Frontline’s Calculus of Caste: C. K. Raju

Guest post by C. K. RAJU

[Frontline carried a historically ill-informed article on Indian calculus which also had mathematical and casteist errors. When the errors were pointed out, the magazine ignored it, contrary to journalistic ethics. Here is Prof Raju’s response to that article.]

Frontline (23 Jan 2015) published an excessively ill-informed article by Biman Nath on “Calculus & India”. The article suppressed the existence of my 500 page tome on Cultural Foundations of Mathematics: the Nature of Mathematical Proof and the Transmission of Calculus from India to Europe in the 16th c. (Pearson Longman, 2007). This suppression was deliberate, for Nath and Frontline ignored it even after it was pointed out to them. They also refused to correct serious mathematical and casteist errors in the article. That is contrary to journalistic ethics. To understand my response, some background is needed.

According to my above book and various related articles, the calculus developed in India and was transmitted to Europe. The second part of the story is lesser known. As often happens with imported knowledge, calculus was misunderstood in Europe. Later that inferior misunderstanding was given back to India through colonial education, and continues to be taught to this day just by declaring it as “superior”. That claim of superiority was never cross-checked to see if it is any different from the other flimsy claims of superiority earlier made by the West, for centuries, for example the racist claim that white-skinned people are “superior”.

Third, in any case, the purported “superiority” of Western mathematics does not add to practical value: all practical value of the calculus today to physics and engineering (such as sending a rocket to Mars) still derives from numerical calculation (today done with computers). That was the way calculus developed in India, not the idealistic way it was misunderstood in Europe. On the contrary, that “superior” Western metaphysics subtracts from practical value. Newton’s misunderstanding of the calculus led to the conceptual error responsible for the failure of his physics. There are also other problems at an advanced level in physics, today, regarding infinities and infinitesimals. Correcting Western errors, and reverting to the original Indian understanding of the calculus using “non-Archimedean” arithmetic together with a different epistemology of zeroism (deriving from realistic sunyavada) to handle infinities and infinitesimals resolves those problems in current physics. It also makes calculus very easy to teach, even to students of social science, as has been demonstrated.

The fourth part of the story is about colonial education: how that impedes any change or even a discussion of it today. It is well known that colonial education bred meek slaves for the empire. That was achieved by copying the church system of creating missionaries. The first step is to instill ignorance. And, the manifest fact is that most colonially educated people today are ignorant about mathematics and science. The ignorant have no choice but to trust authority. Which authority? The second step is to instill blind trust in Western authority, and distrust and contempt for the non-West. This created the “trust only the West” superstition, used for colonial exploitation. That is still propagated by Wikipedia the present-day popular source of “reliable” knowledge. What if a critique gets past the doorkeepers? In that case, status quo is maintained by a deliberate process of debate avoidance, personal attacks, misrepresentation etc., all typical church techniques used to defend otherwise indefensible superstitions such as the belief in virgin birth. Such defences are typically accompanied by striking a pose of “superiority”, which impresses the gullible. This has been going on for the last two decades with my critique of formal mathematics, and Nath and Frontline are only the last straw.

To elaborate a bit more, the calculus developed in India in relation to the two means of wealth: agriculture and overseas trade. Indian agriculture depends upon the rainy season determining which requires a good calendar, hence astronomical models. That required accurate trigonometric values calculated by the 5th c. Aryabhata (precise to 5 decimal places) by numerically solving difference equations (which differ only metaphysically from ordinary differential equations at the heart of the calculus). Those precise trigonometric values were also needed for navigation needed for overseas trade. Techniques of latitude and longitude determination are described in the works of the 7th c. Bhaskara 1. By the 16th c., recursive application of Aryabhata’s techniques led to infinite series, which were used to calculate trigonometric values precise to the 9th decimal place. These precise values were then badly needed for the European navigation problem then regarded as the chief scientific problem confronting Europe, as clear from the large rewards offered by European governments for its solution, across 3 centuries, from 16th until the 18th c. The calculus was hence transmitted to Europe in the 16th c. by Jesuits who had opened a college in Cochin in a deliberate replication of the 12th c. Toledo model, mass translating local texts, and sending them back to Europe. They naturally falsified history by systematically suppressing any non-Christian sources in those days of the Inquisition.

Transmission of knowledge, especially when surreptitious, often involves misunderstanding: a student who copies from another usually does not understand what he has copied, and that lack of understanding is proof of copying or transmission. Indian arithmetic (called algorithms after al Khwarizmi) was earlier transmitted to Europe via Arabs. The very term “Arabic numerals” records how this elementary school stuff was misunderstood by the learned and infallible pope Sylvester. The future pope unaccustomed to algorithms based on place value, and accustomed to the abacus (apices, jetons) used for primitive Greek and Roman arithmetic, hence got a special abacus constructed for “Arabic numerals” in 976.1 That is, he thought there was some magic in the numerals themselves! Likewise, the calculus in India involved a different epistemology of infinity and infinitesimals used to sum infinite series, which epistemology was not understood in Europe. In the case of arithmetic, Europeans eventually accepted the inferiority of Roman arithmetic and abandoned it. However, in the case of the calculus the European misunderstanding (“limits”) was passed off as “superior” and globalised during colonialism. It is taught to this day. The metaphysics of limits has nil practical value, for practical applications of the calculus are today done on a computer using numerical methods similar to those of Aryabhata, since a computer cannot handle the metaphysics of infinity underlying limits or metaphysical “real” numbers.

The bad Western epistemology of calculus creates various problems for present-day math and science. Newton misunderstood the calculus, hence made time metaphysical.2 As I had earlier explained in my book Time: Towards a Consistent Theory (Kluwer, 1994) making time metaphysical was a conceptual error responsible for the failure of his physics. Correcting that error leads to a new physics, as again being explained in my recent series of articles in Physics Education.3 and also to a new theory of gravitation.4University-text calculus fails also in many common situations in physics involving infinities and infinitesimals,5 correcting which leads to an alternative physics.

The other contemporary problem with the bad Western epistemology is that it turns math into a nightmare for millions of students. That metaphysics of infinity is critical to church dogmas about eternity. But that is of negative value to us, the ruled. Eliminating that metaphysics and teaching calculus the way it actually developed historically in India makes it very easy to teach, as I have demonstrated by teaching calculus in 5 days to 8 groups in 5 universities across 3 countries, including a group of social science students in Ambedkar University, Delhi, and one at the Central University of Tibetan Studies, Sarnath, as also various categories of math students in Universiti Sains Malaysia.6 Similar difficulties arise with probability and statistics as explained in my article on “Probability in Ancient India” in the Handbook of Philosophy of Statistics (Elsevier, 2011), and hence I have developed a decolonised course on statistics for social science.7

Macaulay’s infamous minute of 1835 is well known: it brought colonial education to India just by declaring that the West was immeasurably superior in science. Let us cross-check this claim of “superiority”. Is the colonial claim of superiority any better grounded than the earlier racist claim of superiority? Earlier racist historians too had claimed that the Greeks did a “superior” kind of mathematics; hence Euclid was declared the father of “real” mathematics. However, that “real” Western mathematics is a bastard child, for there is no evidence for the existence of Euclid. My challenge prize of Rs 2 lakhs for serious evidence about Euclid has gone abegging for years, like Kovoor’s prize against astrologers. But our experts are united in maintaining that no evidence is needed for Western history, which MUST be accepted on faith, and anyone who deviates from that blind faith and demands evidence must be condemned as a Hindu fundamentalist. That is exactly how the church responded to sceptical critics who were declared heretics and hence killed or banished.

Indeed, the religious connection of mathematics is manifest from its etymological root mathesis which was explicitly related to religious beliefs and the soul by Plato in his Meno and Republic. This religious connection of math persisted for at least 8 centuries with “Neoplatonists” like Proclus who says math leads to the blessed life. That tradition continued in the Islamic theology of reason (aql-i-kalam). During the Crusades, the church morphed aql-i-kalam to the Christian rational theology of Aquinas and the schoolmen. This was just an attempt to rebrand and use Muslim knowledge to convert Muslims by persuasion. By an extraordinary coincidence the philosophy of deductive proof attributed to the mythical Euclid from 1500 years earlier, closely agrees with that Christian theology of reason which developed when “Euclid” first came to Europe in the 12th c. But the gullible colonised mind does not wonder at this strange identity of purpose between “Euclid” and Crusading theologians 1500 years apart. The myth of Euclid allowed the church to wrest “ownership” of reason from its religious opponents, the Muslims, as explained in detail for the layperson in my book Euclid and Jesus.

Those who, for centuries, did not check coarse historical claims like those about “Euclid” cannot be expected to apply their mind to the subtler philosophy of math. The myth of Euclid is related to another myth: the claim that he originated “superior” deductive proof, and the further claim that that is what “real” math is about. As already stated, the contemporary practical value of math relates to calculation, not proof. Further, that claim about proof is also historically false: if we examine the book Elements (wrongly attributed to Euclid) its very first proposition uses an empirical proof, not an axiomatic one. The same applies to the proof of its penultimate proposition (“Pythagorean theorem”). Hilariously, thousands of Western scholars who read the book failed to notice this fact for seven centuries, but kept glorifying themselves using the myth! (I noticed that problem in the 7th standard, and asked my teacher who could not clarify.) When the fact was finally admitted in the 20th c. an apologia was quickly constructed. What matters to the West is neither the existence of “Euclid” nor the actual book Elements he supposedly wrote. All that matters is the myth about it. In the event, Russell and Hilbert rewrote the book to force the facts to conform to the myth! (Though the rewrite does not fit the original, that is how geometry is uncritically taught in our schools today.) Thus, the present-day philosophy of math grew out of concern for myth, not practical value. Why should we share that concern? Our focus should be firmly on practical value, and we should simply reject bunkum Western myths.

Finally, why exactly is deductive proof “superior” to empirical proof? It is true that empirical proofs are fallible: the classical Indian example is that a rope may be mistaken for a snake or vice versa. But that doubt can be easily settled in practice by experiments, repeated if necessary. But why is deductive proof infallible? Any claim of infallibility smacks of religious dogma. Indeed, the dogma among rational theologians was that logic binds God who is free to create the facts of his choice. Hence they thought logic was superior to God who was superior to facts. But why should we believe this dogma? That dogma was excessively parochial, for the fact is that logic is not universal. Buddhist logic of catuskoti and Jain logic of syadavada are counterexamples, and one can conceive of an infinity of different logics. Why should we set aside these facts and believe church dogma? The theorems of mathematics are relative to both postulates and logic, and will change if either is changed. If one chooses 2-valued logic on cultural grounds, then those theorems are mere cultural truths of no value to people from another culture. If one chooses 2-valued logic on empirical grounds, then that choice is fallible.

In either case, the whole claim that deductive proofs are infallible fails. Along with it falls the entire philosophy of formal mathematics, and much of Western philosophy. Western philosophers are dumbstruck and unable to answer this objection: so they have been pretending for two decades that the objection does not exist. If they can’t answer the objection they should change their beliefs. Instead, they hang on to their beliefs in the way missionaries hang on to the literal belief in virgin birth: by debate avoidance, misrepresentation, abuse etc. But why should millions of our students be subjected to this Western dogmatism today? If Western philosophers can’t answer critiques, it is high time the colonised stood up and rejected what the West says as inferior. Once again, we simply need to focus on the practical value of math, which gets along fine without Western metaphysics.

Against this background, let us return to the Frontline article written by Nath, an astronomer, who avoids all these grave matters of concern for millions, which even Frontline‘s sister publication the Hindu has accepted as being of public importance.8

The first thing that strikes one in the article is Nath’s ignorance of calculus even as currently taught. Nath advocates that we should imitate the Western way of doing the calculus, using limits. But he illustrates the concept of limits using rates of change with stock market trajectories. However, this is a well known case in which limits do NOT exist (according to formal mathematics), though numerical calculations may still be done in the way of Aryabhata. Technically speaking, stock market trajectories involve Brownian motion, the sample paths of which are everywhere continuous but nowhere differentiable. Therefore, the example of the stock market is a laughably wrong example to use to illustrate limits, even at the journalistic level. Thus, Nath advocates we ape the Western way of doing calculus, from a position of ignorance, and purely on the colonial superstition that the West is superior. Now, anyone can make a mistake, especially the indoctrinated, but Nath’s mistake is inexcusable since he sticks to that elementary mistake even after it was pointed out.

Indeed, the example given by Nath is a wrong example for another subtle reason. Stock markets involve randomness or probability. Like calculus, probability and sampling too originated in India.9 (There is the famous aksa sukta in the Rgveda, and we are all familiar with the game of dice in the Mahabharata. What is less known is that the romantic story of Nala and Damayanti links the game of dice to the sampling technique of counting leaves on a tree.) Like calculus, probability was transmitted to Europe where it was misunderstood. The practical use of probability is still based on relative frequency which can be empirically observed, and on the theory of permutations and combinations, which too developed in India. However, theoretically, the concept of probability is today defined using Kolmogorov’s axioms, which involve metaphysical limits. Even the well-known normal distribution uses an integral of some sort (either the Riemann-Stieltjes integral or the Lebesgue integral), and the current definition of those integrals involves the metaphysics of limits. Empirical relative frequency is connected to theoretical probability through what is called the “law of large numbers”. The belief is that probability is some sort of limit of relative frequency as the sample size becomes infinite. The problem is that this limit is not the same sort of limit as the limits in university calculus: it is a probabilistic limit, or a limit in measure as it is called. A thousand tosses of an unbiased coin may result in 999 heads. So relative frequency cannot be connected to probability without presupposing the concept of probability. That is, the concept of limits fails at a very fundamental level in the case of probabilities. So, Nath’s example of the stock market is doubly wrong, because the processes in the stock market are random or probabilistic, and limits fail conceptually in that case. Needless to say, practical applications of statistics mostly involve numerical calculations, which don’t need that metaphysics of limits.

Let us grant that Nath, an astronomer does not know any history, and also does not understand the calculus or probability theory. (Why then did he write an article on the history of the calculus?) The blurb tries to establish his authority as a scientist by saying he teaches general relativity. How does he do that without adequate knowledge of calculus? General relativity is formulated using differential equations. That assumes that the functions entering into those equations are differentiable. On university calculus that means the appropriate “limits” must exist (metaphysically). However, that is known to be not the case in many situations. A simple firecracker (or an exploding star) produces shock waves. This is a region of discontinuity where the relevant limits fail to exist (a discontinuous function cannot be differentiated). So, just by clapping one’s hands, one can make the equations of general relativity fail (if understood in the sense of university calculus)! Various common escape routes are blocked. In general relativity, unlike classical fluid dynamics, one cannot just regard the continuum description of matter as some sort of statistical limit due to particles in random motion. Apart from the above problems regarding probabilistic limits, that is also because there is neither a generally covariant statistical mechanics, nor a general relativistic description of atoms and molecules. Another possible escape route is to change the definition of derivative. (Changing definitions is bad practice, for science does not remain refutable, if one keeps doing that in mid-theory.) The Schwartz derivative allows discontinuous functions to be differentiated. However, it does not allow the resulting Dirac delta functions to be multiplied.10 But such multiplication is unavoidable for the equations of general relativity are non-linear. Thus, just trying to make sense of the equations of general relativity, in common situations, unavoidably brings in non-standard analysis or non-Archimedean arithmetic. Even so, a different epistemology of the calculus is needed, as explained in detail in the Appendix on shocks and renormalization in Cultural Foundations of Mathematics.

That “non-Archimedean” arithmetic was the arithmetic used in Indian calculus. The 7th c. Brahmgupta used polynomials (such as 2x+3) which he called unexpressed (avyakta) numbers, for they acquire a value only when the value of x is specified. Naturally, unexpressed numbers led to unexpressed fractions or what are today called rational functions (e.g. (2x+3)/(3x+4)). Such rational functions follow what is called a “non-Archimedean” arithmetic. Using it with zeroism involves just an extension of the common ordinary language usage where a single name pi is used for a multiplicity of slightly differing entities like 3.14, 3.141, 3.1415, etc. (Recall that on realistic Buddhist sunyavada, there is constant change, and a single name used for a person represents such a procession of manifestly varying entities, as the person grows from a child to an adult and old man.)

Enough said about mathematics, which neither Nath nor the editors of Frontline understand. They seem full of missionary fervour to defend false Western history and bad Western philosophy against what they presumably perceive as onslaughts of non-Christian fundamentalists. It does not strike them that they are themselves furthering more dangerous superstitions. However, their attempts to preserve their authority, through a pose of superiority, by refusing to acknowledge mistakes has been carried to ridiculous limits!

Thus, the Frontline article also carries an image of Aryabhata. The image is that of a statue in the Inter University Centre for Astronomy and Astrophysics (IUCAA), Pune, and the same image is also found in Wikipedia. Thus, there is plenty of authority backing that image.

But the real story is as follows. When the statue was first set up, the IUCAA’s public relations officer (PRO) asked me for the original verse where Aryabhata compared the globe of the earth to a kadamba flower. I sent the requested information (Gola 7), but pointed out that the PRO was misspelling the name Aryabhata as Aryabhatta. Slight changes can totally change the meaning, and the changed spelling changes Aryabhata’s caste. As any Sanskrit dictionary will confirm, the word bhata means a slave, soldier etc., whereas bhattais an honorific for a learned Brahmin. The name found in all manuscripts, and used by all commentators and opponents is Aryabhata, never Aryabhatta.

This is historically significant because the followers of Aryabhata in Kerala were the highest caste Namboodiri Brahmins, such as Nilkantha Somasutvan, who, as his name shows, performed the soma yajna. Such a phenomenon of high-caste followers of a lower-caste person from another region (Aryabhata was from Patna) was inconceivable in the India of Ambedkar’s time. Nor was Aryabhata an isolated exception, because he was followed by Aryabhata II after a substantial gap of 5 centuries. This suggests that the nature of the caste system itself was different when Buddhists and Muslims flourished, as we may well expect it to be. This is a significant matter for another reason I have pointed out. While there is ample documentation of many lower-caste religious figures,11 from Valmiki to Ravidas, and there is the British evidence for numerous lower-caste teachers in pre-colonial India, as Dharampal has pointed out, Aryabhata is the first lower-caste scientific figure, involved in “high” science.

Hence, I objected to misspelling the name Aryabhata as Aryabhatta. However, the PRO of IUCAA responded that he was well aware of it, having learnt about it from Jayant Narlikar, the founding director of IUCAA. I, then, enquired why, in the NCERT school text authored by Narlikar, the name continued to be wrongly spelled. There was no reply (that is the stock way of bypassing embarrassing questions) but the spelling in the NCERT text was eventually corrected, after I raised public objections.12

However, the statue of Aryabhata in IUCAA, Pune, changes Aryabhata’s caste using a novel visual technique, which has gone largely unnoticed. It shows him wearing a janeyu, which is a symbol of the twice-born (dvija) Brahmins. It also shows incongruous “caucasian” features, as was the case in the NCERT texts which uniformly depicted “Greeks” from Alexandria in Africa on a caucasian stereotype. The subliminal aim is presumably to impose a racist stereotype to preserve Western authority. The statue in IUCAA and its image in Wikipedia should be remembered as a demonstration of the complete unreliability of pro-Western and Western authority.

Nath and Frontline, in their attempt to defend the indefensible colonial caste system (of regarding the West as superior), have further struck a vain pose of superiority in defence of their mistake, and refused to acknowledge it as a mistake. In the process they have also stuck to their casteist portrayal of Aryabhata. The attempt to force an inferior Western understanding of calculus, without discussion, does a great disservice to millions of math students, while their casteist portrayal of Aryabhata deprives us of an invaluable lower-caste scientific icon.

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1For an image of the pope apices, see C. K. Raju, Euclid and Jesus, Multiversity, 2012.

2C. 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. http://ckraju.net/papers/retarded_gravitation_theory-rio.pdf. Also, “Time: what is it that it can be measured?” Science & Education, 15(6) (2006) pp. 537–551. Draft available from http://ckraju.net/papers/ckr_pendu_1_paper.pdf.

3“Functional differential equations.1: a new paradigm in physics”, Physics Education (India), 29(3), July-Sep 2013, Article 1.http://physedu.in/uploads/publication/11/200/29.3.1FDEs-in-physics-part-1.pdf. “Functional differential equations 2: The classical hydrogen atom”, Physics Education (India), 29(3), July-Sep 2013, Article 2. http://physedu.in/uploads/publication/11/201/29.3.2FDEs-in-physics-part-2.pdf. “Functional differential equations. 3: Radiative damping” Physics Education (India), 30(3), July-Sep 2014, Article 8.http://www.physedu.in/uploads/publication/15/263/7.-Functional-differential-equations.pdf.

4“Retarded gravitation theory”, cited above.

5C. K. Raju, “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. arxiv: 0804.1998.

6C. K. Raju, “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. “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.

7C. K. Raju, “Decolonisation of education: further steps”, paper for the meeting on “Decolonisation and leadership”, Nottingham University, Malaysia Campus, Jan 2015. Draft posted at http://ckraju.net/papers/KL-abstract-and-draft.pdf. Also, “Decolonising math and science education”. Ghadar Jari Hai 8(3), 2014, pp. 5-12. http://www.ghadar.in/gjh_html/?q=content/decolonising-math-and-science-education. “Decolonising math and science”. In: Decolonising the University, ed. Claude Alvares and Shad Faruqi,USM and Citizens International, 2012, pp. 162–195. http://ckraju.net/papers/decolonisation-paper.pdf.

8C. K. Raju, “Decolonising Maths education”, in The Hindu, 24 Oct 2014. Full version posted online at http://ckraju.net/press/2014/Response-to-Glover-Teach-religiously-neutral-math.html.

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. http://www.ckraju.net/papers/Probability-in-Ancient-India.pdf.

10“Distributional matter tensors in relativity”, cited above.

11Sanjay Paswan, Cultural Nationalism and Dalit, Samvad Media, Delhi, 2014.

12C. K. Raju, “Teaching Racist History”, Indian Journal of Secularism11 (2008) 25-28.  http://ckraju.net/papers/Teaching-racist-history.pdf. Also, इतिहास के विचलन,  Jansatta 23 Jan 2008, op-ed page, http://ckraju.net/papers/Jansatta-Euclid.jpg.

Attachments area

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Professor C. K. Raju holds an M.Sc in math from Mumbai and a PhD from the Indian Statistical Institute, Kolkata. He taught formal mathematics at Pune University for several years. He has put forward a new realistic philosophy of math called zeroism (sunyavada) and taught calculus based on it in 5 universities across 3 countries. He has also used it to modify present-day physics and has authored several books, and numerous articles, and campaigns for decolonisation of education, especially math and science education.

34 thoughts on “Frontline’s Calculus of Caste: C. K. Raju

  1. S.Shankar

    What a confused article by a thoroughly confused man! Yes, IUCCA has indeed portrayed Aryabhata a brahmin, but this is hardly surprising in the lowbrow mofussil agraharam that is Indian academia (IISc has put up portraits of Gandhi, Nehru, R.Prasad … every Amar-Akbar-Antony in its library, but not of Dr.Ambedkar!). But why the rant against Macaulay, the man who universalized education in India and brought it to the reach of Dalit children? Is it that the Raju man is cunningly casteist after all? Besides, the western development of calculus is what has resulted in all technological innovations from electricity to powered flight to the web, and is precisely what this author has used to convey his misinformed opinions. I vote for universal western education in science for all children in India, not the dogeared moth-eaten ‘hindu’ science the author roots for.

      1. @Rana Eddy

        Amazing how you jump to wild conclusions, on the strength of a single word yuga/ Are you seriously suggesting it is so easy to change one’s caste, just by using one word or related concept? Please inform the government immediately about your theory, so they can reform their schedules!

        The word yuga as Aryabhata used it relates to the astronomical model underlying the calendar. Can you please explain what is nonsense in that astronomical model, which I doubt you have the foggiest understanding of. On the contrary, the duration of a yuga was used to state a precise duration for the length of the year. In contrast the Julian calendar had an awfully bad length of the year, because the primitive Romans and Greek numerals could not be used to articulate precise fractions or even large numbers, and were hence were eventually abandoned as inferior. The precise value of even the length of the tropical year is still missing in the Gregorian calendar, even after the Gregorian reform. See, for example, http://ckraju.net/papers/presentations/A-Tale-of-Two-Calendars.pdf.

        Your sweeping statements illustrating your ignorance won’t take away from the fact that Aryabhata used a scientific model, and a scientific calendar. That scientific astronomical model was incidentally in wide use. In the case of Arabs and China, we know that the model initially travelled from India to Persia to Arabs (through the Bayt al Hikma), and to Chinese (through the Buddhists, who made the Chinese calendar). Different calendars can be built on similar astronomical models, for example the Islamic calendar is entirely lunar.

        The Indian luni-solar calendar related to Indian economic interests like agriculture and overseas trade (navigation). The need for greater precision in the underlying astronomical model was the reason why the calculus developed in India. For some more details, see, for example, http://ckraju.net/papers/GJH-book-review.pdf, or my book Cultural Foundations of Mathematics. Even the Maya calendar used a similar astronomical model for it begins from -3110 CE compared to the start of the Kali Yuga, at -3102 CE. Their calendrical cycle, equivalent to a yuga, just ended, but it was not associated with the doomsday superstition associated with the Gregorian calendar. The relation of yuga or age to virtue and metals (as in “Golden age”) is a symbolic connection found across the world, including among the Maya. There is nothing specifically Brahmanical about it.

        Yuga relates to the calendar which is a good example of colonial indoctrination. The colonially educated, like you, can tell their date of birth only on the Christian ritual calendar (Gregorian calendar), so, from an early age, their identity is tied to the church superstitions promoted by that calendar. (E.g. to say a date you must acknowledge Jesus as your lord (AD), and your saviour (BC)). That is at least two words you use which makes you a Christian by your logic. That you might be, but on your logic, colonial education is still missionary education, for it turns people Christian, just by teaching them a bad calendar, and its related superstitions. Therefore, also, we should immediately change that education system as I argue.

        As a result of indoctrination, an entire nation failed to understand the linkage of the Gregorian calendar to religious beliefs: hence the two secular Indian festivals are ironically defined only on that inferior and unscientific church calendar. Those superstitions ruin our economic interests because a good calendar, in India, must tell the rainy season, which the unscientific Gregorian calendar cannot. (See the newspaper clips about the repeated annual feature of “delayed” monsoon in the last decade, in the presentation posted at http://ckraju.net/papers/presentations/A-Tale-of-Two-Calendars.pdf.)

        The canine loyalty of the colonised to the West is developed by colonial education which teaches them blind trust in the West and equally blind distrust of the non-West. So, they come to view everything non-Christian (including Maya and Inca) with the deepest suspicion, and always as some strange religious practice, never science. Instilling the Christian calendar from childhood helps in developing that feeling of strangeness and helps to alienate people from their own culture: did you ever learn how to calculate the date of Holi? Nor can you tell its date from the Gregorian calendar with its unscientific months of varying duration. That propaganda that everything non-Christian is unscientific is what makes you arrive at your curious belief that the use of the term yuga makes one a Brahmin.

        Ironically, as in the Gregorian calendar, there are religious superstitions in Newton’s “laws”, as briefly explained in my cited paper on “Islam and science”, posted at http://ckraju.net/papers/Islam-and-Science-kl-paper.pdf, and cited in the OP. However, your indoctrination, while teaching you to defer to the authority of science, never actually taught you even the basic science of the motion of the sun and moon and stars which you see everyday. Therefore, you cannot identify those dogmas and superstitions, and recognize them as such.

        Nor can you understand the alternatives I have proposed, as cited in the OP. You were taught to proceed on blind trust not knowledge, and the “trust only the West” syndrome means you equate science with blind faith in the West. The point of the OP was that Western science is permeated with church dogma from the basic level of mathematics; what the West did was to add metaphysical dogma to the practical value of Indian mathematics, especially the calculus. The practical value of science (still obtained the same way as Aryabhata, as in the OP), is used to spread those nonsense dogmas, to those, like you, unable to separate practical processes from the accompanying metaphysics. Good luck with your faith.

    1. charan

      Sorry, your opinions seems to be biased.
      1)Today, it is accepted that calculus was indeed developed in India, by both historians and mathematicians.
      This is what David Mumford, a Fields medalist and well-known Mathematician. has to say–
      http://www.ams.org/notices/201003/rtx100300385p.pdf
      Reproducing a relevant quote from this article–
      QUOTEThe discovery of the finite difference equation
      for sine led Indian mathematicians eventually to
      the full theory of calculus for polynomials and for
      sine, cosine, arcsine, and arctangent functions, that
      is, for everything connected to the circle and sphere
      that might be motivated by the applications to
      astronomy.UNQUOTE
      And also–
      http://www.ms.uky.edu/~sohum/ma330/sp12/india_calculus.pdf
      2)Please read Dharampal`s research to know about the kind of Education system prevalent in India in pre-British era–
      https://ia601504.us.archive.org/26/items/DharampalCollectedWritingsIn5Volumes/3DharampalJiCollectedWritings-beautifultree.pdf
      (And also, it pays to remember Western Education was not as universal as it is advertised to be! And Universality of Education happened in India before it had happened in the West. In fact, there are documents which show that British Officials, during the colonial era, researched the then prevalent system of Indian Education and later adopted it to Europe.)
      Thank You

      1. @charan

        I agree with you, but why do you cite David Mumford? Or his review of the book written by David Pingree’s student? The existence of the Indian infinite series was publicly known in the West since 1832, but that didn’t stop Macaulay from bragging about Western superiority in 1835. Later, in the 19th c. Bapudeva Sastri struggled in vain to get the West to accept the Indian origin of calculus, citing Bhaskara II, and his notion of instantaneous velocity of planets. That claim was rejected. So what changed now?

        I was the first to investigate the transmission of calculus, to Europe, after a brief encounter with Pingree, way back in 1996. (I initiated an INSA project for that in 1998.) I was the first to point out that Aryabhata used difference equations and that the relevant verse (Ganita 12) could NOT be interpreted as an algebraic equation as even Shukla and Sarma had earlier wrongly done. This is in chp. 3 of Cultural Foundations of Mathematics. I was the first to point out that there is only a metaphysical difference between difference equations and differential equations, for the former is what is still used in practice. I used that to teach calculus. It was only after all this that some people in the West realised that their position had become ridiculous, and hence agreed to the claim that something like calculus originated in India, which claim was not accepted for nearly two centuries earlier.

        That credit is being denied to me is only a small part of my problem. (It is important, because colonialism was anchored in false history, and we should not allow that to develop.)

        The far bigger problem is that if you use a wrong process (relying on Western authority, instead of your own knowledge) you will again fall into the same colonial trap. Thus, if you keep trusting Western authorities, they will continue to misuse that trust to “save” the story by twisting it and retelling it to match their convenience. Look at all the ways in which Pingree twisted the story of Ptolemy, to suggest a Greek origin of trigonometry. His student is an equally slippery historian. David Mumford is not about to give up formal mathematics, or accept my alternative philosophy of mathematics, or change math education. He is not about to give up the practice of promoting Western authority by citing only Westerners. He is only pandering to the Indian ego, hoping to save all that. If you rely on Western authority, you tacitly accept you are inferior (in a bid to persuade the colonised mind perhaps). Naturally, they will use that admission of inferiority to always maintain that what was done in India was something inferior, and not quite the calculus.

        My contention is the opposite that the way calculus was misunderstood in Europe was inferior. That needs to be hammered home, but will never be done by the Mumfords of this world. The West accepted the practical value of the Indian calculus, even in the heyday of the Inquisition; only added some useless metaphysics to it to make it theologically correct. Hence, its teaching needs to be changed, along with the science based on it. That can never be brought out unless one gives up the reliance on Western authority for whatever reason. As the Jains argued against the Buddhists, good intentions don’t matter in this context. Remember, the key issue is the future generation, and how to prevent its miseducation due to blind reliance on Western authority. We need to change that, and it is contrary to that goal to cite Mumford as the source of credibility.

        Let us focus on practical value instead.

        1. charan

          As a person (a Mathematics student) who has read your papers online–can not understand why such VERY important papers are not being discussed openly–I completely understand your intentions.
          Thank You, for your reply sir.
          But,accepted your criticism of citing Western sources(I am just beginning to enlighten about my country).
          Few questions:
          1) I was searching for your book,”Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16th C. CE “since many days, where can I find this ?
          2) There are two Aryabhatas, what is the difference–culturally,geographically and in their contributions–between them?
          3) I have read in Internet,that–“scholars brought by Parashurama after river Saraswati dried up are called namboodiris or namboodiripads.
          Their creamy layer are called bhattathiripads.So there are several Aryan bhattathiris or Arya bhattas.The one who calculated the value of PI, was born in 2700 BC. He was a Malayali from Kerala, NOT a Bihari.”
          And finally, Aryabhatta is Aryan Bhattathiripad, from Kerala.
          If it is true, which Aryabhata–among the two– is it referring to?How much of truth is there in this?(I am unable to verify this claim; and cannot dismiss it as “conspiracy” either.And considering this date–2700 B.C–also explains the high level of technological sophistication,which definitely requires high Mathematical knowledge, that is evident in Ancient India.)
          Thank You

          1. @charan-2
            First, as regards the book, I am sorry that it is out of print. Will strive to bring it back. In the meanwhile why not try the old fashioned method of consulting the book in a library, such as the Sahitya Akademi library in Delhi.

            Second, as regards Aryabhata, the story you have read on the Internet is not correct.

            All manuscripts, all commentators (such as Nilkantha) mention his name as Aryabhata , never Aryabhatta. His opponents (such as Brahmagupta) refer only to the bhata and his disciples. The Internet story is wrong just on this count.

            Secondly, Aryabhata himself gives his date (and since he worked on the calendar, we can accept it as accurate). It is precisely 499 CE. Some of his opinions caused a storm, and we can also corroborate his date from that. Thus, we can also relate him to the date of his opponents Varahamihira and Brahmagupta who wanted to go back to the earlier geometric method of calculating sine values, and use quadratic interpolation (“Stirling’s formula”) instead.

            For more details on this, you may consult the book by Shukla and Sarma, or article by the late K. V. Sarma, who coined the term “Kerala school”, but emphatically denied the “quaint story” that Aryabhata was born in Kerala.

            K. V. Sarma, “Aryabhata: his name, time and provenance”, Indian J. History of Science, 36 (3–4) 2001, pp.~105-115.

            There is a lot of substance to Indian mathematics, and we should not fritter it away by wasting time on outlandish claims without serious evidence. There is no need for wild speculative claims of the sort made by Western historians.

            Regarding Aryabhata II, you could take a look at S. R. Sarma’s translation of the Mahasiddhanta, and commentary on it.

            P.S. Pls do try to avoid using loaded terms like AD and BC.

    2. What exactly is your point? That IISc does not put up portraits of Ambedkar, therefore one should not point out the mistake made by IUCAA in misleading people by Brahamanising Aryabhata? That is an absurd a non-sequitur. In my experience, such irrationality indicates an underlying vested interest, presumably in defending Narlikar.

      Your other opinion is that Macaulay was a liberator. However, the OP made the point that the caste system was less oppressive in pre-colonial times. How else do you explain the prominence of dalit religious leaders from Valmiki to Ravidas? The remarkable fact that two Aryabhata’s could flourish 5 centuries apart and have as their disciples the highest caste Brahmins shows that prominent dalit scientific figures too flourished. This is further supported by Dharampal’s citations of British studies which show that there was widespread dalit participation in education both as teachers and students, in pre-colonial India. This is also a priori credible: while Buddhists and Muslim rulers flourished, conversion was an easy option, available to dalits, as Ambedkar demonstrated. Various Brahmin groups openly aligned with oppressive colonialism, saying British rule was preferable to Muslim rule, and the British naturally encouraged that. Hence, dalit oppression increased during colonialism.

      Macaulay saw education as a means of counter-revolution: this was pointed out in my article in Frontier weekly (http://ckraju.net/papers/Education-and-counter-revolution.pdf) and in more detail in my recent paper at the 38th Social Science Congress in Vizag. Macaulay openly said in British parliament in 1847, just before the Communist Manifesto, that education was the cheapest means of preventing revolts, by indoctrinating the poor and controlling their minds. As an aristocrat he wanted to control the poor, not uplift them as you so absurdly say. He explicitly said the rich were free to educate themselves, but it was the state’s duty to control the poor, by other means than hanging them.

      The mind-numbing power of colonial education arises from its relation to church education. Thus the Western education for which you root was, for centuries, church education and was designed to breed missionaries. Whether education teaches blind loyalty to the church or to empire or to capital is irrelevant. (Anyway, the three are related.) The key point is that Western education always mass produced loyal slaves. It is laughable that you confound this self-confessed slave maker with a liberator.

      Finally, you begin with abuse and proceed to misrepresentation. For example, you falsely say I root for hindu education (sic). Actually, I advocate zeroism in place of formal mathematics: that relates to Buddhist sunyavada as in my book Cultural Foundations of Mathematics, or as in my article on “Probability in Ancient India” for the Elsevier Handbook of Philosophy of Statistics (or the article on zeroism in the forthcoming edition of the Springer Encylopedia of Non-Western science, or on my blog (http://ckraju.net/blog). Such crude misrepresentation, personal attacks etc. are classic missionary tactics passed on to the colonially educated like you! Labelling any critic of the West Hindu fanatic is still the ONLY “argument” that West has: that includes Witzel from Harvard. What a splendid basis you demonstrate of your “superior” knowledge!

      Clearly, the use of such tactics as abuse and misrepresentation is a public confession that your know your opinions are weak, and based only on some vested interest, perhaps your interest in defending Narlikar who repeatedly spouts bad history, in favour of colonialism, or Macaulay who used false history to impose colonialism. Colonial education bred canine loyalty also by offering bribes (the job at the end of education) to create partners in its crimes against humanity across the world.

      Finally, it is an improved contemporary science that I am talking about together with a more honest history! You repeat stories about colonial education and science, but neglect the reality that, as a result of that bad education, you and millions of others know nil science. You root for the colonial education which deliberately instilled ignorance to make you totally helpless and reliant on authority (and taught you to trust only Western authority). Unfortunately, that ignorance also means you are unable to respond substantively to my critique. I understand your frustration, but there is no remedy for it, except to change your opinions and make them compatible with facts, something a missionary will rarely do. So, abuse away, keep misrepresenting for that is the only way left to preserve the stories that colonial education taught you, as you have so clearly demonstrated.

      1. S.Shankar

        Even you should be able to see that in spite of the lack of the ‘practical’ aspects of calculus in its development in Europe that you decry so loudly, it is the ‘West’ that has invented everything in the last 150 years, electricity to the worldwideweb. Of course science did not start in the west, it has a history that goes back to the Arabs, and yes, even to the Hindus, but what use is there in repeatedly shouting out these imagined glories of some ancient hindu civilization, caste and untouchability conveniently erased out of memory, in today’s miserable context? Russia, Japan, and now China have internalized this so called Western Science, have made it their own, have contributed to it in the living present, and become power-houses, but India, thanks to people such as you – no different from the epsilon minuses who think the important issue is to establish that hindu rishis flew in aircraft in vedic times – will remain in the dungheap.

        Now here is a question that is interesting – why did the Kerala school of calculus die out in a few generations, why did not the development of calculus that took place in Europe centuries later, happen here? The answer is clear to anybody who can introspect – education and learning was denied to the vast majority, it was a privilege violently guarded by a few brahmins for their own benefits.

        Universal education in India was a Buddhist innovation – all the universities in ancient India -Taxila, Nalanda, Vikramashila, Somapura,
        Odantapuri, Jagaddala, Vallabi … – were all Buddhist. Hindus never set
        up universities, only ashrams, where only a few “upper” caste could come.
        No wonder the little calculus that was invented here died out double quick.

        A civilization and a society usually gets what it deserves.

        1. While I agree with your saying that “we are doing injustice to ourselves by harping onto old & at times imagined glories” …I do not agree with your confalting Hinduism with Vedanta-Mimamsa-Vedic .Were Hindus & Buddhists two different things organically . When you talk about Nalanda , have you forgotten that it was established by the Guptas. Taxila predates Buddhism . Panini – the teacher at Taxila also predate Buddhism.

          ##Hindus never set up universities, only ashrams, where only a few “upper” caste could come.##

          I think that you are reading too much of Puranas. And BTW , almost all Indian Mathematicians were Brahmins.

        2. First, I pointed out a gross misrepresentation by you. You obviously could not substantiate it, but neither did you apologise. That conclusively establishes your deliberate dishonesty. Coupled with abusiveness to cover up lack of substance.

          You are also unable to contest the point that pre-colonial history, as used by Macaulay, was a fraud: from its tall tales of Greek achievements (originating in Crusading history) to stories of the Copernican and Newtonian revolution (during the Inquisition).

          In desperation you turn to recent history, the last 150 years.

          (a) Certainly the West did something, but was it progress? You “overlook” a key point of the OP that the celebrated formal mathematics and set theory, which developed in the last 150 years, are faulty and deserve to be trashed. The original Indian understanding of calculus was superior, but Westerners were mathematically too challenged to understand it. So, the West did formal math, but it is worthless, and we must change math teaching today. You are unable to engage with that argument. See? That is all colonial indoctrination instilled in you: a pitiable blind faith that the West is superior, and that the superiority can be established just by abusing away. Ha Ha!

          (b) The other point of the OP concerned what the West did NOT do in the last 150 years. It did not correct the bad science arising from bad mathematics. The bad mathematical understanding of calculus led to conceptual errors in Newtonian physics which were only partly resolved a century ago, and persist until today. Science needs to be changed, along the lines I have suggested. But your idea seems to be that science is not about applying one’s mind, but imitating the Japanese who imitate the West! It is all about aping the West somehow, not about being critical!

          (c) Further, the radio invented by J. C. Bose was credited to Marconi. Einstein copied from S. N. Bose, just as a top mathematician, Atiyah, copied from my critique of Einstein. Smallpox vaccination was falsely credited to Jenner, the light bulb was falsely credited to Edison, and not the black Latimer who really invented it. More recently, Britishers serially plagiarised my work even on the transmission of the calculus to Europe. So, false claims of ownership of ideas very much persist.

          (d) Other systematically fraudulent claims of ownership as in the patent for haldi, or in the Western ownership of software built by Indians require a separate discussion.

          (e) This applies also to technology. Are polluting technology and weapons of mass destruction etc. progress? Or is that regress, which too needs to be corrected?

          However, since you are unable to engage with the substantive points in the OP, and only obsessed with establishing your imagined superiority through abuse, this discussion is closed from my side.

        3. charan

          One thing that I understood after reading the comments here is, people have very little, if not nothing, understanding of history; and if they understand it, they mainly use it to satisfy there own personal agendas.

          A well informative lecture, about Indian Mathematics–

          And also, regarding the history and possible causes for it`s “forgotten memories”–
          http://www.ias.ac.in/resonance/Volumes/17/02/0106-0115.pdf
          http://www-history.mcs.st-and.ac.uk/Projects/Pearce/Chapters/Ch10.html

          Today,Science needs a very radical (r)evolution–a new knowledge-base is required, and Indian Philosophy can provide that.This much is accepted today by all(if we forget those petty ideologues).Now it is the job of we Indians to do that; and we should stop speaking for/against few biased opinions–stop chasing the sociological/colonial ideals of India.We should be learned/wise/thoughtful enough to spread “lies” against a 10000 years old civilization–it has a lot to offer to the world.
          And, we should also remember that there are many Women,Jain Mathematicians in India.(and it has not only restricted to brahmins)
          Just, as an example–
          http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol49_3_4_DJadhav.pdf
          C.K.Raju has provided many relevant links to the articles, it would be better if people here go through those and then comment , instead of screaming about personal ideologies.
          Thank You.

  2. v v anand

    (a) S Shankar has hit the nail on the head in describing this article as he does.
    (b) Empirical demonstrations are fallible and no amount of repeated experiments can establish something as indisputably true.You may measure the sum of angles for thousands of triangles and they may sum to 180 degrees, but that is not a proof that the three angles always sum up to 180 deg. Moreover, actual med at easurement will give you figures of, at best, 179.9999 deg or such. Proof is a leap and the ancient Greeks desreve all credit for the method.
    (c) The sad thing is that the real achievements of Indian Math are much more than what this gentleman is talking about. Zero is perhaps the greatest scientfic discovery and it underlies the whole of modern science and technology. That one discovery suffices to establish the robustness of ancient indian science.
    (d) A lot of work is underway which establishes that even calculus originated in the Kerala School , 2 centuries before Newton and Leibniz. Madhava in 14th century Kerala, arrived at the series expression for Tan inverse x, giving it as finite series , with increasing accuracy as you add more terms. Incidentally, even Newton and Leibniz did not get to the classical limit formulation and they began just as empirically as Nadhava before them.The limit formulation, is not “metaphysical” and has power.We need not posit these as contesting ideas.
    (e) Another tantalising path to explore is that Sanskrit texts between , say, 2nd century ad and 15th century AD, have not received the attention they deserve.Much of it is liberating and part of th Bhakti movement. Even in law, 10th century Vigyaneshwara’s Mitakshara, while still patriarchal and hidebound, is a vast improvement on Manu and Ygnavalkya of the 1st and prior centuries. Whatever CK raju may say, even scholarship in these areas is coming more from interested westerners.Shradhavan Labhate Gnanam is practised most dedicatedly by western scxholars. Indian languages,esp. tribal ones have been brought to life by Western enthusiasts. Todas have lived alongside Tamils na Malayalees for millenia, but there is no dictionary from Toda to tamil/Malayalam. Santhals have lived beside Biharis and Bengalis for several millenia, but no dictionary from Bengali or Hindi to Santhali.There does exist, however, a dictionary from English to Santhali.
    (f) Periyar had said that the emptiness of Brahminical culture resulted in absence of the western renaissance in India. Let us recognise that.

    1. Sad to say that you just repeat some bogus myths you acquired through your indoctrination in schools and university. Didn’t even read the OP which seeks to dispel those myths. It is a myth that Greeks invented deductive proof. That myth was used by racist historians to claim that the West is “superior”. Now, first, deductive proof relates to “Euclid”, but Euclid did not exist. To demonstrate this to myth-besotted minds, produced by colonial education, I offered a prize of Rs 2 lakhs for serious evidence about Euclid which prize stands unclaimed like Kovoor’s prize.

      The myth besotted mind does not give up. It invariably tries to “save” the story. A common way to “save” the story is to say it is the book which is important, not the person. However, the fact is that the very first proposition of the Elements uses an empirical proof as does its penultimate proposition: the so-called Pythagorean theorem. Thus, factually, the book did NOT offer deductive proofs of anything. Even more hilariously, blinded by the myth, Western scholars didn’t notice the fact for some seven centuries. You have not noticed it even after reading my article! Hence, I teach this to my students as a simple example of the long-term fallibility of deductive proofs: a mistake between a rope and a snake cannot last that long. If this is what happens at so elementary a level, imagine what happens at more complex levels of metaphysics!

      But you cannot evade that complex metaphysics. For example, consider your own proposition A: “the three angles of a triangle add up to 180 degrees”. Your purported knowledge is not like the knowledge of ordinary human beings, for you reject measurement as a fallible empirical process, but claim your knowledge is different since infallible. So what is it, God-knowledge or soul-knowledge, or some other delusion like a rabbit with horns? Clearly, you are not referring to any REAL triangles, such as those used for practical purposes of navigation. (Indeed, general relativity supposes that spacetime is curved, so perfectly plane triangles do not exist anywhere in reality!) Since you self-admittedly do not refer to ANYTHING empirical, or physical, or real, what you say is metaphysics, and purely imaginary.

      Unfortunately, even that imagination is is not your own. Can you formally (i.e., without referring to any experience) even define what an angle of 1 degree is? That is a challenge, and I am willing to bet a large sum you can’t do it, in one try, even with the help of some IIT professors of mathematics. Note that the NCERT definition of an angle as something (what thing?) made by two straight lines is faulty for many reasons. For example, what are lines? (No thickness?!) You must further define what a straight line is because a perfect straight line does not exist in reality. (Recall that the purported Euclid’s supposed axiomatic definitions are faulty, and actually empirical.) To define a line or do geometry axiomatically, you will actually need to first define numbers axiomatically. Russell and Whitehead tried to do that in their Principia, and took 376 pages just to prove that 1+1=2. (Do you understand even a single line on that page? An image of that page is reproduced in my article on “Decolonising education: further steps” posted at http://ckraju.net/papers/KL-abstract-and-draft.pdf.) Anyway, you will surely need a lot more than just 1+1=2 to define an angle: at least the metaphysical “real” numbers. Further an angle is not primarily related to straight lines, but is actually related to the length of curved lines. So, you must define quadrature to get the length of curved lines, etc. All that will take you at least a thousand pages. I judge that you never did it. An empirical definition, as in the sulba sutra is so much simpler and more robust.

      Anyway, without knowing anything of the complex metaphysics in those thousand pages, you assert it is infallible! Not even a typo? Sorry, I find the claim of infallibility laughable. In fact, that metaphysics is as fundamentally faulty as the church theology on which it is built. This was already stated in the OP. Thus, if you use a different logic, like Buddhist catuskoti, you won’t be able to define an angle the same way, and may never be able to do things like limits (and won’t ever need them). Why is your choice of logic infallible? It is just a church dogma that reason is universal, and you are trying to foist that dogma on me, just on the strength of your indoctrination, and without any knowledge, either of formal mathematics or even of my critique stated in the OP. Limits are bunkum metaphysics, but if they do exist, you have crossed all of them in being gullible. So, we will discuss limits after you are able to define an angle of 1 degree in a way which is 100% non-empirical. Challenge accepted?

      As regards your point (c) I presume you are referring to MY work, serially plagiarised by some Westerners, just as they plagiarised calculus earlier. Of course, those who copy make mistakes. For example, it is not the Kerala school, but the Aryabhata school in Kerala. The difference between the two is the point of my article: that Aryabhata was a dalit from Patna, whereas the Kerala school people were mostly high-caste Brahmins. This is also the point in the OP that Newton made a mistake in understanding the calculus, hence a mistake in his physics.

      As regards your point (e) can you point out a single Indian mathematical text in the time period you name, which I have overlooked in my book Cultural Foundations of Mathematics? Can you point to a single point in that text which is valuable? I am not claiming to be infallible, but just bringing out your inability to engage with anything specific, and your complete ignorance of my writings, including the OP.

      Finally, as regards your point (f), I teach in my course on history and philosophy of science that there was no Western renaissance. First there was no naissance, since the barbaric Greeks were primitive at arithmetic and knew no math or science, as proved by the non-textual evidence of their calendar. The false claims of Greek achievement are all based on manipulation of late texts and connecting the two with wild leaps of prejudiced speculation. The claim that there was a Dark Age in which that knowledge was lost to reappear mysteriously during the Crusades is yet another related myth used to get around the manifest discontinuity (in space, time, and language) in the textual evidence. How do you know that the texts which “reappeared” were the same as those which disappeared? Some infallible authority told you? Why should I believe it? The attempt to establish a continuity by stories of dark age and renaissance is something only a gullible or indoctrinated mind can swallow. Secondly, the claim of scientific work even during the post-Crusade era, in Europe, e.g. Copernicus, is also bogus Crusading and Inquisitional history. Take a look at my booklet Is Science Western in Origin?

      Regrettably you are just pontificating with zero knowledge of either formal mathematics or what I have said about it, even in the OP and are just repeating all sorts of stories which you swallowed uncritically, and expect me to do the same. Hence, I will not respond any further to you, since I expect those who comment must be at least a little informed.

  3. Subramanyam Durbha

    I recently published an article in ‘Mathematics teacher’ the official magazine of the Association of the Mathematics teachers of India,Chennai wherein I was refuting Rajus false claim of Euclid being a myth.The article is titled ‘Stop the tirade against Euclid.Why Euclid should be more emphasized in schools than is currently done’ wherin I have established beyond doubt about Euclid being a genuine historical character, contrary to Rajus arguments that Euclid is a myth,on which most of his attack against present day formal Math and science, hinges.I am not interested in claiming the Rs.2 lakhs prize he has offered since I know him well and for reasons I would let him know later,but I would like your readers to be aware of my article and paper.
    I will send you my article if you would like to post it.Please reply and let me know.

    1. @subramanyam durbha.

      That’s great news! You have proved the existence of “Euclid” beyond doubt? Do send your article to me. The prize is open. What primary historical sources did you use? How did you date them? How did you manage to connect “Euclid” to the book Elements? How do you know me when I don’t at all know you, and can’t find you even on Google!

      BTW, you are completely wrong in what you state above. My arguments against formal mathematics, as summarised in the OP, are independent of the existence of Euclid or any history. I hope your other claims are not similarly wrong.

      The point about “Euclid” is to bring out the connections of formal math to religious beliefs, as explained for the layperson in my book “Euclid and Jesus: How and why the church changed mathematics and Christianity across two religious wars”. Multiversity, 2012. Did you refute all the arguments in that book? Take a look at the section on “He who knows not and knows not that he knows not”, and also the late David Fowler’s comments in the following section (“He who knows and knows he knows”) that NOTHING is known about Euclid.

  4. Subramanyam Durbha

    Raju,
    You do know me,perhaps you dont want to accept it.Ask your friend Mylavarapu Deekshitulu a Reader at kalyan university who is one of your signatories for the petition against Michael Atiyah and perhaps he will remind you.By the way I am subbu who was at ISI calcutta as a research scholar from the year 1979-1980 and we stayed in the same hostel and had several cups of tea together and had interesting discussions on indian philosophy(on which you were dead against and modern day physics on which you did exhibit a good knowledge about) at that time.If you still dont recall me good luck, and let us set that aside.
    you can find me on Google. Hit Subramanyam Durbha.Perhaps it will show Subramanyam Darbha dont hesitate and hit search.You will find me as a professor at Rowan university (where i am not working at present,I am istructor at another college in philadelphia) and a few other citations.
    I do know your arguments against formal math depend on several other things like 2 valued logic and notions of infinity etc .. which I have addressed in the paper.
    the editor of the journal ‘Mathematics teacher’ has told me he will send the journal to your address.You work at Albhukary university in Malaysia,am I right?we got the address from google and you should be receiving it soon.Establishing Euclid as a real historical character is the first step in exposing your false arguments one by one,you will see the other arguments in the paper.
    If you want see the paper in the meanwhile or otherwise(since we cannot depend on Postal mail too much for fast communication) please let me know where I should send it.Give me your e-mail address or any other email address or address and I will send the paper.
    Good luck! and by the way I do not want the 3000 dollar prize, you may be needing it!

  5. Subramanyam Durbha

    Raju,
    Also there is no need to read the Book “Euclid and Jesus” which you are quoting.If you are making a mistake at the first step it would be foolhardy to follow a chain of arguments that follow and depend on it.
    Again you are quoting David Fowler who you think is an authority on Euclid.How come a guy who questions authority, himself relies on authority to propogate his false hoods.
    I have brought all these to the lime light in my article .You can read it and have fun.As I told you give me your email address so that I can send it.

    1. @Subramanyam Durbah-2

      OK, so a Subramanyam Durbah is indeed listed as a part time adjunct instructor at Community College Philadelphia. Still don’t recall you. Are you the astrologer who predicted I would marry late when, in fact, I was already married?

      Anyway, while I am a fallibilist and do change my opinions, your claim that I was “dead against” Indian philosophy in 1979 is factually false. Please take a look at ISI discussion paper No. P&E/E/7904 (or, maybe 7905) from 1979 on “Ancient and Modern Cosmology”. It was published in 1982, and is cited in this paper from 2003/2005 http://ckraju.net/papers/Atman-quasi-recurrence-and-paticca-samuppada.pdf. This point explained in chapter 1 of my book The Eleven Pictures of Time, Sage, 2003. The nature of time/cosmos is central to all early philosophies, including the Indian philosophies of Advaita Vedanta and Buddhism, and its wide contemporary relevance is through the ethical principle in that book now called the “Harmony principle” as recently published in Philosophy East and West and elsewhere. So your claim of personal knowledge from 1979 is not reliable, for whatever reason.

      Coming to “Euclid”, I note the interesting methodology to which you admit: you will refute my points even without reading what I have written in the book Euclid and Jesus. Bravo! This was exactly why I recalled David Fowler to check whether you have read the book. Also out of a sense of deja vu of a similar discussion on “Euclid” 15 years ago still on the net (in discussing history we don’t want to keep repeating the same mistakes do we?) He was the occasional honest Western historian who admitted that NOTHING is known about “Euclid” unlike dishonest Wikipedia which says “almost nothing” is known. His authority is no part of my arguments against Euclid.

      But I am glad that you understand that I don’t accept proofs by authority as most animals do (as explained in the Eleven Pictures of Time). Since you claim your proof is beyond doubt, your evidence is not circumstantial either.

      So, what is your new primary evidence for “Euclid”? You didn’t respond to my question about your new primary historical sources for “Euclid”, from where, in what language, and how dated? Why publish this startling thing in an obscure journal? (About citing obscure references as a deliberate strategy of saving myth, see “proof by expertise” in the Eleven Pictures of Time, an example of which is Toomer’s attempted refutation of R. R. Newton on the mythical Claudius Ptolemy.)

      Anyway, why not simply upload your article somewhere (say, Vixra), and paste a link here. BTW, I am travelling for a few months: am in Boston, right now, and giving a talk in MIT on 26th April at 4.30 p.m. The talk is on the calculus, closely related to the OP, and the abstract is posted at http://ckraju.net/papers/Calculus-story-abstract.html,

  6. Subramanyam Durbha

    Did you read my earlier mail properly.I have no knowledge of you (personally) outside ISI calcutta when I was there during 1979-1980.Where is astrology coming here.I do not know any astrology and never discussed about your marraige.What exactly are you looking for in google?How is any information about me on the google going to ring a bell,on your position of lack of knowledge about me during the said period of time? a photo ? a small snap is there.
    First a few questions.What is OP?I have no knowledge of your books except that I know you have written a few.
    I am currently seriously into Mathematical research which is not your domain.
    I have visited your website qute a few times and occasionally read some articles.I know you have done extensive research on the history of ancient Indian Mathematics I did read your article good bye Euclid and that prompted me to write my article.The reason I chose the Indian magazine is I was connecting the start of greek mathematics with the visit of pythogoras to india.Just as you claim calculus travelled from india to europeI have argued that there was transmission of knowledge from india to greece in ancient times from the time of pythogoras.The only place I part with you, I give full credit for what they did with the knowledge they got from india by developing the method of deductive proof from which arose the elements
    The conversation between you and me will flow smoothly if the clog clears abut your supposedly lost (from memory ) personal acquintace you had with me.
    The existence of Euclid beyond doubt is not a new claim.T.L.Heath knew it.For he asserts in his History of Greek Mathematics One thing is certain about Eucld.He lived and taught at Alexandria.
    As to my proof About the existence of Euclid I am referring to the books of Applonius of Perga on conic sections(which of course heath knew).Four of the original greek books have survived in greek.In book 3 he mentoins Euclid by name and refers to the work Euclid did (actually compiled) on conic seations.
    I wat to stop at this point and will continue later if you respond.

  7. Subramanyam Durbha

    As to your comment about my reluctance to read your book on Euclid and Jesus,its aim is no different from what you wanted to say in the paper,’Good bye Euclid’ and I have addressed the issues in my article.If I have eastablished the truth of statement A( even if it be only to at least to a high level of crediblity) why would any one but a fool try to follow an argument aiming to prove that A is false when I have already read a similar itellectual exercise trying to prove A is false(your paper on good bye Euclid) and exposed the underlying falsehood in those arguments in ‘my’ paper .In this real world logic is 2 valued.(I have addressed your position on other logics in my paper).I have submitted my article to Vixra.As soon as my request is processed and I hear from them I will provide the link to my paper in these columns.
    By other comment seems to have been moderated hence this submission.

    1. Thanks for posting the link to your paper. The kind of ignorance on which your claims are based is now on public display! I downloaded the paper and glanced at it. No primary historical sources. Just a reference to a couple of tertiary sources like dishonest MacTutor. Clearly, you don’t even understand what a primary historical source is and why it is needed in the event of a doubt or contesting claim. No prize for you, whether or not you want it.

      Also, your idea of proof beyond doubt seems to be as follows. You gullibly swallowed all sorts of bogus claims, without ever doubting them, and if I doubt them, you put it beyond doubt by saying “absurd”, “tirade” etc. That is the way religious beliefs are held without doubt.

      How, for example, do you know that the author of the Elements was “Euclid”? Just believed it without doubt, eh? Do you even have a manuscript of the Elements which mentions “Euclid” as the author? Why do you disbelieve the primary manuscripts which say the book was written by Theon or based on his lectures? You have nil understanding of what I am saying.

      Your “knowledge” is good enough only for students of community colleges, it is worthless in the present context, and any further discussion with you would be pointless.

  8. spkc

    for all those people who keep questioning dr raju about his claims about formal mathematics, please take a look a the above video and other videos in the lecture series by dr Wildberger.

  9. Subramanyam Durbha

    Is N. J. Wildberger a joke or a genius when he claims that …
    https://www.quora.com/Is-N-J-Wildberger-a-joke-or-a-genius-when-
    (The above link gives more information about N.J.Wildberger and what others have to say about him.If the above link does not work you can directly access it by googling him)
    After the comment made by the author of the OP about my knowledge, I woudnt have bothered to respond to a comment unless he apologized to me for making such a comment,however since it is a different individual (or a group of individuals who are making this comment,even if it be they were prompted to do so),in the interest of educating them I am posting this comment
    I watched a couple of videos of this guy and have the following comments to make.One of them is the video on sqrt of 2.
    He is totally ignorant of the foundations of Analysis.He does not even know the rigorous definition of the sqrt of 2.Sqrt 2 is not defined as an infinite decimal.It is defined as a section of rational numbers.This is the reason I gave a reference to the book of G.H.hardy ,in my paper, which treats the irrational numbers from this perspective and lays a firm foundation to the real number system by proving all the familiar laws of algebra hold for them.
    His objection to the sentence ‘sqrt 2 is irrational’ is also completely addressed by Hardy in the first few pages of his book ‘A course of pure Mathematics’ which the person posting this comment obviously has no knwoledge of nor the author of OP cared to read after I Posted the references in my paper(or may be is too challenged to understand it).
    In other words Wildbergers understanding of the theory of irrationals,is sufficiently inadequate for us to understand why he is faltering.However there is one thing about this guy,wildberger, he is at least willing to learn if he is at fault and asking for people to come up refuting him, which is not a big deal as I pointed as above .In fact as one of the Mathematicians in the discussion on Quora (which link I have posted above) says,most of the objections he raises are legitimate and have been fixed a hundred years ago.In the early 1900’s the Mathematics text books carried out such rigorous approaches to Mathematics,this guy being educated recently,may be the 60’sand seventies and that too in US has obviously missed such an opportunity for a rigorous Mathematical education.(in fact in Britian and even in India such a rigorous education was offered till the 60’s in the universities)
    Another point I would like to touch on, is about the definition of a function.I understand none of the recent texts carry a rigorous definition of a function.I refer the readers who are interested, to the book by E.G.phillips an old classic on Mathematical analysis,which gives a very satisfactory and rigorous definition of a function.
    All said and done most of the objections that Wildberger raises though legitimate,I am sure, have been addressed in the nineteenth and twentieth centuries,by the eminent Mathematicians of that time (I have indicated how they addressed, two of them, here).His critique of present day Math pertains more to the inadequacies in the educational system and has nothing to do per se with Mathematics.His objections can be addressed,for they have already been fixed.
    Raju’s critique of Math (I hate to use the word formal math,since Math for most part is intended to be formal),on the other hand involves crackpottery and ignorance, the latter as in the case of wildberger can be addressed,the former ,cannot be helped by others. Raju has to help himself.
    To sum up, as the link, I have given above says, Wildberger is neither a joke nor a genius.Same is true of Raju.

  10. Subramanyam Durbha

    I watched a few more videos of Wildberger. Yes, he does talk about Dedikind cuts. That does not in any way improve my rating of the material he presents from, one of reviews, the link I provided above says “may be not Wankish Crackpottery,but interesting crackpottery”.
    His problem seems to be with infinite sets and construction of subsets.
    His mistake is this.Practical infeasibility implies theoretical infeasibilty.
    Wrong.Counter example: I cannot visit the nearest Galaxy,which is several thousand light years away( Physically) but, in thought, I can visit it(feasible in imagination)
    In other words the absence of a practically feasible algorithm for the attainment or realization of a theoretical construct does not in itself imply the non existence of the construct.
    Wildbergers inability to grasp this simple fact ,which I am sure most Mathematicians will agree to, permanently disqualifies him from being taken seriously.

    1. spkc

      His problem is not just with practical vs theoretical feasibility but with the construction of set theory based on assumed notions i.e axioms, whose logical consistency can be questioned outside of basic basic bi variant logic. The analogy you gave about nearest galaxy is flawed because we can measure the distance between earth and the said galaxy experimentally without needing infinite memory(space) and time(unlike the postulates of set theory which need infinite memory and time to be proved/disproved) and hence its not a practically unachievable task(in the future). I suggest that you go through all the videos in the math fundaments playlist of prof Wildberger and then come to a conclusion about his intellect.

  11. Subramanyam Durbha

    This is in response to Skpc’s comment above.Since my earlier comments have been moderated, I am submitting this comment without straying much into the assessment of intellectual capabilities of Wildberger (or the author of the OP),as suggested by Skpc, once again in the interest of educating him, by correcting the misconcepts he has acquired of my arguments presented before, against Wildbergers critique of math and for the painstaking efforts (presumably) he is making to defend Wildberger and C.K.Raju in this context.
    Firstly axioms in Mathematics are never meant to be proved or disproved. They are self evident truths and no further discourse is necessary on their assumptions.
    Secondly skpc has missed my point in my analogy of travel to the nearest galaxy.Whether it is possible in the future is not the point.It is not possible now.The point I was making is that, Wildberger is foolishly embarking on enumerating (constructing) an infinite set which he will never be able to do.What is more foolish is to reject such a set from further consideration because he is not able to construct it. That is one of the reasons the mathematicians in the discussion on Quora,have dubbed his arguments in the way as stated in the link I provided above.
    Thirdly Bivariant logic is just fine in Mathematics.The existence of other logics is well known, including Buddhist ,and jain logics (3 or 4 valued) I suggest you go through my article where I exposed Rajus false arguments regarding other logics. In reference to that argument,would you like to try his 3 or 4 valued logic in trying to put your hand in fire. Clearly it is a wrong application of your arsenal of logics.
    Fourthly the existence of an actual infinity is irrelevant for the foundations of set theory.I suggest you explore some old literature on this (Books) which build set theory without an appeal to an actual infinity(in reality).These books appeared around the time when it was a matter of debate whether an actual infinity exists.(before the end of 19th century and the beginning of 20 th century)
    Fifthly, most scientists believe that the universe is infinite (which is what I believe too) and I believe on the authority of our scriptures which say that time is quasi cylic ( and hence there is no eternity).
    As I pointed as above both these are anyway irrelevant to the foundations of set-theory.
    Finally,since it is missing in the thread of arguments above, Rajus question to me “How for example do you know that Euclid is the author of ‘Euclids Elements’? My response is this.’The oxyrynchus papyrus’ housed in the University of Pennsylvania museum,dated first century A.D. proves the existence of the ‘Elements’ before the time of Theon of Alexandria 3rd century A.D. Therefore theon of Alexandria could not have authored ‘Euclids Elements’ and Euclid is indeed the author of ‘Euclids Elements’.
    A critical examination of the evidence I presented about the existence of Euclid (Applonius of Perga’s reference to Euclid) coupled with the oxyrynchus papyrus conclusively proves that Euclid authored Euclids Elements.I am sure Mathematicians would unanimously agree on this as a valid argument for the existence of Euclid and his authorship of the elements.
    Having already tasted the flavor of Widberger’s arguments in one instance (his trying to construct an infinite set) I do not see a need to proceed further in trying to come to any conclusion of his intellect different from what I observed earlier.

  12. Rohan Kokane

    Dear prof Ck raju,
    I have read Nath’s article on frontline (though did not understand it in its entirety)
    A serious objection which have been put forward by some western historians,including nath,is that indians developed methods of calculus only for trigonometric functions which does not include other important functions.
    I believe that it is wrong to judge Indian calculus from the standards of calculus as used today as knowledge keeps updating.
    Hence,this cannot be an excuse for denying credit for Indians as anything old in science which we give credit to ancient scientists rarely exist in that form anymore,it gets updated and innovated.
    But since I am a layperson in this matters, what is your take on this?

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