**Zeno of Elea** (ca. 490 BC – ca. 430 BC) was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School
founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known for his paradoxes, which Bertrand Russell has
described as "immeasurably subtle and profound".

Little is known for certain about Zeno's life. Although written nearly a century after Zeno's death, the primary source of biographical
information about Zeno is Plato's Parmenides dialogue. In the dialogue, Plato describes a visit to Athens by Zeno and Parmenides, at a time
when Parmenides is "about 65," Zeno is "nearly 40" and Socrates is "a very young man". Assuming an age for Socrates of around 20, and taking
the date of Socrates' birth as 469 BC gives an approximate date of birth for Zeno of 490 BC. Plato says that Zeno was "tall and fair to look
upon" and was "in the days of his youth … reported to have been beloved by Parmenides".

Other perhaps less reliable details of Zeno's life are given by Diogenes Laertius in his Lives and Opinions of Eminent Philosophers, where it
is reported that he was the son of Teleutagoras, but the adopted son of Parmenides, was "skilled to argue both sides of any question, the
universal critic," and that he was arrested and perhaps killed at the hands of a tyrant of Elea.

According to Plutarch, Zeno attempted to kill the tyrant Demylus, and failing to do so, "with his own teeth bit off his tongue, he spit it in
the tyrant’s face". Although many ancient writers refer to the writings of Zeno, none of his writings survive intact.

Plato says that Zeno's writings were "brought to Athens for the first time on the occasion of" the visit of Zeno and Parmenides. Plato also
has Zeno say that this work, "meant to protect the arguments of Parmenides", was written in Zeno's youth, stolen, and published without his
consent. Plato has Socrates paraphrase the "first thesis of the first argument" of Zeno's work as follows: "if being is many, it must be both
like and unlike, and this is impossible, for neither can the like be unlike, nor the unlike like".

According to Proclus in his Commentary on Plato's Parmenides, Zeno produced "not less than forty arguments revealing contradictions", but only
nine are now known.

Zeno's arguments are perhaps the first examples of a method of proof called reductio ad absurdum, literally meaning to reduce to the absurd.
Parmenides is said to be the first individual to implement this style of argument. This form of argument soon became known as the epicheirema.
In Book VII of his Topics, Aristotle says that an epicheirema is "a dialectical syllogism". It is a connected piece of reasoning which an
opponent has put forward as true. The disputant sets out to break down the dialectical syllogism. This destructive method of argument was
maintained by him to such a degree that Seneca the Younger commented a few centuries later, "if I accede to Parmenides there is nothing left
but the One; if I accede to Zeno, not even the One is left."

*Zeno's paradoxes* have puzzled, challenged, influenced inspired, infuriated, and amused philosophers, mathematicians, and physicists
for over two millennia. The most famous are the so-called "arguments against motion" described by Aristotle in his Physics. Some of Zeno's
nine surviving paradoxes (preserved in Aristotle's Physics and Simplicius's commentary thereon) are essentially equivalent to one another.
Aristotle offered a refutation of some of them. Three of the strongest and most famous – that of Achilles and the tortoise, the Dichotomy
argument, and that of an arrow in flight – are presented in detail below. Zeno's arguments are perhaps the first examples of a method of
proof called "reductio ad absurdum" also known as

*proof by contradiction*. They are also credited as a source of the dialectic method
used by Socrates.

The origins of the paradoxes are somewhat unclear. Diogenes Laertius, a fourth source for information about Zeno and his teachings, citing
Favorinus, says that Zeno's teacher Parmenides was the first to introduce the Achilles and the Tortoise Argument. But in a later passage,
Laertius attributes the origin of the paradox to Zeno, explaining that Favorinus disagrees.

In

* the paradox of Achilles and the Tortoise*, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start
of 100 meters, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after
some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the tortoise has run a
much shorter distance, say, 10 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will
have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches
somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach
where the tortoise has already been, he can never overtake the tortoise.

*The dichotomy paradox*. Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he
can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth,
one-sixteenth; and so on. The resulting sequence can be represented as:

.

This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility. This sequence also presents
a second problem in that it contains no first distance to run, for any possible (finite) first distance could be divided in half, and hence
would not be first after all. Hence, the trip cannot even begin. The paradoxical conclusion then would be that travel over any finite
distance can neither be completed nor begun, and so all motion must be an illusion.

This argument is called the Dichotomy because it involves repeatedly splitting a distance into two parts. It contains some of the same
elements as the Achilles and the Tortoise paradox, but with a more apparent conclusion of motionlessness. Some, like Aristotle, regard the
Dichotomy as really just another version of Achilles and the Tortoise.

There are two versions of the dichotomy paradox. In the other version, before Homer could reach the stationary bus, he must reach half of the
distance to it. Before reaching the last half, he must complete the next quarter of the distance. Reaching the next quarter, he must then
cover the next eighth of the distance, then the next sixteenth, and so on. There are thus an infinite number of steps that must first be
accomplished before he could reach the bus, with no way to establish the size of any "last" step. Expressed this way, the dichotomy paradox
is very much analogous to that of Achilles and the tortoise.

In

* the arrow paradox*, Zeno states that for motion to occur, an object must change the position which it occupies. He gives an example
of an arrow in flight. He states that in any one (duration less) instant of time, the arrow is neither moving to where it is, nor to where it
is not. It cannot move to where it is not, because no time elapses for it to move there; it cannot move to where it is, because it is already
there. In other words, at every instant of time there is no motion occurring. If everything is motionless at every instant, and time is
entirely composed of instants, then motion is impossible.

Whereas the first two paradoxes presented divide space, this paradox starts by dividing time—and not into segments, but into points.