Zeno of Elea was recognised for

his string of paradoxes which altogether had a significant influence on

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.

The first of Zeno’s paradoxes

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.

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.

Another of Zeno’s paradoxes was

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.

Zeno’s paradoxes, namely that of

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?

Zeno’s paradoxes also massively

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.

To conclude, Zeno’s paradoxes

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.