Euclidean geometry
Rajju Ganita

C. K. Raju


Workshop about knowledge NOT exams

  • Those interested in exam-oriented results
  • This workshop is about knowledge
  • NOT to enable your children to do better in any exams.

A personal anecdote

  • I studied Euclid when I was in class VIII,
  • and I quite enjoyed it, especially the problems (riders).
  • One day I solved a problem in a particular way
  • but my teacher (our headmaster) said it was wrong.

I asked why

  • and was told it was wrong to use "superposition"
  • and that my proofs must rely only on the previous theorem.
  • I pointed out that the textbook itself used superposition.
  • That particular teacher was honest enough to admit I was right

He also admitted he could not explain

  • why it was right for the book to use superposition
  • but wrong for me to do so.
  • But strongly advised me to not use superposition to avoid losing marks in exams.
  • I hope by the end of the workshop you would have figured out this puzzling position.

Traditional Indian traditional knowledge system was anchored in

  • transparent public debate (शास्त्रार्थ)
  • as a test not only of knowledgeability of a person
  • but also of the validity of knowledge itself.
  • no possibility of doing that in the colonial system of examinations, degrees, and social reputation (institutional ranking).

Tried to initiate a public debate

Repeated attempts to have public debate

Why such a workshop in pandemic times?

  • My aim: to pass on my knowledge before I pass on.😀
  • Bigger goal: to change WHAT math we teach (in schools and universities)
  • not merely HOW we teach it.
  • (This goal may be for you to pursue.)

What is YOUR interest?

  • (1) Making math easy. You or your children found math difficult. You want an easier math.
  • (2) Indic knowledge systems. You are interested in learning about your cultural roots.
  • I suggest there should be a third, more important, reason: critical investigation of current colonial education.
  • If the system does not impart VALID knowledge your children suffer, for generations.

System of "exams"

  • only tests your knowledge about what you are taught.
  • Does NOT test whether what you are taught is valid knowledge.
  • (Academic system full of compromised "experts" and censorship.)
  • Note: Secrecy (SECRETIVE "peer" review) essential to current academic system of deciding valid knowledge. Why?

Preliminaries-2: Some clarifications

Today's workshop is about geometry education

  • Traditionally, in India, geometry was taught as rajju ganita.
  • "Euclidean" geometry came with colonial education.
  • (First Sanskrit translation of "Euclid" from Farsi in 1723, by Samrat Jagannath as रेखागणित, under Jesuit influence on Sawai Jai Singh.)
  • Which is better? No one (Westerner, Indian…) ever PUBLICLY compared the two as we will do today and tomorrow.

Clarification-1: Why compare?

  • Why not just talk about indigenous methods?
  • Because, our current school text (class IX, chp. 5) says Indian geometry was inferior.
  • If it is true it is best abandoned.
  • But if colonial geometry is inferior, be prepared to abandon that too.

Why compare? (continued)

  • On Indian tradition, before elaborating your own position
  • one must begin with the poorva paksha (पूर्व पक्ष) (opponent's position,
  • in my case the poorva paksha is given by current school math texts)

"Euclidean" geometry inferior: a scam

  • I will argue that "Euclidean" geometry is INFERIOR
  • that it is actually a church scam
  • (because colonial education came as church education).

Clarification 2: colonial education was church education

  • NOT specific to British or Macaulay.
  • Colonial education spread across the world, including to Portuguese, Spanish, French, Dutch colonies
  • where neither the British nor Macaulay ever reached.
  • Everywhere, it was the same European system of education.

One phenomenon needs one explanation.

  • Misleading (and intellectually sloppy) to offer 5 different explanations for single global phenomenon.

Church controlled ALL European education

  • Not only mission schools, but ALL the big Western universities
  • Paris, Oxford, Cambridge,… etc., were set up and set up during the Crusades,
  • and controlled by the church, until the 20th century.

Colonial education globalised church education

  • We hence teach "Euclidean" geometry in schools,
  • because this was part of the curriculum for church universities such as Cambridge.
  • [Later (after Sputnik crisis), US School Mathematics Study Group (1961) suggested changes - incorporated as complete hotchpotch in Indian texts since 1970s (to which we will return, LATER.)]

What is the objective of church education?

  • Colonial education is a "gift" from a foreign aggressor. Is it a Trojan horse?
  • Immediate point? Foolish to imagine thoughtlessly that the church set up universities for your benefit.
  • Obviously, the church set up universities for its benefit,
  • and brought this education to you also for its benefit (which you never understood).

Colonial education came as church education

  • Q. Why was the colonizer so keen to "educate" the colonised? (Across the world, not India alone)
  • A. To prevent revolts (by teaching people obedience to authority).
  • As explicitly stated by Macaulay for the case of revolts by the British poor (Speech to the British House of Commons, 18 April 1847.)
  • HENCE, colonial UNIVERSITY education came to India in a big way after revolt of 1857.(But we never understood this.)

Colonial policy of "fool and rule"

  • Colonial policy of "divide and rule" (divide by caste, religion, region well known.
  • But colonial policy of "fool and rule" (in association with the church) not so well known.
  • For "fool and rule" to work, the aggressor captured our education system,
  • and weaponised "education" as a tool to indoctrinate our children
  • and turn them into mental slaves (like well-trained and obedient dogs).

Colonial education due to state-church collusion

  • Obviously, also, there is a quid pro quo: you have to gain in some way.
  • Colonial/church education promised something concrete: a (white collar) job to tempt people.
  • What you lost is something intangible: valid knowledge (not merely traditional knowledge)

How does the church benefit?

  • "Education" helps church to indoctrinate and implant superstitions in young minds.
  • Because the church rules by superstition,

E.g. of superstition? The calendar

  • Your child learns in school to state his birthday on the Christian calendar.
  • This involves an implicit reference to the myth of Jesus, in saying AD (anno domini) and BC (before Christ).
  • Dates are stated often: after repeating this myth thousands of times you start believing that Jesus is a real historical figure,
  • a belief vitally important to the church.

Calendar (continued)

  • That belief is a superstition: What is the historical evidence for Jesus?
  • You were never told, but (because of indoctrination) now likely to be offended by any demand for evidence.
  • Use of this calendar damages our economic interests. Indian economy (still) based on monsoon-driven agriculture: but no rainy season on the Julian/Gregorian calendar,
  • unlike the months of सावन and भादों on the traditional calendar, as known to any child, e.g. through film songs.

Calendar (summary)

Myths of Greeks

  • Foolish to imagine that all church myths are about Jesus and Christianity.
  • most people fail to connect myths about "Greeks", to the church.
  • E.g. your class IX math text says "Greeks" did something superior in mathematics (chp. 5)
  • what all others did was inferior.

Stories of early "Greeks" are a camouflage for the church

  • Long ago (4th c.) Eusebius (first church historian) said the early Greeks were the sole "friends of Christians".
  • Later, during the Crusades, the church concocted a Greek origin of all scientific knowledge in Arabic books
  • which knowledge from across the world including India)
  • to claim that knowledge as a theologically correct Christian inheritance.
  • Hence the first church universities studied "superior" knowledge in "Greek" texts (from Arabic translated to Latin).

The school text likewise says "Greek" geometry was "superior"

  • and the geometry of all other people in the world was inferior,
  • But the school text provide no evidence of "Greeks" to back up these stories.
  • Lack of evidence (and church connection of Greeks) should make you suspicious.

No evidence provided even for actual existence of the "Greeks" such as "Euclid"

  • concocted during the Crusades.
  • NCERT has none.
  • Childhood indoctrination means you believe stories about Euclid
  • like your believe stories about Jesus (without evidence).

"Greeks" and church: What kind of reason?

  • Church too used "reason" in theology, as in its "rational theology" (12th c, onward)
  • (Pope Benedict boasted "We gave reason to the world".)
  • The purported "Greek" use of "reason" in geometry
  • is identical with the church use of reason in theology.
  • Will return to these questions later.

Section summary

  • Colonial education was church education, which continues till today.
  • Handing over education system to the church helps indoctrinate young minds and implant key superstitions.
  • We traded the minds of our children for the promise of a job.
  • Church indoctrination spreads myths (and related superstitions), not only regarding Jesus
  • but also about "the Greeks" regarded as close to the church.

Is colonial math any different?

  • Against that background let us come to
  • today's workshop which is about mathematics.
  • Key question about math: is colonial math different?

A doubt

  • A typical question (from Africa to Iran): is there a different way to do 1+1=2?
  • Does the church/colonial education do it differently?
  • Yes!
  • Have explained this MANY times earlier. This video was part of the reading/viewing list. (But let us go over it again.)


  • Most people think, it is about one orange and one orange making two oranges.
  • This is perfectly true, and that is how you were taught in KG.
  • This is NORMAL mathematics, which existed from ancient times.
  • But that is not how it is done in colonial mathematics taught today,
  • also called FORMAL mathematics or axiomatic mathematics.

Axiomatic/formal proof of 1+1=2

  • For example, Bertrand Russell needed 378 pages for axiomatic/formal proof of 1+1=2.
  • But, this proof applies only to 1, 2 as a "cardinals" or "natural numbers".
  • Often, you need so-called real numbers, explained in chapter 1 of the class IX NCERT text,
  • or used in Birkhoff's axioms for metric geometry (as recommended by SMSG, see reading list).

Axiomatic proof of 1+1=2 in "real" numbers

  • In colonial/formal math, 1 as a natural number is NOT the same as 1 as a "real" number (axioms are different).
  • Axioms for "real" numbers require axiomatic set theory (\(\neq\) Venn diagrams), (see my critique)
  • not studied even by most professional mathematicians.
  • (I tested two HOD's in math (one from an IIT), neither could even define a set correctly.)

But people think they understand

  • Therefore,in my talk, last year, in JNU,
  • I offered a prize of Rs 10 lakhs,
  • to the JNU faculty in math education
  • to formally prove 1+1=2 in real numbers, in 1 day
  • (I required a full proof from first principles.)

And a reduced prize of Rs 1 lakh

  • for the full formal/axiomatic proof, if submitted within one week.
  • Nobody claimed either prize.
  • Why are so many "top" intellectuals ignorant of such an elementary thing as 1+1=2?
  • Because colonial/church education promotes ignorance which is needed to maintain superstitions.

Most of you probably have a job

  • but you traded it for the intangible knowledge of why 1+1=2
  • and allowed your children to be indoctrinated into all kinds of myths (about "superior" "Greeks" etc.)
  • the authenticity of which myths no one among a billion Indians checked in over 200 years,
  • by asking for evidence (especially given the church connection of"Greeks")

Falsehoods, obscurities and alternative

  • Against this general background
  • Will first discuss the falsehoods in the class IX text
  • then the obscurities about geometry
  • and finally the alternative of Rajju ganita.

Section summary

  • Two kinds of math: Normal math (which accepts empirical proof) is different from formal math which rejects empirical proof and asks for axiomatic proof.
  • Almost all people are ignorant of formal/axiomatic math, even at the level of 1+1=2,
  • but our universities accept axiomatic math as "superior" (a superstition as we will see)
  • Widespread ignorance of formal math helps to preserve the superstitions about colonial/church superiority
  • which facilitated colonial rule.

The myth of "Euclid"

Lie No. 0: "Euclid" invented a special kind of geometry

  • Q. What is the evidence for Euclid?
  • A. (1999, Nepal Times) "Raju does not believe in evidence"
  • A. David Fowler (2002) (leading western expert on history of Greek math): NOTHING (no evidence for Euclid).
  • A. NCERT (Hukam Singh, Math head) (2007) Why do you need evidence? We go by a committee.

What is the evidence for "Euclid"?

  • (2011) Challenge prize of Rs 2 lakhs offered before Malaysia deputy education Minister.
  • NCERT (2019): "Western text books speak of Euclid, therefore Euclid exists."
  • (Free translation) "You are still a colonial slave: how dare you question the lies of the Western master and ask for evidence?"
  • But NCERT will NOT even say "No clear evidence for "Euclid".

Does it matter if "Euclid" existed?

  • (Common response since 1999) "There is the book Elements, how does it matter who wrote it?"
  • Problem: myth about the book bigger: myth does NOT agree with the actual book (which people don't read)
  • E.g. on the "Euclid" myth Euclid gave axiomatic proofs
  • but NO axiomatic proofs in the book (as publicly exposed in the 20th c.)
  • Yet the myth of superior "axiomatic proofs" by "Euclid" persists.

Does the "Euclid" myth matter? (another reason)

  • The myth says axiomatic proofs are "superior".
  • Formal mathematics based on axiomatic proof was invented ON this myth by Hilbert and Russell
  • while supplying the "superior" axiomatic proofs missing in Euclid.
  • But formal (axiomatic) reasoning used by the church since Crusading theologian Aquinas
  • as we will see, inferior, NOT superior.

Section summary

  • No evidence for "Euclid". NCERT has only evasive responses.
  • No axiomatic proofs in actual book, but myth says so
  • but such axiomatic proofs central to church theology of reason.
  • and accompanying claim that all other methods of proof are inferior.

Proof (Indian methods of proof)

Lie No. 1: "Indian mathematics had no proof"

While mathematics was central to many ancient civilisations like Mesopotamia, Egypt, China and India,

there is no clear evidence that they used proofs

(p. 287, Class IX, Appendix 1)

Indian methods of proof

  • Indians had a systematic theory of proof, which was used everywhere including mathematics.
  • They had this theory of proof long before "Greeks", whether imaginary (like the mythical 12th c. Aristotle concocted after the Toledo translations of the 12th century),
  • or the historical Aristotle of Stagira (-3rd c.).
  • Indian notions of proof well documented.

Proof (प्रमाण)in the Nyaya Sutra of Gotama

Note use of anumana to REASON that earth is round

  • But class IX school text (p. 79) lies:

In fact, Babylonians and Egyptians used geometry mostly for practical purposes and did very little to develop it as a systematic science. But in civilisations like Greece, the emphasis was on the reasoning behind why certain constructions work. [Emphasis original]

"Greeks" and reason (continued)

The Greeks were interested in establishing the truth of the statements they discovered using deductive reasoning (see Appendix 1)." (p. 79)

  • "Greeks" only an acceptable proxy for "church".
  • The text lies also because Indians did have a philosophical concept of proof, which let us understand better.

Disagreements about proof in Indian tradition

  • Different schools of thought disagreed about what constituted acceptable proof
  • Buddhists accepted only प्रत्यक्ष and अनुमान.
  • Carvaka/Lokayata ("people's philosophers") accepted only प्रत्यक्ष.
  • (Disagreement about proof shows, the Indian notion of proof existed from long before the beginnings of Buddhism, irrespective of the date assigned to the Nyaya sutra.)

Why did Carvaka reject anumana (inference)?

  • Because anuman/deductive reasoning does NOT lead to truth or valid knowledge.
  • Story of "seeing the wolf's footprints" ("वृकपदं पश्यं", from षटदर्शन समुच्चय 81/559–560, pp. 454–55. )
  • leads to the false conclusion that a wolf was around.

Doctrine of relative truth

  • Modern-day Western logicians do not deny the force of the Lokayat objection
  • but try to hide this by saying the non-existent wolf is actually "relative truth"!
  • My "rabbit theorem" illustrates the meaning of "relative truth": any nonsense can be "relative truth".

Rabbit theorem (weakness of axiomatic proof)

  • All animals have two horns (postulate)
  • a rabbit is an animal (postulate)
  • \(\therefore\) a rabbit has two horns (relative truth).
  • As we will see, ALL theorems of formal mathematics are such "relative truths".

Buddhists accepted only pratyaksh (manifest) and anumana (inference)

  • Rejected analogy (upamana) and testimony (shabda) as possibly misleading.
  • Rejection of analogy: story of the elephant and the 9 men blind from birth, खुद्दक निकाय, उदान पालि (6.4), तित्थ सुत्त
  • Bible ("gospel truth") says earth is flat (and tall trees CAN be seen from everywhere).
  • So what some regard as "reliable testimony" ("Bible is the word of God", shabd) is NOT truth.

However, ALL Indian schools of thought accepted pratyaksh (empirically manifest)

Section summary

  • The class IX school text brazenly lies that Indians and others had no method of proof.
  • (Why does teaching of formal math begin with lies?)
  • This lie helps the school text to conflate "proof" with formal mathematical proof (= "proof as used by church")
  • and avoid discussing the advantages or disadvantages of formal mathematical proof.

Proof (axiomatic proof)

Which proof is better?

  • Let us discuss what the NCERT text avoids
  • (by telling lies to make us believe there is only one notion of proof)
  • But, first, we need to understand what is axiomatic proof,
  • Let us understand it in terms of the Indian concepts of proof.

What is axiomatic proof?

  • In terms of Indian concepts,
  • formal/axiomatic mathematical proof accepts ONLY anuman (inference),
  • (and, in a hidden way, shabda paramana [authority] for axioms),
  • but rejects pratyaksh [empirically manifest]

Axiomatic proof rejects pratyaksh

  • Have been writing about this for 20 years (Hawai'i abstract, paper, downloadable paper)
  • but recently realized that
  • (a) people are ignorant about the definition of formal mathematical proof
  • (b) they don't trust my assertion (or check my definition quoted from the texts to which I refer)
  • (Ignorance and trust deficit is how superstitions are maintained.)

Axiomatic proof rejects pratyaksh

  • This is clear from the difference between the proof of 1+1=2 in normal mathematics
  • and the axiomatic proof of 1+1=2 by Russell.
  • Because axiomatic proof rejects prtayksh it is different from ALL Indian systems of proof
  • (since all Indian thought accepts pratyaksh)

Rejection of pratyaksh in formal proof is also stated in the school text

"However, each statement in the proof has to be established using only logic. … Beware of being deceived by what you see (remember Fig A1.3)!" [Appendix 1, p. 301, emphasis original]

That is why your school text lies

  • that Indians etc. had no notion of proof.
  • Easier to lie ("Indians had no proof")
  • than to explain why you must use a strange method of proof
  • different from all Indian methods of proof.
  • They KNOW no Indian asked questions about lies in school texts in 200 years.

Intermission: Questions

Rejecting pratyaksh makes math difficult

  • Disadvantage of rejecting pratyaksh is clear
  • It makes math so difficult that you can't find two persons in the whole country
  • to give the full proof of 1+1=2 in "real" numbers (chap 1, class IX text).

This huge difficulty adds NO practical value

But, if, like the church you want to give

  • "proofs based on reason"
  • to persuade people about imaginary things which don't exist in reality
  • such as angels, God, devil, heaven, hell
  • then beginning with assumptions (axioms/postulates) instead of facts is a terrific advantage.

That is exactly what Thomas Aquinas did

  • Aquinas was a major church theologian during the Crusade
  • who brought about a major shift in church theology
  • towards the Christian theology of reason
  • to compete with the earlier prevalent Islamic theology of reason (from which it borrowed heavily).

Aquinas' theorem

  • In his Summa Theologica, Aquinas REASONED about angels.
  • Since nothing about angels is pratyaksh he assumed whatever he liked about angels
  • to "prove by reason" that many angles can fit on a pin.
  • If "proof by reason" leads to this sort of balderdash,
  • should I say, like Lokayata: "Beware of being deceived by proofs by reason"? 😁

Is a theorem true?

  • The school text says

"The only way to be sure that something is true is to prove it" (Appendix 1, p. 296)

  • It also says, "A mathematical statement whose truth has been established (proved) is called a theorem." (Appendix 1, 293, 305)
  • That is, something which is proved is called a theorem and it is true.
  • So, is theorem = truth? NO?

Truth vs "relative truth"

  • We saw that a theorem proved by deductive reasoning is at best only "relatively true"
  • We also saw that (Wolf theorem, rabbit theorem, Aquinas' theorem) any nonsense can be relatively true.
  • The above easy examples may seem like "toy examples", not real mathematical theorems.
  • So let us take a real-life example: the Pythagorean theorem.

Is the Pythagorean theorem true?

  • Pythagorean theorem is surely a theorem: the class X NCERT text repeats "Pythagorean theorem" 32 times.
  • But is it true in the real world. NO!
  • For example, Pythagorean theorem is FALSE on the surface of the earth
  • (with "straight" lines = shortest distance between two points = geodesics).

Pythagorean theorem is FALSE anywhere in the REAL world

  • (because spacetime is also curved like the surface of the earth, but we won't go into that).
  • The belief that Pythagorean theorem is true is a Western superstition (Indian did not have it)
  • In fact it is a church superstition.

This church superstition (theorem = truth) caused the loss of thousands of lives.

  • How? Long story, but let me cover it quickly.
  • Greeks, Romans, Portuguese, British were bad at math, hence bad at navigation until the 18th c.
  • Vasco da Gama learnt to determine latitude at sea from his Indian navigator
  • who used the pole star altitude (Rajju Ganita text).

Western backwardness in math (elementary arithmetic) led to a bad calendar

  • Hence bad European navigation
  • until the Gregorian calendar reform of 1582
  • (using calendrical info stolen from India)

Why is a good calendar needed for navigation?

  • Calendar which correctly determine equinox needed to determine LATITUDE in daytime
  • from observation of solar altitude at noon (covered in Rajju Ganita text)
  • Not trivial (Protestant countries did not accept the calendar reform until 1752,
  • long after Newton's death).

And serial plagiarists of my calculus transmission work,

  • George Gheverghese Joseph and Dennis Almeida
  • failed to understand this point of mine about the calendar reform,
  • and made foolish mistakes proving they were plagiarists
  • (e.g. confused solar altitude with solar declination speaking vaguely of "angle of the sun".)

Heaving the log to determine longitude

  • Term "blog" comes from web-log. Where did the "log" come from?
  • Europeans used a primitive technique of navigation: "heaving the log" to determine longitude at sea.
  • A heavy log (tied to a knotted rope) was thrown into the sea,
  • and the speed of the ship (in knots) measured by counting how many knots of the rope ran out in a given time.
  • A continuous account of speed was maintained in a log book often just called "log".

From the speed, the distance travelled was calculated.

  • From the distance, and the latitude difference, the longitude difference was calculated.
  • Using the "Pythagorean theorem"
  • applied to the "longitude" triangle
  • (a right triangle, since lines of latitude and longitude meet at right angles).

Indians knew better even a thousand years earlier

  • A similar method described by Bhaskar 1 (7th c.)
  • to determine the longitude of a city
  • if we know its distance from a city on the prime meridian (longitude through Ujjaini)

Specifically, Bhaskar 1 says

Note: Bhaskar 1 is discussing the fact that spherical geometry is "non-Euclidean"

  • long before either "Euclidean" or "non-Euclidean" geometry was known to the West!
  • Wrong belief that Pythagorean theorem is true led to the drowning of thousands of European sailors
  • British parliament in 1711 acknowledged British ignorance by a law setting up a Board of Longitude.

Section summary

  • Pythagorean theorem is FALSE.
  • Belief that "deductively proved" theorems are true is a SUPERSTITION,
  • this false belief led to the death of thousands.
  • To reiterate: theorems only "relative truth", hence often false in the real world.

Section summary (contd.)

  • This false belief (theorem = truth) is actually a CHURCH superstition,
  • supported by Aquinas and his schoolmen (Christian rational theology).
  • This church superstition is peddled by the NCERT class IX school text.
  • (Calendar not the sole superstition peddled by colonial education. )

Is the Pythagorean theorem true? (apologia)

  • Some people say "Pythagorean theorem is approximately true".
  • However, NO concept of "approximate truth" in formal mathematics.
  • (Appendix 1, sec. A1.6 Summary) "1. In [formal] mathematics, a statement is only acceptable if it is either always true or always false." AND
  • "2. To show that a mathematical statement is false, it is enough to find a single counterexample."

Approximate truth

  • However, "approximate truth" found in sulba-sutra-s,
  • used in Rajju Ganita as "Zeroism".
  • will return to it later.

Deduction is fallible

Deductive proof is glorified in the school text

  • But empirical proof (based on pratyaksh) is attacked as fallible in the school text.
  • The text books says "Beware of being deceived by what you see"
  • Now, everyone accepts fallibility of pratyaksh:
  • e.g. रज्जुसर्पन्याय (न्यायावली, 304)(one may mistake a snake for a rope or rope for snake)

Science too accepts this fallibility (possible experimental error)

  • But both Indian tradition and science nevertheless accept pratyakha as the foundational means of proof. Why?
  • Because, in case of doubt you simply repeat the observation/experiment.
  • Prod the rope/snake with a stick.
  • A few iterations usually settle the issue in practice.

But this is an inductive method

  • The school text also points out that inductive proof too is fallible.
  • (A point first made by al Ghazali and plagiarised by Hume.
  • The church/universities learnt about reason from al Ghazali's ineffective opponent, Ibn Rushd/Averroes.)

Is there anything better?

  • Let us accept both points. (1) that pratyaksh is fallible, and (2) that induction too is fallible.
  • But what is the alternative? Is deductive proof infallible?
  • That is the superstition the church/school text peddles.
  • "the only way to be sure that something is true is to prove it." (Appendix 1)
  • Note: we are right now setting aside the question of "true", and looking at the issue of "sure".

But deductive proof too is fallible

  • Quite simply, there are valid proofs and invalid proofs.
  • As a math teacher, I found that students often submitted wrong proofs.
  • If all proof were valid, all students would get 100%.
  • But students usually performed miserably.

How to be SURE whether a proof is valid?

  • As a teacher, I decided (shabd pramana)
  • As a PhD student, I repeatedly rechecked my 100+ pages of calculations/proofs (induction).
  • Key point: deductive proofs are FALLIBLE (they may be invalid)
  • and one cannot be SURE of validity except by induction or authority!

If I ask you is Russell's 368 page proof of 1+1=2 valid?

  • What is your answer? How can you be SURE?
  • Because colonial education made you ignorant by making math difficult (your only answer is "I don't know")
  • You have no choice but to rely on authority (shabd praman, faith-based proof) (say "I TRUST Russell")
  • Whose authority? (Obvious Western authority!)

Formal math has one clear political function

OK, so deduction is fallible. But is it less or more fallible than observation?

  • Two ways to decide.
  • How LONG does it take to correct a wrong axiomatic proof?
  • How OFTEN are there errors in a deductive proof?

How long does it take to correct an invalid proof?

  • For over 750 years, ALL Europeans scholar read Euclid's Elements and
  • and wrongly believed the Euclid text had valid axiomatic proofs.

But no axiomatic proofs in Euclid

  • At the end of the 19th c. it was admitted that this belief is false.
  • There is NOT A SINGLE VALID axiomatic proof in the book "Euclid's" Elements.

Dedekind pointed out that even "Euclid's" first proposition lacks an axiomatic proof

For axiomatic proof, order of propositions is important

So to understand that purported axiomatic proofs in "Euclid" are INVALID

  • took ALL European scholars 750 years to resolve. 🤣🤣🤣
  • But it takes barely ten seconds to resolve the snake/rope issue.
  • But we imitate West and teach our children the nonsense that deductive proofs are infallible!
  • (Two authors of the school text, Jayant Narlikar and Pervin Sinclair got a PhD from Cambridge.)

Given that both deductive proof and observation are fallible

  • Q. which is fallible MORE often?
  • A. Deductive proof is fallible MORE OFTEN than observation.
  • A long and complex proof/calculcation has many sources of errors
  • (for the mind is more easily deceived than the senses).

Simplest example is the game of chess ("a chess problem is genuine mathematics" Hardy p. 11)

  • Based on pure deduction (invented in India!)
  • Error-free game MUST end in a draw.
  • But everyone (even the world champion) almost ALWAYS makes an error
  • hence almost always loses to a computer program, such as AlphaZero.

Deciding validity

  • Many other CURRENT cases of proof where ALL "experts" are unable to decide whether proof is valid:
  • four color theorem, abc conjecture etc.
  • Formal mathematical proof is ultimately only about SOCIAL consensus among
  • TRIBE of formal mathematicians.

Section summary

  • Deductive proof is much MORE often fallible than empirical proof
  • Errors in a deductive proof can only be settled inductively. (Hence deduction MORE fallible than induction.)
  • And may take very long to settle (compared to observational errors).
  • Or they are settled by appeal to authority, making you dependent on Western authority.

Not sure, not true

  • And even if you have a probably valid deductive proof
  • the resulting mathematical theorem does NOT give truth, at best mere "relative truth"
  • at worst complete nonsense.

Obscurities of geometry teaching-1

  • Axioms are metaphysics
  • Distinct and incompatible geometries (Euclidean geometry, Hilbert, Birkhoff, Compass box) are mixed up (no warning to student)

Why is formal math more difficult?

  • We saw that formal math makes 1+1=2 difficult - People often say it is because it is abstract.
  • Wrong!
  • Formal math is not merely abstract, it is metaphysics (= non-physics, non-real).

Axioms and basic notions are metaphysics

Likewise, lines are infinitely long (p. 71)

  • (apart from being infinitely thin)
  • axiom fails for lines drawn on surface of earth
  • Infinity is a prime example of metaphysics
  • Metaphysics is impossible to check on child's own knowledge.

All of school geometry based on straight lines

  • but "straight line" defined only as (infiniute extension of) "line of shortest length" (Class VI text)
  • Even if length assumed known, this requires comparison with length of curved line (empirically using a string)
  • but defined axiomatically for \(y=f(x)\) only by using calculus (a metaphysics of infinity) \[s = \int_a^b \sqrt(1+\frac{dy}{dx}^2) dx\]

All this metaphysics

  • teaches a child to obey authority, have faith in it.
  • But interferes with practical application.
  • Because metaphysics cannot be empirically verified.
  • Formal mathematical theorems are truths relative to unverifiable assumptions.

E.g. Banach-Tarski theorem

At best axioms are "approximately" true

  • but no concept of approximate truth in formal math (as we saw, Appendix 1)
  • Irrefutability of theorems makes them too metaphysics
  • Metaphysics greatly suited the church, but no good for practical applications to science and engineering
  • for which we want something true in this real world.

Obscurities of geometry teaching-2

Colonial education taught us to imitate, so we imitate ALL

  • the original "Euclid" with its figures and SAS theorem (prop. 4)
  • Hilbert's synthetic geometry which corrected "Euclid"
  • Birkhoff's metric geometry which corrected Hilbert
  • compass box geometry (empirical metric geometry)

These distinct geometries are not compatible

  • This point should be clear by now to those who attempted the home assignment
  • Thus, there are no axioms for length measurement in Hilbert's geometry.
  • Since length cannot be measured, Hilbert's geometry is called synthetic.
  • Curiously, though length is not defined area is (to enable the Pythagorean theorem to be stated and proved).

The proof of prop. 1 in "Euclid" is wrong according to Russell

  • since the intersection of the two arcs is suggested by the figure
  • but there is no axiom from which that intersection can be proved.
  • The proof of the Pythagorean proposition is also wrong, according to Russell
  • since it requires prop. 4 (SAS) proved by superposition.

The empirical measure of length

  • on compass-box geometry
  • is NOT the same as the axiomatic definition of length in Birkhoff's geometry.
  • The first is an empirical method (based on pratyaksh)
  • NOT allowed in any axiomatic geometry.

The interesting thing is that

  • the current teaching of geometry includes ALL the above geometries
  • without explaining how they are incompatible, in fact contradictory
  • but stating that contradiction is fatal to mathematics!

"It works" superstition

  • Lastly, I want to take up what I call "it works" superstition,
  • which I have been hearing since 2012.

In 2012 I taught an undergraduate class

  • on the history and philosophy of science,
  • where I criticized formal mathematics.
  • One day, the students said: "we don't disagree with any of the criticism"
  • but, "it works".

More recently, an IIT graduate told me

  • "until now, all applications of mathematics
  • such as rocket technology
  • are done using formal mathematics."
  • it is hard to imagine a more foolish statement.

Formal mathematics came into existence only in the 20th century

  • after Hilbert and Russell etc. "corrected" "Euclid".
  • All mathematics before the 20th c. was normal mathematics.
  • Therefore, all applications of mathematics before the 20th century
  • were done using only normal mathematics.

That includes all bridge building before "Euclid" was heard of in Europe

  • it includes all calculations of planetary orbits done by Newton, etc.
  • for the simple reason that formal mathematics did not exist
  • prior to the 20th century (not even in "Euclid").
  • But this ("practical applications require normal math" is also true of the latest technology.

Thus, I have been pointing out since 1999 that

  • Calculation of rocket trajectories requires calculus
  • which is taught today as formal mathematics in the university, using real numbers.
  • But actual calculation of rocket trajectories by NASA/ISRO etc uses computers
  • which cannot handle metaphysical "real" numbers, but use floating-point numbers instead.

That is, rocket trajectories are NOT calculated using formal mathematics

  • (The same thing is true for the latest technology: algorithms for AI)
  • But the problem is that most people don't understand either rocket technology or AI.
  • The serial plagiarists, George Gheverghese Joseph and Dennis Almeida, who plagiarised my Hawai'i 2001 calculus paper, comically said
  • in a paper published in Race and Class 2004: the "Kerala mathematicians used floating point numbers to sum infinite series".

That is, colonial education kept you ignorant of formal mathematics

  • but taught you immense faith in the master.
  • It is that ignorance + faith which underlies the "it works" superstition.

Therefore let me give a simpler example.

  • Please explain how you will use formal mathematics to measure the area
  • of a given plot of land with irregular boundaries?
  • Can you do it using only axioms, and without ever using pratyaksh to make measurements?
  • Will the tax authorities accept such an axiomatic proof?

Later we will see why with Rajju Ganita

  • it works better.

As the mathematician Hardy (of Ramanujan fame) admitted

mathematics is… unprofitable. Is mathematics… useful, as other sciences such as chemistry…are? …I shall ultimately say No (Mathematician's Apology, p. 8)

  • This is particularly true of number theory ( also the field of Manjul Bhargava)

Mathematics as beautiful

A mathematician, like a painter or a poet, is a maker of patterns…The mathematician’s patterns…must be beautiful; Beauty is the first test(pp 13-14)

  • Heardy went on

It may be very hard to define mathematical beauty, but…that does not prevent us from recognizing… it.

In Russell's time

  • it was understood that (formal) mathematics is metaphysics. (Russell, Mathematics and metaphysicians)
  • So, Hardy is referring to patterns in metaphysics
  • (exactly what Christian rational theology does). BUT

Millions of students immediately recognize beauty in music (without being taught)

  • but reject formal math as UGLY.
  • (Besides if math is art, mathematicians must get support from only department of culture.)
  • Cause of Hardy's confusion?
  • He failed to understand how the church altered Platonic math, making it ugly, while creating formal math during the Crusades.

How and why the church concocted "Euclid" during the Crusades (quick summary)

  • more details in Euclid and Jesus.
  • (Note: Church a powerful political organization \(\neq\) Christianity a religion.)

No evidence for "Euclid"

  • As already stated no evidence that Euclid existed
  • or that he or anyone near his time wrote the book Elements
  • Till today no early (or late) Greek text of Elements
  • which states Euclid is the author.

Ample counter evidence

  • All Byzantine Greek texts say the book Elements (of Geometry) was authored by Theon or based on his work.
  • Theon was from Alexandria, in Egypt, in Africa
  • he subscribed to the Egyptian mystery tradition of geometry as mathesis
  • made public by Plato.

On this Egyptian/Greek tradition of mathesis

  • geometry is about arousing the soul (like music)
  • by using reasoning to drive the mind inward.
  • Same objective as yoga, but a slightly different technique (using reason).

Changing the author from Euclid to Theon changes our understanding of the book

  • However, the church was hostile to this notion of soul
  • Theon's daughter Hypatia, the probable author was raped and lynched in a church.
  • The church pronounced its great curse (anathema) on this notion of soul (553 CE),
  • and closed all schools of philosophy (532 CE) in the Roman Empire.

The commentator Proclus explains at length

  • that the Elements is about geometry as mathesis.
  • But accepting this would leave the church with egg all over it face, therefore it is denied.
  • (An isolated and forged remark in the book by Proclus is the sole "evidence" for Euclid.)
  • This is corroborated by the use of figures in the book (which Russell says are a "dangerous habit")

After the church shut all schools of philosophy in the Roman Empire

  • the philosophers migrated to neighboring Persia (Jundishapur)
  • this tradition of geometry came to be known as aql-i-des (= geometry based on aql or reason = "Uclides")
  • After the conversion of Persia to Islam the "philosophers" (falsafa) became influential in Islam
  • along with the Egyptian/Platonic tradition called "theology of Aristotle" in Islam (Europeans call it "Neoplatonism" = sufi).

The falsafa were destroyed in sunni Islam

  • by the critique of al Ghazali
  • opposed incompetently by Ibn Rushd (Averroes).
  • During the Crusades against Muslims the church stepped into this division within Islam
  • and sided with the philosopher Averroes (from whose texts church universities taught "Aristotle").

Because reasoning had become a key aspect of Islam (aql-i-kalam)

  • The church created Christian rational theology as a counter.
  • The sole church motive was to use reason as a tool to persuade and convert Muslims.
  • But to be able to accept Arabic texts it interpreted aql-i-des as the name of a mythical Greek "Uqlides"= "Euclid"
  • and attributed to him the church motive of providing persuasive ("irrefragable") proofs based conveniently on axiomatic reason.

Propaganda and fool and rule

  • Purported superiority of axiomatic proofs fooled billions and kept this myth alive,
  • even after the absence of axiomatic proofs in "Euclid" was exposed a century ago.
  • and the deficiencies of axiomatic proof have been exposed.

How long will it continue?

  • Europeans were fooled for 8 centuries
  • and Indians for the last 2 centuries
    • 75 years were after independence.
  • Amazing how easily the church fooled Indians,
    • (and is now fooling Africans).

Summary and Conclusions

  • Formal math involves church superstitions
  • spread using the myth of Euclid
  • [concocted by the church for its own purposes (church theology of reason)] during the Crusades
  • Teaching this in schools spreads this superstition widely.

Summary and Conclusions (cotd.)

  • Only possible way to validate this knowledge is by open public debate
  • (anchor of Indian traditional knowledge).
  • But "experts" created by colonial/church education won't permit it to change
  • will also dishonestly dodge public debate.

Summary and Conclusions

  • Common people have the "it works" superstition which is false and dangerous.
  • Metaphysics of formal math does NOT work for applications of math (works only to further church politics)
  • What ALWAYS worked (for science etc.,) since ancient times, and still works is NORMAL math.

Why a hotchpotch is taught?

  • Teaching formal math alone would expose its uselessness.
  • THEREFORE, current math includes normal math, making it a self-contradictory hotch-potch of formal and normal math.
  • formal math as in axiomatic geometry (Hilbert's synthetic geometry, Birkhoff's metric geometry, etc.) AND
  • normal math as in empirical compass box geometry.

What kind of normal math?

  • Alternative is to teach normnal math alone.
  • But what kind of normal math should we teach?
  • Traditional Rajju Ganita shows that an alternative (normal) math,
  • and its teaching, is possible, even at school level.