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

C. K. Raju

*Indian Institute of Advanced Study*

*Rashtrapati Nivas, Shimla*

- 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.

- 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.

- 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.

- 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.

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

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

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

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

- Macaulay v1.0 typically blamed for colonial education in India
- But he said Indians need Western education for science
- because the West is civilizationally superior in science.

- 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.

- but also injected some poison.

- 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.

- 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).

- 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?)

- 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 )

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

- E.g. technique of vaccination from India appropriated and attributed to the West.
- Net result was the current stock history of science: all science due to Christians/whites/West.

- 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.)

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

- 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.

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

- History
- Philosophy

- 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.).

- 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.)

- 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).

- 1+1=2 in Ganita or normal math
- 1+1=2 in formal mathematics (cardinals)
- 1+1=2 in real numbers (1 as "real" number \(\neq\) 1 as natural number in formal math).
- This is my Cape Town challenge with prize of Rs 1 million to JNU faculty to prove 1+1=2 in real numbers.

- 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? 😧

- 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?)

- Have earlier told the story of how the Crusading church changed the philosophy of math - using the "Euclid" text
- (this book now coming out in Spanish translation).
- But this story has not been understood.

- 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.

- 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.

- 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.

- 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.

- 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".

- 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.

- 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.

- 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.🤢

- "Only Greeks used reasoning in math" (pope Benedict and Indian class IX text)
- Nonsense. E.g. Indians INFERRED the earth is round (e.g. Gola 6)
- from the observation that far off trees cannot be seen (Lalla, शिष्यधीवृद्धिद, 20.36).
- Calculated size of the earth from distance to the horizon, as did my students for my HPS course.

- 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.

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

- 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).

- "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).

- 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)

- 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.

- 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")

- 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)

- 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.)

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

- 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

- 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.

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

- 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).

- when the myth of Pythagoras is exposed
- they jump to the myth of 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.

- 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.

- 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

- 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 clarified that even the first proposition lacks an axiomatic proof
- Russell said "Euclid's" proofs are "a tissue of nonsense".
- Hilbert wrote a book on
*Foundations of Geometry*to supply the axiomatic proofs missing in the book - in force-fitting the book to the myth he did great violence to the original (e.g. invented synthetic geometry).

- 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.

- 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?

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

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

- 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.

- 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.

- 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.

- 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.

- 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).

- 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 originated in India with 5th c. Aryabhata (a dalit from Patna)
- who calculated 24 sine values precise to first sexagemisal minute (5 decimal places)
- by numerically solving differential equations (using "Euler's method")
- using elementary rule of 3 for (linear) extrapolation.
- (See Cultural Foundations of Mathematics)

- 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)

- was non-Archimedean
- used by Nilanatha (15-16th c.) to sum infinite geometric series
- together with zeroism (philosophy of inexactitude)

- used 11th-12th order polynomials interpolation/extrapolation
- to derive 24 sine values accurate to the third sexagesimal minute (or about 9 decimal places)
- and derive infinite series such as the "Leibniz series" for \(\pi\).

- 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.

- 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.

- "Euler" method, "Stirling's" formula, Fermat's challenge problem etc.
- On the same dogma even my work on theft of calculus was itself stolen
- serially plagiarized by Christians
- who are still being given credit by our numerous Mir Jafars

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

- 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 "laws" of motion require a clear definition of equal intervals of time
- even to be MEANINGFUL.
- Newton gave no such definition (effectively said God knew about the "even flow" of time)
- and thus retracted an earlier physical definition offered by his mentor Isaac Barrow

- 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.

- 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.

- 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.

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

- 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!)

- (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)

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

- 10 groups in
- 6 universities in
- 3 countries

- 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.

- First in 2009 in the Central University of Tibetan Studies, Sarnath
- Then in 2010 with 4 groups of students in the math department of Universiti Sains Malaysia. See report part 1, part 2. Or graphs.
- Then in 2012 in CISSC, Tehran, see poster and group photo,

- Then in Ambedkar University Delhi, see poster, and group photo
- And in 2017 in SGT University Delhi, poster and group photo.

- 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).

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

- 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

- 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

- can replace all instuments in ritualistic compass box
- is local and eco-friendly (no steel or plastic)
- better for practical purposes
- (such as determining area of a खेत with टेढ़ी मेढ़ी मेढ़)

- Nasik
- Chamrajnagar
- Gundulupete
- Indore (poster, group photo, media reports)

- 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.