Paradoxes: Zeno of Elea

Zeno of Elea was a Greek philosopher famous for paradoxes which assert that motion is nothing but an illusion. It is assumed that Zeno created the paradoxes in support of Plato’s Parminedes.  We will present two of these paradoxes along with a refutation.


1. Achilles and the Tortoise

Achilles was challenged to a race by the Tortoise. The Tortoise boasted the claim that he would win as long as Achilles gave him a small loan of a 10 meter head start. Achilles agreed to this without hesitation and they got into position. They started the race on Achilles’ mark. Once Achilles made it to 10 meters he noticed that the Tortoise had moved 5. Once Achilles made it the extra 5 he noticed the Tortoise had made it an extra 2.5. This continued for a while until eventually Achilles conceded the race knowing that no matter how close he got, the Tortoise would always be ahead.


2. Flight of the Arrow

There was an Acher that stood before his Target. His arrow was loaded and bow was stretched. He took good aim and fired a sure shot. The arrow left the bow but before it could get to the Target, it had to make half the distance.  Then it had to make half of that distance. This continued and the Archer realized that the arrow would never make it to the Target.


So what is going on here? Obviously, from our own everyday experiences, we know that Achilles would run right passed the Tortoise and win the race. We also know (assuming the shot was indeed true) that the arrow would have hit the Target. Yet this line of reasoning seems to indicate otherwise. There have been several refutations of Zeno’s Paradoxes from Aristotle to Bertrand Russel and beyond. Perhaps the calculus version of a refutation is the mathematicians favorite. Suppose we look at the arrow paradox. Let us call the distance from the Archer to the Target to be 1. Then according to Zeno we must make it a distance of 1/2. Then 1/4, 1/8, and so on. So we have a sum \sum_{n=1}^\infty (\frac{1}{2})^n we know from calculus this sum converges to 1. This accompanied by the decreasing amount of time it takes to make it the next distance, properly refutes the paradox and offers a solution. Suppose the arrow is moving v= \frac{1}{16} of the distance per second. Then it takes  8v to get to half the distance. Then it takes 4v more to make it  1/4 more. Then 2v then v. We see then we have a decreasing sequence of time (\frac{16v}{2^n})_{n = 1}^\infty which tends to 0. Thus it is clear that eventually time will become negligible and we will hit our target. It is clear that Zeno is assuming that Space and time are two different entities and are both discrete. These paradoxes also give rise to a philosophical idea that life, in particular time, instead of being one fluid motion is in fact just a series of frames or snapshots if you will. This could lead back to the idea that free will does not exist as one moves according to the next frame. These are fun thoughts to ponder.


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