the article, <b>"From Kerala to infinity", the interview with George G Joseph "Knowledge Travels" (August 20) and the editorial, "This is to certify" (August 14).</b>

On going through AL Basham's books, The Wonder that was India and A Cultural History of India, and the chapter on mathematics in ancient India in Jawaharlal Nehru's Discovery of India, one gets a clear indication as to who were the leading lights of Indian mathematics in the past: Aryabhatt (5th century) Brahmagupta (7th Century) Mahavir (9th century) and Bhaskar (12th Century). There is no reference, even obliquely, to the so-called 'Kerala School'.

Basham says that some progress happened in trigonometry, spherical geometry and calculus chiefly in connection with astronomy. He only mentions the above-mentioned doyens of ancient Indian mathematics. And Will Durant states, "Bhaskar crudely anticipated the differential calculus."

The book, A Cultural History of India, says, "KS Shukla lists a minimum 28 commentaries on it (the reference is to Surya Sidhant) by known authors mostly in Sanskrit but two in Telugu reaching to the early 18th century together with at least 17 works based essentially upon its theory, his recent edition includes the commentary of Parameswar (AD 432) written in Kerala in south India."

Discovery of India talks of Narayan (1150 AD), Ganesh (1545 (AD), and mentions a book, History of Hindu Mathematics, by B Dutta and AN Singh 1935. There were earlier books also Baudhayan (eighth century BC) Apastamb and Katyayan (both fifth century BC).

From the above, it can be inferred that the roots of ancient Indian mathematics go back to Vedic times with the flowering of the genius. <b>Such mathematics reached its zenith from fifth to 12th centuries AD.</b>

The observations in The Wonder that was India and Our Oriental Heritage regarding ancient Indian mathematicians' acquaintance with calculus during the above period was decidedly prior to the claim of so-called Kerala School. Thereafter, very little original work on mathematics was done in India after the 12th century, as opined by Nehru; other works being mere repetitions.

There is no reference to Madhava and Nilakantha of the 'Kerala School', to whom the knowledge of infinite series - one of the basic components of calculus in attributed in the said article.

<!--QuoteEnd--><!--QuoteEEnd-->

Will try to get the Aug 20th article and post here.

<!--QuoteBegin-->QUOTE<!--QuoteEBegin--><b>From Kerala to infinity </b>

In their stunning new research, Dennis Francis Almeida and George Gheverghese Joseph show how mathematicians in Kerala developed the infinite series more than 250 years before Isaac Newton is credited to have done so. It was Jesuit missionaries who carried Kerala's knowledge to Europe

According to literature the general methods of the calculus were invented independently by Newton and Leibniz in the late 17th century after exploiting the works of European pioneers such as Fermat, Roberval, Taylor, Gregory, Pascal, and Bernoulli in the preceding half century.Â

However, what appears to be less well known is that the fundamental elements of the calculus including numerical integration methods and infinite series derivations for 'pi' and for trigonometric functions such as sin x, cos x and tan-1 x (the so-called Gregory series) had already been discovered over 250 years earlier in Kerala.

These developments first occurred in the works of the Kerala mathematician Madhava and were subsequently elaborated on by his followers Nilakantha Somayaji, Jyesthadeva, Sankara Variyar and others between the 14th and 16th centuries. In the latter half of the 20th century there has been some acknowledgement of these facts outside India.

There are several modern European histories of mathematics which acknowledge the work of the Kerala school. However it needs to be pointed out that this acknowledgement is not necessarily universal. For example, in the recent past a paper by Fiegenbaum on the history of the calculus makes no acknowledgement of the work of the Kerala school.

However, prior to the publication of Fiegenbaum's paper, several renowned publications detailing the Keralese calculus had already appeared in the West. Such a viewpoint may have its origins in the Eurocentrism that was formulated during the period colonisation by some European nations.

In the early part of the second millennium evaluations of Indian mathematics or, to be precise, astronomy were generally from Arab commentators. They tended to indicate that Indian science and mathematics was independently derived.

Some, like Said Al-Andalusi, claimed it to be of a high order: "(The Indians) have acquired immense information and reached the zenith in their knowledge of the movements of the stars (astronomy) and the secrets of the skies (astrology) as well as other mathematical studies. After all that, they have surpassed all the other peoples in their knowledge of medical science and the strengths of various drugs, the characteristics of compounds, and the peculiarities of substances."

Others like Al-Biruni were more critical. He asserted that Indian mathematics and astronomy was much like the vast mathematical literature of the 21st century - uneven with a few good quality research papers and a majority of error strewn publications.

Nevertheless a common element in these early evaluations is the uniqueness of the development of Indian mathematics. However by the 19th century and contemporaneous with the establishment of European colonies in the East, the views of European scholars about the supposed superiority of European knowledge was developing racist overtones.

This inclination for ignoring advances in and priority of discovery by non-European mathematicians persisted until even very recent times. For example there is no mention of the work of the Kerala School in Edwards' text on the history of the calculus nor in articles on the history of infinite series by historians of mathematics such as Abeles and Fiegenbaum. A possible reason for such puzzling standards in scholarship may have been the rising Eurocentrism that accompanied European colonisation. With this phenomenon, the assumption of White superiority became dominant over a wide range of activities, including the writing of the history of mathematics.

The rise of nationalism in 19th century Europe and the consequent search for the roots of European civilisation, led to an obsession with Greece and the myth of Greek culture as the cradle of all knowledge and values and Europe becoming heir to Greek learning and values.

While we understand the strength of nationalist pride in the evaluation of the achievements of scientists, we do find difficulty in the qualitative comparison between two developments founded on different epistemological bases. It is worthwhile stating here that the initial development of the calculus in 17th century Europe followed the paradigm of Euclidean geometry in which generalisation was important and in which the infinite was a difficult issue.

On the other hand, from the 15th century onwards the Kerala mathematicians employed computational mathematics with floating point numbers to understand the notion of the infinitesimal and derive infinite series for certain targeted functions.

-- Excerpted from 'Kerala Mathematics and its Possible Transmission to Europe' by Dennis Francis Almeida, University of Exeter & George Gheverghese Joseph, University of Manchester. This was originally published in Philosophy of Mathematics Education Journal.

<!--QuoteEnd--><!--QuoteEEnd-->

and

<!--QuoteBegin-->QUOTE<!--QuoteEBegin--><b>Knowledge travels</b>

In the light of recent research at the University of Manchester which shows that the "infinite series" -- and the "Pi series" therein -- were determined by Kerala scholars 250 years before Isaac Newton is credited to have done so, Nandini Jawli spoke to head researcher Dr George Gheverghese Joseph. Excerpts from the interview:

<b>Q. Did you have an inkling of the fact that the "infinite series" and the "Pi series" must have originally been discovered outside Europe? </b>

A. Certainly there was information about it. While researching on my first edition, I did find out that there were quite a number of papers published in India and elsewhere that discussed this.

Here I would like to specify that when you talk of Infinite series, the Pi series is part of it. They are not separate series. Infinite series involve trigonometric functions - sines, cosines and circular functions.

I don't think Indians then knew something called 'pi', that has come later. They were interested in finding out some sort of a way to calculate the circumference of a circle for a given diameter, which virtually comes to the same thing as pi.

<b>Q. How do you think the knowledge of the Infinite series must have finally reached Sir Isaac Newton and Gottfried Leibnitz? </b>

A. This is a conjecture. <b>We do not have any direct evidence. It is unlikely that Newton or Leibnitz knew anything about Kerala mathematics. But they might have obtained their mathematics from a number of European mathematicians, who lived before them. like Fermat, Wallis, James Gregory, who were at some point in touch with the Jesuits for professional reasons. Some of the prominent Jesuit mathematicians and astronomers - like Matteo Ricci, Antonio Rubino and Johann Shreck - had gone to Kerala between middle of 16th to early 17th century. They were sent on a project to know how Indians constructed calendar and did stellar navigation.</b>

Newton and Leibnitz did not directly know about Kerala mathematics. <b>But they were dependent on some of the mathematical ideas of Fermat and others, who could actually have very well been informed by Jesuits. Probably, Jesuits had got the original idea from Kerala.</b>

Q. While the Jesuit missionaries were studying calendars used in different parts of the world, how come they chanced upon scholarly works on calculus, a completely different field of study? Was it a chance finding? An accident?

A. Certainly, it was! <b>The Jesuits were sent to India on an information-gathering mission to find out about Indian calendar. It was the time when Pope had formed a committee to reform Julian calendar. Clavius, the renowned teacher in Rome, had instructed his students to look out for information on Indian calendars.</b>

So if there was any connection to the infinite series, it was not directly the calendar but stellar navigation, which involves accurate values of sines and cosines.

<b>Q. You have told the media: "There were many reasons why the contribution of the Kerala school has not been acknowledged - a legacy of European colonialism and beyond." What is the solution to this prejudice? </b>

A. This is one reason but there are other reasons as well. <b>The activity of the Kerala school was highly localised, just confined to a small area north of Cochin. Then there was the linguistic problem. Very few people who studied, knew the old Malayalam. Many Western scholars knew Sanskrit but not many knew Malayalam.</b>

Another reason was its being a former colony. It's not been part of any colonial power, not just British, to acknowledge that they owe a debt to anybody, particularly someone from a colony.

The solution: We have to more and more excavate the knowledge, make sure we have strong evidence to support any claims. The solution is to break down this prejudice whether from India , Europe or Islamic countries. We would be making same mistakes like Europeans if we try to monopolise mathematical knowledge.

<b>Q. Would it not have been ethical of Newton to acknowledge the contribution of received wisdom to his works on analytical dynamics? </b>

A. Newton was a great scientist. He will be remembered for his genius for a long time. There is nothing unethical about it. His greatness lies in bringing together a number of strands, putting them in a coherent framework and creating something new. His calculus was the starting point of modern mathematics.

I am simply saying that one particular aspect of something attributed to Newton, may have come from Kerala or India.

<!--QuoteEnd--><!--QuoteEEnd-->

Maybe the info about the Kerala school of mathematics was no known outside Kerala?