Sabtu, 23 April 2011

Zeno of Elea (c. 490 BCE – c. 425 BCE)

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 be
called “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, it
must 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 can
never 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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Sabtu, 23 April 2011

Zeno of Elea (c. 490 BCE – c. 425 BCE)

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 be
called “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, it
must 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 can
never 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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