Sunday, 3 January 2016

Manjul Bhargava on Sanskrit and Mathematics

I attended the Lecture on "Sanskrit and Mathematics" by Fields Medallist Manjul Bhargava at the Kuppuswami Sastri Research Institute, Mylapore, which is part of the Sanskrit college. The following is my collection of notes, which I typed as he spoke. 

----------Begin Notes------

I thought I'd be meeting a small set of students and here is a full house, he begins.

Rich literature in Sanskrit, which is disappearing. Europeans preserving Latin and Greek. Most nations, Germany France Japan South Korea, teach science and math in local languages, one source of their wealth. They use English as second language. It's much easier to learn concepts in local language at young age. Lucky my grandfather was a Sanskrit scholar. At home we had a great library of classical Sanskrit texts. I learnt the Sulba Sutra as a child, before learning from Western books in mathematics. 

Pingala Chanda Sastra. I learnt a lot from Pingala. We have to do this scientifically, good translations, bring these alive in schools in correct accurate way. Repeats the phrase "correct accurate" several times.

(Brief interruption because some people can't hear properly. Actually I can hear, they may have a problem with the accent.)

Lots of treasures in ancient languages in India. Not just scientific, also poetry literature philosophy.

There is an initiative at Harvard, the Murty Classical Library. Which publishes five books each year, mostly translations, in English. Books that have never been translated into any language. Hope we can see them in Tamil Telugu Hindi Bengali, all Indian languages. Most of the translators not Indian because most researchers are not Indian. Yes there is a website (in response to a question). Several mentions of this Murty classical library.

{Some one in audience randomly pops another question. And he is asked to wait until Bhargava finishes.}

I have a great interest in history of mathematics. I learnt quite a bit of math from Indian works and then I would go to school and find out theorems named after some one else!

In most of my research I went to the original sources - Gauss Hemachandra etc. Instead of learning from how people thought about a concept in later centuries you can go to original source and find why that person thought that way and where he got his ideas. Nice to learn in its basic forms. There are insights in original sources that have been forgotten in later references or text books.

Bhargava lecturing at KSRI

I see debates in media about what ancient Indians or mathematicians knew. But they are often two sides just giving opinions with no evidence for what they are saying. Problem is some of these are not available in translation.

Not just translate but connect with the modern way of thinking. Not just Sanskrit but other sources too. How is it different? What inspired a concept? We need interests outside Sanskrit too.

Music and math interested me. Too vast literature, one has to specialise. Somethings I found about math. Only someone who knows Sanskrit and math can understand. And that's a small number. That is not acceptable.

I'll give three examples. "India's contribution to mathematics is zero." True, it's one of the contributions. India created the form in which numbers are used today. It  got transported to Arab world then to Europe who called them Arabic numbers. And now Indians call them Arabic numerals. 

We have to wait for USA to change the terminology.  US mathematics text books now call them Hindu Arabic numerals, because India won't take the lead. Perhaps we will copy from USA. In the Arab world, they are called Hindu numerals.

This system of numerals is incredible and this is one of the greatest achievements in human history. 

When Hindu numerals moved westwards they caused a revolution in mathematics but also in economics. You couldn't think about large numbers or more than a few thousand years. The concept that any number can be written with just ten symbols did not exist anywhere. And once it spread, it changed everything.

I feel ashamed that interest is greater outside India than here. India can help a lot.

There is a fantastic inscription in Gwalior. About 600AD. There is an even older inscription. Shahpur?

We liked to make large numbers and name them. Ten to the power 140. One word for this, in a manuscript I saw.

Phonetics of Sanskrit. Very important. Big revolution in 18th century after Europeans studied it. The Organization of sounds in Sanskrit is amazing. Two variables: one, the organ of speech,  where the sound is produced ; and two, eleven categories of modulation. This is Panini's contribution. You can't say of any other language that it's pronunciation has stayed unchanged for centuries. Basis of modern system of phonetics. Not just Sanskrit, Indian languages.

There was a big debate last year about Pythagoras theorem. Whether it was discovered in India. No shred of evidence that Pythagoras ever proved that theorem, whereas Sulba Sutra has clear evidence of proof.

Text books show no  historical context whatsoever. One gets no understanding of context and conditions under which some new concept was discovered.

Origins of trigonometry. Sine function originates in Aryabhateeya. The notion of jya is the origin of Sine and trigonometry.

Brahma Gupta is one of my great inspirations. One of the greatest mathematicians of all time. Gave the verse that translates to roots of quadratic equation. Every school boy should learn that. (Not integers??! ) . Negative numbers introduced by Brahma Gupta.

Fibonacci numbers. Called Hemachandra numbers in Sanskrit. Mentioned over and over, in Sanskrit texts, long before Fibonacci. Studied in several fields. Some think Fibonacci numbers mentioned in Pingala.

Objective clear history of development of ideas in India has never been written.

Pingala's Meru Prastara is called Pascal triangle in India. Is it clear in Pingala. Commentators before Pascal mention it. Meru Prastara shows one of the most important concepts in math and science, called binomial coefficients.

Yamatarajabhanasa. This sequence is not in Pingala Chanda Sastra but is in the oral tradition. What is the oldest written reference? Earliest reference is an English book in  1882. Balu sir mentions Don Knuth and Bhargava nods in agreement, but expresses frustration about not tracing it back to an older reference.

Calculus. Foundations developed by Madhava in India, which wrote in a mix of Malayalam and Sanskrit. Ramasubramaniam (of IIT Bombay) and his circle have brought this out, he says.

-----End of notes on Manjul Bhargava lecture -----

Gopu's comments

It may have been a slightly difficult lecture to follow for those not familiar with mathematics. The acoustics and the accent exacerbated the communication gap. But I found the speech delightful and ambitious. A Fields medalist with such a deep concern and curiosity about the history of mathematics, such a vivid knowledge of Sanskrit works, a deep passion to correct the fundamental lacunae in text book structure is a breath of fresh air.

His remarks on going to original sources, applies to every single field. I agree here most wholeheartedly. If pursued this is where the greatest good can happen in academia. Reading Aryabhata, VarahaMihira, Bhaskara, Lagadha in the original Sanskrit is a phenomenal experience. Even reading translations of their original works in English is far more informative than reading a book about them. This also applies to other fields. I have thoroughly enjoyed reading Adam Smith, Thomas Malthus, Charles Darwin, Henry Ford, Alfred Russel Wallace, Benjamin Franklin, GH Hardy, Thomas Huxley, in their original words. Even translations of Vitruvius, Plutarch, Al Beruni, Al Khwarizmi, Leonardo da Vinci etc. give us insights, which books about them simply cant.

Lynn Margulis mentions this philosophy of reading original sources in her description of course work at the University of Chicago.

I wrote a lament in September titled, "What did Brahmagupta do?" Bhargava's lecture answered that question  most resoundingly.

Bhargava confined himself to  mathematics and linguistics, leaving aside the Indian accomplishments in  Astronomy and medicine. Indian ignorance about the linguistic accomplishments in Sanskrit is stunning.

Brahmagupta discovered integers. This is a more fundamental breakthrough than even his sloka for the roots of a Quadratic Equation. And the Sulba Sutra of Apstambha gives the first irrational number, the square root of two. Bhargava mentioned Brahmagupta discovering negative numbers, but I don't think the public fully understands the impact.

They are obsessed falsely with Aryabhata gravity and revolution!  The Indian obsession with Pythagoras theorem also puzzles me. We should get a solid understanding of what Indians did rather than try to figure out how some India discovered something before some European - this sentiment reeks of an inferiority complex, not scientific curiosity. I think between Madhava and Jyeshtadeva they discovered infinitesimals. Whether this can be called calculus, I don't know. But I've not read either Madhava or Jyeshtadeva, so I can't judge. Newton and Leibniz discovered calculus after the advent of Cartesian geometry, which to my knowledge Indians did not develop.

Rajagopalan Venkatraman takes a photo of Bhargava after lecture at KSRI campus


  1. Brilllliant, Gopu, Brillliant notes!

    1)*Whether Infinitesimals can be called Calculus. I don't know." I thought you understood it when I told you: Infinitesimals is the germ of Newton-Leibnitz's Calculus.

    2)Newton-Leibnitz Calculus germination of basic cocepts did not have to depend on the birth of Co-ordinate Geometry. The Concept of continuous time variation perceived by Gallileo [acceleration [due to gravity in the first instance] led to 'infinitesimal' as mathematical thought entity even before the advent of graphs of functions.


    1. Yes I understand that infinitesimals are the germ of calculus. But calculus is more than that, we understand it as including differentiation and integration and the whole set of mathematics derived from them.

      I also understand that Newton-Leibnitz calculus did have to depend on Cartesian geometry, but they exploited it, didn't they? I cant imagine an area under the curve without x and y axis...

      I can understand Brahmagupta's Bija ganita as the foundation of modern algebra, without the five basic arithmetic signs + - * / = though I wouldnt try it with Roman numerals. Calculus without co-ordinate geometry would be a more bizarre creature.. I would love to see it fleshed out, though!

  2. Well, I missed the talk..I went to Sanskrit collage, but it was full and from hwere I sttod, I could not hear I walked out. Nicely summed up by you

  3. Gopu: Had some interesting conversation with TV Venkateswaran on Indian Astronomy. He says he is doing some work on this and will send me a draft copy of his article. When he is in Chennai next we should arrange for a discussion.

  4. Badri: Looking forward to it.
    Arkay: Thank you sir.

  5. Very useful. When I wrote my "The number line" article, this history of number origin is something that I lacked. This article seems to give me those pointers. Thank you.

    For your line,
    "I think between Madhava and Jyeshtadeva they discovered infinitesimals. Whether this can be called calculus, I don't know."
    I'm reminded of how I tried to connect between limits and calculus years ago while trying to understand the concept. So, if I may take the liberty of using Infinitesimals and Limits interchangeably, here's how I think of the connection between Limits and Calculus:
    The concept of Limit serves as a tool for two goals :
    1. Limit of f(x) at x=x1 is an extrapolation procedure to find the value of f(x) at x=x1 when the values of f(x) are known at some discrete values of x around the neighbourhood of x1 (both on the left side and the right side of x1). In this sense, it helps you form a continuum from a set of discrete values of function. i.e. This 'Discrete to Continuous' perspective forms the basis of Integral calculus.
    2. On the other hand, Limit also helps you discretize a continuous function to be able to calculate the change of f(x) with respect to minute changes in x. This 'Continuous to Discrete' perspective forms the basis of Differential calculus.

    That said, my mathematical knowledge is bereft of any deep research or book study. I attempt to intuitively figure out. At some point, I did realize I should do a study of the work available already. Hope I get to do that sometime soon.