Sep 24, 2023

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Sep 24, 2023

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*by* * C. K. Raju*

The Indian origin of Calculus and its Transmission to Europe

Abstract

The California education department has introduced a new math framework to meet the projected needs of future technology. Critics argue against it, calling it “cancelling the calculus”. This article explains the alternative of teaching the decolonised calculus which makes calculus easy and enables students to solve harder problems, not covered in usual calculus texts. This has been demonstrated in pedagogical experiments with 8 groups in 3 countries. How is this achieved? The origins of the present-day difficulties in teaching calculus relate to real numbers and limits. Those were invented long after the purported discovery of the calculus by Newton and Leibniz. Why? Because Europeans realized they lacked a clear understanding of the calculus. How did Newton and Leibniz discover the calculus without understanding it? Because Europeans stole the calculus from India, in connection with their navigational problem, and didn’t acknowledge the theft. Does that make a difference to teaching calculus? Yes, because knowledge thieves often fail to understand what they steal. Though the West failed to understand the Indian epistemology of the calculus, Westerners believe they did some “superior” kind of mathematics as in the racist terminology of the “Pythagorean theorem”. We explain that contrary to widespread myths, its actual proof in the “Euclid” book uses the same principles of proof as its Indian proof. This is because, contrary to the myth, axiomatic proofs are NOT actually found in the Euclid book. Indeed, real numbers were invented because even the first proposition of the “Euclid” book lacks an axiomatic proof.

Introduction: California seeks to “cancel” the calculus

The California education department has proposed a major change in the math curriculum for K-12 schools.[1] Though related to the requirements of future high technology the proposed change has led to a huge backlash with critics calling it “cancelling of the calculus”,[2] or killing the calculus,[3] a dumbing down of the math syllabus in schools, and asking for it to be replaced.[4]

The controversial aspect of the change relates to the teaching of calculus, often regarded as the toughest part of math in school and college. The difficulty of teaching and learning calculus is clear from the very size of the typical college calculus texts which run into over 1300 large-size pages.[5] Hence, mastering the calculus, or, rather, performing well in typical calculus tests, came to be seen as a sign of talent. Since the 1980's, in the US, calculus courses at the K-12 level had become a de facto prerequisite for entry to prestigious institutions, the first step to a better job. The situation is similar in India: K-12 calculus is an essential part of the prestigious nationwide IIT entrance examination.

However, educational research in the US[6] showed that these K-12 calculus courses were of little value to the students in *understanding* calculus. Less than 5% of the students benefit. The majority repeat the K-12 calculus course at the college level, or, regress to pre-calculus courses, or bridge courses.

Another key motivation for “cancelling the calculus” is the recent change in the requirements of the high-tech industry which relates to statistics. Recent times have seen an enormous increase in jobs related to machine learning, data science, and statistics, called twenty-first century math. However, calculus is needed for a good understanding of statistics. Indeed, the notion of probability is at the foundation of statistics, which mathematicians understand using “measure theory”.[7] This means the core statistical notion of “average”, in a non-discrete situation, involves an advanced form of the calculus, the Lebesgue integral,[8] not the integral as anti-derivative, nor the Riemann integral.

Indeed, statistics is notorious for reaching incorrect or deliberately misleading conclusions. As the proverbial saying goes, there are “lies, damned lies, and statistics”. A widespread lack of understanding of key aspects of statistics, as might arise from teaching statistics without calculus, could decidedly worsen that.

The aim of this article is to explain that there is an alternative: a way to teach the calculus which makes calculus easy, while preserving all practical value.

The cause of difficulties with the calculus

The Californian proposal to “cancel the calculus” is partly based on the observation that students learning the calculus, at the K-12 level, fail to understand it. There are very good causes for the observed lack of understanding of calculus among students.

Thus, the notions of derivative and integral are at the core of the calculus. Both these notions are today defined using the notion of limit. But limits, hence the core notions of the calculus, are *explicitly left undefined *in Indian school texts. Thus, the current NCERT text for class XI says (chapter 13, p. 281): “First, we give an intuitive idea of derivative (**without actually defining it**). Then we give a naive definition of limit and study some algebra of limits.”[9] [emphasis added] It is perverse to blame the students or teachers (as is commonly done), for failing to understand the calculus, when the text itself is unwilling even to *define* the basic concepts being taught.

The consequence? My own calculus pre-test for science and engineering undergraduates[10] shows that not even a single student manages to get even zero, though they are warned that attempting questions which they don’t understand, hence answer wrongly, would fetch negative marks. That is, not only do beginning undergraduate students not understand anything of the calculus they learnt in school, they have completely wrong ideas based on what they imagine they do “intuitively” understand. If all students perform badly, the students and teachers are not to blame, the text or the subject is.

The problem is not with the Indian NCERT text: the situation is the same, but more deceptively stated, in the usual internationally-used college calculus (“Thomas’ Calculus”) texts. These texts do have a section on the “precise definition of limits”. This is deceptive because the definition given is NOT precise or rigorous; it includes the ritualistic epsilons and deltas,[11] but leaves out an essential ingredient: real numbers are needed for limits, but they are *not* defined in these texts, but left for later texts on “Advanced calculus”[12] or “Real Analysis”.[13] That is, the calculus texts leave out the core definitions of real numbers, hence limits, making it difficult for students to understand. Though, today, “real” numbers are found in the first chapter of the NCERT class IX school text,[14] a precise definition of “real numbers” is *not* given (because it is too hard).

The need for real numbers, on the Western understanding of calculus, can perhaps be driven home for the layperson in another way: Karl Marx failed to understand the calculus[15] and called the “calculus of Newton and Leibniz” as “mystical”. Marx’s lack of understanding of calculus has been correctly attributed, by the celebrated Marxist historian Struik,[16] to the fact that Marx studied calculus *before* the invention of “real” numbers. Therefore, one can expect that it *is* confusing to teach calculus without real numbers, as done in “Thomas’ Calculus” texts. Indeed, before the invention of real numbers, Europeans themselves complained about calculus, and Newton’s own “fluxions” are rightly abandoned as utterly confused.

Now, the stock history of the calculus (e.g., as stated in the NCERT class XI math text[17]) asserts that calculus was “discovered” by Newton and Leibniz. But Newton died 150 years *before* the invention of real numbers by Richard Dedekind. This raises two questions: (1) **Practical:** Without any knowledge of real numbers how did Newton manage to *apply* the calculus to physics or planetary orbits? (2) **Historical:** How did Newton and Leibniz *discover* the calculus without understanding it? Western mathematicians have avoided these question for centuries.

The first (practical) question will be postponed to part 2 of this article which explains that though real numbers and limits are the cause of all the difficulties in teaching and learning calculus, they have nil practical value: Newton successfully applied the calculus to physics just because real numbers are *not* needed for practical application of the calculus to science, and indeed *cannot* ever be used for its practical applications.

This part focuses on the second (historical) question and its consequence for teaching and learning calculus.

On the principle that “phylogeny is ontogeny”, a growing child retraces the entire history of evolution, starting off in a watery womb, then crawling on all fours, before standing erect. Likewise, the teaching of a subject in the classroom retraces its historical development. The calculus, in Europe, initially involved no real numbers or limits for over two centuries. It is exactly that historical, self-admitted, European lack of understanding of calculus which is being replayed in the calculus classroom today, in fast forward mode. Let us enquire deeper into that history.

Contrary to the stock history, there is impeccable evidence that the calculus was known in India for over a thousand years before Newton and Leibniz.[18] It was transmitted to Europe in the 16^{th} c. by Kochi-based Jesuits in connection with the European navigational problem.

So, what difference does that make to the *teaching* of calculus, if the calculus was invented in India and stolen by Europeans to solve their navigational problem? The key point here is my epistemic test: those who steal knowledge, like students who cheat in an examination, do not *fully* understand what they steal. Calculus originated *differently* in India, with a different philosophy, which the West never fully understood.[19] The true inventors of the calculus could hardly have invented it without a proper understanding of it.

What aspect of the Indian calculus did Europeans miss? They certainly understood the practical aspects, whether applications of calculus to navigation or physics, but missed a deep-seated theoretical aspect. **The Indian calculus involved a different arithmetic (of polynomials) which is non-Archimedean (in present-day terminology), hence admits infinities and infinitesimals, therefore lacks limits.** In contrast, the arithmetic of real numbers is Archimedean. This difference helps to eliminate real numbers and limits, the source of calculus difficulties. This is already a key reason why reverting to that original Indian understanding of the calculus makes it easy, without affecting the practical value of calculus which does not require real numbers.

The alternative: decolonised calculus

Reverting to the original Indian philosophy, with which the calculus originated, is the basis of the decolonised calculus.[20] That not only makes calculus very easy, it enables students to solve *harder* practical problems, not covered in the usual school and college calculus courses.[21] This has been demonstrated in teaching experiments with 8 groups in 5 universities in 3 countries, including 3 universities in India (Central University of Tibetan Studies, Sarnath,[22] Universiti Sains Malaysia,[23] CISSC, Iran,[24] Ambedkar University Delhi,[25] and SGT University[26] Delhi NCR).

The calculus was first invented in India by the 5th c. Aryabhata to derive precise trigonometric values, precise to the first sexagesimal minute (kala) or about five decimal places. He used a finite-difference method today called “Euler’s” method of numerically solving ordinary differential equations.[27] Aryabhata’s method was extended by his follower Vateswhar to the second sexagesimal minute (*vikala*). Later (14th c.) his disciples in Kerala extended that to the third sexagesimal minute, precise to about 8-10 decimal places.[28] This was still a few centuries *before* Newton. In the process, Indians also originated “Stirling’s” formula,[29] the infinite “Taylor’s” series for the sine, cosine,[30] and arctangent functions,[31] and hence also the infinite “Leibniz” series for π.[32] Indians certainly were the first to sum the *infinite* geometric series,[33] and they also derived rapidly convergent versions[34] of the various infinite series mentioned above.

There are good reasons to call it the Indian calculus, and not the “Kerala calculus” as it is often wrongly known. First, this is true in a simple sense: the key aspects of the Indian calculus, *especially the three aspects critical to its current-day teaching*, developed centuries before the “Kerala school”. Second, the real originator of the calculus, Aryabhata, was a dalit[35] from Patna in Bihar, so this is totally contrary to the Western demonisation of Indian society as always casteist, a demonisation which accompanied their self-glorification using Newton et al. Even more contrary to that Western demonisation of India is the fact that Aryabhata’s disciples in the “Kerala school”, such as Nilakantha Somasutvan, who wrote a commentary on Aryabhata’s work,[36] were the highest caste (Namboodiri) Brahmins. To the contrary it *is* casteist and chauvinist to keep calling it “Kerala school”, as some chauvinists and present-day serial plagiarists[37] of my work have done.

How and why Newton and Leibniz stole the Indian calculus

So how did Newton, Leibniz and others get to know about the Indian calculus? Because the calculus was used in India to derive precise trigonometric values, which were very important for Europeans in the 16th century. Then, precise trigonometric values were desperately required for accurate navigation, to determine latitude, longitude, *and* loxodromes (needed to prepare navigational charts usable by European sailors, such as the Mercator chart[38]).

European dreams of wealth, then, rested on overseas trade (or piracy) for which proper knowledge of navigation was critical. Unlike other people they called “primitive”, 16th c. Europeans were themselves excessively backward in mathematics (including the arithmetic of fractions), and did not understand (celestial) navigation. Indeed, Europeans recognized their backwardness: the navigational problem was recognized as the biggest scientific challenge in Europe, and many European governments kept offering large rewards for its solution from the 16th until the 18th century. The last reward (of 20,000 UKP) was offered by the British Parliament, which legislated a Board of Longitude, in 1711, when that amount was a fortune. Thus, Europeans had huge motive to steal that Indian knowledge of calculus and its precise trigonometric values for the sake of their means of wealth, dependent on European navigation.

Apart from motive, there was ample opportunity: the first Roman Catholic missionary school was set up in Kochi in 1501, and this later (ca. 1550) became a college which was taken over by Jesuits. The missionary interest was to teach the local Syrian Christians. Naturally, for this purpose, the missionaries learnt the local language Malayalam, which they were teaching in their Kochi college, and, eventually, Jesuits even learnt Sanskrit. Through their local contacts they systematically gathered information and books, which they translated en masse, as they had done with Arabic books centuries earlier in Toledo. These translated books were despatched to Rome. Did these Christian priests, or the recipients of the texts, not have any conscience? No. The doctrine of Christian discovery[39] taught them that stealing knowledge from non-Christians was morally the right thing to do.

There is ample circumstantial evidence of European theft of Indian knowledge of calculus, with the help of their Jesuit college in Kochi. For example, essentially the Indian trigonometric values were published by the Jesuit general Christoph Clavius, in his name, in 1607,[40] to similar precision. Noticeably, Clavius used the *Indian* definitions of the trigonometric functions. Those definitions differed from the current definitions of “trigonometric” functions in that (a) they correctly use a circle, rather than (incorrectly) a right-angled triangle, and (b) they explicitly include the radius of the circle. This much is manifest from the very title of Clavius’ 1607 book listing trigonometric values “for the radius of 10,000,000”. Many Indian texts were then available in the vicinity of the Jesuit college in Kochi, including the Karanapaddhati (VI, 7), for example, which states the circumference of a circle as 31,415,926,536 for a radius of 10,000,000,000.[41] There is documentary evidence that Clavius’ favourite student (and later biographer) Matteo Ricci,[42] trained in navigation, and came to Kochi to collect knowledge of Indian time-keeping (astronomical) techniques, which he sent on to Clavius.

Of course, if all the land in the Americas and Australia could be not only grabbed, but that land grab could be morally and legally justified on the doctrine of Christian discovery,[43] grabbing credit for the calculus was relatively simple. But the truth has unexpected ways of emerging.

On my epistemic test, theft is established by the fact that knowledge thieves, like students who cheat in an exam, don’t *fully* understand what they steal. In history, where “balance of probabilities” is the right standard of proof, lack of understanding of a purported discovery is proof of theft of knowledge. Thus, amusingly, despite such a high-level of precision in “his” trigonometric values, Clavius did not know enough trigonometry to determine the radius of the earth, a critical parameter in navigation, then needed to be able to calculate longitude difference from latitude difference, and, for example, the distance travelled. As the 7th c. Indian mathematician Brahmagupta caustically remarked, “ignorance of the earth’s radius makes longitude [calculations] futile”.[44] However, Europeans were fixated on the wrong value of earth’s radius ever since Columbus grossly underestimated the earth’s size by 40%, to facilitate funding for his project of sailing West to reach India in the East.

Other circumstantial evidence of theft is provided by the evidence for Indian mathematics texts circulating in Europe in the 17th century. For example, Fermat's 1657 challenge problem, which no European mathematician could then solve, was a *solved exercise* in a text of the Indian mathematician Bhaskara II.[45] The numbers involved in the solution to that problem (“Pell’s equation”) are so very large (a nine and a ten digit number *x* =226153980, *y* = 1766319049) that the possibility of “independent rediscovery” is zero, especially given that “dependent discovery” was very much possible, since the 16th c. when many Europeans and missionaries had arrived in India.

Likewise, the infinite series for the sine function, for which Newton falsely claimed credit, and the infinite series for the number π for which Leibniz falsely claimed credit, are found in earlier Indian texts.[46] Newton and Leibniz claimed credit[47] on the same obnoxious doctrine of Christian discovery[48] used to legalise the theft of land from native Americans. These facts are in the public domain for at least the last two decades,[49] and no one has contested them, though they have been repeatedly plagiarised by some Britishers,[50] on the same doctrine of Christian discovery: that Christians have the divine right to legally claim any piece of knowledge or land.

To summarize, there was ample motivation (European navigational problem) and opportunity (Jesuit college in Cochin) for the theft of the Indian calculus. The available circumstantial evidence (e.g., Fermat’s challenge) and documentary evidence (e.g., Ricci’s letter) support this conclusion. The conclusion is sealed by the epistemic test. Though we have used a standard of evidence stronger than that used in criminal law, to prove the theft of calculus beyond reasonable doubt, the “benefit of doubt” does *not* legally apply to history, only to living persons. The correct standard of proof in history is “balance of probabilities”, used in civil cases. On either standard of evidence, Newton and Leibniz stole the calculus from India.

The racist myths of Pythagoras and Euclid

As already summarily explained, the European lack of understanding of the calculus, stolen from India, was due to the epistemic differences between Indian ganita and Western mathematics. Hence, also, the decolonised calculus, by eliminating that lack of understanding, makes the teaching of calculus easy while actually enhancing its practical applications.

**However, the issue is not merely correction of history: the adoption of the decolonised calculus involves overturning deep-seated racist and colonial prejudices**. These prejudices are not even easily recognizable as racist since they involve a false history *and* a bad philosophy of mathematics which was used to assert racist, colonial, and Western civilizational and intellectual superiority. Specifically, racism was bolstered by the belief that Whites/West invented a “superior” kind of mathematics.

Racism, which is popularly understood as color prejudice, is better combated once it is understood that racism is not limited to colour prejudice but is about various inter-related and mutually supportive assertions of superiority, especially through a false history *and* a bad philosophy of math. (I emphasize the bad philosophy part because people are fooled by that combination of history and philosophy and focus only on the false history.) **These interconnections have been explained in detail in two recent articles, on “Euclid must fall”**[51]. While school texts today avoid *explicit* verbalisation of colour prejudice, they still openly propagate the related claim of Western superiority, and make the connection with assertions of White superiority clear, by showing images of e.g. Euclid as white-skinned.

For example, consider the terminology of the “Pythagorean theorem”. The underlying colour prejudice is *implicit* in the justification long-offered for this terminology,[52] that (White) Greeks did something *superior* in geometry compared to (Black) Egyptians, even though the latter built pyramids. Though this “glorification of Greeks” has been strongly contested,[53] the further act of racism, by present-day mathematicians, is to just strike a superior pose and completely ignore that contestation and just keep reiterating the purported superiority of “Greeks”. Thus, though the Californian proposals claim to be rooting for equity, and are purportedly against racism, there is not the slightest hint in them that they plan to abandon or reconsider this terminology of the “Pythagorean theorem” or even take cognizance of this contestation. That is, their concern for racism is sheer hypocrisy. Since the blatant tactic is to preserve that claim by avoiding public discussion of this racist terminology, let us do so.

Why exactly was the mathematics, which the Greeks *purportedly* did, “superior”? For example, the current Indian NCERT class IX text (p. 79) asserts, “In fact,...Egyptians used geometry mostly for practical purposes and did very little to develop it as a systematic science. But in civilisations like Greece, the emphasis was on the *reasoning* behind why certain constructions work. The Greeks were interested in establishing the truth of the statements they discovered using deductive reasoning (see Appendix 1).” [Appendix 1 is about axiomatic proofs.] So, ultimately, the claim is that Greeks did something “superior” in mathematics because they purportedly provided a special and “superior” kind of proof called axiomatic proof.

But where exactly is the *evidence* that “Greeks provided axiomatic proofs”? There is none. Certainly, **there is no evidence that Pythagoras gave any kind of proof of the theorem named after him, and no evidence about what that proof was. Indeed, there is no evidence for even the existence of Pythagoras, only some myths.**

So, why not abandon the racist terminology of the “Pythagorean theorem”? The determined attempts to hang on to that racist terminology are clear from the tactic of myth jumping.

Indeed, when one such myth is challenged, it is typical to offer another *myth* as evidence, and when the second myth is challenged a third myth is offered and so on! This “myth jumping” is just the age-old tactic of defending one lie by telling a thousand more. This tactic is a good way to tire out the questioner seeking to expose the initial lie. Beware! The very act of jumping to a second myth is an indirect admission that the supporters *know* that there is no evidence for the first myth.

In the case of Pythagoras, the jump is to the myth of “Euclid”. Of course, this “Euclid” is depicted as a White male in Wikipedia, and in the Indian NCERT school texts, even though he was supposedly from Alexandria in Africa. Obviously, there is nil actual evidence for the color of “Euclid’s” skin or his gender, but any assertion to the contrary, that “Euclid” was a black woman,[54] is simply vilified and censored.[55] That is the Western way of preserving a fake history of science, exactly the way the church preserved its false dogmas by vilifying and censoring critics.

Indeed, there is no evidence even for the existence of “Euclid”, and my reward of about USD 3000 for primary evidence of “Euclid”[56] has gone unclaimed for a decade. When a demand was made for evidence for “Euclid”, the brazen official stand of the government agency NCERT was that one *must* believe in “Euclid”, because that is what a number of tertiary Western textbooks state,[57] and it is wrong for the colonised to challenge it by demanding *primary* evidence for Western history. For the special case of Western history, everyone *must* believe it without evidence!

This is exactly the stand of Wikipedia, the truth of Western history must be decided, *not* by primary evidence, but *only* by reference to the *beliefs* (or faith) of a “reliable source”, and as everybody knows(!) only White/Western sources are reliable! That is, the racist myths of “Euclid” and “Pythagoras” are preserved by the mere tautological reiteration of racist or civilizational superiority: Whites/Westerners are superior because they say so, and only those Whites/Westerners are reliable judges of superiority!

Pythagorean proposition and Indian ganita

Against this background, let us now examine the similar claim for the purported inferiority of Indian mathematics. It is undeniable that Indians stated the Pythagorean proposition in the sulba sutra-s[58] (e.g. Baudhayana, 1.9,1.12, Manava 10.10, etc.) and this was long before Pythagoras. So, why continue with the terminology of the Pythagorean theorem?

A common argument[59] for this assertion of Western superiority/Indian inferiority is that Indians may have stated it earlier, but they lacked a *proof* of the “Pythagorean” proposition.

*But this “justification” is a complete lie.* Unlike the mythical Pythagoras, there *is* solid evidence that Indians very much did have a notion of proof as stated in the Nyaya Sutra.[60] Examples of the use of Indian methods of proof in mathematics, including a traditional proof of the “Pythagorean” proposition are readily available.[61]

This Indian method of proof *did* use reasoning or *anumana* or deductive inference, contrary to the brazen falsehood in the NCERT class IX school text that Greeks alone used reasoning. An example of such use of reasoning in India is the *deduction* that the earth is a sphere as asserted by the 5th c. Aryabhata in Gola 6.[62] (In contrast, the contemporaneous Bible, a “reliable source”, declares on faith that the earth is flat.) Aryabhata obviously didn’t travel to space to see the earth, and Lalla explains the reasoning:[63] tall trees cannot be seen from afar, ships disappear over the horizon (which is circular). In contrast, the Bible speaks of a tall tree that *could* be seen from the ends of the earth![64]

However, apart from Nyaya, other schools of thought existed in India and had *different* notions of proof. The most interesting case is that of the Lokayata who *rejected* deductive inference as INFERIOR, completely contrary to the Western belief echoed in the NCERT text, that there is something superior about deduction, and the further falsehood that this belief about superiority of the Western method of proof is uncontested. It is clearly contested from long before the purported dates of any Pythagoras or Euclid.

The moment we accept that mere fact of a contestation, the belief about the purported superiority of deduction is subject to scrutiny, and it crumbles.[65]

Indeed, the limited Lokayata argument[66] was that deduction is inferior since it may result in *invalid* knowledge. This *cannot* be denied today, but is buried under a euphemism: that mathematical theorems are “relative truths”, relative to axioms or assumptions. What is not explained is that *any nonsense proposition whatsoever*, such as the proposition that “a rabbit has two horns”, can be a “relative truth”. Even the Pythagorean theorem is *invalid knowledge* on the curved surface of the earth, or anywhere in curved spacetime. When this is pointed out, it is censored[67] to defend the livelihoods of university mathematicians today.

The other interesting case is that of Buddhists and Jains who did *not* accept 2-valued logic dictatorially asserted to be the “universal” basis of reasoning in axiomatic mathematics. If logic varies so will the theorems of mathematics, from the *same* axioms: for example, theorems proved by contradiction would all fail. Under these circumstances, it is completely pointless to prove mathematical theorems. Logic is certainly NOT culturally universal, in fact, and 2-valued logic does NOT apply to the real world (or quantum mechanics[68]). That is, mathematical theorems are at best “relative truths”: relative to BOTH logic and axioms. However, we will set this too aside.[69]

Let us focus instead on a single issue. Comparison with Indian methods of proof makes very transparent a little-known but embarrassing aspect of the currently accepted definition of an axiomatic mathematical proof. **The key difference between all Indian methods of proof and axiomatic proof concerns the empirical.**

This is clear from the very definition[70] of an axiomatic proof as (a) a sequence of statements in which (b) each statement is either an axiom or (c) is derived from two or more preceding statements by some rule of reasoning such as modus ponens. At no stage is one allowed to assert, for example, “I observe this, therefore, it is true.” Why? Because empirical proof is declared inferior since fallible. This prohibition of the empirical is made clear even in the class IX NCERT math text: “However, each statement in the proof has to be established **using only logic**.... Beware of being deceived by what you see...!” [Appendix 1, p. 301].

Note that this divorce of axiomatic proof from the empirical extends to the axioms. The theorems of axiomatic mathematics are supposed “relative truths” relative to the axioms, but the axiom typically cannot be tested for they are metaphysics in the sense of Popper (usually a metaphysics of infinity). Hence, Russell[71] candidly stated: **“mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true”**.

Against this background, let us return to the claim that Greeks did something “superior” in mathematics because they provided axiomatic proofs. But what is the actual evidence that the Greeks provided axiomatic proofs? It is one thing that there is no evidence for the existence of “Euclid” but where is the evidence for the existence of axiomatic proofs in the *book* Elements[72] purportedly written by “Euclid”? Surely it must be in the book (whatever the book attributed to “Euclid”)?

Now, the *myth* was that the “Euclid” book had axiomatic proofs, and this myth was unanimously and persistently asserted by all the most “reliable” Western sources,** **for over 750 years, from around 1125 CE when the “Euclid” book first came to Europe as a Crusading trophy, until 1890. Laughably, the Cambridge University even incorporated this belief (that the “Euclid” book has axiomatic proofs) as part of its exam regulations regarding “Euclid”.[73] Briefly, the Cambridge exam regulations foolishly decreed that the order of propositions in “Euclid” must be followed, though that order is irrelevant if empirical proofs are accepted, as they are in the text book specially commissioned by the University just for these exam regulations! For example, in India the Pythagorean proposition was the first not the last (or second last) as it is in the “Euclid” book.

Can so many “reliable sources” be so collectively wrong for so many centuries about the “Euclid” book, a text book which is right in front of one’s eyes and was read by all educated Europeans? Alas, yes! (Western sources are so utterly unreliable because of church hegemony, as explained later.) By the 20th c., the belief was publicly exposed as totally false: **there is not a single axiomatic proof of the “Pythagorean” proposition (in any manuscript) of the “Euclid” book.**

Many colonised “educators” don’t understand even class IX math, and know nothing of the history of math, but have deeply imbibed the colonial and Wikipedia ideal of blindly trusting only authoritative Western/racist “reliable sources”, *and* blindly distrusting any non-Western critique. But we have this (absence of axiomatic proofs in the “Euclid” book) also on the authority of Bertrand Russell and David Hilbert, the two modern originators of axiomatic mathematics. Russell pointed out that there are NO axiomatic proofs in the “Euclid” book,[74] and Hilbert wrote a whole book[75] (1898) to supply those missing axiomatic proofs (though he badly mangled the original in the process). Still later (1932), Birkhoff[76] gave another set of postulates (which trivialises the original book). After the Sputnik “crisis”, the US School Mathematics Study Group recommended the adoption of Birkhoff’s postulates.

That is, what we have today is a rewrite of the “Euclid” book, a rewrite which **forces the book to conform to the false myth** about it, a rewrite which mangles the “original” book, which is treated as “defective”. The myth is mightier than the actual book!

To reiterate, the reason to go into such great details about the absence of axiomatic proofs even in the mythical “Euclid's” book is that **this is directly connected to a key difficulty of the calculus: real numbers**. (Birkhoff’s postulates use real numbers.) **Real numbers were invented, by Richard Dedekind in the late 19th c, just because the very first proposition of the “Euclid” book lacks a “superior” axiomatic proof, and proofs involving the empirical were regarded as inferior. **

That first proposition is to construct an equilateral triangle on a given line segment AB. To do so, we construct two arcs, one with centre at A and the other with centre at B, both with the same radius AB. If both arcs intersect at the point C, then ABC is the required equilateral triangle. The problem is that this proof involves the empirical: we *see* the two arcs intersecting, but this may be an illusion, just as two seemingly intersecting arcs on a computer screen need not have any pixel in common.

For complete clarity, we emphasize that even the “Pythagorean theorem” is proved in the “Euclid” book in exactly the same way, making *essential* use of empirical inputs. Thus, the “Pythagorean theorem” requires the 4th proposition, also called the side-angle-side proposition, or simply SAS. The SAS is proved empirically, in the “Euclid” book, by picking up one triangle and placing it on top of the other to *see* that the two are equal.

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. The claim of “superior” Greek proofs in the class IX NCERT school text is false (and racist), because the actual proofs in the “Euclid” book are no different (only the false *myth* about “Euclid” says they are different).

**Footnotes**

- https://www.cde.ca.gov/ci/ma/cf/.
- Robert Wood, ‘Calculus Is Canceled: A Legacy Betrayed?’, California Globe, 21 June 2021,

https://californiaglobe.com/section-2/calculus-is-canceled-a-legacy-betrayed/;

Williamson M. Evers, ‘Opinion | California Leftists Try to Cancel Math Class’,*Wall Street Journal*, 18 May 2021, sec. Opinion,

https://www.wsj.com/articles/california-leftists-try-to-cancel-math-class-11621355858. - Daniel Rockmore, ‘Is It Time to Kill the Calculus?’, Salon, 26 September 2020,

https://www.salon.com/2020/09/26/teaching-data-science-instead-of-calculus-high-schools-math-debate/. - https://www.independent.org/news/article.asp?id=13658.
- G. B. Thomas, Maurice D. Weir, Joel Hass, Frank R. Giordano,
*Thomas' Calculus*, Dorling Kindersley, 11th ed., 2008. This has total 1384 pages. James Stewart,*Calculus: early Transcendentals*, Thomson books, 5th ed, 2007. This has total 1327 pages plus a CD. - David Bressoud, Dane Camp, and Daniel Teague, ‘Background to the MAA/NCTM Statement on Calculus’, in
*The Role of the Calculus in the Transition from High School to College Mathematics: Report of the Workshop Held at the MAA Carriage House Washington, DC, March 17–19, 2016.*, ed. David Bressoud (place not specified: MAA, NCTM, 2017), 77–81,

https://www.maa.org/sites/default/files/RoleOfCalc_rev.pdf. - Paul R. Halmos,
*Measure Theory*(New York: Springer, 1974). - W. Rudin,
*Real and Complex Analysis*, 3rd ed. (New York: McGraw Hill, 1987). - https://ncert.nic.in/textbook.php?kemh1=13-16.
- http://ckraju.net/sgt/technical-presentations-faculty/ckr-sgt-tech-presentation-1.pdf. Incidentally, a student from IIT Kanpur, who claimed to have topped his calculus class, took this test and met the same fate (“not even zero”).
- http://ckraju.net/sgt/technical-presentations-faculty/ckr-sgt-tech-presentation-1.pdf.
- D. V. Widder,
*Advanced Calculus*, 2nd ed., Prentice Hall, New Delhi, 1999. - W. Rudin,
*Principles of Mathematical Analysis*, McGraw Hill, New York, 1964. Rudin,*Real and Complex Analysis*. - https://ncert.nic.in/textbook.php?iemh1=1-15, 1.3
- C. K. Raju, ‘Marx and Mathematics 1 Marx and the Calculus’,
*Frontier Weekly*, 28 August 2020,

https://www.frontierweekly.com/views/aug-20/28-8-20-Marx%20and%20mathematics-1.html. - Dirk J. Struik, ‘Marx and Mathematics’,
*Science & Society*12, no. 1 (1948): 181–96. - https://ncert.nic.in/textbook.php?kemh1=13-16. Chp. 13, p. 281.
- C. K. Raju, Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of Calculus from India to Europe in the 16th c, CE (Pearson Longman, 2007) hereafter CFM; See also the abbreviated accounts in C. K. Raju, ‘Calculus’, in Encyclopedia of Non-Western Science, Technology and Medicine (Springer, 2016), 1010–15,

http://ckraju.net/papers/Springer/ckr-Springer-encyclopedia-calculus-1-final.pdf;

C. K. Raju, ‘Calculus Transmission’, in Encyclopedia of Non-Western Science, Technology, and Medicine (Springer, 2016), 1016–22,

http://ckraju.net/papers/Springer/ckr-Springer-encyclopedia-calculus-2-final.pdf. - For a simplified account see C. K. Raju, ‘Marx and Mathematics. 4: The Epistemic Test’,
*Frontier Weekly*, 8 September 2020,

https://www.frontierweekly.com/views/sep-20/8-9-20-Marx%20and%20mathematics-4.html;

C. K. Raju, ‘*Ganit Banam Mathematics*[Ganita vs Mathematics]’,*Himanjali*20, no. July-December (2020): 34–44.

http://ckraju.net/papers/ckr-article-Himanjali-22%e2%80%94Final-18-06-2021.pdf. - C. K. Raju, ‘Decolonising Mathematics’,
*AlterNation*25, no. 2 (2018): 12–43b,

https://doi.org/10.29086/2519-5476/2018/v25n2a2. - For an example of the use of non-elementary elliptic functions, or ballistics with variable air resistance, not covered in college calculus texts see this stock tutorial sheet: http://ckraju.net/sgt/Tutorial-sgt.pdf. For the difficulty of elementary calculus problems such as the simple pendulum with air resistance, or the calculus of variations problem of brachistochrone with air resistance, see C. K. Raju, ‘Time: What Is It That It Can Be Measured?’, Science & Education 15, no. 6 (2006): 537–51

http://ckraju.net/papers/ckr_pendu_1_paper.pdf; Archishman Raju, ‘A Simple Way to Solve the Brachistochrone Problem with Resistance’, Physics Education (India) 28, no. 3 (September 2012),

http://physedu.in/uploads/publication/3/65/Archishman_Brachistochrone13July.pdf. - See photo http://ckraju.net/papers/presentations/images/Sarnath-workshop-photo.png.

The online version of the paper is at

http://ckraju.net/papers/calculus-without-limits-paper-2pce.pdf,

presented at the 2nd People's Education Congress, Homi Bhabha Centre, Mumbai, 2009. (K-12 level.) - C. K. Raju, ‘Teaching Mathematics with a Different Philosophy. 1: Formal Mathematics as Biased Metaphysics’,
*Science and Culture*77, no. 7–8 (2011): 274–79; arXiv:1312.2099;C. K. Raju, ‘Teaching Mathematics with a Different Philosophy. 2: Calculus without Limits’,*Science and Culture*7, no. 7–8 (2011): 280–85. arXiv:1312.2100 (3 undergrad groups, and one post-graduate math group.) - See photo http://ckraju.net/papers/presentations/images/Iran-group-photo.png, and poster http://ckraju.net/papers/presentations/images/Iran-poster-English.png. (Mixed group.)
- See photo http://ckraju.net/papers/presentations/images/aud-group-photo-2.png, and poster http://ckraju.net/papers/presentations/images/AUD-poster.jpg. (Social science post-grads.)
- See photo http://ckraju.net/papers/presentations/images/calculus-sgt-a.jpg, and poster http://ckraju.net/sgt/poster-calculus-without-limits.pdf. (Science and engineering undergrads.)
- CFM, p. 140 et seq.
- For a quick account of these precise trigonometric values, see my talk at MIT, “Calculus: the real story”.

Video: https://www.youtube.com/watch?v=IaodCGDjqzs,

Presentation: http://ckraju.net/papers/presentations/MIT.pdf,

Abstract: http://ckraju.net/papers/Calculus-story-abstract.html. - CFM. p. 141
- C. K. Raju, ‘Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the YuktiBhasa”’,
*Philosophy East and West*51, no. 3 (2001): 325–62,

https://muse.jhu.edu/article/26555/pdf. http://ckraju.net/papers/Hawaii.pdf. - CFM, p. 170.
- CFM, p. 169.
- C. K. Raju, ‘Calculus’.
- CFM, p. 174 et seq.
- C. K. Raju, ‘Aryabhata Dalit: His Philosophy of Ganita and Its Contemporary Applications”’, in
*Theory and Praxis: Reflections on the Colonization of Knowledge*, ed. Murzban Jal and Jyoti Bawane (Routledge, London, 2020), 139–52,

http://ckraju.net/papers/Aryabhata-philosophy-of-ganita-paper-2r.pdf. For a popular-level account, see

https://www.jansatta.com/sunday-column/celebration-of-dalit-achievements-jansatta-column/777664. - K. Sambasiva Sastri, ed.,
*Aryabhatiya of Aryabhatacarya with the Bhasya of Nilakanthasomasutvan*(University of Kerala, Trivandrum, 1930). - "George Joseph: serial plagiarist”, http://ckraju.net/blog/?p=132. Also, “George Joseph serial plagiarist and mathematical ignoramus...”, http://ckraju.net/blog/?p=166.
- CFM, p. 324 et seq.
- C. K. Raju, ‘Marx and Mathematics. 2: “Discovery” of Calculus’,
*Frontier Weekly*, 31 August 2020,

https://www.frontierweekly.com/views/aug-20/31-8-20-Marx%20and%20mathematics-2.html. - Christoph Clavius [Christophori Clavii Bambergensis,
*Tabulae Sinuum, Tangentium et Secantium Ad Partes Radij 10,000,000...*(Ioannis Albini, 1607). - For further details, see Raju, Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of Calculus from India to Europe in the 16th c, CE, 171.
- Matteo Ricci, ‘Letter to Petri Maffei’, in
*Documenta Indica*, vol. XII, 1581, 472–77 For a facsimile of Ricci’s handwritten letter, see my MIT talk.

https://www.youtube.com/watch?v=IaodCGDjqzs. - http://doctrineofdiscovery.org.
- Brahmagupta,
*Brahma-Sphuta Siddhanta*, ed. Ram Swarup Sharma Sharma (New Delhi: Indian Institute of Astronomical and Sanskrit Research, 1966), 679 chp. 11, verse 16, vol. 3. - For more details, see “Calculus transmission” cited above. Fermat's challenge problem is in his Ouvres, p. 332 et seq. Bhaskara’s solution is in
*Bijaganita*(87, trans. Colebrooke, 1816, pp. 176–178). - The infinite “Taylor” series for sine is found in Yuktidipika 2.440-441, and the “Leibniz” infinite series for pi is found in Yukdtidipika 2.271. For further details see
*Cultural Foundations of Mathematics*cited above, and the calculus articles for the Springer Encylopedia cited above. - C. K. Raju, ‘Marx and Mathematics. 2: “Discovery” of Calculus’.
- https://doctrineofdiscovery.org/
- C. K. Raju, ‘Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhasa’,
*Philosophy East and West*51, no. 3 (2001): 325–62,

http://ckraju.net/papers/Hawaii.pdf. - See, for example, the
*Hindustan Times*news reports of 8 Nov. 2004 archived at, http://ckraju.net/Joseph/HT_report_8_Nov_04.pdf, and the*Hindustan Times*retraction of 25 Aug 2007 of a sensational news item, archived at

http://ckraju.net/Joseph/HT_correction_25_Aug_07.jpg.

Further documents and details are accessible at http://ckraju.net/blog/?p=166. - C. K. Raju, ‘“Euclid” Must Fall: The “Pythagorean” “Theorem” and The Rant Of Racist and Civilizational Superiority — Part 1’,
*Arumaruka**: African Journal of Conversational Thinking*1, no. 1 (2021): 127–55; C. K. Raju, ‘“Euclid” Must Fall: The “Pythagorean” “Theorem” and the Rant of Racist and Civilizational Superiority – Part 2’, (to appear),*Arumaruka**: African Journal of Conversational Thinking*1, no. 2 (2021): 57-75,

http://ckraju.net/papers/Tubingen-Pretoria-part-2.pdf. - Richard J. Gillings,
*Mathematics in the Time of the Pharaohs*(New York: Dover, 1972) Appendix 5. - E.g. Martin Bernal,
*Black Athena: The Afroasiatic Roots of Classical Civilization.*, vol. 1: The fabrication of ancient Greece (London: Free Association Books, 1987); C. K. Raju, ‘Black Thoughts Matter: Decolonized Math, Academic Censorship, and the “Pythagorean” Proposition’,*Journal of Black Studies*48, no. 3 (2017): 256–78,

https://doi.org/10.1177%2F0021934716688311. - C. K. Raju, ‘Was Euclid A Black Woman? Sorting Through The False History And Bad Philosophy Of Mathematics | Science 2.0’, 24 October 2016, https://www.science20.com/the_conversation/was_euclid_a_black_woman_

sorting_through_the_false_history_and_bad_philosophy_of_mathematics-180581. - Raju, ‘Black Thoughts Matter: Decolonized Math, Academic Censorship, and the “Pythagorean” Proposition’.
- “Decolonising history: Goodbye Euclid!” at

https://youtu.be/sEK1FCrLHjU?t=3292, or the related presentation at

http://ckraju.net/papers/presentations/Decolonising-history-Goodbye-Euclid.pdf#page=159.

More details at http://ckraju.net/blog/?p=63. - http://ckraju.net/blog/?p=173.
- S. N. Sen and A. K. Bag,
*The Sulbasutras*(Delhi: Indian National Science Academy, 1983). - https://web.archive.org/web/20161102104249/http://ganashakti.com/english/comments/details/156.
- Satish Chandra Vidyabhushana,
*The Nyaya Sutras of Gotama*(Allahabad: Panini Office, 1913) verse 2. Online at http://ckraju.net/papers/presentations/images/Nyaya-Sutra-Gotama-2-proof.pdf. Note that this proof comes from long before any notable Greeks, and its date cannot be fudged by appealing to “reliable (=Western) sources” for the date of the Nyaya Sutra, since this is based on a numerous diverse sources, and disputes regarding this method of proof pre-dated even Buddhism. - http://ckraju.net/papers/presentations/images/Proof-table.html. This is further explained in the video of my workshop on the Indian alternative to “Euclidean” geometry:

https://www.youtube.com/watch?v=ERm25QgyW1w.

Traditionally, in India, the same notion of proof was used for everything (unlike Western notions of proof which accept the empirical in science, but prohibit it in math, and resort to faith in religion, resulting in different categories of truth—mathematical, scientific, and religious truth). - A
*ryabhatiya of Aryabhata*, ed. K. S. Shukla and K. V. Sarma (Delhi: Indian National Science Academy, 1976). - Bina Chatterjee, trans.,
*Si**syadhivrddhida Tantra of Lalla*, 2 vols (Delhi: Indian National Science Academy, 1981) 20.36. - http://ckraju.net/hps-aiu/flat-earth-in-Bible.txt.
- Raju, ‘Decolonising Mathematics’.
- Haribhadra Suri, ed.,
*Shatdarshan Samuchaya*5th ed. (Bharatiya Jnanapeeth, 2000), 454–55. - C. K. Raju,
*Mathematics, Decolonisation and Censorship*, 2017,

https://kafila.online/2017/06/25/mathematics-and-censorship-c-k-raju/. - For quantum logic see chp. 6b, quantum mechanical time in C. K. Raju,
*Time: Towards a Consistent Theory*(Springer, 1994). - Though the false claim of “universality” of 2-valued logic probably derives from the fact that Christian rational theology borrowed heavily from Islamic rational theology, and the related Arabic sources of “Aristotle” were influenced by Nyaya logic which is 2-valued. C. K. Raju, ‘Logic’, in Encyclopedia of Non-Western Science, Technology and Medicine (Springer, 2016 2008), 2564–70.
- L. Mendelson, ‘Introduction to Mathematical Logic, van Nostrand Reinhold’ (New York, 1964).
- Bertrand Russell, ‘Mathematics and the Metaphysicians’, in
*Mysticism and Logic and Other Essays*(London: Longmans, Green, and Co., 1918), 74–96. - Though numerous books are attributed to the mythical Euclid, we will be concerned with only one: The Elements of Geometry, or the Elements vol 1. Since there are no original Greek sources for this, we will take as the stock “original” source the manuscript found in the 19th c. by the racist historian Heiberg in the Vatican, and then translated. (But any Arabic/Byzantine Greek manuscript of the Elements will do for our conclusions.) T.L. Heath,
*The Thirteen Books of Euclid’s Elements*(New York: Dover Publications, 1956); C. K. Raju,*Euclid and Jesus: How and Why the Church Changed Mathematics and Christianity across Two Religious Wars*(Penang: Multiversity and Citizens International, 2012). - H. M. Taylor,
*Euclid’s Elements of Geometry*(Cambridge: Cambridge University Press, 1893). For some more details, see http://ckraju.net/geometry/cambridge-note.html. - B. Russell, ‘The Teaching of Euclid’,
*The Mathematical Gazette*2, no. 33 (1902): 165–67. - David Hilbert,
*The Foundations of Geometry*(The Open Court Publishing Co., La Salle, 1950). - George D Birkhoff, ‘A Set of Postulates for Plane Geometry, Based on Scale and Protractor’,
*Annals of Mathematics*33 (1932): 329–45.

Image (c) istock.com

*11-Dec-2021*

More by : C. K. Raju

A great artical. Next logical step could be to create some content according to indian calculus. May be you should create some youtube video as a beginning point. You must submit your suggestions to framing of National curriculum framework, which is presently under development. This would pave the way for inclusion of this approach of teaching-learning in new syllabus. |

Excellent essay - kudos to Prof. C.K.Raju for busting the age old myths in 'Western' Mathematics. |

The next logical step is to create books of Calculus based on Indian Ganita traditions. |

I was fondly hoping and indeed looking forward to see some such comments coming from some of our own educational establishments. Nonetheless, there are dedicated teachers in our system too. May I therefore request Prof CK Raju to explain his method of teaching Calculus that makes it easy for school children to get a hold of the subject, in another article, for such sharing is sure to better the performance of our students? |

I suggest you create a new simple PowerPoint explaining 'the Indian Calculus' with examples of solved problems from Californian mathematics textbooks. Then, you can submit your formal recommendation for change to the people currently changing the Californian mathematics framework. The email address to send your PPT presentation and recommendations within the California Department of Education is CFIRD@cde.ca.gov Rather than rely on your old calculus material on your website, I recommend a new dedicated presentation addressing current calculus texts in California. Best wishes Jonathan J. Crabtree |

Professor, you made a forceful argument. It should alert our NCERT to revise its text books and initiate action to adopt the method of teaching of calculus you have in mind. For, Indian IT sector, the mainstay of today's employment, which is gearing itself to meet the demands of the changing business environment—Analytics, AI, Machine learning, etc—needs the same skill-sets which California state is aiming at. |

You made an effective argument professor. If my understanding is right, your forceful argument makes it abundantly clear that it’s not only the state of California that has to switch over to the method of teaching of calculus you have in mind but even our NCERT has to heed to your advice immediately. For, Indian IT sector, which is gearing itself to meet the demands of the changing business environment—Analytics, AI, Machine learning, etc—needs the same skill-sets which California state is aimin |