A Recurring Dream...

I wish I were a silent character in Eco's The Name of the Rose..., to live silently and purposelessly somewhere in the abbey witnessing the whole events...,to smell the medieval air, to go inside the library and read the strangest of strange books..., to discover the secret of God's laughing..., and then... to die without doing anything... unknown... unidentified... but knowing everything and identifying everyone....

Russell's Paradox

Probably, I heard the word paradox for the first time when I read about the twin paradox of Einstein. I read this atleast ten years ago. Then I was not matured enough to understand what the word paradox really mean. It was recently, when I was reading Terrence Tao's book on "Analysis", I was once again confronted with another paradox. This was Russell's paradox. I see what a paradox really means.

Russell's paradox: Let's make a statement.

Let x be any object and p(x) be any property of x (i.e., either p(x) is true or is false). Then there exists a set {x : p(x) is true for x}. -----(1)

This seems to be an innocent statement. But actually, it is a very dangerous and fallacious statement. Consider x be a set and p(x) be the statement :-

x does not contain itself ----(2).

Then according to the statement (1), there exists a set S which contains all sets for which property (2) holds (i.e., they do not contain themselves). Does S contain itself?
1) If S does not contain itself, then by its very definition, it must include itself.
2) If S contains itself, then it should not contain itself.
This is called Russell's paradox. And, it disproves the statement given in (1). In fact, until Betrand Russell discovered this paradox, (1) was considered as an axiom of set theory. Now, a set theory which considers (1) as an axiom is called naive set theory.

Even if one does not understand this, one can have a look at this webpage http://www.paradoxes.co.uk/index.htm, at whose end a few exciting, or rather mysterious pictures illustrating paradoxes are given. To know more about Russell's paradox look here: http://en.wikipedia.org/wiki/Russell's_paradox.

A Great Connection!

So that's it. If a set A containing finite number of integers, not all of which are zero, is selected, then those integers generate an ideal of the integer ring. But, any ideal of the integer ring is generated by a single positive integer a, which must be the smallest positive integer in that particular ideal. Then every integer in A is a multiple of a. Then a is a common divisor of them.

Now, if b is a positive integer which divides all elements of A, then the ideal generated by b is either equal to or includes the ideal generated by the elements of A. In the first case, b is the smallest integer in the ideal and hence is equal to a. In the second case, a is included in the ideal generated by b. Hence, b divides a (and a>b). This shows that a is the greatest common divisor (gcd) of the integers in A.

The integer a can be expressed as a linear combination of the integers in A because a is in the ideal generated by them. That's why the gcd of a finite number of integers can be expressed as a linear combination of those integers.

What a simple but great connection!

Boredom

I am bored of living among people whom I don't understand and who don't understand me.

Many believe that there is a predefined way of living and try to make their lives according to those predefined standards. Those standards are set by religious beliefs, social conventions, popular notions, the media, institutions (including parents, they also constitute an institution), and so on. Some resort on more "scientific" notions such as psychology, behaviour theory etc. It is appreciable to read on these topics and to know them. But searching for a method of living in all these is merely a mediocre activity.

I see many people around me who are formidably confident about life. They look so "nice" and "pleasing" to many. But in my case, they generate ultimate abhorrence in me. They represent lack of contemplation and mere pragmatism for me. I suppose that they have agreed upon some predefined nature of human life and believe that they live according to it. This makes them feel triumphant and happy.

ME SANDEEP

By "Me Sandeep" I meant to write about myself. However, I know it is very difficult to write about oneself. Of course, I can start by introducing my name, where I live, and what I do. That's what I do here.

Hi,

My name is Sandeep. I am from Calicut, a district in Kerala state in India. (Know more about Calicut here : http://en.wikipedia.org/wiki/Calicut, Kerala here : http://en.wikipedia.org/wiki/Kerala and India here : http://en.wikipedia.org/wiki/India). It is needless to mention, I am a wiki-lover.

I am currently a PhD scholar in IIT Madras. My area of research is Signal Processing. I don't have anything more to talk on this. I am just keeping quiet now.

So, self-introduction is over? Let me read it once again. Oh! so boring. So ordinary a style: a usual cliche. So, that's it. I realize that I am very ordinary a person. And, my life is more ordinary. I enjoy this ordinary life. Sometimes I am reminded that there is something grandeur in the ordinary. To see that one has to look.