Rajju gaṇita

Rajju gaṇita (string geometry) vs “Euclidean” geometry

C. K. Raju

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

Introduction

Indian string geometry

Found in the śūlba sūtra −800 CE (śūlba=string, rajju=rope)
e.g. “Pythagorean” proposition in Manava 10.10
explicitly uses square roots
needed for actual calculation of diagonal (कर्ण) from sides of a square.

More details in “Black thoughts matter: Decolonized math, academic censorship and the Pythagorean proposition”

Why is \(\sqrt 2\) deaf?

Square roots NOT known to early Greeks
or to Europeans until Toledo (12th c.), Fibonacci (13th c.)

Why is \(\sqrt 2\) deaf? (contd.)

Hence \(\sqrt 2\) = diagonal (कर्ण) of unit square
called “surd” from Latin surdus = deaf (OED)
deaf = silly Toledo translation of “bad कर्ण” as “bad ear”
since कर्ण also means ear!🤣

Message is clear

The West has long claimed false “supremacy” in math
while dodging an open discussion on its history and philosophy for 25 years.
So, now we will laugh!

1.4 billion Indians
+ 1.4 billion Chinese
see “Aristotle: the laughable intellectual superman”

Religious roots of Western math

Egyptians too had a practical (string) geometry.
But they also had a secretive RELIGIOUS “mystery” geometry (mystery=soul-arousing).
And the Greeks copied it, like they copied their gods from Egypt, e.g. Serapis= Ausar+ Apis
just as they also copied their number names from Persian, Sanskrit
to pay tax to their Persian conquerors (tax collector scene from Darius vase −300 CE).

(Because Greeks were the external proletariat to BOTH Persia and Egypt.)
Pythagoreans and “Neoplatonists”(=sūfī-s) followed this Egyptian secretive religious tradition
of mystery geometry which Plato wrongfully made public.
Let’s see the consequences.

Math and soul

Plato in Meno connected mathematics to the soul or mathesis = learning
= arousing the soul to recollect (eternal) knowledge acquired in its previous lives
as Socrates demonstrates by eliciting the untutored slave boy’s innate knowledge of geometry.
Read the full story in my book Euclid and Jesus

Plato/Socrates in Republic VII.527 reasserts this connection of math to soul
hence says math must be compulsory part of education in the Republic
EXPLICITLY not for any practical (utilitarian) value of math
but because math and music, by arousing the soul, make people virtuous.

G. H. Hardy in A Mathematicians’s Apology reinterpreted soul arousal as aesthetics
since, like Plato, he thought the mathematics of his time had little practical value.
Hardy did not explain aesthetics: the fact that millions of school children love music
but hate (current, axiomatic) math because they find no aesthetics in it.

Gaṇita

In contrast, the Indian approach to gaṇita was entirely practical
this can be seen from Mahavira's eulogy
of the variety of PRACTICAL APPLICATIONS of gaṇita.
So, fundamental difference: Indian practical gaṇita vs Western religious math.

BTW

I have NO objection to Plato’s notion of soul
≈ Hindu notion of soul
or “soul arousal” and resulting virtue.

Similar technique in Raj Yoga
or more elementary haṭha yoga āsana of mūrcha.
Aim: to block external influences (by concentration or physically)
to drive mind inward to “arouse soul”.

Crusading intervention

However, openly linking math to soul doomed math.
The (Christian) state-church objected to this “pagan” notion of soul,
also found in early (pre-Nicene) Christianity.
Why? Because this notion of soul is equitable (“slave boy too has a soul”).

Curse on “pagan” notion of soul

Church wanted special privileges for Christians
and to hurt others (e.g. Dante and Paigambar Mohammed).
Hence, the church pronounced its great curse on that “pagan” notion of soul in 6th c.

Start of inequity

Earlier, church mobs smashed all “pagan” temples
in the Roman empire
and lynched Hypatia the real author of “Euclid” book.

Justinian 532 CE shut down all schools of philosophy and math in Roman empire.

During the Crusades church invented axiomatic proof

as part of its Christian rational theology
(as distinct from Islamic rational theology = aql-i-kalām)
De-linked math from soul (“made it soul-less”)
Claimed the function of geometry/math was aligned to use reason to provide metaphysical (axiomatic) “irrefragable” proofs

which proofs church then politically needed to persuade Muslims.
French translation more explicit: this change about dominating the world (with lies)

Axiomatic math “soul-less”

unlike “Platonic” math, hence children hate current math.
Axiomatic math depends on authority
- which axioms to use
- which theorems are valuable etc.
Hence, produces very UN-virtuous people like Atiyah.

Are you complicit?

If you acknowledge this church intervention in philosophy of math
very many people will immediate reject axiomatic math,
If you don’t, you ultimately support inequity.
You are complicit in the resulting trickery and violence of inequity:
genocide in 3 continents, slavery, racism, colonialism.

Aquinas and angels: why church promoted axiomatic proof

Aquinas used this axiomatic method to prove many angels can fit on a pin
since no facts about angels,
so assume whatever needed
to deduce whatever you WANT to “prove”!

Church “reinterpreted” “Euclid” as support for this trick

Axiomatic proof (prohibiting facts) was declared to be the intention of the Elements
a book first brought to Europe as a crusading trophy (by Adelard of Bath, a crusading spy) before Toledo translations, 1125).

No axiomatic proofs in “Euclid”

MYTH of “Euclid” and his “irrefragable” “axiomatic” proofs is central to current Western math.
Prima facie obviously false: the "Euclid" book has lots of diagrams.
diagrams used also by Socrates in Meno. Why
Because figures aid learning, i.e., help arouse the soul.

But figures IRRELEVANT to axiomatic proof,
may hinder it as pointed out by Bertrand Russell ("Teaching of Euclid")
Hence, “Euclid” book aligned to Neoplatonic religious geometry
NOT miraculously aligned to political requirements of Crusading church, 14 centuries later.

The “Euclid” book actually has NO axiomatic proofs

which use only deductive reasoning from axioms,
but prohibit the empirical e.g. Indian class IX text p. 301),
on the church SUPERSTITION that deduced conclusions are infallible truths (whether absolute or relative to axioms AND logic).

Fact 1: there are no axiomatic proofs in "Euclid"

Prop. 1 of Elements gives an empirical proof.
Prop. 4 (SAS) also gives empirical proof (putting one triangle on top of another)
on which proof of prop. 47 (“Pythagorean theorem”) depends.
Hence, claim of axiomatic proofs in "Euclid" a TOTAL lie: not a SINGLE axiomatic proof of any prop. in Elements.

But myth of axiomatic proofs in "Euclid" the belief in Cambridge Univ. until end 19th c.
Empirical proofs can be in any order, e.g "Pythagorean" prop. is FIRST prop. in Yuktibhāṣa, not last.
In axiomatic proofs order of props. is important
Hence, Cambridge U.(1888) superstitiously made the order of props in "Euclid" mandatory for its exams.
To prove a prop. you cannot use a prop. which comes later in “Euclid”.

But this lie about axiomatic proofs in "Euclid" (even in prop. 1)

which the best in the West took 750 years to understand
is NOW acknowledged by West for over a century.
Proof of prop 1 axiomatised by Dedekind real numbers (1873) (used Cantorian set theory, full of paradoxes).
Set theory axiomatised in 1930's.

Attempts to axiomatise “Euclid” book have failed miserably

Hilbert's synthetic axiomatisation (1899) does not fit: it changes “Euclid”,
since it does NOT DEFINE "equal" length
has problem with meaning of EQUALITY of non-congruent areas
Prop. 35: parallelograms on the same base and between the same parallels are EQUAL.

Note the original word is EQUAL (from political equity)
NOT congruence, now used to banish and replace equality.

Since synthetic geometry does not define length,
usually fined by empirical superposition of ruler over line segment.
it has a problem to define areas.
But, of course, everything possible in formal math.

Birkhoff

Birkhoff's (1932)metric axiomatisation also FAILS,
it trivializes proof of “Pythagorean theorem”
does not require 46 intermediate propositions.
but requires Dedekind reals and axiomatic set theory!

Thus no axiomatization of “Euclid” possible to date (and no axiomatic proofs in book)
Current myth: “Euclid” was wrong, but had right “intent”!
Does Wightman’s failure to axiomatise quantum field theory prove it wrong?

US School Mathematics Study Group 1961 chose Birkhoff, after Sputnik “crisis”.
BUT, metric axiomatisation NOT the same as actual (empirical) ruler and compass
which is what you are forced to teach
and is the takeaway, apart from myths.

Big Lie 2:

Also taught e.g. in current Indian class IX text (Chip. 5)
that all OTHER mathematical traditions lacked a notion of proof,
and especially lacked the use of deductive reasoning for prooḟ.

This is completely FALSE for gaṇita

Means of proof explicitly stated in Nyaya sutra 2
and elaboration.

Gaṇita vs math

So, Indian ganita used empirical + deduction, like science.
This comparison makes clear the real difference: it is prohibition of the empirical in axiomatic proof
not “the use of deduction”.

Apart from church myth there are superstitions

Axiomatic math prohibits empirical proof
on the grounds that empirical is fallible.
But fallacious that prohibiting the empirical makes proof infallible.
Why not also make science infallible?😀

Deduction is fallible, MORE fallible than induction. Why?

because the VALIDITY of a complex deductive proof can only be checked inductively
or accepted on authority (more fallible than empirical)
But axiomatic method has the POLITICAL advantage that it puts Western axiom-makers in control of mathematical knowledge
(e.g. asserting that Dedekind reals are essential for calculus).

Instrumental difference

Western geometry based on the straight line as basic
but no straight lines in real world.
Not on curved surface of earth, not on curved sky, or curved space.
Rajju ganita uses a flexible string, hence has curved lines as basic.
But string can be stretched to measure straight lines

Angle

In Western geometry, angle is something (what thing?) made by two straight-lines (rays) from a common point.
In rajju gaṇita it is the relative length of an arc of a circle (चाप))
Relative to the circumference it is in degrees.
Relative to the radius it is in radians.

Children accustomed to a protractor don't understand the last bit.
But angles can be greater than 360° or negative.

String vs geometry box

Flexible string needed to measure curved arc
not provided in current geometry box.
String can replace ALL instruments in a compass box

To draw a D in a soccer or hockey field one uses the string/rope method, not the compass.
To measure a non-straight boundary (common case in agricultural fields) one uses a rope.
+ draw ellipses

2-string kamal vs hack sextant

Geometry box instruments also unsuited to practical task
to measure angles in SPACE, such as the angle between the moon and the sun.
Needed for the scientific definition of a tithi (तिथि)
to teach the Indian calendar.

A tithi is the tine in which the moon moves ahead of the sun by 12°
Hence, a (“synodic”) month is ALWAYS 30 tithis (12° × 30 = 360°)
= cycle of lunar phases
not some 28,29,30, or 31 days as in the UNSCIENTIFIC Gregorian (Christian) calendar.

Angles in space were measured using the 2-stringed kamal
used HARMONIC scale to measure the angular elevation of the pole star
to determine local latitude,
as done by the navigator used by Vasco to discover India

Two-scale (“Vernier”) principle for HARMONIC scales

used by the kamal/rapalagai for accurate
angle measurements (10’)
not understood to date in West.
Hope to include it in the rajju gaṇita kit.

Latitude and Longitude

Measuring latitude and longitude
and size of the earth
included in Rajju ganita course

Stock Indian calendrical calculations, traditionally done for the meridian of Ujjaini
(copied by the meridian of Greenwich),
must be calibrated for the local latitude and longitude

determining which require precise measurement of various angles
such as the angle subtended at the eye by a mountain,
or the angle of dip of the horizon.

The Indian calendar is needed to determine most Indian festivals
(including Buddhist, Jain, and Sikh festivals, not just Hindu festivals)
as opposed to imposing the inferior, unscientific and alienating Christian (Gregorian) calendar.

The immediate question here is of the mathematical pre-requisites needed to teach the Indian calendar
which need to be included in the primary school curriculum.

However, the current geometry box has no instrument such as a flexible string,
or flexible tape needed to measure the length of a curved line,
and directly determine the angle in radians, say.
The axiomatic definition of the length of a curve is very complex.

“Rajju ganita” geometry course

Course has been experimentally taught, to both students and teachers,
in various groups of oIndian schools in several Indian states.
Nasik, Chamrajnagar, Gundlupete, Indore

Media reports
A draft text, contents
and a draft teaching manual are available.

The attempt now is to incorporate the various mathematical pre-requisites
in a structured way into the primary school curriculum,
to enable also the teaching of the traditional Indian calendar to children.

Teaching only the Christian calendar, as done in colonial education,
deliberately alienates people from their culture
to link them to Christian culture.
We wish to undo this.

Further, due to persistent Western arithmetic incompetence
That Christian (Gregorian) calendar has a crude and unscientific definition of months,
delinked from the lunar cycle,

It still uses leap years to avoid fractions
hence gets the tropical year right only on a thousand-year average.

Worst of all, it lacks a way to determine the rainy season
still so critical to the Indian economy.

Wrong Western idea that only d seasons determined only by tropical year.
Tropical year determines ONLY heat balance
or hot and cod seasons.
But rainy season critical for all life in India.

For rainy season we need moisture balance
determined by the wind regime.
Heat balance does not decide wind regime.

Main driver of periodic winds are tidal forces in the atmosphere
tidal forces linked to the phases of the moon,
tracked by Indian calendar which calculates the so-called synodic month
and syncs it with the annual solar cycle.