Indian contributions to math

Introduction

• The Bollywood film पूरब और पश्चिम spread the narrative
• that India’s contribution to mathematics was zero,
• as in its famous song जब ज़ीरो दिया मेरे भारत ने…

• But this is a WRONG narrative.
• Actually, India contributed a LOT more
• Europeans learnt much mathematics from India

• including
  • arithmetic,
  • algebra,
  • trigonometry,
  • calculus, and
  • probability and statistics.

• For a quick summary and references
• see this abstract for the Indology conference, IIAS Shimla

Key point

• Indian gaṇita was SUPERIOR
• European math was inferior
  • but Europe misunderstood and transformed Indian ganita, such as calculus, to math

• Note that gạnita ≠ math
• even at the level of 1+1=2.
• Key difference: gaṇita accepts empirical proof (प्रत्यक्ष प्रमाण)
• mathematics PROHIBITS the empirical (NCERT, class IX, p. 301).

• For more details, see article गणित बनाम मैथमेटिक्स in Himanjali
• or see this article on Decolonising mathematics
• Or see this video on Practical gaṇita vs. religious mathematics
• or this video on How colonial education changed our math teaching

• Some important consequences of the fact that gạnita ≠ math are in different ways of teaching
• arithmetic in primary school(1-5)
• geometry in 6th to 9th
• trigonometry (10th)
• calculus (in XI and XII).

Arithmetic

• Indian arithmetic which used place value
• was far superior to Greco-Roman (pebble) arithmetic
• Q. Can you write 1888 in Roman numerals?

• A. MDCCCLXXXVIII
• as in Boston public lib ray
• needs 13 digits instead of 4 in Indian स्थान-मान प्रणाली , very inefficient.
• (Note: Boston is seat of Harvard, MIT, Radcliffe.)

• Note: addition with Greco-Roman pebble arithmetic very inefficient
• 89 + 79 requires 18 operations instead of 5
• Multiplication very very inefficient
• to do \(89×79\) we must add 89 to itself 79 times or (\(18 × 79\) operations)/

• Thus Indian arithmetic was very efficient ad superior
• Arabs understood the superiority of Indian arithmetic, and

Arabs quickly adopted Indian arithmetic

• al Khwarizmi wrote Hisab al Hind in 9th c.
• (hence word algorithm from his Latin name Algorithmus)

Europeans learnt from Arabs in Spain, and Africa

• some 2 centuries later
• via Gerbert (pope Sylvester II), Toledo translations (Adelard of Bath), Fibonacci.

But Europeans were duffers

• They took some 900 years to grasp primary school arithmetic.

Gerbert

• Gerbert realized that Indian place value system could be used to represent large numbers
• he got an 27 column abacus constructed with apices
• but fail to understand that the use of an abacus kills the efficiency of place value arithmetic

Fibonacci

• a Florentine trader understood the efficiency of Indian arithmetic
• hence its great usefulness for commerce
• Alas, he failed to understand negative numbers (like al Khwarizmi).
• Compare Fibonacci Liber Abaci toc with Mahavira's toc

Europeans failed to understand zero

• 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.

Suspicion of zero

• 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.

This sort of foolishness went on till end 19th c.

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

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

Augustus De Morgan: Dunce

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

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

But modern-day Indians but even more foolish

• in 1835 before de Morgan,
• Macaulay said the West is immeasurably superior in math and science
• we believed him in change our entire education system especially STEM
• now we can change it back

• School teachers and students have no power to change
• Neta-s and babu-s do not know mathematics
• foreign experts will not allow a change.

Geometry

• See

poster for Rajju Ganita workshop

• or see this video on Rajju Ganita vs Euclidean geometry

(part 1, part 2)

• Or see this video on śulba-sūtra geometry: can we teach it in schools today?

Trigonometry

Pocket trigonometry

• 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).

Toledo translations ca. 1125

• 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.😊
• but we are convinced that we must ape the superior West
• Hence, you still use that translation mistake "sine"

• 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 \(92^∘\).)

Calculus

• Calculus originated in India
• and was stolen by Europeans (Jesuits, Gregory, Newton, Leibniz etc.)
• See, e.g. video of my MIT talk

Evidence of theft

• In 1834 Whish pointed out the similarity
• of Gregory, Newton, and Leibniz infinite series
• with series for arctan, sine series, and series for π in various Indian texts
• This remained suppressed for some 175 years.

Key point is that Indian calculus was different

• It had no real numbers, no limits etc.,
• hence very easy
• could solve harder problems as in ongoing IIT course on Calculus as ganita

Real numbers taught in class IX text

• but NOT easy
• Hence California cancelled the calculus
• E.g. my challenge to axiomatically prove 1+1=2 in real numbers
• from first principles without assuming any theorem of axiomatic set theory.

What difference does it make to school teaching

• E.g. simple pendulum taught wrongly because of
• bad teaching of calculus
• See my elder son's school project
• or my younger son's school project on brachistochrone with resistance

Probability and Statistics

• See my article on Probability in Ancient India
• or video of my talk to JNU on statistics

• Game of dice in अक्ष सूक्त (ऋग्वेद 10.34)
• Translation

Mahabharata (Sabha parva)

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

Mahabharata (Van parva 72)

• Story of Nala and Damayanti
• Their separation. Disguised Nala takes a job as a charioteer with Rituparna, king of Ayodhya.
• Damayanti announces swayamvara (widow remarriage).

Counting the fruits on a tree by sampling

• Nala and Rituprana dash to Vidarbha.
• Stop on the way near a Vibhitaka tree (mentioned in the aksa sukta)
• the five-sided fruit of which was used in game of dice.
• Rituparna shows off his knowledge of ganita, by counting the 2095 fruits on the tree.

Conclusions

• Much mathematics originated in India
• Many aspects (rajju ganita, calculus etc.) better than Western math
• but we can't easily teach it in schools
• Because we are still colonised.