To Padmasrinivasan -- Letter from an Asylum

@ Padmini

Your fb was far more erudite than mine. In fact, I don't know much about most of the things you have mentioned. I am very slow in reading others' posts. I am sure one of these days I shall land in front of your door. Incidentally, I was looking at my haiku posts the other day and found that we had exchanged ideas more than once. I had forgotten about them because I was away from IL for a while.

All the best.

oj-da

 

@ Cheeniya

Dear Sri:

It's always an honour to hear back from you. This particular fb was simply delicious. I see it as a colourful piece of art - a rhyme, a limerick and of course the masonic message. Every time I read you, I realize with renewed force why you are so popular and respected. The Freemason bit told me again that you are truly gregarious, as human beings are supposed to be. I know that my greatest shortcoming is my propensity towards reclusion. I am not a developed human.

If you look at the post you have commented on, it is only too clear that I was simply speaking to myself, even though I had given my thoughts the structure of a letter. To tell you frankly, I don't think I believed that the person addressed would really find it useful, even as a funny letter. But I knew I was trying desperately to build a logical structure. In fact, I was trying to convert the square to a circle. I realized that I had failed of course. My stream of consciousness was incomplete. I knew there were too many loose ends that needed to be tied up together.

Whenever I read you, I end up with a feeling of satisfaction, because you define a problem, view it from different perspectives and finally end up with a 'solution'. When I read myself, I simply see a person desperately trying to define a problem. A man in search of a problem! You can't solve a problem before you know what the problem is. As all great researchers say, more than half the research lies in one's search for a problem. A good problem, a tough problem, a problem that you will be solving in a million different ways and yet feel that a million things remain undone.

This observation applies to immortal beings such as Tolstoy, Dostoevsky and others too. I understand that Tolstoy was once discovered sitting in a room with its doors and windows tightly closed during daytime. There was a small chink however (somewhere in a window perhaps) through which a ray of sunshine penetrated the darkness of the room. And Tolstoy was found desperately trying to catch on to the shaft of light with his hands.

The point is of course that he had seen the light and his problem was to find a way of grappling with it. What happens to someone who has never seen any light at all? He has no problem to solve. He sits atop a heap of rubbish, not knowing how to give it a shape.

Anyway, it's too late now. I know that I will never know the answer to my biggest question. And that question is: "What is the question?" It was Henri Poincare who had said this I think. But he knew what the question was, so his struggles were meaningful.

Now that I think about it, in my post I was obviously trying to come up with a humorous portrayal of a problem that has occupied geometers for centuries. Can you convert a square to a circle with equal area, using geometric tools alone and in a finite number of steps? It seems there are partial solutions to the problem, but a complete solution has been proved to be impossible. On the other hand, using arithmetic, we can obviously define a circle and a square to have equal areas. Simply choose the square to have each side equal to the 'square root of pi' and the circle to have a radius equal to 1.

The example proves probably the limits of plane geometry, I mean geometrical proofs. I couldn't make the point clearly enough and ended up beating about the bush! Another example of a contradiction between geometry and arithmetic can be brought up as follows. Let's consider the triangle below and the coloured lines connecting the apex to the base CD. Each line intersects the straight line AB drawn parallel to CD. If we compare the points of intersection of each coloured line with AB and CD, it is clear that AB and CD have exactly the same number of points.

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So, in some sense, AB and CD are equal as far as arithmetic goes. But a geometer would surely claim that AB is smaller than CD. These are the problems Cantor was concerned with and there is nothing new in what I have said above. The resolution of the paradox too has been well-known for many years now.

"In sooth, I know not why I am so sad ..."

Regards.

oj

 

@ Padma

Thanks for the fb. My post is indebted to yours. I have tried to explain to Cheeniya what the connection was. But I am not too satisfied with the letter I wrote. I wanted to be humorous and ended up being far too prosaic! It happens at times. Sometimes one's mind thinks, at others it sleeps. I guess mine sleeps most of the time. And when it wakes up it is surprised to find the world so changed. As was the case with Rip Van Winkle. And then I slip into a delirium. Next time I write a letter to you, I will try to be less delirious.

Have fun!

oj-da

 

@ mindi

That was a fun fb and I truly enjoyed reading it. But I need to disappoint you by a new piece of information now. I didn't discover this. It was all Cantor's work. Well this is the result: The number of points on the real line, i.e. the cardinality of the set of points on the real line, is exactly equal to the number of points on the complex plane!!!!!!!!!

You see, when you are comparing infinities, the usual rules of arithmetic don't work. If you find the time and feel inclined towards it, do please read my response to Cheenia's fb, especially the part where I have drawn a triangle and compared the number of points on two lines, AB and CD. Although AB is smaller than CD, they both contain the same number of points.

As far as the algebra of infinities goes, one really doesn't use the terminology "this is equal to that". We say "this is numerically equivalent to that", or "there is a one to one correspondence between the two objects".

In fact, let me end with a yet more startling statement. The entire complex plane is numerically equivalent to (i.e. contains the same number of points as) a straight line of any chosen length. The line could be, say, .0000000000000000001 inch long. Even then it has exactly the same number of points as the complex plane, or, for that matter, the n-dimensional world you mention, for any finite n.

Infinity is a fascinating and challenging object. The only way you can construct an infinity that is larger than the set of irrationals is to consider the set of all subsets of the set of irrationals.

In the world of infinities, the set of integers and fractions constitutes the smallest of infinities. The next larger infinity is the set of all irrational numbers. The third one is the set of all subsets of the second kind of infinity!! And so on.

oj-da

 

Dear Jaya-san:

Thanks for the lovely end to the fb. Let me add though that "pi" is probably also the first letter in Pythagoras' name written in Greek. And it was the Pythagoras theorem that geometer's used to show the link between arithmetic and geometry!

Happy New Year!

oj