Why the Motionless Runner Parodox FailsDate: 01/01/2005 at 16:34:03 From: Richard Subject: Paradox Below is a problem I found in the archives of Dr. Math. I was wondering if motion is really impossible because there isn't an actual explanation of where the paradox below is faulted. Paradox 1: The Motionless Runner A runner wants to run a certain distance--let us say 100 meters--in a finite time. But to reach the 100-meter mark, the runner must first reach the 50-meter mark, and to reach that, the runner must first run 25 meters. But to do that, he or she must first run 12.5 meters. Since space is infinitely divisible, we can repeat these "requirements" forever. Thus the runner has to reach an infinite number of "midpoints" in a finite time. This is impossible, so the runner can never reach his goal. In general, anyone who wants to move from one point to another must meet these requirements, and so motion is impossible, and what we perceive as motion is merely an illusion. I am now convinced that motion is an illusion. Can you help me prove that this problem is wrong? Date: 01/01/2005 at 21:13:55 From: Doctor Rick Subject: Re: Paradox Hi, Richard. This problem caused the Greek mathematicians much consternation, and I am not sure historically what resolution they settled on in order to move forward with mathematics. At some point the issues raised by Zeno were resolved by concepts you'll learn when you study calculus. For now, let me point out some assumptions included in the paradox as you stated it. One assumption is that space is infinitely divisible. Another is that it is impossible to reach an infinite number of points in a finite time. But if space can be divided infinitely (into infinitesimal, that is, infinitely tiny, distances), then why can't time be infinitely divided too? Then, by the rate equation, distance = speed * time time = distance / speed if you divide an infinitesimal distance by a finite speed, you get an infinitesimal time. As you cut the distance into smaller and smaller chunks, it takes less and less time to reach each point. What you'll learn in calculus is that it's possible for an infinite number of infinitesimal times to total a finite time. But you can see that already, because you've cut up a finite distance into an infinite number of infinitesimal distances. They can be added back up in the same way that the times add up to a finite time. Having said this, some of what we perceive as motion is in fact illusion. Look at the TV screen: you see things moving, but in fact you're seeing a succession of still pictures, about 60 of them each second. A baseball flying across the screen doesn't really pass through every point in its path; it moves a finite distance on the screen between any two frames. Thus Zeno's paradox doesn't apply here. But the mathematical problem is real, yet it has been resolved. I hope you can get up and walk across the room now without thinking that you're imagining the whole thing! ;-) - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ Date: 01/02/2005 at 09:05:42 From: Richard Subject: Paradox Thank you so much for clarifying the issue. I see how your TV example applies. But there is something I'm confused about, which is when you said, "then why can't time be infinitely divided too?" Does this mean that you start with 1 minute, then 30 seconds, 15 seconds, 7.5 seconds, etc.? Date: 01/02/2005 at 17:09:42 From: Doctor Rick Subject: Re: Paradox Hi, Richard. That was exactly my point in the first part of my answer. Time (at least in the mathematical model we are using here) can be infinitely divided; that's why an infinitesimal distance can be traveled in an infinitesimal time, and the times add up to a finite time in the same way that the infinite number of distances add up to a finite distance. Suppose for convenience, that the runner runs 1 meter per second. Then it takes 50 seconds to run the first 50 meters; 25 seconds to run the next 25 meters; 12.5 seconds to run the next 12.5 meters; and so on. The series of distances 50 + 25 + 12.5 + 6.25 + ... adds up to 100 meters, even though there are an infinite number of terms in the series. If this part of the argument is accepted, then the series of times 50 + 25 + 12.5 + 6.25 + ... must add up to 100 seconds. That solves the paradox: the runner does indeed reach the goal in the time you expect, even though the distance is subdivided into an infinite number of parts. There is a question whether time or space in the real world can be infinitely subdivided; this gets into quantum mechanics. But Zeno's paradox has to do with the mathematical abstraction of the real world in which we assume that both are infinitely divisible. My counterexample of the TV screen isn't meant to say that this is how it works either in the real world or in the mathematical world. (If you think TV is the real world, you're in big trouble! ;-) ) It was just to say that in a situation in which time is not infinitely divisible, distance is not either, so Zeno's paradox does not apply. The paradox definitely applies in the mathematical world, and more or less applies in the real world, but as I showed above, it does have a resolution. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ Date: 01/04/2005 at 20:01:25 From: Richard Subject: Thank you (Paradox) Thank you so much! |
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