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== Knowledge and Indian civilization ==

== Knowledge and Indian civilization ==

Most of the amazing science and technology knowledge systems of the modern world are credited to have started around the time of the in Europe in ~ the 15th century. These knowledge systems are generally traced back to roots in the civilization of Ancient Greece, and occasionally, that of Ancient Egypt. Notably both these civilisations were destroyed, their indigenous population decimated or converted to Christianity or Islam by the invaders and their knowledge digested and presented as ancient knowledge. Hence, most of the heroes we are taught about in school and college are European or Greek.

Most of the amazing science and technology knowledge systems of the modern world are credited to have started around the time of the in Europe in ~ the 15th century. These knowledge systems are generally traced back to roots in the civilization of Ancient Greece, and occasionally, that of Ancient Egypt. Notably both these civilisations were destroyed, their indigenous population decimated or converted to Christianity or Islam by the invaders and their knowledge digested and presented as ancient knowledge. Hence, most of the heroes we are taught about in school and college are European or Greek.

 As for India, or even China, it would appear that they have played a minimal role in this magical story. Most Western accounts of the “Ascent of Man” do not devote even a single line to India’s contributions and the world is kept largely ignorant of India's great contribution to  the world in every aspect of knowledge. This is due to the creation of Indology, a system totally unique to colonial India, started by the East India Company to not only digest, but regurgitated as western marvels of science and technology. Post independence, this evil startegy was continued through a dedicated and well-paid bad of brown sepoys wh delineralely neglected any scholarly studies on our history and heritage and specialised in higlihting the <nowiki>''</nowiki>.  Incidentally, this is in contrast to the attitude in almost any other country – people elsewhere have a keen interest and fierce pride and celebrate their own contributions to world knowledge and heritage. Several countries also make a living out of their past through tourism!

 As for India, or even China, it would appear that they have played a minimal role in this magical story. Most Western accounts of the Ascent of Man” do not devote even a single line to India’s contributions and the world is kept largely ignorant of India's great contribution to  the world in every aspect of knowledge. This is due to the creation of Indology, a system totally unique to colonial India, started by the East India Company to not only digest, but regurgitated as western marvels of science and technology. Post independence, this evil startegy was continued through a dedicated and well-paid bad of brown sepoys wh delineralely neglected any scholarly studies on our history and heritage and specialised in higlihting the <nowiki>''</nowiki>.  Incidentally, this is in contrast to the attitude in almost any other country – people elsewhere have a keen interest and fierce pride and celebrate their own contributions to world knowledge and heritage. Several countries also make a living out of their past through tourism!

 However, there does exist, thanks in part to valiant individual efforts, some kind of a background awareness that the Indian civilization is in fact one of the most ancient and glorious, and that India has contributed enormously, perhaps even predominantly, to the growth of world civilization and knowledge in practically every field, ranging from the mundane and practical to the unworldly and spiritual.

 However, there does exist, thanks in part to valiant individual efforts, some kind of a background awareness that the Indian civilization is in fact one of the most ancient and glorious, and that India has contributed enormously, perhaps even predominantly, to the growth of world civilization and knowledge in practically every field, ranging from the mundane and practical to the unworldly and spiritual.

== Some well-known early Indic contributions to Mathematics ==

== Some well-known early Indic contributions to Mathematics ==

In the sciences, seminal contributions have in fact been made by Ancient India to mathematics, astronomy, chemistry, metallurgy, the list is long.

In the sciences, seminal contributions have in fact been made by Ancient India to mathematics, astronomy, chemistry, metallurgy, the list is long.

 Some Indian contributions to mathematics are well known (at least in India) : the zero, the decimal place value system and the commonly used numerals, the so-called “Indo-Arabic” numerals (called Arabic numerals in the West) were discovered in Ancient India. In fact, the importance of these is such that without these, mathematics (and science, commerce, etc.) as we know it would not have even existed!

 Some Indian contributions to mathematics are well known (at least in India) : the zero, the decimal place value system and the commonly used numerals, the so-called Indo-Arabic” numerals (called Arabic numerals in the West) were discovered in Ancient India. In fact, the importance of these is such that without these, mathematics (and science, commerce, etc.) as we know it would not have even existed!

 Further, few are aware that there has been a continuous unbroken tradition of mathematics in India from at least a thousand BCE (and perhaps even several thousand BCE) to ~ 200 years ago, and then again in the modern era.  

 Further, few are aware that there has been a continuous unbroken tradition of mathematics in India from at least a thousand BCE (and perhaps even several thousand BCE) to ~ 200 years ago, and then again in the modern era.  

== The discovery of the Kerala School of Mathematics ==

== The discovery of the Kerala School of Mathematics ==

A relatively recently discovered field is what goes by the name of the “Kerala School of Mathematics” which flourished in a tiny corner of present-'''day Kerala during ~ 1300-1600 CE'''.  Many details about the work of this school and the story of the mathematicians who contributed to it are only now being researched. '''This despite the fact that this work was brought to the attention of western scientists almost 200 years ago'''.  In 1834,  an Englishman named Charles  M. Whish  published an article  entitled <blockquote>"“On the Hindu quadrature of the circle and the infinite series of the proportion of the circumference to the diameter exhibited in the four sastras, the Tantrasangraham, Yukti-Bhasha, Caruna-Padhati and Sadratnamala” in a journal called the ‘Transactions of the Royal Asiatic Society’ of Great Britain and Ireland. '''But the article was long ignored'''.

A relatively recently discovered field is what goes by the name of the Kerala School of Mathematics” which flourished in a tiny corner of present-'''day Kerala during ~ 1300-1600 CE'''.  Many details about the work of this school and the story of the mathematicians who contributed to it are only now being researched. '''This despite the fact that this work was brought to the attention of western scientists almost 200 years ago'''.  In 1834,  an Englishman named Charles  M. Whish  published an article  entitled <blockquote>"On the Hindu quadrature of the circle and the infinite series of the proportion of the circumference to the diameter exhibited in the four sastras, the Tantrasangraham, Yukti-Bhasha, Caruna-Padhati and Sadratnamala” in a journal called the ‘Transactions of the Royal Asiatic Society’ of Great Britain and Ireland. '''But the article was long ignored'''.

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A little more on some of the contributions of the Kerala school

A little more on some of the contributions of the Kerala school

 Calculus is the mathematical study of change, and its essence is the use of  infinitesimals / limits  (and, one of the passages to “limit” is by summing an infinite series).

 Calculus is the mathematical study of change, and its essence is the use of  infinitesimals / limits  (and, one of the passages to limit” is by summing an infinite series).

 The concept of limit as given by Nīlakantha in Āryabhatiya-bhāsya :

 The concept of limit as given by Nīlakantha in Āryabhatiya-bhāsya :

“k+.TMa :pua:naH ta.a:va:de:va va:DRa:tea ta.a:va:dõ Ra:tea .ca ?”

k+.TMa :pua:naH ta.a:va:de:va va:DRa:tea ta.a:va:dõ Ra:tea .ca ?”

How is it that [the sum of the series] increases only up to that [limiting value] and that certainly increases up to that [limiting value]?

How is it that [the sum of the series] increases only up to that [limiting value] and that certainly increases up to that [limiting value]?

Binomial series expansion.

Binomial series expansion.

Taylor series expansion.

Taylor series expansion.

Infinite series expansion of  π (now known as the “Gregory – Leibniz series”).

Infinite series expansion of  π (now known as the Gregory – Leibniz series”).

Discussion of irrationality of  π.

Discussion of irrationality of  π.

Sum of natural numbers

Sum of natural numbers

'''Who were these people ? – some historical details'''

'''Who were these people ? – some historical details'''

Most of these developments took place in temple-villages around a river called Nila in the ancient days (and currently called river Bharatha, the second longest river in Kerala) during ~ 1300-1600 CE. In fact, the area over which this work was carried out was so localized, that some scholars suggest that the school is more appropriately named the “Nila School of Mathematics”. One of the key villages was Sangamagrāma, which was possibly the present-day village of Irinhalakkuta (about 50 km to the south of Nila). (However, there are a few other possible candidates for Sangamagrāma , such as Kudalur and Tirunavaya). What is more certain was the existence of a remarkable lineage of mathematicians in and around Sangama-grama of which the pioneer,  Mādhava (~1340-1420)   seems to be the one who discovered many of the basic ideas of Calculus.

Most of these developments took place in temple-villages around a river called Nila in the ancient days (and currently called river Bharatha, the second longest river in Kerala) during ~ 1300-1600 CE. In fact, the area over which this work was carried out was so localized, that some scholars suggest that the school is more appropriately named the Nila School of Mathematics”. One of the key villages was Sangamagrāma, which was possibly the present-day village of Irinhalakkuta (about 50 km to the south of Nila). (However, there are a few other possible candidates for Sangamagrāma , such as Kudalur and Tirunavaya). What is more certain was the existence of a remarkable lineage of mathematicians in and around Sangama-grama of which the pioneer,  Mādhava (~1340-1420)   seems to be the one who discovered many of the basic ideas of Calculus.

The Kerala school was a culmination of the school of Āryabhata and seems to have been the last bastion of mathematics in India till the modern era. '''The school seems to have died out soon after the arrival of the Portuguese in Kerala for obvious historical reasons. (Prof Ram)'''

The Kerala school was a culmination of the school of Āryabhata and seems to have been the last bastion of mathematics in India till the modern era. '''The school seems to have died out soon after the arrival of the Portuguese in Kerala for obvious historical reasons. (Prof Ram)'''

== Summary ==

== Summary ==

As we have mentioned earlier, the essence of [[Calculus]] is the use of limits. We end this brief article with the following quotes, the first by Charles Seife in “Zero:The Biographyof a Dangerous Idea” (Viking, 2000; Rupa & Co. 2008):

As we have mentioned earlier, the essence of [[Calculus]] is the use of limits. We end this brief article with the following quotes, the first by Charles Seife in Zero:The Biographyof a Dangerous Idea” (Viking, 2000; Rupa & Co. 2008):

 <blockquote>"“The Greeks could not do this neat little mathematical trick. They didn’t have the concept of a limit because they didn’t believe in zero. The terms in the infinite series didn’t have a limit or a destination; they seemed to get smaller and smaller without any particular end in sight. As a result the Greeks couldn’t handle the infinite. They  pondered the concept of void but rejected zero as a number, and they toyed with the concept of infinite but refused to allow infinity – numbers that are infinitely small and infinitely large – anywhere near the realm of numbers. This is the biggest failure in the Greek Mathematics, and it is the only thing that kept them from discovering Calculus.

 <blockquote>"The Greeks could not do this neat little mathematical trick. They didn’t have the concept of a limit because they didn’t believe in zero. The terms in the infinite series didn’t have a limit or a destination; they seemed to get smaller and smaller without any particular end in sight. As a result the Greeks couldn’t handle the infinite. They  pondered the concept of void but rejected zero as a number, and they toyed with the concept of infinite but refused to allow infinity – numbers that are infinitely small and infinitely large – anywhere near the realm of numbers. This is the biggest failure in the Greek Mathematics, and it is the only thing that kept them from discovering Calculus.

"</blockquote>Unlike Greece, India never had the fear of the infinite or of the void. Indeed, it embraced them. Indian mathematicians did more than simply accept zero. They transformed it changing its role from mere placeholder to number. The reincarnation was what gave zero its power. The roots of Indian mathematics are hidden by time. Our numbers (the current system) evolved from the symbols that the Indians used; by rights they should be called Indian numerals rather than Arabic ones. Unlike the Greeks the Indians did not see the squares in the square numbers or the areas of rectangles when they multiplied two different values. Instead, they saw the interplay of numerals—numbers stripped of their geometric significance. This was the birth of what we now know of as algebra.”

"</blockquote>Unlike Greece, India never had the fear of the infinite or of the void. Indeed, it embraced them. Indian mathematicians did more than simply accept zero. They transformed it changing its role from mere placeholder to number. The reincarnation was what gave zero its power. The roots of Indian mathematics are hidden by time. Our numbers (the current system) evolved from the symbols that the Indians used; by rights they should be called Indian numerals rather than Arabic ones. Unlike the Greeks the Indians did not see the squares in the square numbers or the areas of rectangles when they multiplied two different values. Instead, they saw the interplay of numerals—numbers stripped of their geometric significance. This was the birth of what we now know of as algebra.”

And finally, a quote by the famous mathematician John von Neumann:

And finally, a quote by the famous mathematician John von Neumann:

<blockquote>"''“The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.”''"</blockquote>

<blockquote>"''The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.”''"</blockquote>

== Further reading ==

== Further reading ==

Interested readers can find mathematical and historical  details in the following articles (and references therein):

Interested readers can find mathematical and historical  details in the following articles (and references therein):

1) K. V. Sarma, K. Ramasubramanian, M. D. Srinivas and M. S. Sriram, “Ganita-Yukti-Bhasha (Rationales in MathematicalAstronomy) of Jyeshthadeva”, Springer (2008).

1) K. V. Sarma, K. Ramasubramanian, M. D. Srinivas and M. S. Sriram, Ganita-Yukti-Bhasha (Rationales in MathematicalAstronomy) of Jyeshthadeva”, Springer (2008).

2) K. Ramasubramanian and M. D. Srinivas, “Studies in the History of Indian Mathematics”  Ed. by C. S. Seshadri, Hindustan Book Agency, New Delhi, pgs. 201 – 286 (2010).

2) K. Ramasubramanian and M. D. Srinivas, Studies in the History of Indian Mathematics”  Ed. by C. S. Seshadri, Hindustan Book Agency, New Delhi, pgs. 201 – 286 (2010).

 

 

3)T. Padmanabhan, “Dawn of Science : Calculus is  developed in Kerala”, Resonance pgs. 106 -115 (Feb 2012).  

3)T. Padmanabhan, Dawn of Science : Calculus is  developed in Kerala”, Resonance pgs. 106 -115 (Feb 2012).  

4) “Science and Technology in Ancient India”, Ed. Editorial Board, Vijnan Bharati, Mumbai (2006).

4) Science and Technology in Ancient India”, Ed. Editorial Board, Vijnan Bharati, Mumbai (2006).

== References ==

== References ==