Very little is known of the life of Zeno
of Elea. We certainly know that he was a
philosopher, and he is said to have been
the son of Teleutagoras. The main source
of our knowledge of Zeno comes from the dialogue
Parmenides written by Plato.
Zeno was a pupil and friend of the philosopher
Parmenides and studied with him in Elea.
The Eleatic School, one of the leading pre-Socratic
schools of Greek philosophy, had been founded
by Parmenides in Elea in southern Italy.
His philosophy of monism claimed that the
many things which appear to exist are merely
a single eternal reality which he called
Being. His principle was that "all is
one" and that change or non-Being are
impossible. Certainly Zeno was greatly influenced
by the arguments of Parmenides and Plato
tells us that the two philosophers visited
Athens together in around 450 BC.
Despite Plato's description of the visit
of Zeno and Parmenides to Athens, it is far
from universally accepted that the visit
did indeed take place. However, Plato tells
us that Socrates, who was then young, met
Zeno and Parmenides on their visit to Athens
and discussed philosophy with them. Given
the best estimates of the dates of birth
of these three philosophers, Socrates would
be about 20, Zeno about 40, and Parmenides
about 65 years of age at the time, so Plato's
claim is certainly possible.
Zeno had already written a work on philosophy
before his visit to Athens and Plato reports
that Zeno's book meant that he had achieved
a certain fame in Athens before his visit
there. Unfortunately no work by Zeno has
survived, but there is very little evidence
to suggest that he wrote more than one book.
The book Zeno wrote before his visit to Athens
was his famous work which, according to Proclus,
contained forty paradoxes concerning the
continuum. Four of the paradoxes, which we
shall discuss in detail below, were to have
a profound influence on the development of
mathematics.
Diogenes Laertius [10] gives further details
of Zeno's life which are generally thought
to be unreliable. Zeno returned to Elea after
the visit to Athens and Diogenes Laertius
claims that he met his death in a heroic
attempt to remove a tyrant from the city
of Elea. The stories of his heroic deeds
and torture at the hands of the tyrant may
well be pure inventions. Diogenes Laertius
also writes about Zeno's cosmology and again
there is no supporting evidence regarding
this, but we shall give some indication below
of the details.
Zeno's book of forty paradoxes was, according
to Plato [8]:-
... a youthful effort, and it was stolen
by someone, so that the author had no opportunity
of considering whether to publish it or not.
Its object was to defend the system of Parmenides
by attacking the common conceptions of things.
Proclus also described the work and confirms
that [1]:-
... Zeno elaborated forty different paradoxes
following from the assumption of plurality
and motion, all of them apparently based
on the difficulties deriving from an analysis
of the continuum.
In his arguments against the idea that the
world contains more than one thing, Zeno
derived his paradoxes from the assumption
that if a magnitude can be divided then it
can be divided infinitely often. Zeno also
assumes that a thing which has no magnitude
cannot exist. Simplicius, the last head of
Plato's Academy in Athens, preserved many
fragments of earlier authors including Parmenides
and Zeno. Writing in the first half of the
sixth century he explained Zeno's argument
why something without magnitude could not
exist [1]:-
For if it is added to something else, it
will not make it bigger, and if it is subtracted,
it will not make it smaller. But if it does
not make a thing bigger when added to it
nor smaller when subtracted from it, then
it appears obvious that what was added or
subtracted was nothing.
Although Zeno's argument is not totally convincing
at least, as Makin writes in [25]:-
Zeno's challenge to simple pluralism is successful,
in that he forces anti-Parmenideans to go
beyond common sense.
The paradoxes that Zeno gave regarding motion
are more perplexing. Aristotle, in his work
Physics, gives four of Zeno's arguments,
The Dichotomy, The Achilles, The Arrow, and
The Stadium. For the dichotomy, Aristotle
describes Zeno's argument (in Heath's translation
[8]):-
There is no motion because that which is
moved must arrive at the middle of its course
before it arrives at the end.
In order the traverse a line segment it is
necessary to reach its midpoint. To do this
one must reach the 1/4 point, to do this
one must reach the 1/8 point and so on ad
infinitum. Hence motion can never begin.
The argument here is not answered by the
well known infinite sum
1/2 + 1/4 + 1/8 + ... = 1
On the one hand Zeno can argue that the sum
1/2 + 1/4 + 1/8 + ... never actually reaches
1, but more perplexing to the human mind
is the attempts to sum 1/2 + 1/4 + 1/8 +
... backwards. Before traversing a unit distance
we must get to the middle, but before getting
to the middle we must get 1/4 of the way,
but before we get 1/4 of the way we must
reach 1/8 of the way etc. This argument makes
us realise that we can never get started
since we are trying to build up this infinite
sum from the "wrong" end. Indeed
this is a clever argument which still puzzles
the human mind today.
Zeno bases both the dichotomy paradox and
the attack on simple pluralism on the fact
that once a thing is divisible, then it is
infinitely divisible. One could counter his
paradoxes by postulating an atomic theory
in which matter was composed of many small
indivisible elements. However other paradoxes
given by Zeno cause problems precisely because
in these cases he considers that seemingly
continuous magnitudes are made up of indivisible
elements. Such a paradox is 'The Arrow' and
again we give Aristotle's description of
Zeno's argument (in Heath's translation [8]):-
If, says Zeno, everything is either at rest
or moving when it occupies a space equal
to itself, while the object moved is in the
instant, the moving arrow is unmoved.
The argument rests on the fact that if in
an indivisible instant of time the arrow
moved, then indeed this instant of time would
be divisible (for example in a smaller 'instant'
of time the arrow would have moved half the
distance). Aristotle argues against the paradox
by claiming:-
... for time is not composed of indivisible
'nows', no more than is any other magnitude.
However, this is considered by some to be
irrelevant to Zeno's argument. Moreover to
deny that 'now' exists as an instant which
divides the past from the future seems also
to go against intuition. Of course if the
instant 'now' does not exist then the arrow
never occupies any particular position and
this does not seem right either. Again Zeno
has presented a deep problem which, despite
centuries of efforts to resolve it, still
seems to lack a truly satisfactory solution.
As Frankel writes in [20]:-
The human mind, when trying to give itself
an accurate account of motion, finds itself
confronted with two aspects of the phenomenon.
Both are inevitable but at the same time
they are mutually exclusive. Either we look
at the continuous flow of motion; then it
will be impossible for us to think of the
object in any particular position. Or we
think of the object as occupying any of the
positions through which its course is leading
it; and while fixing our thought on that
particular position we cannot help fixing
the object itself and putting it at rest
for one short instant.
Vlastos (see [32]) points out that if we
use the standard mathematical formula for
velocity we have v = s/t, where s is the
distance travelled and t is the time taken.
If we look at the velocity at an instant
we obtain v = 0/0, which is meaningless.
So it is fair to say that Zeno here is pointing
out a mathematical difficulty which would
not be tackled properly until limits and
the differential calculus were studied and
put on a proper footing.
As can be seen from the above discussion,
Zeno's paradoxes are important in the development
of the notion of infinitesimals. In fact
some authors claim that Zeno directed his
paradoxes against those who were introducing
infinitesimals. Anaxagoras and the followers
of Pythagoras, with their development of
incommensurables, are also thought by some
to be the targets of Zeno's arguments (see
for example [10]). Certainly it appears unlikely
that the reason given by Plato, namely to
defend Parmenides' philosophical position,
is the whole explanation of why Zeno wrote
his famous work on paradoxes.
The most famous of Zeno's arguments is undoubtedly
the Achilles. Heath's translation from Aristotle's
Physics is:-
... the slower when running will never be
overtaken by the quicker; for that which
is pursuing must first reach the point from
which that which is fleeing started, so that
the slower must necessarily always be some
distance ahead.
Most authors, starting with Aristotle, see
this paradox to be essentially the same as
the Dichotomy. For example Makin [25] writes:-
... as long as the Dichotomy can be resolved,
the Achilles can be resolved. The resolutions
will be parallel.
As with most statements about Zeno's paradoxes,
there is not complete agreement about any
particular position. For example Toth [29]
disputes the similarity of the two paradoxes,
claiming that Aristotle's remarks leave much
to be desired and suggests that the two arguments
have entirely different structures.
Both Plato and Aristotle did not fully appreciate
the significance of Zeno's arguments. As
Heath says [8]:-
Aristotle called them 'fallacies', without
being able to refute them.
Russell certainly did not underrate Zeno's
significance when he wrote in [13]:-
In this capricious world nothing is more
capricious than posthumous fame. One of the
most notable victims of posterity's lack
of judgement is the Eleatic Zeno. Having
invented four arguments all immeasurably
subtle and profound, the grossness of subsequent
philosophers pronounced him to be a mere
ingenious juggler, and his arguments to be
one and all sophisms. After two thousand
years of continual refutation, these sophisms
were reinstated, and made the foundation
of a mathematical renaissance ....
Here Russell is thinking of the work of Cantor,
Frege and himself on the infinite and particularly
of Weierstrass on the calculus. In [2] the
relation of the paradoxes to mathematics
is also discussed, and the author comes to
a conclusion similar to Frankel in the above
quote:-
Although they have often been dismissed as
logical nonsense, many attempts have also
been made to dispose of them by means of
mathematical theorems, such as the theory
of convergent series or the theory of sets.
In the end, however, the difficulties inherent
in his arguments have always come back with
a vengeance, for the human mind is so constructed
that it can look at a continuum in two ways
that are not quite reconcilable.
It is difficult to tell precisely what effect
the paradoxes of Zeno had on the development
of Greek mathematics. B L van der Waerden
(see [31]) argues that the mathematical theories
which were developed in the second half of
the fifth century BC suggest that Zeno's
work had little influence. Heath however
seems to detect a greater influence [8]:-
Mathematicians, however, ... realising that
Zeno's arguments were fatal to infinitesimals,
saw that they could only avoid the difficulties
connected with them by once and for all banishing
the idea of the infinite, even the potentially
infinite, altogether from their science;
thenceforth, therefore, they made no use
of magnitudes increasing or decreasing ad
infinitum, but contented themselves with
finite magnitudes that can be made as great
or as small as we please.
We commented above that Diogenes Laertius
in [10] describes a cosmology that he believes
is due to Zeno. According to his description,
Zeno proposed a universe consisting of several
worlds, composed of "warm" and
"cold, "dry" and "wet"
but no void or empty space. Because this
appears to have nothing in common with his
paradoxes, it is usual to take the line that
Diogenes Laertius is in error. However, there
is some evidence that this type of belief
was around in the fifth century BC, particularly
associated with medical theory, and it could
easily have been Zeno's version of a belief
held by the Eleatic School.
Article by: J J O'Connor and E F Robertson
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