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.
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 Secularism, 11 (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.
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.