On Kim Plofker's Mathematics in India

That the concept of zero and the now widely used, indispensable decimal numeric system (imprecisely called Arabic numerals) came from India is old news -- a piece of trivia circulated often enough to have little or no surprise value. There is more, much more, to ancient Indian mathematics than these iconic contributions. Though I am trained in the quantitative sciences, I learned all of it – linear algebra, the theory of probability, and combinatorics – from the classrooms and labs of US universities, and very little from India. So it is only appropriate – though also ironic in another way – that the knowledge of ancient India should be made available to me through the painstaking research of a Western historian of mathematics, Kim Plofker.

Plofker has plowed through a range of Indian texts, beginning from the Vedic period – that’s three thousand years ago – to the eighteenth century to produce Mathematics in India. Hers is an achievement as much of linguistics, translation and history as of mathematics. The book sits on my desk, and there it shall be for some years, gathering dust, as I dip in only occasionally. It’s not the type that can be easily read from cover to cover – no book of mathematics is – but Plofker tries her best to make terms and concepts accessible.

Luckily, David Mumford, winner of the Fields Medal – the most prestigious prize in mathematics – has written a six-page essay on the book, which captures, concisely, the major achievements of Indian mathematics. There is the use of Pythagoras’ famous theorem in the construction of pillars, some centuries before the Greek philosopher is said to have postulated it in fifth century BC; Pāṇini’s rules of Sanskrit grammar and recursion, which “without exaggeration…anticipated the basic ideas of modern computer science”; Pingala, whose study of Sanskrit verses led to the binary notation and the development of Pascal’s famous triangle, useful in the calculation of binomial coefficients (which, coincidentally, is what I am teaching now); and Madhava of Sangamagramma (circa 14th century), the genius of the Kerala School, who contributed along with others, to "the discoveries of the power series expansions of arctangent, sine, and cosine" (a text on this in Malayalam has survived).

But what interested me most was the applied orientation of Indian mathematics. Like the ancient Mexicans, who designed sophisticated calendars (and who also used the concept of zero, probably earlier than the Indians did, though such rat races about who did what first are useless beyond a point), the mathematicians of India were spurred by the questions of astronomy. But the applied orientation goes even further than that. Mumford writes:

"It is important to recognize two essential differences here between the Indian approach and that of the Greeks. First of all, whereas Eudoxus, Euclid, and many other Greek mathematicians created pure mathematics, devoid of any actual numbers and based especially on their invention of indirect reductio ad absurdum arguments, the Indians were primarily applied mathematicians focused on finding algorithms for astronomical predictions and philosophically predisposed to reject indirect arguments. In fact, Buddhists and Jains created what is now called Belnap’s four-valued logic claiming that assertions can be true, false, neither, or both. The Indian mathematics tradition consistently looked for constructive arguments and justifications and numerical algorithms [Mumford's italics]. So whereas Euclid’s Elements was embraced by Islamic mathematicians and by the Chinese when Matteo Ricci translated it in 1607, it simply didn’t fit with the Indian way of viewing math. In fact, there is no evidence that it reached India before the eighteenth century."

This is particularly pleasing to me, because my own research, coincidentally, is about using mathematics in computationally feasible ways for various applications, rather than basking in the beauty and abstraction of theorems. (I will admit, however, that I was once obsessed with writing a paper that had only theorems and proofs in it; I did manage something, but only by borrowing significantly from the work of others; and still, months of edits await the paper before it can find its way to print.)

What might be an example of this “applied” orientation? Mumford writes of the “the discovery of the formula for the area and volume of the sphere by Bhaskara II” (circa 12th century):

“Essentially, he [Bhaskara II] rediscovered the derivation found in Archimedes’ On the Sphere and the Cylinder I. That is, he sliced the surface of the sphere by equally spaced lines of latitude and, using this, reduced the calculation of the area to the integral of sine. Now, he knew that cosine differences were sines but, startlingly, he integrates sine by summing his tables! He seems well aware that this is approximate and that a limiting argument is needed but this is implicit in his work. My belief is that, given his applied orientation, this was the more convincing argument. In any case, the argument using the discrete fundamental theorem of calculus is given a few centuries later by the Kerala school, where one also finds explicit statements on the need for a limiting process, like: “The greater the number [of subdivisions of an arc], the more accurate the circumference [given by the length of the inscribed polygon]” and “Here the arc segment has to be imagined to be as small as one wants. . . [but] since one has to explain [it] in a certain [definite] way, [I] have said [so far] that a quadrant has twenty-four chords.”

One of the delightful bits in the excerpt is the choice of “twenty-four chords” – it implies a casual, intuitive approach, yet the same is preceded by knowledge of what needs to be done to achieve exactness. That’s the essence of a “heuristic” or an approximation, which engineers use all the time.

Finally, the closing paragraph of Mumford’s essay:

It is high time that the full story of Indian mathematics from Vedic times through 1600 became generally known. I am not minimizing the genius of the Greeks and their wonderful invention of pure mathematics, but other peoples have been doing math in different ways, and they have often attained the same goals independently. Rigorous mathematics in the Greek style should not be seen as the only way to gain mathematical knowledge. In India, where concrete applications were never far from theory, justifications were more informal and mostly verbal rather than written. One should also recall that the European Enlightenment was an orgy of correct and important but semirigorous math in which Greek ideals were forgotten. The recent episodes with deep mathematics flowing from quantum field and string theory teach us the same lesson: that the muse of mathematics can be wooed in many different ways and her secrets teased out of her. And so they were in India: read this book to learn more of this wonderful story!