Paradoxes: Be Logical, Yet Illogical

I was trying to daydream, but my mind kept wandering. ~Stephen Wright

I love paradoxes. They simply boggle the mind. Despite being built upon sound premises, they ultimately lead to senseless, illogical or self-contradictory conclusions. Here’s one of the best one I have come across.

Ever heard of Zeno of Elea, the bearded Greek guy? Well, it happened that he did not only come up with philosophies but is also accredited with creating several famous paradoxes. I’m sure you have heard of the race between the tortoise and the hare but try this: “The Paradox of the Tortoise and Achilles”. The former anecdote tells of how the hare lost the race due to his arrogance despite having the advantage against the tortoise. In Zeno’s version of the story, the race did not actually take place. But the tortoise did predict he will win the race. He argued with Achilles with seemingly logical reasoning till the simple-minded Achilles eventually believed him! The simplified version of the story goes thus. Judge for youself how ‘logical’ is the Tortoise.

It came to past that the Tortoise challenged Achilles to a race, claming that he would win as long as Achilles gave him a headstart. Being a mighty warrior swift of foot, Achilles was certain he would win.

“How far do you need?” asked Achilles.

“Ten metres.”

Upon hearing this reply, Achilles grew in confidence. Rolling on the floor and laughing away, he replied, “Be my guest then!”

The Tortoise, however, replied seriously. He claimed that he would win the race nevertheless and could prove it by a simple argument. He argued that no matter how fast Achilles could cover the initial ten metre between both of them, he should have covered a distance albeit a short one, say, one metre, during the interval. Now, Achilles would have to cover the one metre. No matter how fast he could go, the Tortoise would have gone a little farther away. Achilles would then have to cover that distance to catch up with the Tortoise. And while he was doing so, the Tortoise should have added another new distance between him and Achilles.

“So you see, everytime you are catching up the distance between us, I would keep moving so that you have to cover a new distance, however small it is, for you to catch up again. The process will go on ad infinitum and you shall never catch up with me!” the Tortoise elucidated.

And so, our mighty hero, Achilles conceded defeat.

While the argument given by the Tortoise seems sound, it does not sound to logic at all. The flaw in the logic, however cannot be easily pointed out for the logic of the situation seems impossibly assailable! The Tortoise would win the race! Yet, we know he woudn’t.

There are also several quotations by famous persons which make good examples of paradoxical statements and not without a touch of humour. American president, Abraham Lincoln wrote, “I’m sorry I wrote such a long letter. I did not have the time to write a short one.” Pablo Picasso, the famous Spanish artist and painter declared that he would like to be “a poor man with lots of money”; while Stephen Wright claimed to be having “amnesia and déjà vu at the same time”.

Believe me or not, it is possible to tell a lie without telling a lie! In philosophies, liar paradoxes are common. Consider this sentence: “This statement is false”. Or imagine someone appraoching you and tells, “I am lying.” Well, think it over… Are you still with me?

I also happened to stumble upon this mind-boggling brain teaser, which serves as an addendum to Zeno’s paradox. Take this paradox of Thompson’s Lamp away with you.

“Consider a lamp initially turned off. Let us imagine there is a being with supernatural powers who likes to play with the lamp as follows. First, he turns it on. At the end of one minute, he turns it off. At the end of half a minute, he turns it on again. At the end of a quarter of a aminute, he turns it off. In one eighth of a minute, he turns it on again. And so on, hitting the switch each time after waiting exactly one-half the time he wated before hiting it the last time.”

It can be easily deduced that all these infinitely many time intervals add up to exactly two minutes. The question is: At the end of two minutes, is the lamp on, or off?

Here the lamp started out being off. Would it have made any difference if it had started out being on?

So, welcome to the world of paradoxes! There are actually a lot of examples of paradoxes available. To include all of them here would be impossible! Maybe some other time!