Zeno

Zeno of Elea Zeno
Born	490 BCE
Died	430 BCE
School tradition	Eleatic
Known for	Zeno's paradoxes

Zeno, or Zeno of Elea, was a Pre-Socratic Greek philosopher who was born in Elea, most likely into a wealthy family, circa 490 BCE. Elea was a Greek colony in southern Italy. Zeno was the most famous of Parmenides' disciples; according to some accounts, he was either the adopted son or lover of Parmenides, though there was a 25-year age gap. In Plato's fictional dialogue ‘’Parmenides’’, Plato wrote that Zeno accompanied Parmenides on his visit to Athens with his book of paradoxes in order to speak with Socrates.

Since the work of Parmenides is fictional and we have very scarce information on Zeno, we have to infer from Plato's fictional account. Zeno is best known today for his paradoxes, which he used to defend Parmenides' philosophy.

Philosophy

Zeno dedicated his time to defending Parmenides' idealist stances and created various paradoxes to defend them. Zeno defended Parmenides' idea that motion is impossible through four paradoxes:

Achilles and the Turtle

This is the most famous paradox; it argues that a fast runner can never overtake a slow one because of a head start. Achilles races a tortoise, but because he is so fast, he gives the tortoise a 100-meter head start. Achilles runs to the 100-meter mark, but by the time he gets there, the tortoise has moved forward about 10 meters. Achilles runs those 10 meters, but by the time he arrives, the tortoise has moved another 1 meter. Zeno states that this creates an infinite series of distances; due to this, Achilles can get infinitely close but never catch the tortoise.

Zeno believed that adding an infinite number of things must always result in infinity. Calculus proves this paradox wrong through the mathematical concept of convergent infinite series, which shows that an infinite sequence of shrinking numbers adds up to a finite total. This means Achilles would end up catching the tortoise in a normal, finite amount of time.

Paradox of Halves

This paradox argues that you can never actually arrive anywhere because motion cannot begin. Zeno explains this by setting up a scenario: A man wants to walk from his starting point to a door. Before the man can even reach the door, he has to cross half the distance. But before he reaches that half, he has to travel a quarter, and before that, an eighth. Zeno conveys that because any distance the man travels can be halved, there is an infinite number of tasks before completing a single step. Since the man cannot complete an infinite number of tasks, motion is impossible.

Zeno incorrectly assumes that completing an infinite number of increasingly smaller divisions requires an infinite amount of time. While they can theoretically be divided infinitely, the sum of their fractions converges to a finite whole.

Paradox of Arrows

This paradox argues that dividing time into static instants means the arrow is stationary in each instance. Since the arrow is static in each instance, it does not move, and motion is just an illusion of a string of stationary instants.

Imagine a man shoots an arrow, and it is flying in the air. Since for time to exist there must be singular moments of time (instants), at each instant of the arrow's flight, the arrow takes up the space of its own length and width. This means that since it is not moving, motion is an illusion.

Zeno didn't understand the nature of time and velocity because he viewed time as a series of isolated pictures. However, motion isn't something that only happens within a single instant; rather, motion is the rate of an object's position changing between instants.

Even if a moment in time is small, the arrow still has some rate of change at that instant.