(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

- Bhata (भट) is dalit, bhatta (भट्ट) is Brahmin (
*जनसत्ता*, 30 Sep 2018. - Wrong spelling breaks the story of ancient India's integration
- across regions and caste.
- (Aryabhata, a dalit from Patna, had disciples who were the highest caste Namboodiri Brahmins from Kerala).
- Raju, C. K. 'Aryabhata Dalit: His Philosophy of Ganita and Its Contemporary Applications'. In
*Theory and Praxis: Reflections on the Colonization of Knowledge*, ed, Murzban Jal and Jyoti Bawane, 139–52. Routledge, London, 2020.

- The approach paper for this meeting mentioned "Vedic math".
- However, there is nothing Vedic in "Vedic math" (Hindu 3 Sep 2014).
- वैदिक गणित में वैदिक कुछ नहीं (
*जनसत्ता*, 10 Aug 2014).

- (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.

- 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.

- 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?

- 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.

- 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.

- 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.

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

- "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.

- 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.

- 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.

- 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)

- Gerbert of Aurillac (later pope Sylvester II) imported it from Cordoba (Muslim Spain).
- He had authored a learned tome on the abacus,
- hence wrongly thought that the abacus was the
**only**way to do arithmetic. - Consequently, he got a special abacus (apices) constructed for "Arabic numerals" in 976.

- 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.

- 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).

- 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.

- 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.

- 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).

- 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 |

- 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.

- 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 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, \(120 \neq 1200 \neq 12000\)

- 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.

- 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.

- 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)

- But India's contribution is not just "zero".
- And we did not teach the world to count.

- 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"?

- 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.

- 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.

- 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 😜.)

- 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 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.

- 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.

- 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.

- 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.

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

- 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.

- 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.

- 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.

- 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.

- On the grounds that the calculus is very difficult,
- and teaching it at the K-12 level serves no particular purpose and is inequitable.
- Hence, critics have bashed it as "cancelling the calculus"
- and asked for it to be revoked.

- 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.

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

- See
*Cultural Foundations of Mathematics…the Transmission of calculus from India to Europe in the 16th c. CE* - For a description of the courses see Decolonising mathematics
- or see the video of my MIT lecture, or the slides, or abstract.

- See my article: Probability in Ancient India, Handbook of Philosophy of Statistics, Elsevier, 2012, 1175–96.
- Or see video of my JNU talk on Statistics…should we teach it using formal math or normal math?
- or see the slides.

- Or see this popular-level account.
- Part 2 to appear on 18 Dec.
- (See under "Education" section at Boloji.com)

- 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).

- 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.)

- 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.

- 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)

- 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.

- 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 refuses to discuss the matter (e.g. see this critique)though they promised to.
- Have talked about the advantages of ganita even to the chair of the parliamentary committee on education.
- That is the most I can do in India, it is NOT my personal agenda.
- If you can do better, do it!

- 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).

- 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.