Indian mathematics and the Kerala School

India's greatest contribution to mathematics is zero - this has been accepted universally. But is it the only major contribution? If you think so, think again - your view is utterly wrong! Well, to prove that Indian contribution to mathematics and astronomy is infinitely more than what we all think, I've done some background work and presented some of the most astonishing findings here. But please don't be lead to believe that this is some figment of my imagination! I have done a proper literature research and this article features the excerpts from the findings of some recognised historians and mathematicians, who are, mostly, Europeans. The main objective of this article is to make known certain facts that, I am ashamed to say, almost all the Indians aren't aware of. While we have been reading and admiring the mathematical and scientific inventions/discoveries of the Europeans, especially the Greeks, developments commensurate if not greater, have happened in our own backyard.

It is said that the winners always write the history...well, sometimes re-write the history. We Indians have been under foreign occupation for a millennium, which has lead to a pathetic state of affairs. The Muslim invaders and the Europeans carried away not only immensely valuable things from India, but also the know-how. Now, we believe that great inventions and discoveries have been made elsewhere, when the truth of the matter is that our ancestors were the ones to have accomplished them!

To a typical historian of mathematics today, if there is one certainty, it is that Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1716) were the first to ‘invent’ a generalised system of infinitesimal calculus, an essential prelude to modern mathematics. Well, they wish! At least two hundred years earlier, the astronomer-mathematicians of Kerala, notably Madhava of Sangamagrama and his disciples, had discovered elements of that calculus, the forerunners of modern techniques used in mathematical analysis. Given the existence of a corridor of communication between Kerala and Europe, especially from the sixteenth century onwards, and the crucial importance of calculus in the growth of modern mathematics, one would have expected that the possibility of the transmission of the Kerala mathematics westwards would be
high on the agenda for historical investigation. That such an investigation has not yet been carried out may reflect the strength and the pervasive nature of Eurocentrism in the history of science.

It is not just calculus alone...Indian scholars made vast contributions to the field of mathematical astronomy and as a result contributed mightily to the developments of arithmetic, algebra, trigonometry and secondarily geometry (although this topic was well developed by the Greeks) and combinatorics. Perhaps most remarkable were developments in the fields of infinite series expansions of trigonometric expressions and differential calculus.

Surpassing all these achievements however was the development of decimal numeration and the place value system, which without doubt stand together as the most remarkable developments in the history of mathematics, and possibly one of the foremost developments in the history of humankind. The decimal place value system allowed the subject of mathematics to be developed in ways that simply would not have been possible otherwise. It also allowed numbers to be used more extensively and by vastly more people than ever before.

The earliest origins of Indian mathematics have been dated to around 3000 BC and this seems a sensible point at which to commence any discussion, while work of a significant nature was still being carried out in the south of India in the 16th century, following which there was an eventual decline. It is hence a vast time scale of almost 5000 years, and indeed it may be greater than that, the estimation of 3000 BC is a slightly crude approximation, and there remains much controversy with regards to the dating of many works prior to 400 AD. It is also worth pointing out that this lack of certainty has allowed several unscrupulous scholars to pick dates of choice for certain Indian discoveries so as to justify suggestions for Greek, Arab or other influences.

We can accurately claim that Aryabhata was born in 476 AD. He was 23 years old when he wrote his most significant mathematical work the Aryabhatiya (or Arya Bhateeya) in 499 AD. He was a member of the Kusuma Pura School, but is thought to have been a native of Kerala (in the extreme south of India), although unsurprisingly there is some debate. It is a concise astronomical treatise of 118 verses written in a poetic form, of which 33 verses are concerned with mathematical rules. It is important here to point out that no proofs are contained with his rules, and this is perhaps a primary reason for the neglect by western scholars. As Indian mathematics is (generally) devoid of proof it is not considered 'true' mathematics in its purest sense. However, adopting this stance is to deny the very origin of remarkable discoveries in mathematics, which may well have been the aim of Eurocentric scholars, as it allowed them to neglect the importance of Indian works in favour of European works. In the mathematical verses of the Aryabhatiya the following topics are covered:

Arithmetic: Method of inversion, various arithmetical operators, including the cube and cube root are though to have originated in Aryabhata's work. Aryabhata can also reliably be attributed with credit for using the relatively 'new' functions of squaring and square rooting.

Algebra: Formulae for finding the sum of several types of series; rules for finding the number of terms of an arithmetical progression; rules for solving examples on interest - which led to the quadratic equation; it is clear that Aryabhata knew the solution of a quadratic equation.

Trigonometry: Tables of sines, not copied from Greek works. The Aryabhatiya is the first historical work of the dated type that uses some of these (trigonometric) functions and contains a table of sines.
Spherical trigonometry: Some incorrect.

Geometry:Area of a triangle, similar triangles, volume rules.

The work of Aryabhata also affords a proof that the decimal system was well in vogue. Of the mathematics contained within the Aryabhatiya the most remarkable is an approximation for p, which is surprisingly accurate. The value given is: p = 3.1416. With little doubt this is the most accurate approximation that had been given up to this point in the history of mathematics. He found it from the circle with circumference 62832 and diameter 20000. Critics have tried to suggest that this approximation is of Greek origin. However with confidence it can be argued that the Greeks only used p = sqrt(10) and p = 22/7 and that no other values can be found in Greek texts.

Aryabhata's work on astronomy was also pioneering, and was far less tinged with a mythological flavour. Among many theories he was the first to suggest that diurnal motion of the 'heavens' is due to rotation of the earth about its axis, which is incredibly insightful (unsurprisingly he was criticised for this).

In the period between 14th and 16th centuries A.D., the south-western tip of India escaped the majority of the political upheaval, which engulfed the rest of the country, allowing a generally peaceful existence to continue. Thus the pursuit of scientific development was able to continue 'uninterrupted'. It has only recently come to light that mathematics (and astronomy) continued to flourish in this area for several hundred years. Kerala mathematics was strongly influenced by astronomy, but this led to the derivation of mathematical results of huge importance. As a result of the recency of these discoveries it is quite probable that there are still further discoveries of 'Kerala mathematics' to be made, and a full analysis has yet to be carried out. However several findings have already been made that show several major concepts of renaissance European mathematics were first developed in India. European scholars may have had first hand knowledge of some Kerala mathematics, as the area was a focal point for trading with many parts of the world, including Europe. There is also some evidence of a transfer of technology between Europe and Kerala. It is true that Kerala was in continuous contact with China, Arabia, and at the turn of the 16th century, Europe, thus transmission might well have been possible. However the current theory is that Keralese calculus remained localised until its discovery by Charles Whish in the late 19th century. There is no evidence of direct transmission by way of relevant manuscripts but there is evidence of methodological similarities, communication routes and a suitable chronology for transmission. Events also suggest it is quite possible that Jesuits (Christian missionaries) in Kerala were 'encouraged' to acquire mathematical knowledge while there.

It is important to bear in mind that the conceptual and epistemological bases of Madhava’s (the most famous Kerala mathematician) mathematics had little affinity with those of early Greek mathematics. Instead, they were founded on the principles elaborated in Aryabhatiya.

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