# The in a different way. When we pull

The most astounding and interesting math problems that troubled mathematicians for centuries, paradoxes in mathematics have always mind boggled many people in the course of thousands of years. These paradoxes defy the logical and fundamental concepts that are familiar to us today. Whether it is the Banach-Tarski Paradox or even a simple one, the Bootstrap Paradox, mathematicians have always been intrigued by these problems. The most popular and interesting of these paradoxes come from Zeno of Elea. Zeno’s paradox are not only of interest to mathematicians like myself, but they are also important to physicists and philosophers around the world. These paradoxes call for infinite processes to end. Today, we are going to shoot a bullet. When we pull the trigger the first time, in reality, the bullet will go zooming fast and will hit the target. However Zeno puts it in a different way. When we pull the trigger in the gun the second time, the bullet will go halfway to the target, and then halfway of that, and halfway of that, and so on. The bullet would not be able to hit the target because there will always be a half of something. In reality, however, the bullet will go zooming fast and will hit the target. There are vast amount of examples that happen in our everyday lives such as reading a book, throwing a ball and even breathing and eating. An effective problem to gun control… The way Zeno of Elea represented these paradoxes were also mind-boggling to the human race. One that proved that Achilles, the fastest man in the iliad losing in a race to a tortoise if the tortoise was given a five meter head start. If Achilles caught up to where the turtle originally started, the turtle would also be running, and would still be ahead of Achilles by about two meters. If Achilles covered the two meters quickly, the turtle would be ahead of him about a meter. So on and so on until the turtle would win the race. This paradox was called “Achilles and the Tortoise Paradox.” The second one proved that an object would never be able to reach its destination, if it was counting a different way. If a man wanted to get to a forest, 50 meters away, it would walk half way which is 25 meters, than half of that which is 12.5 meters, and half of that which is 6.25 meters, and half of that which is 3.125, and so on. The man would never be able to reach the forest because there would always be the half of a distance. This paradox was called “The Dichotomy Paradox.” Zeno of Elea was born around 455 B.C. and he was a very intelligent and brilliant boy in his childhood life. He wanted to prove his teacher wrong when he was being taught the Parmenides thesis of the impossibility of all motion, and he created the “paradox” of Achilles and the tortoise, the solution of which has challenged and troubled mathematicians and philosophers throughout the centuries. Zeno has created many paradoxes, some that have not survived, however others were handed over by other brilliant scholars such as Aristotle. Zeno was also a philosopher and he argued for both sides in court, which got him killed by the king. Zeno was born around 490 B.C. in the Greek colony of Elea in southern Italy. He was a very smart man and he was a student of Parmenides, a famous teacher. He was adopted by his teacher and thought very thoroughly of science, mathematics and philosophy. Zeno is well known in his discoveries of science such as the magnet and other conspiracy theories. He was well known for arguing for both sides, however, he was arrested and killed for doing so. He has written about 40 paradoxes, 8 of which has come from Aristotle, however two have been well known. He was very smart and is very brilliant in Elea, and Greece. In the paradox of “Achilles and the Tortoise” the turtle is given a five meter head start. The race is 30 meters long, and Achilles starts at the 0 mark. When the race begins, Achilles and the Tortoise start running as fast as they can. When Achilles reaches 5 meters, the tortoise was already at 8 meters. When Achilles reached 8 meters, the tortoise was at 9 meters. When Achilles was at 9 meters, tortoise was at 9 and ½ meters. And so on and so on, so Achilles will never be able to reach the tortoise, and the tortoise will win the race. This paradox proves all motion impossible. But it also proves time impossible. How, you may ask? If Zeno has to get from Tree A to Tree B, he has to walk one mile. If he walks one mile per hour, it will take him one hour to get to Tree B. As he walks half the distance for 30 minutes, than half of that for 15 and 7.5 minutes and 3.25 minutes, so on and so on. It will never be one hour, because there will always be half of a minute. Zeno not only made all motion impossible but he also time impossible. Zeno shouldn’t be able to walk from one tree to another, I shouldn’t be able to shoot a gun to a target, we should not be able to read a book or clap our hands. But we know better. What is the flaw in Zeno’s logic? Achilles must travel the following infinite series of distances before he catches the tortoise: first 0.9m, then an additional 0.09m, then 0.009m, … . These are the series of distances ahead that the tortoise reaches at the start of each of Achilles’ catch-ups. Looked at this way the puzzle is identical to the Dichotomy, for it is just to say that ‘that which is in locomotion must arrive before it arrives at the goal’. And so everything we said above applies here too. This paradox has puzzled scientists for centuries. Then, she runs half of half the distance left which is ¼. Then, she runs half of that which is ?. If thought thoroughly, Atlanta will never be able to reach the bus, because she will always be a distance away from the bus. This can also work in time, ½ hour + ¼ hour + ? hour + … ? 1 complete hour. In other words, the message Zeno is trying to get across is that supposedly, ½ + ¼ + ? + … = infinity, but it does not make sense and it is impossible because there cannot be infinite time. The Dichotomy paradox essentially presents an infinite sum of terms of decreasing size , which we can recognize to be . However, unlike a general geometric series, the series implied by The Dichotomy does not start with 1. Consequently, the sum of The Dichotomy series is actually ? 1, which, with x = , equals 1. In other words, the horse makes it from point A to point B.