• Gaṇita accepts empirical proof,
• mathematics prohibits it.
• गणित में प्रत्यक्ष प्रमाण मान्य है,
• मैथमेटिक्स में वर्जित है

• ALL Indian philosophy accepts प्रत्यक्ष (= empirically manifest) as the first means of proof
• e.g., also in ganita: the Nyaya sutra 2, as elucidated here

• e.g. Mendelson, introduction to mathematical logic, page 34
• Also stated in class IX NCERT text(p. 301)

• But reason or inference accepted by all schools of Indian philosophy (except Lokayata)
• e.g (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.

• So reason WAS used in Indian ganita
• but it used reason starting from empirical observation or facts (not axioms)
• foolish caricature that acceptance of empirical means rejection of reason
• Ganita, like science, accepts BOTH empirical and reasoning.

• uses one word "reason" with two very different meanings. (Common church trick.)
• (1) scientific reason =reason PLUS facts, as in ganita (inference from observation, fact)
• (2) the religious reason = reason MINUS facts as in axiomatic mathematics(inference from axiom= postulate= assumption)

• Q. What should we teach? Ganita or math?
• Note: I am not saying and have never said "teach ganita because it is ours"
• (that would limit it to our national boundaries)
• I have said and am saying "teach ganita because it is better and mathematics is inferior"

• science accepts empirical like ganita
• So, please explain:
• Why is religious reasoning (axiomatic reasoning = reason MINUS empirical) better for math/science
• than scientific reasoning (reason PLUS empirical)?

• this claim of Western superiority or Western supremacy during colonialism
• is a direct mutation of earlier claim of White supremacy (used to justify slavery)
• which itself is a mutation of the earlier fanatical claim of Christian supremacy (used to justify genocide)
• all three claims of supremacy were/are justified by the same false history of science with a minor change of labels.

• First, the very NAMES of (small) Greek and Roman numerals copied from Sanskrit from India
• but they were BACKWARD since their counting stopped at (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 started importing Indian ganita ("Arabic numerals")
• To replace its primitive arithmetic ("Roman numerals").
• Let us be clear: my point is not just that the West learnt from us
• but that it had immense difficulties in UNDERSTANDING even elementary Indian arithmetic.

• Imported Indian arithmetic from Cordova (Umayyad Khilafat)
• (Where it had diffused from 9th c. al Khwarizmi of Baghdad who wrote Hisab al Hind in the Abbasid Khilafat)
• However, Gerbert in 976 foolishly got an abacus constructed for "Arabic numerals"
• He FAILED TO UNDERSTAND that algorithms of Indian ganita are very efficient
• And this efficiency is destroyed by an abacus.

• He understood that efficiency of Indian arithmetic
• provided a comparative advantage for commerce.
• But Florentines FAILED TO UNDERSTAND zero
• clear from the very term zero from cipher (from Arabic sifr),

• cipher means mysterious code. (So Europeans FAILED TO UNDERSTAND zero.)
• Why mysterious? Roman numerals additive: xxii = 10+10+1+1
• but in place-value system 10 ≠ 1+0=1.
• Florentines also passed a law against zero and 1299.

• No way to express a general fraction in arithmetic with Roman numerals
• Lack of fractions that you inferior science: the Roman calendar was inferior even after the Julian reform
• This primitive calendar said every fourth year is a leap year
• this obviously defective calendar was adopted as the official Christian calendar in the first Nicene Council
• To determine the date of Easter, then the major Christian festival

• Due to the defect of the calendar, the date of Easter continuously slipped
• Eventually, in 1582 the calendar was again reformed
• Gregorian reform used inputs from India

• But, Gregorian reform of 1582 still used leap years
• saying "every fourth year is a leap year, except every hundredth year which is not a leap year, except every thousandth year which is"
• instead of saying year = 365.241 days.

– Bad math led to bad science

• Hence, reformed calendar still defective:
• right only on a 1000 year average BUT
• equinox does not come on the same day every year.

• was convinced that anything Western is superior
• hence that inferior calendar is today our national calendar
• And our two secular festivals Independence Day, and Republic day are defined
• only on that Christian calendar (and our secularists do not object).

• E.g. De Morgan a very influential professor from University College London
• foolishly declared negative numbers impossible. (Elements of Algebra: Preliminary to the Differential Calculus 1837, p. xi.
• he failed to understand ordering of negative numbers
• and went on to say (1898) belief in witches 10000 times more possible than \(- 9 < 0\). 🤣🤣🤣

• West imported arithmetic from India
• had great difficulties and took a long time to in understand

• (0} large numbers
• (1) efficiency of Indian algorithms vs inefficiency of abacus or pebble arithmetic
• (2) place value system and zero
• (3) general fractions
• (4) negative numbers

• based on the principle "phylogeny is ontogeny"
• classroom teaching reproduces in fast-forward mode the historical evolution of the subject
• colonial education in arithmetic reproduces the European historical difficulties with arithmetic

• these difficulties are not present in Indian Ganita texts
• E.g. all Indian ganita texts begin with place value system and large numbers
• negative numbers described as ऋण (debt) etc.
• a new curriculum for primary schools has been made
• but is yet to be tried out.

• In contrast, Indian geometry was based on the string (रज्जु, शुल्ब )
• angle was defined differently as चाप not कोण
• as the relative length of a curved arc
• (cannot be measured using a compass box, hence students have difficulty with radians)

• This is useful for real-life measurement such as
• calculating the area of an agricultural field with curved or jagged boundaries
• or to teach the Indian calendar
• notion of तिथि involves measuring real life angle between sun and moon
• Not on paper or with a protractor

• and the course has been very successfully tried out with various schools (students and teachers) across India
• Nasik, Indore, Gundlupete, Chamrajnagar etc.

• Major problem is the conflict with Euclidean geometry and compass box geometry currently taught
• e.g. equality NOT congruence,
• angle as curved arc not pair of straight lines etc.
• "Pythagorean" CALCULATION, not theorem
• inexactitude NOT exactitude etc.

• Current plan is to manage this conflict by teaching both
• along with our view that the Western method is inferior

• West learnt trigonometry from India as ganita.
• Failed to fully understand, hence learnt badly.
• but colonized are convinced they must ape the "superior" West

• 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.😊
• The word sine is proof that trigonometry went from India to Europe

• Where it was not correctly understood.
• but we are convinced that we must ape the "superior" West and its foolish mistakes
• Hence, we still use that translation mistake "sine" .

• Word "trigonometry" involves a conceptual error: it is about circles not triangles
• as wrongly taught by NCERT
• Hence, students confused by my pre-test question what is \(\sin 92^∘\)?
• (In a right-angled triangle there cannot be any angle of \(92^∘\).)

• what is \(\sin 1^∘\)?
• Our students can't answer. They are 1600 years behind the times
• because they do Western math, not ganita.
• Values like $sin 1^∘ $ required to calculate the radius of the earth.

• which is easily done using trigonometry
• and which estimate of earth's radius is found in many Indian ganita texts

• West was unable to calculate the radius of the earth correctly
• (Columbus' estimate was off by 40%)
• This led to the famous longitude problem specific to European navigation because
• as Brahmagupta said "ignorance of earth's radius makes longitude calculations futile"

• Recall that the longitude problem was the major scientific challenge facing Europe from 15th to 18th century
• and that as late as 1712 British Parliament passed an act setting up the board of longitude
• to administer a prize of UKP 20,000 for a method of determining longitude at sea.

• On the central principle of colonial teaching "ape the West"
• and on the principle of phylogeny is ontogeny
• our students stay ignorant of this until today
• Calculating \(\sin 1^∘\) is the first step towards calculus,
• as shown by Āryabhaṭa.

• 5th c. Aryabhaṭa's "sine table" has only sine DIFFERENCES
• why differences? Because differences (or change) are what you need to calculate any sine value such as \(\sin 1^∘\)
• using the elementary rule of three (त्रैराशिक, for "unit rate of change")

• not Newton's silly fluxions (= "derivative")
• which Marx rightly called mystical (before the invention of real numbers by Dedekind)]
• No need for ∫ sign either since Aryabhata numerically solved differential equations
• using what is falsely called "Euler's method".

• 900 years later Madhava gave same 24 more sine values to greater precision.
• Jesuits based in Cochin stole and translated the related texts in the 16th c.
• to solve the European navigational problem (latitude, loxodrome, longitude)
• All of which needed precise trigonometric values.

• 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).
• but Europeans/colonized have not understood until today how this was done

• In 17th c. Newton and Leibniz claimed discovery of these series
• On the excessively evil and hugely genocidal doctrine of Christian discovery,
• which says that ownership of any piece of land or knowledge taken from non-Christians
• Belongs to the first Christian to sight it

• By my epistemic test that
• knowledge thieves, like students who cheat in a test, fail to fully understand
• what they steal.

• To reiterate, Madhava's precise trigonometric values were derived using infinite series
• such as the infinite sine series falsely claimed to have been discovered by Newton
• or infinite series for π claimed by Leibniz.
• Newton, a Christian fanatic, was invested in slavery of non-Christians,
• and cannot be expected to have had ANY moral compunctions about stealing knowledge from them.

• real numbers were hence invented by Dedekind (end 19th century
• and the axiomatic set theory needed for that in the 20th century

• 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.

• Thus, rocket trajectories are today calculated on computers (by both NASA and ISRO)
• computers CANNOT use real numbers
• because storing even a single real number requires infinite memory not available on the computer
• Instead computers use floating point numbers

• Associative law for addition HOLDS for reals
• but FAILS for floats:
• (-1+1)+ε = ε ≠ 0 = -1 + (1+ ε)
• if ε < 1E-8 (or 1E-16 on 64 bit systems). Demo)

• Math of "EXACT but अप्रत्यक्ष" UNREAL "real" numbers NEVER used in practice
• All practical value from inexact calculation.

• is Aryabhata's way as numerical solution of differential equations ( what is used in practice).
• To sum infinite series use Brahmagupta's अव्यक्त गणित (of polynomials)
• which involves what is today called non-Archimedean arithmetic
• involving infinities and infinitesimals
• absent in so-called "real" numbers which obey Archimedean arithmetic.

• or शून्यवाद
• to sum infinite geometric series
• we discard infinitesimals instead of small numbers.

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

• Iran (Cissc Tehran), poster, group photo
• Ambedkar University Delhi (poster, group photo, SGT Univ. Delhi (poster, group photo, video

• "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.
• So technique of calculating outcomes in games of chance well known.

• from the time of the Jain Bhagwati sutra, Susruta, and Varahamihira etc.
• E.g. in the common school text ("Slate arithmetic") of Sridhara

• claimed independent rediscovery with Pascal etc.
• and right to understand it using metaphysics of "real numbers and limits"
• as with calculus, but more complex (measure theory, Lebesgue integral, stochastic differential equations etc.)
• But this failed miserably with probability

• 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.