IIT Kanpur talk

Teaching calculus as gaṇita

C. K. Raju

Indian Institute of Education

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

Summary

You are taught calculus badly;
teaching calculus as it originated in India as gaṇita(गणित) makes it EASY, and
BETTER: enables students to solve harder practical problems
not covered in usual calculus courses.

Why are you taught calculus badly?

Because calculus was STOLEN from India
and knowledge thieves, Newton, Leibniz etc. did not fully understand it, e.g. Newton's silly fluxions.
West eventually accepted it had a poor understanding of calculus, but its "remedy" worse than disease.
West invented a complex and scientifically useless but politically useful way (limits, real numbers, axiomatic set theory) to understand calculus.

Can we change calculus teaching?

Major obstacle: your SUPERSTITION that you must blindly APE the West
Central dogma of colonial teaching: "West is superior, TRUST it, REVERE it, ape it".
"Proved" using a FALSE history of science which you never cross-checked in 200 years (taboo)
and related bad (church) philosophy of axiomatic reasoning, which mathematicians unable to defend in public debate.

Can we make India great again?

Not talking about how great India was in past.
Talking of future: how to make India great again!
Can we? Depends on you!
(I did what I could. My life nearly over.)

You are taught calculus badly

A. \(e^x\). Gr8! Everybody knows.
Q. But what is \(e^x\)? Define it!
Q. If an infinite series, how to do an infinite sum?

IITians learn Thomas' calculus

Thomas¹ = 1228 + 34 +80 + 14 + 6 + 6 + xvi (=1384) pages; size \(11 \times 8.5\) inches.
Stewart² = 1168 + 134 + xxv pp. (= 1327) pages; size \(10 \times 8.5\) inches + CD).
A student needs years to read!
At the end what does the student learn?

Surprisingly little!

Hence, US technology hub, California, canceled calculus teaching in schools.1, 2
Why is calculus so difficult?
(Problem with SUBJECT, not with teachers or students.)

The difficulty of limits

Easy to recite calculus statement \[\frac{d}{dx} e^x = e^x,\]
but understanding it needs a definition of \(\frac{d}{dx}\) and of \(e^x\)
Both defined using limits which are NOT defined.

NCERT class XI text on limits

"First, we give an intuitive idea of derivative (without actually defining it). Then we give a naive definition of limit and study some algebra of limits"³

Explicitly admits derivatives AND limits left undefined.

That is, you are INDOCTRINATED to believe limits are needed for calculus,
but (deliberately) never taught what exactly they are.

The formal definition of limits

Formally, for \(f: \mathbb{R} \to \mathbb{R}\)
\[\frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.\]
where \[\lim_{x \to a} g(x) = l\] if and only if \(\forall\: \epsilon > 0, ~\exists\: \delta > 0\) such that \[0 < |x - a| < \delta \: \Rightarrow | g(x) - l | < \epsilon, \quad \forall x \in \mathbb R.\]

The missing element: \(\mathbb R\)

The texts of Thomas and Stewart both have a section called "precise definition of limits".
They have the ritualistic \(\epsilon\)'s and \(\delta\)'s.
But the definitions given are not precise.
They miss out a key element: \(\mathbb R\).

The difficulty of defining \(\mathbb{R}\)

On the same policy of indoctrinating WITHOUT actual knowledge
children in class IX and class X are taught about real numbers
which, of course, are NOT defined in NCERT/Thomas' calculus texts.
Most people, therefore, stay confused all their lives about what exactly the definition of a real number is.
(E.g. Q. 2 of my pre-test question paper.

Dedekind cuts

Formal reals \(\mathbb{R}\) often defined as Dedekind cuts using rationals \(\mathbb {Q}\)
\(\alpha \subset \mathbb {Q}\) is called a cut if
1. \(\alpha \neq \emptyset\), and \(\alpha \neq \mathbb{Q}\).
2. \(p \in \alpha\) and \(q < p \implies q \in \alpha\)
3. \(\not\exists \, m \in \alpha\) such that \(p \leq m\quad \forall p \in \alpha\)
\(+\), \(\cdot\), and \(<\) among cuts defined in the obvious way.
Routine to show that the cuts form an ("Archimedean") ordered field, viz., \(\mathbb{R}\).

Called "cuts" since Dedekind's intuitive idea originated from absence of axiomatic proof
even in first proposition of "Euclid" (Elements 1.1).

Equivalence classes of Cauchy sequences

Alternative approach via equivalence classes of Cauchy sequences in \(\mathbb Q\).
\(\{ a_n \}\), \(a_n \in \mathbb Q\) is called Cauchy sequence if \(\forall \epsilon > 0, \ \exists N\) such that \(|a_n - a_m| < \epsilon, \ \forall n, m > N\).
The decimal expansion of a real number is an example of such a Cauchy sequence.
For \(x \in \mathbb{R}\), \(x \not \in \mathbb{Q}\), the decimal expansion neither terminates nor recurs.
But real number NOT a decimal expansion: a hugely infinite equivalence class.

Completeness

The crucial property of \(\mathbb R\) is that limits exist: every Cauchy sequence in \(\mathbb R\) converges.
e.g. the Cauchy sequence 1.4, 1.41, 1.414… has a limit, converges (to \(\sqrt 2\)) in \(\mathbb R\), though not in \(\mathbb Q\)
(But this is pure metaphysics, nothing to do with reality
e.g. you can never REALLY write down the EXACT value of \(\sqrt 2\) as in my pre-test question paper.
for my course on "Calculus without limits".)

The difficulty of set theory

While texts on real analysis do define real numbers in one of the above ways
On the same policy of education = indoctrination WITHOUT knowledge
They too leave out an essential ingredient: set theory.
Thus, as in my pre-test question paper, school children are indoctrinated into belief in sets.
But not taught the definition of a set.

Difficulty of defining a set

They will foolishly "define" a set as a "collection of objects.
But unable to explain what is "collection" and what is "object" (as in my pre-test question paper)
It is long known that naive set theory may easily result in contradictions hence complete nonsense.
Thus, as we saw above, either definition of \(\mathbb R\) requires hugely infinite sets.
And Cantor's set theory (prevalent in Dedekind's time) was riddled with paradoxes, such as Russell's paradox.

Russell's paradox.

Let \[R = \left\{x | x \notin x \right\}.\]
(This well-formed formula defines a "collection of objects".
If \(R \in R\) then, by definition, \(R \notin R\) so we have a contradiction.
On the other hand if \(R \notin R\) then, again by the definition of \(R\), we must have \(R \in R\), which is again a contradiction.

So either way we have a contradiction.
It is elementary that from a contradiction one can deduce any nonsense conclusion whatsoever: \(A \wedge \neg A \Rightarrow B\)
where \(B\) is an arbitrary statement.

West itself understood that naïve set theory might result in ridicule

Therefore, it formulated axiomatic set theory in the 1930s.
Even axiomatic set theory has many paradoxes (e.g. Banach-Tarski paradox) but won't discuss today.
Imp.: few, even among mathematicians, learn it unless they specialize in mathematical logic (as I did in my MSc).

Calculus difficulties resolved by teaching calculus as gaṇita

as it originated in India.
LIMITS not needed, since Indians used Brahmagupta's अव्यक्त गणित (non-Archimedean arithmetic) in which (unique) limits are impossible.
∴ REAL" NUMBERS not needed (\(\mathbb R\) an Archimedean ordered field)
∴ AXIOMA*TIC SET THEORY not needed to construct \(\mathbb R\) with all its complexities.
ALL practical value of calculus is retained.

Preliminary counter narrative

Why is the Western way to do calculus so difficult?

Because the West stole calculus and hence failed to fully understand it!
Historically, West was persistently inferior in math,
learnt most of its math from India
and failed to understand it fully.

Arithmetic: Indians had advanced arithmetic

In contrast to parardha (\(10^{12}\))found in the YajurVeda 17.2
and \(2^{96}\approx 10^{29}\) found in Jania texts (अनुयोगद्वार सूत्र, part 2, sutra 423)
and tallakshana (\(10^{53}\) and परमाणुरजः प्रवेशानुगता (\(10^{83}\)?) found in the Buddhist text ललित विस्तर: सुत्त (chp. 12)
the measly myriad (10,000) (largest Greek/Roman number) still regarded as virtually "infinite" in English.
See Refutation of ARC for more details. (Hence, fantasized "Aryans" could NOT have created the Veda-s, Indian arithmetic in Veda too sophisticated.)

Arithmetic

Hence, Europeans themselves abandoned their native inferior Greek/Roman arithmetic
and adopted "Arabic numerals" or Algorismus the Latin name of 9th c. al Khwarizmi
who wrote Hisab al Hind ("Indian arithmetic").

European Lack of understanding

Lack of understanding of elementary Indian arithmetic clear from the very term zero from cipher (from Arabic sifr),
cipher means mysterious code.

Why is zero mysterious?

Roman numerals additive: xxii = 10+10+1+1
but in place value system 10 ≠ 1+0=1.
zero an essential part of the place value system (can't write parārdha or tallakshạna w/o 0)
used in India since Vedic times.

European lack of understanding of negative numbers

Zero relates also to negative numbers: e.g. 9-9=0
Roman arithmetic had no negative numbers
Hence, many Europeans confused about negative numbers for CENTURIES.
from Fibonacci (Liber Abaci early 13th c.) to Augustus de Morgan (19th c.)

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

very word 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).

Algebra: Lack of understanding again

Primitive Greek/Roman arithmetic lacked √.
Term for √2 is SURD from Latin surdus = DEAF from Arabic asumu
- Why is √2 DEAF?
Because Europeans learned algebra from India without full understanding.

Deaf roots

In शुल्ब सूत्र √2 = DIAGONAL (कर्ण) of unit square.
But word कर्ण/कान also means ear, as in warrior कर्ण (infant with an earring).
Hence "bad कर्ण" mistranslated as "bad ear" = deaf! 😊
Shows its Indian origin and symbolizes bad European understanding.

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.
jībā misread by Mozharab/Jew 12th c. Toledo mass translators as common word "jaib" = जेब = pocket.😊
Very word "trigonometry" involves a conceptual error: it is about CIRCLES not triangles. (In Indian texts found in chapter on circle.)
Hence my pre-test question what is \(\sin 92^∘\)? (In a right-angled triangle there cannot be any angle of \(92^∘\).)

Probability and statistics: References

"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.
And in my JNU talk and video on statistics teaching.
My book Time: Towards a Consistent Theory Springer, 1994, for quantum probabilities.

Game of dice in Veda-s and Mahabharata

Game of dice mentioned in the अक्ष सूक्त (ऋग्वेद 10.34), Translation (mine)
Very important in Mahabharata (Sabha parva)
Story of Nala and Damayanti (Van parva 72) involves statistical sampling.

Probability pre-requisites: theory of permutations and combinations

first developed in early India.
From the time of the Jain Bhagwati sutra, Susruta, and Varāhamihīra etc.
E.g. in the common school text ("Slate arithmetic") of Sridhara

Western lack of understanding of non-discrete probability

Frequentist understanding fails: probability is NOT a limit of relative frequency in calculus sense: only probabilistic limit. (Begs the question: what is probability?)
Subjectivist understanding fails: quantum probabilities are objective not "degrees of belief".
Measure-theoretic understanding fails: quantum probabilities defined on a quantum logic NOT a boolean lattice or algebra or σ-algebra.
But won't go further into that complex issue today. Let us get back to calculus.

Calculus in India some clarifications

Well know that calculus existed in India
from long before Newton.
But some clarifications needed.

Aryabhaṭa vs Madhava

Today, Indian origin of calculus widely attributed to Madhava and "Kerala school"
This was also the WRONG belief with which I started my research in 1997,
my project was titled "Madhava and the origin of calculus"
but I corrected it after I started teaching calculus without limits in 2009.

Calculus due to Aryabhata

Madhava gives table of 24 sine values precise to 3rd sexagesimal minute (tatpara)
Aryabhata's sine value 1000 years earlier precise only to first sexagesimal minute (kalā)
By Madhava's time, infinite series were prevalent in India
and infinite series easily recognized as calculus as Whish did in 1832.
But there is a problem in attributing calculus to Madhava.

Q. Where are the integrals and derivatives, their symbols?

Today most people learn calculus by doing integrals and derivative
(ONLY of elementary functions: see pre-test question paper Q7d)
Hence, many people identify symbolic computation of derivative and integrals with calculus.

But no integrals or derivatives in Indian tradition.
So, how was it calculus?

A. K. Bag's opinion (2003) while

REJECTING my paper on transmission of calculus from India to Europe for IJHS

"I personally feel that…the question of the transmission [of calculus] from India to Europe is basically a hypothetical issue…"

(Why? Since what was in India was not calculus, since no integrals, derivatives.)

Key point: Madhava gives sine values, Aryabhaṭa's "sine table" has only sine DIFFERENCES
Attributing calculus to Aryabhata resolves the first issue
Indians used finite differences NOT derivatives.
(Do NOT assume derivatives "better" or "superior").

Attributing calculus to Aryabhata also resolves the second issue

Thus, let us ask:
Q. How are Aryabhata's sine differences derived?
By means of a recurrence relation, NOT an algebraic equation
This equivalent to what is today wrongly called "Euler" method
of numerically solving ordinary differential equations (Nīlakaṇṭha corrects it)
(Euler familiar with Indian math texts,hence solved Fermat's challenge problem)

That is, Aryabhata effectively numerically solved a differential equation

to derive his sine values.
Solving differential equations is the heart of calculus
and the essence of ALL problems of Newtonian physics.
Corollary: NO NEED FOR ∫ sign since solution of \(y' = f(x)\) is the indefinite integral \(∫ˣf(t)dt\)

This way of doing calculus vastly superior
for real-life practical applications
since no need to restrict \(f(x)\) to be an elementary function
E.g. first serious science experiment in school, the simple pendulum,
involves non-elementary Jacobian elliptic functions

Usual wrong formula \(T= 2π \sqrt{\frac{l}{g}}\) not compatible with observations
see my son's school project.
Therefore, this is the right way to teach calculus.
Hence, this is key aspect of how I teach calculus as ganita.

PS: Āryabhaṭa dalit

There is a further issue that Aryabhata was a Dalit from Bihar,
while people like Nīlakanṭha of "Kerala school" were the highest caste Namboodiri Brahmins.
To my mind this is splendid testimony of regional integration and that caste was not oppressive prior to colonialism.
But won't go into it further.
(Also Āryabhaṭa 1150 years before Newton.)

So Europeans backward in math till the 15th c.

certainly imported arithmetic, algebra, trigonometry from India
and failed to fully understand many basic concepts.
Were the cases of calculus and probability and statistics any different?
Will not talk today about probability and statistics, so let us get on with the calculus.

What happened in 16th c. Europe?

It was still stuck on primitive pebble arithmetic of Greeks and Romans.
Lacked knowledge of fractions known to Egyptians and Indians from 3000 years earlier.🤣
Hence, the Gregorian calendar reform of 1582 used the inferior technique of leap years
instead of using a precise fraction to correct the duration of the tropical year.

This is undeniable: we still use this crude technique.🤣
Hence, equinox still does not come on a fixed day of the Christian calendar.

Great leap forward

How did Europeans suddenly jump
from ignorance of fractions in 1582 to purported discovery of calculus 1630-1660's?
Reason 1: Europeans came into extensive contact with India and Indian systems of knowledge in the 16th century.
Secondly, calculus had existed in India for over a thousand years since Aryabhata.

On the European history, this increased contact between India and Europe
makes "independent rediscovery" of calculus by Europeans more probable not less!🤣
Just as two students sitting close to each other in an exam are LESS likely to have cheated if their answer sheets are identical!

To understand this paradox

We need to understand Europeans lived under church hegemony; ALL beliefs (including Newtonian physics) colored by church dogmas.
Christian priests formulated a new "moral" dogma of theft and mass murder
called the doctrine of discovery,
under the fanatical bloodlust of the Crusades and the Inquisition.

Thus, Europe was fully Christianised in 15th-16th c.

The Crusades finally succeeded in Europe after 500 years,
the last remaining Muslim rulers of Europe (Granada, Spain) lost to Christians (Castille, Portugal)

Doctrine of discovery

The Bull Romanus Pontifex of 1454

declared that all non-Christians OUGHT to be enslaved or killed by Christians.

After 1492, the bull inter-Caetera declared that the whole world was owned by Christians
and pope divided it between Portugal and Spain.

Discovery doctrine (contd.)

The first Christian to spot a piece of land was declared its discoverer or owner.
It was in this sense that Columbus "discovered" America, or Vasco da Gama "discovered" India despite the millions of people living there for millennia.
It was also in this sense of "Christian discovery" that Newton discovered" calculus
though it was known in India from a thousand years earlier.

Theft of calculus

Arithmetic, algebra, trigonometry were all cases of transmission of math from India to Europe
via Muslims who honestly acknowledged their Indian sources
But the cases of calculus, and probability and statistics were a matter of theft
because they went directly to Christian Europe from India.

Europeans claimed "discovery" based on the UTTERLY EVIL 15th c. church dogma of discovery
that any people OR land OR knowledge "belongs" to the first Christian to sight it!
This dogma recently repealed by the pope since its purpose of justifying land, labor, and knowledge-theft ACCOMPLISHED
(but still part of British and US law based on it).

Since we ape the West, we TEACH it in our school texts (NCERT, class XI) that "Newton discovered calculus"
since our education APES THE WEST and we go by BLIND trust in Western books and REFUSE to examine facts.

Calculus theft: my proof beyond reasonable doubt

Since theft is a criminal offense,
I proposed to use the standard of evidence used in criminal law
Namely, proof beyond reasonable doubt.

(Instead of the usual standard of proof in history, of using "balance of probabilities")

In my 2001 Hawai'i paper I looked at

opportunity : (Jesuits in Cochin in 16th c. translated local knowledge on Toledo mode)
motivation: (navigational problem, needed correct calendar+ precise trigonometric values)
circumstantial evidence (identical infinite series, Fermat's challenge problem, etc)
documentary evidence (e.g. Ricci letter)
(though no law that a murderer/thief can be convicted only on a signed confession of murder/theft) 😀

Note: this summarized in a recent video

and related presentation.
Skip to circumstantial evidence

Opportunity

Let's understand "Toledo model"
The first Western universities (Paris, Oxford, Cambridge) were set up by the church during Crusades (since Christian Europe backward, needed knowledge)
The texts were Latin mass translations of Arabic texts in a library captured at Toledo.

The church earlier burnt "heretical" texts
but needed knowledge to fight Crusades,
hence claimed all these texts from the religious enemy had a "theologically correct" origin in early Greeks"
regarded as the "sole friends of Christians" by Eusebius.

Later copied this same Toledo model in India: steal knowledge, lie about its history.
Church trusts that most people are gullible fools and will never check, as we never did.
Jesuits in Cochin systematically collected and translated Indian texts
with the help of local Syrian Christians whom they taught in their Cochin college since 16th c.
Later attributed it all to Christians(Fermat, Newton, Leibniz etc.) like "Vasco discovered India".

Motivation: Navigational problem

Europe was very poor, all dreams of wealth (whether piracy or conquest) were overseas.
Accurate navigation was needed to bring that wealth home.
Europeans had 3 key navigational problems.
Latitude, loxodromes, longitude at sea.

Latitude

Vasco did not know how to determine latitude at sea,
carried back Indian navigational instrument, kamāl
to have it "graduated in inches" &# 128513;
(Can't be done: not a linear scale it uses a harmonic scale.)

Kamāl tells latitude by the pole star.
Since pole star=kau (Arabic-Malayalam) =teeth, and the string of the instrument is held between the teeth
Vasco pompously and foolishly recorded "the pilot was telling the distance by his teeth".

Despite stealing kamal, there still remained a problem of telling latitude in daytime:
by measuring angular elevation of sun.
That requires an accurate calendar which correctly determines the date of equinox,
as explained in my Rajju Ganita geometry text for class 9.

Hence, Christian priests interested to steal calendrical knowledge
for Gregorian calendar reform of Christian calendar in 1582.

Loxodromes

Europeans navigated by plane charts
They expected that travelling in a fixed direction
will result in straight line motion
but on the curved surface of the earth it results in a logarithmic spiral called a loxodrome = curved line.

Mercator projection

The projection maps the sphere to a plane chart in such a way that straight line course
results in a straight line on the chart (loxodromes are straight lines).
This projection was known in Chinese star maps as Needham points out.

But constructing this projection requires accurate trigonometric values (table of secants).
16th century navigational manuals were full of such tables.
Simon Stevin's tables 1590, derived from Aryabhata via Arabs.

A few years later (1607) Clavius published tables with precision of nine decimal places.
Clavius as the Jesuits general in charge of the Gregorian reform
was the natural recipient of texts stolen by Jesuits.

Longitude

Longitude calculation in Indian tradition requires, as Brahmagupta says, knowledge of the radius of the earth \(R_E\)
easily calculated from elementary trigonometry
by measuring the height of a hill h and the angle of dip from it θ.
But mathematically backward Europeans could not do it.

Therefore, the European navigational problem of determining longitude at sea persisted until 18th c.
Many European nations offered large prizes for its solution,
the last being the British longitude prize of 1711-12, half of it given away in 1762
because the Board of Longitude was only half sure that the prize had been won.
The chronometer for which the prize was given became a reliable instrument only by the mid-19th century.

Why did the Europeans have such a big problem?
because Columbus to get funds for his project of traveling West to go East underestimated the size of the earth by 40%
And the fact is that though Europeans stole trigonometric values they did not know how to use them to measure the radius of the earth.

Today Internet is full of bunkum stories of Eratosthenes. Nobody gives an actual primary source? Why not? (Bcoz source from 19th c.)
Picard's 17th c. measurement was preceded by al Mamun's 9th c. measurement by 800 years as usual.
However, important point is that this is retrospect:
in prospect no navigators were willing to trust Picard
therefore European problem of determining longitude at sea persisted.

Circumstantial evidence

Long list discussed in Cultural foundations of Math

16th c.

Nilkantha's astronomical model= Tychonic model (Tycho Brahe, Royal Astronomer to Holy Roman Empire)
Madhava's values sloka, trans., prec. = Clavius' trigonometric values an interpolated version (Christoph Clavius, Jesuit General, author Practical math (from India) and Gregorian calendar reform of 1582.

ahargaṇa (count of civil days) = Julian day-number (as used with NASA, with AD-BC added) Julius Scaliger a contemporary of Clavius.
Kepler (orbit of Mars from Parameswaran's observations). (Kepler nearly blind, Tycho instruments defective, used Tycho's secret papers.)

17th c.

Galileo doubts meaningfulness of infinite series. His student Cavalieri publishes on calculus
Fermat's challenge problem,
Pascal's triangle (meru prastar, khanda meru)

identical infinite series
- Madhava's sine series = claimed by Newton on doctrine of Christian discovery (also only rigor in calculus, not all calculus)
- Yuktidipika 2.271 = "Leibniz" series for \(π\).

Aryabhata's recursive method = "Euler" method (Euler wrote article on Indian calendar, read Indian texts, hence solved Fermat's challenge)
Brahmagupta, Vateshvara quadratic interpolation = "Stirling's" etc. formula

Doctrine of Christian Discovery: current applications to brazen plagiarism

However, I discovered that evidence does NOT matter for numerous dishonest Western academics
still seeking false credit for "discovery of knowledge".

Atiyah case

Thus, Newton's successor Michael Atiyah REPEATEDLY and brazenly tried to steal my work even after he was informed,
later I forced him to acknowledge my previously published work.
Western historians are shamelessly still defending him.

Brazen plagiarism

My very thesis "that calculus was stolen" was itself stolen BRAZENLY in 2003 and REPEATED in 2007.
by George Joseph author of Crest of the Peacock.
Manchester Univ. supports this fraud: keeps fake news alive on its site with minor erratum though paper reported in it could not be published in 16 years
since it verbatim copied my previously published work.
Many Indians too committed to this Christian "doctrine of discovery".

But immediate concern is with bad calculus teaching (and history)

resulting from such brazen plagiarism by West:
knowledge thieves like exam cheats fail to understand what they steal.

Documentary evidence

Primary sources Ricci's letter. (Ricci was Clavius' student.)

Epistemic test: Europeans failed to understand calculus

Despite all the above evidence West has not admitted theft of calculus.
Dishonest historians like David Pingree and Kim Plofker keeping alive the myth of "independent rediscovery".
Won't even allows it to be discussed it. Any papers would be routinely censored.1, 2, 3

Anyway, in my book I applied my epistemic test

People steal knowledge because they have an INFERIOR knowledge of the subject
like students who cheat in an exam.
(But if not caught in the act, they ALWAYS deny cheating and claim similarity of answers due to "independent rediscovery".)
As a university teacher, I developed a way to catch such cheats AFTER the act

Epistemic test: Even after stealing, knowledge thieves fail to fully UNDERSTAND what they steal

and are unable to explain what they have written in their answer sheets.
I used to ask searching question to suspected cheats about their answers while returning their answer sheets.
Lack of understanding (of their own answers) proves theft or "dependent re-discovery".

Applying epistemic test to history

Much later, I applied this epistemic test to history of calculus.
Cannot interrogate the past, but
under suspicious circumstances (my evidence from criminal law)
failure to understand is clinching proof of theft.
Fact is that Europeans failed to fully understand calculus,

What aspects of calculus did the West fail to understand?

1. How to do an infinite sum
2. Exact definition of derivative

Why needed?

Questions about 1 arose since precise trigonometric values needed for navigation)
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).

West got the formula but failed to understand the method by which it was derived.
Descartes realized that summing an infinite series term by term is physically impossible
it is a supertask or an infinite series of tasks which cannot be done in finite time.
Referring to the infinite ("Leibniz") series for π he said it was "beyond the human mind".

Galileo (who had access to Jesuit texts from Collegio Romano) concurred and left calculus to his student Cavalieri.
Newton defined derivative using his silly fluxions, now abandoned.

West acknowledged its difficulties with understanding calculus (especially "fluxions")

The West (e.g. Berkeley, Karl Marx) acknowledged fluxions were incomprehensible.
Berkeley: on lack of understanding of calculus.
Here is James Jurin's incomprehensible defence of fluxions.
HENCE, mathematician Dedekind proposed "real" numbers at end of 19th c. as a supposedly better way to understand calculus.

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.

Interim summary on Newton and Leibniz

So Newton, Leibniz etc. could not have understood calculus
since "real" numbers came long after them.
and Newton's absurd fluxions stand abandoned
except as proof of theft.

Why is exactitude needed?

derivative as finite difference good enough for all practical purposes
just as finite sums \(π = 3.14159\) etc are ALL that we have in real life.
But Newton linked calculus to physics
and physics to religious belief:
"eternal and universal laws of nature made by the Christian god."

Recap: So what is different about the Indian calculus?

Finite differences (Aryabhata's method = Euler method
Brahmagupta (non-Archimedean) arithmetic, not reals
Zeroism or inexactitude

1. Finite differences (e.g. khanda-jya) but NO LIMITS hence no derivatives

What is today falsely called "Euler's" method of solving differential equations
was invented by 5th c Aryabahata to derive
a table of 24 sine differences
by solving a difference equation for sine.

Problem: Finite differences not unique
but non-uniqueness makes no difference since
aim of calculus is to numerically solve differential equations
and finite differences appear only at an intermediate stage

Specifically aim of calculus is NOT to to learn tricks to calculate symbolic derivatives and integrals (of elementary functions!)
a worthless task best left to computer programs like MAXIMA (earlier MACSYMA).
Solution of differential equations is how all/most applications of calculus are done in practice.
So calculus without limits preserves, indeed enhances the skills of students to solve practical problems related to calculus.

In my calculus without limits teaching program I use my software CALCODE a calculator for ODEs
which accepts symbolic input and provides graphical and numerical output.
which I developed to teach modelling to my children.
But students can develop their own program if they like.

2. Brahmagupta arithmetic

This is the part of calculus which Europeans did not understand.
We saw how the famous but pompous and foolish de Morgan said in 1898:
belief in witches is 10000 times more possible than \(-9 < 0\).🤣🤣🤣
So, even in the late 19th c. a professor of math from University College London did not understand ordering among INTEGERS.

So it is natural that Europeans in Newton's time did not understand ordering of polynomials.
West got algebra from al Khwarizmi who translated from Brahmagupta
but did not consider this ordering
but focused on linear equations.

Polynomials are like integers

In formal math terminology both polynomials and integers are integral domains.
They are also ORDERED integral domains.
But the orderings are fundamentally different:
integral domain of polynomials may be extended to an ordered field of rational functions
just as integers can be extended to rational numbers.

But the ordered field of rational functions is non-Archimedean unlike ordered fields of rationals or reals which are Archimedean.
Why? \(x > n\) for every \(n\).
Brahmagupta arithmetic is non-Archimedean different from arithmetic of reals which is Archimedean.

Unique or exact ε-δ limits NOT possible in a non-Archimedean field
since it has infinites hence infinitesimals.
Hence the title: calculus without limits.
However, inexact limits possible by discarding infinitesimals.

West understood this only in 1960's in its usual ultra-complex way,
using non-standard analysis.
But note: infinitesimals and infinities in non-Archimedean arithmetic are "permanent"
unlike the "intermediate" ones in non-standard analysis used to derive standard results (without infinities and infinitesimals).

In calculus without limits, "approximate" (inexact) limits are possible using zeroism to discard infinitesimals.

3. Zeroism (philosophy of inexactitude)

Apart from the superstition that rigor = banning the empirical
the West had another superstition that math is exact.
Now exactitude is excusable so long as one is restricted to primitive European pebble arithmetic or pennywise accounting.

But the moment one comes to even the "Pythagorean" calculation
(calculating the diagonal given the sides of a rectangle)
exactitude fails since algorithm for \(\sqrt 2\) (diagonal of unit square) does not terminate.
Indian traditional geometry explicitly accepted this inexactitude calling \(\sqrt 2\) सविशेष (= "with something remaining")

This inexactitude commonly handled in everyday life.
E.g. no two dogs exactly the same but you just define a dog ostensively (by pointing) and children figure out how a dog differs from a cat or lion.

Buuddhist philosophy of śūnyavāda (शून्यवाद) points out there is no escape from inexactitude in real life.
a person does not stay exactly the same for two instants (e.g. molecules of your body change with every breath)
changes clearly visible over more time: e.g. from childhood to old age.

Therefore, if math has anything to do with reality
it must have a way to manage inexactitude
and "discard small differences" as inconsequential.

Origin of contrasting European superstition of math as exact

The reasons go back to Plato.
and the idea (of Egyptian mystery geometry) of math as mathesis.
This is where the importance of black Hypatia comes in: the "Euclid" book is on math as mathesis (= Egyptian mystery geometry).

The aim of Egyptian mystery geometry was to arouse the soul by driving the mind inward
e.g. by physically shutting out external sensation as in Indian haṭha yoga,
or just mental concentration and internal meditation as in raj yoga.

However, since the church suppressed "pagan" thought since 4th c., and changed nature of soul, this aspect of nature of soul was lost.
Hence, during the European Dark Ages, they MISunderstood mathesis in terms of sympathetic magic.
They believed: the soul is eternal so whatever best arouses the soul must be eternal knowledge/truth
therefore math must have eternal truths.

Math must hence be exact
since the most minor discrepancy would be exposed as false/changeable in an eternity of time.

Therefore, Europeans found it appealing to cling to the fantasy of exactitude
by doing infinite sums exactly using metaphysical (fantasy) limits
though exactitude can never be obtained in reality, hence has no practical relevance.

With Brahmagupta arithmetic and zeroism

one obtains inexact limits by neglecting infinitesimals: e.g.
instead of saying \(\lim_{n→\infty} \frac{1}{n}=0\),
we say when \(n\) is infinite, \(\frac{1}{n}\) is infinitesimal can be replaced by 0.

One can easily sum infinite geometric series.
Simple algebra tells us \[(1-x)(1+x+x²+...+xⁿ) = 1-x^{n+1}\].

Hence, \[(1+x+x²+....+xⁿ) = \frac{1-x^{n+1}}{1-x}\]
but if \(x<1\) and \(n\) is infinite, \(x^{n+1}\) is infinitesimal, which can be discarded
so the last expression is just \(\frac{1}{1-x}\).

So calculus without limits makes calculus easy and gives

better practical value by teaching calculus as numerical solution of differential equations
including non-elementary functions (e.g. Jacobian elliptic functions)
actual course uses my software CALCODE (Demo)

-developed for my kids as described in this article.

immensely simplified understanding: Brahmagupta arithmetic instead of formal real numbers and limits which few understand.
Realistic (=non-superstitious) everyday philosophy of zeroism instead of false exactitude.

For those who are emotionally attached to the foolish Western idea of teaching calculus as all about calculations of derivatives and integrals of elementary functions
as a sop, I also teach use of MAXIMA (earlier MACSYMAo) to do so.

So, can we change calculus teaching? Not easy!

Colonial education came as church education
stuffed you with superstitions.

Most major European universities (Oxford, Cambridge, Paris)

were set up by the church during Crusades
First British act for secular education in 1870 (only for primary education).
Church power rests on superstitions
and ignorance needed to MAINTAIN superstitions

Church nexus with colonial state

Hence, church TRICK to deliberately teach ignorance.
So you never fully learnt even the formal math of the calculus. (BUT INDOCTRINATED to believe it limits, "real" numbers etc. from early age.)
Why? Ignorant person gropes in dark: FORCED to trust authority. Whose authority?
Key teaching of church/colonial education: "trust the West, mistrust the non-West".

Can there be a bigger superstition?

We were rich they were poor.
Westerners came to loot us and after 250 years did.
British committed genocide of at least 30 million Indians.
But we must trust them!

Misplaced trust misused to promote a false history

to justify Superstition: West is "superior" ape the West".
Classic e.g.: Macaulay (1835) said West immeasurably superior in science.
Why superstition? Since closely related to earlier racist superstition of White superiority
and still earlier superstition of Christian superiority.
Same false history (with change of labels) used as secular argument for all three superstitions.

E.g.

Crusading history of science: all science the work of Christians and their friends the early Greeks
Racist history: all science the work of Whites. Early Greeks were White. (Reason of change: Black slaves converted to Christianity.P
Colonial history: all science the work of West. (Reason for change: Aryan Race Conjecture that Whites had earlier conquered and populated India.)

Whites are superior an entrenched Christian superstition

Racism or the STUPID and SUPERSTITIOUS belief that the color of the skin makes a person superior
based on the Bible "curse of Ham/Kam" that blacks inferior even if Christian
since cursed by God. Hence morally correct for Whites to enslave them.

Our refusal to check false history

Because colonial/church education teaches faith, for faith is the source of church power.
Due to church-state nexus, church taught also faith in West: trust the West, mistrust the non-West
Hence today we refuse to check any of that false history (NCERT's stand: trust West.)
And abuse anyone who demands evidence (like IITK student.)

Present day math teaching based on that false history of Euclid and Newton

The present day teaching of mathematics begins with that false history and
the assertion in the class IX NCERT text that superior way to do mathematics is
to ape a purported early Greek called Euclid (for whom there is no evidence),
whose book purportedly (not actually) contains axiomatic proofs.

False history used to promote a bad philosophy

Axiomatic proof a technique of "reasoning by prohibiting facts"
invented by the church for its theology of reason during the Crusades.
Because it enables one to prove any arbitrary proposition based solely on authority
and to misleading claim (like school text) that it has been "proved by reason"

School text lies

The school text hence lies that Greeks were the only people to use reasoning in mathematics.
Ample evidence for use of reason in India (from long before "Aristotle") in the Nyaya sutra 2, etc.
See video and related presentation on Indian system of proof.
The essence of colonial/church education in mathematics is to slip in such brazen lies to children
to glorify the West and enable it control mathematics.

Prohibiting the empirical

Current mathematics uses axiomatic reasoning. - The difference between axiomatic reasoning and reasoning as used in India is the prohibition of facts.
(NCERT Hindi illiterately uses the word निगमन instead of अनुमान for deduction,
although my dictionary translate निगमन into induction!)🤣

Prohibiting the empirical

The key fact prohibition of the empirical is suppressed, not clearly explained.
Thus, "reason" normally means reasoning PLUS facts, as in (1) gaṇita or (2) science or (3) everyday life.
And this is what the students of class IX assume is meant.

Alas, what is actually meant is axiomatic reasoning or reasoning MINUS facts, or reasoning which prohibits facts.
The class IX text does mention in passing "beware of being deceived by what you see".
This reduces math to metaphysics:
children are taught in Class VI that geometric points are invisible (something wrong if you can see a geometric point!)

But you cannot make any real life measurements without seeing what you are measuring.
To reiterate, in India pratyaksha pramana was accepted as a means of proof. Same is the case in Indian gaṇita.
anuman or deductive inference was based on facts or observations.

Conclusions

Abandon the Christian chauvinist history that "Newton and Leibniz" "discovered" calculus.
Try teaching calculus as ganita, at least as an optional course.
Purported superiority of axiomatic math a silly church superstition;
abandon it or force formal mathematicians to publicly debate it.