• 1. Gaṇita (गणित) differs from math,
• 2. it makes math easy, and
• 3. makes science better.

• Gaṇita is practical
• math is religious
• hence unfit to be taught as a compulsory subject in schools.

• Many people sarcastically ask:
• "Where is the religion in 1+1 = 2?"
• They are thinking of the KG method of 1+1 = 2.
• That is an empirical proof; nothing religious in it.

• since gaṇita accepts empirical proof= प्रत्यक्ष प्रमाण
• accepted by all schools of Indian thought,
• as stated e.g. in the Nyaya sutra 2,
• and elucidated here or in this video.
• So, gaṇita method is practical NOT religious.

• Why not?
• Because empirical proofs prohibited in present-day math
• =formal math = axiomatic math = Western ethnomathematics

• 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. \(B\).)

• "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!

• gaṇita accepts empirical proof
• math (as currently taught) rejects it

• e.g. Russell needed 378 pages in his Principia to prove 1+1=2,
• and few people understand a single sentence on that page 378.
• This extraordinary difficulty of math adds NO practical value in a grocery shop.

• (without loss of practical value).
• But that is not the end of the story…

• 1+1=2 for natural numbers proved using Peano's axioms, but
• 1+1=2 for "real" numbers requires axioms of set theory.
• Very few people (even among professional mathematicians) understand axiomatic set theory
• (different from naïve set theory taught in schools to indoctrinate children)

• E.g., wrong to define a set as a "collection of objects"
• Results in Russell's paradox:
• that there is a set R for which the statement R ∈ R is BOTH true and false.

• If there is a statement which is both true AND false, \(A ∧ ¬ A\)
• then any NONSENSE conclusion \(B\) whatsoever can be MATHEMATICALLY PROVED from it
• using the rule of reasoning called modus ponens
• since \(A ∧ ¬ A \Rightarrow B\) is a tautology whatever be \(B\) (a contradiction implies any statement).

• However, since colonial education (DELIBERATELY) taught students to study only for exam,
• most students think it is fine to repeat any statement made in the NCERT school text,
• e.g., "a set is a collection of objects"!

• To make people take these issues more seriously, I offered a prize of ₹10 lakh in JNU for 1+1=2
• (see this video chaired by the then JNU VC, and current chair of UGC)
• The prize is to anyone who could meet my Cape Town challenge to prove 1+1 = 2 in REAL numbers from first principles (in the manner of Russell's proof of 1+1=2)
• without assuming any theorem of axiomatic set theory and proving everything from axioms.

• real numbers purportedly needed for calculus on current teaching
• and calculus needed for all science.
• Hence, real numbers taught in class IX and class X to indoctrinate children
• without ANY comprehension of the difficulties involved.

• Hence, math of calculus so difficult that
• California (US technology hub) recently cancelled calculus teaching in schools.
• Their aim: to replace it with data science.

• Risky strategy: data science needs statistics which needs calculus.
• Half understood statistics may create bugs in AI programs
• on which the world may depend in future for decision-making.
• So, correct strategy is to make calculus easy.

• This can be done by teaching calculus as ganita
• the way it originated in India
• with the 5th c. Aryabhata
• as the numerical solution of differential equations

• That gaṇita makes calculus easy has been DEMONSTRATED
• by teaching experiments over the last 14 years with 8 groups in 5 universities in 3 countries (India, Malaysia, Iran).
• Central University of Tibetan Studies, Sarnath (2009)
• Universiti Sains Malaysia (2010, 4 groups, see report part 1, and part 2), news report

• Ambedkar University Delhi (2013, PG, social sciences), also poster
• CISSC, Tehran, 2013, also poster
• SGT University Delhi (UG, Science and engineering), 2017, poster.
• After my recent talk in IIT Kanpur (or see abstract) this course on calculus as gaṇita will also be taught there from next semester (Feb 2024)

• Teaching calculus as gaṇita enables students to solve HARDER problems
• not covered in usual calculus courses

• but needed for a correct explanation of the first science experiment in schools

• The simple pendulum
• so, calculus as ganita leads to better science teaching.

• since "real" numbers never used in practice.
• Why? Because most practical applications of calculus to technology
• such as sending a rocket to the moon
• done today on COMPUTERS

• by numerically solving differential equations, the Aryabhata way
• as I learned in C-DAC where my job was
• to implement applications of national importance (space, oil etc.) on the target computer.

• since they have finite memory.
• Hence they use what are called floating point numbers
• which are so different that they do not even obey the associative law for addition (see Hawai'i talk/paper)
• used in ALL axiomatic mathematics, including that of real numbers.
• This applies also to all AI programs done on computers.

• Ganita makes math easy
• enables students to solve harder problems not covered in stock calculus courses
• involves no loss of practical value, for applications to advanced technology.
• So, why not teach ganita?

• we should APE the "Greeks"
• who did a math "SUPERIOR" to what ALL others did (Indians, Egyptians, Babylonians, Maya).
• Since I am taking a critical view, I asked NCERT
• what is the primary evidence for Euclid?

• 1. [2007, Hukam Singh, then math head NCERT] "why do you need evidence? We go by a committee!"
• 2. [2019] "Euclid mentioned in 3 Western math texts. Believe it!"
• BUT, sir, history needs PRIMARY sources for which I asked, you are quoting TERTIARY sources.
• 3. [2019] "Euclid mentioned in 6 Western math texts. Now believe it!"

• i.e., "West is superior, hence trust it (as Wikipedia does"
• ape it because it says it is superior!
• Let us set aside Euclid for whom there is no evidence, and much counter evidence.

• Prop. 1 and prop. 4 of "primary source" from 19th c. has empirical proofs.
• Proof of prop. 47 ("Pythagorean theorem" depends on this, hence empirical, not axiomatic.
• This fact was finally publicly admitted in the 20th century
• when Hilbert (1899) rewrote the "Euclid" book to provide the axiomatic proofs missing in it.

• so, unlike the Jains who reject claims about intentions which can be easily professed
• we are asked to believe in the professed "intentions" of a nonexistent person
• which supposed intentions are NOT reflected in the book which he supposedly wrote,

• to which Hilbert's rewrite does great violence because the author of the "Euclid" book never had those intentions
• as explicitly stated by Proclus etc.
• If you can believe the rubbish in the NCERT text about "Euclid", you can believe anything at all.

• The NCERT textbook tries to establish this claim of "superiority"
• by telling a stream of lies in typical church fashion.

• This is a brazen lie, as we saw (Nyaya sutra 2), gaṇita also uses reason as in inference or anumana
• as used e.g. by Aryabhata to infer that the earth is round (like a kadamba flower)]
• because, as Lalla explains (20.36), far-off trees cannot be seen.
• But there are other tricks.

• one word "reason" which has two very different meanings. (Common church trick.)
• (1) scientific reason or reason PLUS facts, as in ganita (inference from observation)
• (2) the religious reason or reason MINUS facts as in current mathematics.

• Most of you probably did not know until today that axiomatic reasoning prohibits facts or anything empirical.
• So, most people ARE fooled,
• they wrongly assume that by the word "reason", our school text is referring to scientific reason (reason PLUS facts)
• whereas it is actually referring to religious reason (reason MINUS facts).

• So this deception successfully fools children/people by allowing them to conflate the two meanings of reason.
• This is class IX mathematics that we are talking about.
• Can't just blame the NCERT.
• If, in a nation of 1.4 billion people, in 2 centuries, no one else cross-checked or objected to this way of deluding our children
• we are collectively responsible, and our independence is forfeit.

• In the 14th and 15th centuries this claim of Christian supremacy
• was used as the moral justification and imperative for genocide (of 100 million native inhabitants) in the American continent (Bull Inter Caetera)
• AND for the slavery of Africans and other non-Christians (Bull Romanus Pontifex)

• After 2 centuries of the organized and brutal slave trade by Christians,
• many Africans converted to Christianity.
• Therefore, this "moral" directive ("enslave non-Christians") failed.
• Christian superiority was replaced by superstition of White superiority using the superstitious curse of Ham/Kam

• When colonialism replaced slavery as the major means of Western wealth
• The Aryan race conjecture was invented as critically useful to rule Indians by dividing then into North (Aryans) and South (Dravidians)
• as we are witnessing today.
• On this fantasy Whites had earlier conquered and populated India.

• it mutated to Western supremacy, as e.g. used by Macaulay.

• The SAME false history of science, erected during the Crusades, continued to be used as a secular argument for these assertions of Christian/White/Western superiority.
• There was only a change of labels, the early Greeks instead of being called friends of Christians were later called Whites or Western,
• But the same chauvinistic false history continued: that all science was the work of "Christians and friends", or Whites, or West.

• The colonised fanatically trust the church/West (and mistrust the non-West)
• hence never crosscheck the false history used to change our education system and mentally enslave us.
• At any rate that is the NCERT's stated policy: that students must trust any false history of science stated by Westerners.

• Well known that Europeans themselves abandoned their INFERIOR system of arithmetic ("Roman numerals")
• and replaced it with what they called "Arabic numerals"
• or algorismus, since based on the 9th c. text "Hisab al Hind" or Indian arithmetic of al Khwarizmi (of Baghdad, Beyt al Hikma).
• Algorismus or Algorithmus the Latinized name of al Khwarizmi.

• From here to Manoj Kumar (in Purab aur Paschim) Indians keep singing praises of zero.
• But zero, then, was a problem for Europeans not a solution
• so it is NOT the answer to "why?"
• A question which our historians NEVER asked.

• over inefficient native European abacus.
• Hence, Fibonacci wrote Liber Abaci (1203) a Latin translation of Hisab al Hind.
• He was a Florentine merchant, who traded with Arabs in Africa from whom he got this knowledge
• and understood its value for commerce.

• E.g. native Greek-Roman numerals need 12 symbols (instead of 4) for the 4-digit number 1788=MDCCLXXXVIII
• Hence, European pebble arithmetic had difficulties even to represent large numbers
• largest Greek-Roman number is a myriad (10k)
• connotes infinite in English
• compared to parardha (= trillion) in Yajurveda 17.2 and तल्ल्क्षण (=\(10^{53}\) etc. in ललित विस्तर: (chp. 12)

• and their copying from India
• stretches back to antiquity
• The very NAMES of (small) Greek and Roman numerals copied from Sanskrit from India
• Direction of transfer follows knowledge gradient: those who knew only myriad copied from parardha.
• But after some 2000 years, by 13th c. they understood that was not enough.

• 89+79 requires 18 operations.
• \(89 × 79\) requires 1422 operations
• compared to 10 operations (4 single digit "multiplies", 4 single digit "adds" and 2 "carries" in गणित
• More details in my book on Refutation of Aryan Race Conjecture

• Key point: backward Europeans lacked full comprehension of elementary arithmetic
• clear from the very term zero from cipher (from Arabic sifr),
• cipher means mysterious code. Why mysterious?
• Roman numerals additive: xxii = 10+10+1+1
• but in place-value system 10 ≠ 1+0=1.

• Adding zero at the end can inflate a contract
• (not possible with Roman numerals: only III can be added at the end)
• Hence Florence passed a law against zero in 1299.

• Julian calendar hence used leap years
• instead of saying the duration of the (tropical) year is \(365 \frac{1}{4}\) day.
• Since Roman arithmetic lacked large numbers, it also lacked precise fractions.
• Hence, Gregorian reform of 1582 still used leap years instead of saying year = 365.241 days.
• Hence, reformed calendar still defective: equinox does not come on the same day every year.

• Matteo Ricci was the favourite student of
• Clavius who headed the reform committee
• and wrote a book on practical arithmetic.

• 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 (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?

• In Indian शुल्ब सूत्र √2 = DIAGONAL (कर्ण) of unit square.
• But word कर्ण also means ear (=कान),
• 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 \(92^∘\).)

• of arithmetic, algebra, trigonometry
• involved various degrees of incomprehension
• by Europeans while copying from Indians
• but we have declared the duffer as "superior"
• and are playing "follow the leader".

• "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.
• Probability relates to game of dice.
• The first account of the game of dice is in the RgVeda.

• Translation
• Mahabharata (Sabha parva)
• Shakuni wins the game by deceit
• Hence, there was an idea of a "fair (or unbiased) game".

• Mahabharata (Van parva 72)
• Counting the fruits on a tree by sampling
• Permutations and combinations, Binomial theorem etc. part of Indian ganita.

• Frequentist interpretation fails: relative frequency converges to probability only in a probabilistic sense
• Subjectivist interpretation fails: quantum probabilities are objective
• Measure-theoretic axioms fail: quantum probabilities not defined on boolean logic.

• Calculus too originated in India in 5th c.
• and was stolen by Europeans (Jesuits in Kochi)
• in the 16th c.
• Why?

• Indians used calculus to derive precise trigonometric values
• In 16th c. Jesuits based in Kochi(Cochin) in Kerala, translated and sent Indian ganita math texts back to Europe.
• Why? Hoping to solve the European navigational problem
• then the biggest scientific challenge facing Europe.

• Which involved the problem of determining latitude, longitude, and loxodromes at sea
• all of which required precise trigonometric values
• as explained in my school text on Rajju ganita.

• E.g. my 2000 invited plenary at Univ. of Hawai'i and related article)
• and my 2007 book of 500+ pages.
• MIT talk, articles in the Springer Encyclopedia, on calculus, calculus transmission
• Durban keynote, popular-level articles on California and calculus etc.

• Newton and Leibniz did NOT "discover" the calculus, as our school text states:

tḥat is false history

• based on the Doctrine of Christian discovery that any piece of land/knowledge belongs to the first Christian to sight it (Bull Inter-Caetera)
• as in Vasco "discovered" India.

• (Today's talk not about history but about teaching ganita vs math.)
• Europeans stole calculus because they had an INFERIOR knowledge of it.
• Should we ape it?

• Unlike other cases of arithmetic, algebra
• in the case of calculus, the West admitted its failure to understand it
• hence Dedekind invented "real" numbers in late 19th c., 250 years after Newton's death.

• Today "real" number are regarded as essential to understand calculus
• (but, as we saw, almost no one in the country really understands 1+1=2 in real numbers
• and no one has accepted the challenge to prove it)

• and no university mathematician is willing to publicly debate why real numbers are needed for calculus
• when they are never used in practice
• but make calculus so very difficult.

• Consider the number \(\sqrt 2\)
• The number is a surd.
• That means if we apply Aryabhata's 5th c. algorithm (the algorithm you learnt in school) to extract the square root
• the process never terminates.

• That is, $\sqrt 2 = 1.4142135…$
• where the three dots indicate that the process goes on indefinitely.
• Another way to see it is that we have an infinite sum \[\sqrt 2 = 1+ 0.4 + 0.01 + 0.004 + ...\] or \[\sqrt 2 = 1+ \frac{4}{10} + \frac{1}{100} + \frac{4}{1000}+ ...\]

• This fact about \(\sqrt 2\) was known from the earliest times, in India and
• the name for it in the Baudhayan śulba sūtra is सविशेष
• meaning with a remainder (अवशेष)
• (Apastamba calls it sa anitya = impermanent).
• Similar understanding applies to \(π\) for which Āryabhaṭa (Ganita 10) gives an आसन्न (near) value.

• That is these early Indian texts adopted a pragmatic attitude
• In an infinite series, simply sum a finite number of terms to the required precision.
• That is exactly what we still do in practice, for ALL applications of mathematics to science and engineering.
• But the Western attitude was different,
• Westerners wanted to sum an infinite series exactly, let us understand why.

• According to Plato/Socrates all learning is recollection
• as in the PRIMARY source, I just showed you.
• And Proclus explains that "mathematics" etymologically derives from mathesis.
• Proclus further cites Plato's Phaedo to explain that it is ONLY mathematics (and figures) which especially arouses the eternal soul.

• Plato/Socrates in Republic VII.527 reasserts this
• hence says math must be compulsory part of education in the Republic
• because it arouses the soul and that makes people virtuous.

• Note the superstitious belief that mathematics contains eternal truths
• and the belief in sympathetic magic: "like arouses like".
• i.e., the eternal truths of math arouse the eternal soul.

• Because of this superstitious belief ("math contains eternal truths"
• math had to be exact
• infinite series for \(\sqrt 2\) had to be summed exactly
• an anitya (=non-eternal) or inexact value as in the Apastamba śulba sūtra would NOT do
• though it works fine for ALL practical purposes.

• Recall that your school geometry text had numerous diagrams
• as also used by Socrates in Meno (because figures aid learning, i.e., arouse the soul.
• But figures have NO CONNECTION to axiomatic proof,
• as pointed out by Bertrand Russell ("Teaching of Euclid")
• To the contrary, Propositions 1, 4 of Euclid give empirical proof using figures, which proofs are hence rejected today.

• As a student of Cambridge (a church institution)
• Russell subscribed to the Cambridge superstition
• as we do
• that the non-existent "Euclid" INTENDED axiomatic proofs.
• When there was no such intention.

• But the Crusading church said so (that the "Euclid" book was about axiomatic proofs). Why?
• Because Plato used a "pagan" notion of soul
• (Very, very similar to the Hindu notion of ātman.)
• This was an equitable notion
• Cursed by the state-church (4th c. onward) which was committed to the dogma of Christian supremacy.

• Therefore, the church was opposed to the Egyptian/"pagan"/Platonic connection of geometry to the soul.
• This idea of equality was expressed through geometry in many ways.
• For example, through the idea that souls are like spheres (and all spheres are equal)
• Indeed, Justinian's fifth anathema (= great curse) of 532 CE was against this notion of the soul as a sphere

• Again, e.g., all prepositions of the original "Euclid" book are about equality of apparently dissimilar objects.
• But we are blind followers of the West (as NCERT teaches)
• and the West was hegemonised by the church especially during the Crusades, the Inquisition, etc.
• Therefore, Hilbert replaced the term equality by the term "congruence" not found anywhere in the original book
• but we blindly ape it today for that is the dharma of the colonised.

• Therefore, the church rejected any connection of mathematics to the notion of soul it cursed
• Instead it connected the Euclid book to axiomatic proof (= reason MINUS facts)
• Because it wanted to compete with the Islamic theology of reason (aql-i-kalam). Why?

• Because the church failed to convert Muslims by force the way pagans in Europe had been converted earlier
• And Muslims rejected the Bible as corrupted (as did Isaac Newton).
• unable to convert either by force or by scripture, the church was forced to accept "universal" reason which Muslims accepted.

• But the church made the great innovation of reason MINUS facts
• as first used by Aquinas to prove his angel theorem (Summa Theologica, First Part, Q. 52, article 3) .
• All university mathematicians today are committed to this church method of reasoning and proof.

• Axiomatic mathematics adds no practical value, as we saw
• But it has superior POLITICAL value for the West
• because the axioms of mathematics are laid down by the West,
• And that enables the West to control mathematical knowledge
• the way the church controlled its rational theology.

• Also, axiomatic method makes simplest things very difficult
• and widespread ignorance of math reinforces Western control
• on the colonial teaching "trust the West, distrust the non-West".

• So that frauds like Stephen Hawking can easily claim that "science has proved the truth of Christianity".
• Such creep pf religious beliefs into science through mathematics is impossible with gaṇita which starts from प्रत्यक्ष.
• But that is another story. I will stop at this point