Zeno of Elea was recognised for
philosophy and namely, the idea of motion. Zeno was a Greek philosopher who was
an Eleatic, which is a term used when describing an individual or group from
the ancient city if Elea, where there was a school of philosophers. On top of
being a philosopher, Zeno was also a Mathematician, and we can see this
reflected in his paradoxes as he uses mathematics as a point of reference to
further validate his theory. Another philosopher, a fellow Eleatic named
Parmenides also had a significant influence on Zeno, and Zeno happened to be
one of his disciples and built his paradoxes around the work of Parmenides.
Zeno’s paradoxes proceeded to shape philosophy in the aspect of continuity and
infinity; he composed his paradoxes in a way to illustrate that any sort of
idea which differs from the teachings of Parmenides, is unsound. Plato, who
himself was a philosopher also, demonstrated a piece of writing where he spoke about
Parmenides. From this text, we can gather that it wields a source of Zeno’s
intentions that we can regard as the most fitting. Aristotle however, gave
statements for Zeno’s arguments, namely that of motion. From these reports, we
assemble the names of Zeno’s paradoxes; Achilles and the Tortoise and the
Dichotomy. These paradoxes were created in a manner that although both seem
logical when you converge on the conclusion, however, they seem altogether
preposterous.

that I shall cover is that entitled Achilles and the Tortoise. In this, Zeno
furthers this concept of motion by using an array of examples and mathematics.
In this paradox, Zeno adopts the model of Achilles being in a race with a
tortoise. The tortoise is given the advantage of a head start of 100 metres, to
make the race fair. Nonetheless, if we surmise that both Achilles and the
tortoise begin running at differing constant speeds, the tortoise very slow and
Achilles very fast, at some point Achilles will have had to have run 100 metres
or there about to reach the tortoises starting point. However, during the time
the Achilles has run the 100 metres, given the speed at which he is moving
compared to the rate that the tortoise is running, the tortoise would have travelled
a much shorter distance given the same time, say around that of 10 metres.
However, once Achilles has reached the starting point of the tortoise, the
tortoise has now exceeded this point, meaning that Achilles will need to gain
the ten further metres then. It would then take added time for Achilles to run
this distance, and in the time that it takes him to travel this range, the
tortoise has once again exceeded his last point. Achilles then has to reach
this further point by which the tortoise has moved yet again and so on. From
this, we can analyse that, whenever Achilles reaches the certain distance where
the tortoise has been, he still has to advance further before he can achieve
the distance gained by the tortoise.

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To put this into practice, say if
I wanted to reach the other side of a field. For me to get from one side to the
other, I must first reach the halfway point to then be able to traverse the
other half. I must keep covering half the distance and later half of that
length and half of that stretch and so on to infinity. To sum this up, I would
never actually get to the other side of the field because first I must reach an
infinite amount of points before being able to continue to the additional
infinite amount of points and so on. What this means is that all motion is then
seemingly impossible, because for me to reach any distance, I must reach the
infinite amount of distances from A to B, therefore never actually getting to
B, because I first have to cover an endless amount of lengths which may take an
infinite amount of time. If we were to take this paradox for what is it then it
can cause a lot of problems because to think logically we know that given
Achilles running at a faster pace than the tortoise, even with the head start
at some point Achilles will overtake the tortoise. Per contra, if we were to
involve time into the paradox and say that it takes Achilles half the time to
reach 10 metres and it took the tortoise double the time, then eventually
Achilles would overtake the tortoise, or the tortoise would fall behind by
default. However, if we were to believe what Zeno is saying, that to go a
certain distance, I must first go half that length and so on. What we can
deduct from this is, given all the infinite intervals, if we were to add these
up then we would assume that it would give us some infinite number, however it
does not. Given all these sums, the number we get to is distinctly finite, and
as a result of this, we should reach some finite distance, and therefore reach
the finish line. This paradox was given by Zeno to challenge the idea of motion
and to convey that, reality itself can be divided infinitely.

Although Achilles and the
Tortoise is seen to be one of the most popular of Zeno’s paradoxes, it is a
division of a simpler paradox named the Dichotomy. The Dichotomy translates
from ancient Greek into merely meaning the paradox of cutting in two. The
dichotomy argument affirms that, say if you wanted to walk to the other side of
the room, to do so you must first walk halfway. Once you have reached this
halfway point you must then walk half of the remaining distance, and later once
you have completed that you must then walk half of the remaining length and so
on. Each of these intervals can be completed in a finite amount of time, but to
work out exactly how long it would take you to reach the other side of the room
seems simple enough. You just need to add up all the times set of the distances
that you have walked, but the problem with this is that there are infinitely many
to add up. Given this sum, shouldn’t the total be that of infinity as Zeno
thought?

From what we can gather from this
paradox, as depicted above, all motion is seemingly impossible. We can
translate this paradox into a mathematical sum to show that there is a logical
flaw, by doing this we need first to take the distance of one mile. Say that
you are walking at one mile per hour, the mathematical sum of distance divided
by speed would tell us that it would take one hour to walk the mile. However,
if we were to look at this using Zeno’s paradox, then we need to divide the
first half of the journey, stating that it would take half an hour to complete
this. The next part would then take a quarter of an hour and the upcoming an
eighth and so on if we were to create a sum of this where each fraction begins
with a finite number but that the denominator is an infinite number then
surely, we should get infinity. However, mathematicians have since found that
with this equation the answer would not be infinity but equal a finite
number.

named that of the Arrow paradox but also went under the name of Fletcher’s
paradox. What he does is, he imagines that an individual finds that all their
arrows cannot move. Firstly, we need to consider that time itself consists of
an infinite amount of moments, and that in any of these moments the arrow can
be regarded as being motionless, once it has been released. At this moment, the
arrow cannot move at all since it doesn’t have the time in which to do so. To
conclude this, considering all the countless moments that exist, we cannot find
one in which the arrow has the time to move, so regardless of the infinite
amount of moments, in any of these, the arrow can neither fall nor fly.

Aristotle very much disagreed
with Zeno’s paradoxes and said that they were not correct. He took Zeno’s ideas
and created his solution and argued that he had disproved the paradoxes
themselves. His answer was that considering the distance decreases, the time it
takes to cover these distances must also decrease, therefore meaning that
eventually, you would reach the finish. You would not (as Zeno claimed) never
reach the finish, as given all these infinite distances with each of them being
divided in half and then into a quarter and so on, the time it takes you to
cover these shorter distances would not be the same as covering a much larger
distance. For example, it would take you much less time to cover a third of a
mile than the entire mile itself, whereas Zeno never took into consideration
that as the distance decreases the amount of time it takes you to cover them
also does.

A fundamental way in which Zeno’s
paradox helped influence philosophy was with that of space and time. Bertrand Russell
was the first to come up with this, and he named this the ‘at-at theory of
motion,’ Russell created this theory in response to the arrow paradox. The
Arrow paradox was used by Zeno to portray how motion itself is impossible, for
example, when we release the arrow, for every part of the arrows flight it
takes up the exact space it needs for itself. However, the arrow is taking up
just the amount of space as it needs for itself; therefore, the object cannot
be moving at all, so throughout the arrows flight it is at rest and thus in
every interval, not in motion. So regardless of what our senses, such as our
vision tells us, motion is impossible. What Russell stated was that this could
not happen, the arrow (or any object) cannot be at rest or in motion at any
given time. For the arrow to be in motion, it would have to be at a different
point at a different time, to give the impression that it is in fact moving.
For example, if we were to take a specific time and work out the distance at
which the arrow was at that time, and then very slightly change the time by a
millisecond, the arrow would then be at a different location. Therefore, the
arrow is in fact moving, and motion is possible. The way this influenced
philosophy was that it made Russell think to try and come up with a solution to
prove that motion does in fact exist.

Another philosopher named
Grunbaum, who took inspiration from Russell’s ideas, gave himself the task of
illustrating how modern mathematics can seemingly and accurately solve every
one of Zeno’s paradoxes. Grunbaum’s primary aim was to prove that none of
Zeno’s paradoxes created any sort of threat to the concept of infinity, but
also that mathematics when used correctly can give light to an adequate description
of space and time. Whilst conducting his research, Grunbaum found that to solve
Zeno’s paradox, a mathematical solution alone would not be ample as the
paradoxes question the nature of physics as well as the nature of mathematics.
Therefore, given that paradoxes challenge the nature of physics, and that a mathematical
solution by itself would not suffice, the subject of physics would have to be
put in place for the solution to be validated. A mathematical solution alone
would not be enough to solve Zeno’s paradox as it does not involve a useful
description of space, time and motion.

Achilles and the tortoise influenced philosophy in that of Supertasks. This
concept of a Supertask can be explained using the idea of Thompson’s lamp which
was created by James Thompson. We firstly need to imagine that there is some
machine connected up to an ordinary lamp. The primary function of this device
is to turn the lamp on and off, and what is unique about this machine is the
speed at which it can do this. Similarly, the 100-metre runner can complete an
endless number of tasks in a finite amount of time, each by completing each one
faster and faster. The machine begins by turning the lamp on, which takes one
second, half a second later it turns the lamp on, it continues in this pattern.
If we use mathematics to work out the sum of how long the machine would take to
finish turning the lamp on and off we get the answer of two seconds. The
obvious problem with this is that once the machine is done, is the lamp turned
on or turned off? With the way that the sequence is programmed to function,
there is seemingly no last switch on or switch off, regardless of the process
finishing after two seconds.

If we compare this to Zeno’s
paradox where there is no last length for Achilles to cross on the endless
amount of distances to cover in the race, even though the entire race is over
in ten seconds. However, the main difference is that for Achilles, he does not
have to think about the last part of the race, as regardless of what happens he
will always end up at the finish line. The lamp, on the other hand, differs
from this as we are trying to work out and would like to know whether the lamp
is on and off when the machine stops. A simple answer to this question would be
that no such mechanism is possible, as logically there is a limit to the
maximum speed to which the light can be turned on and off, so realistically it
cannot happen that quickly. However, although this seems like a suitable
answer, there could be somebody who one day creates such a contraption which
can complete all of these tasks at such speed, this would then lead to the problem
once again reoccurring without a simple solution.

Returning to the case of the
Achilles and the tortoise, the only difference being that this time an
individual has created a machine which we can apply to the track. It would work
in the manner that, as soon as Achilles starts travelling the said distance the
machines recognise this and creates some impassable object that would be placed
in front of Achilles. The more Achilles moves forward any said distance, the
machine repeats the process and puts in place some impassable object in his
path so that he cannot get past this specific point. With this said, Achilles
can never actually get any said distance since as soon as he moves any
distance, the machine will stop him in his tracks. We can conclude from this
that Achilles can never actually start the race, as for him to start he must
move forward, but as soon as he does so, the machine stops him once again.
Therefore, with the said machine applied to the track, Achilles can never
actually start the race. The main point of Thompson’s lamp was to prove that no
such machine could ever be used as if Achilles cannot start the race then the
machine itself never begins its work; there would be no point in there being
any such machine if it has no purpose. The question still left is that of why
can Achilles not start the race?

influenced mathematics, namely infinitesimals. When we use mathematics, what
was once seen as a critical factor, would eventually convey that infinitesimal
quantities namely anything more substantial than zero but smaller than any
finite number, are not needed. In the late 1600’s Sir Isaac Newton and Leibniz
created something called Calculus, their idea was that any continuous motion
consisted of an infinite number of infinitesimal tasks. Although calculus was
not accepted by many right away, when it was many mathematicians and physicists
believed that motion should consist of real numbers and use time as part of its
argument, which in turn gives the used numbers value. By the early 20th
century, many years after calculus first came about, and many mathematicians
began to make use of it. For example, they argued that to make any sense of
motion, they needed a theory which used real numbers to convey continuity.
Zeno’s paradoxes influenced this idea of infinitesimals as in his paradox of
the Dichotomy, where he claims that for an individual to get to the other side
of the room, he must first reach an infinite number of points before actually
crossing to the other side. Leibniz and Newton, with their use of Calculus, set
out to prove that Zeno’s claim that all motion is impossible was in fact wrong.

were created to challenge the idea of motion and to prove that motion itself is
strictly impossible. I believe that although Zeno gave a convincing argument in
his paradoxes, and if taken at face value they do seem possible; however, once
we use logic and involve the idea of time into the paradoxes, they become
false. I believe that Zeno and his paradoxes had a massive influence on
philosophy, not only with the idea of motion but also that of infinity.