Zeno of Elea
Zeno of Elea (circa 495 BC - circa 430 BC) was an ancient Greek philosopher of Southern Italy, a member of the Eleatic School founded by Parmenides. He is best known for formulating a number of paradoxes based on Eleatic beliefs of the impossibility of motion. They are known to this day as Zeno's paradoxes.
Zeno was stabbed to death, after he tried to attack Nearchus of Elea.
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Mathematical Paradoxes Attributed to Zeno of Elea
- Suppose an object moves from a Point A to a Point B. Before it can reach Point B it must reach the point A1, which is midway between the Points A and B. This is the first step. After completing the first step the object must move from Point A1 to Point A2, which is midway between Points A1 and B. This is the second step. Then it must take the third step to A3, which is midway between Points A2 and B. This process must be continued infinitely to reach Point B. Since no process can be performed an infinite amount of times during a finite period of time, the object will never reach Point B.
- To look at the above problem in a different manner, there is again an object moving from Point A to Point B. Again, to get to Point B the object must first reach the midway point between Points A and B, Point B1. However before this can be done the object must reach Point B2, which is midway between Point A and Point B1. Before it can do this however, it must reach Point B3, midway between Points A and B2. This can be continued on forever, therefore the object cannot even get started in its motion.
- Achilles is trying to overtake a tortoise. Achilles is 100 yards behind the tortoise, but runs 10 times faster then the tortoise. His (Achilles) first step is to reach the point where the tortoise is originally. When Achilles has done this the tortoise is now 10 yards in front of him. Achilles then runs to the spot where the tortoise is occupying. Once he has covered the 10 yards the tortoise is now a yard ahead of him. Once again Achilles runs to where the tortoise is, but once he has reached this point the tortoise is 1/10 of a yard in front of Achilles. In other words whenever Achilles reaches the place the tortoise used to occupy the tortoise will no longer be there, therefore Achilles can never overtake the tortoise.
- Suppose an arrow is flying continuously forward during a certain time interval. Take any instant in that time interval. It is impossible that the arrow is moving during that instant because an instant has a duration of zero, and the arrow cannot be in two different places at the same time. Therefore, at every instant the arrow is motionless, hence the arrow is motionless throughout the entire interval, which means that it can't move during the interval.
Source: (informal...not quoted directly) SATAN, CANTOR, AND INFINITY, AND OTHER MIND-BOGGLING PUZZLES by: Raymond Smullyan Published by Alfred A. Knopf Inc. and Random House of Canada Inc., 1992. ISBN 0-679-40688-3
Note: Zeno of Elea is not to be confused with Zeno of Citium