• 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.

• 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.

• 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.)

• 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?

• 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.)

• 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.

"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.

• 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 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: \(\mathbb R\).

• 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.

• 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).

• 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.

• 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".)

• 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.

• 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.

• 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.

• 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).

• 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.

• 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.

• 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.)

• 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").

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

• 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.

• 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.)

• 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 \(- 9 < 0\). 🤣🤣🤣

• 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).

• 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.

• 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.

• 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).

• 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 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 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.

• 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

• 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.

• 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.

• 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.

• 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").

• 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)

• 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.

• 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.)

• 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.

• 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!

• 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.

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

• 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.

• 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.

• 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")

• 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) 😀

• and related presentation.
• Skip to circumstantial evidence

• 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".

• 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.

• 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.

• 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.

• 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 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.

• Long list discussed in Cultural foundations of Math

• 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.)

• 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

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

• 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.

• 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".

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

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

• 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

• 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".

• 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,

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

• 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.

• 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.

• 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.

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

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

• 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."

• 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.

• 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.

• 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.

• 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.

• 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.

• 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 \(\frac{1}{1-x}\).

• 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.

• 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

• 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".

• 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!

• 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.

• 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.)

• 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.

• 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.)

• 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.

• 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"

• 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.

• 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!)🤣

• 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.