Zeno's Paradox

           This post will focus on the workings of two of Zeno’s paradoxes, and how they relate to mathematics. First, the two paradoxes will be covered, and then the mathematical continuation into limits of infinite sums. Secondly, this post will cover how irrational numbers like Pi, e, and the square root of two, fit into the paradox.

            Zeno was a Greek philosopher that lived roughly around 500 B.C. He proposed a set of paradoxes, and the focus will be on two of those paradoxes, for their similarity. The first paradox starts out with a race between Achilles, and a tortoise. Since a tortoise is much slower than Achilles, the tortoise is given a hundred-meter head start of Achilles. After this head start, Achilles sprints a hundred meters to catch the tortoise. Now during the time it has taken Achilles to reach the hundred meters, the tortoise has move forward 10 meters. Again Achilles sprints ten meters to catch the tortoise, but again the tortoise has moved again another meter. This pattern will continue on forever. The paradox is that Achilles can run faster than the turtle, but cannot beat the turtle.

            The second paradox is similar, but it has to do with shooting an arrow. If an archer shoots an arrow at a target, surely it will travel the full distance to the target. Another way of thinking about it is the arrow must first travel halfway to the target. After traveling halfway, it must travel halfway of the new distance to the target. This process of halving the distance will go on forever, the arrow will just go another half the distance infinitely many times. The Paradox is that if there is an infinite number of halving, then how does the arrow ever hit the target? Surely it is known that the archer can hit the target, but then what happened to the infinite series?

            There is only one explanation that satisfies the two paradoxes. The sum of the infinite series must equal the finite distance traveled.  That may seem impossible, but it is completely possible. This process is used heavily in understanding calculus, the study of rates of change. To explain this answer we will start with partial sums, and then move into limits. To start the arrow travels half the distance, and then it travels half of the remaining distance, which would be a fourth of the total distance. So the sum of the fractional values would look like 1/2 + 1/4 +1/8 +1/16 . . . The sum of the first four terms .94 the total distance. So the more terms that are added to the series the closer the series approaches 1. This series, if summed all the way, with an infinite number of terms would equal one exactly. This idea is the introduction to infinite limits. A limit is a way of saying what happens as something approaches something else. In this case the limit as the number of terms approaches infinity would equal one. Many individuals struggle with the idea that there is a sum associated with an infinite series. If someone tried to put the series into a calculator, they would not ever be able to press enter, because then it would not be all of the infinite number of terms in the series. This idea is hard to understand, but at the same time it must be true because an archer can hit a target, people can clap their hands, people can drive from point A to point B. Every kind of movement requires an infinite sum of fractional parts of that distance, and yet everyone does it.

            This paradox can be seen again in irrational constants such as, pi, or the square root of the square root of two. Looking at pi, if a circle is draw with a compass, having radius of one, then the area of the circle is pi. This is concerning because pi goes on forever, so how can a finite area be an endless decimal? The same goes for the square root of two. Imagine a right triangle with two legs of length one. The hypotenuse is equal to the square root of two, a never ending decimal. Except at the same time the length of the hypotenuse does not go on forever. In both of these cases, it shows that some infinite series can be expressed as a finite number. In general, the numbers of the terms in these series must be getting smaller over time, or else they would just shoot off towards infinity.

Sources http://www.iep.utm.edu/zeno-par/

             https://www.youtube.com/watch?v=u7Z9UnWOJNY

Problems with Infinity

            This post is going to be varying slightly from the standard format of these posts. Instead of focusing on a specific problem, the focus will be shifted toward a number of problems and paradoxes concerning infinity. Infinity is not a number; it is a concept. So when infinity gets involved in problem, it sometimes causes the problem to break. This post will cover four of these problems with Infinity.

            The first problem concerns the Hilbert Hotel. This paradox was proposed by David Hilbert. First, imagine a hotel with an infinite number of rooms. In every room there is a mathematician. The problem starts out with a new mathematician walking in, and wanting a room. Well obviously the hotel is full, but the manager is smart enough to come up with a solution. He asks every mathematician to move to the room next to them, so room one moves to room two, and two moves to room three, and so on. So now room number is empty. Since there is an infinite number of rooms, there is never not enough room. In fact, even if a infinite number of mathematicians walked into the hotel they could all find a room. This goes against the initial understanding that the hotel is full, but at the same time there is always room.

            The second infinite paradox is called Gabriel’s Trumpet. The trumpet part comes from the general shape of the trumpet. A hollow cone like shape that tapers off, except this cone shape tapers off to infinitesimally small. A volume of this cone can be found, but yet the cone would have an infinite surface area. If there was a mathematical paint that had no thickness between molecules an infinite amount of that paint could be poured into the cone, but at the same time the volume of the cone can be calculated with calculus. This is the paradox, how can something take up a given space, but yet have infinite surface area?

            The third paradox concerns a dart board. Imagine a dart with a sharp point, that is in fact a single point thick. The problem arises when the dart is thrown at the dart board. The question is, if there is a hundred percent chance that the dart hits the board, what is the probability of hitting a certain singular point on the board? Since there are an infinite number of point in the surface of a dart board, the probability of hitting a certain point cannot be great than zero of the sum of probabilities would be infinite. Also the probability must be more than zero because if it was zero then the dart would not hit the board. The paradox is that a probability of hitting the board cannot be determined even if there is a hundred percent chance the dart will hit the board.

            The final paradox concerning infinity in this post, is about doubling money. Suppose there is a new casino game that has just arrived. The game is centered around a coin, and a pot of money. The pot starts at one dollar, and if a heads is flipped the money in the pot doubles. If a tails is flipped then the game ends, and the pot is collected. To the average individual it would make sense to put in a few pounds at a time, but mathematics has a different answer. Mathematics says to put as much money in as possible. This is because the sum of every outcome tends towards infinity. The paradox is that like any other casino game there is a chance to lose it all, but at the same there is a mathematical guaranteed chance of walking away with an infinite amount of money.

            As pointed out in these four problems, the concept of infinity creates some problems with mathematics. One of the biggest problems is getting around the idea of infinity. In the universe there is not anything that is infinite, except maybe the space of the universe itself, but that is another paradox in itself. In all of these problem there was something that could not exist in the real world. There are not any infinite hotels, infinite surface areas, and people hit dart boards. The answer is not that infinite needs to be removed, but rather that mathematics as a whole needs to work with infinity in another way. When some new idea is introduced to mathematics there is always a fight to hold on to what is known, and not to change. That will simply limit math as a whole. If no one ever stepped outside the box there would not be the complex number, or zeta functions, or countless other aspects. The point is infinity is not something in mathematics that cannot be dealt with. 

Sources https://www.youtube.com/watch?v=dDl7g_2x74Q

              https://nrich.maths.org/5788

              https://opinionator.blogs.nytimes.com/2010/05/09/the-hilbert-hotel/?_r=0