Zeno's Paradoxes of Motion

Zeno of Elea (c. 490–c. 430 BCE), a student of Parmenides, defended his teacher by showing that the common-sense belief in motion leads to absurd conclusions. In the paradox of Achilles and the tortoise, a fast runner can never catch a slower one with a head start, because each time he reaches where the tortoise was, it has moved on a little further. In the dichotomy, a runner cannot finish a race, since he must first cover half the distance, then half the remainder, and so on forever.

In the arrow paradox, a flying arrow is motionless at every instant, so when does it move? Zeno's method was reductio ad absurdum: assume your opponent's view and show it leads to contradiction. The paradoxes troubled thinkers for centuries; mathematicians later used the theory of convergent infinite series to show how infinitely many steps can add up to a finite distance.