Associated Topics || Dr. Math Home || Search Dr. Math

### Why the Motionless Runner Parodox Fails

```Date: 01/01/2005 at 16:34:03
From: Richard

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.

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

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

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

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

Thank you so much!
```
Associated Topics:
High School Logic
High School Puzzles
Middle School Logic
Middle School Puzzles

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search