Wednesday, August 12, 2009

Zeno's Paradox

In the fifth century BC the Greek philosopher Zeno of Elea posed 4 problems, now known as Zeno paradoxes that were intended to challenge some of the ideas concerning space and time held in his day. Zeno’s second paradox concerns a race between the Greek hero Achilles and a Tortoise that has been given a head start. He said 'The slower when running will never be overtaken by the quicker; for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must necessarily always be some distance ahead.'

Zeno argued, as follows below, that Achilles could never pass the tortoise. Given that Achilles is the fastest Greek hero, Tortoise is one of the slowest animals, the speed difference between is very big, so that the conclusion that Achilles can never pass the Tortoise is contradictory with everyday experience.

Zeno argues as follows: Suppose that Achilles starts at position a1 and the Tortoise starts with the position t1 having a head start in terms of the distance. When Achilles reaches the point a2 = t1, the Tortoise is already further away at position t2. When Achilles reaches the point a3 = t2,

the Tortoise is at position t3. This process continues without end, and Tortoise is always ahead of Achilles. The conclusion is that Achilles can never overtake the Tortoise.

Click here to understand the problem in better way!

Here is the simple explanation:

Let us imagine that the race is 25 miles long with a check point at every 5 miles, so that a1 to a2 = 5 miles., h1 to h2 = 5 miles, and so on. Let’s also imagine Achilles runs 10 mph and the Tortoise runs 5 mph, since Achilles is faster.

Knowing this, we can determine that it will take Achilles 30 minutes to run between each point… Therefore, 30 minutes into the race, Achilles would be at point a2 and the Tortoise would be between h1 and point h2.

An hour into the race, Achilles would be at point a3 and the Tortoise would be at point h2. Note a3 = h2. An hour and a half into the race, Achilles would have passed the Tortoise and be at point a4, while the Tortoise is between the point h2 and the point h3.. Though Zeno’s conclusion that the quicker will never pass the slower (if given a head start) may have been valid during the fifth century B.C., it does not stand true to today’s experiences.

Here is the justification for the fallacy in Zeno's argument:

Metaphysical solution: It's about TIME! Time is not only a physical parameter, but a metaphysical notion as well. Zeno's only take into account Space. Time is completely ignored. In truth, Space and Time are inseparable elements of Cosmos (the Universe). That fundamental truth is valid both physically and metaphysically. Achilles will cover one unit of space in less time than the tortoise. Equivalently, Achilles will cover a longer distance than the tortoise in the same time. By the time Achilles reaches the starting point of the tortoise, the tortoise would have moved a shorter distance. The distance could be so short that Achilles could surpass it in a very short time. Generalizing, the gap the gap reverses. The faster competitor surpasses the slower competitor who had an early start.

1 comment:

  1. Oh my goodness! the arguements are so confusing...and so are the notions about time and space! But one thing is worth applauding, this problem of Zeno's forced everyone to even doubt the capabilities of Archillies when compared to a mere tortoise...I loved the explainations...but I am not convinced as of yet...let me search Archillies and the tortoise to catch hold a more explanatory scene of this paradox!



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