# Paradox of motion implies discrete space?

1. May 24, 2012

### thedragonbook

Came across this video which says that a moving object has to cover infinitely many intervals in order to get from one point to another and because of this motion couldnt really take place and since it does take place, its a paradox. youtube.com/watch?v=u42Y3RbP7JE

Since motion does take place, does this imply that space is discrete?

2. May 24, 2012

### mathman44

Without getting too philosophical, it's a non-sequitur that 'covering an infinite amount of intervals' = 'impossible'. Converging sums contains infinitely many terms yet sum up to a finite number.

3. May 24, 2012

### thedragonbook

Yes, exactly. That is the paradox isn't it? There are an infinite amount of intervals between two points but its still possible to move from one to another which implies that there weren't an infinite number to begin with?

4. May 24, 2012

### mathman44

Why do you think 'traversing an infinite amount of intervals' is impossible?

The infinite sum example should suggest that this IS possible, not that there is something wrong with the math.

5. May 24, 2012

### mathman44

This is Xeno's paradox btw and has been around for about 2 millenia ;)

6. May 24, 2012

### thedragonbook

Do you know what infinite means?

7. May 24, 2012

### ZapperZ

Staff Emeritus
Do you know what converging infinite series mean?

Zz.

8. May 24, 2012

### Staff: Mentor

Suppose you want to move from one point to another. First, you have to move to the point halfway in between... And once you're at the halfway point, you still have an interval ahead of you, so you move the halfway point of that interval, and repeat forever, endlessly splitting the difference between where you are and where you're trying to go... So how can you get to your destination if there are an infinite number of points between you and you destination?

Here's an easy explanation (although mathematical purists may wish to avert their eyes at this point):

Your first step takes you halfway, the second takes you half of a half, the third half of a half of a half, and so forth. So we can write the distance D that you've covered as:

D = 1/2 + 1/4 + 1/8 + 1/16 + .....

2D = 2(1/2 + 1/4 + 1/8 + 1/16 + .....)
= 1 + 1/2 + 1/4 + 1/8 + 1/16 + .....
= 1 + (1/2 + 1/4 + 1/8 + 1/16 + .....)
= 1 + D

So we have 2D=D+1

Solve for D and you'll get D=1; the infinite number of steps we take gets us to our destination eventually. And how long a time is "eventually"?

If we're moving at a constant speed, then the time for each step will be half the time of the previous step. So you can use the same algebra as above to show that the infinite number of steps takes a finite time. We cover the distance and we do it in a finite time, despite the infinity of steps.

9. May 25, 2012

### laughingebony

Convergence of a series has a specific mathematical definition. The fact that, say, $\sum^{\infty}_{n=0}\frac{1}{2^{n}}$ (the series relevant to Zeno's paradox) converges means only that we have a reasonable way of dealing with this series mathematically. It does not imply, for instance, that traveling over the infinite set of intervals described by Zeno's paradox is physically possible.

10. May 25, 2012

### A.T.

It means that the sum of the intervals is finite.
Traveling over a finite distance is trivially possible.

11. May 25, 2012

### laughingebony

It means precisely that for each positive number ε, there exists a natural number K such that for each natural number k≥K, $\left|\left(\sum^{k}_{n=1}\frac{1}{2^{n}}\right)-1\right|<\epsilon$ (I changed the starting index to 1, which is more in line with Zeno's paradox.) If you want to interpret that as meaning that the sum of the intervals is finite, go ahead -- but you're missing the point. I don't think anybody will disagree, even on purely philosophical grounds, that taking k to be "sufficiently large" (in Zeno's paradox, traveling through a large, but finite number of intervals) will get you "close" to distance 1 from the starting point. The issue is whether it is possible to get to exactly distance 1 from the starting point by traveling through the infinite set of intervals.

This is begging the question!

12. May 25, 2012

### Staff: Mentor

No, it's factual reality. And this is what bothered me about the ancient philosophers - they always figured that what was going on in their heads took precedent over reality. But 2000 years later, you have no excuse for making the same mistake. The scientific method is now 500 years old, so you have had ample time to learn it.

Zeno's paradox was never a very good logical problem. It is a very poor (or at least ill-defined) description of what is actually happening. If, for example, every time you moved an interval as described by Zeno's paradox, you stopped and contemplated what you just did for a certain time before moving the next interval, then it would take an infinite time to reach your destination. But a fired arrow does not stop to philosophize. It simply moves.

Now I've heard it said that with their poor understanding of math, the error was excusable, but again, no excuse today.

13. May 25, 2012

### A.T.

I don't have to interpret anything. I can divide any finite interval into an infinite number of positive non-zero intervals. So it's obvious that an infinite number of positive non-zero intervals can sum up to a finite interval.

Why not? The intervals are physically irrelevant. Their number is physically irrelevant. The apparent paradox has nothing to do with motion or physics. It's a purely mathematical counter intuitive issue.

14. May 25, 2012

### ZapperZ

Staff Emeritus
You terribly missed the point.

The OP is arguing that there's something "infinite", and appears to not realize that something CAN be infinite but still have a finite convergence. In other words, he/she doesn't seem to know anything about the existence of finite series, since the argument put forth by someone already as a counter example seemed to have been ignored.

Zz.

15. May 25, 2012

### Staff: Mentor

Clarification/expansion: The ancients believed that philosophy trumped observation in determining what was "real" and as such, some ancients really did believe that Zeno's paradox meant that motion actually is an illusion. The scientific method does not allow for this type of reasoning. In the scientific method, observation is king. If your mathematical model doesn't match reality (after appropriate error checking), then you must assume that it is your mathematical model that is flawed, not your perception of reality. This is why I say that the mathematical issue is secondary here. We don't need fully developed mathematical models of every phenomena in order to accept that they are real, nor does the model precede the observation.

So you see, the fact that Zeno's paradox reflects an underdeveloped mathematical system really is secondary to the problem that it reflects an underdeveloped system of logic. And this flawed system of logic probably hindered scientific advancement until it was corrected. Since the logic/conclusion was assumed to be right, there was no need to attempt to fix it! But of course, now that we've got the math figured out, we can explain the paradox in addition to just dismissing it.

16. May 25, 2012

### A.T.

Aristotle deduced from logic that women have less teeth than men. He didn't bother to check, but if he did it would be another "paradox".

17. May 25, 2012

### Hurkyl

Staff Emeritus
Watching people defend Zeno's paradox, I think the trappiest part is the combination of two things:

• The idea that we can describe things via a sequence of events
• A naïve, and ultimately flawed, intuition about infinite things

In particular, the problem goes like this: suppose we can talk about an infinite sequence of events such as:
• I'm at the starting point
• I'm at the half-way point
• I'm at the three-quarters mark
• ...
Now this is where the naïve view of infinity comes out -- this is an infinite, sequential list of events, so there can't be anything after it. After all, the list is infinite and there's nothing bigger than infinity, right?

So one then concludes that breaking things into an infinite sequence of events is nonsense, and consequently notions of logic, space and time that allows us to break things into a sequence of events must also be flawed. Sometimes this manifests itself as some sort of finitism, or maybe the rejection of the notion of 'point'. Sometimes an insistence on space and time being discrete.

The real problem, of course, is that this sort of 'sequence of events' sort of thing requires transfinite ordinal numbers or even more general order types, but the flawed notion of 'infinite' prevents them from thinking about them.