Zeno of Elea
Zeno of Elea is a Greek mathematician. He would, however, be better described as a
philosopher. Little is known about his life for none of his own writings have survived. Most of
what is known about him comes from the writings of Aristotle. Zeno is said to have lived during
the fifth century BC in the Greek town of Elea in southern Italy. Zeno is important to the study of
Math History because of his major contributions, the paradoxes, and some other minor
The most noted of Zeno's works are his paradoxes. Those dealing with plurality and with
motion. These were written in response to a theory circulating through the region concerning what
we today would call a timespace continuum. The other Greek philosophers and mathematicians
thought that the world was a plurality of points and instants.1 This theory was started by the
Pathagoreans. They said the world had continuity. To contradict them, Zeno took their arguments
for continuity to an extreme at which point they became absurd.
To argue against plurality, Zeno said that if they are a continuous number of points then
there is a given number of them. Yet, if they are finite, they cannot be infinite and plural. The
whole point of his argument is to bring his opponents argument to a contradiction of logic. This
was a typical argument against plurality.
Similar to Zeno's argument against plurality is his argument against motion. He did not
truly believe that motion, as given by the senses, exists. These paradoxes built around the
argument of an infinite being finite make up the majority of his world view.
The first paradox of motion is called the Dichotomy. It basically says that for an object to
travel a given distance, it must first travel half that distance. A runner must run 1/2, then 1/4, then
1/8, and so on ad infinitum. The runner must traverse each given distance in a given time interval,
which is finite. The problem arises in that the runner must pass an infinite number of time
intervals to reach a finite goal.
The second paradox is similar to the Dichotomy. The Achilles is a race between Achilles
and a tortoise. The tortoise has been given a head start. Zeno says that Achilles will never pass
the tortoise even though he runs faster. Achilles must first reach the tortoises starting point. By
the time he does this, though, the tortoise has moved a distance. Achilles must then reach that
starting point. This continues ad infinitum. Therefore, Achilles will never pass the tortoise and as
such lose the race.
The third and fourth paradoxes take the opposite approach as the first two. The third
paradox is the Arrow. The idea involved with this paradox is that of instantaneous velocity. Zeno
says that the world is made of instants. When an arrow that is flying through the air is observed
during an indivisible instant, it has no motion. At each instant, the flying arrow is where it is and
would be in the same place if it were at rest. The Arrow denies that a state of motion exists.
The Stadium is Zeno's fourth paradox of motion. It is set up graphically as follows: There
are three rows of letters. Each row contains four letters of equal size. The first row is A's, the
second B's, and the third row is C's. Each letter is numbered from left to right in the sequence 1,
2, 3, 4. The initial position is B4 under A2 and C1 is under A3. The row of B's are moving to
the right and the row of C's are moving to the left. Each row moves with respect to A which is
stationary. For each A passed the time elapsed is one unit. A unit is the smallest time unit possible
and is indivisible. After two units of time, the columns are A, B, and C, 1 lined up vertically. The
columns are the same for 2, 3, and 4. Seeing then that all four B's and C's crossed in two
indivisible time units, these units are therefore devisable by...