Zeno's Paradoxes

Zeno's paradoxes are a set of four paradoxes dealing with counterintuitive aspects of continuous space and time.

1. Dichotomy paradox: Before an object can travel a given distance , it must travel a distance . In order to travel , it must travel , etc. Since this sequence goes on forever, it therefore appears that the distance  cannot be traveled. The resolution of the paradox awaited calculus and the proof that infinite geometric series such as  can converge, so that the infinite number of "half-steps" needed is balanced by the increasingly short amount of time needed to traverse the distances.

2. Achilles and the tortoise paradox: A fleet-of-foot Achilles is unable to catch a plodding tortoise which has been given a head start, since during the time it takes Achilles to catch up to a given position, the tortoise has moved forward some distance. But this is obviously fallacious since Achilles will clearly pass the tortoise! The resolution is similar to that of the dichotomy paradox.

3. Arrow paradox: An arrow in flight has an instantaneous position at a given instant of time. At that instant, however, it is indistinguishable from a motionless arrow in the same position, so how is the motion of the arrow perceived?

4. Stade paradox: A paradox arising from the assumption that space and time can be divided only by a definite amount.

The dichotomy paradox leads to the following mathematical joke. A mathematician, a physicist and an engineer were asked to answer the following question. A group of boys are lined up on one wall of a dance hall, and an equal number of girls are lined up on the opposite wall. Both groups are then instructed to advance toward each other by one quarter the distance separating them every ten seconds (i.e., if they are distance  apart at time 0, they are  at ,  at ,  at , and so on.) When do they meet at the center of the dance hall? The mathematician said they would never actually meet because the series is infinite. The physicist said they would meet when time equals infinity. The engineer said that within one minute they would be close enough for all practical purposes.

REFERENCES

Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 218-220, 1998.

Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 163-166, 1984.

Grünbaum, A. Modern Science and Zeno's Paradoxes. Middletown, CT: Wesleyan University Press, 1967.

Pappas, T. "Zeno's Paradox--Achilles & the Tortoise." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 116-117, 1989.

Russell, B. Our Knowledge and the External World as a Field for Scientific Method in Philosophy. New York: Routledge, 1993.

Salmon, W. (Ed.). Zeno's Paradoxes. New York: Bobs-Merrill, 1970.Stewart, I. "Objections from Elea." In From Here to Infinity: A Guide to Today's Mathematics. Oxford, England: Oxford University Press, p. 72, 1996.

vos Savant, M. The World's Most Famous Math Problem. New York: St. Martin's Press, pp. 50-55, 1993.