Why the Motionless Runner Parodox Fails

Date: 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!