Mathematics is considered to be the `Queen of Sciences'. Rightly so, as the methodology of expressing facts and exploring their relationships logically are of prime significance to the very existence as well as development of the sciences. Thus the language of Mathematics becomes the universal medium of communication for the scientist -- to the extent that often the possibility of logical formulation becomes the test for whether something is scientific or not. Logical thinking has been the foundation on which all the present edifices of Science and Technology are built. However, when we come to certain realms of knowledge at the very limits of human imagination, we start wondering whether we can venture to unravel or even understand the ultimate frontiers of the unknown by pursuing strictly the methodology of logical thought. Let us look at some examples that baffle our imagination and provide interesting tests for our logical abilities.

North of the North pole....

Human capabilities of logical reasoning are probed in depth by the famous Zeno's paradoxes. (Many books, articles and papers are available on this subject, including an article which appeared in Forays Souvenir of 1996) on the never-moving arrow and Achiles and the tortoise. A relatively modern example where our logical mind gets baffled is that of the Big Bang. Leading cosmologist Stephan Hawking puts it this way: What lies at the north of the North Pole? We all know what the word `North' means. We go North and end up in the Himalayas, or may proceed further north in an expedition to the North Pole. Then? What is the concept of `North' once we are at the Pole? Let take this further: It is believed that the Universe began with a `Big Bang'. Fine. When did this happen? And where? What exactly happened before the beginning? We can always murmur something about `singularity' and also vehemently assert that such questions are meaningless. But we must admit that it is indeed difficult to comprehend and to digest.

Some timely comments

A foreigner was traveling in Englad when he found that his watch had stopped. He framed his question this way to a passer-by: `Sir, what is time? ' The Englishman looked at him for a second and answered: `My dear young gentleman, you have asked a very philosophical question'. What is time? A parameter? A coordinate? You might have heard of this cute little definition of time: `Time is something that keeps changing constantly with time'. That really puts things in perspective. Let us complicate matters by asking whether time has a direction, whether there exists an `arrow of time'. We all have heard of Humpty Dumpty having a good fall, none ever heard of his being put back safe on the wall. Your thermodynamics teacher would have told you that things in the world often go in such a way that entropy increases. That means you can never see a thousand tiny glass pieces rushing onto your table and forming a glass tumbler though we often see the reverse. In other words there seems to be a preferred direction in time, for things to happen. That is why we all shout if a movie is run backwards in a theatre. But then we have to be careful. Can we consider the Universe as a closed system? Is it an isdolated system? Isolated from what ? - the rest of the Universe? Could there be pockets of the universe where entropy could decrease, albeit for a short while? What about the `worlds' where matter is replaced by antimatter and parity is opposite to that of ours? Will their clocks run backwards? The expansion of the Universe, an experimentally verified phenomenon, is also considered to be a sort of pointer of time. But then, we don't know whether it was always expanding. People are not sure whether the Universe goes through shrinking phases and Big Bangs after Big bangs! If there is a shrinking phase, how will the clocks run then?

Don't read this!

Well, I know quite well that you will anyway be reading it. Apart from the psychological ramifications of such statements, these also are of great interest to the Mathematician. These are called self-referential statements. Example: `This is a self-referential statement'. (What about another statement which says exactly the opposite?) We can test our logical capabilities by looking at "THIS STATMENT IS FALSE ". Is it true or false? Observe it closely and analyze it. If the quoted statement is true, then obviously it has to be false. If it is false, then the statement is not false and hence it has to be true. Thus you realize that that statement is true if and only if it is false and vice versa.

There is this favorite story my Maths teacher told me in school: A Maths professor was shipwrecked on an island and was soon captured by cannibals. They were kind enough to give him his choice of the mode of his death. "You may make a statement. If it is true, we shall roast you and if it is false, we shall fry you" they said. Of course he made a statement and they could do neither, they had to set him free. What could that statement have been? Think for a while. Then read on. He said "You will fry me". Now if they were to roast him, the statement would be false and hence they have to fry him Well, what if they fry him? Then, of course, the statement would become true and hence they must roast him. So they can fry him if and only if they roast him. Does this story appear to be logical?

Much ado about nothing?

Then there is our good old sincere village postman who once happened to read some book on logic. He is in terrible confusion afterwards: By definition, a postman is one who collects the letters of those who do not collect their own letters from the post office. Once there was a letter addressed to the poor postman. Should he collect it? If does, then he would belong to those who collect their own letters and hence he should not be taking it, by definition. Okay, then let him leave it. Then he would belong to those who don't collect their own letters and hence he must collect it. Should he or should he not? Your Maths teacher will have a lot to tell you about this.

Math and aftermath

One of my childhood dreams was that I would do penance and make God appear before me. He would then propose a boon as He normally does in stories. He would say: "Well, I will grant you one and only one of your desires; what shall it be ? " (What would you answer in that case?) I also had thought of a plan to outsmart Him. I would say :"My one and only desire is the following: Let all my desires be fulfilled!" But then He turned out to be smarter; He never appeared anywhere near me, at least not so far.

Talking about God, let us try to prove logically that He is not omnipotent. Ask this question: "Can God create such a stone that even God can not lift? " Suppose that He cannot create such a stone. Well, if he can not do something, then he is not omnipotent. Okay, now suppose that he can create such a stone. Then He had made this stone which even He cannot lift - thus proving again that He is not omnipotent! The Mathematician likes to ask: "Consider the set of all sets that do not contain themselves as a subset. Is this set a member of itself? " Argue out logically what happens if it were and also if it were not. Take the help of your Maths teacher.

Forays forever ...

Richard Feynman talked about a huge tree that fell down suddenly in a faraway forest. You might think that it would have made a very huge sound while falling. But alas, there was nobody there to listen to the sound. Does the tree make sound while falling even when nobody is there to hear it? Think about it. Do phenomena occur even when there is nobody to `observe' them, to record them? Let us extend it to the ultimate question: Does the world exist even if `I' am not there? What happens when `I' am switched off? Will this `I' become another `I' and the story go on?

Can we comprehend all these forays into the very edges of logic? Are we comfortable? Some of you might want to read more and your Maths teacher will guide you. The interesting point is: can we at all understand all this? Understanding means explaining the unknown in terms of the known. But then there is no guarantee that everything in the world can be thus explained. When you understand at some level, you feel elated and then try to push your level to a higher hierarchy. This goes on and on. Probably the easiest solution is to ignore all these and go back to sleep. Wait a minute - let me tell you what would happen when you go to sleep after reading this: You might you wake up from a dream (dream at level 1) and realize that it all was a dream. Then, after some time you wake up again and find that that was another dream (dream at level 2). Later you wake up again and find that that was another dream (dream at level 3) and then ...
I am reminded of an old Hindi film song: `Sapne me dekha sapnee ..'

Dr. C. Vijayan
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