Zeno of Elea

Zeno was an ancient Greek philosopher born in Elea (a city in Southern Italy) in 490 BC. He was a friend and student of the philosopher Parmenides, who is famous for his belief that the world is in truth only one thing that cannot be divided, that the senses are deceptive, and therefore that things such as motion and change are an illusion. Zeno was greatly influenced by his teacher’s philosophy, so much so that he came up with his own ways of showing others how beliefs commonly held to be true are actually false. Like Parmenides, he used logical arguments to come up with paradoxes (seemingly unsolvable philosophical puzzles) which today are famously referred to as Zeno’s paradoxes.

A paradox is an argument which starts with statements thought to be true, that are then shown through logical reasoning to be absurd because of a logical contradiction. Zeno’s paradoxes seemed to show that commonly held beliefs in things like motion were absurd, and for this reason created a lot of interest and confusion in ancient Greece. Before Zeno and Parmenides, cosmologists (people who study the world) could just assume that things like motion and change were real. Now, they would have to prove it.

Zeno’s most famous paradox is that of Achilles and the tortoise. Achilles, known for his quickness, decides to race a tortoise. He gives the tortoise a head start. He tries to catch up to the tortoise and pass him, but unfortunately Zeno shows that is logically impossible. This is because since the tortoise is always moving, whenever Achilles tries to reach the tortoise, the tortoise will always be just slightly farther ahead. Because distance is infinitely divisible, there will always be a smaller and smaller distance that the faster Achilles needs to advance beyond in order to get to the new position that the tortoise is in.

Zeno is also known for his paradox of the racetrack that makes use of the same argument. Zeno attempts to show that it is impossible for a runner to run a certain distance, for example 100 meters, because this 100 meters can be divided infinitely into smaller units. As such, how can a runner run an infinite number of distances in a finite amount of time? It should take the runner an infinite amount of time to cover an infinite amount of distance, and therefore the runner will never reach his goal. Logically speaking, the same thing is true of his first step. Since the first step involves covering an infinity of small distances, how can he do even that?

Zeno also shows the paradox of an arrow moving through the sky to a target. How can an arrow be in motion if it needs to be in a particular position in space at every moment? To have a position in space, the arrow must be at rest in that position. If the arrow is always at rest in each new position in it’s trajectory, what is motion other than an illusion?

Aristotle will come to say that Zeno’s paradoxes are rooted in the use of false assumptions based on an improper use of language. For example, he explains that Zeno is mistaking a potential infinity for an actual infinity. A distance can be potentially divided infinitely, but that does not mean that it contains an actual infinity of distances. In this way, Aristotle shows that such paradoxes can be refuted if one understands which of its arguments premises are actually incorrect. As such, Zeno demonstrates that although logic is very powerful for coming up with conclusions, it doesn’t guarantee the truth. Meaningful knowledge is the result not only of logic but also sound premises.

Zeno’s thoughts would be greatly influential on future logicians, philosophers and mathematicians. George Berkeley, Immanuel Kant, and Henri Poincare would all come to defend the concept of potential infinity, while Leibniz would accept the existence of actual infinities. In the twentieth century, the philosopher Bertrand Russell would exclaim “Zeno’s arguments, in some form, have afforded grounds for almost all theories of space and time and infinity which have been constructed from his time to our own.”