• But no one able to produce evidence for Pythagoras or his connection to the proposition
• Claim based on utterly FALSE history of math (and science)
  • e.g. “Euclid” gave axiomatic proofs
  • Newton “discovered” calculus

• Adds no EPISTEMIC value (e.g. my Durban keynote in AlterNation)
• deduction MORE fallible than induction, empirical proof.
• SUBTRACTS practical value by making math difficult
• e.g. Bertrand Russell's silly 378 page proof of 1+1=2 useless in a grocery shop

• Similar difficulty of calculus with Dedekind (axiomatic) reals bad for current tech:
• California canceled calculus in schools (1,2).
• No proper statistics or data science without calculus.
• But formalism adds political value, enabling West to control mathematical knowledge.

• the false history/myths of math
• (e.g. West “discovered” calculus”)
• AND the bad philosophy/superstitions of math (axiomatic proofs, axiomatic reals “superior”, like White skin)

• 1. Calculus began in India in the 5th c.; was stolen in the 16th c.
• Theft proved beyond reasonable doubt: opportunity, motivation, circumstantial evidence
  • Term “stolen” used to emphasize that stolen knowledge is usually poorly understood,
  • e.g. Newton’s pitiable flux ions with no understanding of infinitesimals.

• 2. Eventual Western solution to its poor grasp of calculus, ­ used metaphysical or unreal “real” numbers,
  • globalized by the power of colonialism.
  • (Decolonisaton asserts this is an INFERIOR solution.)

• 3. Teaching calculus as it first originated
  • using Āryabhaṭa’s (5th c.) numerical finite difference methods +
  • Brahmagupta’s (7th c.) “non-Archimedean” arithmetic of polynomial (with infinitesimals +
  • a secular philosophy of zeroism

• makes calculus very easy
• enables students to solve harder practical problems using calculus
• as demonstrated by pedagogical experiments in last 15 years in 5 univs in 3 countries.

• Accepting this involves great loss of face, power, and money for West (like end of slavery, apartheid)
• and is also contrary to Western religious beliefs linked to math since Plato and Crusading church.
• But if West fails to switch from formal to normal math
• it will fall behind in technology development.

• "To decolonize math stand up to its false history and bad philosophy"
• To reiterate: BOTH false history
• and BAD philosophy of math used to claim supremacy, BUT
• Related article published in Conversation, went viral, THEN censored
• since no valid intellectual response to my arguments.

• See Was Euclid a black woman?,
• "Black thoughts matter: Decolonised math, academic censorship…", J. Black Studies 48, no. 3 (2017): 256–78.
• Mathematics and censorship,
• Article also in Rhodes must fall-Oxford, chp. 26.

• Stephen Hawking’s co-author G.F.R. Ellis started the conspiracy theory polemic against me in Cape Town
• as also the Bantuization polemic suggesting that my decolonised calculus course involves a dumbing down
• when it actually enables students to solve harder problems not included in usual calculus courses.

• Similar deliberate misrepresentation by John Armstrong of King’s College London
• claiming my decolonized course teaches only software!
• The West has not understood it is fast losing all credibility
• and such “joker responses” only accelerate the loss of credibility.

• My personal interest: to check if the West is at all capable of an intellectual response
• based on facts and arguments.

• I was a formal mathematician, teaching/doing real/functional analysis.
• Was invited by ANU math department
• and lectured on a key DEFECT of CALCULUS as taught today.
• Problem: can't differentiate discontinuous functions such as \[\theta (x) = \begin{cases} 0 & \text {if}\ x < 0 \\ 1 &\text {if}\ x > 0 \end{cases}.\]

• Why do we need to do so?
• E.g. to make sense of the differential equations of physics at a physical discontinuity
• such as a shock wave or blast wave in a fluid.

• BUT, even if one defines the derivative (at a discontinuity), e.g., as a Schwartz distribution
• \(\theta ' = \delta\)
• one is UNABLE to multiply Schwartz distributions (e.g., \(\delta \cdot \delta\))
• (required since differential equations of physics are nonlinear).

• My 1982 product used (Robinson's) non-standard analysis
• a key feature of which is “non-Archimedean” arithmetic
• which has infinities and infinitesimals (hence, no limits),
• very different from (“Archimedean”) real numbers.

• But these infinites and infinitesimals disappear in the final result
• as in my product applied to shocks in fluids, discontinuities in relativity
• and shocks and singularities in general relativity
• (Won't go into technical details. See appendix to my CFM book.)

• My points: (1) Calculus originated in India with 5th c. Āryabhaṭa
  • over 1150 years before Newton,
  • as a numerical technique (“Euler’s” method of solving ODEs) using finite differences.

• (2) calculus (and infinite series) best understood with “non-Archimedes” arithmetic (+ zeroism)
  • which has infinities and infinitesimals (hence no limits)
  • (NOT formal real numbers and limits + non-standard analysis)

• this Indian technique since 7th c. Brahmagupta's “non-Archimedean” arithmetic of polynomials.

• Brahmagupta's (7th c.) अव्यक्त गणित = “unexpressed arithmetic” (Brāhmasphuṭasiddhānta vol. 4)
• = arithmetic of polynomials = “unexpressed numbers”
• “unexpressed” since \(2x+3\) acquires a value only when a value is assigned to \(x\).
• Later badly translated as “algebra” (al jabr waal Muqabala) by al Khwarizmi.

• Brahmagupta solved both linear and quadratic (multivariable) equations (source, trans.).
• Later Indians expressed rational functions \(\frac{4 x^2 - 4}{4 x^3 - 4x}\) in usual place-value notation
• (e.g. Kriyākramakarī p. 388) as \[ \frac {[4|0|4°]}{[4|0|4°|0]}. \]

• Arithmetic of polynomial fractions or (“rational functions”)
• very similar to arithmetic of ordinary fractions (rational numbers).
• In terms of formal algebra both rational numbers and rational functions constitute an ordered field
• EXCEPT that ordering among polynomial fractions is “non-Archimedean”.

• As a number \(x-n\) positive or negative depending on value of \(x\)
• but as a polynomial \(x-n>0\) for every \(n\)
• since number \(x-n\) positive for sufficiently large \(x\) for any \(n\)(e.g. Moise (1963) ¹)
• hence \(x > n\) for every \(n\), i.e., \(x\) is infinite.

• Formally, since rational functions are a field, we also have
• \(0 < \frac{1}{x} < 1/n\) for every \(n\), so \(\frac{1}{x}\) is infinitesimal.
• No limits possible with infinities and infinitesimals in Brahmagupta’s “non-Archimedean”) arithmetic

• Avyakt gaṇita is NOT non-standard analysis. But works similarly:
• infinitesimals disappear from the final results.
• E.g. sum of infinite geometric series (Nīlakanṭh, 15th-16th c.)
• infinitesimals drop out using a philosophy of inexactitude called zeroism (inexactitude earlier called śūnyavāda).

• So, we already have some fundamental differences between original Indian calculus and current Western calculus:
• (1) Brahmagupta's (“non-Archimedean”) arithmetic vs (“Archimedean”) real numbers
• (2) Zeroism (exactitude impossible in real world) vs Western belief in (metaphysical) exactitude in math,
• West believes in exactitude since it always connected math to religious belief

• In the 16th c. Jesuits in Kochi systematically translated many Indian texts
• on the Toledo model and sent them to Europe.
• Math+astronomy texts went to Clavius-Tycho Brahe-Kepler-Galileo-Cavlieri-Fermat-Pascal-Gregory-Newton-Leibniz
• So the story that “Newton discovered calculus” is COMPLETELY BOGUS history.

• Newton called "discoverer" of calculus
• on the genocidal church dogma of “Christian discovery”.
• in which Newton believed, hence he heavily invested in evil transcontinental slave trade
• “morally justified” by that dogma.

• According to the dogma the first Christian to spot a piece of land
• or knowledge, is declared its “discoverer” hence OWNER
• regardless of prior inhabitation (of land) or prior knowledge. (US Supreme Court judgment.)

• So, Columbus “discovered” America, (regardless of millions of prior inhabitants who could hence be “morally” murdered)
• Vasco “discovered” India, Cook “discovered” Australia,
• Newton “discovered” calculus, regardless of its prior invention by other non-Christians.
• This was brazen theft, Newton was a calculus thief.

• Epistemic test: Knowledge thieves fail to FULLY understand the knowledge they steal,
• like students who cheat in an exam fail to FULLY grasp what they copy
• and can't explain their answer-sheet (They grasp some part.)

• Because calculus was stolen knowledge from India, Europeans failed to FULLY understand it.
• Newton could not explain the infinitesimals he used for his fluxions.
• Europeans did not understand how to sum Indian infinite series for centuries after Newton.

• widely ADMITTED till 19th c.
• e.g. by bishop Berkeley (1734)
  • “why not set infinitesimals to zero at the start of a calculation not at the end”
• Karl Marx (ca. 1860),“Newton’s calculus mystical”,

• Dedekind (1871) “calculus lacks rigor”(when he invented Dedekind reals).
• But today Dedekind reals are the socially accepted solution.

• Many superstition e.g. racism too socially accepted in the West for centuries,
• that no reason why WE should accept it.
• My point Dedekind reals a BAD solution
• add NIL practical value
• epistemc value only on Western superstitions.

• On my second visit to Australia and New Zealand, in 2005,
• I already spoke on the Indian origin of calculus at UNSW and at Auckland (1,2),
• but was also invited by Sydney University (and Melbourne U.)to talk on my theory of time.

• On the morning I left, I was attacked by a racist dog,
• in the park in front of Sydney University.
• Took all my knowledge of dog psychology to fend off the dog without even a stone,

• till its master leisurely sauntered up and said
• "he is a good dog, it is just your clothes".
• Both racist master and dog apparently believed that Western clothes are universal:
• the norm is everyone OUGHT to wear them.

• This mandatory “ought to” is what I call “normative universality”
• in my decolonized curriculum on HPS, part 2
• Normative universality is double speak for a racist sense of SUPREMACY
• “you OUGHT to do calculus with reals and limits, as we do”.
• even if our understanding is INFERIOR (like De Morgan’s 19th c. understanding of elementary arithmetic?)

• On day of the racist dog, Sir Michael Aliyah, Fields Medalist + Abel Laureate + President of Royal Society,
• gave his 2005 Einstein Centenary lecture to the AMS
• in this lecture he plagiarised the thesis in my 1994 Time book

• that correcting Einstein's mistake
• and using functional differential equations in electrodynamics and relativity
• could explain quantum mechanics
• But did NOT cite my book published a decade earlier,
• instead added "Don't forget that I suggested this".

• it is unethical to claim ignorance of past work even ONCE:
• especially a Springer book ten years in print.
• Yet Atiyah was immediately informed and acknowledged the emails.

• But he AGAIN repeated his claim a 2nd time
• "Don't forget that I suggested this"
• in an article he got published in the Notices of the AMS 53(6) 674-78 (2006)
• prominently reporting Atiyah's “Einstein lecture”.

• Eventually Atiyah was forced to admit my priority, Notices of the AMS 54(4) (2007) p. 472.
• Atiyah was held prima facie guilty by an ethics committee (case no. 2 of 2007)
• But no apology, BRAZEN pretense of accidental oversight was maintained by
• Notices which REFUSED to publish my letter pointing out that this plagiarism happened twice
• 2nd time after Aliyah was personally informed!

• Clear case of brazen dishonesty at the top
• supported by key math institutions.

• Cultural Foundations of Math
• on Indian origin of calculus and how Europeans (Newton, Leibniz etc.) stole it

• was itself repeatedly stolen from India (from me)
• by the serial plagiarists George Joseph, Dennis Almeda and others.
• 2007 press release from Manchester 2 months after my book on calculus was published
• claiming Joseph and Almeida authored a paper on calculus transmission

• The paper was a near-verbatim copy of my 1999 paper
• submitted to Joseph for a conference he organized in 2000 in Trivandrum.

• Again it was done a third time after two very irregular ethics committees
• one in Exeter had warned one of them in 2003.
• This is clear from the retraction of the 2007 front page news by Hindustan Times.

• Manchester University still keeps on its website the fake news release which made headlines
• though the paper it mentions was NEVER published in 17 years since 2007.
• It could not be published because it VERBATIM plagiarizes most of my paper
• published in the proceedings of a conference organized by Joseph in Trivandrum Jan 2000
• without crediting me.

• Please write to Manchester and ask;
• where is that paper mentioned in your 17 year old fake news?
• Why do you brazenly preserve the fake news on your website
• despite an ethics committee?

• with Stephen Hawking’s co-author G. F. R. Ellis
• responding with an abusive racist mob to my decolonization proposal.

• for West has a long record of systematically dishonest history as a WEAPON to assert supremacy
• such as Western supremacy during colonialism
• (this diagram from my Pretoria-Tübingen keynote "Euclid must fall, part 1")

• Actual history: West was PERSISTENTLY inferior even in elementary arithmetic
• from early Greek times to 19th c.

• Gerbert made a monstrous abacus with 27 columns to represent large numbers
• with a 1000 apices
• Gerbert understood large numbers need place value, BUT
• FAILED to understand efficiency of Indian arithmetic for arithmetic operations
• DESTROYED by his abacus.

• E.g. \(89 + 89\) needs 18 operations on Graeco-Roman abacus but only 3 on gaṇita
• \(89 \times 89\) needs 1602 operations on abacus instead of 8 on place-value gaṇita

• Took another 2 centuries for Christian Europeans to grasp the efficiency of Indian gaṇita.
• Then Florentine merchant Fibonacci wrote Liber Abaci (13th c.).
• based on al Khwarizmi's 9th c. Hisab al Hind.
• Both (Fibonacci and al Khwarizmi) gave a simplified treatment of Indian gaṇita

• As found in common Indian texts emphasizing commercial uses of gaṇita (instead of astronomy)
• Compare contents of 9th c. Mahavira's गणित सार संग्रह (Gaṇita sāra sangraha),
• with Fibonacci's contents (square/cube roots etc. only in chp. 14)
• Note Fibonacci's chp. 4 on subtraction (only LESSER numbers)

• That is, still mired in the paradigm of pebble arithmetic like Gerbert
• Fibonacci blundered: has no negative numbers (like al Khwarizmi)
• because in primitive Graeco-Roman calculus (pebble arithmetic)
• subtraction means removing pebbles from an existing hoard
• and you can't remove more pebbles than there are on the board!

• OF COURSE Fibonacci numbers stolen from India, he made NO original contribution.
• Fibonacci's influence largely limited to Florence and neighbouring states
• since zero (= sifr=cipher) puzzled Europeans] (and Florence passed a law against zero)
• Finally, in 16th c. Gregor Reisz declared victory of algorismus over abacus

• followed by Adam Riese.
• Most importantly Jesuit general Christoph Clavius again imported gaṇita direct from India
• wrote a text on practical mathematics,
• introduced it in the Jesuit syllabus ca. 1575.

• Fibonacci's confusion about negative numbers persisted in Europe
• Three quick indicative examples
• Pascal (17th c.), Euler 18th c.), De Morgan (19th .)

• Conclusion: Western math was PERSISTENTLY INFERIOR,
• even in the matter of primary-school arithmetic
• from early Greek times till end of 19th c.
• contrary to colonial tale of Western supremacy.

• On the principle that “phylogeny is ontogeny”
• these Western difficulties with elementary math are reproduced even in the primary school classroom today.
• Hence, aping the West makes math difficult even in the matter of primary school arithmetic.
• So the decolonial solution is to revert to the original methods of teaching ganita.

• E.g. Assume child has learnt representation of numbers by objects at home.
• START teaching with place value, teach large numbers early.
• Teach practical value of arithmetic as in Mahavira etc.
• not ass. law com. law etc.

• Calculus originated with 5th c. Āryabhaṭa
• who obtained sine values (trans.) and value of π
• accurate to the the first sexagesimal minute
• by solving finite difference equations.

• Āryabhaṭa used a simple recursive numerical technique falsely called “Euler method” today.
• Euler knew of it since he studied Indian math texts (wrote an article on Indian calendar in 1738)
• hence magically solved Fermat's 1657 challenge problem
• (solve \(Nx²+1=y²\) for \(N=61\)) a century after it was posed

• That problem a solved exercise in Bhaskara II (Bījagaṇita 87, trans. Colebrooke)
• Numbers in solution (\(x\) =

226153980, \(y\) = 1766319049) too large for miracle of “independent rediscovery”

• but Western Christian-chauvinist history is full of such miracles.

• Tycho Brahe's astronomical model a carbon copy of the model of Nīlakanṭha (Āryabhaṭa's commentator)
• Clavius' 1607 sine table, an interpolated version of Madhava's sine table
• will skip “pre-calculus” of Cavalieri, Fermat, Pascal

• “Newton's” sine series carbon copy of Madhava's infinite sine series
• like “Leibniz” series for π
• “Stirling's” formula copies Brahmagupta’s (7th c.) quadratic interpolation formula
• he used to revert to traditional table of 6 sine values 15° apart.

• Vaṭeshwara (904 CE) used PAST (backward) and FUTURE (forward) differences to get
• Newton-Gauss formula and sine values precise to seconds.
• Western history of math chock full of miracles.
• But such a long series of improbable “coincidences” or “independent rediscovery” has probability zero.

• I see this as impeccable circumstantial evidence that
• Indian math/calculus texts imported direct from India (not via Arabs)
• were circulating in Europe since Clavius and Tycho).

• Note: Like calculus, West also stole probability, and sampling theory
• along with theory of permutations and combinations from India.

– See my article “Probability in Ancient India”, Handbook on Philosophy of Statistics (2011) 1175–96. Elsevier.

• My 1999/2001 Hawai’i paper gave proof beyond reasonable doubt on criminal law (see extract)
• there was opportunity, motivation, circumstantial, and also documentary evidence

• Roman Catholic missionaries in Kochi (Kerala) since 1500
• set up a mission school + college to try to convert local Syrian Christians
• Using their help to locate and translate local texts in Toledo mode
• and sending the translations back to Rome.

• Europeans had a major navigational problem
• e.g. British law of 1711 constituting board of longitude.
• Solution of longitude problem needed precise trigonometric values (hence Clavius' 1607 trigonometric tables).

• Epistemic test: BUT, though Europeans (Clavius etc) stole from India and lied they were not smart enough
• to USE these precise trigo values to calculate the radius of the earth!
• A value found in every Indian math text.(Next talk on rajju gaṇita).

• Hence, Europeans could not use Indian techniques of celestial navigation for longitude
• Brahmagupta: "भूव्यासस्य अज्ञानाद् व्यर्थं देशान्तरं" (BSS, 11,15-16)
• "ignorance of earth's radius makes longitude [calculations] futile"

• There is also documentary evidence, though signed confession not expected from thieves.
• Matteo Ricci, Clavius pet student and biographer
• wrote from Kochi in 1580 that he was collecting knowledge of Indian methods of time keeping
• from “an intelligent Brahmin or an honest Moor”

• by opportunity, motivation, circumstantial and even documentary evidence
• that calculus was stolen from India
• and dishonestly attributed by Western historians to Newton etc.

• Clear case of ACTIVE dishonesty because Whish article of 1832 suppressed for two centuries
• like my assertion of calculus theft suppressed for 25 years
• and my courses misrepresented and maligned.
• (But West has lost its iron grip on information in last 20 years.)

• Central University of Tibetan Studies (2009)
• Universiti Sains Malaysia (4 groups, 2010)
• CISSC Tehran (2012)
• Ambedkar Univ. Delhi, 2012 (social science post grads)
• SGT Univ Delhi (2017) (Science/Engineering undergrads)

• Claim that calculus was “discovered” by Newton etc. is bogus.
• Europeans didn't understand Indian calculus. just as they failed to understand Indian primary-school arithmetic for centuries.
• That in 19th-20th c. they hence invented the metaphysics of unreal real numbers
• which should be rejected since it adds difficulty but no practical value.

• That axiomatic math, like racism, is based on religious (church) superstitions
• which allow West to control content of math
• by controlling axioms and deciding validity of theorems.
• That limits and real numbers are not needed.

• Calculus about numerically solving differential equations,
• not about formulae for integration and differentiation of elementary function.
• Non-elementary functions arise in the simplest applications such as the simple pendulum.

• Derivative as limit not needed finite differences enough for solution.
• Integral not needed, solution of differential equations enough.
• Seroism and Brahmagupta arithmetic better substitute (even for discontinuities)

• Sample tutorial sheet
• Sample results (UG-pure-math, UG-appl-math, UG-non-math, PG-math)
• Sample software my Calcode (demo)
• Decolonization reading list: https://tinyurl.com/decol-list-new

• it is about prohibiting empirical in proof suited church (since facts fatal to church dogmas)

• This method of proof was read into the Euclid book (which arrived during the Crusades)

• though “Euclid” book has not a single axiomatic proof
• (if you think it has, accept my prize, show me one, don't tell me stories why it is missing).

• Apart from myths about axiomatic proofs
• there is also the superstition that prohibiting the empirical
• makes proof infallible.

• Church superstition that prohibition of empirical makes proof infallible.
• if true, prohibit empirical in science too to make it infallible?
• Fallacy: As math teachers do you give 100% marks to all proofs given by your students?
• No! You discriminate between valid and invalid deductive proof.

• Is Russell’s 378 page proof of 1+1=2 valid?
• Not even a typo in those 378 pages? Did you check? Or go by trust?
• During my PhD, I had a 110 page derivation.
• Re-derived repeatedly. Each check took a month.

• So, (a) deductive proofs ARE fallible.
• (b) Validity of a deductive proof can only be decided inductively by repeat checks
• or by authority.
• Therefore deductive proofs MORE fallible than inductive proof or proof by authority.
• Fallacy from Hume a racist who copied al Ghazālī without understanding.

• As in Russell’s silly proof of 1+1=2
• but no practical value
• which comes from calculation
• (or computational math in which the associative law for addition fails for floats)

• silly church SUPERSTITION of Crusading theology of reason
• that 2-valued is universal,
• since it binds even God.
• “God cannot create an illogical world but can create the facts of his choice"

• Indian Nyāya school accepted 2-valued logic
• (from which Aristotelian syllogism derives)
• but Buddhist logic of catuṣkoṭi,
• Jain logic of syādvāda,

• and quantum logic (e.g. AND not distributive over OR)
• are all quasi-truth-functional