Practical gaṇita vs religious math

Practical gaṇita vs religious mathematics

C. K. Raju

Indian Institute of Education

c.k.raju@ganita.guru, ckr@ckraju.net

Summary

1. Gaṇita (गणित) differs from math,
2. it makes math easy, and
3. makes science better.

Key difference

Gaṇita is practical
math is religious
hence unfit to be taught as a compulsory subject in schools.

Pūrva paksha (पूर्व पक्ष)

Many people sarcastically ask:
"Where is the religion in 1+1 = 2?"
They are thinking of the KG method of 1+1 = 2.
That is an empirical proof; nothing religious in it.

KG method of 1+1=2 is gaṇita method

since gaṇita accepts empirical proof= प्रत्यक्ष प्रमाण
accepted by all schools of Indian thought,
as stated e.g. in the Nyaya sutra 2,
and elucidated here or in this video.
So, gaṇita method is practical NOT religious.

But KG/gaṇita method NOT allowed in math

Why not?
Because empirical proofs prohibited in present-day math
=formal math = axiomatic math = Western ethnomathematics

Empirical PROHIBITED in math

People find this hard to believe.
But stated in any stock text on mathematical logic (e.g. Mendelson, introduction to mathematical logic, page 34)
i.e., A mathematical proof is a sequence of statements in which each statement is either an axiom, or is derived from preceding statements by some rule of reasoning
(e.g.,modus ponens 1. \(A \Rightarrow B\), 2. \(A\), ∴ 3. \(B\).)

Also stated in class IX NCERT text(p. 301)

"each statement in a [mathematical] proof has to be established using only logic…Beware of being deceived by what you see …!"
i.e., math allows only अनुमान NOT प्रत्यक्ष.
This text is compulsory reading for all (math compulsory up to class X),
if you didn't read it, it is your problem!

This proves my first preposition: gaṇita is different from math

gaṇita accepts empirical proof
math (as currently taught) rejects it

A consequence: prohibiting the empirical makes 1+1=2 VERY difficult in math

e.g. Russell needed 378 pages in his Principia to prove 1+1=2,
and few people understand a single sentence on that page 378.
This extraordinary difficulty of math adds NO practical value in a grocery shop.

This already proves my second proposition: ganita makes math easy

(without loss of practical value).
But that is not the end of the story…

In axiomatic math no unique concept of number 1

1+1=2 for natural numbers proved using Peano's axioms, but
1+1=2 for "real" numbers requires axioms of set theory.
Very few people (even among professional mathematicians) understand axiomatic set theory
(different from naïve set theory taught in schools to indoctrinate children)

E.g., wrong to define a set as a "collection of objects"
Results in Russell's paradox:
that there is a set R for which the statement R ∈ R is BOTH true and false.

If there is a statement which is both true AND false, \(A ∧ ¬ A\)
then any NONSENSE conclusion \(B\) whatsoever can be MATHEMATICALLY PROVED from it
using the rule of reasoning called modus ponens
since \(A ∧ ¬ A \Rightarrow B\) is a tautology whatever be \(B\) (a contradiction implies any statement).

However, since colonial education (DELIBERATELY) taught students to study only for exam,
most students think it is fine to repeat any statement made in the NCERT school text,
e.g., "a set is a collection of objects"!

JNU prize of ₹10 lakh

To make people take these issues more seriously, I offered a prize of ₹10 lakh in JNU for 1+1=2
(see this video chaired by the then JNU VC, and current chair of UGC)
The prize is to anyone who could meet my Cape Town challenge to prove 1+1 = 2 in REAL numbers from first principles (in the manner of Russell's proof of 1+1=2)
without assuming any theorem of axiomatic set theory and proving everything from axioms.

The case of "real" numbers is important since

real numbers purportedly needed for calculus on current teaching
and calculus needed for all science.
Hence, real numbers taught in class IX and class X to indoctrinate children
without ANY comprehension of the difficulties involved.

Hence, math of calculus so difficult that
California (US technology hub) recently cancelled calculus teaching in schools.
Their aim: to replace it with data science.

Risky strategy: data science needs statistics which needs calculus.
Half understood statistics may create bugs in AI programs
on which the world may depend in future for decision-making.
So, correct strategy is to make calculus easy.

This can be done by teaching calculus as ganita
the way it originated in India
with the 5th c. Aryabhata
as the numerical solution of differential equations

That gaṇita makes calculus easy has been DEMONSTRATED
by teaching experiments over the last 14 years with 8 groups in 5 universities in 3 countries (India, Malaysia, Iran).
Central University of Tibetan Studies, Sarnath (2009)
Universiti Sains Malaysia (2010, 4 groups, see report part 1, and part 2), news report

Teaching experiments and reports (contd)

Ambedkar University Delhi (2013, PG, social sciences), also poster
CISSC, Tehran, 2013, also poster
SGT University Delhi (UG, Science and engineering), 2017, poster.
After my recent talk in IIT Kanpur (or see abstract) this course on calculus as gaṇita will also be taught there from next semester (Feb 2024)

Important point

Teaching calculus as gaṇita enables students to solve HARDER problems
not covered in usual calculus courses

but needed for a correct explanation of the first science experiment in schools

The simple pendulum
so, calculus as ganita leads to better science teaching.

There is NO loss of practical value

since "real" numbers never used in practice.
Why? Because most practical applications of calculus to technology
such as sending a rocket to the moon
done today on COMPUTERS

by numerically solving differential equations, the Aryabhata way
as I learned in C-DAC where my job was
to implement applications of national importance (space, oil etc.) on the target computer.

Computers CAN'T use real numbers

since they have finite memory.
Hence they use what are called floating point numbers
which are so different that they do not even obey the associative law for addition (see Hawai'i talk/paper)
used in ALL axiomatic mathematics, including that of real numbers.
This applies also to all AI programs done on computers.

Interim summary

Ganita makes math easy
enables students to solve harder problems not covered in stock calculus courses
involves no loss of practical value, for applications to advanced technology.
So, why not teach ganita?

GREAT COLONIAL SUPERSTITION

Because of SUPERSTITIOUS FEAR
due to the KEY superstition instilled by colonial/CHURCH education
APE the West or risk catastrophe.
Simple trick: say the West is superior, hence ape it to begone superior.

E.g. class IX school text math (chp. 5) says

we should APE the "Greeks"
who did a math "SUPERIOR" to what ALL others did (Indians, Egyptians, Babylonians, Maya).
Since I am taking a critical view, I asked NCERT
what is the primary evidence for Euclid?

NCERT answers

1. [2007, Hukam Singh, then math head NCERT] "why do you need evidence? We go by a committee!"
2. [2019] "Euclid mentioned in 3 Western math texts. Believe it!"
BUT, sir, history needs PRIMARY sources for which I asked, you are quoting TERTIARY sources.
3. [2019] "Euclid mentioned in 6 Western math texts. Now believe it!"

i.e., "West is superior, hence trust it (as Wikipedia does"
ape it because it says it is superior!
Let us set aside Euclid for whom there is no evidence, and much counter evidence.

"Euclid" book has no axiomatic proofs

Prop. 1 and prop. 4 of "primary source" from 19th c. has empirical proofs.
Proof of prop. 47 ("Pythagorean theorem" depends on this, hence empirical, not axiomatic.
This fact was finally publicly admitted in the 20th century
when Hilbert (1899) rewrote the "Euclid" book to provide the axiomatic proofs missing in it.

so, unlike the Jains who reject claims about intentions which can be easily professed
we are asked to believe in the professed "intentions" of a nonexistent person
which supposed intentions are NOT reflected in the book which he supposedly wrote,

to which Hilbert's rewrite does great violence because the author of the "Euclid" book never had those intentions
as explicitly stated by Proclus etc.
If you can believe the rubbish in the NCERT text about "Euclid", you can believe anything at all.

What is "superior" about "Greek" (=axiomatic) math?

The NCERT textbook tries to establish this claim of "superiority"
by telling a stream of lies in typical church fashion.

Key lie: "Greeks alone used reason".

This is a brazen lie, as we saw (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.
But there are other tricks.

Our class IX school text misleading uses

one word "reason" which has two very different meanings. (Common church trick.)
(1) scientific reason or reason PLUS facts, as in ganita (inference from observation)
(2) the religious reason or reason MINUS facts as in current mathematics.

Most of you probably did not know until today that axiomatic reasoning prohibits facts or anything empirical.
So, most people ARE fooled,
they wrongly assume that by the word "reason", our school text is referring to scientific reason (reason PLUS facts)
whereas it is actually referring to religious reason (reason MINUS facts).

So this deception successfully fools children/people by allowing them to conflate the two meanings of reason.
This is class IX mathematics that we are talking about.
Can't just blame the NCERT.
If, in a nation of 1.4 billion people, in 2 centuries, no one else cross-checked or objected to this way of deluding our children
we are collectively responsible, and our independence is forfeit.

The myth of superiority (a quick detour)

Right from the 4th c. when the church first married the Roman state
asserting Christian superiority became a key part of church dogma
and its revised doctrine of the soul
against the earlier belief in equity.

By the 5th c.the church was already using a secular argument from false history (Orosius) to buttress this claim of superiority.
During the Crusades this technique of using false history false history went viral
with all knowledge from Arabic texts from the Muslim enemy being attributed to early Greeks
regarded as the sole friends of Christians.

This false history made that knowledge in Arabic texts theologically correct,
so it would be used as texts in the first Christian universities
such as Oxford Cambridge, Paris all set up by the church during the Crusades
(to encourage knowledge absorption needed to win the Crusades).

Christian supremacy

In the 14th and 15th centuries this claim of Christian supremacy
was used as the moral justification and imperative for genocide (of 100 million native inhabitants) in the American continent (Bull Inter Caetera)
AND for the slavery of Africans and other non-Christians (Bull Romanus Pontifex)

From Christian to White supremacy

After 2 centuries of the organized and brutal slave trade by Christians,
many Africans converted to Christianity.
Therefore, this "moral" directive ("enslave non-Christians") failed.
Christian superiority was replaced by superstition of White superiority using the superstitious curse of Ham/Kam

From White to Western supremacy

When colonialism replaced slavery as the major means of Western wealth
The Aryan race conjecture was invented as critically useful to rule Indians by dividing then into North (Aryans) and South (Dravidians)
as we are witnessing today.
On this fantasy Whites had earlier conquered and populated India.

Since White supremacy could no longer be used,

it mutated to Western supremacy, as e.g. used by Macaulay.

Same false history of science reused

The SAME false history of science, erected during the Crusades, continued to be used as a secular argument for these assertions of Christian/White/Western superiority.
There was only a change of labels, the early Greeks instead of being called friends of Christians were later called Whites or Western,
But the same chauvinistic false history continued: that all science was the work of "Christians and friends", or Whites, or West.

The colonised fanatically trust the church/West (and mistrust the non-West)
hence never crosscheck the false history used to change our education system and mentally enslave us.
At any rate that is the NCERT's stated policy: that students must trust any false history of science stated by Westerners.

This claim of White/Western origin of science is blatantly false

science needs mathematics
And the fact is that the West was far behind India in mathematics for millennia.
THEREFORE, the West imported ALMOST all aspects of school mathematics from India
arithmetic, algebra, trigonometry, calculus, probability and statistics

(The sole exception is geometry to which I will return later.)
Let us start with the case of arithmetic.

Arithmetic

Well known that Europeans themselves abandoned their INFERIOR system of arithmetic ("Roman numerals")
and replaced it with what they called "Arabic numerals"
or algorismus, since based on the 9th c. text "Hisab al Hind" or Indian arithmetic of al Khwarizmi (of Baghdad, Beyt al Hikma).
Algorismus or Algorithmus the Latinized name of al Khwarizmi.

Key question: WHY did Europeans abandon their native arithmetic?

From here to Manoj Kumar (in Purab aur Paschim) Indians keep singing praises of zero.
But zero, then, was a problem for Europeans not a solution
so it is NOT the answer to "why?"
A question which our historians NEVER asked.

Efficient Indian गणित gave a comparative advantage in commerce

over inefficient native European abacus.
Hence, Fibonacci wrote Liber Abaci (1203) a Latin translation of Hisab al Hind.
He was a Florentine merchant, who traded with Arabs in Africa from whom he got this knowledge
and understood its value for commerce.

Efficient Algorismus vs inefficient Greek/Roman abacus

E.g. native Greek-Roman numerals need 12 symbols (instead of 4) for the 4-digit number 1788=MDCCLXXXVIII
Hence, European pebble arithmetic had difficulties even to represent large numbers
largest Greek-Roman number is a myriad (10k)
connotes infinite in English
compared to parardha (= trillion) in Yajurveda 17.2 and तल्ल्क्षण (=\(10^{53}\) etc. in ललित विस्तर: (chp. 12)

Inferiority of European arithmetic

and their copying from India
stretches back to antiquity
The very NAMES of (small) Greek and Roman numerals copied from Sanskrit from India
Direction of transfer follows knowledge gradient: those who knew only myriad copied from parardha.
But after some 2000 years, by 13th c. they understood that was not enough.

Inefficiency of Greek/Roman pebble arithmetic (coin-counter system)

89+79 requires 18 operations.
\(89 × 79\) requires 1422 operations
compared to 10 operations (4 single digit "multiplies", 4 single digit "adds" and 2 "carries" in गणित
More details in my book on Refutation of Aryan Race Conjecture

The problem with zero

Key point: backward Europeans lacked full comprehension of elementary arithmetic
clear from the very term zero from cipher (from Arabic sifr),
cipher means mysterious code. Why mysterious?
Roman numerals additive: xxii = 10+10+1+1
but in place-value system 10 ≠ 1+0=1.

Suspicion of zero

Adding zero at the end can inflate a contract
(not possible with Roman numerals: only III can be added at the end)
Hence Florence passed a law against zero in 1299.

Roman arithmetic lacked FRACTIONS

Julian calendar hence used leap years
instead of saying the duration of the (tropical) year is \(365 \frac{1}{4}\) day.
Since Roman arithmetic lacked large numbers, it also lacked precise fractions.
Hence, Gregorian reform of 1582 still used leap years instead of saying year = 365.241 days.
Hence, reformed calendar still defective: equinox does not come on the same day every year.

Gregorian reform used inputs from India

Matteo Ricci was the favourite student of
Clavius who headed the reform committee
and wrote a book on practical arithmetic.

De Morgan's folly

E.g. De Morgan a very influential professor from University College London
foolishly declared negative numbers impossible. (Morgan, Augustus de. Elements of Algebra: Preliminary to the Differential Calculus, 2nd ed. London: Taylor and Walton, 1837, p. xi.
and went on to say (1898) belief in witches 10000 times more possible than \(- 9 < 0\). 🤣🤣🤣

"Algebra" from "al Jabr waal muqabala" of al Khwarizmi

who partly translated 7th c. Brahmagupta's unexpressed arithmetic (अव्यक्त गणित)of polynomials
and linear and quadratic equations.
Primitive Greek/Roman arithmetic lacked √.
Term for √2 is SURD from Latin surdus = DEAF from Arabic asumu
- Why is √2 DEAF?

Deaf roots

In Indian शुल्ब सूत्र √2 = DIAGONAL (कर्ण) of unit square.
But word कर्ण also means ear (=कान),
hence bad कर्ण mistranslated as bad ear = deaf! 😊

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

Key point: ALL these cases

of arithmetic, algebra, trigonometry
involved various degrees of incomprehension
by Europeans while copying from Indians
but we have declared the duffer as "superior"
and are playing "follow the leader".

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.

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.

Calculus

Calculus too originated in India in 5th c.
and was stolen by Europeans (Jesuits in Kochi)
in the 16th c.
Why?

Why was calculus stolen?

Indians used calculus to derive precise trigonometric values
In 16th c. Jesuits based in Kochi(Cochin) in Kerala, translated and sent Indian ganita math texts back to Europe.
Why? Hoping to solve the European navigational problem
then the biggest scientific challenge facing Europe.

Which involved the problem of determining latitude, longitude, and loxodromes at sea
all of which required precise trigonometric values
as explained in my school text on Rajju ganita.

Reading list

E.g. my 2000 invited plenary at Univ. of Hawai'i and related article)
and my 2007 book of 500+ pages.
MIT talk, articles in the Springer Encyclopedia, on calculus, calculus transmission
Durban keynote, popular-level articles on California and calculus etc.

Interim summary

Newton and Leibniz did NOT "discover" the calculus, as our school text states:

tḥat is false history

based on the Doctrine of Christian discovery that any piece of land/knowledge belongs to the first Christian to sight it (Bull Inter-Caetera)
as in Vasco "discovered" India.

So what difference does Western theft of calculus make to calculus TEACHING today?

(Today's talk not about history but about teaching ganita vs math.)
Europeans stole calculus because they had an INFERIOR knowledge of it.
Should we ape it?

Unlike other cases of arithmetic, algebra
in the case of calculus, the West admitted its failure to understand it
hence Dedekind invented "real" numbers in late 19th c., 250 years after Newton's death.

Today "real" number are regarded as essential to understand calculus
(but, as we saw, almost no one in the country really understands 1+1=2 in real numbers
and no one has accepted the challenge to prove it)

and no university mathematician is willing to publicly debate why real numbers are needed for calculus
when they are never used in practice
but make calculus so very difficult.

Let us understand the problem

Consider the number \(\sqrt 2\)
The number is a surd.
That means if we apply Aryabhata's 5th c. algorithm (the algorithm you learnt in school) to extract the square root
the process never terminates.

That is, $\sqrt 2 = 1.4142135…$
where the three dots indicate that the process goes on indefinitely.
Another way to see it is that we have an infinite sum \[\sqrt 2 = 1+ 0.4 + 0.01 + 0.004 + ...\] or \[\sqrt 2 = 1+ \frac{4}{10} + \frac{1}{100} + \frac{4}{1000}+ ...\]

This fact about \(\sqrt 2\) was known from the earliest times, in India and
the name for it in the Baudhayan śulba sūtra is सविशेष
meaning with a remainder (अवशेष)
(Apastamba calls it sa anitya = impermanent).
Similar understanding applies to \(π\) for which Āryabhaṭa (Ganita 10) gives an आसन्न (near) value.

That is these early Indian texts adopted a pragmatic attitude
In an infinite series, simply sum a finite number of terms to the required precision.
That is exactly what we still do in practice, for ALL applications of mathematics to science and engineering.
But the Western attitude was different,
Westerners wanted to sum an infinite series exactly, let us understand why.

Mathematics and religion

Plato tells the story of how Socrates visits his friend Meno.
Since Meno is not convinced of Socrates' theory of the soul
Socrates offers to demonstrate it by calling Meno's slave boy
to demonstrate the slave boy's innate knowledge of geometry

Math first connected to religion through mathesis = learning

According to Plato/Socrates all learning is recollection
as in the PRIMARY source, I just showed you.
And Proclus explains that "mathematics" etymologically derives from mathesis.
Proclus further cites Plato's Phaedo to explain that it is ONLY mathematics (and figures) which especially arouses the eternal soul.

Plato/Socrates in Republic VII.527 reasserts this
hence says math must be compulsory part of education in the Republic
because it arouses the soul and that makes people virtuous.

Note the superstitious belief that mathematics contains eternal truths
and the belief in sympathetic magic: "like arouses like".
i.e., the eternal truths of math arouse the eternal soul.

Because of this superstitious belief ("math contains eternal truths"
math had to be exact
infinite series for \(\sqrt 2\) had to be summed exactly
an anitya (=non-eternal) or inexact value as in the Apastamba śulba sūtra would NOT do
though it works fine for ALL practical purposes.

Diagrams/figures

Recall that your school geometry text had numerous diagrams
as also used by Socrates in Meno (because figures aid learning, i.e., arouse the soul.
But figures have NO CONNECTION to axiomatic proof,
as pointed out by Bertrand Russell ("Teaching of Euclid")
To the contrary, Propositions 1, 4 of Euclid give empirical proof using figures, which proofs are hence rejected today.

Russell's superstition

As a student of Cambridge (a church institution)
Russell subscribed to the Cambridge superstition
as we do
that the non-existent "Euclid" INTENDED axiomatic proofs.
When there was no such intention.

The church and its notion of soul

But the Crusading church said so (that the "Euclid" book was about axiomatic proofs). Why?
Because Plato used a "pagan" notion of soul
(Very, very similar to the Hindu notion of ātman.)
This was an equitable notion
Cursed by the state-church (4th c. onward) which was committed to the dogma of Christian supremacy.

Therefore, the church was opposed to the Egyptian/"pagan"/Platonic connection of geometry to the soul.
This idea of equality was expressed through geometry in many ways.
For example, through the idea that souls are like spheres (and all spheres are equal)
Indeed, Justinian's fifth anathema (= great curse) of 532 CE was against this notion of the soul as a sphere

Again, e.g., all prepositions of the original "Euclid" book are about equality of apparently dissimilar objects.
But we are blind followers of the West (as NCERT teaches)
and the West was hegemonised by the church especially during the Crusades, the Inquisition, etc.
Therefore, Hilbert replaced the term equality by the term "congruence" not found anywhere in the original book
but we blindly ape it today for that is the dharma of the colonised.

Church rational theology

Therefore, the church rejected any connection of mathematics to the notion of soul it cursed
Instead it connected the Euclid book to axiomatic proof (= reason MINUS facts)
Because it wanted to compete with the Islamic theology of reason (aql-i-kalam). Why?

Because the church failed to convert Muslims by force the way pagans in Europe had been converted earlier
And Muslims rejected the Bible as corrupted (as did Isaac Newton).
unable to convert either by force or by scripture, the church was forced to accept "universal" reason which Muslims accepted.

But the church made the great innovation of reason MINUS facts
as first used by Aquinas to prove his angel theorem (Summa Theologica, First Part, Q. 52, article 3) .
All university mathematicians today are committed to this church method of reasoning and proof.

Axiomatic mathematics adds no practical value, as we saw
But it has superior POLITICAL value for the West
because the axioms of mathematics are laid down by the West,
And that enables the West to control mathematical knowledge
the way the church controlled its rational theology.

Also, axiomatic method makes simplest things very difficult
and widespread ignorance of math reinforces Western control
on the colonial teaching "trust the West, distrust the non-West".

So that frauds like Stephen Hawking can easily claim that "science has proved the truth of Christianity".
Such creep pf religious beliefs into science through mathematics is impossible with gaṇita which starts from प्रत्यक्ष.
But that is another story. I will stop at this point

Conclusions

We made the strategic BLUNDER of not studying the colonial enemy
And the church-state nexus during colonialism
which looted us for 2 centuries
by telling us lies which we REFUSE to check even 75 years after the British left.
That has made us (and our children) mental slaves.

Macaulay argued that colonial education was needed for science.
And decolonisation must begin from gaṇita/math needed for science,
and from a critical re-examination of its related history AND philosophy.
The PHILOSOPHY of gaṇita vs math is critical to correct present-day math teaching.