IIT: Mandi Colloquium

Teaching calculus as gaṇita

C. K. Raju
c.k.raju@ganita.guru

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

Relevant self intro

Selected, IIT-JEE in 1970 (declined)
Left CAS in Math/TIFR courses to join IIT: Delhi (1976) for PhD in Math.
Attended 2 lectures asked 2 easy questions which went unanswered.(see blog "Idiots and IIT").
One prof candidly said: "so what if I don't know math, don't I have a right to earn my living?" 😲
I left IIT in 3 months (since math faculty knew no math) for PhD from ISI (Delhi/Kolkata).

Taught formal math

In Stats/Math DeptS of Pune University (1981-88) Taught real analysis from Baby Rudin
and Advanced Functional Analysis from Chacha Rudin (Topological Vector Spaces, Schwartz Distributions).
Difference? On baby Rudin a discontinuous function is not differentiable
on chacha Rudin it is INFINITELY differentiable if it is Lebesgue integrable.😀
E.g. derivative of discontinuous Heaviside function is the Dirac δ, 2nd derivative δ′ etc.

A key topic of my research then:

which derivative is BETTER for nonlinear PDEs of physics at a discontinuity such as a shock wave:
college calculus derivative (undefined at a discontinuity)
or the Schwartz derivative
- for which multiplication (e.g. δ.δ etc.) not defined (hence does not work) for nonlinear PDEs
So what fails at a discontinuity: physics or calculus?

But at a lecture in IIT:BHU some years ago,

a professor (sic 😄) walked out saying a discontinuous function cannot be differentiated (see blog, 😠)
He knew ONLY college calculus or baby Rudin, not chacha Rudin. 🤣
From IIT:Delhi (1976) to IIT:BHU(2019), my experience of over 40 years: IIT math faculty has exhibited extreme ignorance of math.
My worry: If the faculty lacks knowledge of basics, how will students ever learn calculus properly?

Anyway I joined C-DAC, 1988, initially headed Applications Development

abandoned formal/axiomatic math as COMPLETELY useless for practical applications of math
which require calculation NOT axiomatic proof
just as in a grocery shop, you need arithmetic for CALCULATION, NOT axiomatic proof.
Bertrand Russell's 378 page axiomatic proof of 1+1=2 has nil practical value
(Its believed epistemic value ("rigor") is mere church SUPERSTITION.)

Wrote articles and book since 1998 on Indian calculus

on its THEFT from India in 16th c. AND
And on the Christianization of math (axiomatic math) through the myth of "Euclid".
Worked on decolonization of math in various countries for last 15 years.
See e.g. video of lecture on calculus at MIT, Cambridge, Mass,
and last year's presentations (1,2) at 9th Pacific Rim Conference on Math in Australia.
Bigger reading list here.

Extended abstract of today's talk (with refs) online

But will deviate from that script. Why?
Useless to talk of advanced things when most don't know basics.😀
So, let us start from fundas and another story.

Minister S&T at Science Congress 2015

repeated an old claim.
"We did Pythagorean theorem FIRST"
Hindu report, Times report

Neta did not read class IX math text, chp. 5, p. 79

which says "many (Egyptians, Babylonians, Indians…) did geometry EARLIER, but Greeks did it BETTER"

Point 1: First or Better?

Saying "We did it FIRST" is a foolish response to the pūrva pakṣa which claims
"many did it earlier, but Greeks did it BETTER"
Many Indians repeatedly make the same mistake: happy just to claim "we did it first".
Why? Deciding which is BETTER requires knowledge of BOTH math and gaṇita which are being compared.

Our neta-s certainly don't have knowledge of math or its history. Do you?
Does your faculty have it?

Which calculus is better?

(1) Indian calculus, or
(2) the calculus you teach/learn in IIT?
Obvious you implicitly BELIEVE (2) is better
that is why you teach it!

Implicitly bcoz for 10 years, mathematicians in India have dodged public debate on it.
Repeated calls for debate: e.g. 2015 ISSA, also on X: J. V. Narlikar, P. Sinclair, Nandita Narain, 2023 Delhi U.
Bcoz mathematicians cannot defend publicly
the SUPERSTITIOUS belief in "rigor" they teach in the classroom.

Which calculus is better (contd)?

My point: Indians did calculus first,
also DIFFERENTLY AND BETTER.

BUT West failed to understand for centuries the different Indian way to do calculus,
or that it was better,
just as West collectively stupid, failed to understand for 9 centuries
even Indian primary-school arithmetic which too it "imported"

West arithmetically backward since early Greek times
first imported Indian arithmetic in 10th c. ("Arabic numerals").
But West was STUPID and failed to GRASP primary-school arithmetic
for 9 centuries till the 19th c.
But the colonised blindly believe Western supremacy from early Greek times.

10th c. Gerbert (pope Sylvester II) accepted that

Graeco-Roman PEBBLE arithmetic is inferior
since their counting stopped at myriad ($10^4$) foolishly regarded as uncountable.🤣
Hence, Gerbert brought Indian-origin "Arabic numerals" to compete with Muslims
but failed to understand EFFICIENCY of Indian algorithms
destroyed by his silly abacus/apices.😄

Fibonacci (13th c.) too blundered:

his Liber Abaci lacks negative numbers (like al Khwarizmi)🤣
Why? Because in primitive Graeco-Roman (pebble) arithmetic
subtraction means removing pebbles from an existing hoard
and you can't remove more pebbles than there are on the counting board!

Western foolishness about Indian arithmetic persisted till 19th c.

Pascal (17th c.), Euler 18th c.), De Morgan (19th c.) 🤣🤣🤣
but the colonized cannot accept these celebrated Western mathematicians were utter fools.

My Q: how could the West have "discovered" calculus centuries before it fully understood primary school arithmetic?
A. Like Vasco [Christian] discovered India, or Columbus [Christian] discovered America!

Macaulie

de Morgan comes after Macaulay
BUT we did not check facts, but blindly accepted Macaulay's lie about
"immeasurable Western supremacy" in math and science
Hence, calculus before arithmetic.

पादरीवाद

Hence we adopted the colonial education system = church education system.)

Our पादरीवादी colonial education system
designed by पादरी-s from mission schools to major Western universities
indoctrinates us into the WOW syndrome (WOW=Worship-Of-West).
E.g. term "Pythagorean theorem repeated 32 times in class X text to emphasize worship of the west.

(More details about पादरीवाद and colonial education in my school-text on Indian calendar.)

WOW-syndrome

So, some go WOW, see the greatness of Pascal : he said $ 0-4 = 0 $!
WOW, see the greatness of Euler: he said $ -1 < 0 $, BUT $ -1 > ∞ $!
WOW, see the greatness of De Morgan: he said $ 10-11 $ is not meaningful!

De Morgan's foolishness went further
He edited and got the East India Company to finance the publication of a book
by an Indian Yesudas [=Jesus slave] Ramchundra
on maxima and minima WITHOUT using calculus or negative numbers

Story NOT over: our supposedly leading historian of math Dhruv Raina in JNU
(a 3rd class in chemistry who knows no math)
did his PhD on Ramchundra, because of the WOW syndrome (personal communication, "de Morgan praised it")
despite my advice that it was nonsense.
Naturally, I and my pet monkey laugh at the WOW fanaticism of the colonially educated!

Anyway, if math faculty in IITs does not even understand (e.g. at IIT:BHU)
different notions of derivative WITHIN axiomatic math
How can THEY debate WHICH calculus (math or gaṇita) is better?
So back to elementary school stuff.

Ganita vs math (for Pyth. proposition)

In 2015, Minister should have said, as I have done
that Indian way of doing Pythagorean calculation (NOT theorem)
as in the Manava (NOT Baudhāyana) śulba sūtra different
and BETTER than anything Greeks COULD do.

Why is gaṇita better (in geometry)?

Because practical applications need calculation NOT axiomatic proof
Manava SS 10.10, unlike Baudhayan, calculates diagonal,
uses square ROOTS. It says:
In a rectangle with sides $a$, $b$, and diagonal $d$, \[d = \sqrt {a² + b²}.\]

Square roots unknown to early Greeks, who were BACKWARD in arithmetic
(square roots first came to Europe with 13th c. Fibonacci, from India via Arabs)
but needed to CALCULATE $d$ from knowledge of $a$, $b$ as required for practical applications.
So, gaṇita better for practical applications.
What about epistemic "rigor"?̣

Point 3: class IX math text brazenly LIES

that only Greeks used reasoning, Indians clearly did from earlier.
Methods of proof in gaṇita DO use reasoning
but as in SCIENCE (= reasoning PLUS empirical)
NOT as in axiomatic proof (= reasoning MINUS empirical.

To deceive students, school text deliberately
uses only one word "reasoning" for both.

Why such a BIG lie at the foundations of math?

To hide that axiomatic proof NOT found among early Greeks OR in "Euclid" book;
but IS found in Crusading church theology.
If you believed/understood that "rigorous" "axiomatic" proof invented for politics of crusading church
would you still use it?

Reasoning/deduction USED for proof in gaṇita

अनुमान in Nyāya sūtra means logical INFERENCE
not conjecture as it means in current Hindi.
(Did Aristotle's logic derive from Nyāya? Skipped.)

"Euclid" book has NO axiomatic proofs

uses empirical proof in its 1st and 4th proposition (SAS)
needed to prove Pyth. proposition.
Superstitious West understood it only after 750 years at the end of 19th c.

Indian gaṇita method of proof BETTER and EASIER:

"Pythagorean" proposition proved in 1 step in e.g. Yuktibhāṣā,
instead of prolix 47 steps as in "Euclid" book
Hence, clearly easier.

To reiterate the key point

the class IX school text HIDES the fact that axiomatic proof originated NOT with Greeks but with Crusading church theology of reason
by falsely suggesting that "Greeks" ("Euclid") gave axiomatic proofs.

Formal mathematicians are corrupt; to save their livelihood ad please West.
They mislead children and defend this lie,
but too scared to meet my public challenge to provide any example of an axiomatic proof in math
in "Euclid" book or prior to Hilbert 1899.

A question

Why is it essential to tell lies to children about the foundations of math?
Lies which cannot be publicly defended?
Govt standard of proof ("this is stated in Western texts so you MUST believe it") NOT acceptable.
(Western univs were church madrases, which spread church propaganda. Reject this rubbish.)

E.g. Cambridge blunder

Cambridge University blundered and believed this church SUPERSTITION for 750 years
(that the "Euclid" book has axiomatic proofs,
In 1887 it introduced an exam regulation to follow order of theorems in "Euclid" book)
just BEFORE it was accepted 125 years ago, that there are NO axiomatic proofs in the "Euclid" book.
Hence, David Hilbert, Bertrand Russell, G. Birkhoff etc., all rewrote the "Euclid" book to provide the "missing axioms and axiomatic proofs

This rewrite fundamentally changed the book
(e.g. Hilbert's synthetic geometry lacks notion of distance/length needed for Pyth. theorem.
since "Euclid" book was a Neoplatonic religious text
never intended axiomatic proof.

Won't explain in more detail. My point today

If you don't understand 9th std. geometry
you cannot possibly UNDERSTAND calculus.
Possible remedy? Take my course in rajju-gaṇita (for 9th std. as, when and if I offer it).

Point 4: scientific vs theological reasoning

In 2015, Minister S&T could also have said, as I say
that gaṇita uses the scientific method of proof (प्रत्यक्ष + अनुमान = empirical + reasoning)
far better than the Christian theological method of axiomatic proof (= reasoning MINUS empirical ).

Point 4 (axiomatic proof prohibits empirical)

To reiterate axiomatic proof = reasoning MINUS empirical
Clarified explicitly in class IX text p. 301: "Beware of what you see".
AND implicitly in standard texts on mathematical logic (Mendelson, p. 34).
Class IX text also explains, axiom=postulate=assumption (axioms ≠ "self-evident truth").

Axiomatic proof an INFERIOR form of proof. Why?

Axiomatic proof does NOT lead to VALID knowledge.
E.g. Rabbit theorem
This is a valid axiomatic proof.
But the conclusion ("theorem") is invalid knowledge, in fact, nonsense.
So, Rabbit theorem shows that axiomatic proof may lead to nonsense conclusions/theorems
unlike proofs in gaṇita (which may only be approximately wrong, like science)

Why did valid axiomatic proof (of Rabbit theorem) lead to nonsense conclusions?

Because axiom 1 is false.
BUT it is EMPIRICALLY false: we can see so many animals right here who don't have two horns.
But if you accept empirical observation and seeing things as a a valid means of prof,
first expunge from the class IX text the statement: "Beware of what you see".
AND change the definition of axiomatic proof.

So, are all axioms in math empirically tested?

No. They CANNOT be.
Take the assertion in class VI text that geometric points are invisible.
How do you test the location of an invisible point?
You can't because invisible points are metaphysics (as suits theology) NOT practical math.

Axiomatic math = metaphysics, axioms can't be empirically tested

Another e.g.: take the axiom that there is a unique (invisible) line through 2 (invisible) points?
How do you empirically determine the location of an INVISIBLE point?
And how do you empirically test the axiom/superstition that every line extends infinitely in both directions?
(Does even the cosmos extend infinitely in all directions?)

E.g. axiom of choice

Or take the axiom of choice for an UNCOUNTABLY infinite collection of sets.
(Can't be empirically tested, since uncountable infinity = pure metaphysics, like church theology.)
or any other "principle of transfinite induction" such as Zorn's lemma, Hausdorff maximality, or even continuum "hypothesis" \[2^ℵ₀ = c = #(ℝ)\]

Since axioms of math are metaphysics

their validity CANNOT be decided by empirical observation.
Validity of axioms can only be decided by authority: शब्द प्रमाण.
Whose authority? The West.
Theorems depend on axioms. So, math knowledge controlled by Western authority.

Prohibiting empirical suited theology since theorems decided purely by authority (शब्द प्रमाण)

In fact, any nonsense proposition whatsoever, e.g. Aquinas' angel theorem, can be and was proved axiomatically in theology.
To prove A just assume A or a B which is logically equivalent to (or implies A) as an axiom.
This greatly suited church theology which aimed to make people believe whatever was politically convenient.
But for us, it shows INFERIORITY of Christian theological axiomatic proof.

Ganita tied to प्रत्यक्ष, hence can't prove "anything you like"

This "prove anything you like" does NOT happen with gaṇita or scientific proof
that a valid proof (of e.g. Rabbit theorem) leads to nonsense conclusions.

Fallibility of empirical

Now it is accepted in Indian tradition that the empirical is fallible, that observation may lead to errors
रज्जु-सर्प न्याय, in bad light you may mistake a rope for a snake or vice versa.
Science too accepts this as in experimental errors.
But neither gaṇita nor science BANS empirical observation as axiomatic proof does.

What the school text ALSO hides: axiomatic proof highly fallible

People OFTEN make mistakes in complex deductive tasks.
Hence, students often make mistakes and flunk in math.
Game of chess is pure deduction
But EVERY human being almost EVERY time makes a mistake hence loses to a machine (mobile phone).
Far MORE frequent that OCCASIONAL mistake in observation (रज्जु सर्प न्याय)

So, "rigorous" = axiomatic proof INFERIOR because

1. Any NONSENSE proposition WHATSOEVER can be axiomatically proved as a THEOREM by selecting convenient axioms (e.g. Rabbit theorem, Angel theorem).
2. Axioms of math are metaphysics: CANNOT be empirically tested (e.g. invisible points, infinite lines, axiom of choice etc.)
3. Axiomatic math makes West the boss.
- IITs dare not even TRY my course "calculus without limits"
- since math faculty scared boss would be angry.

Some people now change tactics

switch from claim of superior epistemology ("rigor")
to claim of superior practical value.

Objection: "they have sent a man to the moon"

FALSE, worthless objection.
Practical value of math does NOT come from the math which is taught.
E.g. practical value of geometry comes from measuring land to pay taxes, or estimate produce.
No one measures the area of a खेत using invisible points and invisible lines.

Floating point vs "real" numbers

Like wise to go the moon requires CALCULATION of rocket trajectories, done on a computer (or by hand)
a computer users floating point numbers
which do NOT obey even the most basic axioms of unreal axiomatic "real" numbers
such as the associative LAW for addition.

So, floating point numbers ≠ "reals"

But "reals" regarded as essential to teach calculus today.
So, calculus as taught today ≠ what is used for sending a man to the moon.

Rocket trajectories calculated using Āryabhaṭa's gaṇita method

of numerically solving ODEs.
as I also teach in my calculus course.
I realized the useless of teaching axiomatic "real" numbers
when I joined C-DAC in 1988 and actually computed rocket trajectories.
(Recall that earlier I used to teach real analysis and advanced functional analysis.)

Are IITians smart enough to see the difference between theory and praxis of calculus?

and knowledgeable enough to decide which is better even for practical value:
the metaphysics of West-dominated formal math
or the practical way of gaṇita?
I have serious doubts: Do you even know the axioms of "real" numbers?

Cape Town/JNU challenge: 1+1=2 test

"Axiomatically prove 1+1=2 in REAL numbers from first principles (WITHOUT assuming any theorem from axiomatic set theory)"
like Bertrand Russell's 378 page proof of 1+1=2 in cardinals.
In JNU I offered Rs 10 lakhs; in IIT, I offer 1 lakh if you turn in the proof by the end of the lecture.
BUT if you try and blunder like the Cape Town mathematician
You agree to leave IIT immediately, and forever.

Colonial education being church education stuffs you with lies from childhood

Lies about "Euclid" and the foundations of math
but also the lie that floating point numbers = real numbers
And now their weapons Wikipedia and Grok feed you numerous lies about history:
they know you are well trained gullible lie consumers
and intellectually too lazy to check facts.

Calculus origin and theft by Europeans

Calculus originated in India with the 5th century dalit Āryabhaṭa
who used finite differences to solve ordinary differential equations
and obtain sine and cosine values accurate to the first sexagesimal minute.

Q. Why no signs for integral and derivative in Indian calculus?

A. Not needed. Calculus about solving differential equations
calculus NOT about evaluating integrals and derivatives (ONLY of elementary functions) as IITians learn.
Don't learn non-elementary (e.g. Elliptic) function e.g. from pre-test for my calculus course.

To numerically solve differential equations finite differences (e.g. खंड्ज्या) enough.
No need for integral sign either: solution of $y'=f(x)$ is $∫f(x)dx.$
Note: on the gaṇita method neither $f$ nor $∫ f$ need be an elementary function

Calculus theft

since theft is a criminal offense
my 2000/2001 keynote/paper,
and my 2007 book
used the standard of evidence in criminal law

viz. proof beyond reasonable doubt,
a standard never used earlier for history.
(History goes by the standard of proof in civil law: balance of probabilities.)

Calculus existed in India from the time of the 5th century dalit Āryabhaṭa

AND 7th century Brahmagupta
centuries before the Kerala school
and a th```ousand years before Newton.

There was

OPPORTUNITY to steal it from the 16th century (e.g. Jesuit college in Cochin)
MOTIVATION (the European navigational problem)
CIRCUMSTANTIAL EVIDENCE (e.g. Fermat's challenge problem a solved exercise in Bhaskara II; numbers large to be and "independent rediscovery")
DOCUMENTARY EVIDENCE (e.g. Matteo Ricci's letter for 1582 calendar reform, needed to determine latitude in daytime).

Epistemic test

Those who steal from others,
like students who cheat in an examination
do not FULLY understand what they steal,
proves "independent rediscovery" FALSE.
Thief Newton failed to FULLY understand the calculus he stole.

That Newton did not fully understand calculus is obvious

Newton used fluxions,
today we reject them and use axiomatic "real" numbers
first proposed by Dedekind after 1872
just because he realized that calculus was not properly understood.
(Actually axiomatizing "real" numbers had to wait till axiomatic set theory of the 1930s.)

Because axiomatic "reals" so difficult

Hence my 1+1=2 challenge
Hence, California recently cancelled the teaching of calculus in schools (1,2)
Don't know what Trump will do!

Earlier critiques

Berkeley: "if you put something to zero at the end of a calculation, why not make it zero at the beginning"
Berkeley: "if you arrive at the right answer without knowing why, that is not science, it could be due to TWO mistakes, one canceling the other"
Karl Marx: "Newton's calculus is mystical" (= incomprehensible).

West extremely annoyed with my claim of calculus theft

summarized at a math conference last year in Australia.
Many newspaper attacks, in India, Soutb Africa etc. BUT
in 25 years not a single published academic response to my evidence.

Wikipedia and Grok etc. lie brazenly.
They KNOW that Indians trained to be gullible, will NEVER check facts.
WOW teaches to blindly "trust the West and distrust the non-West"
(also a Wikipedia principle) on which Grok trains.
they know that Indians foolishly changed their education system without checking the Macaulie in 2 centuries

What exactly did the West fail to understand in Indian calculus?

Failed to understand meaning of infinitesimal
and failed to understand how to sum infinite series.
(Understood how to numerically solve differential equations= Newtonian physics.)

Indian sum of infinite geometric series

Finite geometric series very old
found in Yajurveda 17.2 and
in Egyptian series of fractions called eye of Horus.

but infinite geometric series 1st summed by 16th c. Nīlakanṭh (Kerala school, commentary on Gaṇita 17)
involves Brahmagupta's notion of infinitesimal,
and śūnyavāda/zeroism principle of discarding small numbers or infinitesimals
śulba-sūtra: सविशेष (with an अवशेष for $\sqrt 2$), Āryabhaṭa आसन्न (near value for π etc.)
on the gaṇita philosophy that math is about APPROXIMATE CALCULATION
not exact and eternal truth (a religious belief which does not exist in reality)

What are infinitesimals?

Since there are many WOW fanatics here I will first give the WOW viewpoint.
in the 1960s Robinson introduced nonstandard analysis
this makes calculus easy, but learning nonstandard analysis is HARD
I learnt it as a kid for my PhD, used in paper on Dirac delta in my thesis
to improve calculus to understand nonlinear partial differential equations of physics at a discontinuity (shock wave)

Nonstandard transfer principle

Any standard result
derived using nonstandard analysis
could have been derived without using it
Non-standard analysis makes it easier, NOT different.

Non-Archimedean arithmetic

Essential feature of non-standard analysis
which I used in my PhD thesis was non-Archimedean arithmetic.
In axiomatic math this means arithmetic in ordered fields without "Archimedean" property.
(Name "Archimedean" is bogus, since early Greeks lacked fractions.)

"Archimedean" property (AP)

In an ordered field $F$:
for any $x ∈ F$, $∃ n ∈ N$ s.t. $x < n$
($n= 1+1+...+1$ $n$ times, where 1 is the multiplicative identity in $F$.)
Axiomatic "reals" $ℝ$ LARGEST ordered field with "Archimedean" property.

Infinities and infinitesimals (for WOW fanatics)

Nonstandard analysis defines

$^*R$ a proper field extension of $R$.

Hence, Archimedean Property FAILS in $^*R$
Hence $^*R$ has infinities: $∃ x ∈ ^*R$ s.t. $x>n$ for every $n∈ N$
and infinitesimals:$0 < \frac{1}{x} < \frac{1}{n}$ for every $n∈ N$
Hence, ε - δ limits NOT POSSIBLE.

Infinitesimals for DIFFERENT reason in Brahmagupta approach

Brahmagupta (7th c.) introduced polynomials
as अव्यक्त गणित (Brāhma Sphuṭa Siddhānta chp. 18)
later BADLY copied by al Khwarizmi in his al Jabr wa'al Muqabala
(=Algebra=forcible (jabr) solution by means of a contest (muqabala=equation).

e.g. Brahmagupta first solved quadratic equations BSS 18.44 (text, trans.)
But apart from algebra, there is POLYNOMIAL ARITHMETIC (class IX)
Polynomials can be added, subtracted, multiplied, and divided like ordinary numbers
that is avyakt gaṇita; avyakt bcoz numbers are not expressed.

For WOW fanatics

Polynomials constitute an integral domain
No zero divisors, since $p(x).q(x)≡ 0$ only if $p(x)≡ 0$ or $q(x) ≡ 0.$
This integral domain can be extended to a field of quotients in the usual way
whether coefficients of polynomials are "rationals" or "reals".

Ordering polynomials (defining a positive cone)

In this field of quotients (polynomial arithmetic)
we can define order as usual (WOW ppl see Moise).
As a NUMBER $x-a$ may be both positive and negative
depending on the value of $x$
But as a POLYNOMIAL $x-a$ positive if it is positive for sufficiently large $x$.

Non-"Archimedean" arithmetic (Brahmagupta)

With the above formal definition, Archimedean property fails for polynomial arithmetic
since $x-n > 0$ for every $n ∈ N$
hence $x$ is infinite nd \[0 < \frac{1}{x}<\frac{1}{n} \] so \[\frac{1}{x}\] is infinitesimal.

With infinitesimals ε-̣δ limits impossible.
Hence, Indian calculus was calculus without limits.
Hence also completeness of polynomial arithmetic irrelevant
We can junk all courses on real analysis as irrelevant Western metaphysical FOOLISHNESS
like De Morgan's extreme foolishness about -9<0.
To reiterate: junk both baby Rudin and chacha Rudin.

Because of WOW syndrome we are afraid

of innovation, especially AGAINST the West.
Failure without trying reminds me of
E.g. C-DOT's failure in parallel supercomputing
E.g. Bharat GPT failure to use Indian calculus and Indian notions of probability.

Remember I am saying we should use gaṇita bcoz it is BETTER

NOT bcoz it is Indian.
So anyone else not suffering from WOW syndrome (China, Africa…)
can use it to develop better tech
and we will be left just saying "We did it first".

Concluding remarks

2 questions and a suggestion
Q1. Why did the Europeans/West make such fools of themselves over primary-school arithmetic?
Till 20th c. (Hall and Knight, text on algebra "negative numbers impossible in arithmetic but possible in algebra".
Note Muslims grasped it relatively quickly ("Arabic numerals")

Q.1 Why European foolishness in arithmetic?

A.1: HUBRIS
They started believing their own self-aggrandizement:
Christian supremacy (4th c.) → White supremacy (17th c.) → Western supremacy (19th c.)
supported by false history ("Newton discovered calculus") etc.)

Hence clung to the belief that the pebble arithmetic
early Greeks learnt from Persians
was "superior".

Macualie

Q. 2. Why were WE so foolish as to believe the Macaulie
that the West was always "immeasurably superior" in math and science?
Refused to check it in 2 centuries!
Why are we still so foolish as to believe falsehoods from Wikipedia and AI bots
without checking them?

Suggestion

Ask your math faculty a Q. like I did
Why can't the course on calculus without limits even be TRIED in IIT?
When it has been taught in 5 univs in 3 countries?

They (math faculty) should
try it, or
PUBLICLY debate it
or quit their jobs!