Created: 2025-09-05 Fri 11:47

Busting Macaulay:
(1) The epistemic test,
(2) the theft of calculus by “Christian discovery”, and
(3) the myths and superstitions in the present-day teaching and practice of axiomatic math

C. K. Raju, Visiting Professor, IIT Mandi

Introduction: Macaulay

Macaulay

Macaulay's claim that the West was "immeasurably superior" in math and science
was but a further MUTATION of the old 4th c. fanatical padreist dogma of Christian supremacy
and its later mutation to the religious dogma of RACIST supremacy (see also article)

Macaulay also proposed free and compulsory church education/indoctrination for the British poor.
(It was 100% church education then, for there was no secular education in Britain in Macaulay's time.)
Why "education"? He asserted in British parliament that education was the cheapest means to COUNTER REVOLTS by the poor (such as French revolt).
See, my article on education and counter-revolution.

Padres and church, NOT Macaulay

Obviously, the very same colonial/church education spread also in the French, Dutch, Spanish, and Portuguese's colonies
where Macaulay never reached.
That is, church education was globalized (not limited to India) by padres, NOT Macaulay.
So, declaring "Macaulay" as the CAUSE of colonial education shows extreme lack of शत्रु बोध.

Doctrine of supremacy and false history

Well known that padres and church totally transformed Christianity in the 4th c.
after the padres grabbed power and started ruling the Roman empire.
They modified and weaponised Christian dogmas (doctrine of sin, eternal damnation, etc.) into instruments of power.

Doctrine of Christian supremacy in this very world (not merely in the next)
was another weaponisation (that the Christian god rewarded and punished people also in this very world)
supported by the padreist trick of systematically false history, making history a tool of Christian propaganda
since 5th c. Orosius' "History AGAINST the pagans"

Later "upgraded" to an utterly false Crusading, then Inqusitional, then racist, and colonial history of SCIENCE (also back cover)
which claimed a theologically-correct Greek origin of all scientific Arabic texts translated at Toledo
(to enable their use as text books in padreist universities (such as Oxford, Cambridge, Paris)
with a view to grab knowledge needed to win the Crusades.

Later texts were declared of Christian origin
on the Inquisitional Model (e.g. Copernicus) of denying heretical origin.

Or the genocidal religious doctrine of Christian discovery that any land (such as Americas, Australia)
or knowledge (e.g. calculus)
is "owned" by the first Christian to "discover" it, REGARDLESS of prior knowledge (or occupancy, see cited US Supreme Court judgment)
so that most land in the world came to be owned by Christians.

That is also how Newton "discovered" calculus
exactly like Vasco "discovered" India or Columbus "discovered" Americas
regardless of earlier occupants of India or Americas.
And eventually all science was claimed to originate with "Christians and friends".

Theft of calculus

Whish' 1832/34 paper well known
Modern presentation in my book Cultural Foundations of Mathematics
Madhava's earlier infinite sine series claimed by Newton
The infinite series for π and arctan claimed by Leibniz and Gregory

These infinite series led to precise trigonometric values using 12th order interpolation

How to explain the stark similarity?

between the EARLIER Indian infinite series
and the LATER infinite series today attributed to, Gregory, Newton, and Leibniz?

Onus of proof

On current academic norms, onus of proof,
to prove "independent rediscovery" is on those who came later
especially when they are habitual criminals: thieves and dacoits
who have stolen 3 whole continents and committed the most massive genocides known to humanity.

The double coincidence

The West also needs to prove that
the simultaneous discoveries by several people in Europe were independent of each other
Roman Catholics Cavalieri, Pascal, Fermat, …
Anglican Gregory (who brought Indian calculus from Padua)
to Protestants Newton, Leibniz who eventually claimed credit .

Nevertheless, though the onus of proof is on West, I anyway provided proof of theft.
That is, Newton did not actually discover calculus: he STOLE it.
Theft is a criminal offence.

Hence, 25 years ago, in my Hawai'i talk (2000)+article (2001),
I proposed a new standard of proof in history

based on the standard of current CRIMINAL law ("proof beyond reasonable doubt")
(as distinct from the usual standard in history/civil law of "balance of probabilities").
Though such a high standard of proof was never before or after used in history.
Contrary to usual Wikipedia lies, I DID provide ample evidence.

Initial evidence for theft of calculus

As in a murder case, I pointed out
(1) opportunity,
(2) motivation,
(3) circumstantial evidence, and
(4) documentary evidence.

Opportunity

Vasco fled from Kozhikode to Kochi to escape the wrath of the Samudri (Zamorin)
The first Roman Catholic mission (1501) in Kochi and later school and Jesuit college
not only taught the local Syrian Christians
but used them as informants to gather and translate Indian math, and astronomy texts available near Kochi.

So, there was contact, and ample opportunity for Europeans to steal Indian texts starting 16th c.
These Indian texts were mass-translated to Latin
mimicking the Toledo model (1125 CE) of mass translating Arabic texts.

Motivation: Navigational problems specific to backward Europeans

Why translate these texts? To solve the specifically European navigational problems of determining (at sea)
- Loxodromes
- Latitude
- Longitude
All of which required accurate trigonometric values.

Determining latitude in day
by measuring solar altitude at noon also required declination or
at least an accurate date of equinox or accurate calendar.
Hence 1582 Gregorian reform of the hopelessly inaccurate earlier Christian calendar (≠ Julian calendar).

Loxodromes

Problem: 16th c. European navigators FOOLISHLY thought moving in a fixed direction,
e.g. set by compass or stellar rhumb line,
would move them in a straight line.
But on the curved earth (in non-cardinal directions) it results in a logarithmic spiral
= loxodrome (loxos = curved, dromos = line)

The Mercator chart uses a conformal map which maps loxodromes to straight lines
but bloats areas at high latitudes
e.g. Greenland seems larger than Africa which is actually some 14 times larger;
area scale factor is \(\sec²φ\) which goes to infinity near poles and is 1 for the equator.

Latitude

Columbus did not know how to determine latitude
recorded he was near Cathay(🤣) (41°) when he was off Cuba (20°).

Vasco was equally ignorant; crept along African coast;
hired a navigator to go from Malindi in Africa to Khozikode
deprecatorily called him (Mualim Kanak, "Malemo Cana") a "pilot".

Kanak said he was telling the latitude by kau (=pole star)
using a kamāl (wooden board with two knotted strings,
held between teeth to determine the angular elevation of pole-star = kau)
Vasco recorded the "pilot told the distance by his teeth" (=kau)😲

Vasco carried back a copy of the kamāl to have it "graduated in inches" 😜
(can't be done, distance between knots is in HARMONIC proportion, NOT linear).

No Westerner understood, to date, that the two-scale principle ("Vernier principle") can also be applied to harmonic scales to get high accuracy
Drunk in their sense of racist supremacy, they lacked commonsense even to ask how Lakshadweep islanders routinely navigated to small islands
without accurate instruments.

Longitude

For navigation in the Indian ocean (in recent centuries) it could be easily determined
by measuring the angular elevation of the pole star
as done by the navigator (Mualim Kanak) who brought Vasco from Melinde to Calicut.

Longitude could also be determined by one of the several Indian techniques
by solving a longitude triangle (Laghu Bhaskariya 1.32, p. 11).
But this required an accurate knowledge of the radius of the earth
not known to Europeans then.

Its value is stated in all Indian ganita texts.
Brahmagupta (7th c.) "ignorance of the earth's radius makes longitude calculation futile."

Bhaskara I (7th c.) also points out that using the "Pythagorean" proposition to solve the plane triangle is
described as a gross method since it neglected the curvature of the earth
according to भटस्य शिष्या: (Bhaskara 1, Mahabhaskariya, 2.5)

Crude European method of heaving the log

But a thousand years later, backward Europeans were still applying this gross method.
Further, they measured distance at sea by the crude method of "heaving the log" and using a log-book
to record the ship's speed in knots (on the rope tied to the log).
This, was a notoriously unreliable method of speed determination.

Of course, they still needed accurate trigonometric values to solve even the plane sailing triangle
from course angle and distance travelled.
Naturally, they grabbed knowledge from every available source.
There is ample circumstantial evidence they did so.

Circumstantial evidence

Long list discussed in Cultural foundations of Math. E.g.
meridian of Greenwich a copy of Meridian of Ujjaini called yamayottari
identical infinite series
- "Newton's" sine series (Newton claimed sine series, and only rigor in calculus)
- "Leibniz" series for π,
Fermat's challenge problem,
"Pascal's" triangle for binomial coefficients \(̱̱\binom{n}{r}\) called khanda meru/meru prastar
in the theory of permutations and combinations first developed in India for the Vedic metre (see my article Probability in Ancient India).

Aryabhata's recursive method = "Euler" method (will cover this in detail, later)
Nilkantha's planetary model= Tychonic model (see my book on Indian calendar) (Tycho as Royal astronomer to the "Holy Roman empire" was a natural recipient of stolen Indian texts)
Madhava's sine values = Clavius' trigonometric values (which were an interpolated version)
Brahmagupta-Vateshvar(="Stirling's") formula

Documentary evidence

Primary sources [Ricci's letter. (Ricci was Clavius' favourite student and biographer.)
He was naturally searching for Indian calendrical texts in Kochi
since this was right before the Gregorian calendar reform of 1582
authored by Clavius whose favorite he was.

But in those days of the Inquisition
it was impossible for the padres to admit that the reform of the Christian RELIGIOUS calendar was based on superior HERETICAL knowledge.
(Note it was the Christian religious calendar,
officially adopted at the first Nicene council to fix the date of Easter
NOT the Julian calendar).

As usual the chief padre, the pope, lied about the 1582 calendar reform telling the story that it was
based on the "Alfonsine tables" from centuries ago
and due to one Alyosisus Lilius a shadowy figure trained as a physician (not astronomer).
But the story "established" that the knowledge had a valid Christian lineage.
Why, then, was Ricci searching for knowledge of the calendar among heretics?😄

Anyway, Wikipedia says I provided no evidence in my book
so just go with that blind belief.😆

Epistemic test

In my 2007 book, Cultural Foundations of Mathematics
I introduced a new method of proof: the epistemic test.
Those who steal knowledge, like students who cheat in an exam,
do NOT fully understand the knowledge they steal.

This is important because if Newton's fluxions and
Dedekind's axiomatic real numbers
do not incorporate a correct understanding of calculus
then today calculus has to be taught differently as I am currently doing in IIT, Mandi

Example of epistemic test from arithmetic

Early Greeks and Romans were BACKWARD in arithmetic

Names of small numbers similar in Greek, Latin, Persian and Sanskrit
obvious e.g. सप्त (September), अष्ट (October), नव (November), दश (December)
More details

But NO large numbers in Greek and Latin

Greek and Latin numbers stopped at a myriad (10000).
Though this is a puny number,
in the English language
it still connotes an infinitely large number!😀

But Sanskrit numbers go on

till a trillion (परार्ध, \(10^{12}\)) in the Yajurveda 17.2
and तल्लक्षण (\(10^{53}\)) and परमाणुरजःप्रवेशानुगता (\(10^{108}\) in Lalitavistara sutta [Life of Buddha] chp.12).
(Buddha was asked to name numbers after 100 crores,
Greeks, Romans would have flunked.😀)

Greeks learnt Indian number-names

due to the Persian (Achaemenid) conquest of Greece.
(No Aryans, no common origin, read my book on Aryan Race Conjecture, won't go into it today.)
The Dara (Darius) Vase shows how Greeks learnt arithmetic from Persians
to pay taxes to them.!

But Persians too stopped at a myriad
="beavan" in Avesta (Ervad Rooyinton P. Peer, personal communication 25 July 2021)
Same small numbers BUT no large numbers = Greeks learnt from Indians via Iran
but learnt little.

Primitive pebble arithmetic

Gerbert's apices

Apices were Gerbert's striking innovation!😀
That is, instead of having, say, 7 pebbles,
he had one pebble with the number 7 written on it,
using Arabic notation, in 976 CE.

Fibonacci's blunder

Most importantly Fibonacci did not fully understand subtraction
Hence, NEGATIVE numbers are missing in Fibonacci.
Mahavira's TOC mentions positive and negative numbers
Fibonacci's TOC says that only a SMALLER number can be subtracted from a LARGER number.😀

Confusion about negative numbers in modern times

Thus, West was backward in elementary arithmetic
from early Greek and Roman times until 10th c.
Then, Europeans imported Indian arithmetic ("Arabic numerals")
but had great difficulty in understanding it.

Did Europeans overcome their difficulty with arithmetic by 13th c?

No! Their confusion about negative numbers etc. persisted until 20th c.
Whole story too long to tell here
Will give just two examples.

Euler: two kinds of \(-1\)?

Wallis, Euler etc. argued that \(-1 < 0\) but \(\infty < -1\) 🤣🤣
Proof? \(-1 < 0 <1\), hence \(\frac {1}{1} < \frac {1}{0} = ∞ < \frac {1}{-1} = -1\)
Euler cheated! Euler method to solve ODEs copied from Aryabhata/Nilakantha
His soln. of Fermat challenge problem a solved exercise in Bhaskar II (Bijaganita 87).(Wrote on Indian calendar in 1740.)
Cartoon summary.

Augustus De Morgan: Dunce

A very influential mathematician of the 19th c.
Professor of math at University College London.
part of an influential group which rejected negative numbers
E.g.1 Algebra and Calculus [p. xi]
E.g.2 Study of Difficulties in math [p. 72]

Confusion about negative numbers persisted in Europe
from Fibonacci till Dunce de Morgan end 19th century.
(in fact till my Algebra school text in 1960's)

To support theft of calculus

and bury the Whish paper
De Morgan got East India company to finance
publication of a foolish book by one Yesudas (= Jesus slave) Ramchandra
on how to do maxima and minima without calculus or negative numbers.

Not sole case of Institutuional complicity with serial plagiarists

This is also important because of Western institutional complicity with serial plagiarists of my work
such as the complicity of AMS with Sir Michael Atiyah, ex-President of Royal Society
who, starting from his Einstein centenary lecture of 2005,
serially plagiarised my earlier published books and papers etc correcting Einstein's mistake.

Or the complicity of Univs of Exeter and Manchester
by retaining for 18 years a fake news release
about a paper submitted to a conference Joseph organized which paper he plagiarised verbatim
hence it could not be republished as the Manchester fake news item claimed
George Gheverghese Joseph and his gang of Christian academic thieves who verbatim serially plagiarised my work

Earlier his gang of academic thieves plagiarised from my paper
without understanding the need for the 1582 calendar reform for latitude determination
or even my 2000 point that
practical value of calculus (sending a man to the moon) comes NOT from real numbers
but from floating point numbers which do not even obey the associative law for addition.

These buffoons wrote in an article in Race and Class (DOI: 10.1177/0306396804043866)
that floating-point numbers (IEEE standard 754 of 1985)
were used by "Kerala mathematicians"
to derive infinite series.

Back to Newton and Leibniz

Leibniz accused Newton of stealing his "discovery"
and complained to the Royal Society.
Today people say both "discovered" calculus independently (How?)

In response to Leibniz, Newton as President of the Royal Society appointed a committee
which he himself headed to investigate the charge against himself.
He wrote the report exonerating himself,
published it and also anonymously published a review of it.

In this he called Leibniz the "second inventor"
since Leibniz did not understand how to sum the infinite series he claimed
adding that "second inventors have no rights".

So, by his own standard, how does Newton as "second discoverer", have any right to claim calculus
or even Madhava's sine series?
As Newton himself rightly says, even dividing the credit for calculus would be an act of injustice.

Newton's term for calculus was Theory of Fluxions a book published posthumously.
Fluxions were thought of as small particles of the flow of time which many people found incomprehensible.
Recall that the foolish idea (Sriharsha/McTaggart's paradox) that time itself flows was critical to Newtonian physics
and conceptual confusion about time was the reason for the failure of Newtonian physics

Conceptual Problems in Newtonian Physics (skip physics)

Newton’s laws of motion and gravitation are metaphysical individually
Take Newton's first "law" of motion:
"in the absence of external forces the body stays in its state of rest or uniform motion in a straight line".
What is meant by "uniform" motion?

Uniform motion supposedly means the "body" (= point mass) covers equal distances in equal times.
Set aside equal distances.
What is meant by equal times?
To measure it a clock is needed.
But which clock to use? Your heartbeats or mine?

Barrow's definition

Newton's predecessor and mentor Isaac Barrow recognized the need for physicists to define equal intervals of time
saying they are otherwise quacks. He defined
"equal causes take equal times to produce equal effects"
hence the times measured by inverting a sand glass are equal.

But Newton overturned it

as also the "law" of gravitation.
Newton’s belief in “mathematical time” was theological
Only by combining both laws we get refutability
e.g., Newtonian ballistic trajectories are elliptic while Galilean are parabolic

Relativity and the Fall of Newtonian Physics

These conceptual flaws—not experiments—led to the downfall of Newtonian physics.
Electrodynamics brought in waves which travel at \(c\) the speed of light
How would \(c\) and Maxwell's equations change if the reference frame was moving?

Berkeley's critique

Berkeley was a bishop, a padre, and defender of the church
The fear was that after Newton's death his venomous writings against the padres might leak out.

Newton's Religious Hostility

Newton, though a fanatic Christian,
was totally anti-church for the way the post-Nicene corrupted Christian beliefs for power.
His writings were suppressed;
an appraiser from the Royal Society called them "foul papers related to church matters, unfit to be published".

Newton called (post-Nicene) Christian priests "spiritual fornicators"
"the most evil sorts of men ever to have inhabited the earth"
His 7-volume history of the church is suppressed to this day.
Hence, Berkeley's discourse was against an "Infidel mathematician"

Our concern

Newton's "fluxions" are today abandoned as utterly confused.
Berkeley criticized Newton's lack of understanding,
of what he called the theory of fluxions, = derivatives = tiny particles of "flowing" time
stating science must involve genuine comprehension.

Berkeley quote (skip)

And. . . there be other Fluxions, which Fluxions of Fluxions are called second Fluxions.
And the Fluxions of these second Fluxions are called third Fluxions: and so on,
fourth, fifth, sixth, &c. ad infinitum.

Berkeley quote (contd)

Now as our Sense is strained and puzzled with the perception of Objects extremely minute,
even so the Imagination, which Faculty derives from Sense, is very much strained and puzzled
to frame clear Ideas of the least Particles of time,
or the least Increments generated therein. . .

Berkeley quote (Contd)

And it seems. . . to. . . exceed, if I mistake not, all Humane Understanding.
The further the Mind analyseth and pursueth these fugitive Ideas, the more it is lost and bewildered;
the Objects, at first fleeting and minute, soon vanishing out of sight.
Certainly in any Sense a second or third Fluxion seems an obscure Mystery….

the nascent Augment of a nascent Augment, i.e. of a thing which hath no Magnitude:
Take it in which light you please, the clear Conception of it will…be found impossible.

Berkeley's other argument

He pointed out that the fluxions/infinitesimals could neither be finite quantities
(for that would destroy the purported perfection of the mathematical theory),

nor could they be infinitely small quantities
(since they could then be neglected without fear of error),
nor could they even be zero (for

all the derivations would then fail).

Finally, he pointed out that mathematicians of his time
were unable to pin down the nature of infinitesimals
which always disappeared from the final

result.

This led to Berkeley's famous polemic

"And what are these same evanescent Increments?
They are neither finite Quantities

nor Quantities infinitely small,

nor yet nothing.
May we not call them the

Ghosts of departed Quantities?"

Irrespective of Berkeley’s political motivation to pull down Newton
(due to Newton's hostility to the church and padres)
and irrespective of his polemics,
Berkeley's arguments are valid.

He is careful to point out that he is not challenging Newton’s scientific conclusions,
but only his understanding.
He explains in detail how one can arrive at a right result in a wrong way,
through a double mistake each of which cancels the other.

Berkeley's point was that such a process,
lacking understanding, cannot be called science:
"For Science it cannot be called, when you proceed blindfold, and arrive at the Truth not knowing how or by what means."

Berkeley's Valid Critique

So, despite Berkeley, despite political motives, and polemics,
correctly argued that Newton's calculus lacked conceptual clarity.

Jurin’s feeble response

Newton was defended by James Jurin, who explained it thus
If this makes the slightest sense, go ahead explain it to me.

Newton's conceptually confused notion of flowing time

led to the failure of Newtonian physics,
and its replacement by relativity.
we go into this very briefly (more details in my book Time: Towards a Consistent Theory.

Newton’s Character and Ethics

Newton was a Christian fanatic who believed in biblical creationism.
Newton was deeply involved in human trafficking
Falsely claimed credit for Indian calculus using dogma of “Christian discovery”
Spent most of his life secretly trying to repair the Bible.

Theological Origins of belief in “Laws of Nature”

Term “laws of nature” stems from Thomas Aquinas’ Crusade-era theology
a political counter to Muslim beliefs
contrary also to Hindu, and Buddhist views, but won't go into all that.
Simple point: belief in eternal laws is NOT refutable, hence unscientific.
If refutable, then refuted when Newtonian physics itself failed and was replaced by relativity over a century ago.

However, not only Berkeley

but also Karl Marx (no religious hostility to Newton)
and Dedekind (and many others) found Newton's fluxions unacceptable.
Hence, fluxions were replaced by Dedekind cuts (axiomatic real numbers)
which lack the notion of infinitesimal.

This proves two things.
Newton did not understand calculus
all the practical consequences of Newtonian physics (before Dedekind) were obtained without the use of real numbers.

Basically, Europeans failed to understand two aspect of Indian calculus
the notion of infinitesimal
how to sum an infinite series

Lacunae in Real-analysis defn of derivative

The college calculus, or the notion of derivative as defined in real analysis
does not work for all of physics
e.g., it does not work for shock waves
where discontinuity may arise
so that the differential equations of physics become meaningless on that definition.

Note that the Schwartz derivative too cannot be used
Although it defines the derivative of a discontinuous function
the Schwartz theory is a linear theory
in which the products such as δ.δ are not defined.

– A similar problem arises in the context of renormalization in quantum field theory.

This shows that the existing understanding of calculus is still not good enough for all current physics.

The solution to this problem in my PhD thesis

within axiomatic math
was to apply non-standard analysis to Schwartz distributions

Later, I abandoned non-standard analysis

realizing that the only thing needed was
what is today called non-Archimedean arithmetic (in an ordered field)
which has infinitesimals and infinities.
Brahmagupta's polynomial arithmetic (अव्यक्त गणित = Bhaskara's बीज गणित) is naturally non-Archimedean).

Have also explained how this can be used to sum
infinite geometric series as Nīlakanṭha first did
but this involves a philosophy of math (zeroism) different from that of axiomatic math.

Note that this is better than using hyperreal numbers
which makes calculus "easy" AFTER making its beginning impossibly complicated.

Myths and superstitions of axiomatic math

Dedekind cuts expose the myth that "Euclid" gave axiomatic proofs
In fact, there are no axiomatic proofs in the "Euclid" book
not even in its first proposition.
This is an acknowledged fact for over a century since Hilbert's (1899) synthetic geometry

Of course there is no Euclid either and no takers for my "Euclid" challenge prize (video, presentation, book)
I teach children that teaching brazen lies about math is essential to teaching Western/padreist math.

Superstition about superiority of axiomatic proof

Children today are taught that axiomatic proofs are superior
in this way of doing math axiomatically (Western ethnomath) has been globalized.

Difference between axiomatic math and ganita

Gaṇita accepts empirical proof= प्रत्यक्ष प्रमाण
accepted by all schools of Indian thought,
as stated e.g. in the Nyaya sutra 2.

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!

A question

If math is done for science and technology which accepts empirical
why does rejection of empirical make math better?

Does make it better for theology
as in Aquinas angel theorem
from his summa Theologica
Axiomatic proof was first used by the Crusading Christian theology of reason
since it can be used to prove any nonsense proposition whatsoever by anyone authorized to lay down the axioms.

As such, believe in its superiority ("infallibility") is a foolish superstition
much more foolish than racism.
This superstitions benefits axiomatic mathematicians
and hurts the nation.

Busting Macaulay: (1) The epistemic test, (2) the theft of calculus by “Christian discovery”, and (3) the myths and superstitions in the present-day teaching and practice of axiomatic math