What Is Zeno's Paradox?

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Zeno of Elea (c. 490-c. 425 B.C.), a Greek philosopher and mathematician, is famous for his paradoxes which deal with the continuity of motion. (A paradox is a statement that runs counter to common sense, but may actually be true.) For instance, Zeno pointed out that at each point in time, an object occupies one particular location. This means that it must be at rest. Following from that, he claimed, motion is theoretically impossible.

Zeno also explained that if an object moves with constant speed along a straight line from point 0 to point 1, the object must first cover half the distance (½), then half the remaining distance (¼), then half the remaining distance ( ) , and so on without end. In this scenario, he concluded, since there is always some distance to be covered, the object never reaches point 1.

Zeno illustrated this paradox with the story of a race between a tortoise and the swift runner Achilles. Achilles could run 100 times faster than the tortoise; however, the tortoise was given a lead of 10 rods (16.5 feet or 5.03 meters) ahead of Achilles. The tortoise always advanced Moo of the distance that Achilles advanced in the same time period. Zeno claimed that it was theoretically impossible for Achilles to catch up to the tortoise because each time Achilles reached the place where the tortoise began, the tortoise would have moved on to another point.

Greek philosopher and scientist Aristotle (384-322 B.C.) provided an answer to Zeno's paradox. Aristotle pointed out that Zeno's subdivisions of motion (i.e. covering half the distance) were theoretical, not real. Aristotle argued that a distance would only be subdivided if the object in motion actually stopped.

Sources: "Achilles Paradox." Encyclopaedia Britannica CD 97; Dictionary of Scientific Biography, 14, pp. 607-8; Hofstadter, Douglas R. Godel, Escher, Bach: An Eternal Golden Braid, pp. 28-32; Lloyd, and Jefferson Hane Weaver. Conquering Mathematics, pp. 47-48; "Paradoxes of Zeno." Encyclopaedia Britannica CD 97.

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