Ancient India's contribution to mathematics
(and its relevance to modern technology)

C. K. Raju
Indian Institute of Education
G. D. Parikh Centre, J. P. Naik Bhavan
Mumbai University Kalina Campus

Introduction (some clarifications)

Aryabhata (NOT Aryabhatta)

Vedic math

No connection to Veda-s

  • (a) NO connection whatsoever to the Veda-s
  • Bharti Krishna Tirtha gave no source when challenged.
  • The book Vedic Math itself says (p. v, p. xxxv)

obviously these formulas are not to be found in the present recensions of the Atharvaveda

  • Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaja, Vedic Mathematics, Motilal Banarsidass, New Delhi, Rev. ed. 1992, reprint 1997.

Actual math found in the Veda-s is different

  • M. D. Pandit, Mathematics as Known to the Vedic S amhitâs , Sri Satguru Publications, New Delhi, 1993.
  • Hence, title of the "Vedic math" book is grossly misleading.
  • The Veda-s are sacred, wrong to use them for misleading marketing.

Not even ancient

  • Some promoters of "Vedic math" quibble: "Vedic" not about actual "Veda-s".
  • (This should be put on the cover.)
  • They say "all ancient knowledge is Vedic".
  • Q1. How do you know this is ancient knowledge?

Probably modern

  • Source hidden, probably modern since source not in Veda-s but also
  • these sutras NOT mentioned by any traditional Indian mathematician across over 3000 years.
  • from Vedanga Jyotisa (\(-1500\) CE), to Nilakantha (+1500 CE).
  • Later-day katapayadi system IS mentioned in the "Vedic math" book.

Q.2 Is everything ancient Vedic? NO!

  • Many people in ancient India were ANTI-Vedic

"The agnihotra, &c., are only useful as means of livelihood, for the Veda is tainted by the three faults of untruth, self-contradiction (व्याघात), and tautology (पुनरुक्ति);

…the three Vedas themselves are only the incoherent rhapsodies (प्रलाप) of knaves (धूर्त) …"

  • Sarva-Darshan-Samgraha, trans. E. B. Cowell and A. E. Gough (London: Trubner, 1882), p. 4.

No relevance to S&T

  • Since marketing label "Vedic" is misleading and invalid
  • we should discard it. But what is left?
  • Merely some tricks of elementary mental arithmetic
  • of nil value to science and technology today in these days of computers.

And what does this "Vedic math" seek to replace?

  • Genuine Indian arithmetic algorithms, which went to Europe
  • and returned to India during colonialism.
  • Why abandon the real Vedic for the fake "Vedic"?

Summary

  • "Vedic math" has nothing to do with Veda-s.
  • Not even ancient (and everything ancient in India not Vedic).
  • Only some simple tricks of mental arithmetic, of no relevance to S&T.
  • Seeks to replace genuine Indian math, with key contributions from dalits, such as Aryabhata,
  • and makes us a laughing stock.

How and WHY Indian math went to Europe

Background: Al Khwarizmi

  • Word "algorithm" from 9th c. al Khwarizmi's Latinized name "Algorismus" or "Algorithmus".
  • He wrote Hisab al Hind from which Europe learnt arithmetic.
  • This arithmetic returned to India during colonialism.
  • Arithmetic we teach in schools today is of Indian origin.

Three phases of European import

  • Indian arithmetic went to Europe in three phases.
  • Phase 1: Gerbert of Aurillac (10th c.), later Pope Sylvester II, imported it from Cordoba, Muslim Spain.
  • Phase 2: Fibonacci (12th c.) Florence, Liber Abaci, imported it from Africa (trans. of Hisab al Hind)
  • Phase 3: Christoph Clavius (16th c.), Practical Arithmetic, Rome, imported it direct from Kochi.

KEY ISSUE: Europeans failed to understand elementary Indian arithmetic of algorithms (for 6 centuries!)

  • Why? Were they stupid?
  • Basically because their native arithmetic ("Roman numerals") used the abacus
  • a system quite different from the Indian arithmetic of algorithms (after al Khwarizmi)

The first European importer of Indian arithmetic

This shows Gerbert understood the Indian place-value system

  • But NOT the the efficiency of algorithms over the abacus.
  • The very act of constructing an abacus (for "Arabic numerals")
  • which does arithmetic on the primitive coin-counter system
  • destroyed this efficiency of Indian arithmetic.

Indian decimal place value system is more EFFICIENT

  • E.g. 1788 written as MDCCLXXXVIII in Roman numerals
  • requires 12 symbols instead of the 4 used in 1788.
  • Large numbers such as \(10^{12}\) (found in Yajurveda 17.2)
  • or तल्लक्षण (\(10^{53}\), ललित विस्तर सुत्त chp. 12) cannot be written at all in Roman numerals.
  • Myriad (=10,000) was the largest Roman number (regarded as almost infinitely large).

Algorithm vs abacus for addition

  • 89 + 89 = 178 (trivial using algorithm for addition)
  • But let us do it in "Roman numerals" (abacus method)
  • 89 = LXXXIX in Roman numerals
  • Step 1: Write in full as LXXXVIIII.

Abacus method for addition

  • Think of L, X, V, I as coins or counters of 50, 10, 5, and 1.
  • Method is to pool all coins/counters and simplify. (C = count, R = replace)
  • Step 2: Pool all the "counters" in the numbers to be added, namely LXXXVIIII, and LXXXVIIII.
  • Step 3: Simplify, starting with the smallest.
  • (8 C) We count there are 8 I's which we simplify to 1 V and 3 I's.
  • (1 R) The 3 I's are left as they are, and the 1 V is "carried".
  • (3 C+1 R) Next there are 3 V's (including the carried V) which simplify to 1 X (carried), and 1 V.
  • (7 C and 1 R)Next there are 7 X's (including the carried X) which simplify to 1 L (carried) and 2 X's.
  • (3 C and 1 R) Finally, the 3 L's simplify to 1 C and 1 L.
  • Thus the final answer is CLXXVIII = 178
  • but obtained in a very inefficient way
  • which involves far more steps
  • 21 counts and 4 replacements = 25 operations.

Abacus method of multiplication

  • The inefficiency of native European arithmetic is more marked in multiplication and division.
  • Multiplication = repeated addition: to do \(89 \times 89\) add 89 to itself 89 times!
  • That is, the previous 25 operations must be repeated 89 times = 2225 operations.
  • (Likewise division = repeated subtraction).

Algorithm for multiplication

  • In contrast the usual (Indian) algorithm for \(89 \times 89\) involves
  • 4 single digit multiplications + 6 single digit additions = 10 single digit operations
  89
\(\times\) 89
  801
  7120
  7921

Native European arithmetic of abacus

  • very inefficient compared to (Indian) algorithms
  • For multiplying two two-digit numbers abacus needs 2225 operations compared to 10 for algorithms.
  • Efficient Indian arithmetic gave comparative advantage in commerce
  • to Arab traders.

Hence Europeans again imported (Indian) arithmetic

  • Fibonacci was the son of a Florentine merchant.
  • Realized the comparative advantage of efficient arithmetic ("algorithms") for commerce,
  • while dealing with Arab traders in Africa.
  • Learnt it, and enthusiastically wrote about it: Liber Abaci (ca. 1202 CE)
  • Some Florentines accepted the new system.
  • But many were suspicious of the "new-fangled" "Arabic numerals".

The story of zero

  • The reason for suspicion was ZERO.
  • Roman numerals are additive: XII means 10+1+1
  • But, the place value system is not additive,

\(120 \neq 1+2+0 = 3\).

  • Therefore, \(120 \neq 1200 \neq 12000\)

Using zero for financial fraud

  • Florentines complained that "zero has no value in itself,
  • but adds any amount of value to the preceding number."
  • This led to financial frauds: financial contracts could be changed by adding trailing zeros.
  • 120 could be changed to 12000.

Demonising zero

  • Therefore, in 1300 Florence passed a law:
  • financial contracts written in "Arabic numerals" should also be written in words.
  • We still follow this law ( e.g. in writing cheques).
  • Zero from sifr = cipher = secret code.

Dying for Western recognition

  • Foreigners told us a story with a grain of truth:
  • "contribution of Ancient India to mathematics was zero"
  • Their aim: to HIDE the real story of their own arithmetic incompetence.
  • We happily accepted this "small praise" (without investigating) and there is even a film song on it (Manoj Kumar, Purab Paschim)
  • Ancient India's contribution was EFFICIENT ARITHMETIC.
  • But that story shows Europeans as arithmetically challenged, as they really were.
  • Hence, Western historians hide that history,
  • but why do WE erase that real history by talking only of zero and "Vedic math"?

Fractions and the bad European calendars

  • Greek/Roman arithmetic had no way to represent general fractions.
  • Romans understood only simple fractions (parts of 12 on which their coinage was based e.g. 12 pence = 1 shilling)
  • Because Europeans were arithmetically challenged
  • they could not even SAY the right length of the year.

Julian reform

  • HENCE, original Greek and Roman calendar were so awfully bad.
  • After Julius Caesar's conquest of Egypt,
  • under Egyptian guidance
  • the Roman calendar was reformed into the Julian calendar with great fanfare.

Leap year system

  • To avoid fractions (year = \(365 \frac{1}{4}\) days) the Julian calendar used the system of leap years.
  • In the first 20 years, Romans did not understand this system of leap years.
  • Instead of every fourth year as a leap year, they miscounted every THIRD year as a leap year.
  • (They counted the present year as 1 😜.)

July and August

  • Augustus Caesar who corrected this mistake had a month named after him, for this reason.
  • Because the central issue was not scientifc timekeeping
  • but the vanity of the two Caesars, he increased August from 30 to 31 days like July!
  • (And we swallow it all, neat.)

The church/colonial calendar

  • The church adopted the Julian calendar as the official Christian calendar in the 4th c.
  • Church/colonial education brought this calendar to India
  • and it is the ONLY (unscientific) calendar most colonially educated people know.

Gregorian reform of calendar

  • Fractions were first introduced in the European (Jesuit) syllabus in 1570s
  • by Christoph Clavius author of the Gregorian reform of the calendar
  • The Gregorian reform of 1582 corrected the duration of the (tropical) year in the Christian calendar to 365.241 (tropical) days.
  • BUT it used LEAP YEARS, not a precise fraction.

Disadvantage of converting fractions to leap years

  • Every 4th year is a leap year, every 100th year is not, every 1000th year is
  • \(365.241 = 365 + 0.25 - 0.01 + 0.001\)
  • However, this conversion from fraction to leap years accurate only over a 1000 year average.
  • Not from year to year—equinox still does not come on the same day each year.

Summary

  • Efficient Indian arithmetic (basic algorithms for \(+,~ \times\) etc.) went to Europe,
  • repeatedly from 10th to 16th c. for its use in commerce and navigation (calendar).
  • The real story of zero is this:
  • Europeans took over 6 centuries to understand and accept Indian arithmetic.

Relevance to modern technology

Theme of today's seminar

  • "Ancient Indian science: implications for Modern Technology".
  • So, how is the story (I just told) about ancient Indian math relevant to modern technology?

Technology about people, not machines

  • Those who only buy technology think it is about machines
  • but whose who BUILD technology, know it is about knowledgeable people.
  • Multiple failures of technology missions in India due to lack of knowledgeable people.

California is a key hub of this modern technology

  • especially Silicon Valley.
  • We think technology is all about MACHINES.
  • But, ever since the "Sputnik crisis" of 1958, - the US has understood that winning the technology race is about knowledgeable PEOPLE.

Machine learning and data science

  • The cutting edge of modern technology is machine learning and data science.
  • The corresponding math has been called "21st c. math".
  • Properly trained people are needed.

California K-12 proposals

  • Therefore, the California education department has revamped its K-12 school math syllabus
  • to meet the projected needs of future technology.
  • The aim is to teach statistics and data science.
  • But the catch is that this is an ALTERNATIVE to the calculus.

Cancelling the calculus

Statistics needs calculus

  • Core notions of statistics such as probability "measure", average (= mathematical expectation)
  • involve an advanced theory of the (Lebesgue) integral.
  • Further, statistical inference is proverbially misleading (or misused):
  • "there are lies, damned lies, and statistics"
  • Without a proper understanding of statistics, it is easy to arrive at wrong conclusions.
  • A wrong algorithm in artificial intelligence (or an algorithm without an understanding of its limitations)
  • can spell disaster in the future
  • where people will depend more on decisions made by machines.

The alternative

  • A simple alternative is to teach decolonised calculus
  • and
  • the way they originated in India as ganita.

Decolonized calculus

Decolonised statistics

Popular-level account

So, why didn't I talk about this?

  • Because the thesis is that
  • the calculus is difficult because Europeans (Newton and Leibniz) failed to understand it
  • because they stole the calculus from India
  • (and we blindly imitate their lack of understanding).

Epistemic test

  • Knowledge thieves often fail to fully understand what they steal
  • Like students who cheat in an examination.
  • Or the way Europeans took six centuries to understand even elementary Indian arithmetic.
  • (But difficult to explain the same point aboutIndian calculus.)

The solution

  • So, the solution to calculus difficulties is to reject its Western understanding, and
  • to teach calculus the way it originated in India.
  • That makes calculus easy while preserving (actually enhancing) all its practical value.
  • But how can I talk about Aryabhata's philosophy of ganita in calculus (in 30 mins!)
  • to those who don't even know calculus and unsure of the correct spelling of Aryabhata?
  • Read my books and articles, particularly the article appearing today.

Colonial mentality

  • Our understanding of calculus is worse than that of West
  • for colonial education only teaches the colonised to imitate the West,
  • and copy its mistakes including its miserable calendar.
  • So, no point in talking about calculus when you don't know why 1+1=2 in real numbers (and think 30 mins is enough)

One point: Public debate essential

  • WRONG to consult a mathematician privately.
  • Because ganita is so different from mathematics,
  • from which the mathematician derives his livelihood
  • Public debate is essential; but no mathematician in willing for it.

Prerequisite for debate: 1+1=2 test

  • If you know anyone willing to participate in such a PUBLIC debate please let me know.
  • Prerequisite: First meet my "Cape Town challenge", repeated in JNU with a prize of Rs 10 lakhs:
  • "Axiomatically prove 1+1=2 in REAL numbers from first principles (without assuming any result from axiomatic set theory)"
  • like Bertrand Russell's 378 page proof of 1+1=2 in cardinals.
  • Don't judge someone as a "math expert" just by his foreign degrees, judge him by the above 1+1=2 test.

NCERT etc.

Conclusions

  • Indian ganita (calculus and statistics etc.) is very relevant to modern technology (AI etc.)
  • because it makes math easy while preserving all practical value (California math framework).
  • But first we need to understand what real Indian ganita is
  • and reject false stories about "Vedic math", or spelling Aryabhata wrong (in a way which changes history).

Conclusions (contd.)

  • That requires respect for our own knowledge, which the Indian society has lost,
  • because colonial/church education taught us that we are inferior
  • and must hence blindly imitate and trust only the West.