### The Kerala school vs calculus teaching today

Wednesday, August 16th, 2017As pointed out in the previous post, calculus started not with the Kerala school but with the dalit Aryabhata, of Patna, in the 5^{th} c. The Aryabhata school in Kerala acknowledged him as their master, and Nilakantha somasutvan wrote a commentary (*bhashya*) on the *Aryabhatiya*. To repeat, the Indian calculus was a pan-India development, and NOT a product of the Kerala school alone.

In particular, though infinite series are an easily recognized aspect of calculus, the emphasis on them is misleading, especially for the purpose of teaching Indian calculus in universities today. The above quote in the earlier blog post continues:

“Further, if we teach the Indian calculus today in universities (as I do) the focus will be only on what Aryabhata did. So, the plagiarists’ false understanding of history also prevents us from reforming calculus teaching today. Neither of the plagiarists understands the calculus well enough to teach it.”

Another quote, in the earlier blog post, closely related to this is the following.

“Like all plagiarists, Joseph and Almeida made horrible blunders while restating my thesis (stated in the Hawai’i paper,

^{1}that the calculus developed in India with a different epistemology). For example, in one of their papers inRace and Classthey asserted “the Kerala mathematicians used the floating point numbers”, used in modern-day computers. Ha! Ha! Ha! What a joke! Only complete ignoramuses like the two plagiarists could have thus misunderstood my thesis about floating point numbers stated in my Hawai’i paper, which was part of a course on C-programming that I was then teaching as Professor of computer science.”

No doubt this was a blunder, but why was this a *horrible* blunder? Because a different number system was at the heart of the Indian method of summing infinite series, but Europeans did not understand it (and did not understand how to sum infinite series), and Western historians like Plofker do not understand it till today.

This lack of understanding of Indian calculus by Europeans had serious consequences: it led to the failure of Newtonian physics. I have analysed Newton’s error in understanding the Indian calculus, the consequent conceptual error in the notion of time in his physics, responsible for the failure of his physics, and proposed a corrected theory of gravitation.^{2} (An expository account of the new theory of gravitation is also available.^{3}) The point about floating point numbers used to do calculus on computers is further explained in the course of this analysis, as is the point about *avyakt ganit*. Floats are a finite set, *smaller* than formal reals, with no recognizable algebraic structure, because the associative law fails even for addition; *avyakt ganit* results in a “non-Archimedean” ordered field *larger* than formal reals. Calculus can be done with number systems smaller or larger than formal reals, university calculus as taught today is not the only way to do calculus as some foolish historians assume.

The matter is simple, the Indian use of *avyakt* numbers very naturally led to *avyakt* fractions which are today called rational functions and correspond to the use of so-called non-Archimedean arithmetic. (more…)