In defense of decolonisation

C. K. Raju

Indian Institute of Education
G. D. Parikh Centre, J. P. Naik Bhavan
University of Mumbai, Kalina Campus
Santacruz (E), Mumbai 400 098

Summary: False history+ bad philosophy of math

  • This boast was used to trick the colonised into accepting colonial education
  • for “superior” Western math and science.
  • Church education used for childhood INDOCTRINATION of the colonised
  • e.g. “Pythagorean theorem” term repeated 32 times in current Indian class X text

Evidence for Pythagoras etc?

Colonialism also spread a BAD philosophy of axiomatic math

FORMAL (axiomatic) math INFERIOR

  • 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.

To repeat: Decolonisaton challenges both

  • 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)

Negating false history: Indian calculus

  • 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.

The difficulty

  • 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.

Decolonisation prescription simplified

More on the censorship

Other response is to misrepresent and malign

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

Introduction

When I first visited Australia (1988),

  • 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.

Indian calculus

  • 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.

Indians not only did calculus FIRST, but they did it DIFFERENT

  • 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”.

Polynomial ordering

  • 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
  • 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

Calculus theft: Quick background summary

  • 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.

Is calculus THEFT relevant to math and math education? Yes! Why?

  • 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.

Lack of Western understanding of calculus

  • 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.

However, the racist dog

  • 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.

Normative universality

  • 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?)

Personal examples

  • A couple of personal examples
  • on how false history continues to be concocted
  • and preserved.
  • Show why West will never willingly accept decolonisationm agenda.

The dishonorable knight

  • 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".

On AMS ethics

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

Similar serial plagiarism + brazen dishonesty with my 3rd book

The thesis that the calculus was stolen

  • 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 institutional dishonesty which is still alive

  • 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?

Similar events in South Africa

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

So, nothing shocking that Newton stole calculus

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

Graeco-Roman calculus: Arithmetic revolution

  • Contrary to triple miracle of Archimedes + 1200 year Dark Age + verbatim reappearance fantasy.
  • Fact 1: From Athenian inscriptions thru Darius vase to Herodian, Greeks used only Attic numerals with no fractions.
  • Fact 2: Graeco-Roman arithmetic with no general fractions CONTINUED till 10th c.
  • Took smartest of Europeans over 1000 years to understand simple fact
  • that Graeco-Roman pebble arithmetic is primitive/INEFFICIENT
  • compared to sophisticated place-value of Indian gaṇita.
  • Small number 1888 needs 13 figures MDCCLXXVIII not 4
  • Large numbers need place value.
  • Then Gerbert made a great innovation of apices
  • learning the system of "nine Indian figures" from Spain
  • Wrote them on the back of counters/jetons
  • i.e., using a counter with 9 at the back instead of using 9 counters!

Failed to grasp efficiency of "algorithms"

  • 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

Fibonacci

  • 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

Fibonacci's blunder

  • 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.

Persistent confusion about subtraction and negative numbers

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

Interin conclusion

  • 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.

Phylogeny is ontogeny

  • 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

Āryabhaṭa

  • 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.

“Euler” method

  • 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.

A series of 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
  • 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).

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

Proof beyond reasonable doubt

  • 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

Opportunity: Jesuits in Cochin

  • 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.

Motivation: European navigational problem

  • 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"

Documentary evidence

Proved beyond reasonable doubt

  • 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.)

Decolonised course on calculus without limits

  • But what difference does that theft of calculus make?
  • First a question:
  • Q. When Europeans made such persistent asses of themselves with elementary arithmetic (zero, subtraction, negative numbers etc.)
  • how could they have understood calculus?
  • A. They didn't; as clarified at the beginning of this talk.
  • Europeans acknowledged difficulties in understanding calculus until Dedekind and axiomatic reals.
  • But Dedekind reals only reflect Western social consensus.
  • Many issues on which the West had a social consensus for centuries
  • such as “moral” genocide,“moral” slavery, “moral” racism, and “moral” colonialism.
  • We won't accept such social consensus which we regard as the height of evil
  • in fact, aim of decolonisation is to kick out such remnants of colonialism.
  • We, the former colonised, will accept it only if it has practical value:
  • but formal reals have none.
  • When I helped to build the first Indian supercomputer
  • one of my tasks was to implement calculation related to rocket trajectories etc. on it.

My course on calculus without limits

  • based on Āryabhaṭa's method of numerical solution of differential equations
  • Can't teach calculus just with theory of infinite series
  • most applications involve solutions of differential equations.
  • Āryabhaṭa used linear interpolation/extrapolation (elementary “rule of 3”)
  • Later 11th/12th order interpolation/extrapolation used by Āryabhaṭa school of Kerala.

Course tried in

Course contents: history and philosophy of calculus

  • 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.

Clarifications

  • 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 pre-test question papers (UG-Engg, PG-math)

Normal math vs Formal math

  • Normal math(e.g., Indian gaṇita) accepts empirical proofs + deductive inference
  • as does science
  • as explicitly stated in Nyaya sutra 2 and elaboration
  • i.e., 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

  • (e.g.,modus ponens 1. \(A \Rightarrow B\), 2. \(A\), ∴ 3. \(B\).)

So the difference NOT about use of deductive reasoning as trumpeted

  • 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.

Fallibility of deduction

  • 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.

How do you know a given proof is valid?

Deduction MORE fallible than induction

  • 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.

Axiomatics adds huge difficulty

Even logic is not certain:

  • 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"

Logic NOT culturally universal

Logic NOT empirically certain