Tuesday, November 23, 2004

The Man Who Knew Infinity:

A Life of the Genius Ramanujan
by Robert Kanigel

Srinivas Ramanujan (1887-1920)In the past few decades, we have witnessed how Ramanujan's contributions have made such a profound impact on various branches of mathematics. The book, "The man who knew infinity", by Robert Kanigel reached out to the general public the world over by describing the fascinating life story of Ramanujan. And now, in the form of a play, the public is made aware, once again, of this wonderful story. This is a very impressive play and I had the pleasure of seeing it with Prof. George Andrews, the world's greatest authority on Ramanujan's work and on partitions.

The more I read about Ramanujan I was attracted to the idea of him being able to predict outcomes devised in some general mathematical way that seemed easy to him, but in the case of the Taxi Cab and Hardy's question, what was the nature of the taxi cab's number?

When looking through this infomration I come across interesting perspectives about such a man whose culture was far away from the society of currents math and physics. Who developed logically, through his own study. This is a intersting case to me of what could be brought to a world steep in mathematical structures. CRanks or not, whohave found joy in peering into a world that few would cosider in their day to day lives.

No account of Ramanujan is complete without the Taxi Cab episode. There is a charming scene of Hardy meeting Ramanujan in a hospital, and when Hardy mentions that he arrived by the taxi numbered 1729, Ramanujan immediately points out that 1729=10^3+9^3=12^3+1^3, the smallest positive integer that can be expressed as a sum of two cubes in two different ways. Of course, the Ramanujan taxicab equation x^3+y^3=z^3+w^3 yields Fermat's equation for cubes by setting w=0, but it is to be noted that the taxicab equation has positive integer solutions, whereas Fermat's does not.

There is something deeper here that has caught my attention. The Harmonical nature permeates my thoughts, about the extra dimensions and how we could look at the bulk?

If such thinking went beyond the two points and our focus was drawn to the space in between, what would such a nature exemplified by such harmonical attributes? Would it not have made one wonder how the world would seem in ways that our current perceptions are not accustomed too? How would Ramanujan fit here, as a emergent property of strings?

When strings vibrate in space-time, they are described by a mathematical function called the Ramanujan modular function.26 This term appears in the equation:27

[1-(D - 2)/24]
where D is the dimensionality of the space in which the strings vibrate. In order to obey special relativity and manifest co-variance), this term must equal 0, which forces D to be 26. This is the origin of the 26 dimensions in the original string theory.

In the more general Ramanujan modular function, which is used in current superstring theories, the twenty-four is replaced by the number eight, making D equal to 10.28

In other words, the mathematics require space-time to have 10 dimensions in order for the string theory to be self-consistent, but physicists still don’t know why these particular numbers have been selected.

I was interested in how this man came to think and I place this here for consideration from another poster.

Dick Well, in Ramanujans case we have some clues.

He spent his teenage years studying and mastering one book, on analytical function theory and analytical number theory (which are joined at birth). The format of the book was step by step: it started with (x-y)(x+y)=x2 - y2. You proved that and then came a slightly harder theorem of the same kind, in which the proof uses the first theorem, and so on one result building on another up to the state of the art as it was when the book was published (1880s). The book was intended to prepare or "cram" Cambridge students for the math exam known as the Tripos.

Ramanujan became not just familiar with the math in this book, it became his environment.

A recent author has suggested that math ability derives from the brain abilities used in social understanding. Think of living in a tribe or small town where "everybody knows everybody". By growing up in such an environment you know not only everyone else's name, but their preferences and personal characteristics. You are freely able to think what so-and-so and such-and-such would talk about if they had a conversation. And it is proposed that mathematicians have this same ability, only with the abstract things they think about and discuss, rather than people.

And so Ramanujan's "town" was the complex number system and the various peculiar things that could happen there. This was the focus of his imagination throughout his growing up and it is scarcely surprising that he was able to see relationships that more lazily prepared mathematicians (including great ones like Hardy) could not.

You don't have to postulate extra dimension, the depth of the human capabilities is sufficient.

As you can see Dick rejected the idea that such information could have settle in any mind, from a fifth dimensional consideration, and the extra dimensions, as he has stated. It just made sense to me, that solid things, had other information that it concealed. The harmonical nature was not only limited to the numbers and oscillations?

The Planck Epoch to now, contained interesting information. Should we find some structure to contain it all, and yet, find that this structured world is not limited to such structures alone? It was just a pattern, and one of many?

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