## The Kerala school vs calculus teaching today

As 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. This arithmetic is useful also in situations (such as shock waves) where current calculus (whether based on formal reals, or the Schwartz theory) fails. I have explained this in several places,^{4} including in expository lectures in 2010 at the math department of the Universiti Sains Malaysia (see the second lecture posted here, especially the section “Why R?” et seq. Ignore the typos, e.g. with the delta function, it should be “un-physical”, not “physical”.) These lectures preceded my teaching of the calculus in USM, to 4 groups, including one group of post-graudate students, along these lines, as partly reported in two papers.^{5}^{, }^{6}

This is undoubtedly a better way to teach calculus than using the metaphysics of formal real numbers because practical value derives from methods of calculation, not metaphysics. Aryabhata’s method of numerically solving differential equations is what results in practical value even today. That is how Newtonian physics managed to deliver practical value, even though neither he nor other Europeans understood how to sum infinite series. (The metaphysics of Dedekind’s formal reals, or equivalence classes of Cauchy sequences, needed Cantorian set theory, which was full of paradoxes, supposedly eliminated by formal set theory, completed only in the 20^{th} c.)

Unfortunately, we have almost no proper historians of science, and immense trust in the duplicitous West, and keep relying on charlatans and Western apologists for our history. And feel so proud when they say something nice, and in the process, keep damaging our heritage further. Indians should stop boasting about a history they are unwilling to invest in, for they have only second and third-rate knowledge, either from those unfamiliar with the primary sources, or those who cannot interpret them correctly because they lack knowledge of science.

### References

1 “Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the YuktiBhâsâ”, *Philosophy East and West*, **51**(3), 2001, pp. 325–362. http://ckraju.net/papers/Hawaii.pdf.

2 “Retarded gravitation theory” in: Waldyr Rodrigues Jr, Richard Kerner, Gentil O. Pires, and Carlos Pinheiro (ed.), Sixth International School on Field Theory and Gravitation, American Institute of Physics, New York, 2012, pp. 260-276. http://ckraju.net/papers/retarded_gravitation_theory-rio.pdf.

3 “Functional Differential Equations. 4: Retarded gravitation”, Physics Education (India) 31(2) April-June, 2015, http://www.physedu.in/uploads/publication/19/309/1-Functional-differential-equations-4-Retarded-gravitation-(2).pdf.

4 “Distributional matter tensors in relativity”, *Proceedings of the Fifth Marcel Grossman meeting on General Relativity*, D. Blair and M. J. Buckingham (ed), R. Ruffini (series ed.), World Scientific, Singapore, 1989, pp. 421–23. See the slightly revised version in arxiv: 0804.1998, where I make the point about non-Archimedean arithmetic as a replacement for the non-Standard analysis which I had used earlier in 1980’s. .

5 “Teaching mathematics with a different philosophy. Part 1: Formal mathematics as biased metaphysics.” *Science and Culture* **77**** **(7-8) (2011) pp. 274–279. http://www.scienceandculture-isna.org/July-aug-2011/03%20C%20K%20Raju.pdf, arxiv:1312.2099.

6 “Teaching mathematics with a different philosophy. Part 2: Calculus without limits”, *Science and Culture* **77 **(7-8) (2011) pp. 280–85. http://www.scienceandculture-isna.org/July-aug-2011/04%20C%20K%20Raju2.pdf. arxiv:1312.2100.