Free Outer Limits of Reason by Noson S. Yanofsky

mathematician is not adding each of the infinite terms to a running sum. She is simply displaying the first few terms and then indicating with an ellipsis that there are an infinite number of terms. She is doing a trick that shows what the sum of all of them would be if she summed them. If one were to sit down and add all infinite terms, it would indeed take an infinite amount of time.
    A better solution is to say that the problem with Zeno’s reasoning is that he assumes that space is continuous. That means that space looks like the real-number line and is infinitely divisible—that is, between every two points lies an infinite number of points. Only with this assumption can one describe the dichotomy paradox. In contrast, imagine that we are watching the slacker go to the door in an old-fashioned television made up of millions of little pixels. Then as he is moving, he is crossing the pixels. He crosses half of the pixels and then he crosses half of the rest of the pixels. Eventually the TV slacker will be one pixel away from the door and then he will be at the door. There are no half pixels to cross. A pixel is either crossed or not crossed. On the TV screen there is no problem with the slacker getting to his destination and Zeno’s paradox evaporates. Maybe we can say the same thing with the real world. Perhaps space is made up of discrete points each separated from its neighbor and that between any two points there is at most a finite number of other points. In that case we would not have to worry about the dichotomy paradox. If we assume such a discrete space, then we can understand why our lazy slacker makes it to the door: he only has a finite number of points to cross. At a certain point, the intervals could no longer be split into two. Objects move in this type of space by going from one discrete point to the next without passing between them.
    In the language of chapter 1 , we can say that this is a paradox because we are assuming that space is continuous:
    Space is continuous ⇒ movement is impossible.
    Since there is definitely movement in this world, and our assumption led us to a false fact, we conclude that space is not continuous. Rather, it is discrete, or separated into little “space atoms.”
    Such ideas of discrete space are familiar to people who study quantum mechanics. 8 Physicists discuss something called Planck’s length , which is equal to 1.6162×10 −35 meters. Something smaller in length cannot be measured. To some extent, nothing smaller than that exists. Physicists assure us that objects go from one Planck’s length to another. In high school chemistry it is taught that electrons fly in shells around a nucleus of an atom. When energy is added to an atom, the electrons make a “quantum leap” from one shell to the next. They do not pass in between the shells. Perhaps our lazy slacker also makes such quantum leaps and hence can finally reach the door.
    Let us reconsider figure 3.3 . The square is infinitely divvied up as illustrated. But this is only possible if we think of the square as a mathematical object. In mathematics every real number that represents a distance can be split into two, hence we can continue chopping forever. In contrast, let us think of the square as a piece of paper. We can start cutting paper into smaller and smaller pieces using finer and finer scissors. This will work for a while, but eventually we will reach the atomic level where no further cutting will be possible. This is true for any physical object made of atoms. We are forced to conclude that the square depicted in figure 3.3 is not a good model for the physics associated with the paper square. The real numbers can be infinitely divided but the paper cannot be. What Zeno is forcing us to do is to ask the question of whether space (which is not made of atoms) can be infinitely divvied up. If it can be, the slacker will not reach his goal. If it cannot be, there must be