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Zeno was an Eleatic philosopher, a native of
Elea (Velia) in Italy, son of Teleutagoras, and the favorite disciple
of Parmenides. He was born about 488 BCE., and at the age of forty
accompanied Parmenides to Athens. He appears to have resided some
time at Athens, and is said to have unfolded his doctrines to people
like Pericles and Callias for the price of 100 minae. Zeno is said to
have taken part in the legislation of Parmenides, to the maintenance
of which the citizens of Elea had pledged themselves every year by
oath. His love of freedom is shown by the courage with which he
exposed his life in order to deliver his native country from a
tyrant. Whether he died in the attempt or survived the fall of the
tyrant is a point on which the authorities vary. They also state the
name of the tyranny differently. Zeno devoted all his energies to
explain and develop the philosophical system of Parmenides.
We learn from Plato that Zeno was twenty-five years younger than
Parmenides, and he wrote his defense of Parmenides as a young man.
Because only a few fragments of Zeno's writings have been found, most
of what we know of Zeno comes from what Aristotle said about him in Physics, Book 6,
chapter 9.
Zeno's contribution to Eleatic philosophy is
entirely negative. He did not add anything positive to the teachings
of Parmenides, but devoted himself to refuting the views of the
opponents of Parmenides. Parmenides had taught that the world of
sense is an illusion because it consists of motion (or change) and
plurality (or multiplicity or the many). True Being is absolutely
one; there is in it no plurality. True Being is absolutely static and
unchangeable. Common sense says there is both motion and plurality.
This is the Pythagorean notion of reality against which Zeno directed
his arguments. Zeno showed that the common sense notion of reality
leads to consequences at least as paradoxical as his
master's.
Table of Contents (Clicking on the links below will take you to that part of this article)
1. Paradoxes of Multiplicity and Motion
Zeno's arguments can be
classified into two groups. The first group contains paradoxes
against multiplicity, and are directed to showing that the
'unlimited' or the continuous, cannot be composed of units however
small and however many. There are two principal
arguments:
- If we assume that a line segment is
composed of a multiplicity of points, then we can always bisect a
line segment, and every bisection leaves us with a line segment
that can itself be bisected. Continuing with the bisection
process, we never come to a point, a stopping place, so a line
cannot be composed of points.
- The many, the line, must be both limited
and unlimited in number of points. It must be limited because it
is just as many (points) as it is, no more, and less. It is
therefore, a definite number, and a definite number is a finite or
limited number. However, the many must also be unlimited in
number, for it is infinitely divisible. Therefore, it's
contradictory to suppose a line is composed of a multiplicity of
points.
The second group of Zeno's arguments concern
motion. They introduce the element of time, and are directed to
showing that time is no more a sum of moments than a line is a sum of
points. There are four of these arguments:
- If a thing moves from one point in space to
another, it must first traverse half the distance. Before it can
do that, it must traverse a half of the half, and so on ad
infinitum. It must, therefore, pass through an infinite number of
points, and that is impossible in a finite time.
- In a race in which the tortoise has a head
start, the swifter-running Achilles can never overtake the
tortoise. Before he comes up to the point at which the tortoise
started, the tortoise will have got a little way, and so on ad
infinitum.
- The flying arrow is at rest. At any given
moment it is in a space equal to its own length, and therefore is
at rest at that moment. So, it's at rest at all moments. The sum
of an infinite number of these positions of rest is not a
motion.
- Suppose there are three arrows. Arrow B is
at rest. Suppose A moves to the right past B, and C moves to the
left past B, at the same rate. Then A will move past C at twice
the rate. This doubling would be contradictory if we were to
assume that time and space are atomistic. To see the
contradiction, consider this position as the chains of atoms pass
each other:
A1 A2 A3 ==>
B1 B2 B3
C1 C2 C3 <==
Atom A1 is now lined up with C1, but an instant ago A3 was lined
up with C1, and A1 was still two positions from C1. In that one
unit of time, A2 must have passed C1 and lined up with C2. How did
A2 have time for two different events (namely, passing C1 and
lining up with C2) if it had only one unit of time available? It
takes time to have an event, doesn't it?
Both groups of Zeno's arguments, those
against multiplicity and those against motion, are variations of
one argument that applies equally to space or time.
For simplicity, we will consider it only in its spatial sense. Any
quantity of space, say the space enclosed within a circle, must
either be composed of ultimate indivisible units, or it must be
divisible ad infinitum. If it is composed of indivisible
units, these must have magnitude, and we are faced with the
contradiction of a magnitude which cannot be divided. If it is
divisible ad infinitum, we are faced with the contradiction
of supposing that an infinite number of parts can be added up to
make a merely finite sum.
2. Kant's, Hume's, and Hegel's Solutions to Zeno's Paradoxes
According to Kant, these contradictions are
immanent in our conceptions of space and time, so space and time
are not real. Space and time do not belong to things as they are
in themselves, but rather to our way of looking at things. They
are forms of our perception. It is our minds which impose space
and time upon objects, and not objects which impose space and time
upon our minds. Further, Kant drew from these contradictions the
conclusion that to comprehend the infinite is beyond the capacity
of human reason. He attempted to show that, wherever we try to
think the infinite, whether the infinitely large or the infinitely
small, we fall into irreconcilable contradictions.
As might be expected, many thinkers have
looked for a way out of the paradoxes. Hume
denied the infinite divisibility of space and time, and declared
that they are composed of indivisible units having magnitude. But
the difficulty that it is impossible to conceive of units having
magnitude which are yet indivisible is not satisfactorily
explained by Hume.
Hegel believed that any solution which is to
be satisfactory must somehow make room for both sides of the
contradiction. It will not do to deny one side or the other, to
say that one is false and the other true. A true solution is only
possible by rising above the level of the two antagonistic
principles and taking them both up to the level of a higher
conception, in which both opposites are reconciled. Hegel regarded
Zeno's paradoxes as examples of the essential contradictory
character of reason. All thought, all reason, for Hegel, contains
immanent contradictions which it first posits and then reconciles
in a higher unity, and this particular contradiction of infinite
divisibility is reconciled in the higher notion of
quantity. The notion of quantity contains two factors,
namely the one and the many. Quantity means
precisely a many in one, or a one in many. If, for example, we
consider a quantity of anything, say a heap of wheat, this is, in
the first place, one; it is one whole. Secondly, it is many, for
it is composed of many parts. As one it is continuous; as many it
is discrete. Now the true notion of quantity is not one, apart
form many, nor many apart from one. It is the synthesis of both.
It is a many in one. The antinomy we are considering arises
from considering one side of the truth in a false abstraction from
the other. To conceive unity as not being in itself multiplicity,
or multiplicity as not being unity, is a false abstraction. The
thought of the one involves the thought of the many, and the
thought of the many involves the thought of the one. You cannot
have a many without a one, any more than you can have one end of a
stick without the other.
Now, if we consider anything which is
quantitatively measured, such as a straight line, we may consider
it, in the first place, as one. In that case it is a continuous
divisible unit. Next we may regard it as many, in which case it
falls into parts. Now each of these parts may again be regarded as
one, and as such is an indivisible unit; and again each part may
be regarded as many, in which case it falls into further parts;
and this alternating process may go on for ever. This is the view
of the matter which gives rise to Zeno's contradictions. But it is
a false view. It involves the false abstraction of first regarding
the many as something that has reality apart from the one, and
then regarding the one as something that has reality apart from
the many. If you persist in saying that the line is simply one and
not many, then there arises the theory of indivisible units. If
you persist in saying it is simply many and not one, then it is
divisible ad infinitum. But the truth is that it is neither
simply many nor simply one; it is a many in one, that is,
it is a quantity. Both sides of the contradiction are,
therefore, in one sense true, for each is a factor of the truth.
But both sides are also false, if and in so far as, each sets
itself up as the whole truth.
3. The Contemporary Solution to Zeno's Paradoxes.
Kant's, Hume's and
Hegel's solutions to the paradoxes have been very stimulating to
subsequent thinkers, but ultimately have not been accepted. There
is now general agreement among mathematicians, physicists and
philosophers of science on what revisions are necessary in order
to escape the contradictions discovered by Zeno's fruitful
paradoxes. The concepts of space, time,
and motion have to be radically changed, and so do the
mathematical concepts of line, number, measure, and sum of a
series. Zeno's integers have to be replaced by the contemporary
notion of real numbers. The new one-dimensional continuum, the
standard model of the real numbers under their natural (less-than)
order, is a radically different line than what Zeno was imagining.
The new line is now the basis for the scientist's notion of
distance in space and duration through time. The line is no longer
a sum of points, as Zeno supposed, but a set-theoretic union of a
non-denumerably infinite number of unit sets of points. Only in
this way can we make sense of higher dimensional objects such as
the one-dimensional line and the two-dimensional plane being
composed of zero-dimensional points, for, as Zeno knew, a simple
sum of even an infinity of zeros would never total more than zero.
The points in a line are so densely packed that no point is next
to any other point. Between any two there is a third, all the way
"down." The infinity of points in the line is much larger than any
infinity Zeno could have imagined. The non-denumerable infinity of
real numbers (and thus of points in space and of events in time)
is much larger than the merely denumerable infinity of integers.
Also, the sum of an infinite series of numbers can now have a
finite sum, unlike in Zeno's day. With all these changes,
mathematicians and scientists can say that all of Zeno's arguments
are based on what are now false assumptions and that no Zeno-like
paradoxes can be created within modern math and science. Achilles
catches his tortoise, the flying arrow moves, and it's possible to pass an infinite number of places in a finite time, without
contradiction. And one need not accept that a person can perform an infinite
number of actions in a finite time, if actions have first points and last
points, or beginnings and endings and next actions. No single person can be credited with having
shown how to solve Zeno's paradoxes. There have been essential
contributions starting from the calculus of Newton and Leibniz and
ending at the beginning of the twentieth century with the
mathematical advances of Cauchy, Weierstrass, Dedekind, Cantor,
Einstein, and Lebesque. Philosophically, the single greatest
contribution was to replace a reliance on what humans can imagine
with a reliance on creating logically consistent mathematical
concepts that can promote quantitative science.
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