# All movement should be impossible (?)

1. Aug 2, 2012

### Edi

So I found this video, in witch it was said and explained that all movement should be impossible .. it was explained somehow like this (too bad i can not find the video now) : imagine a arrow (or bullet.. or anything else) shooting/ moving from start to its target - in order to get to the target, the object has to reach the trajectories/ paths mid-point. To reach this mid-point, it has to reach the mid-point between start and first mid-point. To reach this new mid point, it has to reach the mid point between start and mid points mid point.. and so on to infinity.
So, according to this, all movement should be impossible - it would take infinite amount of time to traverse any infinitely small distance.
Even although the time to traverse each sub-mid-point would decrease, no matter how small the time gets, it would have to be multiplied by infinity, as there would be infinite .. steps.
That is .. weird.
It got me thinking and what I thought up was - space must be quantized. In that case, there would be a finite amount of steps and time required would be finite.

.. but that got me thinking even further - if space is quantized, then there is a actual limit of how round a circle can be - with the smoothness increasing with size! (basic geometry)

And so, to conclude, if no circle can be a perfect circle and the roundness (for the lack of a better word) is limited, then the constant pi (3.14159...) does not have to be irrational and it can be finite..

2. Aug 2, 2012

### arivero

This is known as "Zeno's Arrow paradox", albeit it is also mentioned in a chinese text, in a list of paradoxes of the chinese "logicians", whatever they are (it sound as a taoist paradox to me).

It you do deeper into Zeno, you will see that the naive quantisations of space are also critiquised by this guy.

Most probably, the right exit is to quantize a product of speed and position, such as the angular momentum of the arrow.

3. Aug 6, 2012

### Sjm757

There are alot of motion paradoxes by Zeno, but in my opinion that one is solved like 1/3 is equal to .333, but 3/3 is equal to 1, so we know .999=1.

4. Aug 6, 2012

### Quantum Kid

This is solved using calculus. You can add an infinite sum and still get a finite answer. Here's how:

Your series can be described as follows:

1/2 + 1/4 +...+1/2n with n approaching infinity

This is a geometric series where each term is multiplied by the same term: 1/2.

The terms get smaller and smaller so it converges or sums by the following equation:

S∞=a/1−r (|r|<1), where a = the initial value of the series and r = the constant each term is multiplied by
=.5/(1-0.5)
=1 (a finite number!)

Thus even through the arrow's flight can be divided into infinite sub-intervals motion is still possible because the sum of these sub-intervals yields a finite answer. This is the magic of calculus.

5. Aug 7, 2012

### crissyb1988

Exactly what quantum kid said. Just because there is an infinite amount of points between the start and finish it doesn't mean that the distance between the two points is infinite. This implies that it is possible in certain cases like this to traverse and infinite amount of points in a finite amount of time.

Just on a side note in reality there may not be and infinite amount of points between distances because on the quantum level distance may be discrete and once again zenos paradox will fail

6. Aug 7, 2012

### H2Bro

The history of atomism is interesting, because it touches on these very ideas. Democritus had a good thought experiment to show why he thought the world was "discrete" or "atomic" , or equivalently for your purposes, had quantized space.

Imagine slicing a thin piece of an apple with some perfectly sharp and thin knife. the apple has a certain curvature in the area that you slice, so the cross-sectional area is decreasing smoothly. Now you have two apple pieces, and two cross sections where you cut. Are the cross sections the same area, or is one slightly smaller owing to the curvature?

Democritus hypothesized that at the very lowest level, this "perfect apple" (he was actually probably talking about conic sections) would have a certain 'grainyness', so no matter how finely you sliced it, one cross section would always be slightly smaller than the other, again due to curvature.

You might recognize this is similar to finding the derivative in calculus, or measuring the 'infintesimal' change in things. The atomists were generally highly unpopular in ancient times, and most people thought by geometrical necessity that the two cross sectional areas of a cone would be equivalent, despite the slope.

This is what leads to paradox's like Zeno's. The interesting thing about paradoxes is, they don't actually happen in reality. Next time you encounter a paradox you can instantly infer that one or more premises are simply incorrect, by dint of knowing that no true 'paradox' has or ever will occur.

7. Aug 7, 2012

### Cody Richeson

I was surprised this has only been mentioned once. I'm by no means an expert, but if Planck lengths exist, then wouldn't it be accurate to say that the arrow travels one Planck unit at a time to its target?

8. Aug 7, 2012

### Cerlid

Seen this many times, it is an issue integral calculus solves rather neatly as did H2bro. Take the decay of a collection of atoms, is there ever a point where the half life denotes nothing exists, or is 0 an asymptote? Does matter decay to nothing? That would break all known laws, so...

Impressively mathematicians in ancient Greece solved the so called paradox using something akin to calculus as H2bro mentioned, although it was not formalised until Newton and Liebniz came along.

Suffice to say this finds parallels in does a bouncing ball ever stop bouncing also. And the answer is both mathematically yes and yes of course it does, it's mathematically concerned with a decay constant. Energy is not infinite, and hence it has a limit of action it may perform over time (t): time is exactly related to position, but it is of course exactly related to an action at which it eventually closes off x or the position, something Xeno didn't understand but calculus does. Divide something into enough slithers and the graph is perfect but we must understand we can not physically do that and hence the ball always stops, the arrow always hits a target, it's why limits are important, finite quantities don't quite half in relation as we would think intuitively in our mundane world.

Well yes and no, beyond that limit nothing can be said about the travel of any object it is undefined, 'cause we cannot measure it, or even get close to doing so, which does not mean it does not exist just that approaching "infiinitesimals" are subject to reality, and we are subject to what we can observe. 1 Planck length is not equivalent to one exact energy concern is I think the point. 1 unit of infinite definition in relation to another is impossible and made so by simple calculus laws. Simple but true. Physically observable basic science and logic. If we can ever measure the size of nothing or everything that might change, but I doubt those limits are going to be approached exactly, because everything is bigger than we can know, and nothing smaller than we can conceive.

Last edited: Aug 7, 2012
9. Nov 15, 2012

### Shaky

I believe the paradox arises from the following. An arrow at rest occupies a space equal to itself. Zeno reasoned that what is in motion neither moves in its current place, nor in the place where it is not, thus the flying arrow is really motionless.

10. Dec 8, 2012

### Shaky

Aristotle disagreed with Zeno by arguing "time is not composed of instantaneous moments".

11. Dec 8, 2012

### AlephZero

No. It's "solved" by shooting an arrow at something, if you want to stick with ancient greek technology.

If you want to be a scientist and your mathematical model of something doesn't match the real world, don't ever conclude that "the real world is wrong".

12. Dec 9, 2012

### Shaky

Against the doctrine of continuous change that Heraclitus propounded -- The founder of the Eleatic school from which Zeno belonged said this: "Come, then, listen to my word and take heed of it: I will tell you of the two roads of inquiry which offer themselves to the mind. The one way, that IT IS and cannot not-be, is the way of credibility based on truth. The other way (the way of opinion), that IT IS NOT and that not-being must be, cannot be grasped by the mind; for you cannot know not-being and cannot express it." In other words, the universe only appears to change and move, but in reality, it's a perfect sphere homogeneous in all directions.

13. Dec 9, 2012

### DeIdeal

As a side note, since no one else has explicitly commented to it: You can rest assured that pi has infinite decimals. It doesn't matter if there are no circles in the Real World that aren't perfect. Pi is still an irrational mathematical constant which happens to be the ratio of the circumference and the diameter of a perfect circle -- even if one does not exist.

14. Dec 9, 2012

### Darwin123

Zeno's claim was not that the real world was wrong. He was claiming that all mathematical models were wrong. Much of Euclid's geometry relies on line segments being divided into small line segments. The contemporary discovery of irrational numbers depended on an infinite series.

Zeno would say there is no such thing as fundamental science. There are only physical paradigms waiting for their time to come. Zeno was a sophist. Sophists were a lot like postmodernists, except that they wore togas.

Sophists believed things like "Man was the measure of all things," that "the majority is always right," etc. etc. Therefore, his claim was really aimed at ethics and morality. He claimed that logic had nothing to do with ethics or morality.

I don't believe the Sophists were entirely correct either in the scientific or the ethical sphere. It is obvious that they are wrong in the scientific sphere. There is a little more uncertainty refuting them in the ethical sphere. They aren't completely right there, though sometimes they are.

With regards to forum guidelines, Zeno's paradox belongs more to the mathematical section than the physics section. All physicists agree that the arrow gets there. Or at least the experimental physicists do. The real question is the logical consistency of the "limit" concept. There is a lot to discuss with regards to that.

Logic is a measure of some things, even if man isn't always a good measure. Physics needs good logic, in addition to experimental results. I don't think one can go very far without understanding how to use the concept of limit.

15. Dec 9, 2012

### K^2

The series that Zeno worked with converge on rationals, though. So while he had the right idea, his argument was invalid either way.

16. Jan 10, 2014

### MightyKaykoher

This depends on what time is. Instantaneous moments or something that relies on motion itself to flow/(insert word here). In both cases, motion is possible.

Proving motion is possible, is simple.
Let's take an irrational number like pi and multiply it by a real number (lol not a fake number hehe). 2 times pi will never 7. never, can't happen. If there is an infinite amount of something/some object between two objects, then their lengths must be 1/infinite. Multiply them. Yeah, the product is 1.

Still a paradox here. That's infinite. Infinite messes up everything these days. Infinite time, infinite space. The real question here is how to get rid of infinite.eg If I travel at an infinite speed for any amount of time will time end? Time speeds up as you move faster right? So if time is infinite and you wait infinite time (possible through moving at a infinite speed) time will end and nothing will exist. Fortunately, we can go that fast.

Or is it dimensions that are our problem. In 1 dimension something can be infinitely small.

More math here but, is infinite - infinite = 0? Is inf times inf = inf squared or just inf?
And the biggest question, is inf/inf =1?

Please help.

____________________________________ E=mc^2in other words,m is E's cousin.

17. Jan 12, 2014

### Khashishi

In the path integral formulation of quantum mechanics/electrodynamics, you cannot think of an arrow that goes from A to B as stepping through each point between them. The arrow sort of teleports to point B, and you cannot be sure the arrow was at some point between A and B unless you actually measure it there. In fact, the arrow can be thought of as taking every possible path from A to B, including paths that violate the speed of light. This is why Zeno's paradox doesn't apply. But, if you can continuously measure the arrow using quantum measurements, then you do get a sort of Zeno's paradox, called the Quantum Zeno Effect. In this case, the arrow cannot move!

18. Jan 12, 2014

### nickbob00

Fortunately for mathematicians, they work in a dream world where space is perfectly euclidian, no funny quantisation business going on!

19. Jan 13, 2014

### D H

Staff Emeritus
Thread closed pending moderation.

We have had far too many threads on Zeno's paradoxes. We do not need another.

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