ZENO'S PARADOXES (450 B.C.)

Megan Hennessy
ma163sbb@sdcc14.ucsd.edu
Written and Presented on 5/8/95

	Zeno of Elea, born about 490BC, is famous for his four paradoxes 
of motion and his paradox of plurality.  He is a student of Parmenides 
and his paradoxes were designed to show the absurdity of views of those 
who made fun of Parmenides.  Zeno tries to cut off all possible avenues 
of escape from the conclusion that space, time, and motion aren't real 
but illusory.  

Paradox of Plurality

	Zeno argues that if extended things exist, they must be composed 
of parts; thus there is a plurality of parts.  Furthermore, these parts 
must themselves have parts.  Since the process of subdivision is 
indefinitely repeatable, there must be an infinity of parts.  Allen 
explains two difficulties which arise: 1) the ultimate parts must not 
have a magnitude, because if they did, they can be further subdivided.  
But the whole object can't be made up of parts which have no magnitude, 
for no matter how many of them are put together, the result will have no 
magnitude, and 2) yet the parts must have magnitude.  But the addition of 
an infinite number of magnitudes, all greater than zero, will yield an 
infinite magnitude.  Hence, whole objects, if they exist, are "so small 
as to have no magnitude and so large as to be infinite."

Achilles and the Tortoise

	Achilles, the fastest of Greek warriors, is to run a footrace 
against a tortoise.  The tortoise gets a headstart.  Achilles can never 
catch up with the tortoise, no matter how fast he runs.  In order to 
overtake the tortoise, Achilles must run from his starting point A to the 
tortoise's original starting point T.  While he is doing that, the 
tortiose will have moved ahead to T1.  Now Achilles must reach the point 
T1.  While Achilles is traveling the new distance, the tortoise moves 
still farther to T2, and again Achilles must reach this new point.  And 
so it continues; whenever Achilles arrives at a point where the tortoise 
was, the tortoise has already moved a bit ahead.  Achilles can narrow the 
gap between him and the tortoise, but he can never actually catch up with 
him.

The Dichotomy

	(1st Form) Achilles can never reach the end of the race course, 
much less the original starting point of the tortoise.  Zeno argues that 
before Achilles can cover the whole distance he must cover the first 
half.  Then he must cover the first half of the remaining distance, and 
so on.  Therefore, he is always somewhat short of his goal.  Hence, he 
can never reach it.
	(2nd Form) Achilles can never actually get started.  Before he 
can complete the full distance, he must run half of it.  But before he 
can complete the first half, he must run the first quarter, and so on.  
In order to cover any distance, no matter how short, the runner must 
already have completed an infinite number of runs.  Since the sequence of 
runs completed form a regression, it has no first member, and hence, 
Achilles can never get started.

The Arrow

	An arrow in flight is always at rest.  At any given moment, the 
arrow is where it is, occupying a portion of space equal to itself.  
During the instant it cannot move, for that would require the instant to 
have parts, and by definition, and instant cannot have parts.  For the 
arrow to move during the instant would require that during the instant it 
must occupy a space larger than itself, for otherwise it has no room to 
move.   

The Stadium

	Consider these two positions of rows:

	First Position			Second Position
        A1   A2   A3                     A1   A2   A3
   B1   B2   B3                          B1   B2   B3
             C1   C2   C3                C1   C2   C3

	While row A remains at rest, rows B and C move in opposite 
directions until all three rows are lined up.  In the process, C1 passes 
twice as many Bs as As; it lines up with the first A to its left, but 
with the second B to its left.  Zeno concluded that "double the time is 
equal to half."


Cauchy's work on the summing of an infinite series converging, shed some 
light on the Dichotomy and Achilles paradoxes.  Some philosophers argue 
that the running of a race as "completing an infinite sequence of tasks" 
is not a praper way to describe it.  The use of infinity machines, like 
"The Desk Lamp," show that its impossible to say consistently that the 
machine has completed an infinite sequence of operations, and likewise, 
that the runner has completed an infinite number of runs.

Exercise

B. Russell disposed of the Arrow paradox definitively on the assumption 
of the mathematical continuity of space, time, and motion.  Zeno has been 
accused of the inability to distinguish between instantaneous rest and 
instantaneous motion.  The Stadium paradox depends on disregarding the 
relative motions of the bodies.  What is meant by these statements and 
how do they help explain the paradoxes?  

Resolution of the Paradox (A Philosophical Puppet Play)
by Abner Shimony

Characters: Zeno, Pupil, Lion
Setting: The school of Zeno at Elea

Pupil: Master! There is a lion in the streets!

Zeno: Very good.  You have learned your lesson in geography well.  The 
fifteenth meridian, as measured from Greenwich, coincides with the high 
road from the Temple of Poseidon to the Agora-- but you must not forget 
that it is an imaginary line.

Pupil: Oh no, Master!  I must humbly disagree.  It is a real lion, a 
menagerie lion, and it is coming toward the school!

Zeno: My boy, in spite of your proficiency at geography, which is 
commendable in its way-- albeit essentially the art of the surveyor and 
hence separated by the hair of the theodolite from the craft of a slave-- 
you are deficient in philosophy.  That which is real cannot be imaginary, 
and that which is imaginary cannot be real.  Being is, and non-being is 
not, as my revered teacher Parmenides demonstrated first, last, and 
continually, and as I have attempted to convey to you.

Pupil: Forgive me, Master.  In my haste and excitement, themselves 
expressions of passion unworthy of you and of our school, I have spoken 
obscurely.  Into the gulf between the thought and the word, which, as you 
have taught us, is the trap set by non-being, I have again fallen.  What 
I meant to say is that a lion has escaped from the zoo, and with 
deliberate speed it is rushing in the direction of the school and soon 
will be here!

The lion appears in the distance.

Zeno: O my boy, my boy!  It pains me to contemplate the impenetrability 
of the human intellect and its incommensurability with the truth.  
Furthermore, I now recognize that a thirty-year novitiate is too brief-- 
sub specie aeternitatis-- and must be extended to forty years, before the 
apprenticeship proper can begin.  A real lion, perhaps; but really 
running, impossible; and really arriving here, absurd!

Pupil: Master...

Zeno: In order to run from the zoological garden to the Eleatic school, 
the lion would first have to traverse half the distance.

The lion traverses half the distance.

Zeno: But there is a first half of that half, and a first half of that 
half, and yet again a first half of that half to be traversed.  And so 
the halves would of necessity regress to the first syllable of recorded 
time-- nay, they would recede yet earlier than the first syllable.  To 
have traveled but a minute part of the interval from the zoological 
garden to the school, the lion would have been obliged to embark upon his 
travels infinitely long ago.

The lion bursts into the schoolyard.

Pupil: O Master, run, run!  He is upon us!

Zeno: And thus, by reductio ad absurdum, we have proved that the lion 
could never have begun the course, the mere fantasy of which has so 
unworthily filled you with panic.

The pupil climbs an Ionic column, while the lion devours Zeno.

Pupil: My mind is in a daze.  Could there be a flaw in the Master's argument?


Bibliography

Allen, R.E.  Plato's Parmenides: Translation and Analysis.  Minneapolis: 
	Univ. of Minnesota Press, 1983.

Grunbaum, Adolf.  Modern Science and Zeno'x Paradoxes.  Middletown: 
	Wesleyan University Press, 1967.

Meinwald, Constance C.  Plato's Parmenides.  New York: Oxford University 
	Press, 1991.

Rossvaer, Viggo.  The Laborious Game: A Study of Plato's Parmenides.  
	Norway: Universitetsforlaget, 1983.

Salmon, Wesley C.(Editor)  Zeno's Paradoxes. New York: Bobbs-Merrill Co. 
	Inc., 1970.

Sternfeld, Robert and Harold Zyskind.  Meaning, Relation, and Existence 
	in Plato's Parmenides. New York: Peter Lang Publishing Inc., 1987.

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