Zeno was born in Elea (or Velia, to give it its Latin name), southern Italy about 494 BC. Little is known about the life of Zeno, but we do know he
was a disciple of the philosopher Parmenides, founder of the Eleatic school of thought. When Zeno moved with Parmenides to Athens, he
proposed the four paradoxes that have made him famous. Zeno died in around 435 BC.
Zeno’s paradoxes concern the ambiguity between the continuous and the discrete, an ambiguity that has remained an issue for
mathematicians throughout the ages.
A contribution of Zeno to mathematics
The first of Zeno’s paradoxes is known as The Dichotomy. To give a modern example, this paradox can be described in terms of a 100 metre
runner. To complete the race, the runner must first reach the 50 metre mark; to reach the 50 metre mark, she must first reach the 25 metre
mark; to reach the 25 metre mark, she must first reach the 12.5 metre mark; and so on indefinitely. Thus the runner has to reach an infinite
number of midpoints before she can complete the race. Since it is impossible to reach an infinite number of points, it is also impossible for the
runner to complete the race.
The second paradox is known as The Achilles. Achilles is chasing after a crawling tortoise.
Before he can catch the tortoise, he must first reach the point where he first saw the tortoise. However, by the time he reaches that point, the
tortoise has already moved forward and so is still ahead. Repeating the argument, it is clear that Achilles will never catch up with the tortoise.
These first two paradoxes have shown that, if we subdivide space and time up into an infinite number of parts, then motion is impossible.
Zeno’s other two paradoxes show that motion is also impossible if we divide space and time up into a finite number of parts.
The third paradox is called The Arrow and argues as follows. An arrow in flight is at any instant either at rest or moving. If the instant is
indivisible, then the arrow cannot be moving since, if it did move, the instant would be divided. Thus the arrow is always at rest.
The fourth paradox is known as The Stadium. This one is more difficult to describe. To do so, I will use a modern example. A, B and C are
three trains, each with four carriages, on parallel tracks.
Train A is stationary, train B is moving at a constant speed towards the right, and train C is moving with the same constant speed towards the
left. At the first instant of time, the three trains are as shown in the first diagram. An instant later, their positions are as shown in the second
diagram. In that instant of time, train C will have passed two carriages of train B. This means we now have a smaller interval of time, which is
the time it takes train C to pass one carriage of train B.
Zeno’s paradoxes have encouraged a lot of debate over the centuries and were not fully resolved until Georg Cantor developed his theory of
infinite sets in the late nineteenth century.
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