Gaṇita (गणित) vs math

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

Personal Introduction: my lessons in math

Physics was my first love. After an honours degree in physics,
I switched to math M.Sc. to better understand physics.
Became a UGC JRF at Centre for Advanced Study, in math in the univ,
was banished to TIFR for graduate course work by my math dept head S. S. Shrikhande

who told me to not "show my face in the Univ"
He a thorough gentleman, who thought it best for me that I should stay in TIFR
He organized a place for me to sit and study in TIFR.
He meant it quite literally, but I disobeyed.
One day I bumped into him in the univ. canteen and Shrikhande asked, "why are you here? Didn’t I tell you not to show your face in the Univ.?"

Anyway, I hated the completely useless math courses they taught in TIFR,
since my aim in doing math was to USE it for physics. But learnt two lessons
Lesson 1. Most math is useless.
Lesson 2. Most advanced level axiomatic math is ugly and repulsive.

Left Mumbai within six months,

Joined IIT Delhi for PhD in math,
Left after attending one class, and asking a question which was not answered.
Lesson3: most of the mathematicians in this country are incompetent.

After a PhD from ISI,

I made a strange career choice:
Did not go abroad or to an elite (well funded) institution in India like all others colleagues who did PhD from ISI.
Joined state level Pune university.
Taught real analysis, and advanced functional analysis especially Schwartz distributions.

In Real Analysis I taught that a differentiable function must be continuous
hence a discontinuous function cannot be differentiated.
In the functional analysis course I taught that the discontinuous Heaviside function can be differentiated
infinitely many times and its derivatives are the Dirac delta function and its derivatives.

Lesson 4: Math theorems have little to do with truth, as Russell agreed.
Formal math is metaphysics, completely subjective,
which can be twisted to conclude almost anything one wants
based on community opinion and popularity.

Many of our incompetent mathematicians don't even know there are multiple definitions of the derivative
in my talk at IIT BHU one incompetent professor of mathematics walked out when I said a discontinuous function can be differentiated,
No one knows which definition of the derivative is needed for the differential equations of physics or
why both the real analysis definition and the Schwartz definition of derivative fail in physics.
Won’t go into that. I don’t think any mathematician or physicist in the DU understand that.

Later joined C-DAC

my job was to implement applications of national importance on the target machine.
Formal mathematicians coveted the funding we had (a million dollars a month)
but were unable to contributing to a solution.

Lesson 6: real life problems have much simpler solutions and the theorems published by formal mathematicians are of no use.
those simple techniques work even when there are no theorems
e.g. stochastic differential equations driven by Levy motion (used in finance)
my last demonstration problem in C-DAC

PHISPC (Project of History of Indian Science, Philosophy, and Culture)

Some influential intellectuals decided we should tell our own stories
And not be bound by stories about us told by others. I was part of the initial group.
I discovered that much of the current history of math (especially calculus) is bunkum.
Here is my 500 page volume on that. Let us discuss it today.

Also discovered that many lies are told about the philosophy of math

No axiomatic proof in geometry in "Euclid" book or any early Greek work
As falsely taught by our class IX school text.
But axiomatic proof found in church theology of reason from the Crusades (Aquinas)
hence prohibits facts or observations which are all contrary to church dogma.

Both these points a key to discussion today

but please stick to the point
produce an axiomatic proof in the Euclid book
and explain why Hilbert's 1899 book giving axiomatic proofs was nonsense and not needed.
Show that the church did not use axiomatic proofs in its theology of reason.

The central question

Q. What should we teach? Ganita or math?
NCERT teaches gaṇita (गणित) is the Hindi translation of mathematics.
So my first point: gaṇita (गणित) ≠ math

What is the difference?

Gaṇita accepts empirical proof,
mathematics prohibits it.
गणित में प्रत्यक्ष प्रमाण मान्य है,
मैथमेटिक्स में वर्जित है

How do we know?

ALL Indian philosophy accepts प्रत्यक्ष (= empirically manifest) as the first means of proof
e.g., also in ganita: the Nyaya sutra 2, as elucidated here

But mathematics prohibits the empirical

Our NCERT text lies: "Greeks alone used reason".

But reason or inference accepted by all schools of Indian philosophy (except Lokayata)
e.g (Nyaya sutra 2), gaṇita also uses reason as in inference or anumana
as used e.g. by Aryabhata to infer that the earth is round (like a kadamba flower)]
because, as Lalla explains (20.36), far-off trees cannot be seen.

So reason WAS used in Indian ganita
but it used reason starting from empirical observation or facts (not axioms)
foolish caricature that acceptance of empirical means rejection of reason
Ganita, like science, accepts BOTH empirical and reasoning.

Our class IX school text deliberately misleads:

uses one word "reason" with two very different meanings. (Common church trick.)
(1) scientific reason =reason PLUS facts, as in ganita (inference from observation, fact)
(2) the religious reason = reason MINUS facts as in axiomatic mathematics(inference from axiom= postulate= assumption)

so, let us get back to the central question

Q. What should we teach? Ganita or math?
Note: I am not saying and have never said "teach ganita because it is ours"
(that would limit it to our national boundaries)
I have said and am saying "teach ganita because it is better and mathematics is inferior"

We teach math mostly for science

science accepts empirical like ganita
So, please explain:
Why is religious reasoning (axiomatic reasoning = reason MINUS empirical) better for math/science
than scientific reasoning (reason PLUS empirical)?

The claim of Western superiority

Central to colonial teaching: APE the West
since the West is superior.

As explained in my Tübingen-Pretoria keynote and related article

this claim of Western superiority or Western supremacy during colonialism
is a direct mutation of earlier claim of White supremacy (used to justify slavery)
which itself is a mutation of the earlier fanatical claim of Christian supremacy (used to justify genocide)
all three claims of supremacy were/are justified by the same false history of science with a minor change of labels.

But math West was backward and INFERIOR even in elementary arithmetic

First, the very NAMES of (small) Greek and Roman numerals copied from Sanskrit from India
but they were BACKWARD since their counting stopped at (myriad (10k)
connotes infinite in English
compared to parardha (= trillion) in Yajurveda 17.2 and तल्ल्क्षण (=$10^{53}$ etc. in ललित विस्तर: (chp. 12)

After a few thousand years, the West woke up to its inferior arithmetic

And started importing Indian ganita ("Arabic numerals")
To replace its primitive arithmetic ("Roman numerals").
Let us be clear: my point is not just that the West learnt from us
but that it had immense difficulties in UNDERSTANDING even elementary Indian arithmetic.

Gerbert (10th c. later Pope Sylvester II)

Imported Indian arithmetic from Cordova (Umayyad Khilafat)
(Where it had diffused from 9th c. al Khwarizmi of Baghdad who wrote Hisab al Hind in the Abbasid Khilafat)
However, Gerbert in 976 foolishly got an abacus constructed for "Arabic numerals"
He FAILED TO UNDERSTAND that algorithms of Indian ganita are very efficient
And this efficiency is destroyed by an abacus.

Fibonacci (13th c.) a Florentine merchant again imported Indian ganita from Africa

He understood that efficiency of Indian arithmetic
provided a comparative advantage for commerce.
But Florentines FAILED TO UNDERSTAND zero
clear from the very term zero from cipher (from Arabic sifr),

cipher means mysterious code. (So Europeans FAILED TO UNDERSTAND zero.)
Why mysterious? Roman numerals additive: xxii = 10+10+1+1
but in place-value system 10 ≠ 1+0=1.
Florentines also passed a law against zero and 1299.

Europeans FAILED TO UNDERSTAND fractions

No way to express a general fraction in arithmetic with Roman numerals
Lack of fractions that you inferior science: the Roman calendar was inferior even after the Julian reform
This primitive calendar said every fourth year is a leap year
this obviously defective calendar was adopted as the official Christian calendar in the first Nicene Council
To determine the date of Easter, then the major Christian festival

Due to the defect of the calendar, the date of Easter continuously slipped
Eventually, in 1582 the calendar was again reformed
Gregorian reform used inputs from India

But, Gregorian reform of 1582 still used leap years
saying "every fourth year is a leap year, except every hundredth year which is not a leap year, except every thousandth year which is"
instead of saying year = 365.241 days.

– Bad math led to bad science

Hence, reformed calendar still defective:
right only on a 1000 year average BUT
equinox does not come on the same day every year.

But colonised fanatic Meghnad Saha

was convinced that anything Western is superior
hence that inferior calendar is today our national calendar
And our two secular festivals Independence Day, and Republic day are defined
only on that Christian calendar (and our secularists do not object).

West FAILED TO UNDERSTAND negative numbers

E.g. De Morgan a very influential professor from University College London
foolishly declared negative numbers impossible. (Elements of Algebra: Preliminary to the Differential Calculus 1837, p. xi.
he failed to understand ordering of negative numbers
and went on to say (1898) belief in witches 10000 times more possible than $- 9 < 0$. 🤣🤣🤣

Interim summary

West imported arithmetic from India
had great difficulties and took a long time to in understand

(0} large numbers
(1) efficiency of Indian algorithms vs inefficiency of abacus or pebble arithmetic
(2) place value system and zero
(3) general fractions
(4) negative numbers

Relevance to math education

based on the principle "phylogeny is ontogeny"
classroom teaching reproduces in fast-forward mode the historical evolution of the subject
colonial education in arithmetic reproduces the European historical difficulties with arithmetic

these difficulties are not present in Indian Ganita texts
E.g. all Indian ganita texts begin with place value system and large numbers
negative numbers described as ऋण (debt) etc.
a new curriculum for primary schools has been made
but is yet to be tried out.

Geometry

Europeans did NOT take any geometry from India
they took religious geometry from Egypt.
Hence their teaching of geometry involved mystical notions
such as invisible points, invisibly thin lines etc.
in accord with the core principle of math that the empirical must be prohibited

Which are extremely puzzling to students when first encountered
because of the prohibition of the empirical
Further, colonial teaching uses the compass box or geometry box
which suggests to the student that geometry is about straight lines
and to be done on paper and not in real life.

Rajju ganita

In contrast, Indian geometry was based on the string (रज्जु, शुल्ब )
angle was defined differently as चाप not कोण
as the relative length of a curved arc
(cannot be measured using a compass box, hence students have difficulty with radians)

This is useful for real-life measurement such as
calculating the area of an agricultural field with curved or jagged boundaries
or to teach the Indian calendar
notion of तिथि involves measuring real life angle between sun and moon
Not on paper or with a protractor

A textbook is ready

and the course has been very successfully tried out with various schools (students and teachers) across India
Nasik, Indore, Gundlupete, Chamrajnagar etc.

Major problem is the conflict with Euclidean geometry and compass box geometry currently taught
e.g. equality NOT congruence,
angle as curved arc not pair of straight lines etc.
"Pythagorean" CALCULATION, not theorem
inexactitude NOT exactitude etc.

Current plan is to manage this conflict by teaching both
along with our view that the Western method is inferior

Trigonometry and calculus

Europeans copied and failed to fully understand

West learnt trigonometry from India as ganita.
Failed to fully understand, hence learnt badly.
but colonized are convinced they must ape the "superior" West

Pocket trigonometry

Word "Sine" from sinus=fold from Arabic jaib (जेब) = pocket (OED).
What has trigonometry to do with POCKETS?
From Sanskrit term for it ardh-jyā
(half-chord) or जीवा
rendered in Arabic as jībā (no v sound in Arabic).

Toledo translations ca. 1125

Written as consonantal skeleton "jb" (without nukta-s) like "pls" in SMS.
Misread by Mozharab/Jew 12th c. Toledo mass translators as common word "jaib" = जेब = pocket.😊
The word sine is proof that trigonometry went from India to Europe

Where it was not correctly understood.
but we are convinced that we must ape the "superior" West and its foolish mistakes
Hence, we still use that translation mistake "sine" .

Word "trigonometry" involves a conceptual error: it is about circles not triangles
as wrongly taught by NCERT
Hence, students confused by my pre-test question what is $\sin 92^∘$?
(In a right-angled triangle there cannot be any angle of $92^∘$.)

The other question I ask is

what is $\sin 1^∘$?
Our students can't answer. They are 1600 years behind the times
because they do Western math, not ganita.
Values like $sin 1^∘ $ required to calculate the radius of the earth.

which is easily done using trigonometry
and which estimate of earth's radius is found in many Indian ganita texts

Because the West did not properly understand trigonometry

West was unable to calculate the radius of the earth correctly
(Columbus' estimate was off by 40%)
This led to the famous longitude problem specific to European navigation because
as Brahmagupta said "ignorance of earth's radius makes longitude calculations futile"

Recall that the longitude problem was the major scientific challenge facing Europe from 15th to 18th century
and that as late as 1712 British Parliament passed an act setting up the board of longitude
to administer a prize of UKP 20,000 for a method of determining longitude at sea.

On the central principle of colonial teaching "ape the West"
and on the principle of phylogeny is ontogeny
our students stay ignorant of this until today
Calculating $\sin 1^∘$ is the first step towards calculus,
as shown by Āryabhaṭa.

Āryabhaṭa's sine table

5th c. Aryabhaṭa's "sine table" has only sine DIFFERENCES
why differences? Because differences (or change) are what you need to calculate any sine value such as $\sin 1^∘$
using the elementary rule of three (त्रैराशिक, for "unit rate of change")

Note: Aryabhata uses only finite differences

not Newton's silly fluxions (= "derivative")
which Marx rightly called mystical (before the invention of real numbers by Dedekind)]
No need for ∫ sign either since Aryabhata numerically solved differential equations
using what is falsely called "Euler's method".

Theft of calculus

900 years later Madhava gave same 24 more sine values to greater precision.
Jesuits based in Cochin stole and translated the related texts in the 16th c.
to solve the European navigational problem (latitude, loxodrome, longitude)
All of which needed precise trigonometric values.

were derived in India (14th c.) using infinite series for sin and cosine and arctangent functions.

Summing infinite series

Indians had summed infinite geometric series by 15th c.
(finite geometric series known since ancient time, e.g. Egyptian "Eye of Horus" fraction).
but Europeans/colonized have not understood until today how this was done

In 17th c. Newton and Leibniz claimed discovery of these series
On the excessively evil and hugely genocidal doctrine of Christian discovery,
which says that ownership of any piece of land or knowledge taken from non-Christians
Belongs to the first Christian to sight it

Independent rediscovery is ruled out

By my epistemic test that
knowledge thieves, like students who cheat in a test, fail to fully understand
what they steal.

Summing infinite series: what Europeans FAILED TO UNDERSTAND

To reiterate, Madhava's precise trigonometric values were derived using infinite series
such as the infinite sine series falsely claimed to have been discovered by Newton
or infinite series for π claimed by Leibniz.
Newton, a Christian fanatic, was invested in slavery of non-Christians,
and cannot be expected to have had ANY moral compunctions about stealing knowledge from them.

The West recognized its failure to understand infinite sums

real numbers were hence invented by Dedekind (end 19th century
and the axiomatic set theory needed for that in the 20th century

Real numbers and limits

Today all calculus texts, including Indian class XI school texts, teach that limits are essential for calculus.
Indeed, infinite sums, derivatives, integrals, and even functions all understood as "limits".
But limits need "real" numbers: e.g. sequence of partial sums of \[ \sqrt 2 = 1.414... = 1+\frac{4}{10} + \frac{1}{100} + ... \]
has NO limit in rational numbers, since $\sqrt 2$ NOT a rational number.

So Newton, Leibniz etc. could not have understood calculus
since "real" numbers came long after them.

But "real" numbers do NOT "work"

Thus, rocket trajectories are today calculated on computers (by both NASA and ISRO)
computers CANNOT use real numbers
because storing even a single real number requires infinite memory not available on the computer
Instead computers use floating point numbers

Floats ≠ reals

Associative law for addition HOLDS for reals
but FAILS for floats:
(-1+1)+ε = ε ≠ 0 = -1 + (1+ ε)
if ε < 1E-8 (or 1E-16 on 64 bit systems). Demo)

Interim summary

Math of "EXACT but अप्रत्यक्ष" UNREAL "real" numbers NEVER used in practice
All practical value from inexact calculation.

Correct way to teach calculus

is Aryabhata's way as numerical solution of differential equations ( what is used in practice).
To sum infinite series use Brahmagupta's अव्यक्त गणित (of polynomials)
which involves what is today called non-Archimedean arithmetic
involving infinities and infinitesimals
absent in so-called "real" numbers which obey Archimedean arithmetic.

And the philosophy of zeroism

or शून्यवाद
to sum infinite geometric series
we discard infinitesimals instead of small numbers.

Course has been taught in 5 universities in 3 countries

I conducted teaching experiments on calculus without limits
with 8 groups (incl. 1 Post Graduate group in 5 universities in 3 countries
Central University of Tibetan Studies, Sarnath
Universiti Sains Malaysia (4 groups, see report part 1, and part 2)

Iran (Cissc Tehran), poster, group photo
Ambedkar University Delhi (poster, group photo, SGT Univ. Delhi (poster, group photo, video

Probability and statistics

Indian origins of probability and statistics as ganita

"Probability in Ancient India", Handbook of Philosophy of Statistics, Elsevier, 2012, pp. 1175-96.
"Probability", Encyclopedia of Non-Western science…, Springer, 2016, pp. 3585–3589.
Probability relates to game of dice.
The first account of the game of dice is in the RgVeda.

अक्ष सूक्त (ऋग्वेद 10.34)

Translation
Mahabharata (Sabha parva)
Shakuni wins the game by deceit
Hence, there was an idea of a "fair (or unbiased) game".

Mahabharata (Van parva 72)
Counting the fruits on a tree by sampling
Permutations and combinations, Binomial theorem etc. part of Indian ganita.
So technique of calculating outcomes in games of chance well known.

The theory of permutations and combinations found in many ancient Indian texts

from the time of the Jain Bhagwati sutra, Susruta, and Varahamihira etc.
E.g. in the common school text ("Slate arithmetic") of Sridhara

West to get from India

claimed independent rediscovery with Pascal etc.
and right to understand it using metaphysics of "real numbers and limits"
as with calculus, but more complex (measure theory, Lebesgue integral, stochastic differential equations etc.)
But this failed miserably with probability

Key point: probability NOT understood with axiomatic math

Frequentist interpretation fails: relative frequency converges to probability only in a probabilistic sense
Subjectivist interpretation fails: quantum probabilities are objective
Measure-theoretic axioms fail: quantum probabilities not defined on boolean logic.

Conclusions

For swaraj in education, teaching ganita is the first step,
to free the colonized mind from indoctrination
into false belief in Christian/White/Western superiority,
to stop aping it, and to view our current education system critically.

Formal mathematicians are the most mentally enslaved,
since formal mathematics is totally dependent on Western authority.
Let them debate publicly, not exercise authority privately.
Mathematics is NOT their monopoly.