Western appropriations of Indian gaṇita:
contemporary consequences

C. K. Raju

Indian Institute of Education


Introduction: Background

PHISPC story

  • 30 years ago, a group of influential Indian intellectuals
  • (D. P. Chattopadhyaya, Ravinder Kumar, Daya Krishna, A. Rahman, Kapila Vatsyayana, G. C. Pande… all dead now)
  • decided: Let us tell our own stories.
  • The West has been telling our stories for us for too long.

My authored volume took time: was the 50th in the PHISPC series

  • I am one of the few from the original group still alive since I was then the youngest in the group. 😄
  • Can't cover full book, just some highlights
  • and future developments.

History about FUTURE not past

  • 1st lesson from initial PHISPC discussions:
  • history is about the FUTURE not the past.

Also my key point

Most people don't understand this

  • or that there can be DIFFERENT ways to do math.
  • (Colonial education teaches myths and rote learning to make the colonised ignorant.)
  • Myth says math is universal, even for extraterrestrials.
  • "Isn't 1+1=2?" people ask, imagining their kindergarten lesson.

Proof of 1+1=2

  • But see Bertrand Russell's 378 page proof of 1+1=2 (in cardinals) in his Principia. Do you understand it?
  • Or my "Cape Town challenge" to prove 1+1=2 in real numbers from first principles (without assuming any result from set theory)
  • (real numbers needed for calculus as currently taught);
  • (offered reward of Rs 10 lakhs for this in JNU).
  • Why is 1+1=2 so difficult in formal/axiomatic math?

Formal math PROHIBITS the empirical

  • Because axiomatic (=formal) math prohibits the empirical. (It is 100% metaphysics.)
  • The KG proof of 1+1=2 is REJECTED in formal math just because you can see the oranges/apples!
  • "Beware of what you see" STATES NCERT class IX text (or any text on mathematical logic).
  • The prohibition of empirical in current math little known, poorly understood.
  • Shows it is WRONG to assume colonial/formal math is universal.

Indian gaṇita (गणित) had a DIFFERENT notion of proof

Common illiterate Western caricature

  • "Use of empirical = non-use of reasoning". ("Reasoning unique to West.") ❌
  • Reasoning used WITH empirical (facts) in Indian gaṇita
  • exactly as in science (or Sherlock Holmes) ✅
  • from before Aristotle
  • (real one from Stagira not the fake from 12th c. Toledo)
  • e.g. Gola 6, that earth is a sphere
  • DEDUCED from the OBSERVATION (Lalla 20-36) that far-off trees cannot be seen
  • (ships disappear/appear over the horizon, which is circular).

Normal vs formal math

  • In a word gaṇita is NORMAL math
    • (uses NORMAL reasoning = reasoning + facts)
    • (e.g. I SEE smoke and infer fire.)
  • Axiomatic math is FORMAL math
    • (uses formal reasoning = reasoning - facts)

Q. WHY is the empirical prohibited in (axiomatic) math?

  • Claim: "empirical proofs are fallible".
  • Yes, रज्जुसर्पन्यायः (Nyayavali 304)
  • Science too accepts experimental error
  • but still accepts empirical proof (experiment).
  • And science is still our BEST means of knowledge.
  • And if people study math for science, why prohibit empirical in math?

Superstition of infallibility of deduction

  • Problem is with the related UNSTATED claim ("contrapositive") that banning the empirical results in INFALLIBILITY,
  • i.e., that pure deduction (minus facts) is infallible.
  • Stated explicitly, this sounds just like the silly church SUPERSTITION
  • that the pope is infallible.

Let us again ask: Q. Why prohibit empirical in math?

  • A. Because it suited the CHURCH.
  • Church dogmas CONTRARY to facts (e.g. virgin birth).
  • Hence, church prohibited facts in reasoning when it accepted reasoning in 13th c.
  • Church invented (formal/axiomatic) reasoning (minus facts) for its Crusading theology of reason.

Church origins of (axiomatic) math (= Western ethnomath)

  • During Crusades church set up Christian theology of reason
  • to compete with Islamic theology of reason (aql-i-kalam)
  • Church accepted reason minus facts = axiomatic reason.
  • since it could not accept reason+facts(=gaṇita=science)

E.g. Aquinas' theorem (Summa Theologica)

  • Many angels can fit on a pin.
  • Since no facts about angels just assume anything.
  • Aquinas' assumption (=axiom) about angels (that they occupy no space)
  • Because persuasive proof key Church concern proof (not calculation) became central to axiomatic math.

Which is better? Normal math (gaṇita) or formal math?

Axiomatic mathematical proofs INFERIOR since HIGHLY fallible

  • E.g. students err in proofs hence flunk in math
  • Authorities too fallible — wrong proofs of Riemann hypothesis etc. published.
  • No error (not even a typo) in Russell's 378 page proof of 1+1=2? How do you know?
  • (a) blindly trust Russell's authority (b) inductively recheck the proof yourself.
  • Hence deductive proofs MORE fallible than proof by authority or inductive proofs.

Deductive proofs more FREQUENTLY fallible

  • Chess pure deduction. But every human makes a mistake every time hence loses to a computer.
  • Hence deductive proofs more frequently fallible.
  • So why prohibit empirical proofs in math?

From the 4th c. church claimed "Christians are privileged, superior"

  • "Superiority" proved by appeal to (secular) various models of false history since 5th c.
  • This false history reused by racist and colonial historians to "prove" White/Western superiority.
  • Main lesson of colonial/church education since Macaulay: West is superior imitate it.
  • Our education minister boasted "we have not changed a single comma, full stop".

Globalization of colonial math

  • E.g. "Greeks" ("Euclid") did something "superior" in math.
  • This (axiomatic) math globalized by colonial education.
  • My point: truth should be decided by PUBLIC debate, NOT myths or manipulative secretive refereeing.
  • Why is axiomatic math (Western ethnomath) "superior" to गणित or any normal math.
  • Any mathematician willing for PUBLIC debate today?

Affects current teaching of math

Why should you worry?

  • (a) Because it makes math difficult for your children with adding practical value.
  • (b) Epistemic dependence: forces them to accept Western authority as truth even for 1+1=2
  • (c) This math a compulsory school subject, but NOT secular unlike gaṇita.

Not even secular

  • Word "mathematics" derives from mathesis (Plato, Proclus),
  • involves a doctrine of soul
  • (same as Hindu ātman and Egyptian/"pagan" notion of soul) cursed and changed by the 6th c. church.
  • Math further changed by the Crusading theology of axiomatic reason (falsely attributed to "Euclid").

Making math difficult impedes technology development. E.g.

Colonialism globalized, entrenched this false history and related bad philosophy

  • Colonial education designed to create awe of the West and enslave your mind
  • made this false history an essential aspect of childhood indoctrination
  • to teach uncritical imitation of the West (as "superior")

Herculean task

  • Task of correcting 1600 years of FILTH in history
  • over 50 times bigger than Herculean task of cleaning the Augean stables (only 30 years of shit ).
  • Against this long background let me come to the topic of today's talk
  • the Indian alternative to Western transmogrified calculus.

Today's talk: calculus origin and theft

  • My talk today will centre around my 2007 PHISPC volume, and subsequent developments
  • 1. That the calculus originated in India.
  • BUT the story does NOT end there.
  • 2. Calculus was stolen by Europeans,
  • 3. who failed to FULLY understand it

Contemporary consequences

  • 4. Europeans added church metaphysics and returned an INFERIOR version of calculus through colonial education
    • which version makes calculus difficult without adding an iota to its practical or epistemic value,
  • 5. We therefore need to change current teaching of calculus, indeed all math
  • in the interests of your children who find calculus (or math needlessly) difficult
  • in the interests of future technology development (AI, data science)
  • which needs a workforce with a proper understanding of math.
  • In the interests of secularism since church metaphysics adds a religious bias.

Who invented calculus?

NCERT class XI text attributes calculus to Newton

  • Teaches limits are essential to define derivative
  • but does NOT define derivative!
  • Does not even define real numbers needed for limits.
  • Students stay totally confused.
  • Most students flunk my calculus pre-test, with NEGATIVE marks; no one gets even zero

Āryabhaṭa (आर्यभट, 5th c.) gave a table of 24 sine values


  • अर्धज्या = half chord = sine
  • (jya -> jiva -> jiba ->jaib -> sinus ->sine)
  • कला = first sexagesimal minute (विकला = 2nd , तत्परा = 3rd)
  • = accuracy of about 5 decimal places (e.g. Āryabhaṭa's value of \(\pi = \frac{62832}{20000} = 3.1416 \approx 3.14159\))
    • prolixity due to need to preserve metre in verse.
    • al Khwarizmi (9th c.) and Simon Stevin (16th) verbatim repeat this prolixity showing their source

Kerala school?

Madhava's values more accurate

  • from some 900 years later.
  • But can one honestly say Kerala school invented calculus, not Āryabhaṭa?
  • And why not Vaṭeśvara? (10th c.) who derived sine valued accurate to the seconds?

Infinite series

However, summing infinite series NOT essential to practical applications of calculus

  • Newton did not know how to sum infinite series, but obtained practical results from his physics.
  • A computer cannot sum infinite series (or use real numbers) but is used for all practical applications of calculus today.
  • No one knows how to sum the infinite S-matrix expansion of quantum field theory
    • used to calculate all experimental consequences of qft.

My "calculus without limits" course stressed 4 key aspects of (Indian) calculus

  • 1. Derivative (no limits, finite difference [खंडज्या] including backward [गतखंड]and forward difference [भोग्यखंड] (Brahmagupta 7th c.)
  • 2. Integral = numerical method to solve a difference/differential equation
    • ("Euler" method, first used by Āryabhaṭa as elementary "rule of three" त्रैराशिक).
    • (Āryabhaṭa's sine table has ONLY sine DIFFERENCES.)
  • 3. Non-Archimedean arithmetic of Brahmagupta's polynomials (अव्यक्त गणित) (instead of "real" numbers)
  • 4. A philosophy of zeroism (शून्यवाद) or INEXACTITUDE (instead of exactitude).

Note that 3 and 4 can be used to sum infinite series

  • such as the infinite geometric series. Thus,
  • \(1+x+x²+...+xⁿ = \frac{1-x^{n+1}}{1-x}\) (simple multiplication of polynomials)
  • Non-Archimedean arithmetic has infinities and infinitesimals.
  • If \(x<1\), then when \(n\) is infinite, \(x^{n+1}\) is infinitesimal.

However, these key aspects of calculus

  • all antedate the "Kerala school".
  • Conclusion: credit for originating calculus goes to Āryabhaṭa (finite differences, "Euler method")
  • or Āryabhaṭa + Brahmagupta (ODE solver+non-Archimedean arithmetic)
  • though Āryabhaṭa school in Kerala contributed to its development.

Why Europeans stole calculus

  • Basically to solve their navigational problem.
  • Since sources of European wealth (piracy, trade) all overseas.
  • European governments hence announced large prizes for its solution.
  • Last being the British Longitude prize through an act of 1711.

Latitude, longitude and loxodromes

  • Despite the terrible Crusading/racist/colonial lies about "Greeks"
  • fact is that early Greeks and Romans were VERY BACKWARD in math
  • E.g. they lacked efficient arithmetic,
  • hence repeatedly imported Indian arithmetic ("Arabic numerals") from 10th to 16th c.

Non-textual evidence: fractions and calendar

  • Early Greeks and Roman (and Europeans) lacked
  • knowledge of elementary fractions (known in Egypt)
  • Fractions introduced in Jesuit syllabus ca. 1572.
  • Therefore, they had a miserably bad calendar.
  • Irrefutable non-textual evidence.
  • Solar and lunar cycles, both, are not an integer number of days.
  • both year and month in fractions of days.
  • Accurate calendar (date of equinox) needed for navigation
  • (to determine latitude at sea from observation of solar altitude at noon).

Mercator map needed trigonometric values

Determining latitude needed a good calendar

  • Measuring latitude in daytime through observation of solar altitude at noon
  • requires knowledge of declination \(δ\), say \[δ =(23+\frac{27}{60})\sin(\frac{360d} {365.25})\]
  • \(d\) = number of days since equinox.
  • Needs accurate date of equinox.
  • Julian calendar = official Christian calendar since 4th c. had WRONG date of equinox
  • Hence, Gregorian calendar reform of 1582 to fix date of equinox accurately.
  • Could be done only after contact with India (because traditional Indian calendar was accurate. Europeans lacked fraction. Of course the pope told a story.)

Determining Longitude needs trigonometric values

  • (Plane navigation) Longitude can be calculated by solving the nautical triangle
  • Europeans did by dead reckoning (heaving the log to observe ship speed in knots on a rope tied to the log).
  • Bhaskar 1 (7th c,) said Mahābhāskarīya 2.5, text, trans) disciples of "the bhaṭa" (भटस्य शिष्या:) say this method is gross
  • (coarse distance, "Pythagorean theorem" fails due to earth's curvature).

Size of the earth

  • Nautical triangle can also be solved using latitude difference.
  • But that requires knowledge of the size of the earth
  • Earth size available in all Indian texts (in yojana-s).
  • Accuracy of method checked using al Birunī's figure in Arabic miles carefully related to English mile.


European longitude problem

  • Bible says earth is flat (say tall trees CAN be seen from "ends" of earth).
  • Columbus reduced earth size by 40%, to facilitate funding for his project,
  • resulting in later navigational disasters
  • and Portuguese law banning globes aboard ships.
  • But size of earth REMAINED unknown to Europeans until late 17th c.
  • (But no sailor trusted Picard's measurement that then.)
  • Hence European longitude problem persisted until end 18th c. (at least).
  • Why all these details?

Standard of evidence for theft of calculus

  • Since theft is a criminal offence I used the standard of evidence in criminal law
  • 1. Motive,
  • 2. opportunity,
  • 3. circumstantial evidence and
  • 4. documentary evidence


  • European navigational problem provides the all-important MOTIVE
  • for theft of calculus.
  • Eince good navigation was the key to dreams of wealth of poor Europeans then
  • and precise trigonometric values obtained from Indian calculus the key to the solution.
  • E.g. every navigational theorist from the 16th c. Simon Stevin gave tables of secants (for the Mercator chart).


  • While calculus did not originate in Kerala
  • Kerala was the key to its theft
  • First Roman Catholic mission started in Kochi by Vasco in 1501
  • for the local Syrian Christians.

Jesuits in Kochi

  • Later (ca. 1550) Kochi mission turned into a Jesuit college.
  • Portuguese and "Kerala school" had a common patron.
  • Missionaries naturally learnt local languages.
  • Translated Indian texts in Toledo mode.

Toledo mode

  • During the Crusades a huge Arabic library at Toledo came under Christian control
  • These Arabic texts were MASS translated into Latin starting 1125.
  • Q. Since, the church policy was to burn heretical books how could texts from the religious enemy be translated?
  • A. The origin of the world knowledge in these Arabic books (including Indian knowledge) was wholesale attributed to Greeks, then regarded as "friends of Christians".

Racist and colonial historians

Interim summary

  • Anyway, point is that Kochi Jesuits had the opportunity
  • and inside info about Indian texts from Syrian Christian who were their close friends till 16th c.

Circumstantial evidence

  • Jesuit general Clavius's sine table interpolated version of Madhava's
  • His contemporary Julius Scaliger used Indian ahargaṇa (called Julian day numbers)
  • Tycho Brahe (Astronomer Royal to the Holy Roman Empire) claimed an astronomical model identical to Nīlakanṭha, etc.
    • Tycho kept his "observations"(/heretical imports) secret from even his assistant Kepler.

Circumstantial evidence (contd.)

  • We have seen similarity of Stirling's formula
  • "Euler" method. (Euler wrote an article on Indian astronomy.)
  • Fermat's challenge problem to European mathematicians,
  • remained unsolved for long (and was eventually solved by Euler)

Fermat's challenge problem

  • Thus, in Feb 1657, Fermat (Ouvres, p. 332 et seq.) asked European mathematicians to solve the problem
  • \(Nx^2 + 1 = y^2\) for \(N=61\), and \(N=109\).
  • the smallest solutions are \(x = 226153980,\ y = 1766319049\) given by Bhaskara II centuries earlier. (Bījagaṇita (87, Colebrooke 1816, pp. 176–178)
  • "Independent" rediscovery? No. Numbers too large.

Documentary evidence

The epistemic test

First international article to state these 4 legal criteria of evidence

  • was my Hawaii paper of 1999/2001.
  • However, it attracted thieves.
  • If theft of Indian calculus not understood by Indians in so many centuries
  • then easy to steal more from Indians
  • especially since everybody knows "trust the West" is the main lesson of colonial education.

Theft of the transmission thesis

Theft of the transmission thesis (contd)

Western and colonial support for continued plagiarism

  • Plagiarists counted on support from Indian journalists and intellectuals.
  • Subhash Kak was the first to celebrate the plagiarists in sulekha.com ("British approval for calculus theft")
  • sent it PHISPC which has an appendix on the first (known) act of plagiarism.

Subsequently I used the EPISTEMIC TEST

  • While plagiarizing, Almeida and my former post-doc John made gross errors:
  • confounded solar altitude and declination
  • stated (thrice) solar declination was measured at sea.
  • Contrary to European need for good calendar and Gregorian reform to determine latitude.

Floating point numbers

On the epistemic test

  • Those who copy (like students to cheat in an exam)
  • do NOT fully understand what they copy
  • Applied also to Einstein (who plagiarized from Poincaré,
  • and to Michael Atiyah who plagiarized my correction to Einstein (at the same time as Joseph).

Epistemic test applies ALSO to Newton and Leibniz

  • Neither understood how to sum infinite series (Leibniz series, since series, which Newton claimed).
  • Newton called Leibniz the "second discoverer" since Leibniz did not know how to sum the series today called Leibniz series.
  • Berkeley rightly made fun of Newton's silly and meaningless fluxions.
  • How did Newton get practical value then?
  • Because practical value depends only on solving differential equations.
  • Numerical solution fine for all application: does not require limits, fluxions, real numbers.
  • So called "analytic solutions" even for the harmonic oscillator
  • do NOT give better value since even the sine function can at best be numerically evaluated.

Real numbers are Western admission of lack of understanding of calculus

  • "Real" numbers were invented (ca. 1880) by Richard Dedekind {{color(red,BECAUSE)}}} of difficulties in understanding calculus
  • (also require axiomatic set theory invented in 20th c.)
  • Partial understanding = Lack of full understanding = clinching proof that Newton and Leibniz did NOT "discover" calculus
  • except in the sense of the vile principle of "Christian discovery"

Follow up

  • It is 15 years since I wrote the PHISPC volume.
  • A second edition and a French translation are due.
  • Conducted a series of formal courses on calculus without limits
  • with 8 groups in 5 universities in 3 countries.

Calculus without limits

Rajju gaṇita (string geometry)

  • Teaching Indian calculus requires preparation
  • Indian/Egyptian geometry used a flexible string (रज्जु): can directly measure curved lines
  • impossible with current geometry box/compass box.
  • See videos of workshop on Rajju ganita, day 1, day 2.

Teaching experiments

Probability and statistics

  • Current teaching requires (advanced) calculus
  • At lest Riemann-Stieltjes integral (or Lebesgue integral and probability as measure)
  • Too difficult for most students, hence California recently canceled the calculus.

Probability and statistics too originated in Ancient India

However, teaching statistics without calculus is risky and dangerous

  • My decolonised calculus course easy even for social science students
  • leads up to decolonised statistics curriculum.

Summary and conclusions-1: math philosophy

  • Math NOT universal. Formal/axiomatic math a Crusading church construct,
    • falsely attributed to Euclid (a Crusading myth)
  • Formal math differs from gaṇita = normal math
    • adds religiously biased metaphysics of nil practical (or epistemic) value

S&C -2: Calculus origins

  • Calculus originated as gaṇita with the 5th c. Āryabhaṭa
  • since accurate trigonometric values needed for agriculture and navigation–two key sources of Indian wealth.
  • Developed over next 1000 years
  • Kerala school gave accuracy to thirds, added infinite series, recognizable as part of current calculus
  • but infinite series NOT critical to teaching calculus.

S&C-3: Calculus theft

  • Calculus (and calendrical info) STOLEN by Kochi-based Jesuits in 16th c. for solution to European navigational problem (16th-18th c.)
  • since determining the 3 ells, latitude, longitude, loxodromes required accurate trigonometric values obtained using calculus
  • and then European dreams of wealth depended on a good method of navigation.

S&C-4 Western and colonised response

  • West (and colonised) hell bent on defending this plagiarism and stories of White/Western "superiority".
  • They defend plagiarism (by Christians) from non-Christians, non-Whites, non-West) tooth and nail (Joseph, Atiyah)
  • just as no hope returning land in Americas and Australia's to original inhabitants
  • or reparations to Black slaves whose labor created initial American wealth.

S&C-5: Epistemic test

  • Hence epistemic test essential
  • Real numbers (regarded as essential to calculus teaching)
  • invented in 19th/20th c. just because of admission that West did not understand the calculus.
  • No practical or epistemic value: see S&C-1.

S&C-6: Decolonised courses exist, tried out

  • in calculus
  • rajju gaṇit
  • statistics
  • Now it depends on will of colonised to decolonise.


  • So is it a matter of great PRIDE that Indians invented calculus?
  • Now church invented false history as secular proof of Christian "superiority"
  • Later reused by racists and colonizers to claim White/Western "superiority".
  • Therefore, this rhetoric of pride seems a great antidote.

Matter of great shame!

  • But I see it as a matter of great shame!
  • 75 years after independence we keep crowing "Kerala school"
  • and keep studying Thomas' calculus for IIT, (and real analysis later).

Matter of great shame (contd)

  • Shows we have no understanding of calculus or Indian achievements.
  • Infinite series easily recognizable aspect of current calculus but little use for calculus for teaching.
  • We have not even understood the source of false history is the church NOT Macaulay
  • Obvious: since same colonial education was globalised.

Insufficient to understand just one church trick

  • E.g. the church demonised Indian society as casteist and iniquitous
  • this campaign very active today in US.
  • But caste system existed also in Africa,
  • and Africans had no problems with it. (Cheikh Anta Diop, 1987, Precolonial Black Africa).
  • Therefore, also, Aryabhata NOT Kerala school

Āryabhaṭa (NOT Āryabhaṭṭa)

Another example

  • E.g. if you don't understand false Crusading history of Euclid
  • or the real origins of axiomatic reasoning in church theology of reason.
  • will still keep doing the wrong math and the wrong calculus.
  • But no change possible, since "experts" not publicly accountable.

And no public debate possible in 6 years

  • In 6 years I could not find one mathematicians willing to discuss publicly the math we teach in the classroom
  • (wait a few more years and no one will be left to debate!)
  • Even asked the union education minister of state
  • just to organize a one-day one-on-one debate on math teaching.
  • But we can't even discuss it any more than people could discuss heresy during the Inquisition.

Collective failure of colonised society

  • Not only neta-s, babu-s, "experts", but ordinary people could do nothing.
  • In tech industry, Google CEO Sundar Pichay did this nonsense Thomas's calculus for IIT.
  • But not willing to invest in change, not for moral reasons,
  • not even for Google's benefit in better trained manpower.
  • Well: will result in poetic justice of falling behing in the technology race!

Decolonisation possible only if you have the desire to be free

  • if not you voluntarily remain a (mental) slave
  • So, you may celebrate "Kerala calculus", and do IIT calculus! 😄
  • (Future generation will laugh at your limited understanding.)
  • Or the amazing fear to rectify or even discuss the much bigger problem of math teaching
  • PS. I have no problems with Kerala, my mother was from Kerala.

Created: 2022-03-27 Sun 08:27