David Mumford reviews Kim Plofker’s Mathematics in India:

Did you know that Vedic priests were using the so-called Pythagorean theorem to construct their fire altars in 800 BCE?; that the differential equation for the sine function, in finite difference form, was described by Indian mathematician-astronomers in the fifth century CE?; and that “Gregory’s” series … was proven using the power series for arctangent and, with ingenious summation methods, used to accurately compute π in southwest India in the fourteenth century? If any of this surprises you, Plofker’s book is for you.

Her book fills a huge gap: a detailed, eminently readable, scholarly survey of the full scope of Indian mathematics and astronomy (the two were inseparable in India) from their Vedic beginnings to 1800.
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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. 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.

Secondly, this scholarly work was mostly carried out by Brahmins who had been trained since a very early age to memorize both sacred and secular Sanskrit verses. Thus they put their mathematics not in extended treatises on parchment as was done in Alexandria but in very compact (and cryptic) Sanskrit verses meant to be memorized by their students. What happened when they needed to pass on their sine tables to future generations? They composed verses of sine differences, arguably because these were much more compact than the sines themselves, hence easier to set to verse and memorize.
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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 semi-rigorous 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!