गणित vs formal math

C. K. Raju

14 word summary of book:

  • Ganit (गणित) differs from (formal) math,
  • it makes math easy, and
  • makes science better.

Hard to communicate

  • Can only be communicated to an educated person,
  • and educated people today are colonially educated.

The colonially educated are mostly

  • mathematically illiterate
  • hence scientifically illiterate.
  • And colonial education blanked out indigenous knowledge.

  • Hence, the colonially educated do not understand any of the three:
    • गणित, or
    • formal math or
    • science.

  • First a recap of my first presentation

Recap-1 Colonialism and church

Colonial education was church education.

  • Not only mission schools but also universities.
  • Paris, Oxford, Cambridge were all set up by the chuch during the Crusades

Western education was a church monopoly

  • The church controlled universities at least until the 20th c.
  • (First secular education in Britain only AFTER 1871 act only for primary schooling.)

Clarification: church \(\neq\) Christianity

  • E.g. scientist Isaac Newton was a devout Christian but secretly and abusively anti-church.
  • E.g. he wrote church priests were "spiritual fornicataors", and
  • "the most evil men ever to have lived on the earth."

  • (Hence, his 7-volume anti-church writings suppressed till today as
  • "foul papers related to church matters, unfit to be published")

  • Every graduate of Cambridge was required to sign an oath of service to the church.
  • Newton refused but was given a special dispensation by Isaac Barrow.
  • For references to original sources (Newton's own writings etc.), see "Newton's secret" chp. 4 in Eleven Pictures of Time, Sage, 2003.

Key point

  • Being against church does not mean being anti-Christian.

Church a political organization tied to the state

  • I have every right to contests its politics,
  • and an ethical duty to do so.

The church committed to genocide

  • Did you ever publicly condemn this evil?
    • Silent support?
    • (On principle of proportionate condemnation, you should condemn it 20 times as often as Hitler.)

  • Did you ever condemn popes for their evil proclamations?
  • And their refusal to withdraw those bulls (fatwas) until today?

Cultural genocide

  • Church also committed to cultural genocide and
  • destruction of indigenous knowledge.

The church propaganda was always just this:

  • Everything inidgenous is inferior (reject it)
  • Everying Western is superior (blindly accept it)
  • Therefore imitate the West (as we do)

Trivially false

  • E.g. Christian (Gregorian) calendar is decidedly inferior to indigenous calendar
  • See my "Tale of two calendars", video or article in Multicultural Knowledge and the University, 2013.

My stand (Decolonisation)

Colonial education supposedly came for science

  • but it made most people mathematically and scientifically illiterate.
  • Don't blame yourself. Not the fault of the individual teachers and students,
  • but the fault of the colonial education system.

Education system was designed to create insular missionaries.

  • Hence, it deliberately taught ignorance
  • Ignorance forces you to depend on others.
  • Which others? Those in authority.

Stories of Western glory

  • Therefore, colonial education ONLY taught you lots of STORIES about SCIENCE to glorify the West
  • and inculcate the formula "trust (only) the West".

  • These stories taught at a young age to indoctrinate gullible children who do not check facts.
  • (Suspension of disbelief).
  • Then fight when the story changes. (Insularity.)

  • We continue to accept these stories even as adults.
  • e.g. "Copernicus road" in Delhi.

  • What exactly did Copernicus do beyond Ibn Shatir?
  • Primary facts please, no stories.
  • No "trust the West".

Stories a way to dissolve facts

  • Principle of HPS. Any theory can be defended against any facts for any length of time by piling on the hypotheses.
  • Any lie can be defended against any facts by telling a thousand more lies.
  • Any story can be defended against any facts by telling another hundred stories.

  • So, please don't tell me: "I have heard a story about Tycho Brahe",
  • "and let me finish, I have another ten stories to repeat"
  • (and another ten facts to dissolve).

Did you check any facts?

  • What is the Tychonic model?
  • How does it differ from Nilakantha's model? Or the heliocentric model?

  • What was the precision of Tycho's masonry instruments?
  • How did the nearly-blind Kepler arrive at accurate data about Mars? etc.

Facts can only be checked one at a time

  • My right to stop you and check facts of one story MORE IMPORTANT
  • than your right to dissolve multiple facts by telling a long chain of stories.
  • And using one story as proof of another.

  • So, if you tell me a story,
  • I have a right to stop you and demand what you know about facts.

Recap 2: How गणित travelled to the West and back

First presentation explained historically

  • Most school math (arithmetic, algebra, "trigonometry", calculus, probability and statistics)
  • went from India to Europe between 10th to 16th c. for its PRACTICAL value.

Cultural dissonance between Indian गणित and Western math

  • In Europe its practical value was accepted
  • but aspects of गणित were not understood.

E.g. (from elementary arithmetic for commerce)

Eventually, गणित was changed to formal math

  • packaged with a false history (e.g. "Newton invented the calculus")
  • and returned through colonial education.

  • This talk will explain the philosophical differences
  • between गणित and formal math
  • which led to the cultural dissonance.

  • Explanation will be pitched at the easiest possible level.

Math is not universal but varies with culture

E.g. of angle

Strange term

Meaning of उभयनिष्ठ

  • उभयनिष्ठ not in Hindi (or Sanskrit) dictionary.
  • Student must infer its meaning from उभय and निष्ठ (in Sanskrit): "loyal to both".

  • it refers to the common initial point of
  • an angle made by two straight lines (rays).

Hindi word for angle?

  • Use of such a complicated term suggests that the corresponding concept is absent in Hindi.
  • Is it? What is the Hindi word for angle?

Common answer

  • Common answer कोण.
  • Wrong!

Word कोण a late invention

  • Word कोण not found in India before 18th c.
  • (Coined by Samrat Jagannath 1723 in Rekhaganit, a Sanskrit translation of "Euclid's" Elements from Farsi.)
  • (Translated at behest of Sawai Jai Singh of Jantar Mantar fame.)

Earlier concept of angle

  • So, did Indians have no concept of angle earlier?
  • They did.

E.g. in RgVeda

E.g. in Vedanga Jyotish

  • The Vedanga Jyotish (Rk 10-11) (trans.) has even finer divisions (bhamsha-s भांशा: \(= 0.1°\)).
  • (Note: jyotish \(\neq\) astrology, jyotish = time measurement = angle measurement)

चाप

  • The concept of angle in India was different.
  • चाप (chord) is the correct traditional term for "angle".
  • E.g., famous निहत्य चाप वर्गेण चापं from Yukitidipika, 441 (Infinite "Taylor's" series for sine).
  • Angle = the relative length of an arc.

Instrument to measure angle

  • If an angle is about two straight lines, why is a semi-circular protractor needed to measure it?
  • And, what instrument in a compass box (geometry box) can be used to measure angle in the sense of चाप?

Instrumentally different concepts

  • None.
  • The two concepts of angle require different instruments for their measurement.
  • The चाप definiton requires a flexible string (or a clock).

Church influence on Western math

How was Western math affected by culture

  • So, math varies with culture.
  • How did Western math vary with culture?

Hegemonic church a dominant influence

  • Key cultural influence on Western math was obviously the church.
  • The medieval church taught the quadrivium: which included "sacred geometry"

Definition of point etc?

  • E.g. what is the definition of a point, line, plane?
  • "Sacred geometry" of Church quadrivium taught a foolish definition.
  • What you have learnt starts from this foolish medieval church geometry. (No protests against church medievalism?)

Church adopted "reason" to persuade Muslims

  • Later, the church adopted "reason" as the ONLY way to persuade Muslims to convert.
  • (Muslims accepted aql-i-kalam, but rejected the Bible as corrupted.)
  • Crusades (after the first) aimed to convert by force, but failed militarily for centuries.

"Euclid" used as church text to teach reasoning

  • The church adopted the Christian theology of reason of Aquinas and schoolmen.
  • Between the 12th and 15th c., the church adopted the text of "Euclid's" Elements (reinterpreted)
  • to teach reasoning to its priests for persuasion.

Western ethnomath about proof (like formal math)

  • Since persuasion was the church goal,
  • therefore, "proof" is today taken as the main ingredient of math (= Western math = formal math)
  • Formal math focuses on proof.

"Reason why" requires a proof

  • That \(1+1=2\) not so important in formal math.
  • Important thing is why \(1+1=2\).
  • In formal math, the "why" requires a "proof".

We teach that गणित is inferior

  • Our class IX NCERT math school text says
  • the geometry Indians did was "practical"
  • meaning it lacked proof, hence गणित inferior

Pythagorean controversy of 2015

  • E.g., in 2015, Harsh Vardhan said, "India gave the Pythagorean theorem to the West".
  • Now Pythagorean theorem may have been stated in शुल्ब सूत्र (E.g. Baudhayana 1.12, Apastamba, 1.4, Katyayana, 2.7.)
  • this may be before Pythagoras

Purkayastha's response

Taboo against history of science

  • History of science taboo for ALL our historians, and
  • hence they never understood the trick of using philosophy to distort history. (Part of the syllabus for my HPS course).
  • Just blaming Marxists won't do.

Our current class IX text claims

"But in civilisations like Greece, the emphasis was on the reasoning behind why certain constructions work. The Greeks were interested in establishing the truth of the statements they discovered using deductive reasoning" NCERT, class IX, Math, pp. 78-79).

A dozen falsehoods

  • This statement in our school text contains about a DOZEN FALSEHOODS.

Demonising गणित in our school texts

"Indian math had no proof"

"Indians did not use reasoning"

  • False. Above proof involves reasoning.
  • But it is argued that it is an empirical proof.
  • Further caricature: Indians used the empirical, hence did not use reasoning.
  • but that too is false.

Indians accepted empirical proofs

  • It is true that Indian गणित explicitly accepts the empirical
  • E.g. आर्यभट, गणित 13: साध्या जलेन समभूरधऊर्ध्वं लम्बकेनैव
  • "Determine level ground by water, verticality by a plumb"

But also used reasoning

  • Above proof of pf Pythagorean theorem uses both reasoning and empirical.
  • E.g., Arabhata used both reason and facts (far off trees cannot be seen, horizon is round) to deduce that the earth is round.
  • like a kadamba flower (Gola 7)

गणित like science

  • गणित uses BOTH facts (empirical proofs) and reasoning exactly like science.
  • So why is it declared inferior?
  • Is science inferior?
  • Will science become superior if you prohibit experiment?

Prohibiting the empirical

A church technique of reasoning

  • Church accepted reasoning, but not facts, for facts contrary to its dogmas
  • (reasoning + facts = science) so church declared mathematical proof must prohibit facts.

Formal math imitates church

  • Formal math imitates this and prohibits empirical proof.
  • (Empirical proof = proof based on evidence of senses.)

Not allowed to point

  • E.g. Formal math does NOT allow you to "prove" \(1+1=2\) by pointing.
  • Why not? Because you can see the fruits.
    • That is empirical and prohibited.

Begin with axioms

  • If you prohibit facts what should you do?
  • Begin with axioms

What is an axiom?

  • An axiomatic proof begins with axioms not facts.
  • Today, axiom = postulate(= assumption) NOT uncontested truth,
  • as our class IX text admits.

Definition of axiomatic proof

  • A sequence of sentences.
  • Each sentence must be either an axiom
  • or derived from preceding sentences by a rule of reasoning.

E.g.

  • A implies B
  • A
  • Therefore, B.

Early Greeks were the only "friends of the church" (Eusebius)

  • No evidence that Concerned Greeks "Pythagoras", "Euclid" existed.
  • (Purkayastha repeats silly church myth that Pythagoras had a proof. What was that proof?)
  • Much counter evidence: hence my Rs 2 lakh challenge prize for serious evidence about Euclid.

Essence of colonial education

Does the person matter?

  • Apologists say the person doesn't matter, the book does.
  • :) They never read the book Elements. Just believed the myth about it.

Greeks never used axiomatic reasoning

  • Further, Greeks gave axioms and proofs, SEPARATELY but
  • NO axiomatic proofs (i.e., proofs without using the empirical).

The fish figure and cross

  • E.g. First proposition of the Elements gives an empirical proof.
  • One sees the two arcs intersecting.
  • But they may or may not have a point in common.

SAS

  • Also the 4th proposition (Side-angle-side theorem).
  • Original proof was by putting one triangle appropriately on top of the other to see that the two triangles are equal.
  • (Not an axiomatic proof. Anything one can see is NOT formal math, therefore, today we teach SAS postulate.)

Needed for the whole book

  • 4th proposition needed for proof of penultimate "Pythagorean theorem"
  • So Greek proofs were no different from Indian proofs only more prolix.

A church method

  • Method of reasoning without facts invented by the church by misinterpreting Euclid's Elements
  • E.g. used by Aquinas

Aquinas' Theorem

  • Aquinas theorem: There can be several angels on a pin.
  • from Aquinas' axiom: angels occupy no space. Summa I. Q.52(3).
  • Do angels exist? Can one check this empirically?

Foolish Cambridge math syllabus

We compound that foolishness

  • Length measurement is empirical: so let us reject also the compass box (geometry box) given to every student.
  • If empirical is accepted, then SAS is a theorem, not a postulate (as in the original).

A nation of idiots?

  • Are we a nation of 1.3 billion idiots that,
  • even after 70 years of independence, only one person can take a PUBLIC stand on this foolishness in our school texts?
  • This is proof that colonial education (or missionary training) makes you a mental slave: you are unable to challenge the West.

Hilbert's synthetic geometry

  • First fully axiomatic proof of Pythagorean theorem in 20th c. with synthetic: geometry of David Hilbert.
  • Synthetic means non-metric: to avoid empirical superposition, length measurement is disallowed.

Defining area without defining length

  • But area is still defined (to be able to prove the Pythagorean theorem which is about equal areas).
  • Defining area without defining length first is beyond school texts.

Conflating two types of reasoning

  • To further fool school students, one word "reason" used for both
  • scientific or normal reasoning (reasoning with facts), and
  • axiomatic of formal reasoning (reasoning without facts)

Misleading children about "reason"

  • Using one word,"reason" confounds issues and misleads as in our school text.
  • one meaning of "reason" relates to science (and गणित)
  • the other to metaphysical dogma.

Summary: A key difference

  • Shorn of doublespeak about reason,
  • and false myths about Greeks,
  • a key difference between गणित and formal math is that
  • गणित accepts empirical proofs, formal math rejects them.

Which is superior?

  • So which is superior?
  • Can we teach geometry without practical/empirical length measurement?
  • Or teach both: geometry is theoretically (according to the church) about non-empirical proofs (from metaphysical axioms).

What is wrong with empirical proof?

Is science inferior because it accepts experiment

  • What is wrong with seeing something?
  • Is science bad because it accepts the empirical?

  • Claim is that the empirical is fallible:
  • E.g. one may mistake a snake for a rope or rope for snake.

Rope/snake

  • This claim and example (rope/snake) is accepted in India (न्यायावलिः 3041, रज्जुसर्पन्यायः)
  • Nevertheless, ALL Indian systems (without exception) न्याय-वैशेषिक, सांख्य-योग, मीमांसा (incl. अद्वैत वेदांत), लोकायत, बौद्ध, जैन, etc.
  • accept empirical proof (प्रत्यक्ष प्रमाण) as the first means of proof.

  • First lesson in lab science: experiments involve errors.
  • But empirical is not prohibited for that reason.

So are axiomatic proofs superior?

  • Claim is that deductive/axiomatic proofs are infallible.
  • But is this merely a superstition?

Are axiomatic proofs infallible?

  • Moreover, Russell only proved 1+1=2 for "integers" (actually "natural numbers").
  • In formal math, a separate proof is required for 1+1=2 in formal "real" numbers.

Cape Town challenge

  • Cape Town challenge: Give a full axiomatic proof of 1+1=2 in "real" numbers (without assuming any axiomatic set theory).
  • May need a 1000 pages or more (no one did it).
  • Try this challenge on some IIT professors of math or anyone you think is knowledgeable in math.

Is Russell's proof error-free?

  • But how exactly do you know that Russell's proof is valid?
  • That it does not contain 22 mistakes?

  • Only through ignorance of math, and superstitious trust in Western authority

  • Authority is MORE fallible than empirical proof.
  • Therefore, deduction is MORE fallible than empirical proof.

Chess: Another example of fallibility of deduction

  • The game of chess is pure deduction.
  • You always lose to machine because even the top grandmaster always makes mistakes.

  • Belief in the infallibility of deduction is mere SUPERSTITION
  • like the belief in the infallibility of the pope.

Summary: Deduction is fallible

  • The theorems of math are not even certain RELATIVE truths relative to the axioms.
  • At best fallible relative truths.

Axioms as metaphysics

Lokayata critique:

  • Wrong axioms/assumptions lead to wrong conclusions.
  • Lokayata explained this using the story of the wolf's pug marks

E.g. in modern terms

  • Axiom 1. All animals have two horns.
  • Axiom 2. A rabbit is an animal.
  • Theorem. Therefore, a rabbit has two horns.

  • So an axiomatically proved mathematical theorem need not be even remotely valid knowledge.
  • We know that axiom 1 above is false only empirically.
  • Should we also reject the idea the axiom that a unique straight line connects two points.

  • How does one know axioms are valid?
  • Usually, no way to test the axioms of mathematics empirically.
  • That is, axioms of formal math are pure metaphysics (irrefutable in the sense of Karl Popper).

E.g. axiom from school text

  • Our class IX school text says geometric points are invisible (no size). (Remember: If you can see it, it is not formal math.)
  • Is that a con-trick like the Emperor's new clothes?

  • Hence, a line which consists of points is also invisible. (Class V text defines line as a breadthless length.)
  • The text states an axiom: a unique (invisible) straight line connects any two points.

Note 1: Difference between metaphysics and abstraction

  • People wrongly say math is abstract.
  • The word "dog" is abstract: dogs come in many shapes, sizes, and colours.
  • But no child has a difficulty with this abstraction.

  • For we can point to several instances of dogs (that is how a child learns the meaning of dog).
  • We do not define the word dog, but children learn to discriminate between dogs and cats by seeing.
  • However, we cannot point to an invisible point. (It is metaphysics, not mere abstraction.)
  • (Definitions by pointing are prohibited in formal math.)

  • If the axioms of formal math are metaphysics, so are the theorems.
  • Mathematical theorems are at best, fallible relative, metaphysical truths (relative to both axioms and logic).
  • They are true only in a fantasy world, not the real world.

  • At best, one may choose the axioms so that the fantasy world of formal math somehow "approximates" the real world.

Calculation not proof

Calculation has practical value

  • E.g. in a grocer's shop calculation of \(74 \times 12\) more important than proof on Russell's line.
  • E.g. Columbus "proved" earth was round
  • but could not calculate its radius
  • resulting in navigational disasters.

"Pythagorean theorem" in the Manava शुल्ब सूत्र

Rectangle and diagonal

  • Note, first that this statement is about a rectangle and its diagonal.
  • (This is not altogether trivial since, according to Heath, Egyptians did not know what a right-angled triangle was.)

Uses square roots

  • That is instead of \(d^2 = l^2 + b^2\)
  • It states the result in the form \(d = \sqrt{l^2 + b^2}\).

Calculation, not proof

  • This tells us how to CALCULATE the diagonal.
  • Calculation NOT available in the Western way
  • (because the West learnt about square roots very late.)

Inexactitude

Square root of 2

  • If we take \(l=b=1\)
  • then we get \(d = \sqrt 2\).

  • Western religious superstition that math is eternal truth
  • led to the foolish religious belief that math is exact.

  • But where is the "Pythagorean theorem" ever exact (in the real world)?

  • Not on the curved surface of the earth
  • (as pointed out by Bhaskar 1)
  • Not in curved space

Infinite series

  • \(\sqrt 2 = 1.414\dots\)
  • \(= 1 + \frac{4}{10} + \frac{1}{100} + \frac{4{{1000} + \dots\)

Precise (सटीक) vs exact (त्रुटिहीन)

  • Precision is needed, and we can calculate the sum precisely (as done in the शुल्ब सूत्र, in Iraq (Babylon) and Egypt.
  • But never EXACTLY (except in a fantasy world of metaphysics).

  • e.g. Berkeley against Newton
  • "It is said, that the minutest Errors are not to be neglected in Mathematics" (The Analyst).

  • Led to metaphysical overload in present-day math.

Religious origins of math since Plato

Exact since eternal truth

  • West had a religious view of math right since Plato

Math from mathesis

  • Word "mathematics" derives from mathesis meaning learning
  • or arousal of the soul to make it recollect its eternal knowledge from past lives. (Plato, Meno, 4th occurrence of "soul".)

  • Proclus explained this happens by sympathetic magic,
  • because the eternal truths of math arouse the eternal soul. (Commentary).
  • (Don't trust tertiary source Wikipedia that math from mathema.)

    • This superstition ("math has eternal truths" persisted with the church.
    • the major cultural influence on Europe.

Aquinas on eternal laws

  • Aquinas said: God rules the world with eternal laws of nature (Summa Theologica First part of Second Part, 91, 1)
  • naturally written in the language of eternal truth: mathematics.

  • Led to related superstition that math is exact
  • (Exactitude possible only in a fantasy world of metaphysics.)

Metaphysics can be religiously biased

  • especially because the colonised blindly accept Western authority
  • E.g. axioms about infinity allied to church dogmas of eternity. (Won't go into details, see cited article.)
  • This metaphysics creeps into science. Resulting in a religiously biased science, E.g. Stephen Hawking.

Hiding the religious connection of formal math

E.g. by talking of Greeks, to hide the church connection to formal math.

E.g. by talking of "aesthetics" to hide lack of practical value of formal math

Plato said both math and music arouse the soul.

  • but today children love music but most hate math.
  • Hence, no aesthetics in formal math except for experts with vested interests.

Summary गणित vs formal math

गणित (normal math) accepts empirical proofs, formal math prohibits them

  • Both use reasoning.
  • Deductive reasoning is fallible: theorems of formal math are FALLIBLE relative truths relative to axioms.
    • More fallible than empirical proofs, since dependent on authority.
  • Axioms of formal math are metaphysics (of infinity allied to church dogmas of eternity).

गणित emphasizes calculation.

  • All practical value comes from calculation.

गणित accepts inexactitude.

  • is practical and applies approximately to the real world.
  • Formal math claims to be exact (Pythagorean theorem) onward applies exactly only to a fantasy world.

गणित is secular.

  • Western math related to religion since Plato.

गणित is practical

  • formal math is church dogma and whether it approximates the real world is problematic.
  • It works politically. (E.g. using science for creationism.)
  • What "works" for real science is (and always was) only गणित or normal math.

Pedagogical front

  • Told you last time about pedaogical experiments 2009-2018.
  • Rajju Ganit text ready.
  • Pune University seems inclined to accept a regular course on calculus without limits.

Science front

Retarded Gravitation Theory v. 2.0

  • \[F = -\frac{kc^3}{(X.U)^2} \left\{ \frac{X}{(X.U)} - \frac{V}{(V.U)} \right\} . \]
  • \(F\) = four force,
  • \(X = (ct_r, \vec{w}(t_r))\) = the retarded position of the "attracting body",
  • \(V = \gamma_v (c, \vec{v}(t_r))\) retarded velocity 4-vector at retarded time \(t_r\),
  • \(U\) is the 4-velocity of the "attracted body"

Difference from Newtonian gravitation

Important issue

  • RGT 2.0 definitely gives an extra \(3\frac{v-u}{c}\) radial acceleration over Newtonian gravitation.
  • Too large to be explained also by GRT (general relativity theory).
  • (RGT 1.0 gave only smaller \(\frac{v^2}{c^2}\) term also given by GRT.)