Abstract: Calculus originated with Aryabhata 5th c. NOT Madhava or the Kerala school which refined it a thousand years later. Related video excerpt of Indica talk, “How and why Europeans stole the calculus in the 16th c.” )
It is nearly two centuries since Whish wrote his seminal paper of 1832/1835 bringing to the notice of Britishers the occurrence of infinite series in India.[1] However, the West has yet to acknowledge that the calculus originated in India (and that they stole it). This is because the West did not steal only money during colonialism, but from pre-colonial times it also stole knowledge[2] and glorified itself by writing systematically false history down the ages.[3]
Colonial education taught the colonised to ape the West, hence the colonised thought likewise. Now, finally some colonised have come around to accept that something like the calculus existed in India. But because the West never understood the Indian calculus properly, the credit goes to the texts that Whish described, namely the texts of the Kerala school and Madhava on infinite series.
The following excerpt from my Indica talk on “How and why Europeans stole the calculus in the 16th century” explains (again!) why this is wrong. The calculus was actually invented a thousand years earlier in the fifth century by Aryabhata. The related video is posted at https://youtu.be/_eEkHKLKnaA.
The key point to understand is that infinite series though a part of calculus, do not by themselves constitute the calculus, if our intention is to teach calculus. It is true that the key epistemic difficulties that Europeans encountered with the Indian calculus related to infinite series, just as a key epistemic difficulty that Europeans encountered with Indian arithmetic related to zero. But zero is not the essence of Indian arithmetic, that is a Eurocentric position. To teach Indian arithmetic we would not just teach zero but would emphasize the efficiency of Indian arithmetic “algorithms” based on the place value system. Likewise we have to consider how we would teach calculus, and not merely what difficulties Europeans had with it and with so many other aspects of elementary math.
So did Europeans steal calculus from India?
The evidence for theft of calculus been summarised many times, starting from my Hawaii paper,[4] and is explained in detail in my book on this.[5] But let us look at the issues once more.
- Q1. Was it calculus?
- Q2. Did Europeans steal?
Obviously, we first need to establish that what developed in India was actually the calculus. There are some serious doubts here.
Q1. If what developed in India was the calculus where are the integral and derivative signs?
Today most people learn calculus by learning formulae doing integrals and derivatives (ONLY of elementary functions: see pre-test question paper Q7d). Hence, many people wrongly identify calculus with symbolic computation of derivatives and integrals. Especially the foolish colonised mind taught to ape the “superior” West.
But there were no integrals or derivatives in Indian tradition. So, how was it calculus?
Many historians of science hence thought that what developed in India was not the calculus. For example, the historian of science A. K. Bag’s offered the following opinion in 2003 while
REJECTING my paper on transmission of calculus from India to Europe for Indian Journal of History of Science.
“I personally feel that…the question of the transmission [of calculus] from India to Europe is basically a hypothetical issue…”
Why? Since he thought what developed in India was not calculus, since it had no integrals, derivatives. (Indeed, I was the first to actually call the work of the “Kerala school” by the name “calculus” in my project for the same Indian National Science Academy.)
Aryabhaṭa vs Madhava
Today, Indian calculus (whether or not it was calculus) is widely attributed to Madhava and “Kerala school”, because of the 1832/35 paper by Whish This was also the WRONG belief with which I started my research in 1997. My project was titled “Madhava and the origin of calculus”. But I corrected it first when I wrote my book and more explicitly after I started teaching calculus without limits in 2009.[6]
Calculus due to Aryabhata
Madhava used 11th/12th order interpolation from infinite series to give table of 24 sine values precise to 3rd sexagesimal minute (tatpara). In contrast, Aryabhata’s sine value 1000 years earlier were precise only to first sexagesimal minute (kalā).’
However, the key point is this: Madhava gives sine values, Aryabhaṭa’s “sine table” has only sine DIFFERENCES. That resolves the first issue regarding calculus in India: Indians used finite differences NOT derivatives.
Karl Marx struggled to understand the stolen calculus in the 19th century (before the invention of real numbers and limits by Dedekind)[7]. He correctly referred to Newton’s fluxions as mystical[8] (meaning incomprehensible), which is why they have been totally abandoned today. On the other hand, finite differences are straightforward and easy to understand, though Marx wrongly attributed them to the D’Alembert.
Though, today, derivatives are defined using limits and real numbers one should not assume automatically that derivatives “better” or “superior” to finite differences (an attitude emphasized in current courses in numerical anaysis) The West has always boasted of its superiority based on a variety of superstitions and exceptionally silly beliefs (like skin color). The idea of a superior formal mathematics is another one of these silly church superstitions.[9]
How were Aryabhata’s sine differences derived?
But finite differences by themselves are not calculus. But let us also take into account the way Aryabhata derived them. He did so by means of a recurrence relation, NOT an algebraic equation. The key point: this is equivalent to what is today wrongly called “Euler” method of numerically solving ordinary differential equations/ (Nīlakaṇṭha corrects Aryabhata’s formula giving exactly Euler’s method.) Finite differences plus the Euler method almost constitute calculus. (There are some finer refinements[10] such as Brahmagupta’s non-Archimedean arithmetic and the philosophy of zeroism, but we won’t go into those issues here.)
Needless to say, Euler was familiar with Indian math texts,hence solved Fermat’s challenge problem, which was a solved exercise in Bhaskara II. Euler wrote an article on the Indian calendar in 1700, naturally had to consult Indian math and astronomy texts for that purpose, and Bhaskara’s work and the commentaries on it were very widely distributed. But it is the habit of the West to lie and to pile on the lies, and not acknowledge its sources to glorify itself.
That is, Aryabhata effectively numerically solved a differential equation
to derive his sine values. Solving differential equations is the heart of calculus and the essence of ALL problems of Newtonian physics.
Corollary: NO NEED FOR ∫ sign since solution of \(y’ = f(x)\) is the indefinite integral \(∫ˣf(t)dt\)
This way of doing calculus vastly superior for real-life practical applications since no need to restrict \(f(x)\) to be an elementary function.
E.g. first serious science experiment in school, the simple pendulum, involves non-elementary Jacobian elliptic functions. The usual wrong formula \(T= 2π \sqrt{\frac{l}{g}}\) that is taught to avoid non-elementary functions is not compatible with observations. See my elder son’s school project.
Therefore, this is the right way to teach calculus. (It also makes calculus very easy.) Hence, this is key aspect of how I teach calculus without limits.
Brahmagupta’s critique: quadratic interpolation
Brahmagupta critiqued Aryabhata for increasing the number of sine values/differences to 24. His argument: earlier traditional 7 values 15° apart were adequate for the same precision (to the minute) with quadratic interpolation.
This idea was carried forward by Vateshwar who used quadratic interpolation (“Stirling’s formula”) + a table of 96 values. This achieved precision to the 2nd sexagesimal minute (vikala). This was further, carried forward by Madhava and the later Aryabahta school in Kerala who used 11th/12th order interpolation to achieve precision to the third sexagesimal minute(tatpara). So the work of the “Kerala school” was the culmination of a thousand-year old continuous effort, not its beginning.
Conclusion
Aryabhata (and Brahmagupta) invented calculus, the Aryabhata school in Kerala refined it.
PS
There is a further issue that Aryabhata was a Dalit from Bihar,[11] while people like Nilakantha of the Kerala school were the highest caste Namboodiri Brahmins. To my mind this is splendid testimony of regional integration and the fact that caste was not oppressive prior to colonialism. But won’t go into it further.
References
1Whish, Charles M, ‘On the Hindu Quadrature of the Circle and the Infinite Series of the Proportion of the Circumference to the Diameter Exhibited in the Four Shastras, the Tantrasamgraham, Yukti-Bhasa, Carana Padhati and Sadratnamala”’, Trans. R. Asiatic Soc. Gr. Britain and Ireland 3 (1835): 509–23.
2C. K. Raju, ‘Precolonial Appropriations of Indian Ganita: Epistemic Issues’ (International round table on Indology, IIAS, Shimla, 2020), http://ckraju.net/papers/ckr-indology-abstract.pdf.
3C. K. Raju, ‘“Euclid” Must Fall: The “Pythagorean” “Theorem” and The Rant Of Racist and Civilizational Superiority — Part 1’, Arụmarụka: Journal of Conversational Thinking 1, no. 1 (2021): 127–55, https://doi.org/10.4314/ajct.v1i1.6; C. K. Raju, Is Science Western in Origin?, Dissenting Knowledges Pamphlet Series (Multiversity, 2009).
4C. K. Raju, ‘Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhāṣā’, Philosophy East and West 51, no. 3 (2001): 325–62, http://ckraju.net/papers/Hawaii.pdf. For the relevant excerpt on history of calculus, see http://ckraju.net/papers/HAWAIIpp26_32.pdf.
5C. K. Raju, Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of Calculus from India to Europe in the 16th c, CE (Pearson Longman, 2007).
6C. K. Raju, ‘Teaching Mathematics with a Different Philosophy. 1: Formal Mathematics as Biased Metaphysics’, Science and Culture 77, no. 7–8 (2011): 274–79; C. K. Raju, ‘Teaching Mathematics with a Different Philosophy. 2: Calculus without Limits’, Science and Culture 7, no. 7–8 (2011): 280–85; C. K. Raju, ‘Calculus without Limits: Report of an Experiment’, in 2nd People’s Education Congress (TIFR, Mumbai: Homi Bhabha Centre for Science Education, 2009). http://ckraju.net/papers/calculus-without-limits-paper-2pce.pdf.
7C. K. Raju, ‘Marx and Mathematics-1 Marx and the Calculus’, Frontier Weekly, 28 August 2020, https://www.frontierweekly.com/views/aug-20/28-8-20-Marx%20and%20mathematics-1.html.
8C. K. Raju, ‘Marx and Mathematics. 4: The Epistemic Test’, Frontier Weekly, 8 September 2020, https://www.frontierweekly.com/views/sep-20/8-9-20-Marx%20and%20mathematics-4.html.
9To see why this boast of a superior formal math is a mere church superstition, see, presentation of lecture at Sol Plaatje U, 1 June 2023, http://ckraju.net/papers/presentations/points-sol-plaatje.html. Or see, C. K. Raju, ‘The Church Origins of (Axiomatic) Math’, Medium, 21 June 2022, https://medium.com/@c_k_raju/the-church-origins-of-axiomatic-math-e08036dbe29d.
10C. K. Raju, ‘California, Indian Calculus and the Technology Race. 1: The Indian Origin of Calculus and Its Transmission to Europe’, Boloji.Com, 11 December 2021, https://www.boloji.com/articles/52924/california-indian-calculus; C. K. Raju, ‘California, Indian Calculus and the Technology Race. 2: Don’t Cancel the Calculus, Make It Easy!’, Boloji.Com, 24 December 2021, https://www.boloji.com/articles/52950/california-indian-calculus-and.
11C. K. Raju, ‘Aryabhata Dalit: His Philosophy of Ganita and Its Contemporary Applications”’, in Theory and Praxis: Reflections on the Colonization of Knowledge, ed. Murzban Jal and Jyoti Bawane (Routledge, London, 2020), 139–52, http://ckraju.net/papers/Aryabhata-philosophy-of-ganita-paper-2r.pdf.
