गणित vs formal math

Re-examining the philosophy of mathematics, its pedagogy, and the implications for science

C. K. Raju

Indian Institute of Advanced Study

Rashtrapati Nivas, Shimla

An anecdote

  • Was selected for IIT JEE, did not join.
  • Joined IIT: Delhi for math PhD,
  • attended one class, asked one question, which went unanswered
  • and left (for ISI), since teacher admitted he knew no math.

Hence advised my children to NOT appear for IIT:JEE

  • My elder son represented India for the Physics Olympiad (Padova 1999) and got a special prize equivalent to a gold medal
  • Though the method he used has a Latin name–Regula Falsi–it is a ganita method which I taught him.
  • All other Olympiad team members were IIT:JEE toppers (top 6).
  • But he joined St Stephen's and was very unhappy that his fellow physics students were all IIT rejects.

Anecdote (contd)

  • As an Olympic gold medallist, after graduation
  • he got admission with full scholarship in all top US Universities: Harvard, MIT, Caltech
  • I was reluctant (thought he was too young) but he said
  • "No one in DU knows any physics", all they understand is where you did your PhD from.
  • I well knew he was right, and had no answer.

Anecdote (concluded)

  • Two centuries ago, colonial education came supposedly for science
  • but, even today, almost nobody, even in our top universities, knows any math or science,
  • all they understand are certificates of Western approval,
  • the story (that colonial education came for science) differs from facts: any child knows that.

Story about colonial education diverges wildly from facts

  • Why? What really happened?
  • This is a key problem addressed in my book.

Twitter summary of book

  • 1. Ganita (गणित) differs from (formal) math,
  • 2. it makes math easy, and
  • 3. makes science better.
  • 4. This is an obituary of formal math.

Slogan formulation

  • Formal math is dead,
  • long live normal math (गणित)

Book has four parts

  • Part 1: Introduction
  • Part 2: The false history and superstitions of Western (formal) mathematics
  • Part 3: The alternative (pedagogy)
  • Part 4: The alternative (science)

Part 1 has two chapters

  • Chp. 1: Colonial education as church education, and the propaganda of civilizational superiority
  • Chp. 2: Ganita versus formal mathematics: an outline

Chp. 1. Colonial education (and dogma of civilizational superiority)

Wrong to blame Macaulay (it was church education)

  • Colonial education went to ALL colonies whether French, Portuguese, or Dutch….
  • in Macaulay's time Western education including higher education was a 100% church monopoly.
  • No historian or educationist has put this simple fact on the table:
  • colonial education was church education which came to all colonies on the strength of church propaganda.

Church education provided some practical value,

  • but also injected some poison.

Macaulay v2.0 makes that poison clear

  • He said, Britain should educate its poor (for free) to stop the threat of revolt.
  • Church education was designed to create missionaries,
  • it makes the educated very submissive, and receptive to propaganda
  • hence prevents revolt.

Dogma of "Civilizational superiority" like racism

  • a mutation from earlier dogmas of racist and religious superiority.
  • All three evil dogmas organically linked,
  • since based on the same false (church) history of science (used also by Macaulay).

False (Christian chauvinist) history a traditional church weapon

  • ever since 4th c. church married state and sought power through superstitions and lies.
  • The church started writing false history (Eusebius and Orosius) since 4th c.
  • But false history of science went ballistic during the Crusades (12th c. to 15th c.)
  • (No other historian in India EVER talked of this "history of history". No guts? No knowledge?)

False history-1: "Greeks"

  • Knowledge in captured/imported Arabic texts was APPROPRIATED
  • by indiscriminately attributing it to early Greeks, real or imaginary
  • Greeks initially declared as "friends of Christians" (religious superiority), later as Whites (racist superiority),
  • then as West (civilizational superiority )

False history-2: "Doctrine of Christian discovery"

  • Subsequently (during and after "renaissance") - indigenous scientific knowledge across the world was APPROPRIATED
  • by attributing it to Christian "discoverers".
  • (see video "Discovery of India")

Is science Western?

An example

  • Newton and Leibniz "discovered" calculus
  • just as Vasco da Gama "discovered" India. 🤣
  • Prior occupancy of land by millions, or prior knowledge by non-Christians makes no legal difference to "discovery"
  • (US Supreme Court judgment, currently valid; we too accept this rotten British law.)

False history is written by would-be victors

  • because liesof history have power
  • (of the "non-violent" sort)
  • greatly needed by militarily weak Crusading church
  • AND colonialisers who feared revolt.

Summary of chp. 1

  • Church education was exported to all colonies
  • claiming it was needed for science
  • using the dogma of civilizational/racist/religious superiority.
  • "Secular" justification for the dogma was a false history of science erected by the church.

Misled and enslaved

  • This church "education" misled and mentally enslaved the colonized,
  • through stories of science
  • without teaching them science.

Chp. 2: Ganita vs formal math an outline

  • History
  • Philosophy

Chp 2, part 1: History

  • Most present-day school/practical math
  • arithmetic, algebra, trigonometry, calculus, probability, and statistics
  • was imported by Europeans from India between 10th and 17th c.
  • for its practical value (commerce, navigation, gambling etc.).

Due to cultural differences in math

  • Europeans made hilarious blunders (🤣) for centuries about imported Indian ganita. E.g.
  • Pope's abacus. 976 CE abacus for "Arabic numerals"/"algorithms" based on the foolish notion that arithmetic requires an abacus.
  • Zero. Florentine law against zero (1300): "write numbers also in words".
  • Fluxions. Newton's "fluxions" etc. (17th-19th c.)

Chp. 2, part 2: Philosophy

  • Mathematics varies with culture, hence ganita \(\neq\) math.
  • Key cultural tension between ganita and (Western, formal) math is this
  • Ganita was always practical and secular
  • Western math was always religious (since Plato, and Crusades).

Example of 1+1=2

So let us face it: colonial education has made you ALL mathematically illiterate

  • Those loyal to the colonial master
  • believe the story that colonial education came for science (for our benefit)!
  • My aim NOT to humiliate anyone, but to wake you up to some unpleasant facts:
  • if you didn't learn why 1+1=2, what science did you learn? 😧

Superstition is being firmly convinced though totally ignorant.

So, why is 1+1=2 so difficult in formal math?

  • Is there any practical or scientific advantage from this difficulty?
  • Did you ever have any difficulty in your daily life from NOT knowing the "axiomatic proof" of 1+1=2?
  • If it has no practical value why do we teach formal math?
  • Because colonial education is church education which teaches church math. (What is church math?)

Church math

Church theology of reason

  • Militarily weak Crusading church erected Christian theology of reason (of Aquinas)
  • to compete with the Islamic theology of reason (aql-i-kalam), to persuade Muslims to convert
  • since it failed to convert them by force, and Bible did not work with Muslims.

Facts are contrary to church dogmas

  • Since facts are very embarrassing to church dogmas.
  • E.g. what facts about virgin birth? About heaven and hell? God?
  • Hence, church rejected empirical proof
  • It invented metaphysical reasoning = reasoning MINUS facts = axiomatic reasoning.

Axioms are more convenient than facts

  • Church accepted axiomatic reasoning, since anything can be assumed as an axiom.
  • E.g. no facts about angels, but Aquinas axiomatically assumed they occupy no space.

Church intervention in math

  • As its new "holy book", the church adopted an Arabic text which came to Europe ca. 1125 during the Crusades
  • this was attributed to an unknown "Greek" "Euclid".
  • It was claimed that this text used axiomatic reason in geometry
  • using axioms and metaphysical reason instead of facts, exactly as the church wanted.

This "Euclid" text

  • or rather its church "interpretation"
  • became the norm for Western mathematics.
  • Western/formal math became metaphysics
  • as Russell also explained in "Mathematics and the Metaphysicians".

Back to history

  • Recall that Europe imported most practical math from India.
  • To reconcile the practical value of imported Indian ganita
  • with their native religious (church) understanding of math.
  • Eventually, Europeans added metaphysics (of nil practical value) to imported Indian ganita.

The package for colonial math education

  • Together with a false history (e.g. "Newton discovered calculus"),
  • this package (false history + bad philosophy) was declared "superior" and returned through colonial education
  • and is still being taught today in our schools and universities.
  • Our historians won't contest the false history, nor our philosphers the bad philosophy.

We need to understand and reject both

  • the false history, by examining facts
  • and the bad (church) philosophy underlying Western (church) math
  • and its false claims of superiority.
  • Great shame that we have not done either in two centuries, and refuse to do so today.🤢

Propaganda to maintain the package (false history + bad philosophy)

Propagandist misrepresentations of ganita-1

Ganita accepts empirical proof

  • Indians (all schools of thought) accepted empirical proofs (प्रत्यक्ष प्रमाण), as does science.
  • Hence, empirical proof accepted also in ganita.
  • Fallacious to imagine that acceptance of empirical means rejection of reason
  • as fallacious as asserting that Kosambi did mathematics and therefore could not do history.


  • "Indian ganita lacked proof".
  • Nonsense.
  • E.g. Indian proof of Pythagorean proposition.
  • but this proof involves empirical proof: we move the triangles, we superpose to see that the two areas are equal.

Doublespeak about "reason"

  • People think "reason" = normal reason(= reason PLUS facts)
  • when the formal mathematician means reason= church reason (= reason MINUS facts) .

Theorems NOT valid knowledge

  • Mathematical theorems are NOT valid knowledge if only axiomatically proved
  • as Aquinas' "angel theorem" demonstrated.
  • ANY nonsense proposition WHATSOEVER can be proved axiomatically (just make it, or something equivalent, an axiom).
  • E.g. Pythagorean theorem is NOT valid knowledge on the curved surface of the earth (or anywhere the cosmos—space is curved).

No use saying it is "approximate" knowledge

  • "Approximate knowledge" without an error estimate is worthless.
  • Like telling a drowning sailor he is "approximately" near land
  • where "approximate" might mean 3O m (life)
  • or 300 km (death).

Indian ganita preferred inexact calculations

  • and it certainly understood these imitations
  • E.g. Bhaskara I (7th c.) Mahabhaskariya II.5 states that calculating distance using the "Pythagorean" theorem is gross
  • (according to the disciples of the bhata)
  • on account of the curvature of the earth (also Laghubhaskariya, 1.27)

European Navigational disasters due to ignorance

  • Backward Westerners did not know how to measure the radius of the earth (until the 17th c.)
  • hence kept having navigational disasters until the 18th c.)
  • for they had no error estimate with its use.

The two "Pythagorean" calculations known to Indian ganita were

  • to calculate the diagonal of a rectangle from a knowledge of its sides
  • to calculate the sides from a knowledge of the diagonal and the angle it makes one of the sides ("trigonometry")

The Manava sulba sutra (10.10) states the first

  • this involves calculation of the square ROOT known to Egyptians, Iraqis, and Indians
  • but unknown to the West until the 12th c. bad karna ->? bad ear = deaf = surdus = surd
  • Square roots usually can be precisely but NOT exactly calculated
  • hence sulba sutra name for \(\sqrt 2\) is सविशेष (with an अवशेष)(Baudhayana 2.12)

Arayabhata in his table of precise sine values (Gitika-10)

  • uses only one Sanskrit word कलार्धज्या
  • (sine values precise to the first sexagesimal minute)
  • jya -> jiva -> jiba -> jaib -> (OED) sinus
  • (Sine values too cannot usually be exactly calculated.)

Ganita preferable to formal math

  • since inexact calculation
  • = inexact knowledge in the real world
  • preferable to theorem which pretends to be exact
  • = exact "knowledge" in an unreal world.

Part 2: The false history and superstitions of Western (formal) mathematics

  • Chp. 3: False history of math-1: Pythagoras, Euclid, and geometry
  • Chp. 4: False history of math-2: Newton, Leibniz, and the calculus
  • Chp. 5: Superstitions of formal math-1: Politics of reason and the fallibility of deduction
  • Chp. 6: Superstitions of formal math-2: Politics of eternity and the metaphysics of infinity

Chp. 3: False history of math-1: Pythagoras, Euclid, and geometry

  • Dogma of Western civilizational superiority in mathematics
  • is anchored on the myth of Pythagoras and his theorem.
  • To indicate the linkage to the dogma of racist superiority
  • our class IX school text in math shows images of white-skinned males as Pythagoras and Euclid.

To break the dogma (of civilizational/racist superiority)

  • it is necessary to smash these two myths of Pythagoras and Euclid

Nil primary evidence for Pythagoras or the color of his skin

  • or that he proved any theorem
  • or how he proved it.
  • Plenty of counter evidence that Pythagoreans were disinterested in proving metaphysical theorems
  • and interested in Egyptian mystery geometry only for its connection to the soul as elucidated by Plato (Meno).

Western "historians" are myth-jumpers

  • when the myth of Pythagoras is exposed
  • they jump to the myth of Euclid.

But nil primary evidence for Euclid

  • (My "Euclid" challenge prize of Rs 2 lakhs still standing after ten years.)
  • Nil evidence that he was the author of the book attributed to him.
  • Nil evidence that the book was written anytime near the date attributed to him.
  • Nil evidence that the author was a white male.

Again plenty of counter evidence

  • The book was written by a black woman from Africa.
  • It was written in the "5th c. (800 years after the supposed date of Euclid),
  • as a book on Egyptian mystery geometry which so enraged the church,
  • that the author was lynched and raped in a church.

The myth of the book

  • Myth jumping Western historians now jump to the myth of the book
  • they say (as was said in the Indology round-table last March) "the author does not matter, the book is there".
  • Yes, but these Greediots never read the book (just assumed church myths about it must be true).
  • As did all Western scholars under church hegemony for 750 years
  • as shown by Cambridge University foolish exam regulations (1888) about Euclid

No axiomatic proofs in "Euclid"

  • The myth ABOUT the book is COMPLETELY FALSE.
  • Actually, the "Euclid" book does NOT have a SINGLE axiomatic proof in it.
  • Moreover, this is publicly known for over a century.

Dedekind, Russell and Hilbert

But false church myths of civilizational superiority

  • have so deeply penetrated the psyche of Western historians
  • that the best of them—Gillings, Needham, Clagett—
  • Kept reciting the myth of "superior" axiomatic proofs in "Euclid" even a century after its public exposure 😜
  • Anyway, the "Pythagorean theorem" is inferior knowledge, as already pointed out.

Chp. 5: Superstitions of formal math-1: Politics of reason and the fallibility of deduction

  • Dedekind, Russell, Hilbert accepted the absence of axiomatic proofs in "Euclid"
  • but founded formal math on the belief that axiomatic proofs are "superior" 😜
  • So what exactly is "superior" about axiomatic proofs?

Such proof served church purposes,

  • hence the church glorified them and
  • hegemonised Europeans and colonised minds believed it.

Philosophical claim: deductive proofs are infallible

  • This is pure superstitious nonsense, but the basis of formal math.

Any math teacher knows

  • students frequently make errors in proofs and hence flunk in math tests.
  • Authorities also make errors, as in wrong claims of proof of Riemann hypothesis.

Dodging the inevitable

  • No use saying "a valid deductive proof is infallible".
  • That is an irrefutable tautology also true of a valid empirical proof.
  • Whole problem is to determine validity.

Given that errors in deductive proof are possible

  • its validity can only be decided by
  • repeated rechecking (induction)
  • or by trusting authority (such as infallible pope! 😀)
  • When authorities differ, go by their social reputation (😜 some infallibility!)
  • In either case, deduction WEAKER than induction or empirical proof.

Deduction MORE frequently fallible (than empirical proof)

  • because human mind more easily deceived than the human senses.
  • A complex task of deduction almost invariably involves error.
  • E.g. game of chess is pure deduction; error-free game must end in a draw.
  • However, EVERY human being almost always errs, hence loses to a machine.

Logic not culturally unique

  • Church learnt about logic from Arabic texts (Ibn Rushd, AND al Ghazali)
  • hence wrongly thought that logic is unique and binds God.
  • Hence, 2-valued logic used in mathematical proof today.
  • However, [[http://ckraju.net/papers/Nonwestern-logic.pdf][Indian culture knows of logics which are not even truth-functional (e.g. Buddhist catuskoti, Jain syadvada).

Logic not empirically unique

  • Physics at the micro physical level modeled by quantum mechanics.
  • Quantum logic too not even truth-functional.
  • Anyway, empirically logic not certain.
  • If logic itself decided empirically, deductive proof is decidedly weaker than empirical proof.


  • Purported infallibility of deductive proof a mere church superstition.
  • Metaphysical proofs more convenient to the church
  • though decidedly inferior for practical applications.


Calculus, Newton and gravitation (Chps. 4 and 11)

Brahamagupta (7th c.) who criticised Aryabhata

  • suggested quadratic interpolation and a return to earlier 6 sine values.
  • Vateshvar used quadratic extrapolation ("Stirling's formula")
  • and 96 values to get accuracy to the second (sexagesimal minute)

Brahmagupta's polynomial arithmetic (अव्यक्त गणित)

Aryabhata school in Kerala

Precise sine and cosine values ("tables of secants") were needed for navigation

  • to determine latitude, longitude, and loxodromes, and size of the earth
  • then (15th-18th c.) the major scientific challenge before Europe.
  • Hence Cochin-based Jesuits stole them in the 16th century (with the help of local Syrian Christians)
  • They translated them and sent them to Europe.

On the "doctrine of Christian discovery"

  • all these mathematical results were shamelessly attributed to Christians.
  • Precise trigonometric values to Clavius.
  • Nilakantha'a astronomical model to Tycho Brahe.
  • Madhava sine series to Newton, \(\pi\) series to Leibniz.

Doctrine of Christian discovery (contd)

The epistemic test exposes these intellectual thiefs

  • Those who steal knowledge,
  • like students who cheat in an exam and copy,
  • do NOT fully understand what they falsely claim as theirs

Newton too failed to fully understand the calculus

  • He thought of the derivative as a "fluxion"
  • on the foolish idea that time "flows",
  • an idea destroyed by the 8th c. Sriharsa (but "discovered" by McTaggart only in the 20th c.)
  • "Fluxions" left Europeans puzzled from Descartes (17th c.) to Bishop Berkeley (18th c.) to Karl Marx (19th c.)

Newton's physics HENCE failed

Newton's laws HENCE failed

  • and were replaced by special relativity
  • (Poincaré 1904, NOT Einstein 1905).
  • However, Newton's laws of motion, come as package deal with his law of gravitation.
  • Therefore, Newtonian gravitation also needs to be modified.

General theory of relativity (GRT) provided one modification

  • Newtonian gravitation worked very well within the solar system
  • because it was back calculated from ancient (Indian) observations of planetary movement
  • "discovered" by Tycho Brahe and his assistant (the nearly blind) Kepler.
  • But Newtonian gravitation prima facie fails for the galaxy.

Possible to save the theory by accumulating hypotheses

  • Assume there is dark matter in the Galaxy
  • But this dark matter must be peculiarly distributed
  • with its density reaching a peak where the luminous matter thins out to zero.
  • It must be exotic matter (non-baryonic) etc.

GRT differs very little from Newtonian gravitation for the galaxy

  • and is too complex so the many body problem could not be solved in a century.

My retarded gravitation theory (RGT) proposed in this context

  • as a minimal modification of Newtonian gravitation
  • to make it consistent with special relativity.
  • RGT much simpler than GRT: many-body problem is solvable.
  • Question about the exact form of the RGT force.
  • (This question has taken most of my time!)

Possible/observed departures from Newtonian gravitation

  • (1) Perihelion advance of Mercury (\(\frac{u^2}{c^2}\))
  • (2) Galactic rotation curves (inexplicable on GRT except by accumulating hypotheses)
  • (3) Flyby anomaly (\(\frac{v}{c}\), too large for GRT, \(v\) rotation velocity of earth)
  • (4) Oumuamua (\(\frac{u}{c}\), too large for GRT)

RGT 1.0 focused on 3 (first presentation)

RGT 2.0 focuses on 1 and 2

  • Perihelion advance of Mercury
  • and galactic rotation curves
  • involves velocity-dependent gravitational effects for latter which are TOO LARGE for GRT.



Course on "Calculus without limits" conducted with

  • 10 groups in
  • 6 universities in
  • 3 countries

It uses

  • Aryabhata's numerical methods (calculus as numerical solution of differential equations)
  • Brahmagupta's non-Archimedean अव्यक्त गणित (polynomial arithmetic), hence no limits in
  • Zeroism (शून्यवाद) or dealing with inexactitude and non-uniqueness.

Course on calculus without limits has been tried out

Course on calculus without limits (contd)

Advantages of calculus without limits course

  • Conceptual clarity: no unreal "real" numbers, limits etc. which few understand
  • Teaches real life applications (e.g. ballistics with air resistance)
  • Teaches non-elementary integrals omitted from calculus courses (e.g. correct theory of simple pendulum using elliptic functions).
  • See e.g. tutorial sheet
  • Teaches that it is silly to memorise formulae by using MAXIMA (earlier MACYSMA).


Course on empirical string geometry (of शुल्ब सूत्र) for schools

  • as a replacement for nonsense of Euclid
  • and related dogma of civilizational superiority

Geometry currently taught by confounding 5 distinct and mutually contradictory geometries as one

  • 1. Religious geometry (Egyptian mystery geometry, Plato, beauty etc.)
  • 2. Church interpretation of "Euclid" as metaphysics (invisible geometric points etc.)
  • 3. Hilbert's synthetic geometry (axiomatic proof)
    • length measurement disallowed,
    • "congruence" instead of equality

Confounding distinct geometries

  • 4. Birkhoff's axiomatic metric geometry (Yale SMSG recommendation after Sputnik crisis)
  • 5. Empirical compass-box geometry
    • ritualist instruments such as set squares
    • no instrument to measure legth of curved lines

String superior

Rajju Ganita tried out with 4 groups

Draft text book for class 9 ready

  • Egyptians/Africans also used string geometry (harpedonaptae)
  • African today perhaps more willing to oppose
  • dogma of civilizational superiority
  • which mutated from dogma of racist superiority.