Calculus: the real story

C. K. Raju

It is by now well known that the calculus and its infinite series originated in India across a thousand year period, starting from the 5th c. Aryabhata. It was needed for agriculture and overseas trade, the two key sources of Indian wealth.1 Indian monsoon-driven agriculture requires a good calendar, which requires good astronomy (hence precise trigonometric values) needed also for navigation.

Europeans then were backward in navigation and hence European governments offered large prizes for a solution to the navigational problem from the 16th to the 18th c. In the 16th c., Jesuits had turned their Cochin college into a centre for mass translation of Indian texts (on the 12th c. Toledo model of mass translation of Arabic texts). The content of these Indian texts started appearing in Europe in the later 16th c. and early 17th c. and was used to solve the latitude problem (Gregorian reform) and the problem of loxodromes (Mercator's chart). There is other circumstantial evidence, as in the works of Tycho Brahe (“Tychonic model”, identical to Nilakantha's), Christoph Clavius (trigonometric values, interpolated version of Indian values), “Julian” day numbers (ahargana), Kepler (Parameswaran's observations), Cavalieri, Fermat and Pascal (challenge problem, including probability), and finally Leibniz (“Leibniz” series) and Newton (sine series).

However, like Indian arithmetic earlier, Europeans did not understand Indian methods of summing infinite series using “non-Archimedean” arithmetic, and a different philosophy, now called zeroism. They tried to fit it into their religious beliefs about mathematics as “perfect” and error-free.

Newton thought, as in his theory of fluxions, that this could be done by making time metaphysical2 (“mathematical time which flows equably”). The error about time was the reason why his physics failed.3 This history has contemporary value. Correcting Newton's mistake in understanding calculus leads to a reformulation of physics, and, in particular, the theory of gravitation.4 This also corrects various problems of infinity that arise from the inadequacy of university calculus, or the Schwartz derivative for quantum field theory,5 general relativity,6 and electrodynamics,7, 8 as also the Lebesgue integral for probability,9 especially in quantum mechanics.

The other contemporary value is pedagogical. Calculus with add-on metaphysics makes math very difficult and was globalised during colonialism. Eliminating that redundant metaphysics in math makes math easy to teach.10, 11

About the author

Professor C. K. Raju holds an M.Sc in math and a PhD from the Indian Statistical Institute, Kolkata. He taught and researched in formal math (functional analysis) for several years. He was responsible for porting applications on the first Indian supercomputer Param, and that experience led him to abandon formal math. He has authored several books. In Cultural Foundations of Mathematics (Perason Longman, 2007) he proposed a new philosophy of math, called zeroism, and compiled evidence for the development of calculus in India and its transmission to Europe. In Time: Towards a Consistent Theory (Kluwer, 1994) he proposed a new physics, using functional differential equations. In the Eleven Pictures of Time (Sage, 2003) he proposed a new way to relate science and religion through time. He has developed and taught decolonised courses on math, and the history and philosophy of science. His shorter books include Is Science Western in Origin? (Multiversity 2010), Ending Academic Imperialism (Citizens International, 2011) and Euclid and Jesus (Multiversity, 2012). He has wide-ranging interests and is also Vice-President of the Indian Social Science Academy.

1C. K. Raju, Cultural Foundations of Mathematics: the nature of mathematical proof and the transmission of the calculus from India to Europe in the 16th c. CE, Pearson Longman, 2007.

2C. K. Raju, “Time: What is it that it can be measured”, Science & Education, 15(6) (2006) pp. 537–551. Draft available from

3C. K. Raju, Time: Towards a Consistent Theory, Kluwer Academic, Dordrecht, 1994.

4C. K. Raju, “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.

5C. K. Raju, “On the Square of x-n J. Phys. A: Math. Gen. 16 (1983) pp. 3739–53. Also, “Renormalisation, Extended Particles and Non-Locality.” Hadronic J. Suppl. 1 (1985) pp. 352–70.

6C. K. Raju, “Distributional matter tensors in relativity”. In: 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. arxiv: 0804.1998.

7S. Raju, and C. K. Raju, “Radiative Damping and Functional Differential Equations”, Mod. Phys. Lett. A, 26 (35) (2011) pp. 2627-2638. arxiv:0802.3390.

8For a non-technical account of the above problems with infinities, see C. K. Raju, Functional differential equations. 3: Radiative damping”, Physics Education (India), 30(3), July-Sep 2014, Article 8.

9C. K. Raju, “Probability in Ancient India”, chp. 37 in Handbook of the Philosophy of Science, vol 7. Philosophy of Statistics, ed, Dov M. Gabbay, Paul Thagard and John Woods. Elsevier, 2011, pp. 1175-1196.

10“Teaching mathematics with a different philosophy. Part 1: Formal mathematics as biased metaphysics.” Science and Culture 77 (7-8) (2011) pp. 274–279., arxiv:1312.2099.

11Teaching mathematics with a different philosophy. Part 2: Calculus without limits”, Science and Culture 77 (7-8) (2011) pp. 280–85. arxiv:1312.2100.