Zeno's paradoxes

Zeno's Paradoxes are a famous set of thought-provoking stories or puzzles created by Zeno of Elea in the mid-5th century B.C. Philosophers, physicists, and mathematicians have argued over how to answer the questions raised by Zeno's Paradoxes for two and a half millennia. Although the specifics of each paradox differ from one another, they all deal with the tension between the apparent continuous nature of the universe and the discrete or incremental nature of mathematics and physics.

A Most Ingenious Paradox

The most famous of Zeno's Paradoxes is called Achilles and the Tortoise. The story goes that if Achilles (the famous hero from The Iliad) were to race a tortoise, but the tortoise were given a head start, Achilles could never catch the tortoise no matter how fast or how long Achilles ran. This is of course at odds with common sense and observation, which conclude that Achilles will eventually catch up with and pass the tortoise. However, the paradox lies upon a purely mathematical inspection of this scenario.

In order for Achilles to catch the tortoise, he must of course cover some distance A between where he began the race and where the reptilian racer began ahead of him. However, in the time Achilles traversed A, the shelled shlepper traveled a second distance B. Of course Achilles is undaunted, because B is less than A, and brings himself to the point where he has gone the distance A + B. To the champion's chagrine, the tortoise has not yet given up, and in the time Achilles took to cover B, has continued to travel to C. This progression can be carried out ad infinitum, and it can be shown that, mathematically speaking, there is an infinite number of time intervals and/or distances which Achilles must endure to catch his intrepid opponent.

This then is where Zeno's paradox lies: both pictures of reality cannot be true at the same time. Hence, either:

1. There is something wrong with the way we perceive the continuous nature of time, 2. In reality there is no such thing as a discrete, or incremental, amounts of time, distance, or perhaps anything else for that matter, or 3. There is a third picture of reality that unifies the two pictures--the mathematical one and the common sense or philosophical one--that we do not yet have the tools to fully understand.

A Most Famous Paradox

Suppose someone wishes to get from point A to point B. Well, first they must move halfway. Then, they must go half of the remaining way. And so forth, never actually reaching the ending. So, motion from any point A to any different point B is impossible. A common response to such a problem is to point at calculus: we can add up infinite series like 1+1/2+1/4+1/8+1/16... to get 2. However, the basic question Zeno is asking is how one can deal with an infinite progression with doing each element individually. Using calculus does not actually involve adding up infinite numbers one number at a time. Instead, it adds up a large group of numbers all at once. The quantum physicists respond to this paradox by saying that Zeno's premise was incorrect. That is, it is not always necessary that for a point A and a point B, that one can go halfway between them before getting to point B.