Zenon's paradox

The Greek philosopher Zeno, who lived in the fifth century B.C., decades before Socrates, dedicated his life's work to showing the logical paradoxes inherent to the idea of indefinite divisibility in space and time -- i.e., that every line is composed of an infinite number of points. One of these paradoxes is known as the arrow paradox: If the motion of a flying arrow is divided ad infinitum, then during each of these infinitesimal moments the arrow is at rest. The sum of an infinity of zeroes remains zero, hence the arrow cannot move. One can imagine how someone giving a flying arrow quick, repeated glimpses can actually freeze it in place. Zeno inferred from this that movement cannot happen. Indeed, he was a true follower of Parmenides, his teacher and mentor, who advocated that any change in nature is but an illusion.

Of the 40 arguments attributed to Zeno by later writers, the four most famous are on the subject of motion:

The Dichotomy: There is no motion, because that which is moved must arrive at the middle before it arrives at the end, and so on ad infinitum.
The Achilles: The slower 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 always be some distance ahead.
The Arrow: If everything is either at rest or moving when it occupies a space equal to itself, while the object moved is always in the instant, a moving arrow is unmoved
The Stadium Consider two rows of bodies, each composed of an equal number of bodies of equal size. They pass each other as they travel with equal velocity in opposite directions. Thus, half a time is equal to the whole time.

From Mathpages.com.