# History of Indian mathematics: IIT Gandhinagar (2017)

There’s a nice series of videos on YouTube, here: https://www.youtube.com/playlist?list=PLRfu94TCePTtgPLfyysqQPMqdwyJvy0Vf

According to the playlist title, it’s from “International Conference on Exploring the History of Indian Mathematics | December 4-6, 2017”

All the talks are interesting if you are interested in Indian mathematics.^{1} Many are of general interest.

Below are just some things I noted down / found worth remarking on. They are not *not* summaries of the talks! In fact I was not paying complete attention when watching these videos; some I just listened to while doing other stuff. Also they are notes to myself so may not be very readable to anyone else, sorry.

(BTW I think the order of videos in the playlist is, for some reason, *not* the order in which they were presented. E.g. video 9 by Ramasubramanian refers to talks that must have happened earlier, which are later in the playlist.)

Also, sometime in the middle of watching these, I had a couple of thoughts about the Indian mathematical tradition, which surely must have occurred to many people, but I haven’t seen written down anywhere (or maybe I have and I’ve forgotten – also, in the last video Clemency Montelle touched on the second point):

Often, it is algorithmic (gives procedures). The right translation in such cases IMO is not into notation with subscripts etc., but into pseudocode (accompanied by an animation of the procedure being carried out and the numbers changing etc). After writing as code, understanding it becomes analysis of algorithms. This probably explains the comment of Knuth (sorry can’t find a reference right now), that he was reading some translation of a Sanskrit text of mathematics and felt he could perfectly resonate with the Sanskrit author, as if he was talking to him directly, and felt he understood him better than the translator – Knuth built his career on “analysis of algorithms”, basically making it up as an academic discipline and inventing the name.

Mathematics is universal. You find mathematics in works on prosody (how many metres can there be of a certain length), on music (how many combinations of notes), on food (how many combinations of six rasas), etc. Similarly Shulba-sutras on ritual practice necessitated its own mathematics, etc. Each tradition developed its relevant mathematics. And once a question was raised, the tradition developed on its own (see Alsdorf article on Pratyayas). (Later mathematicians did realize that the same question was discussed in different fields.) So it is that say poor Kedara-bhatta writing the Vrtta-ratnakara is forced to exercise his mind and come up with mathematics, even though the problem has no relevance to aesthetics. Maybe this is a good thing; the expectation of the Indian tradition was that you were required to be competent in all things (see comments about Bhaskaracharya etc).

Anyway, comments about the videos:

1:11:32

01| Prof. P.P Divakaran | Two masters: Aryabhata and Madhava, a reassessment

First, P. P. Divakaran (who was my teacher at CMI! but I didn’t know at the time he worked on this…) gave a good talk on Aryabhata and Madhava. One of the things he had is a good perspective on the “calculus-like” mathematics that we find in India, in Bhaskara etc. and most prominently among the Kerala group: he said “it’s not pre-calculus or proto-calculus; it *is* calculus just specific to trigonometric functions” (because that’s all they cared about). This makes a lot more sense and is a better description of the facts (e.g. it makes complete sense how Madhava had the π/4 series earlier than Gregory-Leibniz) – the ideas were not the ideas that come “before” calculus, but the only difference with “European” or “modern” calculus is that it’s not about general functions. (And frankly, one would find it hard to argue that Newton or Leibniz had “general” functions in mind either, but well…)

37:00

02 | Prof. Kim Plofker | Heliocentrism in 18th century Sanskrit mathematical astronomy

We know that Sanskrit works on mathematics and astronomy were translated into Arabic etc. in early centuries, and there were also translations in the reverse direction in the 18th century. She had some mention of how these were translated: a Persian/Arabic scholar and a Sanskrit scholar would sit in the same court, the former would read a section and translate into the local language (think something like Hindi), then the Sanskrit scholar would put that into Sanskrit. Though not all.

18:37

03 | Prof. Anil Narayanan. N. | Relevance of Paramesvara’s commentary on Lilavati

I unfortunatately don’t remember much of this talk except being impressed by the speaker. It is about a particular commentary on Lilavati, which has many commentaries – I think this is the talk in which it was mentioned that after the Bhagavad Gita and another work, the Lilavati seems to have the most number of commentaries.

56:32

04 | Prof. Amartya Kumar Dutta | Antiquity of the Arithmetic Mean

This is a fantastic talk. First, he argues that the simple-to-us idea of the arithmetic mean (not the average of two numbers, but of many numbers) is not found in Europe until very recent centuries (17th or something). He has a very persuasive argument. One may think that it’s a natural thing to combine N observations by taking their mean, but he points out that it is counter-intuitive: you would rather throw away the “poor” observations and keep the “most reliable” one, than just average them. So how come it was found in India? It seems to be in the context of averaging not multiple observations of the same quantity, but of different quantities: finding the average depth of a ditch by taking the depths at different points (there are some hints of the Riemann integral here).

He ends with some questions about the popular conception (among Indians) about the history of Indian mathematics.

36:11

05 | K. Mahesh and Aditya Kolachana | Contributions of lesser known mathematicians in Munisvara

Like talk 3, the speaker seemed impressive but I don’t remember much else.

18:02

06 | Aradhana Anand | sin 18’ based on Bhaskaracharya’s Siddhanta Shiromani

This is a young girl still in school (see https://www.youtube.com/watch?v=okVRbUK5hoY and https://www.youtube.com/watch?v=FAY7oOpHR2U), who gave the talk in Sanskrit and then translated into English. One thing that was not clear (maybe I was not paying attention) is to what extent this is based on Bhaskaracharya’s Siddhanta Shiromani and to what extent her own intelligence (with only the value of sin 18 degrees coming from the ancient work and not the derivation). Either way, in one way she has truly engaged with the tradition in a manner no one else has: she composed an Arya verse for sin 54 degrees in a very similar style, with bhutasankhya and everything.

48:06

07 | Prof. M.S. Sriram | Spherical trigonometry in medieval Indian texts on astronomy

I should write down notes closer to the talk :-(

1:01:34

08 | Prof. Parthasarathi Mukhopadhyay | ZERO - An Eternal Enigma

A great talk by a “zero enthusiast” summarizing the state of historical research about 0: what’s non-obvious about it, what was different in India versus elsewhere, etc. (He starts with the observation that although historians and scholars don’t know any specific person who “invented” zero, the average Indian seems to confidently know the answer. :P)

1:01:30

09 | Prof. K. Ramasubramanian | Evolution of the style of composition in Indian mathematics

Great talk as always. (A bit wide-ranging as usual maybe. :) The core point is just that we see sutra style, then simple metres like arya, then more elaborate/beautiful metres like shardulavikridita. But the speaker touches so many more things.)

26:26

10 | Aditya Kolachana & K. Mahesh | Novel mathematical modelling of pranakalantara by Madhava

Was quite technical (astronomy, declension/ascension and longitude, spherical trigonometry, ecliptic, bhujAphala, koTIphala, antyaphala, etc). Apparently Madhava derives some quantity in a few different ways, just for the joy of it – shows his genius and worthy name of “golavid”.

1:14:51

11 | Prof. M.D. Srinivas | The Chakravala compared with the Euler-Lagrange method

Starts with some remarks (drawing from Andre Weil) about history of science being for scientists (see Knuth’s talk on a similar topic). Then goes into details of the Chakravala method. I would like to re-watch this video as I’d like to learn the method.

14:14

12 | S. Agarkar & R. Gondhalekar | Using Lilavati to Create Interest in Mathematics

Results from some workshops run for high-school students. The students enjoy it. Something addressed “to a mathematician” seems to interest them. Expressions like “O doe-eyed one” are fascinating to the girls. The students are surprised to see mathematics teachers who are smiling: generally the opinion is that mathematics is a difficult subject and must be studied seriously. Often the school administration, before the workshop, think students won’t be interested for 1.5 hours (especially a mathematics lecture), but it turns out that students are.

31:45

13 | Prof. Venketeswara Pai R. | Mathematical basis for the Vakya system in Karanapaddhati

(Technical)

53:12

14 | Prof. S.G. Dani | Syamadas Mukhopadhyaya : his life and works

About a mathematician 1866–1937. One interesting titbit from the talk is that Lobachevksy and Bolyai, having independently come up with hyperbolic (non-Euclidean) geometry, only learned of the existence of each other about 20 years later – so unheralded and “weird” were these results. It took Beltrami and Poincare for non-Euclidean geometry to gain some respectability.

42:34

15 | Prof. Clemency Montelle | Computing Sines: Trigonometry in Nitynanda’s Sarvasiddhntarja

A nicely presented talk. Even a technical topic can be made interesting if you put some effort into giving the talk well. Despite her pronunciation she reads out Sanskrit verses with enthusiasm, which is nice to see :-)

39:20

16 | Jambugahapitiye Dhammaloka | Direction, place, and time in Sripati’s Siddhantasekhara

Seems to be a Sri Lankan monk who was doing his PhD with Clemency Montelle and Kim Plofker. Another technical talk. Aside: I think what some of these research talks miss is what Simon Peyton Jones said (see page, video, slides, paper): a research talk should not be your paper’s contents but be an advertisement for your paper (get people excited to read your paper).

50:31

17 | Prof. Atul Dixit | Ramanujan’s formula for ζ(2m + 1) and subsequent developments

Wish I was watching the video as well, as the material sounded quite interesting. Was a very nice talk even just audio-only. (See Zeilberger’s “Music and Lyrics”) A fan of Ramanujan :-)

1:00:50

18 | Prof. Michel Danino | India’s Crosscultural Exchanges in Mathematics and Astronomy

This is the first time I’ve seen/heard (as opposed to just read) Michel Danino, and he does seem to be quite more nuanced than I had imagined. For example he points out that the 60-minute-hour is not an Indian idea (which preferred the 24-minute ghatika, of which there were 60 in a day, or the 48-minute muhurta), similarly the 7-day week (rather than the fortnight paksha).

Aryabhata was translated into Arabic and thence into Latin, was known as Ardubarius.

This is a great talk – for a general audience this is one of the talks I’d surely recommend.

18:45

19 | Aradhana Anand | Soothram Sulabham Sundaram - A method to find Rationals between Surds

Was happy to see another talk from her. She has really understood and digested the Baudhayana shulba-sutra method from kalpa-shastra. Then she’s able to solve a problem from her high-school textbook using a method “based on” the Shulba-sutra. She has a good guess for how the shulba-kara got the famous 1 + ⅓ + 1/(3*4) - 1/(3*4*34) approximation.

Seeing talks like these two by her, and this by Siddhartha, it’s really wonderful to see how these young people are really entering into the intellectual/cultural tradition, not just speaking about it but from within it. That’s what we need more of.

35:58

20 | Priyamvada. N. | Indian Mathematics in the mainstream

Apparently she volunteers at her local temple in the USA, and teaches Indian-origin American kids about mathematics in Sanskrit texts. Shows how mathematics was done in India, the actual words and notation used.

Another nice talk, suitable for generic audience.

25:29

21 | Prof. Sivanadan D.S. | The rope technique to lay out fire altars in the Kerala Vedic tradition

The *real* Shulba-sutras, how Vedic yaga practice in reality uses them today in Kerala, etc. Including a legend about a person named Agnihotri who performed 99 athi-rathras, each time increasing the area (as required), until stopped by Indra himself. Photos of rope etc. There’s something super interesting casually dropped as the last sentence: He says these days people are using a modern tape measure even though they know the old method.

1:22:15

22 | Panel discussion: “What is Indian about Indian mathematics”

PPD has an interesting point: if Indian mathematics really was non-rigorous and had no proofs, then we’d expect to see a lot of incorrect results stated. But in the entire history only two are known, and both by the great Aryabhata, for the volume of a cone and tetrahedron (who probably reasoned by analogy in those cases).

Plofker: Greek mathematics is rudimentary in estimation, approximation, iterative procedures, etc., because it was constrained by the axiomatic method of Euclid. (You can’t give a geometric proof of an estimation method, or draw a static diagram of a dynamic iterative process.) Similarly the Greek concern with prime numbers don’t appear till late in Indian texts, though they are concerned with divisibility, etc – prime numbers as a category show up later. (Narayana I think?)

A K Datta:

- The algebraic character of trigonometry for example, is Indian in character. (Even though it’s geometry of triangles, worked out algebraically.)
- He also mentions something about the Indian tendency of building up big things from little things.
- Also, people who write on the decimal system tend to focus on the notation, missing the system’s presence in oral traditions.
- On theorems: No one credits Eilenberg with inventing topology, even though he gave axioms, and no one denies that Poincare had theorems, even though they came before axioms: his proofs are not considered “empirical”.

K Ramasubramanian: Terms are well-defined and memorable, etc.

Clemency Montelle: Mentions that unlike Greek tradition where mathematics was privileged / central, in India mathematics is found in many unlikely places (and grammar is central).

After this there are some people asking questions to the panel… the word “blowhard” comes to mind; let me see if I can find a more polite word to describe (many of) them. Anyway, I think they are from the history department or something. At every Q&A there’s at least one person who needs to be reminded to ask a question rather than make comments, but here almost everyone is like that!

On the comments from earlier: see Knuth comments on notation for algorithms / computational processes – from *The Early Development of Programming Languages* (1976), as reprinted in Chapter 1 of *Selected Papers on Computer Languages*:

Before getting into real programming languages, let us try to set the scene by reviewing the background very quickly. How were algorithms described prior to 1965?

The earliest known written algorithms come from ancient Mesopotamia, about 2000 B.C. In this case the written descriptions contained only sequences of calculations on particular sets of data, not an abstract statement of the procedure; it is clear that strict procedures were being followed (since, for example, multiplications by 1 were explicitly performed), but they never seem to have been written down. Iterations like ‘

fori := 0step1until10’ were rare, but when present they would consist of a fully-expanded sequence of calculations. (See [KN 72] for a survey of Babylonian algorithms.)By the time of Greek civilization, several nontrivial abstract algorithms had been studied rather thoroughly; for example, see [KN 69, p. 295] for a paraphrase of Euclid’s presentation of “Euclid’s algorithm”. The description of algorithms was always informal, however, rendered in natural language.

Mathematicians never did invent a good notation for dynamic processes during the ensuing centuries, although of course notations for (static) functional relations became highly developed. When a procedure involved nontrivial sequences of decisions, the available methods for precise description remained informal. and rather cumbersome.

…

Mathematicians would traditionally present the control mechanisms of algorithms informally, and the computations involved would be expressed by means of equations. There was no concept of assignment (i.e., of replacing the value of some variable by a new value); instead of writing $s \leftarrow -s$ one would write $s_{n+1} = -s_n$, giving a new name to each quantity that would arise during a sequence of calculations.

- “Indian mathematics” in the sense of the mathematics historically done in the Indic tradition. What used to be called “Hindu mathematics”, as in “Hindu-Arabic numerals”, but is no longer called that because these days “Hindu” is often not taken to include Bauddha/Jaina.
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