The Achilles: Achilles races against a tortoise, which, being a slower runner, is given a headstart. By the time Achilles reaches the tortoise's starting point, it will have moved some distance; by the time Achilles reaches its new position, it will have moved a little farther; and so on. The process continues indefinitely, with the result that Achilles, no matter how swift, can never overtake the tortoise ... but common sense tells us he ought to.
Suppose, on the other hand, that time and space are not infinitely subdivisible; that is, there are smallest possible (and indivisible!) units of distance and time. Then motion is still impossible for the following reason:
The Arrow: An arrow flies through the air. An object in flight always occupies a space equal to itself, but it cannot move an indivisible unit of distance while it does occupy a space equal to itself. Hence, the flying arrow must actually be at rest and its motion an illusion ... but common sense tells us it does move.
Either way, motion is impossible ... but somehow it happens anyway!
In problem 3, credit will be given for creativity and cleverness. Don't just look this up -- we already know about limits!
Department of Mathematics
Trent University