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California, Indian Calculus and the Technology Race - 1 |
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by C. K. Raju |
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The Indian origin of Calculus and its Transmission to Europe Abstract The California education department has introduced a new math framework to meet the projected needs of future technology. Critics argue against it, calling it “cancelling the calculus”. This article explains the alternative of teaching the decolonised calculus which makes calculus easy and enables students to solve harder problems, not covered in usual calculus texts. This has been demonstrated in pedagogical experiments with 8 groups in 3 countries. How is this achieved? The origins of the present-day difficulties in teaching calculus relate to real numbers and limits. Those were invented long after the purported discovery of the calculus by Newton and Leibniz. Why? Because Europeans realized they lacked a clear understanding of the calculus. How did Newton and Leibniz discover the calculus without understanding it? Because Europeans stole the calculus from India, in connection with their navigational problem, and didn’t acknowledge the theft. Does that make a difference to teaching calculus? Yes, because knowledge thieves often fail to understand what they steal. Though the West failed to understand the Indian epistemology of the calculus, Westerners believe they did some “superior” kind of mathematics as in the racist terminology of the “Pythagorean theorem”. We explain that contrary to widespread myths, its actual proof in the “Euclid” book uses the same principles of proof as its Indian proof. This is because, contrary to the myth, axiomatic proofs are NOT actually found in the Euclid book. Indeed, real numbers were invented because even the first proposition of the “Euclid” book lacks an axiomatic proof. Introduction: California seeks to “cancel” the calculus The California education department has proposed a major change in the math curriculum for K-12 schools.[1] Though related to the requirements of future high technology the proposed change has led to a huge backlash with critics calling it “cancelling of the calculus”,[2] or killing the calculus,[3] a dumbing down of the math syllabus in schools, and asking for it to be replaced.[4] The controversial aspect of the change relates to the teaching of calculus, often regarded as the toughest part of math in school and college. The difficulty of teaching and learning calculus is clear from the very size of the typical college calculus texts which run into over 1300 large-size pages.[5] Hence, mastering the calculus, or, rather, performing well in typical calculus tests, came to be seen as a sign of talent. Since the 1980's, in the US, calculus courses at the K-12 level had become a de facto prerequisite for entry to prestigious institutions, the first step to a better job. The situation is similar in India: K-12 calculus is an essential part of the prestigious nationwide IIT entrance examination. However, educational research in the US[6] showed that these K-12 calculus courses were of little value to the students in Another key motivation for “cancelling the calculus” is the recent change in the requirements of the high-tech industry which relates to statistics. Recent times have seen an enormous increase in jobs related to machine learning, data science, and statistics, called twenty-first century math. However, calculus is needed for a good understanding of statistics. Indeed, the notion of probability is at the foundation of statistics, which mathematicians understand using “measure theory”.[7] This means the core statistical notion of “average”, in a non-discrete situation, involves an advanced form of the calculus, the Lebesgue integral,[8] not the integral as anti-derivative, nor the Riemann integral. Indeed, statistics is notorious for reaching incorrect or deliberately misleading conclusions. As the proverbial saying goes, there are “lies, damned lies, and statistics”. A widespread lack of understanding of key aspects of statistics, as might arise from teaching statistics without calculus, could decidedly worsen that. The aim of this article is to explain that there is an alternative: a way to teach the calculus which makes calculus easy, while preserving all practical value. The cause of difficulties with the calculus The Californian proposal to “cancel the calculus” is partly based on the observation that students learning the calculus, at the K-12 level, fail to understand it. There are very good causes for the observed lack of understanding of calculus among students. Thus, the notions of derivative and integral are at the core of the calculus. Both these notions are today defined using the notion of limit. But limits, hence the core notions of the calculus, are The consequence? My own calculus pre-test for science and engineering undergraduates[10] shows that not even a single student manages to get even zero, though they are warned that attempting questions which they don’t understand, hence answer wrongly, would fetch negative marks. That is, not only do beginning undergraduate students not understand anything of the calculus they learnt in school, they have completely wrong ideas based on what they imagine they do “intuitively” understand. If all students perform badly, the students and teachers are not to blame, the text or the subject is. The problem is not with the Indian NCERT text: the situation is the same, but more deceptively stated, in the usual internationally-used college calculus (“Thomas’ Calculus”) texts. These texts do have a section on the “precise definition of limits”. This is deceptive because the definition given is NOT precise or rigorous; it includes the ritualistic epsilons and deltas,[11] but leaves out an essential ingredient: real numbers are needed for limits, but they are The need for real numbers, on the Western understanding of calculus, can perhaps be driven home for the layperson in another way: Karl Marx failed to understand the calculus[15] and called the “calculus of Newton and Leibniz” as “mystical”. Marx’s lack of understanding of calculus has been correctly attributed, by the celebrated Marxist historian Struik,[16] to the fact that Marx studied calculus Now, the stock history of the calculus (e.g., as stated in the NCERT class XI math text[17]) asserts that calculus was “discovered” by Newton and Leibniz. But Newton died 150 years The first (practical) question will be postponed to part 2 of this article which explains that though real numbers and limits are the cause of all the difficulties in teaching and learning calculus, they have nil practical value: Newton successfully applied the calculus to physics just because real numbers are This part focuses on the second (historical) question and its consequence for teaching and learning calculus. On the principle that “phylogeny is ontogeny”, a growing child retraces the entire history of evolution, starting off in a watery womb, then crawling on all fours, before standing erect. Likewise, the teaching of a subject in the classroom retraces its historical development. The calculus, in Europe, initially involved no real numbers or limits for over two centuries. It is exactly that historical, self-admitted, European lack of understanding of calculus which is being replayed in the calculus classroom today, in fast forward mode. Let us enquire deeper into that history. Contrary to the stock history, there is impeccable evidence that the calculus was known in India for over a thousand years before Newton and Leibniz.[18] It was transmitted to Europe in the 16 So, what difference does that make to the What aspect of the Indian calculus did Europeans miss? They certainly understood the practical aspects, whether applications of calculus to navigation or physics, but missed a deep-seated theoretical aspect. The alternative: decolonised calculus Reverting to the original Indian philosophy, with which the calculus originated, is the basis of the decolonised calculus.[20] That not only makes calculus very easy, it enables students to solve The calculus was first invented in India by the 5th c. Aryabhata to derive precise trigonometric values, precise to the first sexagesimal minute (kala) or about five decimal places. He used a finite-difference method today called “Euler’s” method of numerically solving ordinary differential equations.[27] Aryabhata’s method was extended by his follower Vateswhar to the second sexagesimal minute ( There are good reasons to call it the Indian calculus, and not the “Kerala calculus” as it is often wrongly known. First, this is true in a simple sense: the key aspects of the Indian calculus, How and why Newton and Leibniz stole the Indian calculus So how did Newton, Leibniz and others get to know about the Indian calculus? Because the calculus was used in India to derive precise trigonometric values, which were very important for Europeans in the 16th century. Then, precise trigonometric values were desperately required for accurate navigation, to determine latitude, longitude, European dreams of wealth, then, rested on overseas trade (or piracy) for which proper knowledge of navigation was critical. Unlike other people they called “primitive”, 16th c. Europeans were themselves excessively backward in mathematics (including the arithmetic of fractions), and did not understand (celestial) navigation. Indeed, Europeans recognized their backwardness: the navigational problem was recognized as the biggest scientific challenge in Europe, and many European governments kept offering large rewards for its solution from the 16th until the 18th century. The last reward (of 20,000 UKP) was offered by the British Parliament, which legislated a Board of Longitude, in 1711, when that amount was a fortune. Thus, Europeans had huge motive to steal that Indian knowledge of calculus and its precise trigonometric values for the sake of their means of wealth, dependent on European navigation. Apart from motive, there was ample opportunity: the first Roman Catholic missionary school was set up in Kochi in 1501, and this later (ca. 1550) became a college which was taken over by Jesuits. The missionary interest was to teach the local Syrian Christians. Naturally, for this purpose, the missionaries learnt the local language Malayalam, which they were teaching in their Kochi college, and, eventually, Jesuits even learnt Sanskrit. Through their local contacts they systematically gathered information and books, which they translated en masse, as they had done with Arabic books centuries earlier in Toledo. These translated books were despatched to Rome. Did these Christian priests, or the recipients of the texts, not have any conscience? No. The doctrine of Christian discovery[39] taught them that stealing knowledge from non-Christians was morally the right thing to do. There is ample circumstantial evidence of European theft of Indian knowledge of calculus, with the help of their Jesuit college in Kochi. For example, essentially the Indian trigonometric values were published by the Jesuit general Christoph Clavius, in his name, in 1607,[40] to similar precision. Noticeably, Clavius used the Of course, if all the land in the Americas and Australia could be not only grabbed, but that land grab could be morally and legally justified on the doctrine of Christian discovery,[43] grabbing credit for the calculus was relatively simple. But the truth has unexpected ways of emerging. On my epistemic test, theft is established by the fact that knowledge thieves, like students who cheat in an exam, don’t Other circumstantial evidence of theft is provided by the evidence for Indian mathematics texts circulating in Europe in the 17th century. For example, Fermat's 1657 challenge problem, which no European mathematician could then solve, was a Likewise, the infinite series for the sine function, for which Newton falsely claimed credit, and the infinite series for the number π for which Leibniz falsely claimed credit, are found in earlier Indian texts.[46] Newton and Leibniz claimed credit[47] on the same obnoxious doctrine of Christian discovery[48] used to legalise the theft of land from native Americans. These facts are in the public domain for at least the last two decades,[49] and no one has contested them, though they have been repeatedly plagiarised by some Britishers,[50] on the same doctrine of Christian discovery: that Christians have the divine right to legally claim any piece of knowledge or land. To summarize, there was ample motivation (European navigational problem) and opportunity (Jesuit college in Cochin) for the theft of the Indian calculus. The available circumstantial evidence (e.g., Fermat’s challenge) and documentary evidence (e.g., Ricci’s letter) support this conclusion. The conclusion is sealed by the epistemic test. Though we have used a standard of evidence stronger than that used in criminal law, to prove the theft of calculus beyond reasonable doubt, the “benefit of doubt” does The racist myths of Pythagoras and Euclid As already summarily explained, the European lack of understanding of the calculus, stolen from India, was due to the epistemic differences between Indian ganita and Western mathematics. Hence, also, the decolonised calculus, by eliminating that lack of understanding, makes the teaching of calculus easy while actually enhancing its practical applications.
Racism, which is popularly understood as color prejudice, is better combated once it is understood that racism is not limited to colour prejudice but is about various inter-related and mutually supportive assertions of superiority, especially through a false history For example, consider the terminology of the “Pythagorean theorem”. The underlying colour prejudice is Why exactly was the mathematics, which the Greeks But where exactly is the So, why not abandon the racist terminology of the “Pythagorean theorem”? The determined attempts to hang on to that racist terminology are clear from the tactic of myth jumping. Indeed, when one such myth is challenged, it is typical to offer another In the case of Pythagoras, the jump is to the myth of “Euclid”. Of course, this “Euclid” is depicted as a White male in Wikipedia, and in the Indian NCERT school texts, even though he was supposedly from Alexandria in Africa. Obviously, there is nil actual evidence for the color of “Euclid’s” skin or his gender, but any assertion to the contrary, that “Euclid” was a black woman,[54] is simply vilified and censored.[55] That is the Western way of preserving a fake history of science, exactly the way the church preserved its false dogmas by vilifying and censoring critics. Indeed, there is no evidence even for the existence of “Euclid”, and my reward of about USD 3000 for primary evidence of “Euclid”[56] has gone unclaimed for a decade. When a demand was made for evidence for “Euclid”, the brazen official stand of the government agency NCERT was that one This is exactly the stand of Wikipedia, the truth of Western history must be decided, Pythagorean proposition and Indian ganita Against this background, let us now examine the similar claim for the purported inferiority of Indian mathematics. It is undeniable that Indians stated the Pythagorean proposition in the sulba sutra-s[58] (e.g. Baudhayana, 1.9,1.12, Manava 10.10, etc.) and this was long before Pythagoras. So, why continue with the terminology of the Pythagorean theorem? A common argument[59] for this assertion of Western superiority/Indian inferiority is that Indians may have stated it earlier, but they lacked a
This Indian method of proof However, apart from Nyaya, other schools of thought existed in India and had The moment we accept that mere fact of a contestation, the belief about the purported superiority of deduction is subject to scrutiny, and it crumbles.[65] Indeed, the limited Lokayata argument[66] was that deduction is inferior since it may result in The other interesting case is that of Buddhists and Jains who did Let us focus instead on a single issue. Comparison with Indian methods of proof makes very transparent a little-known but embarrassing aspect of the currently accepted definition of an axiomatic mathematical proof. All Indian schools of thought accept the pratyaksh, or the empirically manifest, as the first means of proof, as does science. But appeal to the empirical or to observations is prohibited in axiomatic proof.This is clear from the very definition[70] of an axiomatic proof as (a) a sequence of statements in which (b) each statement is either an axiom or (c) is derived from two or more preceding statements by some rule of reasoning such as modus ponens. At no stage is one allowed to assert, for example, “I observe this, therefore, it is true.” Why? Because empirical proof is declared inferior since fallible. This prohibition of the empirical is made clear even in the class IX NCERT math text: “However, each statement in the proof has to be established Note that this divorce of axiomatic proof from the empirical extends to the axioms. The theorems of axiomatic mathematics are supposed “relative truths” relative to the axioms, but the axiom typically cannot be tested for they are metaphysics in the sense of Popper (usually a metaphysics of infinity). Hence, Russell[71] candidly stated: Against this background, let us return to the claim that Greeks did something “superior” in mathematics because they provided axiomatic proofs. But what is the actual evidence that the Greeks provided axiomatic proofs? It is one thing that there is no evidence for the existence of “Euclid” but where is the evidence for the existence of axiomatic proofs in the Now, the Can so many “reliable sources” be so collectively wrong for so many centuries about the “Euclid” book, a text book which is right in front of one’s eyes and was read by all educated Europeans? Alas, yes! (Western sources are so utterly unreliable because of church hegemony, as explained later.) By the 20th c., the belief was publicly exposed as totally false: Many colonised “educators” don’t understand even class IX math, and know nothing of the history of math, but have deeply imbibed the colonial and Wikipedia ideal of blindly trusting only authoritative Western/racist “reliable sources”, That is, what we have today is a rewrite of the “Euclid” book, a rewrite which To reiterate, the reason to go into such great details about the absence of axiomatic proofs even in the mythical “Euclid's” book is that That first proposition is to construct an equilateral triangle on a given line segment AB. To do so, we construct two arcs, one with centre at A and the other with centre at B, both with the same radius AB. If both arcs intersect at the point C, then ABC is the required equilateral triangle. The problem is that this proof involves the empirical: we For complete clarity, we emphasize that even the “Pythagorean theorem” is proved in the “Euclid” book in exactly the same way, making Let us understand one easy implication of this.
- https://www.cde.ca.gov/ci/ma/cf/.
- Robert Wood, ‘Calculus Is Canceled: A Legacy Betrayed?’, California Globe, 21 June 2021,
https://californiaglobe.com/section-2/calculus-is-canceled-a-legacy-betrayed/; Williamson M. Evers, ‘Opinion | California Leftists Try to Cancel Math Class’,*Wall Street Journal*, 18 May 2021, sec. Opinion, https://www.wsj.com/articles/california-leftists-try-to-cancel-math-class-11621355858. - Daniel Rockmore, ‘Is It Time to Kill the Calculus?’, Salon, 26 September 2020,
https://www.salon.com/2020/09/26/teaching-data-science-instead-of-calculus-high-schools-math-debate/. - https://www.independent.org/news/article.asp?id=13658.
- G. B. Thomas, Maurice D. Weir, Joel Hass, Frank R. Giordano,
*Thomas' Calculus*, Dorling Kindersley, 11th ed., 2008. This has total 1384 pages. James Stewart,*Calculus: early Transcendentals*, Thomson books, 5th ed, 2007. This has total 1327 pages plus a CD. - David Bressoud, Dane Camp, and Daniel Teague, ‘Background to the MAA/NCTM Statement on Calculus’, in
*The Role of the Calculus in the Transition from High School to College Mathematics: Report of the Workshop Held at the MAA Carriage House Washington, DC, March 17–19, 2016.*, ed. David Bressoud (place not specified: MAA, NCTM, 2017), 77–81, https://www.maa.org/sites/default/files/RoleOfCalc_rev.pdf. - Paul R. Halmos,
*Measure Theory*(New York: Springer, 1974). - W. Rudin,
*Real and Complex Analysis*, 3rd ed. (New York: McGraw Hill, 1987). - https://ncert.nic.in/textbook.php?kemh1=13-16.
- http://ckraju.net/sgt/technical-presentations-faculty/ckr-sgt-tech-presentation-1.pdf. Incidentally, a student from IIT Kanpur, who claimed to have topped his calculus class, took this test and met the same fate (“not even zero”).
- http://ckraju.net/sgt/technical-presentations-faculty/ckr-sgt-tech-presentation-1.pdf.
- D. V. Widder,
*Advanced Calculus*, 2nd ed., Prentice Hall, New Delhi, 1999. - W. Rudin,
*Principles of Mathematical Analysis*, McGraw Hill, New York, 1964. Rudin,*Real and Complex Analysis*. - https://ncert.nic.in/textbook.php?iemh1=1-15, 1.3
- C. 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. - Dirk J. Struik, ‘Marx and Mathematics’,
*Science & Society*12, no. 1 (1948): 181–96. - https://ncert.nic.in/textbook.php?kemh1=13-16. Chp. 13, p. 281.
- C. 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) hereafter CFM; See also the abbreviated accounts in C. K. Raju, ‘Calculus’, in Encyclopedia of Non-Western Science, Technology and Medicine (Springer, 2016), 1010–15,
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), 1016–22, http://ckraju.net/papers/Springer/ckr-Springer-encyclopedia-calculus-2-final.pdf. - For a simplified account see C. 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; C. K. Raju, ‘*Ganit Banam Mathematics*[Ganita vs Mathematics]’,*Himanjali*20, no. July-December (2020): 34–44. http://ckraju.net/papers/ckr-article-Himanjali-22%e2%80%94Final-18-06-2021.pdf. - C. K. Raju, ‘Decolonising Mathematics’,
*AlterNation*25, no. 2 (2018): 12–43b, https://doi.org/10.29086/2519-5476/2018/v25n2a2. - For an example of the use of non-elementary elliptic functions, or ballistics with variable air resistance, not covered in college calculus texts see this stock tutorial sheet: http://ckraju.net/sgt/Tutorial-sgt.pdf. For the difficulty of elementary calculus problems such as the simple pendulum with air resistance, or the calculus of variations problem of brachistochrone with air resistance, see C. K. Raju, ‘Time: What Is It That It Can Be Measured?’, Science & Education 15, no. 6 (2006): 537–51
http://ckraju.net/papers/ckr_pendu_1_paper.pdf; Archishman Raju, ‘A Simple Way to Solve the Brachistochrone Problem with Resistance’, Physics Education (India) 28, no. 3 (September 2012), http://physedu.in/uploads/publication/3/65/Archishman_Brachistochrone13July.pdf. - See photo http://ckraju.net/papers/presentations/images/Sarnath-workshop-photo.png.
The online version of the paper is at http://ckraju.net/papers/calculus-without-limits-paper-2pce.pdf, presented at the 2nd People's Education Congress, Homi Bhabha Centre, Mumbai, 2009. (K-12 level.) - C. K. Raju, ‘Teaching Mathematics with a Different Philosophy. 1: Formal Mathematics as Biased Metaphysics’,
*Science and Culture*77, no. 7–8 (2011): 274–79; arXiv:1312.2099;C. K. Raju, ‘Teaching Mathematics with a Different Philosophy. 2: Calculus without Limits’,*Science and Culture*7, no. 7–8 (2011): 280–85. arXiv:1312.2100 (3 undergrad groups, and one post-graduate math group.) - See photo http://ckraju.net/papers/presentations/images/Iran-group-photo.png, and poster http://ckraju.net/papers/presentations/images/Iran-poster-English.png. (Mixed group.)
- See photo http://ckraju.net/papers/presentations/images/aud-group-photo-2.png, and poster http://ckraju.net/papers/presentations/images/AUD-poster.jpg. (Social science post-grads.)
- See photo http://ckraju.net/papers/presentations/images/calculus-sgt-a.jpg, and poster http://ckraju.net/sgt/poster-calculus-without-limits.pdf. (Science and engineering undergrads.)
- CFM, p. 140 et seq.
- For a quick account of these precise trigonometric values, see my talk at MIT, “Calculus: the real story”.
Video: https://www.youtube.com/watch?v=IaodCGDjqzs, Presentation: http://ckraju.net/papers/presentations/MIT.pdf, Abstract: http://ckraju.net/papers/Calculus-story-abstract.html. - CFM. p. 141
- C. K. Raju, ‘Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the YuktiBhasa”’,
*Philosophy East and West*51, no. 3 (2001): 325–62, https://muse.jhu.edu/article/26555/pdf. http://ckraju.net/papers/Hawaii.pdf. - CFM, p. 170.
- CFM, p. 169.
- C. K. Raju, ‘Calculus’.
- CFM, p. 174 et seq.
- C. 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. For a popular-level account, see https://www.jansatta.com/sunday-column/celebration-of-dalit-achievements-jansatta-column/777664. - K. Sambasiva Sastri, ed.,
*Aryabhatiya of Aryabhatacarya with the Bhasya of Nilakanthasomasutvan*(University of Kerala, Trivandrum, 1930). - "George Joseph: serial plagiarist”, http://ckraju.net/blog/?p=132. Also, “George Joseph serial plagiarist and mathematical ignoramus...”, http://ckraju.net/blog/?p=166.
- CFM, p. 324 et seq.
- C. K. Raju, ‘Marx and Mathematics. 2: “Discovery” of Calculus’,
*Frontier Weekly*, 31 August 2020, https://www.frontierweekly.com/views/aug-20/31-8-20-Marx%20and%20mathematics-2.html. - Christoph Clavius [Christophori Clavii Bambergensis,
*Tabulae Sinuum, Tangentium et Secantium Ad Partes Radij 10,000,000...*(Ioannis Albini, 1607). - For further details, see Raju, Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of Calculus from India to Europe in the 16th c, CE, 171.
- Matteo Ricci, ‘Letter to Petri Maffei’, in
*Documenta Indica*, vol. XII, 1581, 472–77 For a facsimile of Ricci’s handwritten letter, see my MIT talk. https://www.youtube.com/watch?v=IaodCGDjqzs. - http://doctrineofdiscovery.org.
- Brahmagupta,
*Brahma-Sphuta Siddhanta*, ed. Ram Swarup Sharma Sharma (New Delhi: Indian Institute of Astronomical and Sanskrit Research, 1966), 679 chp. 11, verse 16, vol. 3. - For more details, see “Calculus transmission” cited above. Fermat's challenge problem is in his Ouvres, p. 332 et seq. Bhaskara’s solution is in
*Bijaganita*(87, trans. Colebrooke, 1816, pp. 176–178). - The infinite “Taylor” series for sine is found in Yuktidipika 2.440-441, and the “Leibniz” infinite series for pi is found in Yukdtidipika 2.271. For further details see
*Cultural Foundations of Mathematics*cited above, and the calculus articles for the Springer Encylopedia cited above. - C. K. Raju, ‘Marx and Mathematics. 2: “Discovery” of Calculus’.
- https://doctrineofdiscovery.org/
- C. K. Raju, ‘Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhasa’,
*Philosophy East and West*51, no. 3 (2001): 325–62, http://ckraju.net/papers/Hawaii.pdf. - See, for example, the
*Hindustan Times*news reports of 8 Nov. 2004 archived at, http://ckraju.net/Joseph/HT_report_8_Nov_04.pdf, and the*Hindustan Times*retraction of 25 Aug 2007 of a sensational news item, archived at http://ckraju.net/Joseph/HT_correction_25_Aug_07.jpg. Further documents and details are accessible at http://ckraju.net/blog/?p=166. - C. K. Raju, ‘“Euclid” Must Fall: The “Pythagorean” “Theorem” and The Rant Of Racist and Civilizational Superiority — Part 1’,
*Arumaruka**: African Journal of Conversational Thinking*1, no. 1 (2021): 127–55; C. K. Raju, ‘“Euclid” Must Fall: The “Pythagorean” “Theorem” and the Rant of Racist and Civilizational Superiority – Part 2’, (to appear),*Arumaruka**: African Journal of Conversational Thinking*1, no. 2 (2021): 57-75, http://ckraju.net/papers/Tubingen-Pretoria-part-2.pdf. - Richard J. Gillings,
*Mathematics in the Time of the Pharaohs*(New York: Dover, 1972) Appendix 5. - E.g. Martin Bernal,
*Black Athena: The Afroasiatic Roots of Classical Civilization.*, vol. 1: The fabrication of ancient Greece (London: Free Association Books, 1987); C. K. Raju, ‘Black Thoughts Matter: Decolonized Math, Academic Censorship, and the “Pythagorean” Proposition’,*Journal of Black Studies*48, no. 3 (2017): 256–78, https://doi.org/10.1177%2F0021934716688311. - C. K. Raju, ‘Was Euclid A Black Woman? Sorting Through The False History And Bad Philosophy Of Mathematics | Science 2.0’, 24 October 2016, https://www.science20.com/the_conversation/was_euclid_a_black_woman_
sorting_through_the_false_history_and_bad_philosophy_of_mathematics-180581. - Raju, ‘Black Thoughts Matter: Decolonized Math, Academic Censorship, and the “Pythagorean” Proposition’.
- “Decolonising history: Goodbye Euclid!” at
https://youtu.be/sEK1FCrLHjU?t=3292, or the related presentation at http://ckraju.net/papers/presentations/Decolonising-history-Goodbye-Euclid.pdf#page=159. More details at http://ckraju.net/blog/?p=63. - http://ckraju.net/blog/?p=173.
- S. N. Sen and A. K. Bag,
*The Sulbasutras*(Delhi: Indian National Science Academy, 1983). - https://web.archive.org/web/20161102104249/http://ganashakti.com/english/comments/details/156.
- Satish Chandra Vidyabhushana,
*The Nyaya Sutras of Gotama*(Allahabad: Panini Office, 1913) verse 2. Online at http://ckraju.net/papers/presentations/images/Nyaya-Sutra-Gotama-2-proof.pdf. Note that this proof comes from long before any notable Greeks, and its date cannot be fudged by appealing to “reliable (=Western) sources” for the date of the Nyaya Sutra, since this is based on a numerous diverse sources, and disputes regarding this method of proof pre-dated even Buddhism. - http://ckraju.net/papers/presentations/images/Proof-table.html. This is further explained in the video of my workshop on the Indian alternative to “Euclidean” geometry:
https://www.youtube.com/watch?v=ERm25QgyW1w. Traditionally, in India, the same notion of proof was used for everything (unlike Western notions of proof which accept the empirical in science, but prohibit it in math, and resort to faith in religion, resulting in different categories of truth—mathematical, scientific, and religious truth). - A
*ryabhatiya of Aryabhata*, ed. K. S. Shukla and K. V. Sarma (Delhi: Indian National Science Academy, 1976). - Bina Chatterjee, trans.,
*Si**syadhivrddhida Tantra of Lalla*, 2 vols (Delhi: Indian National Science Academy, 1981) 20.36. - http://ckraju.net/hps-aiu/flat-earth-in-Bible.txt.
- Raju, ‘Decolonising Mathematics’.
- Haribhadra Suri, ed.,
*Shatdarshan Samuchaya*5th ed. (Bharatiya Jnanapeeth, 2000), 454–55. - C. K. Raju,
*Mathematics, Decolonisation and Censorship*, 2017, https://kafila.online/2017/06/25/mathematics-and-censorship-c-k-raju/. - For quantum logic see chp. 6b, quantum mechanical time in C. K. Raju,
*Time: Towards a Consistent Theory*(Springer, 1994). - Though the false claim of “universality” of 2-valued logic probably derives from the fact that Christian rational theology borrowed heavily from Islamic rational theology, and the related Arabic sources of “Aristotle” were influenced by Nyaya logic which is 2-valued. C. K. Raju, ‘Logic’, in Encyclopedia of Non-Western Science, Technology and Medicine (Springer, 2016 2008), 2564–70.
- L. Mendelson, ‘Introduction to Mathematical Logic, van Nostrand Reinhold’ (New York, 1964).
- Bertrand Russell, ‘Mathematics and the Metaphysicians’, in
*Mysticism and Logic and Other Essays*(London: Longmans, Green, and Co., 1918), 74–96. - Though numerous books are attributed to the mythical Euclid, we will be concerned with only one: The Elements of Geometry, or the Elements vol 1. Since there are no original Greek sources for this, we will take as the stock “original” source the manuscript found in the 19th c. by the racist historian Heiberg in the Vatican, and then translated. (But any Arabic/Byzantine Greek manuscript of the Elements will do for our conclusions.) T.L. Heath,
*The Thirteen Books of Euclid’s Elements*(New York: Dover Publications, 1956); C. K. Raju,*Euclid and Jesus: How and Why the Church Changed Mathematics and Christianity across Two Religious Wars*(Penang: Multiversity and Citizens International, 2012). - H. M. Taylor,
*Euclid’s Elements of Geometry*(Cambridge: Cambridge University Press, 1893). For some more details, see http://ckraju.net/geometry/cambridge-note.html. - B. Russell, ‘The Teaching of Euclid’,
*The Mathematical Gazette*2, no. 33 (1902): 165–67. - David Hilbert,
*The Foundations of Geometry*(The Open Court Publishing Co., La Salle, 1950). - George D Birkhoff, ‘A Set of Postulates for Plane Geometry, Based on Scale and Protractor’,
*Annals of Mathematics*33 (1932): 329–45.
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