• You had a revolution in South Africa, 3 decades ago, against apartheid.
• That revolution asserted the truth that Whites are NOT superior
• Similarly, my talk asserts the RELATED truth that White/Western MATH is NOT superior.
• You need this revolution 2.0 to fully emerge from the grip of White/Western domination.

• Europeans were historically backward and primitive in elementary math.
• Hence imported most of their elementary math.
• Key point: they had difficulties in understanding the math they imported and made blunders.

• Europeans themselves abandoned their native 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 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 relates to negative numbers: e.g. 9-9=0
• Roman arithmetic had no negative numbers
• Hence Europeans confused about negative numbers.
• 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 \(- 9 < 0\). 🤣🤣🤣

• who partly translated 7th c. Brahmagupta's unexpressed arithmetic 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?

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

• 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.
• Misread by Mozharab/Jew 12th c. Toledo mass translators as common word "jaib" = जेब = pocket.😊
• Word "trigonometry" involves a conceptual error: it is about circles not triangles.
• Hence my pre-test question what is \(\sin 92^∘\)? (In a right-angled triangle there cannot be any angle of \(92^∘\).)

• See this article for more details (e.g probability) and references
• but enough jokes, lets get down to business

• E.g., California State Board recently cancelled calculus teaching in schools
• because most US school kids find calculus too difficult.
• California board aimed to switch to data science needed for AI and big data (expected key tech for 21st c.)

• a proper understanding of which requires calculus.
• (Building AI bots like ChatGPT without a proper understanding of statistics potentially disastrous for the world.)
• So, correct solution is to make math easy (especially calculus).
• How?

• E.g. my 2000 invited plenary at Univ. of Hawai'i and related article)
• and my 2007 book of 500+ pages.
• as summarised in book title: calculus is NOT of Western origin
• Newton and Leibniz did NOT "discover" the calculus, tḥat is false history.

• How? Why?
• Briefly: Indians developed calculus 5th c. onward and used it to derive precise trigonometric values.
• In 16th c. Jesuits based in Kochi(Cochin) in Kerala, translated and sent Indian math and calendrical texts back to Europe.
• Why? Hoping to solve the European navigational problem
• then the biggest scientific challenge facing Europe (from 16th-18th c.)

• I now say calculus "stolen" (not transmitted) because West did NOT acknowledge true origins of calculus. And still doesn't.
• Why not? Because claim of "discovery" of calculus critical to boast of Western superiority in math and science.

• 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

• Calculus developed in India across a 1000 years (5th to 16th c.)
• but came overnight in Europe.

• Europeans with their inferior Roman arithmetic lacked fractions: introduced in the Jesuit syllabus around 1572
• but known to Egyptians and Indians from at least 3000 years earlier.
• Proof: Gregorian calendar reform of 1582 still described tropical year using leap years not fractions.

• It said every 4th year is a leap year, every 100th year is not a leap year, every 1000th year is
• instead of saying year = 365.25 - 0.01 + 0.001 = 365.241 days.
• See my book or this article for Shakespeare quote

• Sudden leap from ignorance of fractions (1582) to "discovery of calculus" (mid 16th c.)
• VERY suspicious but not proof of theft.

• so I followed standard of criminal law in my 2000/2001 Hawaii paper:
• opportunity (first Roman Catholic mission in Cochin 1500)
• motivation (European navigational problem, determining loxodromes, latitude, longitude at sea, radius of earth)
• circumstantial evidence (e.g. Fermat's challenge problem a solved exercise in Bhaskar II)
• documentary evidence (e.g. Matteo Ricci's 1581 letter from Cochin related to Gregorian calendar reform of 1582))

• so in my 2007 book I applied the epistemic test
• Fact: West failed to FULLY understand the calculus at least until the late 19th c.
• That is proof it stole calculus.

• 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
• 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.
• 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.

• "Limits" need "real" numbers, which are omitted from the usual fat college calculus texts (e.g. "Thomas" calculus (1384 pages, A4 size)
• and taught only in courses on "Advanced calculus" or "real analysis".
• (that and "Advanced functional analysis" were the subjects I taught in Poona University for years).

• But even these advanced courses omit an essential component needed to define "real numbers "set theory".
• which involves "supertasks" or a metaphysics (fantasy) of infinity
• which turns formal mathematicians into supermen as in Superman fantasy!

• Net results, as my pre-test for my calculus course shows no student can correctly define a set
• or resolve Russell's paradox which applies to Cantorian set theory/Naive set theory.
• Its resolution needs axiomatic set theory developed in the mid-20th c.
• which is not taught to calculus students
• and known to only a few even among professional mathematicians whom you are asked to trust.

• So most people just blabber "real numbers, limits"
• without understanding what a real number is.

• This demonstrated by my Cape Town challenge:
• "to axiomatically prove 1+1=2 in real numbers from first principles"
• (direct from axioms of set theory without assuming any theorem of set theory on trust).
• Mathematician in Cape Town debate BLUNDERED, said "use Peano's axioms" which do NOT apply to real numbers.
• Did not know enough set theory to ever give the proof.

• I offered this Cape Town challenge (1+1=2) with a prize of Rs 10 lakhs (≈US$ 12000) at the leading Indian university JNU
• and e.g. in my talk to Indian army to anyone who could prove it in 1 day
• just to show how difficult a proper understanding of the theory of real numbers is
• and how much it requires us to "trust" others about the metaphysics (fantasy) of infinity.

• West stole the calculus hence failed to understand it.
• Acknowledged its failure to understand calculus
• but developed a very complicated way to "understand" which cannot be and is not taught
• but is socially approved as the correct way.

• Worse:it adds nothing to the practical applications of calculus to science and tech.
• BUT ADDS TO THE POLITICAL POWER OF THE WEST since for everything (from 1+1=2) you must "ask the West"

• E.g. Newton got practical value from calculus before real numbers and limits.
• Why? Because the EXACT value of a real number such as \(\sqrt 2\) can never be written down
• since writing it down would take infinite time.
• Humans HAVE to work with rational numbers.

• E. g. Today to send a rocket to the moon
  • rocket trajectories are calculated by NASA/ESA/ISRO using computers
  • which use floating point numbers and
  • CANNOT(repeat CANNOT) use metaphysical (=unreal) "real" numbers.
• so, ANY application of calculus one can do with computers
• can be and is done WITHOUT "real" numbers.

• That is the key lesson I learnt as part of the C-DAC team
• charged with porting applications of national importance on the C-DAC supercomputer.
• So real numbers add NOTHING to the practical value of calculus
• but add hugely to the DIFFICULTIES of calculus.

• it certainly invented the present-day difficulties of calculus (real numbers, limits)! 😊
• Those math difficulties a source of power for the West.

• But during colonialism (during which it totally controlled education and globalised colonial education)
• West sold/imposed its inferior way of doing math (especially calculus) as "superior".
• You superstitiously believed it and imitate the West today
• therefore are plagued by those math/calculus difficulties.

• I say "Superstition" because you are TERRIFIED of even TRYING something different:
• You think you must first ask a Western certified mathematician: "Please sir, can we TRY teaching calculus differently?"
• And he will say: No!
• That's mental slavery! A trap from which there is no exit.

• Since I had long been a formal mathematician I didn't ask. Instead I thought:
• Whoever really invented the calculus (Āryabhaṭa, 5th c., Brahmagupta, 7th c., Madhava 14th c., Nīlakaṇṭh 15th c.)
• HAD to have understood it, and in a SIMPLE way
• NOT involving the huge complexities of formal "real" numbers and limits (and underlying set theory).

• by teaching in that simple way in which the calculus (and a variety of other math) was originally invented
• by refusing to accept the false claim of Western superiority
• and refusing to imitate the West in teaching AND UNDERSTANDING math, especially calculus
• not just rejecting Western way of TEACHING
• but by REJECTING the entire Western PHILOSOPHY of math (=formal math, axiomatic math)

• To implement that in the current (colonial) educations system one must FIRST uproot the rhetoric of Western superiority in math:
• and the resulting White/Western control over math teaching (as I realized later.)
• But initially I just wanted to establish an alternative curriculum and teaching methodology.

• I conducted teaching experiments on calculus without limits
• with 8 groups (incl. 1 Post Graduate group in 5 universities in 3 countries (India, Malaysia, Iran).
• Central University of Tibetan Studies, Sarnath
• Universiti Sains Malaysia (4 groups, see report part 1, and part 2, and related video "Goodbye Euclid!)

• Ambedkar University Delhi (PG, social sciences), also poster
• CISSC, Tehran, also poster
• SGT University Delhi (UG, Science and engineering), poster.
• Students were DELIGHTED at how easy the real calculus is.

• Also explained what I was doing in many Western univs.
• including in this video of talk on "Calculus: the real story" at MIT, Cambridge, Mass. (also abstract)
• and these articles [0,1,2,3,4] in the Springer encyclopedia (or free access here [0,1,2,3,4]

• I separated out the way of teaching geometry with the curved line as basic
• and using the string instead of the compass box as the basic instrument.
• Hence, also did pedagogical experiments at school level (where curriculum tightly state-controlled)
• on teaching geometry with an alternative philosophy of math (rajju ganit or string geometry.

• These experiments were done in Nasik (group of 40 schools)
• Chamrajnagar teachers, students
• Gundlupete
• Indore, poster, media reports
• Once again students and even many teachers very happy.

• Based on these teaching experiences and workshops wrote a New school text on string geometry
• + a teacher's manual for teaching it.
• For content of these workshops see home assignment
• Also videos of a 2-day workshop (day1, day2) comparing string geometry with "Euclidean" geometry.
• A bigger reading list of alternative curricula etc. is available here.

• since 2 centuries of colonial education has entrenched Western political control over mathematical knowledge
• AND over mathematicians
• and entrenched the SUPERSTITION "Whites/West superior in math, so all must imitate them.
• Attempt to decolonise was supported by sections of the Malaysian and Iranian press, e.g. [1], [2]
• and of the Indian press, e.g. [1], [2 ]

• When I first visited South Africa in 2016
• the "Rhodes must fall" agitation was on
• and there was much talk of decolonisation.
• So, I thought it would help to share my ideas and experiences on making math easy.

• "To decolonise math stand up to its false history AND BAD PHILOSOPHY"
• The article went viral and was reproduced world-wide
• but then was CENSORED by the South Africa editor of Conversation
• and was taken down worldwide. (Science 2.0 didn't take it down, Wire, India, put it back.)
• Clear example of global Western political control of mathematical knowledge.

• because what I say is true though very annoying to Whites/West.
• Thus, censored article was reproduced in entirety in (peer reviewed) J. Black Studies
• and in Rhodes Must Fall (Oxford)
• Q. Why did this truth so enrage Whites in South Africa?

• 1. 'Euclid' was a Black woman
• 2. What mathematics did "dead White men" create when they were ignorant even of elementary FRACTIONS (until 1572, hence some 3000 years behind Black Egyptians)?
• These claims centrally attacked the dogma of White/Western supremacy in math
• at the core of math teaching today.

• And no one had a valid academic response to my well-researched facts and arguments (so the rage of Whites/West was even greater).
• Will return to these points later in this talk, so keep any objections ready!

• on decolonising math and science.
• But again amazing racist response: no academic response (obviously), only censorship (stopped from speaking in the math department) personal abuse ("conspiracy theorist"), and lies ("Bantuization").
• By then I should have expected that; what was surprising was the amazing STUPIDITY of the lies and abuse
• so absurd they lacked prima facie credibility.

• A prof from UCT (a student of GFR Ellis) brazenly lied to the SA press that I make math easy by Bantuization.
• South African press never checked back with me for my response.

• But I gave my response in a scholarly article (Keynote, Higher Education Conference 11,Durban)
• I pointed out that my tutorial sheet for my calculus course included non-elementary Jacobian elliptic functions
• Currently covered ONLY in advanced calculus courses but
• included in my calculus course to show that making math easy enables students to easily solve HARDER math problems.

• the simple pendulum.
• to compare theory of pendulum with observation
• as explained in detail in my elder son's school project document
• and in this 2005 article (or Sydney plenary) on how I first taught calculus this way to my children:
• (Elder son PhD in physics from Harvard, younger one (did more difficult problem of Brachistochrone with resistance); PhD from Cornell.)

• To compare theory with observation/ experiment so essential to science.
• Simplified theory of pendulum as simple harmonic motion
• taught in schools says time period independent of amplitude \[T = 2 π \sqrt {\frac{l}{g}}\]
• Contrary to easy observation: time period of simple pendulum VARIES with amplitude.

• As explained in my 2005 pendulum article (or Sydney plenary)
• students and even teachers wrongly believe the THEORY in texts
• and reject the actual observations.
• so bad teaching of calculus leads to a bad first lesson in science,
• believe the authority of the text: not the observation.

• Comparison of right theory (elliptic functions) with wrong theory (simple harmonic motion) easily possible with my software CALCODE (Demo)
• and done in my son's school project

• But these same rascals who introduced Bantuization now say teaching correct science is Bantuization! 🤣🤣🤣
• They want you to learn only bad calculus and bad science to stay under their thumb
• and believe any stupid lie will be believed by all people for all time.

• Astonishing that this brazen lie of math prof of UCT accepted by the South African press without cross-checking
• Even a person sympathetic to me believed it and thought I didn't know math and was just a historian of math

• If the alternative philosophy of math is accepted,
• mathematicians will lose their jobs (or be forced to retrain).
• Much of their life-work might become useless.
• And they KNOW they can exploit your ignorance of math to fool you even with the most brazen lies.

• I don't say, "trust me" not even to my students,
• my formula to them: "check, check, cross check" for only then the knowledge becomes yours.
• If you don't cross check, and blindly trust the very people who exploited you for centuries,
• you have only yourself to blame for mental slavery.

• Another common polemic used by many White faculty of Univ of Cape Town (including the then White VC)
• to prevent me from speaking in their math department (on my long published work against their top dog GFR Ellis)
• was to call me a conspiracy theorist.
• Later, a White Kenyan reporter writing in an MIT magazine repeated this polemic of calling me a conspiracy theorist.

• How exceptionally FOOLISH, even by standards of racist foolishness,
• to call as "conspiracy theory"
• an alternative PHILOSOPHY (like saying Plato or Immanuel Kant was a conspiracy theorist!)
• Or a new PEDAGOGY of math which enables the correct science of the pendulum.

• Again the general point I am making is that if someone responds to
• the alternative philosophy and pedagogy of math
• only through such wild personal attacks
• you know they lack the wits to reasonably contest the new points being made.

• But explaining why not requires academic engagement,
• and the whole point of the polemic of "conspiracy theory" was to desperately avoid academic engagement.

• e.g. Witzel a prof from Harvard did a similar thing earlier (2012)
• in response to my paper on "Probability in Ancient India"
• BRAZENLY lying about it and hurling inquisitional abuses, but not engaging academically.
• Same pattern with Whiteside (1999) a professor in Cambridge, etc.
• They have shown neither the brains to academically address the issues I raise
• nor the honesty to accept what I say.

• Whites/West greatly fear my critique of formal math
• but are unable to produce even a moderately credible, reasoned academic response in last 25 years.

• Q. Why are Whites/ West (and mathematicians) so frightened of what I am saying?
• What is so fearful about making math easy?
• What is so fearful about an alternative philosophy of math that even its existence cannot be acknowledged?

• The fear is because I am attacking the dogma of Western superiority in math and science
• which dogma is just a mutation (or a disguised version) of the earlier dogma of White supremacy
• which ensures White/Western dominance and power.

• Apartheid/racism involved a stupid and SUPERSTITIOUS claim of White supremacy.
• (Superstitious since Bible [curse of Ham/Kam] used to morally justify earlier huge profits from slavery.)
• During colonialism, claim of White supremacy MUTATED to closely related claim of Western supremacy
• as explained in my keynote "'Euclid' must fall" at Univs of Tübingen-Pretoria (video)and related article part 1

• During colonialism West declared itself "immeasurably superior" in math and science (e.g. Macaulay (Minute on Education, 1832)
• who later (1848) revealed his real intentions of using the policy of colonial/church education in India AND Britain as a counter-revolutionary measure.
• And due to other lesser known political reasons explained in my book on Aryan race conjecture and related colonial divide-and-rule policy
• since profits from colonialism much larger than profits from slavery.
• Won't go into details, pls read the above book and the article "Euclid must fall".

apartheid : colonialism :: White supremacy : Western supremacy

• So, decolonisation = overthrow of claim of Western supremacy (simple definition).
• which persists even after the end of White supremacy and direct colonial rule.
• (That is what is revolutionary about this talk.)

• Colonial education is STILL with us.
• Did NOT end with end of apartheid or end of colonialism.
• It still openly asserts Western supremacist view (disguised form of White supremacy) :
• that what the West (Whites) did in math and science was superior and we must all imitate it.

• did NOT mean end of supremacist claims,
• still being asserted in math and science.
• (Will restrict myself to math and science, and focus on math before coming to science.)

• Start by putting that Western supremacist dogma on the table,
• so we can clearly see it and openly discuss it:
• this Western supremacist dogma often stated,
• but there has never been an open academic discussion on it
• So, force this discussion (but watch out for the propagandist tricks of personal abuse, lies, and censorship, and apologetics).

• Snip from the CURRENT Indian class IX official school text (NCERT, Mathematics), chp. 5 "Introduction to Euclid’s geometry",
• It asserts: Greeks did geometry in a way SUPERIOR to what ALL others did — Indians, Egyptians, Babylonians.
• Whole point of false Euclid story is to assert Greeks did a superior math
• which West does today and
• which all are asked imitate.

• If there is a SUPERIOR and an INFERIOR math
• there must be at least TWO types of math:
• formal math as done by the West since 20th c., and as supposedly done by purported early Greeks
• normal math as traditionally done by the rest for thousands of years.

• Why? Because (according to the text) it ALONE uses reasoning in math like the "Greeks".
• This involves several blatant lies.

• Deductive reasoning used in math by others from long before any known Greeks.
• E.g. Indian tradition explicitly listed reason (=deductive inference) as the second means of proof in the Indian Nyaya Sutra, verse 2
• reasoning as means of proof accepted since pre-Buddhist times by all but one school of Indian philosophy.
• This note explains how these means of proof were used in math, in India.
• A striking use of deductive reason in math is by Aryabhata (5th c.) to infer that the earth is round
• AND deduce its radius

• No such explicit listing of reasoning as a means of proof is found in any Greek math text.
• But Plato, Republic, Book VII, explicitly denies that Greeks mathematicians used reasoning
• But we know how to handle it, vilify and demonise the critic, to "prove" that the propagandas in the school text is right and the primary source is false.

**

• My limited point: whether or not Greeks used reasoning in math,
• they were certainly not unique in using reason.
• Other cultures such as Indians were using reasoning as a means of proof long before Greeks.
• Therefore "use of reasoning" is not a valid reason to assert the superiority of Greek/Western/formal math.

• Let us look instead at the definition of proof in present-day formal math
• (Definition given in any text in mathematical logic)

• Thus, a proof (in formal math) is a sequence of propositions(wffs) in which
• each proposition is either an axiom
• or is derived from previous propositions by means of a rule of reasoning
• such as modus ponens (\(A, A ⇒ B, ∴ B\))

• At no stage in a formal proof is there any possibility to say "I see this, ∴ it is true".
• Or "I observe this, ∴ it is true".
• In my Hawai'i talk/paper of 2000 I explained this by saying: "formal math is divorced from the empirical".
• The Indian class 9 math text (p. 301) puts it in a way that children can understand: it says: beware of what you see!

• To fix ideas, let us take a concrete example of 1+1=2 in both normal and formal math
• To prove 1+1=2 in NORMAL mathematics we use empirical means of proof.
• One orange plus one orange makes two oranges and any child in kindergarten can understand this.
• But formal/axiomatic mathematics PROHIBITS THE EMPIRICAL; so it rejects this way of proof as wrong, it has a different way to prove 1+1=2.

• E.g. Bertrand Russell (and Whitehead) in Principia, took 378 pages to prove 1+1=2.😅
• Q. Does that 378 page proof increase the USE VALUE of 1+1=2 in a grocer's shop?
• Same situation for all science and technology (e.g. sending a rocket to the moon)
• what works is normal math
• formal math adds much complexity but zero use value.

• E.g. one cannot exactly write down a real number such as \(\sqrt 2\) (pre-test question paper)
• therefore real numbers irrelevant for practical applications of calculus
• anyway done using computers, which use floating point numbers and CANNOT use real numbers.
• So which math is better? Normal math or formal math?

• To hide the fact that formal math adds huge complexity but zero use value to normal math
• trick is to confound useless formal math with useful normal math by using the myth of the universality of math.
• When asked about the use value added by formal math
• pretend that math is universal so useless formal math = useful normal math which is what "works"

• But if you say traditional math must be universal, let us teach it,
• the response is "formal math is superior, you must teach that.
• But how can Western math be superior if math is universal?

• In 2017, I visited the University of Cape Town to talk about decolonization of mathematics and science.
• At the end of visit I went to meet the VC and the then deputy VC (and current VC) Mamagetty Phakeng (math educator).
• She asked "if mathematics were wrong, why are our bridges still standing?"
• GROSS misrepresentation of what I say
• especially coming from a math educator.

• Hope you understand this TRICK of switching between contradictory premises
• (1)"math is universal" (so normal math=formal math) BUT
• (2) "Western math is superior" (so teach formal math)
• to derive the pre-desired conclusion.

• If you push hard enough, and long enough it would be admitted that
• formal math adds nil use value.
• But then the claim would be that its adds epistemic value ("rigor")
• as in the school text quote ("only Greeks used reason").

• Use of reason NOT unique to "Greeks" or formal math
• Other cultures (e.g. Indians) used reason as means of proof, science uses it.
• But prohibition of the empirical IS unique to
  • (a) formal math
  • (b) Christian theology of reason adopted by the church during Crusades
  • (adapted from earlier Muslim theology of reason, aql-i-kalam)

• but note how today "prohibition of empirical" is misleadingly equated with "use of reasoning" on the Crusading polemic.
• (Why is such sophistry needed in a school math text?)

• very first proposition uses an empirical proof
• essential use of empirical proof in proof of "Pythagorean" theorem in the book.
• My substantive point about Euclid as the Black woman Hypatia was that
• when the book first arrived in Europe during Crusades the church reinterpreted the book to align it with its theology.

• Axiomatic proofs were never intended by the author of the "Euclid" book
• which is about math as mathesis in Plato's sense as correctly asserted by the commentator Proclus (will skip more on that in this talk).

• because ALL church dogmas (God, heaven, hell, angels…) are CONTRARY to facts
• e.g. Aquinas' angel theorem based on reasoning about angels (Suuma Theologica)
• but since there are no facts about angels he assumed whatever he liked!
• (Assumptions=postulates=axioms).

• because God can create the facts of his choice
• but is bound by logic hence Inuit create an illogical world.

• Comically failed to ask: WHICH logic binds God?
• (e.g. Buddhist logic, quantum logic NOT 2-valued, not even truth functional.)
• Likewise, school text says "proofs should be based only on logic" but is silent on the question of WHICH logic to use.
• from among the infinitely many available.
• Or how that question "which logic" can be settled without reference to empirical facts, such as quantum behavior.

• not the church politics of reason.

• How does prohibition of the empirical make formal/Western math "superior"?
• For applications to SCIENCE which accepts the empirical as its first principle.
• The claim is that what you see or observe (the empirical) is fallible. (OK)
• But the purported superiority of formal math comes from the usually unstated part of claim: that what you deduce is infallible.(NOT OK).
• Unstated since false; it can easily be exposed.

• 1. Process of deduction is fallible:
• Any math teacher knows that many students make mistakes in a proof, hence flunk
• Even authorities publish wrong proofs (e.g. about Riemann hypothesis etc)
• and Riemann himself deduced wrong conclusions for shock waves.

• Most people never carefully read those 378 pages: how do you know it is a valid proof?
• Indeed, for most people, just that page 378 is pure gobbledygook.
• They simply rely on authority: Russel got a Nobel prize etc.
• So by banning the empirical one can use polemic of reason to force reliance on authority.

• because in a complex task of deduction
• such as a game of chess
• EVERY human being almost always makes a mistake
• hence looses to a machine.

• so a validly deduced mathematical theorem may NOT be valid knowledge.
• e.g. Banach-Tarski theorem or my simplified rabbit theorem.
• Worse, no way to test the axioms of e.g. set theory which are a metaphysics (fantasy) of infinity.
• Must be accepted on(Western) authority.

• (E.g. mistaking a rope for a snake, experimental error, respectively.)
• Nevertheless, both accept the empirical as the FIRST (and STRONGEST) means of proof.
• Therefore, also, formal math is INFERIOR since it prohibits the empirical and relies solely on deduction which is MORE fallible, and may lead to invalid conclusions etc
• More details in my Durban article.

• Formal math adds neither use value to normal math
• nor any epistemic value
• Only adds POLITICAL value
• giving power to the West to control mathematcal knowledge.

• 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 rigour = 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)
• 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.

• On the college calculus definition of derivative as a limit
• a differentiable function must be continuous
• so a discontinuous function cannot be differentiated.
• but the need to differentiate discontinuous functions arose in science.

• E.g. a shock wave or blast wave represents a hypersurface of discontinuity across which the variables such as pressure,density vary sharply
• across a distance of a mean free path.
• pressure etc. ares are statistical averages not defined on such short lengths.
• So what happens to the equations of physics? Do they fail?

• which permit a discontinuous function to be differentiated
• Sobolev, Mikusinski, Schwartz, Gelfand and Shilov
• Since formal math goes by Western social consensus Schwartz derivative accepted!
• Q. Not equivalent or a generalization of \(ε- δ\) definition,
• Which derivative to use?

• \[ θ² = θ, 2θ· θ' = δ, \theta · δ=\frac{1}{2}δ\]
• BUT \[ θ³ = θ, 3θ²· θ' = δ, \theta · δ=\frac{1}{3}δ\]
• so \[\frac{1}{2}=\frac{1}{3}\]
• (Seems West has still not learnt fractions properly!🤣)

• in general relativity
• and in real fluids
• for details see my book (appendix) and cited references.

• Both shocks and singularities concern understanding of PDEs of physics when
• \(ε-δ\) understanding of calculus fails (infinities arise).
• One can change calculus or physics.
• But Stephen Hawking in his serious book,
• linked it to God and creation.

• This connection of singularities to God and creation
• was made quite explicit in his popular-level book.
• what breaks down is not calculus, or physics, but the very "laws of God"
• hence it must be a moment of Creation.🤣

• put it more bluntly in American style.
• He even wrote several books
• That's where \(ε-δ\) calculus leads!🤣
• In a 2-day debate with Roger Penrose on singularity theory (Delhi 1997) I took this up
• but Penrose was evasive.

• 1988 article on Distributional matter tensors in relativity
• 2003 book Eleven Pictures of Time on the similarity between Stephen Hawking and Augustine
• articles in newspapers
• and a long form article (Singular Nobel?) after Nobel prize to Penrose for singularity theory.

• My quarrel was with Hawking and Penrose,
• Ellis was just a sidekick until I came to Univ of Cape Town, where he was the top dog in the math department.
• having got the million dollar Templeton award for putting together science and religion (we say how) with Hawking.

• In addition to my general talk,
• I offered to take up the more technical aspects of singularity theory in the math department.
• This frightened Ellis to death that
• so to protect his million dollar scam he started a smaller "conspiracy theory" scam
• with the help of the local church.

• if you trust a formal mathematician
• there are many willing to fool you in a variety of ways.

• Europeans stole calculus, failed to understand it
• developed an impossibly difficult and scientifically useless (but politically powerful) way to understand it
• but boasted that it is "superior" and globalised its teaching through colonial education.

• The original Indian calculus is far easier
• improves science teaching
• but West (and mathematicians subordinate to it) will not allow its teaching even on a trial basis
• since they are afraid of loss of face and loss of control over mathematics.

• Try teaching math in a way different from that of the West
• calculus as normal math as I have explained.
• West always lied that colonialism was for the benefit of the colonised.

• Stole all the diamonds in Kimberley, reduced India from a wealthy to a poor country
• killed 100 million
• all for OUR good (!) not theirs!
• Don't trust the West!

• Now teaches formal math saying it improves understanding (only increases their power)
• Hammer home that real numbers are no good and NOT NEEDED for "understanding" calculus since you are never properly taught real numbers.

• If you don't even TRY something different you will forever will remain a Western mental slave.
• (Even if you decide to reject AFTER trying, it will be your own knowledge.)

• brazen lies that e.g. the "Euclid" book has axiomatic proofs or that such proofs were ever intended by the (REAL) author.
• Also challenge propagandist obstructions like claims that doing better science is Bantuization
• and ask the concerned UCT prof to resign for misleading people.

• that the West is very AFRAID that its claim of supremacy in math and science is being challenged.
• Hence Nature wrote an editorial that "there is nothing to fear in decolonisation of math"
• Of course there is (for the West)! But they hope to mislead and misguide the decolonisation efforts that are on.

• In 2012 a mullah in Qom asked me: what is there to decolonise in math? Isn't 1+1=2?
• Also almost the first question I was asked in South Africa in 2016.
• Now throw my Cape Town challenge at them: to prove 1+1=2 in real numbers from first principles.

• of the "Phakeng fallacy" ("it works" fallacy).
• Always ask: "WHICH math works? normal math or formal math?"
• Ask: "How can NASA use computers to calculate rocket trajectories, when a computer cannot use real numbers?"

• And if normal math is what works
• why, then, not teach the normal math?