Zeno of Elea was recognised forhis string of paradoxes which altogether had a significant influence onphilosophy and namely, the idea of motion. Zeno was a Greek philosopher who wasan Eleatic, which is a term used when describing an individual or group fromthe ancient city if Elea, where there was a school of philosophers. On top ofbeing a philosopher, Zeno was also a Mathematician, and we can see thisreflected in his paradoxes as he uses mathematics as a point of reference tofurther validate his theory. Another philosopher, a fellow Eleatic namedParmenides also had a significant influence on Zeno, and Zeno happened to beone of his disciples and built his paradoxes around the work of Parmenides.

Zeno’s paradoxes proceeded to shape philosophy in the aspect of continuity andinfinity; he composed his paradoxes in a way to illustrate that any sort ofidea which differs from the teachings of Parmenides, is unsound. Plato, whohimself was a philosopher also, demonstrated a piece of writing where he spoke aboutParmenides. From this text, we can gather that it wields a source of Zeno’sintentions that we can regard as the most fitting. Aristotle however, gavestatements for Zeno’s arguments, namely that of motion. From these reports, weassemble the names of Zeno’s paradoxes; Achilles and the Tortoise and theDichotomy. These paradoxes were created in a manner that although both seemlogical when you converge on the conclusion, however, they seem altogetherpreposterous.

The first of Zeno’s paradoxesthat I shall cover is that entitled Achilles and the Tortoise. In this, Zenofurthers 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 atortoise. The tortoise is given the advantage of a head start of 100 metres, tomake the race fair. Nonetheless, if we surmise that both Achilles and thetortoise begin running at differing constant speeds, the tortoise very slow andAchilles very fast, at some point Achilles will have had to have run 100 metresor there about to reach the tortoises starting point. However, during the timethe Achilles has run the 100 metres, given the speed at which he is movingcompared to the rate that the tortoise is running, the tortoise would have travelleda much shorter distance given the same time, say around that of 10 metres.However, once Achilles has reached the starting point of the tortoise, thetortoise has now exceeded this point, meaning that Achilles will need to gainthe ten further metres then.

It would then take added time for Achilles to runthis distance, and in the time that it takes him to travel this range, thetortoise has once again exceeded his last point. Achilles then has to reachthis further point by which the tortoise has moved yet again and so on. Fromthis, we can analyse that, whenever Achilles reaches the certain distance wherethe tortoise has been, he still has to advance further before he can achievethe distance gained by the tortoise. To put this into practice, say ifI wanted to reach the other side of a field. For me to get from one side to theother, I must first reach the halfway point to then be able to traverse theother half. I must keep covering half the distance and later half of thatlength and half of that stretch and so on to infinity. To sum this up, I wouldnever actually get to the other side of the field because first I must reach aninfinite amount of points before being able to continue to the additionalinfinite amount of points and so on.

What this means is that all motion is thenseemingly impossible, because for me to reach any distance, I must reach theinfinite amount of distances from A to B, therefore never actually getting toB, because I first have to cover an endless amount of lengths which may take aninfinite amount of time. If we were to take this paradox for what is it then itcan cause a lot of problems because to think logically we know that givenAchilles running at a faster pace than the tortoise, even with the head startat some point Achilles will overtake the tortoise. Per contra, if we were toinvolve time into the paradox and say that it takes Achilles half the time toreach 10 metres and it took the tortoise double the time, then eventuallyAchilles would overtake the tortoise, or the tortoise would fall behind bydefault. However, if we were to believe what Zeno is saying, that to go acertain distance, I must first go half that length and so on. What we candeduct from this is, given all the infinite intervals, if we were to add theseup then we would assume that it would give us some infinite number, however itdoes not. Given all these sums, the number we get to is distinctly finite, andas a result of this, we should reach some finite distance, and therefore reachthe finish line.

This paradox was given by Zeno to challenge the idea of motionand to convey that, reality itself can be divided infinitely. Although Achilles and theTortoise is seen to be one of the most popular of Zeno’s paradoxes, it is adivision of a simpler paradox named the Dichotomy. The Dichotomy translatesfrom ancient Greek into merely meaning the paradox of cutting in two. Thedichotomy argument affirms that, say if you wanted to walk to the other side ofthe room, to do so you must first walk halfway. Once you have reached thishalfway point you must then walk half of the remaining distance, and later onceyou have completed that you must then walk half of the remaining length and soon. Each of these intervals can be completed in a finite amount of time, but towork out exactly how long it would take you to reach the other side of the roomseems simple enough. You just need to add up all the times set of the distancesthat you have walked, but the problem with this is that there are infinitely manyto add up. Given this sum, shouldn’t the total be that of infinity as Zenothought? From what we can gather from thisparadox, as depicted above, all motion is seemingly impossible.

We cantranslate this paradox into a mathematical sum to show that there is a logicalflaw, by doing this we need first to take the distance of one mile. Say thatyou are walking at one mile per hour, the mathematical sum of distance dividedby 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 thefirst half of the journey, stating that it would take half an hour to completethis. The next part would then take a quarter of an hour and the upcoming aneighth and so on if we were to create a sum of this where each fraction beginswith a finite number but that the denominator is an infinite number thensurely, we should get infinity.

However, mathematicians have since found thatwith this equation the answer would not be infinity but equal a finitenumber. Another of Zeno’s paradoxes wasnamed that of the Arrow paradox but also went under the name of Fletcher’sparadox. What he does is, he imagines that an individual finds that all theirarrows cannot move. Firstly, we need to consider that time itself consists ofan infinite amount of moments, and that in any of these moments the arrow canbe regarded as being motionless, once it has been released. At this moment, thearrow cannot move at all since it doesn’t have the time in which to do so.

Toconclude this, considering all the countless moments that exist, we cannot findone in which the arrow has the time to move, so regardless of the infiniteamount of moments, in any of these, the arrow can neither fall nor fly. Aristotle very much disagreedwith Zeno’s paradoxes and said that they were not correct. He took Zeno’s ideasand created his solution and argued that he had disproved the paradoxesthemselves. His answer was that considering the distance decreases, the time ittakes to cover these distances must also decrease, therefore meaning thateventually, you would reach the finish. You would not (as Zeno claimed) neverreach the finish, as given all these infinite distances with each of them beingdivided in half and then into a quarter and so on, the time it takes you tocover these shorter distances would not be the same as covering a much largerdistance. For example, it would take you much less time to cover a third of amile than the entire mile itself, whereas Zeno never took into considerationthat as the distance decreases the amount of time it takes you to cover themalso does.

A fundamental way in which Zeno’sparadox helped influence philosophy was with that of space and time. Bertrand Russellwas the first to come up with this, and he named this the ‘at-at theory ofmotion,’ Russell created this theory in response to the arrow paradox. TheArrow paradox was used by Zeno to portray how motion itself is impossible, forexample, when we release the arrow, for every part of the arrows flight ittakes up the exact space it needs for itself. However, the arrow is taking upjust the amount of space as it needs for itself; therefore, the object cannotbe moving at all, so throughout the arrows flight it is at rest and thus inevery interval, not in motion. So regardless of what our senses, such as ourvision tells us, motion is impossible. What Russell stated was that this couldnot happen, the arrow (or any object) cannot be at rest or in motion at anygiven time. For the arrow to be in motion, it would have to be at a differentpoint 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 atwhich the arrow was at that time, and then very slightly change the time by amillisecond, the arrow would then be at a different location.

Therefore, thearrow is in fact moving, and motion is possible. The way this influencedphilosophy was that it made Russell think to try and come up with a solution toprove that motion does in fact exist. Another philosopher namedGrunbaum, who took inspiration from Russell’s ideas, gave himself the task ofillustrating how modern mathematics can seemingly and accurately solve everyone of Zeno’s paradoxes. Grunbaum’s primary aim was to prove that none ofZeno’s paradoxes created any sort of threat to the concept of infinity, butalso that mathematics when used correctly can give light to an adequate descriptionof space and time. Whilst conducting his research, Grunbaum found that to solveZeno’s paradox, a mathematical solution alone would not be ample as theparadoxes question the nature of physics as well as the nature of mathematics.Therefore, given that paradoxes challenge the nature of physics, and that a mathematicalsolution by itself would not suffice, the subject of physics would have to beput in place for the solution to be validated. A mathematical solution alonewould not be enough to solve Zeno’s paradox as it does not involve a usefuldescription of space, time and motion. Zeno’s paradoxes, namely that ofAchilles and the tortoise influenced philosophy in that of Supertasks.

Thisconcept of a Supertask can be explained using the idea of Thompson’s lamp whichwas created by James Thompson. We firstly need to imagine that there is somemachine connected up to an ordinary lamp. The primary function of this deviceis to turn the lamp on and off, and what is unique about this machine is thespeed at which it can do this. Similarly, the 100-metre runner can complete anendless number of tasks in a finite amount of time, each by completing each onefaster and faster. The machine begins by turning the lamp on, which takes onesecond, 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 tofinish turning the lamp on and off we get the answer of two seconds. Theobvious problem with this is that once the machine is done, is the lamp turnedon 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 processfinishing after two seconds.

If we compare this to Zeno’sparadox where there is no last length for Achilles to cross on the endlessamount of distances to cover in the race, even though the entire race is overin ten seconds. However, the main difference is that for Achilles, he does nothave to think about the last part of the race, as regardless of what happens hewill always end up at the finish line. The lamp, on the other hand, differsfrom this as we are trying to work out and would like to know whether the lampis on and off when the machine stops. A simple answer to this question would bethat no such mechanism is possible, as logically there is a limit to themaximum speed to which the light can be turned on and off, so realistically itcannot happen that quickly. However, although this seems like a suitableanswer, there could be somebody who one day creates such a contraption whichcan complete all of these tasks at such speed, this would then lead to the problemonce again reoccurring without a simple solution. Returning to the case of theAchilles and the tortoise, the only difference being that this time anindividual has created a machine which we can apply to the track. It would workin the manner that, as soon as Achilles starts travelling the said distance themachines recognise this and creates some impassable object that would be placedin front of Achilles. The more Achilles moves forward any said distance, themachine repeats the process and puts in place some impassable object in hispath so that he cannot get past this specific point.

With this said, Achillescan never actually get any said distance since as soon as he moves anydistance, the machine will stop him in his tracks. We can conclude from thisthat Achilles can never actually start the race, as for him to start he mustmove 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 neveractually start the race.

The main point of Thompson’s lamp was to prove that nosuch machine could ever be used as if Achilles cannot start the race then themachine itself never begins its work; there would be no point in there beingany such machine if it has no purpose. The question still left is that of whycan Achilles not start the race? Zeno’s paradoxes also massivelyinfluenced mathematics, namely infinitesimals. When we use mathematics, whatwas once seen as a critical factor, would eventually convey that infinitesimalquantities namely anything more substantial than zero but smaller than anyfinite number, are not needed. In the late 1600’s Sir Isaac Newton and Leibnizcreated something called Calculus, their idea was that any continuous motionconsisted of an infinite number of infinitesimal tasks.

Although calculus wasnot accepted by many right away, when it was many mathematicians and physicistsbelieved that motion should consist of real numbers and use time as part of itsargument, which in turn gives the used numbers value. By the early 20thcentury, many years after calculus first came about, and many mathematiciansbegan to make use of it. For example, they argued that to make any sense ofmotion, they needed a theory which used real numbers to convey continuity.Zeno’s paradoxes influenced this idea of infinitesimals as in his paradox ofthe Dichotomy, where he claims that for an individual to get to the other sideof the room, he must first reach an infinite number of points before actuallycrossing to the other side. Leibniz and Newton, with their use of Calculus, setout to prove that Zeno’s claim that all motion is impossible was in fact wrong. To conclude, Zeno’s paradoxeswere created to challenge the idea of motion and to prove that motion itself isstrictly impossible. I believe that although Zeno gave a convincing argument inhis paradoxes, and if taken at face value they do seem possible; however, oncewe use logic and involve the idea of time into the paradoxes, they becomefalse.

I believe that Zeno and his paradoxes had a massive influence onphilosophy, not only with the idea of motion but also that of infinity.