Zeno

I'm a great fan of paradoxes. A paradox uses a rational argument to arrive at an impossible conclusion. The point, I think, is not to prove that the conclusion is wrong. The wrongness of the conclusion is a requirement to be called a paradox. The point is to find the flaw in the seemingly rational argument, to think deeply about the subject, and maybe in a way you haven’t before. The result may expose flaws in our assumptions and improve the rigor and depth of our science and philosophy.

Probably the most famous set of paradoxes are the three known as Zeno’s paradoxes.

1. Achilles and the Tortoise - Achilles and a Tortoise have a footrace. Obviously, Achilles is going to give it a head start, but then he’s off! Soon he’s reached the spot where the tortoise was when he started, but by then the tortoise is farther along, and by the time Achilles reaches that spot the tortoise has gone further still! It seems he can never catch up!

2. The Dichotomy Paradox - A similar situation: in order for you to go from point A to point B, you must first go half way. Then you must traverse half the remaining distance, then half of that distance and so on forever. It seems you can only approach, not reach, your destination.

In fact, can you ever get started? to make it to that first half way point, you’d have to go half-way there...

See Meg Ryan in IQ.

3. The Arrow Paradox - Pick a point in time where an arrow is flying through the air. At that moment, the arrow is not moving. Well, we could pick any moment of the arrow’s flight and observe the same. If at every point in time the arrow is not moving, how can it get from point A to point B?

These have been very stubborn and resisted satisfactory explanation for thousands of years. We've since developed mathematical tools like calculus to deal with infinite quantities of infinitely small things and get results that make sense, but providing an alternate explanation that works is not quite the same as showing what was wrong with the original paradoxical explanation.

It may be that relatively new ideas in physics show us something that may be wrong with Zeno’s assumptions. It looks like space can not be divided into arbitrarily small pieces.

For example, the idea of the Planck length. If the distance in the Dichotomy paradox is one meter, after about 116 half-way trips, the distances you would be traversing would be so small that the type of Newtonian  linear motion we’re talking about isn't valid anymore and quantum effects dominate. 

Also It looks like an arrow, on a quantum level (you may notice philosophy often vaguely waves it's arms and mutters something about quantum when it's in trouble), can not be said to be frozen in place with no motion at a point in time - you run into the Heisenberg Principle. At a point in time, there’s a limit to how precisely you can know the position and speed of a thing, so maybe Zeno can’t assume that it’s not moving at that point in time.