Zeno (450 B.C.), a disciple of Parmenides, was an ancient Greek philosopher who proposed the following arguments for the impossibility of motion (no motion).

First argument: motion is impossible, says Zeno. A moving object must first reach the midpoint of whatever distance remains, and this repeats endlessly, without end. Suppose an object travels from 0 to 1; after n moves, its position is 1 - 1/2^n. We can never find an n for which this equals 1 exactly, and therefore the object never truly moves.

Modern physics introduces the concept of “infinity” to resolve this: like Zeno, it still assumes a continuously changing process, but holds that the object passes through infinitely many midpoints — that is, there’s no 1 - 1/2^n through which the object fails to pass. Zeno denied infinity, and so he denied motion.

Second argument: Achilles races a tortoise. Achilles starts behind but moves twice as fast. Suppose the starting points are 0 and 1, and the finish line is 2. For any integer n, when Achilles reaches 2 - 1/2^n, the tortoise is already at 2 - 1/2^(n+1). Therefore, Achilles can never catch up to the tortoise.

From the standpoint of modern physics, an object’s motion across an interval of time is made up of infinitely many positions across infinitely many moments. If you deny the existence of infinity, motion becomes discontinuous.

Third argument: the flying arrow does not move. At every single instant, the arrow’s position is fixed, and therefore the arrow never moves.

That the arrow doesn’t move at any single instant doesn’t mean it doesn’t move across an interval made up of infinitely many instants. In calculus, for instance, it’s easy to find examples where 0 × infinity = 1.

Fourth argument: there are three rows — row A stands still, while rows B and C move toward each other, each at top speed. Then B’s speed relative to C would be twice the top speed — which is impossible (it exceeds the maximum speed), and therefore motion cannot occur.

The general formula for the no-motion argument: deny infinity + other conditions = no motion. Its equivalent form: (accept) motion + other conditions = accept infinity.

Freely translated from Mathematics: A Concise History and Philosophy — click here for the original text