In the fifth century B.C.E., Zeno of Elea offered arguments that
led to conclusions contradicting what we all know from our
physical experience–that runners run, that arrows fly, and that
there are many different things in the world. The arguments were
paradoxes for the ancient Greek philosophers. Because most of
the arguments turn crucially on the notion that space and time
are infinitely divisible—for example, that for any distance
there is such a thing as half that distance, and so on—Zeno was
the first person in history to show that the concept of infinity
is problematical. In his Achilles Paradox, Achilles races to
catch a slower runner–for example, a tortoise that is crawling
away from him. The tortoise has a head start, so if Achilles
hopes to overtake it, he must run at least to the place where
the tortoise presently is, but by the time he arrives there, it
will have crawled to a new place, so then Achilles must run to
this new place, but the tortoise meanwhile will have crawled on,
and so forth. Achi