Mast Kalandar

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Fri, 12 Oct 2007

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The Mathematics of the "Kerala School"

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1. One the use of "natural" vs. "formal" language for the communication of mathematics.

The advantage of natural language is that it is easier to understand in the context of the communicator's daily life.

The advantage of formal language is that it is easier to transmit over time and space without siginificant distortion of the content.

In both cases the advantage is also a disadvantage! Natural language texts become significantly harder to read if one is not embedded in the same context as the author (think of a book on statistics that refers to baseball!). On the other hand, formal texts are extremely dense and need substantial "unzipping" in order to understand.

Clearly, one aspect of this was understood by those who decided that Sanskrit verse be used as the formal language for the transmission of mathematical (and other) knowledge.

What they probably missed was another difficulty which one can see in a modern context given our inability to play LPs which were invented only 50 years ago! Even accurately transmitted data is of no use if there is no one who can read and understand it. So if everyone becomes a "blind transmitter" of knowledge and the tribe of those who understand, interpret and more importantly, criticise and modify the knowledge is steadily diminished, the knowledge dies.

Perhaps, by writing part of their mathematics in Malyalam and part in Sanskrit the "Kerala school" was probably trying to get the best of both worlds --- just as all good exposition (of mathematics) does.

2. Is the use of "algebra" or notation the same as the use of abstraction?

This issue could be confused with the previous one, especially by philosophers!

If the readers of a mathematical text are familiar with the algebraic notation used, then this use of notation is of great computational convenience. For example, once decimal notation for numbers is employed, it is much more convenient to carry out calculations in it than it is to carry out calculations in "natural language" --- and so decimal notation is part of natural language for most people nowadays.

Abstract concepts are quite different. In formulating the notion of a group one is taking as giant a leap as in formulating the notion of counting. It is like creating a new "word" (I almost mis-typed "world"!).

It is clear that the Kerala school had many of the abstract concepts for calculus but did not succeed in developing the notation for it. This might seem paradoxical given that Bhaskara had developed so much algebraic notation. However, it is the natural cycle of things that abstract notions often develop before appropriate notation for them. Look at how Riemann developed the ideas behind the theory of manifolds well before we could find the appropriate notation to use them conveniently.

Now mathematicians (or more specifically, algebraists and analysts) seem to be more than happy to deal with the formal aspects and leave the process of abstraction to "others"; at the same time physicists (or geometers) and other scientists (biologists!) often spend most of their time in the creation of concepts and relegate the actual computations with these concepts to "mere algebra".

However, really remarkable people are able to fluently combine whatever resources they have on both sides to achieve something remarkable --- like the Kerala school did. Bhaskara did too since he invented the concept of a "variable" and was able to put it to good use --- or perhaps he invented the concept of a variable since he already had seen how useful it is!

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