Pallavi . Talks

Sunday, July 06, 2008

"Paradoxes" @ 2:32:00 pm writes:

This seems like a fascinating topic for me to blog about. At least, I think so, after reading a few articles on paradoxes. I recently Stumbled on www.paradoxes.co.uk , which had excellent references to paradoxes, and I was completely intrigued by the whole concept. Those of you for whom the word seems alien, worry not:). I have compiled togather a complete article (written mostly by me; the problems have been copied shamelessly from the net; after all, I didn't create them!)

Paradoxes

A Paradox, in my opinion is a wonderfully twisting, mind-bending problem which has no obvious answers, or that which contradicts itself leaving no possible answer.

Wikipedia defines a Paradox as -

'An apparently true statement or group of statements that leads to a contradiction or a situation which defies intuition; or it can be, seemingly opposite, an apparent contradiction that actually expresses a non-dual truth'

The Chambers Dictionary, 2006 defines it as-

'Something which is contrary to received conventional opinion; something which is apparently absurd but is or may be really true; a self-contradictory statement'

I hope you get the general idea- a paradox is an impossible situation. Still confused? Allow me to enlighten you with a basic (rather silly) example-

What would you do if you see this notice which said-

PLEASE IGNORE THIS NOTICE

Would you ignore it? If you ignore it, you would actually be noticing it. If you notice it, you would actually not be ignoring it.

That was example number one. Now let's move on to more complex stuff (am I starting to sound like a teacher?? If so, you have the absolute liberty to give me a good whack on my head).

Example number 2 goes like this-
Consider this sentence-

This sentence is false.

This is popularly known as the Liar Paradox. Is that sentence true or false? If it is false then it is true, and if it is true then it is false...

Boggling? Trust me, it gets worse...

Consider this, also known as the famous 'Hilbert's Hotel Paradox', which is based on the concept of infinity...

Imagine a hotel with a finite number of rooms, and assume that all the rooms are occupied. A new guest arrives and asks for a room. "Sorry" - says the proprietor - "but all the rooms are occupied." Now let us imagine a hotel with an infinite number of rooms, and all the rooms are occupied. To this hotel, too, comes a new guest and asks for a room. "But of course!" - exclaims the proprietor, and he moves the person previously occupying room N1 into room N2, the person from room N2 into room N3, the person from room N3 into room N4, and so on... And the new customer receives room N1, which becomes free as a result of these transpositions.

Let us imagine now a hotel with an infinite number of rooms, all taken up, and an infinite number of new guests who come in, and ask for rooms.

"Certainly, gentlemen," says the proprietor, "just wait a minute." He moves the occupant of N1 into N2, the occupant of N2 into N4, the occupant of N3 into N6, and so on, and so on...

Now all odd numbered rooms become free and the infinity of new guests can easily be accommodated in them.

How is this a paradox?

The proprietor's "just wait a minute" seems optimistic; it would surely take him an infinite time to shift the guests around.
This implies that the job will take an infinite amount of time, therefore the infinite number of guests will have to be kept waiting for infinity...

Phew...

Here's a brilliant one I found(please, readers, have the patience to go through each and every word of the post....it'll pay off, be guaranteed about that!)-

The Unexpected Hanging

A man condemned to be hanged was sentenced on Saturday. "The hanging will take place at noon," said the judge to the prisoner, "on one of the seven days of next week. But you will not know which day it is until you are so informed on the morning of the day of the hanging."

The judge was known to be a man who always kept his word. The prisoner, accompanied by his lawyer, went back to his cell. As soon as the two men were alone, the lawyer broke into a grin. "Don't you see?" he exclaimed. "The judge's sentence cannot possibly be carried out."

"I don't see," said the prisoner.

"Let me explain They obviously can't hang you next Saturday. Saturday is the last day of the week. On Friday afternoon you would still be alive and you would know with absolute certainty that the hanging would be on Saturday. You would know this before you were told so on Saturday morning. That would violate the judge's decree."

"True," said the prisoner.

"Saturday, then is positively ruled out," continued the lawyer. "This leaves Friday as the last day they can hang you. But they can't hang you on Friday because by Thursday only two days would remain: Friday and Saturday. Since Saturday is not a possible day, the hanging would have to be on Friday. Your knowledge of that fact would violate the judge's decree again. So Friday is out. This leaves Thursday as the last possible day. But Thursday is out because if you're alive Wednesday afternoon, you'll know that Thursday is to be the day."

"I get it," said the prisoner, who was beginning to feel much better. "In exactly the same way I can rule out Wednesday, Tuesday and Monday. That leaves only tomorrow. But they can't hang me tomorrow because I know it today!"

He is convinced, by what appears to be unimpeachable logic, that he cannot be hanged without contradicting the conditions specified in his sentence. Then on Thursday morning, to his great surprise, the hangman arrives. Clearly he did not expect him. What is more surprising, the judge's decree is now seen to be perfectly correctly. The sentence can be carried out exactly as stated.

A similar, easier one-

On a Monday morning, a professor says to his class, "I will give you a surprise examination someday this week. It may be today, tomorrow, Wednesday, Thursday, or Friday at the latest. On the morning of the examination, when you come to class, you will not know that this is the day of the examination."

Well, a logic student reasoned as follows: "Obviously I can't get the exam on the last day, Friday, because if I haven't gotten the exam by the end of Thursday's class, then on Friday morning I'll know that this is the day, and the exam won't be a surprise. This rules out Friday, so I now know that Thursday is the last possible day. And, if I don't get the exam by the end of Wednesday, then I'll know on Thursday morning that this must be the day (because I have already ruled out Friday), hence it won't be a surprise. So Thursday is also ruled out."

The student then ruled out Wednesday by the same argument, then Tuesday, and finally Monday, the day on which the professor was speaking. He concluded: "Therefore I cannot get the exam at all; the professor cannot possibly fulfil his statement." Just then, the professor said: "Now I will give you your exam." The student was most surprised!

Alright, let's move on to Mathematics (do I hear groans??).

Interesting and Uninteresting Numbers

The question arises: Are there any uninteresting numbers? We can prove that there are none by the following simple steps. If there are dull numbers, then we can divide all numbers into two sets - interesting and dull. In the set of dull numbers there will be only one number that is the smallest. Since it is the smallest uninteresting number it becomes, ipso facto , an interesting number. We must therefore remove it from the dull set and place it in the other. But now there will be another smallest uninteresting number. Repeating this process will make any dull number interesting.

Now the following must be one that has reached you long before you even understood any advanced math. Anyway, have a look at it and its obvious solution....nostalgia....

Assume that

a = b. (1)

Multiplying both sides by a,

a² = ab. (2)

Subtracting b² from both sides,

a² - b² = ab - b² . (3)

Factorizing both sides,

(a + b)(a - b) = b(a - b). (4)

Dividing both sides by (a - b),

a + b = b. (5)

If now we take a = b = 1, we conclude that 2 = 1. Or we can subtract b from both sides and conclude that a, which can be taken as any number, must be equal to zero. Or we can substitute b for a and conclude that any number is double itself. Our result can thus be interpreted in a number of ways, all equally ridiculous.

The paradox arises from a disguised breach of the arithmetical prohibition on division by zero, occurring at (5): since a = b, dividing both sides by (a - b) is dividing by zero, which renders the equation meaningless. As Northrop goes on to show, the same trick can be used to prove, e.g., that any two unequal numbers are equal, or that all positive whole numbers are equal.

Absurd? It sure is.

Now I move on to a much more exciting type of paradoxes- an Ontological Paradox. Before you start freaking out seeing the zsize of the word, chill. It's fascinating and reading this sets the dullest person's brains into clockwork.

So what the hell is the onto-something paradox???

Wikipedia(yes I know, I'm shamelessly copying....but I have no choice...at least I credit dear old Wiki!)-

An ontological paradox is a paradox of time travel that questions the existence and creation of information and objects that travel in time.

In simpler words, it challenges the logic behind the seemingly impossible concept of time-travel.

Let's look at the most famous example for this type of a paradox (again...kill me if I'm being teacher-ish, 'coz I totally know how that feels...).

It's called the Grandfather Paradox.

Suppose a man traveled back in time and killed his biological grandfather before the latter met the traveller's grandmother. As a result, one of the traveller's parents (and by extension, the traveller himself) would never have been conceived. This would imply that he could not have travelled back in time after all, which in turn implies the grandfather would still be alive, and the traveller would have been conceived, allowing him to travel back in time and kill his grandfather. Thus each possibility seems to imply its own negation, a type of logical paradox.

Though who would ever do such a ridiculously stupid, not to mention cruel act, God knows.

The grandfather paradox has been used to argue that backwards time travel must be impossible. However, a number of possible ways of avoiding the paradox have been proposed, such as the idea that the timeline is fixed and unchangeable, or the idea that the time traveler will end up in a parallel timeline, while the timeline in which the traveler was born remains independent.

The concept of a parallel universe is what excites me tremendously, but I won't bore you further by delving into whatever I know about that(at the moment, very limited).

However, I encourage everyone(of my age....don't want Ph.D graduates to laugh themselves silly about how puerile this post is) to read it up....MUCH more interesting than the Physics chapters I'm currently enduring in school(in my opinion the Ontological part comes under Physics).

Anyway, here is music for your eyes(if there ever is such a thing)...in the form of more paradoxes, represented pictorially. Also called Optical Illusions...