Ganita (गणित) differs from (axiomatic) math. It makes math easy (especially calculus). Enables students to solve harder scientific problems.

Summary

Plato/Socrates connected mathematics to soul, using a (“pagan”) notion of soul later cursed by the church, which hence initially banned mathematics in 532 CE. During the Crusades, the church accepted back mathematics, but refashioned it to fit the politics of its Crusading theology of (axiomatic, religious) reason (MINUS facts). In contrast, Indian ganita was always practical and religiously neutral. It used scientific reason (reason PLUS facts).

Historically, the West was excessively backward in mathematics for millennia, and learnt from Indian ganita almost all current school mathematics (arithmetic, algebra, trigonometry, calculus, probability and statistics). However, Europeans had tremendous difficulties in fully understanding what they copied or stole from India. The very words like zero, surd, sine, etc. tell this story. Specifically, the West (though it chauvinistically claims to have “discovered” calculus) failed to fully understand calculus, which it hence transformed in the 20th century by adding axiomatic (metaphysical, unreal) real numbers. This adds no practical value, but has made calculus so difficult that California, the US tech hub, recently cancelled calculus teaching in schools. In contrast, as demonstrated by my teaching experiments across 14 years, with 8 groups in 5 universities in 3 countries, teaching calculus as ganita makes it easy and enables students to solve harder problems, not covered in usual calculus courses.

The sole exception to the about pattern (“West copied math but failed to grasp it”) is geometry (which the West copied from Egypt via Plato). Though, axiomatic reasoning (MINUS facts) was first invented and used by the Crusading church, it is attributed in our school texts to a mythical Euclid. NCERT has no evidence for “Euclid” and the “Euclid” book does not have a single axiomatic proof. This was publicly acknowledged only in the 20th century, when attempts by David Hilbert and Bertrand Russell to “save” the Euclid myth led to the axiomatisation of all mathematics. Axiomatic math is asserted to be “superior” and was globalized by colonialism. However, axiomatisation (or formalization) is not superior; it adds no practical value to the original normal mathematics or ganita. It also does not add epistemic (or aesthetic) value. It however does add political value by (a) making math impossibly difficult even at the level of 1+1 = 2, thereby reducing most people to a state of abject ignorance of math hence making them dependent, and (b) giving control over mathematical knowledge to the West.

Therefore, it is almost impossible even to publicly discuss the possibility of changing teaching from religious mathematics to practical ganita, because the common person is ignorant of even 1+1 = 2 in axiomatic mathematics, and wrongly trusts the university professors of mathematics to give an unbiased judgment, though they are under the thumb of the West, and earn their livelihood from teaching religious/axiomatic mathematics. (And they have fled from public debate in the last 10 years, unethically refusing to defend in public what they teach to a captive and threatened audience of students in the math classroom.)

Contents

Introduction
1+1=2
JNU/Cape Town challenge
Does axiomatisation add any practical value?
Phakeng fallacy (“it works” fallacy)
Aping the “superior” West
The pope’s apices (986 CE)
Fibonacci and zero
European ignorance of fractions and the Gregorian reform
19th c. London prof didn’t understand negative numbers
Western inferiority or superiority?
Western theft of Indian calculus
Our blunder and the church trap
Changing math teaching: INFERIOR Western understanding
European failure to understand infinite sums
Which is better? (The brazen lies in the school text)
Does axiomatic math have superior epistemic value?
Egyptian/Platonic mystery geometry
Christian theology of reason
Cambridge’s laughable blunder
Is axiomatic math superior?
Political advantage of axiomatic proofs
Conclusions

Introduction

Did you or your children find math difficult? If so what is the solution? Change the teacher or change the child? This talk/article offers a novel solution. The fault lies with the very subject of mathematics, which was messed-up by the West by mixing it with religious politics during the religious fanaticism of the Crusades. While colonialism taught us only Western mathematics, it is NOT universal: Indian ganita (गणित) was different, it was always practical. The religious aspects of mathematics make it unsuitable to teach such math as a compulsory subject in Indian schools, compared to ganita which is religiously neutral.

Teaching ganita makes math easy and enables students to solve harder scientific problems, not covered in stock calculus texts,[1] as has been demonstrated in teaching experiments across 14 years with 8 groups in 5 universities in 3 countries.[2] Now, teaching calculus as ganita,[3] is planned as a course in IIT Kanpur.

This is part of a broader effort, as put together in my forthcoming book “Ganita vs Math: the implications for pedagogy and science.[4] (This was my book project as a Tagore Fellow at the Indian Institute of Advanced Study, Shimla.) Here is its 12 word summary: 1. Ganita differs from math, 2. it makes math easy, and 3. makes science better. The book also explains how religious mathematics has infiltrated frontline science:[5] but those issues are much harder to explain, and will be omitted here.

1+1=2

Most people are astonished at the thought of religion in mathematics. “Where,” they sarcastically ask, “is the religion in 1+1 = 2?” They are thinking of 1+1 = 2 as they learnt it in kindergarten as one orange plus one orange equals two oranges.

That is an empirical proof; nothing religious in it.

But empirical proof, or प्रत्यक्ष प्रमाण, is accepted (only) in Indian ganita (and by all Indian schools of philosophy) as the first means of proof. As stated in Nyaya Sutra 2,[6]

Here is an elucidation of how these four means of proof were used in ganita, or see the related video.

Warning: acceptance of empirical proof in ganita does NOT mean rejection of reasoning as is so often glibly caricatured. Thus, the above statement from the Nyaya Sutra also clearly accepts deductive reasoning (अनुमान) as the second means of proof. [7] The important point to note is that अनुमान (anumana) is reasoning PLUS facts, as in science, which too accepts observations and experiments (empirical proof), but also accepts reasoning.

But the KG way to do 1+1=2 is NOT the way 1+1=2 is done in present-day mathematics, also called formal mathematics or axiomatic mathematics or better called Western ethnomathematics. Why not? Because empirical proofs are prohibited in present-day mathematics (=Western ethnomathematics). Many laypersons have difficulty in believing it. However, this can be easily checked in any elementary text on mathematical logic which defines a formal mathematical proof.[8]

A mathematical proof is a sequence of statements in which each statement is either an axiom, or is derived from preceding statements by some rule of reasoning [such as modus ponens[9]].

On the above definition, there is no room in a formal mathematical proof to introduce a statement based on empirical observations, such as “I now see two oranges” which is neither an axiom nor an inference. This prohibition of the empirical in present-day math (=Western ethnomath) is also explained in the current NCERT class IX mathematics text (p. 301), “each statement in a [mathematical] proof has to be established using only logic…Beware of being deceived by what you see…! [Emphasis original] This text is compulsory reading for all, so everyone ought to know it (but few do because this portion is never asked in the examination). Anyway, though many laypersons have difficulty in grasping it, the fact is that the KG method of proving 1+1=2 is NOT accepted by present-day (axiomatic, formal) math or Western ethnomathematics, though it is acceptable in ganita.

A key effect of reasoning MINUS facts (= axiomatic reasoning) is that it makes math (even 1+1 =2) extraordinarily difficult. Thus, Bertrand Russell, in his Principia, took 378 pages to prove 1+1 = 2. Most people in the world (including most professors of mathematics) would be unable to explain what is there in those 378 pages, or to decipher even a single sentence on that page 378.

JNU/Cape Town challenge

Worse, in formal mathematics there is no unique entity “1”: the natural number 1 differs from the “real” number 1 because different axioms are needed for the two: Peano’s axioms used for natural numbers do NOT apply to “real” numbers. Therefore, the proof of 1+1 = 2 in “real” numbers is far more difficult, since it requires knowledge of axiomatic set theory, which again few (even among professional mathematicians) know. To bring out this difficulty, I offered (as shown in this video) a prize of Rs 10 lakhs to anyone who could give a proof of 1+1 = 2 in “real” numbers from first principles (i.e., without assuming any theorem from axiomatic set theory, as in my Cape Town challenge). This prize was offered during a lecture in our leading university JNU.[10] Nobody accepted the challenge. (And nobody resigned out of shame because our “leading” math educators are very comfortable with their own ignorance of 1+1 = 2!)

So, my first key point to is that prohibiting the empirical, as done in formal math, makes it impossibly difficult for most people to understand even 1+1=2. In comparison, the ganita way of doing 1+1=2 empirically is very easy.

Does axiomatisation add any practical value?

Next, let us ask: does this extreme complexity of axiomatic mathematics or Western ethnomathematics add any practical value to our earlier knowledge of 1+1 = 2? NO! Most people are unaware of Russell’s axiomatic proof of 1+1 = 2, so there is no question of them applying it to everyday shopping, which certainly needs empirical inputs prohibited in axiomatic math. People continue to shop for groceries using the traditional empirical understanding of arithmetic used for thousands of years across the world.

However, because most people are ignorant of mathematics, they superstitiously fear that some aspects of axiomatic/formal/Western ethnomathematics (which they don’t know) may be essential for advanced technology. This is not true! A very simple argument is as follows. For example, to send a rocket to the moon we need to calculate its trajectory. This is done using calculus. According to the current university teaching of calculus, real numbers (as in my Cape Town/JNU challenge) are essential for the calculus. But they are never used in actual practice, cannot be. Indeed, today, actual calculation of rocket trajectories is done on computers, using the ganita system of numerically solving differential equations due to Aryabhata.[11] Real computers have finite memory. Therefore, the cannot representing even a single real number which requires infinite memory. Therefore, computers never use real numbers. Instead they use what are called floating-point numbers, represented by 32 or 64 etc. bits. These numbers do NOT[12] obey even the associative law for addition which is required for all axiomatic numbers. The same considerations (computers cannot use real numbers) apply also to machine learning or AI programs, which use statistics. Though statistics is taught using real numbers, machine learning is done using computers and real numbers cannot be represented on computers running any ML program. Therefore, unreal “real” numbers of Western ethnomath are NEVER used in applications to advanced technology today.

To summarise, formal math adds immense complexity without adding any practical value.

Phakeng fallacy (“it works” fallacy)

This point should be remembered in the context of the Phakeng fallacy: ignorant people say of existing math, ”it works so let us continue with the status quo”. They naïvely assume that there is only one “universal” kind of math, whereas there are two kinds: formal math and normal math (ganita). It is normal math which works and has always worked, for applications of math to science and technology, not formal math or Western ethno-math, which only provides a deceptive covering for normal math. So, whenever people say of math “it works”, and try to infer from such a vague statement that formal math must be taught, do ask them to explain exactly how Russell’s axiomatic proof of 1+1 = 2 “works” in a grocery shop.

The other argument in support of formal math claims that formal math adds epistemic value, even if it does not add any value to applications of math. We will return later on to this argument: the claim of added epistemic value is actually a religious claim.

Aping the “superior” West

First let us examine another very strong superstitious driving force instilled by indoctrination through colonial/church “education”: the desire to ape the West. This desire to ape is especially strong among those ignorant of mathematics who superstitiously fear that failure to imitate the West may result in a catastrophe. But how real was that assertion of Western superiority two centuries ago when Macaulay first assert it and persuaded us to adopt colonial/church education?

As explained in my Tübingen-Pretoria keynote, and this related article,[13] the assertion of Western superiority is just a mutation of the earlier superstitious assertion of white superiority, which itself is a mutation of the still earlier assertion of Christian superiority. When the assertion of Christian superiority could no longer be used to morally justify large-scale slavery of Africans (because Black Africans converted to Christianity), the superstitious claim of the inferiority of Blacks was invented to preserve the wealth accruing from the slave trade. After Plassey, colonial loot replaced slavery as the major source of wealth for the West.

But to help rule India on the colonial policy of divide and rule, the Aryan race conjecture was invented[14]: the fantasy that India had earlier been conquered and populated by whites. Hence the assertion of white supremacy could no longer be used in India, and was replaced by the assertion of Western supremacy, as used by Macaulay. The important thing to understand is that a secular argument from false history of science was used to support all three closely related claims of Christian/White/Western superiority. This false history of science went viral during the Crusades when all science was wildly attributed to Christians and their sole friends the early Greeks. This false history was elaborated and reused by racist and colonial historians with a mere change of labels, e.g. by calling the Greeks as Whites or the West. The entire evolution of this false history of science is shown in this diagram.

As regards mathematics the claim of Christian/White/Western superiority is completely false. In fact, a solid case can be made out for persistent Christian/White/Western INFERIORITY in mathematics.

The simple fact is that science needs mathematics, and the early Greeks and the Romans, and the Europeans at least until the 16th century were very backward in mathematics, and learnt almost all that mathematics from India. The list of Europeans learning math from India is very long. First, the very names of numbers in Greek and Latin are copied from Sanskrit.

The direction of transmission is clear from the fact that Greeks and Romans stopped at the puny myriad (=10,000), whereas the Indian Yajurveda, from long before any noted Greeks, has numbers as large as a parardha (= trillion, Yajurveda 17.2), and Buddhist sources (Lalita Vistara sutta[15], chp. 12) list numbers as large as a tallakshana (10⁵³ ) and far beyond.

The pope’s apices (986 CE)

This persistent inferiority of Europeans in math, and the process of Europeans consequently learning math from India, continued for thousands of years. Romans/Europeans arithmetic was tied to a primitive system of pebble arithmetic or the abacus. As is well known they eventually abandoned that primitive system in favour of sophisticated “Arabic numerals” from India. This happened in three phases spread over many centuries. First Gerbert (10th century, later Pope Sylvester) imported “Arabic numerals” from Cordoba. The important point, never mentioned in Western history is of mathematics, is that the future pope failed to understand the efficiency of Indian arithmetic, based on algorithms (after al Khwarizmi’s Latinized name). This is clear from the fact that he got an abacus (shown below) constructed for Arabic numerals in 986 CE.

An abacus is based on primitive principle of pebble counting for all arithmetic operations, and destroys the efficiency of algorithms which made Indian arithmetic superior. Indeed, the very terminology of “Arabic numerals” shows that Europeans foolishly thought there was some magic in the shape of the numbers, not a difference in the ways of doing arithmetic.

Fibonacci and zero

In the second phase, Fibonacci a 13th c. Florentine merchant trading with Arabs in Africa understood that the efficiency of (Indian) arithmetic provides a comparative advantage in commerce, and that Roman arithmetic based on the abacus is terribly inefficient and inferior. Hence he imported Indian arithmetic, from Arabs via Africa, and wrote a book on it called Liber Abaci. But Florentines failed to understand the place-value system स्थान-मान प्रणाली (which is the first lesson in all Indian ganita texts), on which Indian arithmetic algorithms are based. HENCE, also Europeans had difficulties in understanding zero. This is clear from the very word zero from zephyr/cipher (= Arabic sifr) today meaning mysterious code.

Basically, the difficulty with zero arose because primitive Greek and Roman numerals are additive like pebbles: XVII means 10+5+1+ 1 = 17 (and the entire European tradition is from Greek and Latin). This additivity is not true on the place value system 17≠ 1+7 = 8. So Florentines complained that zero has no value in itself but adds any amount of value to the preceding number.

A contract for 25 could easily be changed into one for 250, which is not possible with Roman numerals. Accordingly, Florence passed a law against zero in 1299, to which we still adhere: in a cheque any quantity written in (“Arabic”) numbers must also be written in words.

European ignorance of fractions and the Gregorian reform

This process of Europeans persistently importing Indian math continued in the 16th century, when Jesuits in Cochin mass translated and exported Indian knowledge back to Europe, on the Toledo model of appropriating knowledge. Christoph Clavius, the Jesuits general, wrote a textbook on practical mathematics,[16] incorporating Indian techniques. On my epistemic test,[17] this European theft of knowledge was accompanied by lack of understanding. An easy-to-understand example is the European difficulty with fractions. Fractions were known to Indians and Egyptians from at least -1500 CE, and probably long before that. But they were not understood by Europeans even in the late 16th century. This is clear from the fact that the Gregorian calendar reform of 1582 (authored by Clavius} used leap years and not precise fractions (which common Europeans did not then understand), Hence, the reformed calendar still gets the tropical year wrong (it is only right on a thousand year average), so that equinox does not come on a fixed day of the Gregorian calendar.

19th c. London prof didn’t understand negative numbers

This is hardly the end of the story. Zero is associated with negative numbers. But in the late 19th century Augustus de Morgan, a very influential professor of mathematics from University College London, said that negative numbers are impossible.[18]

This “eminent” Western mathematician (of “De Morgan’s laws” fame) naturally also failed to understand ordering among negative numbers that e.g., −9 < 0.

He further asserted:

“Above all, he [the student] must reject the definition still sometimes given of the quantity, that it is less than nothing. It is astonishing that the human intellect should ever have tolerated such an absurdity as the idea of a quantity less than nothing; above all, that the notion should have outlived the belief in judicial astrology and the existence of witches, either of which is ten thousand times more possible.” – — Augustus De Morgan, 1898[19], On the Study and Difficulties of Mathematics, pp. 70–71

De Morgan cited Indian ganita texts on negative numbers, first used by Brahmagupta, but is consumed by his misplaced sense of colonial and racist superiority, resulting in his laughable blunder.

Western inferiority or superiority?

This evidence of persistent primitiveness and backwardness of Greeks, Romans, Europeans in elementary arithmetic should be repeatedly contrasted with Macaulay’s contemporary boast of immeasurable superiority of the West, believing which (without checking it) we changed our entire education system and put the minds of our impressionable and gullible children in the hands of the church.

For simplicity and clarity (for the layperson) the above story of how Europeans imported mathematics from India, but persistently failed to understand it, is restricted to elementary arithmetic. But the very same story is true of various other branches of mathematics such as algebra, trigonometry, calculus, probability and statistics. The failure of Europeans to understand Indian algebra and trigonometry is clear from the very words such as surd and sine, as described in this video, and in this summary article, covering all areas. (The only exception is geometry, which Greeks like Plato and Pythagoras learned from Egyptians, and which mystery understanding of geometry was adapted by the crusading church to suit its political purposes, with the addition of various fantasy stories, which the colonized implicitly believe without ever checking facts.)

Western theft of Indian calculus

The issue of European theft of calculus[20] is of special interest since all present day science is based on the calculus: almost all physics is formulated using differential equations, to understand which calculus is needed. The related issue of probability and statistics[21] is also of special interest since it is the basis of the widely anticipated future technology of machine learning and artificial intelligence.

As regards the calculus, it is obviously not possible to cover in one article or talk my 500-page book[22] on its Indian origins and its transmission to Europe, and numerous prior and subsequent writings and talks on the issue across two decades.[23] As regards the Indian origin of calculus, here is the video of a key talk at MIT, Cambridge, Mass.[24] As regards its transmission, or rather theft without acknowledgment, see the video of this recent talk.

For most Indians the critically important issue here is to say with pride, “we did it first!” However, let us note a key point. Our immediate issue is not two rewrite history to bring out our achievements. Our immediate issue is the teaching of calculus. Therefore, it is important to explain whether this change of history affects the teaching of calculus. It does, as explained in this recent talk at IIT Kanpur. But how to EXPLAIN this to common people who are shaky about 1+1=2? An explanation is needed because, for such a major change in our teaching, many people must understand the reasons for the change.

Our blunder and the church trap

But as we saw the vast majority of people in India (including all decision-makers) do not understand even 1+1 = 2 as it is done in Western ethnomathematics (or axiomatic mathematics) which we teach. They consider it a waste of time to understand, because this axiomatic way of doing mathematics adds no practical advantages for the applications of mathematics in everyday life. (Their blunder is in not understanding its political implications, and, from a position of complete ignorance, not believing that mathematics has political implications.)

Thus, the ordinary person wrongly thinks that university professor of mathematics or mathematics education has an understanding of this, which is false, as demonstrated by my prize of Rs 10 lakh at JNU. Indeed, it is rather foolish to trust university mathematicians to give an unbiased judgment in the matter of ganita vs math, given that they have long been indoctrinated into axiomatic mathematics, and their livelihood depends upon it. So why should they support a change? Why should they even debate it since they are in power? Proof of the total dishonesty of formal mathematicians is clear from the fact that they have persistently fled from my decade-old call for an open public debate to justify what they teach in the calculus classroom.[25] Therefore, whether we take the route of appealing to common people (who are ignorant) or to those currently recognized as “experts” (who are biased, indoctrinated and subservient to the West), no change seems possible in math teaching.

This is a deep church trap. A trap laid by the church through colonial education. A trap which we completely failed to understand in 200 years, because we made a strategic blunder: we did not study the colonial enemy or the church-state nexus during colonialism. Therefore, we entirely failed to understand the church technique of controlling knowledge (including math and science) into which it injected its religious dogmas. Therefore, also, the church grip on colonially educated minds continues unabated even 75 years after the British were forced to leave India.

Changing math teaching: INFERIOR Western understanding

However, we absolutely must change our math teaching. The first reason is that math is difficult today. In recognition of this fact, California State board recently canceled calculus teaching in schools.[26] California, which is the US technology hub, is obviously concerned about preparing students for the widely anticipated future technology of machine learning and data science. However, data science needs statistics, a proper understanding of which needs calculus. Therefore, teaching data science without teaching calculus is an extremely risky strategy and may lead to subtle bugs in machine learning programs, which may put the whole world at risk a couple of decades from now, when the world gets more dependent on such technology. The correct solution therefore is not to cancel the calculus but to make calculus easy by teaching it as ganita.[27] The US and West may never do it, because it will mean that they have to eat humble pie and reject all the false stories they told for centuries to glorify themselves. But we are not compelled to ape them: let them fall behind!

The story of how Europeans imported or stole Indian mathematics without fully understanding it, of course, applies also to the calculus. To reiterate, the West stole the calculus from India, and like all knowledge thieves, it failed to fully grasp what it stole. But the difficulty is this: How to explain this to the colonized mind ignorant of 1+1=2 (so it is forced to rely on authority of those it trusts), and indoctrinated by colonial/church education to distrust the non-West and trust the West (like Wikipedia)?

Let us try to do so in terms some might understand. Step 1: the undeniable fact is that all of our schools and universities today teach calculus using axiomatic real numbers, considered essential for an understanding of calculus. Step 2: the undeniable fact is that axiomatic real numbers did not exist in Newton’s time, therefore, on its present teaching, Newton could not have properly understood the calculus. Step 3: it was in recognition of this fact (that calculus was not quite understood by Europeans (even in the 19th c.) that Dedekind invented axiomatic real numbers at the end of the 19th century 250 years after Newton’s death[28] Step 4: it is an undeniable fact that axiomatic real numbers need set theory, and axiomatic set theory emerged only in the 1930s. So, by its own admission, the West was doing calculus without properly understanding it until the 1940s. When it did evolve what it regarded as a “proper understanding” of calculus the fact is it proved so difficult that California is today forced to cancel calculus teaching in schools.

In contrast, the 5th c. Aryabhata invented the calculus by numerically solving differential equations[29] to compute accurate sine values. He used a very simple and elementary but useful arithmetic technique called the rule of three (त्रैराशिक). This was the easy part of calculus which Europeans understood. Hence, Newton could derive practical value from his physics, long before the invention of real numbers.

European failure to understand infinite sums

What Europeans did NOT understand was how to do an infinite sum, i.e., sum an infinite series. Recall that Europeans stole the calculus for size trigonometric values needed for navigation, and the followers of Aryabhata in Kerala had derived precise trigonometric values by using infinite series. So this summing infinite series was a critical issue for Europeans. Descartes declared that doing an infinite sum was beyond the human mind. Galileo (who had access to the Jesuit Collegio Romano) concurred, hence left the credit or discredit for calculus to a student Cavalieri.

There are two issues here. First, for all practical purposes one could sum just a finite number of terms. For example, π = 3.14159…= 3 + (1/10) + (4/100) + (1/1000) + _ can be written as just 3.14. this was the process used in India to get an approximate (सविशेष) value of √2, or used by Aryabhata to get a near (आसन्न) value of π. However, accepting inexact values was contrary to the Western ethnomathematical/religious belief in mathematics as having eternal truths hence exact. Therefore, to be quite precise, the European difficulty with Indian calculus was how to find the EXACT sum of an infinite series.

This brings us to the part which Europeans did not understand. To sum infinite series, such as the infinite geometric series first summed by the 15th c. Nilakantha, Indians used Brahmagupta arithmetic (अव्यक्त गणित) of polynomials, nowadays called non-Archimedean arithmetic, which has infinities and infinitesimals[30] (unlike real numbers whose arithmetic is Archimedean).

Anyway, it was this problem of how to obtain the exact sum open infinite series which the invention of axiomatic real numbers apparently solved. However, it did so in a metaphysical way, which hence has application.

As we have already seen, the 20th c, axiomatisation of math added enormous difficulty to ganita but offers no added practical value. An obvious argument is this: most applications of calculus today are done on computer which cannot implement the metaphysics of real numbers.

Which is better? (The brazen lies in the school text)

Anyway, the question before us today is which is better? The easy ganita understanding of calculus or its complicated understanding using axiomatic math? To give a serious answer to this question, we need to guard against the Western rhetoric of superiority.

For example, the class IX school text asserts that “Greeks did a form of mathematics which was ‘superior’ to what all others (Indians, Egyptians, Babylonians. Maya) did”.

Sadly, to “prove” this claim of superiority, the school text uses several lies. 1. the school text lies “only Greeks used reason”, no others (Indians etc.) did whereas as we have already seen the use of reason was quite explicit in Indian ganita. 2. The school text uses a church trick of playing with words: it uses just one word reason, for both scientific reason (reason PLUS facts) and religious reason (reason MINUS facts), thus encouraging schoolchildren to conflate and confound the two. This is particularly obnoxious since this confusion is being planted in the minds of gullible children. 3. There are no axiomatic proofs in the book attributed to “Euclid”, but the text never tells children about it. It is common for the church to use this technique of a stream of lies, to confound and confuse its opponents, and the apex Indian educational body thinks this is what education is about.

This stream of lies occurs in chapter 5 of the NCERT class IX text on “Euclid’s geometry”. The claim about “Euclid” is itself a lie. Upon being asked to provide primary sources for the belief in Euclid, the NCERT could not provide any such sources. Instead, it referred to three Western tertiary mathematics texts which mention Euclid. When reminded that the historical evidence must be from primary sources, which were hence demanded, the NCERT increased the number of cited tertiary texts to six! In other words, the apex body for Indian education implicitly asserts that all the nonsense generated during the fanaticism of the Crusades and the Inquisition should be unquestioningly accepted by Indian schoolchildren.

Does axiomatic math have superior epistemic value?

Setting aside the NCERT advocacy and propagation of church lies, let us now return to the basic question: “what is this purported “superior” epistemic value of axiomatic math which so few can grasp, and which offers nil practical value?”

At this point we absolutely need to understand the religious roots of Western ethnomathematics which compels the West (and the colonized who ape them) into such absurdities.

Egyptian/Platonic mystery geometry

Recall that geometry is the one exception which the West did not copy from India, but took from Egypt. And later twisted it to add the political agenda of the church during the Crusades. Hence, geometry is the cornerstone of Western ethnomathematics.

While Egyptians had a very practical geometry very similar to Indian string geometry (of sulba sutra), they also had religious or mystery (or mystic) geometry. This first came to the early Greeks, Pythagoreans, and the Platonists. In Plato’s Meno,[31] Socrates uses elementary geometry to expound the (Egyptian/pagan) doctrine of mathesis (=learning): to expound the Platonic doctrine that all learning is recollection of eternal ideas in the soul which the soul learnt in its previous lives. Indeed Proclus in his commentary[32] on “Euclid’s” Elements etymologically derives the word mathematics from mathesis, and explains the religious significance of mathesis, as expounded by Pythagoreans, that mathesis or arousing the soul “leads to the blessed life”.

As is well known, the church after it married the state, in the fourth century, propounded a doctrine of Christian supremacy, hence turned against this equitable notion of soul in which it had earlier believed. Accordingly, it fought a violent religious war with the pagans, smashed all their temples in the Roman Empire, lynched Hypatia,[33] shut down all schools of philosophy in the Roman Empire, and pronounced its great curse (anathema) on this equitable notion of soul.[34]

Therefore, Egyptian mystery geometry was always anathema to the church.

The remaining philosophers were forced to flee the Roman Empire, and many sought refuge in neighbouring Iran (Jundishapur) where the Egyptian/Neoplatonic traditions survived as Sufism, and crept into a nascent Islam. Many famous early Muslims were Sufis. Hence, also, Egyptian mystery geometry survived in Arabic texts and influenced the Islamic theology of reason (aql-i-kalam). Note that even al Ghazali, the cornerstone of Sunni orthodoxy, wrote a text on logic.

During the Crusades, a crusading spy Adelard of Bath, travelled disguised as a Muslim student, and brought back a book on mystery geometry which he translated. In a famous dialogue with his nephew[35]Adelard commented that Muslims used reason, whereas Christians were authoritarian, hence “mere brutes”. Subsequently, this Arabic book was again translated in 1125 as part of the Toledo mass translations. As usual, the Toledo translators blundered, and translated the subtitle aql-i-des[36] meaning “rational geometry”, as the name of the Greek Uclides, nowadays known as Euclid. As is well known, this was an easy church trick of attributing knowledge in Arabic books to early Greeks. This made the knowledge theologically correct, since the early Greeks were regarded as the sole friends of Christians. Thus, instead of burning those Arabic books (the usual church practice for “heretical” books) they could be mass translated unto Latin, and Christians could learn from them: these were used as a text-books[37] in the first Western universities such as Oxford, Cambridge, and Paris,[38] all set up by the church to digest knowledge grabbed from Muslims. This was done with a view to win the Crusades, because Christian Europe then was very backward compared to Muslim Europe, and knowledge is needed to win wars. The “Euclid” text continued to be used in Cambridge[39] until the 20th century.

There is no known early Greek text of the book Elements which attributes its authorship to Euclid; the name comes only from Latin translations from Arabic. All early Greek texts attribute the book to Theon or a mysterious author, very probably his daughter Hypatia,[40] who came in the short interval between Theon and Proclus, who wrote a commentary[41] on the Elements, clearly asserting that it was a book on mathesis or mystery geometry.

In fact, Proclus etymologically derives the word mathematics from mathesis.

Christian theology of reason

During the Crusades the Christians were militarily weak, hence the church needed to fight the Crusades by other means. Earlier, the only basis of church propaganda had been to cite the scriptures. However, the Muslims rejected the Bible as corrupted, and force, which had succeeded with European “pagans”, then failed with Muslims. With the failure of both force and gospel, the church was forced to invent a new way to convert Muslims who accepted only reason. Therefore, the church sought a way to support its dogmas using reason. Since facts are almost always contrary to church dogmas, it hit upon the idea of using reason MINUS facts (religious reason).

This method (reason MINUS facts) can be used to derive any predetermined conclusion whatsoever by making appropriate assumptions (axioms, postulates), which need to be justified only on authority. Aquinas’ angel theorem[42] is an example of this church theological technique. He simply assumed that angels occupy no space.

To hide the fact that this novel and politically convenient notion of religious reason (MINUS facts) was a church invention (and different from scientific reason [PLUS facts]), the church attributed this method of axiomatic reasoning to the recently arrived “Euclid” book. This killed two birds with one stone: for it also eliminated the connection of (mystery) geometry to the “pagan” notion of soul, cursed by the church.

Cambridge’s laughable blunder

Of course, there are no axiomatic proofs in the actual “Euclid” book, but the church never bothered about facts. Westerners buried under church hegemony are brought up on church myths. Anyway, they dared not articulate the slightest dissent or scepticism in the days of the Crusades and the Inquisition, It was only after 775 years that this fact (no axiomatic proofs in “Euclid’s” Elements) was finally publicly admitted in the West.

But, by then, centuries of blind belief in the superiority of axiomatic proof had taken their toll. For example, Cambridge University made the laughable mistake of prescribing the order of the propositions in “Euclid” as essential part of its exams. The order of propositions is important only if the proofs are axiomatic (empirical observations are prohibited). Amusingly, the specially rewritten version of “Euclid” prepared by Cambridge,[43] to fit this syllabus, freely uses empirical proofs!

In contrast, Indian geometry (ganita of the sulba sutra or rajju ganita) explicitly accepts empirical proof, therefore the order of propositions is unimportant, and hence the “Pythagorean” proposition is the first proposition in Indian texts, such as Yuktibhasha,[44] instead of the last or penultimate (47th) proposition in the “official” “Euclid” book.

Hilbert[45] (1899) rewrote the book to provide the axiomatic proofs missing in it. Hilbert’s rewrite of “Euclid” is the clearest explicit acknowledgment that there are no axiomatic proofs in the actual book, a (seemingly secular) delusion asserted by the church from crusading times (1125). Note also, how this delusion was believed superstitiously believed for all those centuries by all top Western intellectuals for whom “Euclid” was a compulsory part of study. Hilbert, too, could only partly shake such myths and superstition: for this great Western mathematician (who, rather than Einstein, invented general relativity[46]) foolishly thought the mythical “Euclid” somehow intended such axiomatic proofs not found in the actual book. Of course, one can assume whatever one likes about the intentions of a non-existent person.

This assumption was false for it is a book on mystery geometry. What are the consequences of this false assumption? Hilbert’s rewrite of the book, based on the false myth that the “Euclid” book is about axiomatic proofs, does great violence to it, and does not fit the actual book. Thus, to try and explain the seeming prolixity of the Elements, Hilbert’s geometry is synthetic:[47] that means measurement of any kind (e.g. with the scale or protractor) is forbidden. This artificial hypothesis makes geometry needlessly difficult, but it explains the seeming prolixity of the Elements (like the extreme prolixity of Russell’s axiomatic proof if 1+1=2).

The manuscripts of Elements (such as they are) all use the term “equality” in their theorems. This term “equality” has a simple physical interpretation in terms of superposition: put one triangle on top of the other and see (from both sides) that they are equal. But this is a metric term, for one measures a line segment by physically superposing a ruler on it. Hence, it was disallowed by Hilbert.

The colonised nod their heads as if this was an act of great wisdom, rather than extreme foolishness. They never pause to think how they would do geometry without the ruler and the protractor in the compass box. Before Hilbert, this term “equality” was prevalent in all earlier “Euclid” textbooks such as those of Taylor[48] or earlier texts such as those of Todhunter.[49]

Recall that the equality of all beings was the central issue (contrary to belief in Christian supremacy which the state-church wanted to peddle, hence it cursed the pagan notion of soul. The colonially (mis) educated might imagine that I am confounding two notions of equality: political equality and geometric equality. It is one thing that they never read Plato on the relation of geometry to the soul. It is quite another that the colonized blundered in not studying the colonial enemy, and the church-state nexus of the colonial state. Had they studied church history, they would have immediately realized that geometry was at the forefront of the first religious war that the church fought against pagans and their leaders the (Neo)Platonic philosophers, whom Justinian banned from the Roman Empire in 532 CE when he shut down all schools of philosophy in the Roman Empire. Because geometry (and the related pagan/Platonic notion of soul) were at the centre of this conflict, Justinian’s fifth anathema[50] condemns the belief that souls are like spheres. The related mystery-geometry point about spheres is that all spheres are equal.

Anyway, the fact is that all theorems of the original Elements are about equality. Hilbert’s rewrite eliminates the term equality (because it is metric, and involves superposition) and replaces it with the synthetic term congruence, today used in all math texts as if it were the original term. Russel[51]and others joined in this effort, and later on the Bourbaki group joined this bandwagon, so that in the 20th century all of Western ethnomathematics became axiomatic.

So far as our school texts are concerned, the change from “equality” to “congruence” came about in the 1970s. Thus, Birkhoff,[52] in contrast to Hilbert, gave an axiomatic but metric reformulation of geometry. This, of course, is only superficially similar to compass box geometry,[53] since Birkhoff’s metric reformulation assumes metaphysical real numbers. After the Sputnik crisis, the US revised its mathematics syllabus. The school geometry study group recommended the teaching of Birkhoff’s metric geometry (which still uses the word congruence from Hilbert). We aped it because the central teaching of colonial education was “ape the West” and we never understood that this teaching makes us follow the church as the West did for so long.

Anyway, as noted earlier, axiomatization is of little value because it makes the simplest statements of mathematics impossibly difficult to understand, without adding the slightest practical value to what was known for thousands of years earlier.

Is axiomatic math superior?

Before concluding, let me explain why the claim that axiomatisation of mathematics provides superior epistemic value is pure religious bunkum.

Here are some arguments that are offered for the claim that “axiomatic proof is superior”. Thus, the belief (widespread among the colonially educated) is that proofs based solely on logic are “stronger” than proofs which use empirical facts. This was based on the theological superstition that God is bound by logic[54] (hence cannot create an illogical world) but is not bound by facts (hence can create a world with the facts of his choice). In modern Western philosophy, this superstition is mirrored in possible-world semantics, where “world” now means a logical world in the sense of Wittgenstein, rather than a world created by God. On this semantics, a logical truth is necessary truth, since true in all possible worlds, whereas facts are contingent truths, since true in only in some possible worlds.

The glaring flaw in this way of looking at things is obvious: let us say for the sake of argument, that logic does binds God. The question is which logic binds God? Medieval theology, like Western philosophy and Western ethnomathematics, never thought of this question, and just naïvely assumed that logic is universal.

However, as I have long been pointing out,[55] logic is NOT culturally universal (e.g., Buddhist logic of catuskoti, or Jain logic of syadavada are quasi-truth-functional, not 2-valued). Nor is logic empirically certain (e.g. quantum logic[56]). Obviously, a change of logic will change all the mathematical theorems derivable from the same set of axioms. For example, all proofs by contradiction would fail with a quasi-truth-functional logic.

The same argument (“logic stronger than facts”) is often repeated in a different form: empirical proofs are fallible, logical proofs are not. This is a half-truth. It is true that empirical proofs are fallible, as acknowledged in rajju-sarpa nyaya,[57] or mistaking a rope for a snake or a snake for a rope. This is also acknowledged in science that experiments may have errors. That does not mean science abandons the experimental method.

The theological/Western fallacy is imagining that deductive proofs are infallible.[58] The fact is that deductive proof are fallible. Mathematics students fail because they frequently give wrong proofs. Even eminent mathematicians have published wrong proofs e.g. about the Riemann hypothesis. Even Riemann gave a wrong derivation of conditions for shock-waves.[59] In fact, in any complex task of deduction (such as a game of chess, or Russell’s 378 page proof of 1+1 = 2) human beings almost always make a mistake. While a mistake of perception (e.g. mistaking a snake for a rope) can be quickly corrected , mistakes in deductive proofs linger. For example, how exactly does one know that Russell’s 378 page proof is infallible? Not even a single typo? In fact, if one understands the proof the only way to be sure that there is no mistake is to check the proof repeatedly, an inductive process which is fallible. If one does not understand the proof (most people), then one has to rely on authority, or shabda praman which is highly fallible. In either case, deduction is MORE fallible than empirical proofs, regardless of any Western superstitions to the contrary.

Further, as the Lokayata pointed out, even a valid deductive proof, does not mean that the conclusion (the mathematical theorem) is true. Mathematical theorems are at best true relative to the axioms or assumptions which may be false as in the case of supposing that the wolf’s artificial footprints[60] what made by real wolf. In current mathematics an example of this sort (a validly proved mathematical theorem which is not true) is provided by the Banach-Tarski theorem. In fact, most axioms of mathematics involve infinity in some form and hence are not refutable or empirically testable. For example, consider the axiom that a unique invisible straight line connects two invisible geometric points. Or the axiom that a straight line has infinitely many points etc. as such these axioms are metaphysics which have to be accepted on authority. All current axioms of mathematics are laid down by the West.

Political advantage of axiomatic proofs

That brings us to our final point. Though the church sold its method of religious reasoning (MINUS facts) as superior, this method of reasoning actually was POLITICALLY superior (for the church, not for us).[61] Why? Because this technique, unconstrained by facts, can be used to draw any wild pre-desired conclusion, and say that it is a “reasonable” conclusion since based on reason (MINUS facts).

This Western control over axioms enables the West to control the content of axiomatic mathematics, just as the church controlled the contents of theology using its method of reasoning MINUS facts. This is useful because it enables one to manipulate science and say for example, as e.g. Tipler[62] did, that science has proved the truth of Christianity and has disproved other beliefs. This happens even at the level of the Templeton prize and the Nobel prize, given to Penrose for proving the existence of a singularity, which is not empirically verifiable, and is proved by assuming certain contentious beliefs about calculus.

To summarise, axiomatic reasoning or religious reasoning MINUS facts is epistemically inferior to ganita reasoning or scientific reasoning PLUS facts.

Conclusions

Mathematics (=Western ethnomathematics) is both practically and epistemically INFERIOR to ganita. It makes math needlessly difficult.

But it offers a POLITICAL advantage to the West, and puts the West control of the content of mathematics. This is beneficial to the West, but not to the colonized.

It is high time we demolished this vast web of deceit, and seized back control of mathematical knowledge from those who have been persistently inferior in mathematics. High time we started teaching superior ganita.

(The article was published on Medium.com on September 05, 2023 and has been reproduced here)

References

[1]See, e.g., the problem related to the elliptic functions needed for the first science experiment (the simple pendulum) in this tutorial sheet: http://ckraju.net/sgt/Tutorial-sgt.pdf.

[2]C . K. Raju, ‘Decolonising Mathematics’, AlterNation 25, no. 2 (2018): 12–43b. https://doi.org/10.29086/2519-5476/2018/v25n2a2.

[3]See the abstract of this recent talk at IIT Kanpur. http://ckraju.net/papers/teaching-calculus-as-ganita.pdf.

[4]C. K. Raju, ‘Ganita vs Formal Math: An Obituary of Formal Math’ (Shimla, 25 March 2021), https://www.youtube.com/watch?v=0UrMypVN6c0; C. K. Raju, ‘गणित बनाम मैथमेटिक्स [Ganita vs Mathematics]’, Himanjali 20, no. July-December (2020): 34–44; http://ckraju.net/papers/ckr-article-Himanjali-22-Final-18-06-2021.pdf. C. K. Raju, ‘Practical Ganita vs Religious Mathematics (व्यावहारिक गणित और मज़हबी मैथमेटिक्स)’, https://www.youtube.com/watch?v=k5UsF18cbno. The book is yet to be finalized and published.

[5]e.g. C. K. Raju, ‘A Singular Nobel?’, Mainstream 59, no. 7 (30 January 2021), http://www.mainstreamweekly.net/article10406.html.

[6]Satish Chandra Vidyabhushana, The Nyaya Sutras of Gotama (Allahabad: Pāninī Office, 1913).

[7]अनुमान in Sanskrit means deductive inference, and NOT अंदाज़ा as it has come to mean in Hindi, a meaning it acquired due to the constant Lokayata critique of deductive inference as an inferior means of proof. The wrong term निगमन used by NCERT Hindi texts ironically meant induction, not deduction!

[8]L. Mendelson, ‘Introduction to Mathematical Logic’ (New York: van Nostrand Reinhold, 1964), 29. The definition given above is simplified and paraphrased, not directly quoted.

[9]Modus ponens is the Latin name for the rule that from the two statements 1. A implies B, and 2 A, one can infer 3. B. For the view that the “Aristotelian” syllogism is derived from the Nyaya syllogism, obtained from Arabs, see C. K. Raju, ‘Logic’, in Encyclopedia of Non-Western Science, Technology and Medicine (Springer, 2016 2008).http://ckraju.net/papers/Nonwestern-logic.pdf.

[10]Raju, C. K., ‘Statistics for Social Science and Humanities: Should We Teach It Using Normal Math or Formal Math?’, lecture at JNU, 11 Aug 2020. https://www.youtube.com/watch?v=A9Og1k-Z5O4.

[11]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).

[12]C. K. Raju, ‘Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhāṣā’, Philosophy East and West 51, no. 3 (2001): 325–62, http://ckraju.net/papers/Hawaii.pdf.

[13]C. K. Raju, ‘“Euclid” Must Fall: The “Pythagorean” “Theorem” and The Rant Of Racist and Civilizational Superiority — Part 1’, Arụmarụka: Journal of Conversational Thinking 1, no. 1 (2021): 127–55, https://doi.org/10.4314/ajct.v1i1.6.

[14]C. K. Raju, Refutation of the Aryan Race Conjecture: The Arithmetic Evidence and Conquest-of-Greeks Theory (Delhi: Kant Academic Publishers, 2022).

[15]Rajendralal Mitra, ed., The Lalita Vistara (ललित विस्तर: सुत्त) (Calcutta: Asiatic Society of Bengal, 1877); Vaidya P. L., ed., Lalita-Vistara (ललित-विस्तर:), Buddhist Sanskrit Texts 1 (Darbhanga: Mithila Institute, 1958).

[16]Christoph Clavius, Arithmeticae Practicae [Practical Arithmetic] (Rome: Dominici Basae, 1583).

[17]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.

[18]Augustus de Morgan, Elements of Algebra: Preliminary to the Differential Calculus, 2nd ed. (London: Taylor and Walton, 1837).

[19]Augustus de de Morgan, On the Study and Difficulties of Mathematics, New Ed. (Chicago: Open Court Pub, 1898), 70–71.

[20]Raju, Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of Calculus from India to Europe in the 16th c, CE.

[21]C. K. Raju, ‘Probability in Ancient India’, in Handbook of Philosophy of Statistics, ed. Paul Thagard Dov M. Gabbay and John Woods, vol. 7, Handbook of Philosophy of Science (Elsevier, 2011), 1175–96, http://ckraju.net/papers/Probability-in-Ancient-India.pdf; C. K. Raju, ‘Probability’, in Encyclopedia of Non-Western Science, Technology and Medicine, ed. Helaine Selin (Dordrecht: Springer, 2016), http://ckraju.net/papers/Springer/Probability-springer.pdf.

[22]Raju, Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of Calculus from India to Europe in the 16th c, CE.

[23]https://tinyurl.com/decol-list-new.

[24]For the abstract see http://ckraju.net/papers/Calculus-story-abstract.html, and for presentation, see http://ckraju.net/papers/presentations/MIT.pdf.

[25]See, e.g. Invite to Nandita Narain to publicly debate her teaching of real analysis in the math classroom. https://twitter.com/CKRaju14/status/1495195375459651587. Or Invite to Parvin Sinclair, author of the NCERT school math text, and former Director of NCERT. https://twitter.com/CKRaju14/status/1498133388850384896.

[26]C. K. Raju, ‘California, Indian Calculus and the Technology Race. 1: The Indian Origin of Calculus and Its Transmission to Europe’, Boloji.Com, 11 December 2021, https://www.boloji.com/articles/52924/california-indian-calculus.

[27]C. K. Raju, ‘California, Indian Calculus and the Technology Race. 2: Don’t Cancel the Calculus, Make It Easy!’, Boloji.Com, 24 December 2021, https://www.boloji.com/articles/52950/california-indian-calculus-and.

[28]Raju, ‘Marx and Mathematics. 4: The Epistemic Test’.

[29]On the church dogma of discovery, all discoveries must be named after Christians, so Aryabhata’s method is called Euler’s method after Euler who studied Indian texts to write an article on the Indian calendar in 1700. C. K. Raju, ‘The Meaning of Christian “Discovery”’, Frontier Weekly 47, no. 29 (2015): 25–31, http://www.frontierweekly.com/archive/vol-number/vol/vol-47-2014-15/47-29/47-29-The%20Meaning%20of%20Christian%20Discovery.html.

[30]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.

[31]Plato, Meno, trans. Benjamin Jowett, http://classics.mit.edu/Plato/meno.html.

[32]Proclus, A Commentary on the First Book of Euclid’s Elements, trans. Glenn R. Morrow (Princeton, New Jersey: Princeton University Press, 1970).

[33]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).

[34]C. K. Raju, The Eleven Pictures of Time: The Physics, Philosophy and Politics of Time Beliefs (Sage, 2003) Chp 2, ‘The curse on “cyclic” time’.

[35]Hermann Gollancz, Dodi Ve-Nechdi (Uncle & Nephew) : From the Work of Berachya Hanakdan and Adelard of Bath (London: H. Milford Oxford University Press, 1920).

[36]This insight is due to the late Martin Bernal (personal communication). I earlier subscribed to the belief that it was ucli-des or “the key to geometry”, as discussed in the Farsi translation of Is Science Western in Origin? (Iran University press, 2012),

[37]C. K. Raju, ‘How to break the hegemony perpetuated by the university: decolonised courses in mathematics and the history and philosophy of science (Arabic)’, in Culturalisation of Humanities: Vision and Experiments. (Proceedings of the International Conference on Culturalization of the Humanities, held in Beirut on 20–21 November 2018.) (Beirut: Al Maaref University, 2019), 77–114, http://ckraju.net/papers/Beirut-paper%20for%20iias%20journal.pdf; Raju, C. K., ‘How to Break the Hegemony Perpetuated by the University: Decolonised Courses in Mathematics and the History and Philosophy of Science’, Studies in Humanities and Social Sciences 26, no. 2 (2019): 86–109.

[38]trans Dana C. Munro, Translations and Reprints from the Original Sources of European History, №3, The Medieval Student, vol. II: №3 (Philadelphia: University of Pennsylvania Press, 1897).

[39]H. M. Taylor, Euclid’s Elements of Geometry (Cambridge: Cambridge University Press, 1893).

[40]Raju, Euclid and Jesus: How and Why the Church Changed Mathematics and Christianity across Two Religious Wars.

[41]Proclus, A Commentary on the First Book of Euclid’s Elements.

[42]Thomas Aquinas, Sumnma Theologica, n.d., First Part, Q. 52, article 3, http://www.newadvent.org/summa/1052.htm#article3.

[43]Taylor, Euclid’s Elements of Geometry.

[44]Raju, ‘Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhāṣā’.

[45]David Hilbert, The Foundations of Geometry (The Open Court Publishing Co., La Salle, 1950), http://ckraju.net/geometry/Hilbert-Foundations-of-Geometry.pdf.

[46]Raju, The Eleven Pictures of Time: The Physics, Philosophy and Politics of Time Beliefs chp. 5, ‘In Einstein’s shadow:’

[47]E. A. Moise, Elementary Geometry from an Advanced Standpoint (Addison Wesley, 1990).

[48]Taylor, Euclid’s Elements of Geometry, 22.

[49]I. Todhunter, The Elements of Euclid for the Use of Schools and Colleges; (London: Macmillan, 1869), 9, https://ia800202.us.archive.org/35/items/elementsofeuclid00todhuoft/elementsofeuclid00todhuoft.pdf.

[50]Raju, The Eleven Pictures of Time: The Physics, Philosophy and Politics of Time Beliefs, 39.

[51]B. Russell, ‘The Teaching of Euclid’, The Mathematical Gazette 2, no. 33 (1902): 165–67; Bertrand Russell, An Essay on the Foundations of Geometry (Cambridge, 1897); A. N. Whitehead and B. Russell, Principia Mathematica (Cambridge University Press, 1927).

[52]George D Birkhoff, ‘A Set of Postulates for Plane Geometry, Based on Scale and Protractor’, Annals of Mathematics 33 (1932): 329–45.

[53]See the home assignment on geometry posted at http://ckraju.net/geometry/http://ckraju.net/geometry/Geometry-workshop-home-assignment.html.

[54]C. K. Raju, ‘The Religious Roots of Mathematics’, Theory, Culture & Society 23 (March 2006): 95–97.

[55]Raju, ‘Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhāṣā’; Raju, The Eleven Pictures of Time: The Physics, Philosophy and Politics of Time Beliefs; Raju, Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of Calculus from India to Europe in the 16th c, CE.

[56]C. K. Raju, Time: Towards a Consistent Theory (Kluwer Academic (Springer), 1994).

[57]Nyayavali, 304, https://sanskritdocuments.org/doc_z_misc_major_works/nyaayaavalii.html.

[58]Raju, ‘Decolonising Mathematics’.

[59]C. K. Raju, ‘A Singular Nobel?’

[60]Haribhadra Suri, ed., षटदर्शन समुच्चय, 5th ed. (Bharatiya Jnanapeeth, 2000) ksrika 81, commentary 559, p 452, and 454–6.

[61]C. K. Raju, ‘Why (Axiomatic) Math Is Racist’, Medium (blog), 28 February 2022, https://medium.com/@c_k_raju/why-axiomatic-math-is-racist-f1d81d716f13.

[62]F. J. Tipler, The Physics of Immortality: Modern Cosmology, God, and the Resurrection of the Dead (London: Macmillan, 1996); F. J. Tipler, The Physics of Christianity (New York: Doubleday, 2007).