Marx and mathematics-2:
“Discovery” of calculus

C. K. Raju

How could Newton and Leibniz have discovered the calculus when there was no understanding of it in Europe, not in their time, and certainly not even in Marx’s time, even unto the end of the 19th c.? Western historians have never asked this question and never supplied an explanation, which is actually quite simple.

The long tradition of false Western history
As a first step, we need to understand that the West has traditionally produced an enormous amount of systematically false and chauvinistic history which obscures our understanding of the past. Why? Because, as the church discovered long ago, falsehoods are a major source of political power.

Many people understand church falsehoods about the future: about doomsday, and how the Christian god will extract eternal revenge from non-Christians and reward Christians after death. And many will, today, laugh at such falsehoods. However, few have realized how the church told falsehoods about the past, falsehoods that are not prima facie lacking in credibility and are still being believed.

This process of systematically false church history started since the 5th c., when the second church historian Orosius wrote his History Against Pagans. The book curiously begins with a discussion of the psychology of dogs: how they can be easily trained to obey the will of the master through rewards and punishments. With human beings, mere stories of rewards and punishments are enough! And when the rewards and punishments are only imagined, they can just as well be imagined in the past as in the future. Accordingly, Orosius used false history to glorify Christians and denigrate (and frighten) non-Christians with concocted stories about the rewards and punishments of the Christian god in this very life, in the past. The Orosian model of history persisted until the Crusades, by which time the pagans were physically liquidated, and the church acquired a new enemy in Muslims.

This church technique of using systematically false history went ballistic during the Crusades, when the origin of all scientific knowledge (from around the world) in Arabic books, captured at Toledo, was indiscriminately attributed to the early Greeks (the sole “friends of Christians” as the first church historian Eusebius called them; the later Greeks were persecuted by the church). By attribution to “Greeks”, all that knowledge in Arabic books was appropriated as a theologically-correct Christian inheritance. It was hence declared as fit to be translated to Latin, as was done since 1125, even though the immediate source of that knowledge was from the books of the religious enemy, the sort of books the church had earlier burnt for centuries.1

Even today, stories of Greeks, real or imaginary, are a matter of great pride in the West. Those Arabic books incorporated world knowledge including Indian knowledge which was all indiscriminately attributed to Greeks on faith, though the fact is that Greek knowledge of even elementary arithmetic was rudimentary, compared to say, the Egyptians a thousand years before them.2 Therefore, the Greeks could not have done any serious science. But for the believers who never cross-checked any of those claims, it is the faith that counts, not the facts. Mind you, this is faith in false history, faith about the past, not faith in a particular afterlife.

Later, when the “New World” was “discovered”, the church created an even more novel understanding of “discovery” as a method of appropriating land and knowledge. A genocidal church doctrine, the doctrine of Christian discovery, was used to twist the term “discovery”, and give it a technical meaning in Christian theology. I have earlier explained3 this theological meaning of the term “discovery” as commonly used in the utterly absurd claims of Western-propped history that “Vasco da Gama discovered India”, or “Columbus discovered America” etc. Only Christians (whether Roman Catholic or Protestants) can be discoverers, as explained at length by the US Supreme court, in a judgment which still stands (cited in the above article).

What applies to appropriation of land also applies to appropriation of more intangible knowledge. That is, Christians are to be deemed “discoverers” hence owners (of land, knowledge) notwithstanding the prior knowledge among “natives” or prior occupancy of the land by them. It is the first Christian who came across that land or knowledge who counts as the “discoverer”. This doctrine was used to attribute all “native” knowledge to Christians as discoverers (as in “Newton discovered the calculus”). This enabled the utterly chauvinistic history that all science was first discovered by Christians after their sole “friends” the early Greeks.

Though this false history was initially written not by victors but by losers—those who persistently lost the military conflict in the Crusades for centuries—those church lies had political power. Hence, this church technique of building power on lies (not only about the future afterlife, but also about the past) was later picked up by racists, to glorify Whites (Greeks were now categorised as Whites), and denigrate Blacks. The claim was now that all science was the work of Whites. This was used to justify an extremely vicious form of slavery, and its economic advantage to Whites, against the natural repugnance that people have for it.

These racists included very many key Western intellectuals of Marx’s time, such as Hume, Kant,4 and Hegel,5 and numerous others who uncritically believed in Western intellectual superiority based on this false history. It was because of this prevailing atmosphere of racism amongst Western intellectuals in Marx’s time that Marx never doubted the innately strange claim that Newton and Leibniz had “discovered” the calculus without understanding it.

We should not under-estimate the legacy of this racism on history: many later-day historians, including J. D. Bernal and Joseph Needham assumed the validity of this history (about “Greeks”, and “Euclid” etc.) without bothering to cross check it. Many Marxists today believe it must be true because J. D. Bernal wrote it, even if his own son, Martin Bernal, did not believe in that false history of the Greeks. It is all about whom one trusts, not the facts.

As if this religious, racist, and “philosophical” context were not enough, these lies about “superior” Western intellectual achievements were very important also to colonisers, to amplify their power and offset their military weakness far from home. Hence, colonial historians again modified the categories slightly to glorify the West (Greeks were now regarded as culturally part of West) and denigrate the non-West. As I have argued, this false history is still the key basis of continuing racism today.6 It is also the basis of the continuing mental subjugation of the colonised to the ex-coloniser, and their squalid faith in Western authority (e.g. Wikipedia), induced by colonial education, exactly as Orosius had visualised for dogs.

These false beliefs include the belief that a particular philosophy of mathematics and calculus, as practised today in the West, is universal or “superior”, and the only right way to understand the calculus. It is a blind belief because the vast majority of those who hold this belief would be unable to meet my Cape Town Challenge: prove 1+1=2 in formal real number (Dedekind cuts etc.) from first principles, in the manner of Russell and Whitehead’s proof of 1+1=2 in cardinals, without assuming any result of axiomatic set theory. Naturally, they would not be able to explain why this is extraordinarily complex way of doing 1+1=2, which has nil practical value, as Hardy boasted, is “superior”.

Indeed, the huge political power of this false history is demonstrated by the fact that the entire Indian education system was changed, and (church) education system was brought to India by Macaulay with the aid of a false history.7 Not only that, those educated in this system never themselves checked that history for two centuries, but also object to anyone else cross-checking it today, since the principal thing they learnt is that only Westerners are “reliable”, as also assumed by Wikipedia. But cross-check we must, irrespective of the damage it does to Western idols.

The false history of the calculus
Let us now apply this general background to the particular case of Marx and mathematics. The relevant false history (which Marx gullibly accepted) is that of the calculus: the false claim that it was “discovered” by Newton and Leibniz. Newton, himself,8 was more circumspect. He was well aware that the calculus had been circulating around in Europe from long before him. Accordingly, he claimed credit only for (a) rigor (his fluxions vs Cavalieri’s indivisibles) and (b) the infinite “Taylor” series expansion for the very specific case of the sine function. But in the ordinary (as distinct from the twisted Christian theological) meaning of the term “discovery”, even this minimal claim was absolutely false.

Let us first look at the simpler, historical part (b) of the claim. It is false because the infinite series for sine was known outside Europe, from long before Newton, some three centuries earlier. It is found in numerous texts of the Aryabhata school of Kerala (NOT “Kerala school”), particularly the 16th c. Yuktibhasa, the 15th c. Yuktidipika, or the Tantrasangrahavyakhya. It is attributed to the 13th -14th c. Madhava of Sangamgrama, and is part of a continuous thousand-year development since the 5th c. Aryabhata. While Aryabhata used linear interpolation/extrapolation to obtain sine values precise to the first sexagesimal minute (about 5 decimal places), this was soon extended to quadratic interpolation /extrapolation by Brahmagupta and Vateshwar who stated sine values precise to the second sexagesimal minute. The Aryabhata school in Kerala extended this to obtain sine values to the third sexagesimal minute. Nilakantha also explicitly gave the sum of infinite geometric series.

Specifically, the exact sine series, which Newton falsely claimed to have “discovered” can be found in two verses. Where? Since palm leaf manuscripts can vary, and are difficult to access, I use the critically edited and printed text of the Yuktidipika 2.441 and 2.442. The Sanskrit verses, the English and mathematical translation of these two verses can be found in, for example, my Hawai’i paper,9 my more recent articles on calculus for the Springer Encyclopedia,10 available online, or my book11 etc. Likewise, the infinite series for π, today called the “Leibniz series” in stock Western chauvinistic history, is found long before Leibniz in the Yuktidipika 2.271, and explained in my 2007 book (p. 168). The verse and its translation may also be accessed online in the video12 and presentation of my MIT talk.

Of course, to understand my position that Newton was nevertheless trying to be honest, we need to understand that Newton claimed discovery in the Christian theological sense. It is also in this sense that he used the word, when he called Leibniz the second “discoverer” of the infinite series today falsely credited to him. Newton was a victim of his times. But, today, we must understand clearly the Christian theological sense of the term “discovery” as frequently used in those days. Only then can we understand the peculiar European phenomenon of “discovery without understanding”.

Did Newton know that non-Christians had knowledge of the series before him? The short answer is: Yes. How? Because knowledge of the Indian infinite series was circulating in Europe after they were translated by Jesuits in their Cochin college and taken to Europe in the 16th c., long before Newton. Galileo had access to the Jesuit archives, but was afraid to dabble in infinite series, hence left the task to this student Cavalieri. Of course, Newton knew about that, and acknowledged Cavalieri’s work, while rejecting it.

Why exactly were Europeans so keen to discover this Indian knowledge of calculus?

(To be continued)

1  C. K. Raju, Is Science Western in Origin?, Dissenting Knowledges Pamphlet Series (Multiversity, 2009).
2  C. 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; C. K. Raju, “Not out of Greece (5 Lectures)” (University of South Africa, Pretoria, January 2017), http://ckraju.net/unisa; C. K. Raju, “To Decolonise Math Stand up to Its False History and Bad Philosophy,” in Rhodes Must Fall: The Struggle to Decolonise the Racist Heart of Empire (London: Zed Books, 2018), 265–70; C. K. Raju, “Not out of Greece,” Frontier Weekly, January 28, 2014, http://www.frontierweekly.com/archive/vol-number/vol/vol-47-2014-15/47-25/47-25-Not%20Out%20Of%20Greece.html.
3  C. K. Raju, “The Meaning of Christian ‘Discovery,’” Frontier Weekly 47, no. 29 (2015) http://www.frontierweekly.com/archive/vol-number/vol/vol-47-2014-15/47-29/47-29-The%20Meaning%20of%20Christian%20Discovery.html.
4  Immanuel Kant, Observations on the Feeling of the Beautiful and the Sublime and Other Writings, ed. P. Frierson and P. Guyer (Cambridge: Cambridge University Press, 2011); Raju, Euclid and Jesus: How and Why the Church Changed Mathematics and Christianity across Two Religious Wars.
5  Robert Bernesconi, “Hegel’s Racism,” Radical Philosophy 119 (2003): 35–37; C. Neugebauer, “The Racism of Kant and Hegel.,” in Indigenous Thinkers and Modern Debate on African Philosophy, ed. H. Odera Oruka (Leiden: Brill, 1990), 259–272.
6  C. K. Raju, “History Shows How Our Education System Inculcates Racist Values,” TheQuint, July 7, 2020, https://www.thequint.com/voices/opinion/history-of-racism-in-education-system; C. K. Raju, “नस्लवाद और आधुनिक सामाजिक सोच,” Jansatta, July 12, 2020, sec. रविवारीय स्तम्भ, https://www.jansatta.com/sunday-column/peoples-thinking-impact-of-education-and-world-attitude-towards-racism-in-modern-social-system/1465188/.
7  C. K. Raju, Ending Academic Imperialism: A Beginning (Penang: Citizens International, 2011).
8  Isaac Newton, “An Account of the Book Entituled Commercium Epistolicum Collinii Et Aliorum, de Analysi Promota”,” Philosophical Transaction of the Royal Society of London, no. 342 (February 15, 1714): 173–224. Newton provided this account as an anonymous review of his own report on the priority dispute with Leibniz.
9  C. K. Raju, “Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhāṣā,” Philosophy East and West 51, no. 3 (2001): 325–362. http://ckraju.net/papers/Hawaii.pdf.
10  C. K. Raju, “Calculus,” in Encyclopedia of Non-Western Science, Technology and Medicine (Springer, 2016), 1010–1015, http://ckraju.net/papers/Springer/ckr-Springer-encyclopedia-calculus-1-final.pdf; C. K. Raju, “Calculus Transmission,” in Encyclopedia of Non-Western Science, Technology, and Medicine (Springer, 2016), 1016–1022, http://ckraju.net/papers/Springer/ckr-Springer-encyclopedia-calculus-2-final.pdf.
11  C. 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).
12  C. K. Raju, Calculus: The Real Story, 2015, video: https://www.youtube.com/watch?v=IaodCGDjqzs. Presentation: http://ckraju.net/papers/presentations/MIT.pdf.

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Aug 31, 2020