**Ze no**

Greek philosopher and mathematician

**Zeno of Elea**(c. 490 BCE – c.425 BCE) was the first great philosophical skeptic. He is famous for his

paradoxes, which deal with the continuity of motion. He made a series

of arguments in which he purported to prove by logical means that

motion and plurality are impossible. In his view all human knowledge

is based on an unprovable hypothesis: that time and space are

continuous. In

*A History of Mathematics*, Carl B. Boyer wrote: “The arguments of Zeno seem to have had a profound influence on the development of Greek

mathematics, comparable to that of the discovery of the incommensurable, with which it may

have been related…. The realm of number continued to have the property of discreteness, but

the world of continuous magnitudes (…) was a thing apart from number and had to be treated

through geometric method. It seemed to be geometry rather than number that ruled the world.

This was perhaps the most far-reaching conclusion of the Heroic Age, and it is not unlikely that

this was due in large measure to Zeno of Elea and Hippasus of Metapontum.”

Zeno was born and died in Elea, which is in southern Italy, and he is identified by his place of birth to

distinguish him from other Zenos of antiquity. According to E.T. Bell (

*The Deve lopment of**Mathemat ics*), by tradition Zeno of Elea was:

“a pugnacious dialectician with a passion for being different from everyone else….And it

is told that his uncompromising intellectual honesty finally cost him his life. He had conspired

with the political faction which lost, and met his death by torture with heroic fortitude.”

According to Plato, Zeno’s now lost treatise, in which he indirectly argued against the reality of

multiplicity and of motion, consisted of several discourses. In each, Zeno made a supposition and then

gave an argument that led to an absurd consequence. He may have been the first to employ this method

of indirect proof, now referred to as

*reductio ad absurdum.*Bell wrote that the paradoxes should becalled “sophistries,” which are logical arguments that some cannot accept but neither can they refute

them. Aristotle argued against Zeno’s paradoxes, calling them “fallacies,” without being able to show

where they were false.

Originally there were forty paradoxes, but only eight have survived. Zeno’s arguments concerning

motion introduced the element of time, and revealed that time cannot be considered merely the sum of

moments. Briefly the four arguments are as follows:

1.

*The Dichotomy*: Motion cannot exist because if something moves from one place to another, itmust first reach the midpoint of the distance to be traveled, but before it can do that it has to reach

the midpoint of the first half, and before it can do that it must reach the midpoint of the first

fourth, and so on ad infinitum. It must, therefore, pass through an infinite number of points, and

this is impossible in a finite amount of time.

2.

*The Ach illes*: In a race between the running Achilles and the crawling tortoise, the former cannever overtake the latter if the tortoise has a head start. Before Achilles reaches the point from

which the tortoise started, the tortoise will have move ahead a little way and Achilles must run to

this new position but by the time he reaches it the tortoise has moved ahead again, and ad

infinitum. English mathematician and writer Charles Dodgson, better known as Lewis Carroll,

used the characters of Achilles and the tortoise to illustrate his paradox of infinity.

**3.**

*The Arrow*: An arrow shot in the air is either in motion or at rest. An arrow cannot move, because

for motion to take place, the arrow would have to be in one position at the beginning of an instant

and at another at the end of the instant. But as time is made up of instants, which are time’s

smallest measure and are not further divisible, this is a contradiction. Hence, the arrow is always

at rest.

**4.**

*The Stadium*: This is the most awkward of Zeno’s paradoxes to describe. It concludes that half the

time is equal to twice the time. Suppose that there are three rows of soldiers, A, B, and C [Figure

7.4].

First Position Second Position

A * * * A * * * B * * * B * * * C * * * C * * * Figure 7.4

In the smallest unit of time A remains stationary while rows B and C move at equal speeds in

opposite directions. When they have reached the second position, each B has passed twice as

many C’s as A’s. Then it takes row B twice as long to pass row A as it does to pass row C.

However, the time for rows B & C to reach the position of row A is the same. Thus half the time is

the same as twice the time. The paradox arises from the assumption that space and time can be

divided only by a definite amount.

Mathematicians, physicists, and philosophers came to realize that to escape the contradictions found in

Zeno’s paradoxes, it was necessary to radically reinterpret the concepts of space, time, and motion, as

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