The Indic Mathematical Tradition 6000 BCE To ?

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There was a thread that I initiated in BR,during the days when political correctness had not run amuck there and I was still tolerated. Iwilll resurrect someof my posts from there since there is a virtual goldmine of information in one place. The URL is

http://www.bharat-rakshak.com/phpBB2/vie....php?t=233


<!--QuoteBegin-->QUOTE<!--QuoteEBegin-->I will try to reconstruct this thread from what i have saved,Kaushal


posted by Gganesh

I came across this great link that provides an index of Ancient Indian Mathematicians. It basically made me want to read more. I did a google search and came up with a few more useful links. I would greatly appreciate any relevant input.

History Topics: Index of Ancient Indian mathematics
http://www-gap.dcs.st-and.ac.uk/~history/I...es/Indians.html

Astronomy and Mathematics in Ancient India
http://www.cerc.utexas.edu/~jay/india_science.html

Indian Mathematicians
http://www.ilovemaths.com/ind_mathe.htm

HOW ADVANCED WERE WE?
http://www.lifepositive.com/mind/culture/i...ncientindia.asp

Vigyan: A website of Indian Science and Technology
http://www.vigyan.org.in/mathlinks.html

History of Mathematics in India
]http://members.tripod.com/~INDIA_RESOURCE/mathematics.htm]

Equations and Symbols
http://www.gosai.com/chaitanya/saranagati/...ath/math_5.html
http://www.ourkarnataka.com/vedicm/vedicms.htm

http://www1.ics.uci.edu/~rgupta/vedic.html

http://www.astrocommunity.com/VM/sutras.php

Kaushal
Member
Member # 138

posted 15 August 2002 10:37 AM
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The use of concise mathematical symbolism (<,>,=,*,(), inf., sup.,the integral sign, the derivative sign, the sigma sign etc is a relatively recent revolution in mathematics, When calculus was invented by Newton and Leibniz independently, they used different notations. Newton used dots on top of the quantity and called them fluxions, while Leibniz used the 'd' notation. Unfortunately when all the major advancements in science and mathematics were happening in Europe in the 17th and 18th century, India was stagnant, caught in interminable wars and conquests.

Ironically it was the Indian place value notation that triggered the advance in Europe. Prior to this it was common to express all problems in wordy sentences and verse.

As to the value of PI (cannot be expressed as a fraction) it is impossible to calculate it without the use of some variant of the 'method of exhaustion'. In this particular instance it was a matter of using increasingly larger number of isosceles triangles( forming an n sided polygon) within the circle. This is the germ of the method of series expansions. The ancient Indians were well aware of this technique and used it for several trigonometric calculations. The felicity with which Indians did series expansions extended to Ramanujam. He was the incomparable master and there may be none like him on the face of this earth again.

Again there is no claim that the Indians did everything. For example there is no evidence that the Indians were familiar with the representation of complex numbers and complex variables or advanced topics such as the calculus of variations.

The ancient vedic indians were interested in very practical aspects of mathematics, namely the positions of the stars, developing a panchanga (calendar), ordinary mathematics for everyday use, measurements, such as that of land and weights etc.. There is no evidence that they were familiar with the science of mechanics for instance, which developed in Europe in the 18th century after Newton.

Kaushal

posted 15 August 2002 11:32 AM
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http://www.udupipages.com/book/hindhu.html

a History of PI (does not give details)

http://www.math.rutgers.edu/~cherlin/Histo...000/wilson.html
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http://www.geocities.com/ifihhome/articles/sk003.html

The Beautiful Tree


Subhash Kak


Sulekha Columns, May 22, 2001


As a young boy raised in small towns of Jammu and Kashmir, I often came across people who could not read or write. The school books said that literacy in all of India was low, perhaps 30 percent or so, and this was despite the introduction of the British education system more than 100 years earlier. The books implied that before the arrival of the British the country was practically illiterate. This thought was very depressing. Perhaps I shouldn't have believed the story of India's near total illiteracy in the 18th century so readily. India was rich 250 years ago when the British started knocking at the door for a share of its trade. Paul Kennedy, in his highly regarded book, The Rise and Fall of the Great Powers : Economic Change and Military Conflict from 1500 to 2000 estimates that in 1750 India's share of the world trade was nearly 25 percent.



To understand this figure of 25 percent, consider that this is USA's present share of the world trade, while India's share is now only about half a percent. India was obviously a very prosperous country then, and this wealth must have been mirrored in the state of society, including the literacy of the general population.



Unfortunately, education in medieval India is not a subject that has been well researched. But thanks to the pioneering book, The Beautiful Tree by Dharampal, we now have an idea of it before the coming of the British. The book uses British documents from the early 1800s to make the case that education was fairly universal at that time. Each village had a school attached to its temple and mosque and the children of all communities attended these schools.



W. Adam, writing in 1835, estimated that there were 100,000 schools in Bengal, one school for about 500 boys. He also described the local medical system that included inoculation against small-pox. Sir Thomas Munro (1826), writing about schools in Madras, found similar statistics. The education system in the Punjab during the Ranjit Singh kingdom was equally extensive.



These figures suggest that the literary rate could have approached 50 percent at that time. From that figure to the low teens by the time the British consolidated their power in India must have been a period of continuing disaster.



Amongst Dharampal's documents is a note from a Minute of Dissent by Sir Nair showing how the British education policy led to the illiteratization of India: "Efforts were made by the Government to confine higher education and secondary education, leading to higher education, to boys in affluent circumstances... Rules were made calculated to restrict the diffusion of education generally and among the poorer boys in particular... Fees were raised to a degree, which, considering the circumstances of the classes that resort to schools, were abnormal. When it was objected that minimum fee would be a great hardship to poor students, the answer was such students had no business to receive that kind of education... Primary education for the masses, and higher education for the higher classes are

discouraged for political reasons."



According to Dr Leitner, an English college principal at Lahore, "By the actions of the British the true education of the Punjab was crippled, checked and is nearly destroyed; opportunities for its healthy revival and development were either neglected or perverted."



Dharampal's sources appear unimpeachable and the only conclusion is that 250 years ago the Indian basic education system was functional. Indeed, it may have been more universal than what existed in Europe at that time.



One might, with hindsight, complain that the curriculum in the pathshalas was not satisfactory. Dharampal's book lists the texts used and they appear to have provided excellent training in mathematics, literature, and philosophy. Perhaps the curriculum could have had more of sciences and history. I think the school curriculum was not all that bad in itself. Judging by the standards of its times, it did a good job of providing basic education.



What was missing was a system of colleges to provide post-school education. After the destruction of ancient universities like Taxila and Nalanda, nothing emerged to fill that role. Without institutions of higher learning, the Indian ruling classes did not possess the tools to deal with the challenges ushered in by rapid scientific and technological growth.



The phrase ‘the beautiful tree’ was used by Mahatma Gandhi in a speech in England to describe traditional Indian education. Gandhi claimed that this tree had been destroyed by the British. Dharampal's book provides the data in support of Gandhi's charge.



The Macaulayite education system, put in place by the British, almost succeeded in erasing the collective Indian memory of vital, progressive scientific, industrial and social processes. But not all records of the earlier history were lost. Dharampal has authored another important book, Indian Science and Technology in Eighteenth Century: Some Contemporary European Accounts which describes the vitality of Indian technology 250 years ago in several areas.



It is not just colonialist ideas that are responsible for the loss of cultural history. The need to pick and choose in today's information age is also leading to an erosion of cultural memory. The scholar and mathematician C. Muses from Canada did his bit to counter it by writing about Ramchundra (born 1821 in Panipat), a brilliant Indian mathematician, whose book on Maxima and Minima was promoted by the prominent mathematician Augustus de Morgan in London in 1859. Muses's work appeared in the respected journal The Mathematical Intelligencer in 1998. Ramchundra had been completely forgotten until Muses chanced across a rare copy of his book.



Muses called me over a year ago, just before he died, to tell me how he got interested in India. He said that he wanted to make sense of why Indians had not developed science, as colonialist and Marxist historians have long alleged. But the deeper he got into the original source materials, he found an outstanding scientific tradition that had been misrepresented by historians who were either biased or plain incompetent.



Although Muses did not so speculate, one might ask if de Morgan's own fundamental work on symbolic logic owed in part to the Indian school of Navya Nyaya. De Morgan, in his introduction to Ramchundra's work, indicates that he knew of the Indian tradition of logic, "There exists in India, under circumstances which prove a very high antiquity, a philosophical language (Sanskrit) which is one of the wonders of the world, and which is a near collateral of the Greek, if not its parent form. From those who wrote in this language we derive our system of arithmetic, and the algebra which is the most powerful instrument of modern analysis. In this language we find a system of logic and metaphysics."



Finally, there is the loss of memory taking place due to the carelessness with which we are preserving our heritage. This is a process of permanent loss, although on a few lucky occasions long-forgotten documents are found. One example of this latter event is the recovery of the lost notebooks of Srinivasa Ramanujan (1887-1920), who may have been the greatest mathematical genius of all time. Ramanujan had been called a second Newton in his own lifetime, yet the full magnitude of his achievements was appreciated only when his [lost] notebooks, full of unpublished results, were discovered in the eighties.



You can read a fine biography of Ramanjuan by Robert Kanigel titled The Man Who Knew Infinity. I also recommend Ramanujan: Letters and Commentary, edited by Bruce Berndt and Robert Rankin.

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Kaushal
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Joined: 31 Dec 1969
Posts: 481

Posted: 18 Aug 2002    Post subject: Re: Ancient (and recent) Indian Mathematics  

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ALEXANDRIA, VA - Statistician and American Statistical Association (ASA) member C.R. Rao, Eberly Professor Emeritus of Statistics and Director of the Center for Multivariate Analysis at Pennsylvania State University, will be honored at the White House with the National Medal of Science on May 29, 2002.

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C. R. Rao, one of this century's foremost statisticians, entered statistics quite by chance and went on to become a household name in the field. He is currently at Penn State as Eberly Professor of Statistics and Director of the Center for Multivariate Analysis

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Kaushal
BRFite


Joined: 31 Dec 1969
Posts: 481

Posted: 18 Aug 2002    Post subject: Re: Ancient (and recent) Indian Mathematics  

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More on Ramchundra

http://www.mfo.de/Meetings/Documents/1998/...eport_41_98.doc

http://www.dcs.warwick.ac.uk/bshm/abstracts/R.html

Raina, Dhruv, ‘Mathematical foundations of a cultural project or Ramchandra’s treatise "Through the unsentimentalised light of mathematics"’, Historia mathematica 19 (1992), 371-384
The 19th century witnessed a number of projects of cultural rapprochement between the knowledge traditions of East and West. In his Treatise on the problems of maxima and minima, the Indian polymath Ramchundra tried to render elementary calculus amenable to an Indian audience in the indigenous mathematical idiom. The "vocation of failure" of the book is discussed within the context of encounter and the pedagogy of mathematics.

http://xerxesbooks.com/cats/alge534.mv

34. Ramchundra. TREATISE ON PROBLEMS OF MAXIMA AND MINIMA SOLVED BY ALGEBRA London 1859 Wm. H. Allen. 8vo., 185pp., 8 plate, original cloth. Owner signed and owner bookplate (Edward Ryley) . Good, cloth faded, cover and spine ends worn, one tiny chip in spine cloth. $235.00

http://mcs.open.ac.uk/puremaths/pmd_resear...md_resnews8.htm

My own lecture, entitled ‘Multiculturalism in history: voices in 19th century mathematics education east and west’, was on the history of interaction between European and Indian mathematics education in the 19th century, discussing the contributions in particular of Henry Colebrooke, Yesudas Ramchundra, and Mary Everest Boole





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