## Zeno's paradoxes refer to a set of paradoxes that ...

Zeno's paradoxes refer to a set of paradoxes that were formulated by Zeno of Elea so as to support the doctrine of Parmenides that "all is one" and that in contradiction of the evidence of individual’s senses, both the belief in plurality as well as change is flawed. Zeno’s paradoxes were presented by the Greek philosopher by the name Zeno of Elea. He provided a lot of paradoxes in support of the hypothesis of Parmenides that “all is one.” However, the three paradoxes in relation to the “motion” are the most well-known. They include the paradox of Achilles and the tortoise, the arrow paradox, as well as the dichotomy paradox (Sorensen 23).

Zeno of Elea put forward the paradoxes in order to challenge the acknowledged notions of both space as well as time, which he came across in different philosophical circles. Zeno paradoxes have puzzled mathematicians for many years (Hellerstein 124), and it was until Cantor developed the theory of sets, that they could finally resolve the paradoxes fully. These paradoxes places emphasis on the relationship between the discreet and the continuous, a matter that is imperative in the mathematics field. According to Lechte, “Zeno created these paradoxes since there were other philosophers who opposing the point of view of Parmenides” (42). Mazur (45) asserts that the arguments by Zeno are probably the primary examples of the approach of proof that is referred to as proof by contradiction. Hellerstein states that “to mathematicians as well as historians, the paradoxes by Zeno are merely mathematical problems that the contemporary calculus can provide a mathematical solution” (98).

Sorensen (65) states that more than 40 paradoxes that are attributed to Zeno in a book he wrote acted as a defense to his teacher’s, Parmenides, philosophies. Parmenides based his belief in monism, which implied that reality is a single, continuous, static thing that is called ‘Being.’ In the defense of this far-reaching belief, Zeno formulated 40 arguments to illustrate that motion as well as plurality are not possible. According to Mazur, “the Zeno’s paradoxes have drawn a lot of interest from philosophers, scientists, as well as mathematicians” (76). Recently, Mazur (212) published a new resolution towards the paradoxes. They claim that the mathematics formalism that was used in earlier resolutions of the paradoxes is not adequate. The most well-known of these arguments by Zeno is the Achilles:

The Achilles and tortoise paradox is about a race between the two; with the tortoise being given a 10 meters lead. The assumption is that the tortoise is going to be slower in comparison to Achilles. The paradox argues that Achilles cannot outdo the tortoise in the race. The logic in this argument is that Achilles, sprinting at a speed of 10 m/s, will reach the starting point of the tortoise in one second, whereas the tortoise, at a speed of 1 meter per second, will have covered 1 meter. Once Achilles arrives at every point where the tortoise was initially, the tortoise will have covered another distance from that point. The Achilles and tortoise paradox is attacking the notion that many philosophers held, claiming that space was substantially divisible, as well as that motion was, therefore, continuous (Mazur 32).

There is a solution to Zeno’s paradox of the Achilles as well as the Tortoise. The distance intervals that are involved are presumed to continue infinitely, that is, become permanently recurring without ending, from the 10 meters down to feet, after that to inches, then centimeters, followed by millimeters et cetera, concluding that Achilles will never catch up with the tortoise.

Suppose John wishes to cross a field that has a distance of 200 meters. First, of course, he has to cover half of the distance, which is 100 meters. Then, he must cover half of the distance that is remaining, that is, 50 meters. After that, John must cover half of the 50 meters remaining, that is 25 meters, then half of 25 meters, which is 12.5 meters, and the process goes on and on until forever. The effect is that John will never reach the other side of the field.

What this in fact does is that it makes every motion impossible, because for John to cover half of the half of the distance, as well as prior to doing that, he must cover half of the half of the half of the distance that is remaining, and this procedure keeps repeating itself so that in fact he can never move whatever distance at all, since by so doing, it entails first covering an infinite number of little middle distances.

Well, assuming that John can be able cover all the infinite number of tiny distances, how far will he have walked? Two hundred meters! In other words,

[200=\frac{200}{2}+\frac{200}{4}+\frac{200}{8}+\frac{200}{16}+\frac{200}{32}+ frac{200}{64} +…]

Summing up an infinite figure of positive distances is supposed to provide a distance that is infinite from the sum. However, it does not -in John’s case it gives a number that is finite; in reality, all these small distances sum up to 200. The logic is that if one can divide a distance that is finite into a number of little distances that are infinite, then summing up all those distances collectively ought to give back the finite distance that one started with.

Therefore, the solution to the Zeno’s paradox is simple. Apparently, it will take John some fixed time so as to cover half of the field’s distance, for instance, 10 minutes. How long will John take to cover half of the 100 meters that is remaining? Half as long – just 5 minutes. In order to cover half of the remaining 50 meters, it will take him 2.5 minutes, and so on. Once John has covered all the numerous infinite distances, as well as sum up the time he took to cross them. Finally, John will reach the other end of the field. Therefore, in the paradox of Achilles and the tortoise, Achilles will have won that race.

In conclusion, there are various flaws that exist in Zeno’s paradox, and which are still debatable today. The solution in this paradox is analogous. Although the solution to the paradoxes of Zeno took many years in order to materialize, the philosophical problem can be solved. In case there exists different alternative treatments to the Zeno's paradoxes, then there arises the issue of if there is a distinct solution to these paradoxes or a number of solutions. From the viewpoint of Zeno’s paradoxes, the most noteworthy lesson that has been learned by those trying to resolve these paradoxes is that the only way out needs revision of many old theories as well as their concepts.