 Paradoxes don't exist, but they can teach us something about our ideas. Whenever we discover a contradiction in our beliefs, we're forced to make revisions. Paradoxes can help us elicit nestled contradictions in our worldview that we wouldn't see otherwise. Some of the most famous and oldest paradoxes in the world are Zeno's paradoxes. They came up with several, but I just want to focus on the most famous one and defend it. Usually paradoxes need to be carefully resolved. They bury subtle logical errors within them, but with Zeno, his logic is actually sound. What isn't sound is his conclusion. He thought he proved that motion is impossible, but instead he proved something unique about the physical world. I think his arguments are brilliant, and they revolve around the difficulties that infinity presents us with. Modern philosophy frequently overcomes Zeno's paradoxes by saying, ah, calculus solves the problem. But this is mistaken. Logic actually solves the problem, not mathematics. I've written extensively about some of the problems that I see in modern mathematics, specifically the presupposition of Platonism and the treatment of infinities, in set theory in particular. So before I explain the paradox, I want to commend Zeno on his commitment to philosophic rigor. I agree with him. We must follow logical argumentation wherever it leads, as he did, regardless of how unintuitive the conclusions appear. We do not gain insight into the world by believing what appears to be true, rather what necessarily follows from what, with logic as the final judge. So the paradox goes like this. Let's take two examples. In the first, consider a train traveling to its destination. It has to travel, let's say, 100 feet. Before it completes its journey, it must first pass the halfway point, the 50-foot mark. Then it must travel through the next halfway point, leaving 25 feet remaining. Then again, it must go through the next halfway point, leaving 12 and a half feet remaining. Then again, and again, and again, and so on, just down to inches. Then half inches, then half, half inches. This can continue ad infinitum. There's no possible distance which can't be divided in half, and the train must pass through each point. Because it is impossible to fully complete a non-ending or infinite set of tasks, the train will never arrive. So work through the logic of this argument. It's broken down into four parts. One, it is necessarily impossible to complete an infinite set of tasks. Two, at every point along the path, there is always a halfway mark remaining which must still be crossed. Three, an infinite amount of halfway marks cannot be crossed. Conclusion four, therefore, the path can never be fully completed. If one and two are true, then three and four necessarily follow. So Zeno concluded, and therefore, motion is impossible. He went even further than this, in fact, outright rejecting all divisions between things and claiming that the most logical position is mysticism, that all is one. So one more example, and then I'll present my alternative resolution to the paradox, consider a race between Fred and Mary, where Mary has a small head start, but Fred is much faster. Zeno argued that Fred can never actually win a race between them, because he'll never catch up to Mary, and it's a matter of pure logic. So in order for Fred to overtake Mary, he must first pass the point, which Mary started from. Then in the time it took for Fred to do this, Mary has moved a little bit forward. Fred must then again make up that new distance. But when he arrives, Mary has again moved forward in that period of time. Each time Fred arrives at Mary's previous point, time has elapsed, which means Mary has covered new distance. Because there's an infinite number of points which Fred must reach, where Mary's already been, he cannot overtake her. Again, work through the logical power of this argument. Before Fred can overtake Mary, he must cross every point that she has crossed. Two, in order for Fred to reach Mary's previous point, time must elapse. Three, every time that time elapses, Mary creates new distance. Four, Fred must then cross that new distance, which also takes up a certain amount of time, creating new distance to cross. Five, no distance can be crossed without time elapsing, and therefore, conclusion six, Fred will never be able to get ahead of Mary. These are profound arguments, concluding that, well, motion must be impossible is not actually as ridiculous as it first sounds. But the modern response is to lazily say, oh, calculus solves it. And sure enough, calculus gives us incredible precision in predicting events like Fred passing Mary, it conforms to our daily experiences, as we witness motion all the time so we don't treat it with much skepticism. But mathematical calculations, in this case, do not have the final say in matters of logic. An infinite series of events cannot be completed by logical necessity of what we mean by infinite. It doesn't matter how well the calculations work, and in fact, it points to a problem with how people think of calculus. Calculus is about the asymptotic relationship between quantities which are never allowed to be equal. So where does that leave us? Fortunately, I have a resolution which preserves logical necessity, preserves a sensible understanding of calculus, and it clears up a definition of infinity. So the answer is very simple. The reason that motion is possible is because physical reality is not infinitely divisible. There is a base unit. Revisit the train example. As the train is nearing its destination, it has to travel over the half-inch point, the half-half-inch point, the half-half-half-inch point, and so on. But this doesn't continue infinitely if there is a base unit. When there are, say, two base units remaining, it still has to go halfway and travel over one of them. Then there's only one unit of distance left. Since there's no meaningful half-a-base unit, at least such a concept cannot be represented in physical reality, the train could arrive at the station without having to cross any more halfway points. This means that dividing a base unit in half is merely a mental exercise. It's word play. It's not something that's possible in physical reality. In fact, this must be true by virtue of what we mean by the term base unit. It can't have some parts. So when looking at the four-part breakdown of the train argument, the mistake is with premise number two, quote, at every point along the path, there will always remain a halfway mark, which must be crossed. Now that premise is simply not true. At some point, you are one individual unit away from your destination. Therefore, conclusions three and four do not follow, and we eject infinity from the argument. Now let's revisit the race between Fred and Mary. It too gets resolved by a base unit in physical reality. Infinity is harder to see, but it's buried in premise number three, quote, every time that time elapses, Mary creates new distance. Now this premise presupposes that space and time are infinitely divisible. If we reject this idea, then we can see there must be some amount of time in which Mary would not move even one base unit. If this is true, well, that's all that's required in order for Fred to pass Mary, assuming that he's moving faster than she is. You can think of it this way. Instead of their speeds being calculated in MPH, miles per hour, imagine they were in IB UPS, iBUPS, individual base units per second. Fred might be moving a trillion iBUPS while Mary only 100 billion. Well, then it becomes trivially easy to see that Fred will overtake Mary in X seconds, depending on how far away he is. So this idea of a base unit is not something exclusive to philosophy. In physics, these are sometimes referenced as plank units. There's plank time, plank length, et cetera. And to the extent that motion is possible, I'd say that their existence is necessary for the reasons that Zeno pointed out. Though, of course, I realize there's subtleties with plank units in physics, but that's beside the point. In this view, calculus is also preserved perfectly while avoiding infinities altogether. A base unit explanation of calculus allows us to purge all infinities, in fact, from our calculations. If we think of the train needing to cross an infinite series of points in order to arrive at its destination, well, we've already doomed that train to never arrive. With a base unit, we needn't deal with any kind of infinitesimal distances or such nonsense, infinite limits or anything like that. Instead, we're left with concrete, smallest required numbers for all of our calculations. There's only so many decimal places you need to go in order to be perfectly precise. I'll offer one more theory which can explain motion while preserving Zeno's logic. I call it the pixel theory of reality. It requires that we tweak our idea of motion slightly. Imagine that the physical world is made up like your computer screen. It's a fixed structure containing a zillion little spaces for tiny pixels. When you see your mouse cursor move across the computer screen, you're not actually seeing motion as we think about it. You're seeing a kind of progression of little dots that are firing in succession. It's like one of those LED signs with ticker symbols on it, something you might see at a baseball game. When you see the letters move across the sign, you're actually just seeing individual light bulbs firing in a particular pattern. There's no letters moving at all. So motion could simply be the firing of little pixels in succession, based on particular inputs in the world. In fact, if you think about it, this is precisely what happens whenever you're watching a movie on TV. You see what appears to be regular motion in the physical world, people moving around and such, but it's actually not that at all. It's just a motionless screen giving you the illusion of movement. You might even say, motion is the transmission of energy across the screen of physical reality. The pixels just pop in and out of existence. They don't move anywhere. So this explanation also preserves logical sense, explains the phenomena we experience, it avoids infinities, and in fact, Zeno himself might like the conclusion. As far as I can tell, either scenario is logically possible, but they both require the same thing, a base unit of physical reality either fixed or in motion. This should not be controversial, however, when you think about it. Imagine an empty container. Does an infinite amount of space reside within it? Of course not. But if it were true that physical reality were infinitely divisible, it would mean that infinity resides within the finite. It doesn't matter how little the particles are. If there's an infinite amount, there's a non-finite amount of space occupied by them. Take two points, A and B, and draw a line between them. Are you looking at an infinitely long line? Of course not. You're just looking at two finite points being joined by a finite line, all of which are divisible down to their fundamentals. Now, if you respond, oh, but mathematical points do not take up space, then I would respond, then your point is a concept. It's not something within the line. You cannot meaningfully reference a point which does not take up space. If it doesn't take up space, it has no spatial existence and therefore cannot be used as a unit of length. This is a confused way of thinking about the nature of space which comes from Euclidean geometry, my criticisms of which are a topic for another time. So all of these conclusions follow from simple premises. You cannot progress over an infinite distance in a finite period of time. You cannot add up infinity to get a finite number. You cannot contain an infinite amount of space within a finite amount of space. You cannot complete an infinite series of events because an infinite series is necessarily non-completable. From my perspective, Zeno correctly understood the logical power of infinities in argumentation even though he got his ultimate conclusion wrong. His reasoning was sharp and he inadvertently proved to the extent that motion is possible we necessarily live in a finite world. Logic requires it.