C. K. Raju

"This plays on the psychology of the colonized, who are persistently taught that only Western sources are reliable, a cardinal principle of Wikipedia even today."

"Zeroism is an alternative philosophy of mathematics,1 based on śūnyavāda, a realistic philosophy often ascribed to the Buddhist teacher Nagarjuna (2nd c. CE).2 It is now called zeroism to emphasize that the concern is with the practical and contemporary benefits of that śūnyavāda philosophy, as distinct from fidelity to this or that interpretation of the textual sources of śūnyavāda, which have often been misunderstood and mangled by scholars unfamiliar with the idiom. Indeed, the whole idea of relying on the authority of textual sources is a practice of scriptural traditions, and contrary to śūnyavāda, which denies the validity of proof by authority."

"In Indian philosophy only empirical proof (प्रतरक प्रमार) was universally accepted, by all schools of philosophy. Further, the Lokayata accepted only empirical proofs; specifically they rejected deductive proof (अिम ाि) as inferior. As the Lokayata critique of deductive proofs shows, and as even formal mathematicians today admit, deductively proven theorems are, at best, true relative to postulates. Hence, mere deductive proof does NOT lead to valid knowledge (the goal of Indian philosophy) until the postulates are empirically validated, as in science."

"While there is nothing Vedic in “Vedic mathematics”, there is church dogma in formal mathematics."

"But, the original Indian understanding of calculus as a method of numerically solving differential equations,22 together with non-Archimedean arithmetic and the philosophy of zeroism makes calculus easy enough to teach in five days.24 This ease enables students to solve harder problems such as elliptic integrals required for the first science experiment with the simple pendulum."

"The key thing to recognise here is that false history was (and remains) a key source of colonial power."

"Of course, formal Western mathematics (and indeed much of Western philosophy) is likely to be a long-term casualty of any departure from 2-valued logic. In fact, the very idea that logic (or the basis of probability) is not culturally universal, and may not be empirically certain, unsettles a large segment of Western thought, and its traditional beliefs about induction and deduction."

"The linguistic error of translation in the term “sine” was accompanied by a conceptual error, as in the very word “trigonometry” where the functions relate to the circle, not the triangle. That error persists to this day in the teaching of “trigonometry” which is stuck in the pre-Āryabhat.a era. The word “trigonometry” is in quotes, since this geometric method wrongly suggests that the concepts of sine and cosine relate to the triangle, whereas they actually relate to the circle."

"The translated Indian texts would naturally have gone first to the Jesuit general. There is ample circumstantial evidence that did happen. Christoph Clavius, who authored the Gregorian calendar reform, also published in his name a table of sines in 1607. Curiously, these were the so-called Rsines, in that they explicitly involved the radius of the circle. Simon Stevin follows the same practice for his secant tables. Curiously, Clavius used the same large number for the radius as used in Madhava’s values (Clavius 1607). Documentary evidence of a connection comes from Clavius’ student Matteo Ricci who visited Cochin just prior to the Gregorian reform to get information about Indian methods of timekeeping (Ricci 1581). The Indian timekeeping or astronomy texts near Cochin contained detailed accounts of the calculus. On the epistemic test, those who copy don’t fully understand what they copy. This is also evidence of transmission: Clavius got the imported sine values explicitly interpolated to build a larger table, but did not know enough trigonometry to calculate the size of the earth. Recall that this size was routinely mentioned in Indian texts, and that the size of the earth was a key parameter needed for determining longitudes. (The calendar reform only settled the problem of latitudes.)"

"There is other circumstantial evidence of transmission of calculus to Europe. Clavius’ contemporary, Julius Scaliger, is credited with Julian day-number system which is the same as the Indian ahargan.a. Likewise, another contemporary Tycho Brahe, Royal Astronomer to the Holy Roman Empire, produced the Tychonic astronomical model (in which all planets go round the Sun, which itself goes round the earth) which is just a carbon copy of the astronomical model of Nı̄lakant.ha, stated in his Tantrasangraha. Tycho’s masonry instruments (copied from Ulugh Beg’s Samarkand observatory) were not accurate enough to make accurate observations of Mars, such as made by Parameswaran over a 50 year period. Nevertheless, Tycho, in those days of the Inquisition, kept some secret documents with which his assistant Kepler decamped, after Tycho’s untimely death or murder. Why did Tycho keep mere observations such a secret from his own assistant? How did Kepler, a nearly blind person, arrive at those super-accurate observations, without appropriate instruments?"

"Specifically, time machines are impossible, since realistic time travel implies spontaneity (different from chance). The novel features of this model can be expected to be especially prominent at the microphysical level of biological macromolecules and single particles."

"But how trustworthy is that authority? Are mathematicians more honest than church priests? Uncritical acceptance of authority always invites abuse of authority, and mathematicians are no exception."

"Since there is no evidence for “Pythagoras”, the terminology of the “Pythagorean theorem” is defended by “myth jumping” successively to each of the myths of the person “Euclid” and the myth of the “Euclid” book, that it has axiomatic proofs. But both those myths are false; there is ample counter-evidence against both myths."

"But Hoodbhoy declares the belief in “laws” to be the basis of physics because of his ideological and colonial commitment to slavish imitation of Christian superstitions about laws of nature, an ideology he wants to force on people using the authority of science, just like Macaulay. What he is using is just a modification of the preacher’s doomsday argument (“Covid is round the corner; repent and uncritically accept the authority of science”). Scientists are not more honest than other humans: there are any number of scientists who were and are rascals, just as there are any number of doctors today who are commercialised and dishonest. One uncritically trusts their authority at one’s peril. One can understand why Imran Khan, in a televised debate, got irritated enough to ask Hoodbhoy what he was paid for his propaganda!"

"Let us understand one easy implication of this. The claim that the “Greek” proof of the “Pythagorean theorem” in the “Euclid” book is “superior” to its Indian proof is complete balderdash, though lots of “reliable sources” have asserted it. The actual “Euclid” book uses the same principles of proof as the Indian notion of proof, but is only a lot more prolix."

"It was much later that I realized that most people conflate formal mathematics with the kindergarten mathematics they learned. Thus, they are ignorant of this divorce of formal mathematics from the empirical, though this is stated even at the level of the class IX Indian school text."

"To put matters bluntly, I come from a tradition where, even over 2500 years ago, in supposedly barbaric pre-Christian times, the followers of Buddha and Mahavira were ferociously debating whether it was ethical to unintentionally step on an ant and kill it (hence Jain monks wear masks to avoid unintentionally swallowing any tiny creature, or insect, and carry brooms to sweep aside any ants etc. in their path, to avoid “unintentionally” stepping on them and killing them). So the question is really: what was the ethical or moral justification in the West for mass murder and mass slavery of human beings?"

"However, there is a further problem with deductive proofs. Mathematical theorems, even if validly proved, are invalid knowledge. Hence, the people's philosophers (Lokayata) from India rejected deduction as fallible thousands of years before the church declared it as infallible! The Lokayata objection was simple: deduction may begin from false premises. The classic Lokayata example (SURI 2000) was that observing a wolf’s footprints, people wrongly inferred that a wolf was around, when in actual fact the wolf’s footprints were made at night by a man to demonstrate the fallibility of deductive inference."

"But the primary rule of Western faith-based history of math, as one should well understand by now, is that myth is evidence, and all evidence contrary to the myth, even if this is evidence in front of one's eyes, should be thrown out to preserve the myth."

"There is not the slightest doubt that every piece of empirical evidence can be explained away by one who wants to hang on to the myth of Euclid, just as every piece of evidence against astrology can be explained away by those who make a living from it."

"Such forgeries were common enough... So unenviable was the reputation that priests had acquired in this matter that Isaac Newton spent 50 years of his life trying to undo the forgeries that he thought various priests had incorporated into the Bible, to serve their temporal ends. And the only answer to his scholarly and voluminous accusations was to hide them for some 250 years—in fact they still remain secret."

"note here that the long-standing claims of Euclid's existence, and the surprisingly flimsy evidence on which they are based, also provide an example of the de facto standards of evidence in historiography—standards to decide origin and transmission that should either be uniformly applied elsewhere or rejected here as well."

"The khichdi geometry in the NCERT text for Class 9 is indigestible because it has mixed up the Elements by mixing up elements that ought not to be taken together—like diazepam and alcohol—unless the object is to induce a comatose state."

"It would be rather pointless and confusing to retain in these books information that was incorrect or defective or inaccurate. That is to say, books on science and mathematics would naturally be propagated accretively, with the addition of numerous anonymous updates, though no one maintained a revision history. Certainly Arab authors in Baghdad, for example, were actively disinterested in verbatim translations, but were interested rather more in useful paraphrases and creative reworking."

"Nevertheless, this laughable hypothesis is exactly what has been adopted with the 12th and 16th c. sources of “Greek” or “Hellenic” tradition." Hence, virtually all the knowledge prevalent in the 11th c. world, as known to Indians and Arabs, is attributed to Greeks like Aristotle, Archimedes, and Ptolemy. The fact is that the knowledge in these 11th c. texts accurately reflects the knowledge that then prevailed—as is naturally to be expected. However, Western historians explain this fact not by the simple and natural hypothesis of accretive up- dating of the texts, but by the extraordinary claim that all (or most of) the contemporary knowledge of the 11th c. world was derived by transmission from the Greeks, who had anticipated these developments. There is no other, or direct, evidence that these Greek authors wrote anything at all. Thus, by way of evidence, this extraordinary‘theory of transmission simply begs the question! To complete the story, it is thought enough to supplement it with a speculative chronology, attached to Greek names, based on stray remarks of doubtful authenticity in late texts. This sort of story-telling may be perfectly consonant with the standards of theology (and most early Western historians were priests), but is completely unconvincing from a somewhat more sceptical and down-to-earth point of view."

"Attributing a book to a famous early source added not only to-the authority of the book, but also to its market price in what was evidently a flourishing book bazaar in Baghdad. That many books were fakes and falsely attributed to famous early sources is evident from the Eihrist of al Nadim, a Baghdad shopkeeper of the 10th c., who hence prepared this fihrist or list of books he regarded as genuine. Of course, al Nadim was a shopkeeper, not a scholar, and his concerns about genuineness were limited to saleability—so, common hearsay was good enough for him— and he is unlikely to have been bothered by a well-established fake."

"“Ptolemy’s” Almagest begins (as natural for an 11th c. text) with what look like paraphrases of controversies from the history of Indian astronomy, “Aristotle’s” syllogisms are remarkably similar to the Nyàya theory of syllogisms, "Aristotle" uses theories like those of "action by contact" and the same words like "aether" (= sky = dkdsa) long used in India, and his physics is as similar to Arabic physics as "Archimedes" is to 11th c. Arabic mathematics."

"So, in practice, Western history has used two standards of evidence for transmission: one ultra-lax standard of evidence for transmission from "Greeks", and another ultra-strict standard for transmission to the West. For cases of alleged transmission from the Greeks, mere speculations—a speculative chronology combined with speculative attribution—are regarded as ample evidence of transmission. In the other direction, similarity with a real earlier work, by a real author, together with a clear channel of transmission, do not prove anything, for there is always the possibility of repeated miracles by which any number of people in the West may independently reinvent things just when they could be transmitted."

"The Doctrine of Christian Discovery, which instigated the subsequent triple genocide in three continents of South and North America and later Australia—the only known successful cases of genocide in a literal sense—was explicitly proclaimed in papal bulls (Romanus Pontifex, 1454, and Inter Caetera 1493), which declared it the religious duty of Christians to kill and enslave all non-Christians. The first-hand descriptions of the genocide in the Americas provided by Las Casas (who accompanied Columbus) clearly show that it was religiously motivated, and that those engaged in the genocide thought they were doing their Christian duty by eliminating non-Christians and carrying out God’s will here on earth as it would be in hell."

"Therefore, it is hardly a matter of surprise that there is much similarity between Indian knowledge, and knowledge that has been attributed to the early Greeks based on late Arabic texts: for example, the astronomical model attributed to “Ptolemy” is remarkably similar to Indian astronomical models, "Aristotle's" theory of action by contact, using aether (sky —àkàsa) is as similar to the Nyaya theory as his syllogisms are to Nyàya syllogisms, etc."

"In the many centuries, since Toledo, that Western historians have been talking of transmission from the Greeks, who ever produced a Sanskrit manuscript of Ptolemy? Who ever proved that Aryabhata had seen such a Sanskrit manuscript? Yet every Western reference work on the subject asserts that Indian astronomy is transmitted from the Greeks. So is it the case that these reference works are all out of date, and that the standard of evidence for transmission has now changed? Does Owen Gingerich now deny transmission from the Greeks on the grounds that there is no evidence? Not at all; in the very same article he sticks to the entire fairy tale about transmission from the Greeks. So, it is not so much that the standards of evidence have changed, but that there are (even as of today) two simultaneous standards of evidence for transmission. One for transmission to the West, and another for purported transmission from the West. Not only is the judge biased, the very rules of evidence are biased!"

"So, similarity and precedence do not always establish transmission. Whether or not they establish transmission depends upon the direction of transfer. Thus, in practice, there are two standards of evidence for transmission: an ultra-lax standard for transmission from Greeks, and an ultra-strict standard for transmission to the West."

"In support of the West’s physical claim to the whole world, the Western history of science sought to establish an intellectual claim to all knowledge in the world, especially all scientific knowledge. To situate this claim in its proper perspective, we need to probe a little deeper to understand a bit of the unstated logic behind colonialism. According to the religious beliefs of the colonialists, such an intellectual claim of discovery, in turn, established the colonialist’s moral claim to the whole world. It was these “moral” claims that distinguished colonialism from a simple project of robbing the world by physical force."

"But the mysterious source of Mercator's precise trigonometric values, and his technique, remains unknown to this day. Mercator, who worked with Gemma Frisius at the Catholic University of Louvain, obviously had privileged access to information brought in by sailors and priests returning from India and China, via Antwerp. So it is hardly surprising that the "Mercator" projection is identical with a projection used in maps of the celestial globe from China from at least five centuries earlier—and the same principle could obviously be applied to the terrestrial globe. How- ever, since Mercator was arrested by the Inquisition, and was lucky to escape with his life, it is also not surprising that he kept his "pagan" sources of information a closely guarded secret. The tables of trigonometric values published by Clavius, in 1608, used the Indian de- finition of sines and cosines, and the then common Indian value for the radius of the circle. Hence, these tables far exceeded in accuracy the "tables of secants" provided by earlier nav- igational theorists like Stevin for calculation of loxodromes, which were (at the accuracy of) Aryabhata's values, known to the Arabs. It is hard to see how such accuracy (unprecedented for Europe) could even have been attempted without calculus techniques. Clavius, who au- thored the calendar reform proclaimed by pope Gregory, certainly had access to every bit of information brought in by the Jesuits, but could hardly be expected to be truthful enough to acknowledge his “pagan” sources. Since Clavius’ tables were published several years be- fore the first hint of the calculus “officially” appeared in Europe in the works of Kepler, and since Clavius provides no explanation of his method, it remains a mystery how these high- precision trigonometric values were calculated. The only reasonable explanation is that like his contemporaries, Tycho Brahe, who merely articulates Nilakantha’s astronomical model, or Scaliger, whose “Julian” day number system copies the Indian ahargana system, Clavius obtained his trigonometric values from India."

"We have seen that calculation of loxodromes involved the solution of a problem equivalent to the fundamental theorem of calculus. But that theorem was unknown to Europeans in the 16th c. How, then, did Mercator draw the chart? The abiding nature of the Mercator mystery is due to the fact that it cannot be appropriately solved within the framework of the Western historical narrative about the calculus. The mystery can be resolved by changing that narrative. It is hard to believe that Mercator drew his chart through sheer skill in draftsmanship. It is rather more likely that he had access to information from India or China, which he kept a secret. That this information was adequate to enable the calculation of loxodromes is evident from the fact that loxodromes were earlier used to map the zodiac, and a Chinese [Dunhuang] star map from ca. 950 follows the very same principle of isogonal cylindrical projection that has come to be known as the “Mercator” projection. This chart is reproduced in Needham’s volume."

"The trigonometric values published by Clavius, who was at the centre of the Jesuit web, provide further circumstantial evidence that the Jesuits had obtained the latest Indian texts on mathematics and astronomy, and had studied them. Thus, Clavius’ trigonometric values use exactly the Indian definition of the sine and also the same value of the radius?? used by Indian sources in stating Madhava’s sine values. Further, Clavius was unable to give any explanation for the way those trigonometric values were derived, and, obviously enough, the derivation of such precise values required essentially calculus techniques. Had Clavius himself discovered a striking new procedure, by which to obtain more precise trigonometric values, would he not have announced it, to establish his priority, especially since this was towards the end of his life? In fact, Clavius, though he published sophisticated trigonometric tables in his name, lacked a proper understanding of even elementary trigonometry, since he was unable to use trigonometry to determine a key navigational parameter—the size of the globe."

"When Indian astronomy works, translated by Jesuits in Cochin, started arriving in Europe, Tycho, as one of the most famous astronomers of his day, and the Mathematician of the Holy Roman Empire, would naturally have been chosen as the person to whom they were referred. Nilakantha's model was what later came to be called the “Tychonic” model, which Tycho was trying to check against observations. Why, after all, was Tycho so secretive about his papers, not even allowing his trusted assistant Kepler to see them? In any case, on Tycho's sudden death, Kepler obtained not just Tycho's observations, but also the rest of his papers which contained the underlying theory."

"Galileo's access to Jesuit sources at the Collegio Romano is well documented. Galileo did not himself take up the calculus because he did not quite understand it, as is clear from the difficulties and the various paradoxes of the infinite that he raised in his letters to Cavalieri. Thus, this state of affairs is better explained by supposing that there was a common body of Indian work related to the calculus, known to both Galileo and Cavalieri, and that Galileo was not satisfied with Cavalieri's interpretation of it, and not willing to risk his reputation, while Cavalieri was. Nevertheless, out of deference for his teacher, he waited five years before staking his claim."

"The influence of Cavalieri’s work on Torricelli and Roberval is well known. Roberval was a member of Mersenne’s discussion group, and was involved, along with Fermat and Pascal, in debating with Descartes, the validity of these new methods. There is a clear chain of influence from Cavalieri to Torricelli, to Wallis to Gregory and Newton. As is well known, while Newton acknowledged the influence of Wallis, Leibniz acknowledged the influence of Pascal on their respective works relating to the calculus. A diffusionist model for the calculus in Europe is, therefore, rather more appropriate than the simplistic Eurocentric model which gives all credit to Newton and/or Leibniz just because the two had a nasty priority dispute! There is further circumstantial evidence of transmission. The calculus methods of Cavalieri, Roberval, Fermat and Pascal are very similar to those of the Yuktsbhasd, TantrasangrahaVyakhya, Kriyakramakari."

"The issue of transmission does not end with the receipt of the calculus in Europe. Because of the epistemological differences between Indian and European mathematics, actual assimilation of the calculus took a long time. It is worthwhile trying to understand this assimilation process, since this sheds light on the historical as well as the contemporary mathematical situation, and since such a task seems never before to have been attempted by historians of mathematics, who have not acknowledged or understood the historical existence of epistemological differences within mathematics."

"Briefly, Europe inherited not one but two mathematical traditions: (i) from Greece and Egypt a mathematics that was spiritual, anti-empirical, proof-oriented, and explicitly religious, and (ii) from India via Arabs a mathematics that was pro-empirical, and calculation-oriented, with practical objectives.' Much mathematics taught at the K-12 level is of Indo-Arabic origin: (1) arithmetic, (2) algebra, (3) trigonometry, and (4) calculus. Despite the obviously different philosophical orientations of these two streams of mathematics Europe recognized only a single possible philosophy of a "universal" European mathematics, into which it forcibly sought to fit both mathematical streams."

"... formal mathematics is no more than a culturally-dependent system of aesthetics, ... it may continue to be taught like Western music..."

"The linkage of time perceptions to ethics applies also to Buddhism. The relevant notion of time here is the notion of paticca samuppada , an understanding of which was equated by the Buddha with an understanding of the dhamma. This is a deep and tricky point about Buddhist ethics"

"Unfortunately, there are double standards in the matter: one standard for Greek history, another for Indian."

"Witzel’s way of arguing, by concocting a false position for the opponent and attacking it, is unethical, whether it was done deliberately or because of lack of understanding."

"We have no truthful account of what happened next, for the written accounts that have come down to us are all from the viewpoint of the Christian priests. This is rather like describing a rape and murder from the viewpoint of the rapist and the murderer on the grounds that there is no other reliable source of evidence."

"Students need to be taught that belief in Einstein or Stephen Hawking is not less superstitious than the belief in Sai Baba, just because those figures are endowed with high scientific authority in the West. They should be taught to use right means of validating knowledge, without relying on authority. (This applies also to decision makers who should not rely on the privately expressed opinions of “experts”, since this may involve a conflict of interests, but should use public discussions.)"

"People have been indoctrinated to believe that any attempt to correct Western history is necessarily chauvinistic. This latter belief has been greatly helped along by the more extreme elements in the non-West who have often made wild claims."

"In sharp contrast, all Indian systems of philosophy, without any exception, accept the empirical (pratyaksa) as the first means of proof (pramana) while the Lokayata reject inference/deduction as unreliable. So, Indian philosophy considered empirical proof as more reliable than logical inference.Thus, the contrary idea of metaphysical proof as “stronger” than empirical proof would lead at one stroke to the rejection of all Indian systems of philosophy. This illustrates how the metaphysics of formal math is not universal but is biased against other systems of philosophy."

"However, those precise trigonometric values were calculated by Indian mathematicians using infinite series expansions (today called “Taylor's” expansion, “Leibniz” series, etc.), and sophisticated techniques to sum infinite series. These techniques were not comprehended by European mathematicians (who were, then, still struggling at the level of decimal fractions introduced by Stevin, only in 1582). The key difficulty was with the notion of infinite sums, as in the non-terminating, non- recurring decimal expansion for the number pi. The notion of infinity brought religious beliefs prominently into play."