Is Motion Possible in a Continuous Space?

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Discussion Overview

The discussion revolves around the concept of motion in a continuous space, questioning how movement is possible when, theoretically, there are infinitely many points between any two given points. Participants explore implications of Zeno's Paradox and the nature of space, considering whether space might be discrete rather than continuous.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant suggests that if space is continuous, motion is paradoxical because one cannot define a "next" point to move to, as there are infinitely many points between any two points.
  • Another participant argues that this reasoning restates Zeno's Paradox, which has been invalidated in various ways.
  • A participant expresses skepticism about solutions that rely on convergent series, asserting that they fail to address the core issue of crossing infinitely many points in finite time.
  • Some participants propose that the assumption of continuous space may need to be reconsidered, suggesting that a discrete model could resolve the paradox.
  • Another viewpoint emphasizes that while the assumption of continuity leads to a contradiction with observed motion, it does not inherently imply that motion is impossible.
  • One participant points out that there are infinitely many slices of time available to traverse the infinite slices of space, suggesting a potential resolution to the paradox.
  • Another participant mentions that some quantum gravity approaches attempt to quantize space, introducing a natural cutoff that resembles a discrete space model.
  • Disagreement arises regarding the validity of calling certain arguments nonsensical without providing counterarguments.

Areas of Agreement / Disagreement

Participants express multiple competing views on the implications of continuous versus discrete space, with no consensus reached on the validity of the original argument or the nature of motion in a continuous space.

Contextual Notes

Some participants note that the discussion hinges on the definitions of motion and space, and that assumptions about continuity may lead to contradictions with real-world observations. The complexity of Zeno's Paradox and its interpretations remains a focal point of contention.

JK423
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I have quite a naive question, which doesn't really go deep into physics/mathematics.. :)

Let's take seriously the idea that space is continuous. The questions is, how are we able to move in such a space? We know that in a continuous space (real numbers), between two points there are infinitely many other points. That's true for any two points, no matter how close they are, and this seems to be a kind of a paradox.
It seems to me that we have to impose a cut off, i.e. the existence of a smallest number that below which you cannot go, since in that case between two points there will be only finitely many points to cross, not infinitely many, hence motion will be in principle possible.

Does this make any sense?

EDIT: (I rephrase the problem in a more proper way, so if anyone wants to attack an approach, prefer the following one:)

The problem can be also rephrased in this way: You are sitting on a mathematical point x0 and you want to move. Motion means that you want to go from x0 to x1, where x1 is the next mathematical point. So the question is, what is the next mathematical point? The answer is, since space is assumed to be continuous, there is no next mathematical point. The exact next point can only be x0 itself, since, if you choose any point x other than x0, you can always find another point y that is closer to x0 than x is. Hence motion is not possible. If the problem is rephrased this way, you can simply ignore all these proofs that involve convergence series thinking that they have solved the problem.
 
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No, you have just given a restatement of Zeno's Paradox which "proves", in an invalid way, that motion is impossible
 
phinds thank you for the reply.

So, where is the flaw in the argument?
 
JK423 said:
phinds thank you for the reply.

So, where is the flaw in the argument?

Do you not know how to Google? Check out Zeno's Paradox
 
According to Wikipedia about the article on Zeno's paradox: "Today there is still a debate on the question of whether or not Zeno's paradoxes have been resolved.".

I found some solutions that use convergent series to prove that the distance between two points is finite, but they are missing the point.. The crucial point in this paradox is that in order to get from any point to any other point (no matter how close they are) you have to cross infinitely many points, irrespectively of whether the distance is finite or infinite. In the aforementioned solutions, they assume that you can actually move between points, hence they remove the paradox by hand. That's not a solution. If you want to offer a solution, you have to explain how you can cross infinitely many points in finite time.

The problem can be also rephrased in this way: You are sitting on a mathematical point x0 and you want to move. Motion means that you want to go from x0 to x1, where x1 is the next mathematical point. So the question is, what is the next mathematical point? The answer is, since space is assumed to be continuous, there is no next mathematical point. The exact next point can only be x0 itself, since, if you choose any point x other than x0, you can always find another point y that is closer to x0 than x is. Hence motion is not possible. If the problem is rephrased this way, you can simply ignore all these proofs that involve convergence series thinking that they have solved the problem.

The only solution that i can see is to accept that space is discrete.

So, phinds, what's the flaw in the argument?
 
If you don't think you can move, then don't. Personally, I have no trouble moving.
 
Haha! Exactly! Which probably means that the hypothesis of space continuity is wrong! ;)

Isn't it amazing that you can be lead to quantum gravity ideas by the very fact that you can move?
 
JK423 said:
Haha! Exactly! Which probably means that the hypothesis of space continuity is wrong! ;)

Isn't it amazing that you can be lead to quantum gravity ideas by the very fact that you can move?

I don't see why/how it leads to either on of those conclusions. Zeno's Paradox is meaningless in those terms. The question of the quantization/non-quantization of time and space and gravity are all very interesting open questions in physics today.
 
JK423 said:
So, phinds, what's the flaw in the argument?
Fortunately, there are also infinitely many slices of time in which to cross the infinite slices of space. However you slice up the distance, you can slice up time in the same way - then you always have exactly the right infintesimal time to cross each infintesimal slice of space.

Incidentally, the references on Wikipedia "for" Zeno's paradox the last time I looked were frightening in thelr inanity. One builds an infinitely complex machine, is unable to predict what it will do, and purports to draw conclusions from that...
 
  • #10
When you make an assumption (i.e. space is continuous) which leads to prediction (i.e. motion is impossible) that is in contrast to the real world (i.e. motion is possible), then it's natural to re-consider your assumption.

In some QG approaches they try to quantize space. In these approaches there is a natural cutoff on the smallest possible length. The idea of the cutoff is very similar to saying that space is discrete, solving Zeno's paradox.
 
  • #11
Ibix, here is the problem rephrased a little bit more properly,

JK423 said:
The problem can be also rephrased in this way: You are sitting on a mathematical point x0 and you want to move. Motion means that you want to go from x0 to x1, where x1 is the next mathematical point. So the question is, what is the next mathematical point? The answer is, since space is assumed to be continuous, there is no next mathematical point. The exact next point can only be x0 itself, since, if you choose any point x other than x0, you can always find another point y that is closer to x0 than x is. Hence motion is not possible. If the problem is rephrased this way, you can simply ignore all these proofs that involve convergence series thinking that they have solved the problem.

As you see time doesn't even get into the picture, we are not talking about speed and motion. We are talking about whether you can even define motion in the first place. If you cannot, then it's cheating to talk about speed etc. As far as i can tell, space itself is the problem.
 
  • #12
JK423 said:
When you make an assumption (i.e. space is continuous) which leads to prediction (i.e. motion is impossible) that is in contrast to the real world (i.e. motion is possible), then it's natural to re-consider your assumption.

In some QG approaches they try to quantize space. In these approaches there is a natural cutoff on the smallest possible length. The idea of the cutoff is very similar to saying that space is discrete, solving Zeno's paradox.

This is utter nonsense! Your assumption that space is continuous in no way makes the prediction that motion is impossible. The resolution to Zeno's paradox is simply that an infinite series can have a finite sum
 
  • #13
phyzguy first of all *relax*, we are chatting here, peacefully!

Second of all, it's not legitimate to call something non-sense without arguing against it. Above i have phrased the problem of defining motion in a continuous space, check post 11. As you will see if you actually read what I've written there, just saying that "infinite series can have a finite sum" has nothing to do with the actual problem.
 
  • #14
JK423 said:
phyzguy first of all *relax*, we are chatting here, peacefully!

Second of all, it's not legitimate to call something non-sense without arguing against it. Above i have phrased the problem of defining motion in a continuous space, check post 11. As you will see if you actually read what I've written there, just saying that "infinite series can have a finite sum" has nothing to do with the actual problem.

But once again you are presenting a non-nonsensical argument, so why do you feel that "non-sense" is not a valid way to describe it. You KNOW it's nonsense because it says motion is not possible, but you do move.
 
  • #15
JK423 said:
Let's take seriously the idea that space is continuous. The questions is, how are we able to move in such a space? We know that in a continuous space (real numbers), between two points there are infinitely many other points. That's true for any two points, no matter how close they are, and this seems to be a kind of a paradox.

You can take time to be continuous too. If there is continuous time, there can be continuous motion in continuous space.

Then you can ask, well is there motion in continuous spacetime? It is sensible to answer "no". http://fqxi.org/data/essay-contest-files/Nikolic_FQXi_time.pdf
 
  • #16
phinds said:
But once again you are presenting a non-nonsensical argument, so why do you feel that "non-sense" is not a valid way to describe it. You KNOW it's nonsense because it says motion is not possible, but you do move.
Where is the "non-sensicality" phinds?
 
  • #17
atyy said:
You can take time to be continuous too. If there is continuous time, there can be continuous motion in continuous space.

Then you can ask, well is there motion in continuous spacetime? It is sensible to answer "no". http://fqxi.org/data/essay-contest-files/Nikolic_FQXi_time.pdf
atyy see my post 11, the problem as i phrase seems to be independent of time.
 
  • #18
JK423 said:
phyzguy first of all *relax*, we are chatting here, peacefully!

Second of all, it's not legitimate to call something non-sense without arguing against it. Above i have phrased the problem of defining motion in a continuous space, check post 11. As you will see if you actually read what I've written there, just saying that "infinite series can have a finite sum" has nothing to do with the actual problem.

OK, let me counter your argument.

You say "Motion means that you want to go from x0 to x1, where x1 is the next mathematical point." This is a meaningless definition, since, as you yourself pointed out, the concept of "the next mathematical point" is ill-defined. So let's change your definition to say, "Motion means that you want to go from x0 to x1, where x1 is a finite distance away." Now I can ask how long it takes to move this finite distance, and I can show that, even if I break the distance up into an infinite number of time steps, the sum of this infinite series is finite, so that I can move from x0 to x1 in a finite time. So your whole problem results from a meaningless initial definition of the concept of motion.
 
  • #19
JK423 said:
Where is the "non-sensicality" phinds?

How do you not see that it is nonsensical to say that motion is impossible? It doesn't MATTER what the argument is, if it says motion is impossible then it is nonsense.
 
  • #20
phyzguy said:
OK, let me counter your argument.

You say "Motion means that you want to go from x0 to x1, where x1 is the next mathematical point." This is a meaningless definition, since, as you yourself pointed out, the concept of "the next mathematical point" is ill-defined. So let's change your definition to say, "Motion means that you want to go from x0 to x1, where x1 is a finite distance away." Now I can ask how long it takes to move this finite distance, and I can show that, even if I break the distance up into an infinite number of time steps, the sum of this infinite series is finite, so that I can move from x0 to x1 in a finite time. So your whole problem results from a meaningless initial definition of the concept of motion.
Ok, i agree that one solution is to attack the definition of motion that i gave. But my definition of motion is intuitive: It says that in order to go from a point x0 to a point x1, you have to pass from all the points in-between. Do you agree with this statement?
I) If yes, then how can you avoid my conclusion?
II) If no, then you are basically discretizing space yourself without realizing it, since you propose to "jump" from one point to the other somehow. Or you are denying the existence of mathematical points, have you got something else to propose?

Your definition ""Motion means that you want to go from x0 to x1, where x1 is a finite distance away."" is inadequate, you don't give an exact definition of what motion means, you just describe what you want to do.
 
  • #21
phinds said:
How do you not see that it is nonsensical to say that motion is impossible? It doesn't MATTER what the argument is, if it says motion is impossible then it is nonsense.

I thought you were calling my argument non-sensical. But no, you are calling my conclusion non-sensical. Ok, I agree with you. That's why i conclude that the assumption of space continuity that i used has to be wrong since it leads to absurd conclusions!
 
  • #22
phinds said:
How do you not see that it is nonsensical to say that motion is impossible? It doesn't MATTER what the argument is, if it says motion is impossible then it is nonsense.

To be fair, JK423 is not claiming that motion is impossible, he is claiming that motion is only possible if we make an additional assumption, namely that space is "discretized".

That claim has been pretty solidly rejected, and phyzguy has provided a pretty good reason why.
 
  • #23
JK423 said:
atyy see my post 11, the problem as i phrase seems to be independent of time.

If there is no time, you cannot move. And this is true whether space is discrete or not.

When time is continuous, because there are an infinite number of time points between any two times, just as there are an infinite number of space points between any two places, you can match each different point in space to a different point in time, and so move continuously across a finite distance in a finite time.
 
  • #24
JK423 said:
But my definition of motion is intuitive: It says that in order to go from a point x0 to a point x1, you have to pass from all the points in-between. Do you agree with this statement?
I) If yes, then how can you avoid my conclusion?

Your conclusion might be valid if the points you're talking about were something physical, like little tollbooths that the object is required to check in at as it moves. But they aren't. They're a mathematical construct that allows us to attach numbers (which we call coordinates) to the positions of a moving object. That is, the moving object came first, and the notion that there are "points" at which it is located is our invention layered on top of that.

We choose the math to describe how the world works; we don't choose the math and then demand that the world conform to it.
 
  • #25
JK423 said:
I thought you were calling my argument non-sensical. But no, you are calling my conclusion non-sensical. Ok, I agree with you. That's why i conclude that the assumption of space continuity that i used has to be wrong since it leads to absurd conclusions!

But it DOESN'T. Space can be continuous or discontinuous and in either case, you can still move.

EDIT: Uh, wait. I see that once again I am perhaps being unclear by categorizing your CONCLUSION as though I were categorizing your argument. I'm going to shut up now :smile:
 
  • #26
JK423 said:
Ibix, here is the problem rephrased a little bit more properly,
As you see time doesn't even get into the picture, we are not talking about speed and motion. We are talking about whether you can even define motion in the first place. If you cannot, then it's cheating to talk about speed etc. As far as i can tell, space itself is the problem.

As has been pointed out by others, motion is the change of position with time. So the fact that time doesn't appear in your argument is a strong hint that it is incomplete.

If you complete the argument by putting time into it, you'll find that you have an infinite number of tasks but an infinite number of slices of time in which to complete them. If you formalise that argument in terms of calculus, you just define velocity.
 
  • #27
atyy said:
If there is no time, you cannot move. And this is true whether space is discrete or not.

When time is continuous, because there are an infinite number of time points between any two times, just as there are an infinite number of space points between any two places, you can match each different point in space to a different point in time, and so move continuously across a finite distance in a finite time.

I'm not saying there is no time, I'm saying that time is irrelevant. Assume that time is flowing in what I've been saying. When you say "you can match each different point in space to a different point in time" you assume that can go from one point a different point. In the post 11 i argue that you cannot go from one point to a different one, regardless of whether time is flowing or not. So what you are saying is blocked by my argument.

The question is: Is my argument flawed? If yes, why?

Note that the same argument can be applied to time as well, to show that if you assume time to be continuous then it wouldn't be able to flow. But we have no idea what time is so let's leave that. Motion in space is more intuitive.

As a general remark, it's quite interesting that continuous spaces in general seem to be unable to deal with "flow".
 
  • #28
Ibix said:
As has been pointed out by others, motion is the change of position with time. So the fact that time doesn't appear in your argument is a strong hint that it is incomplete.

If you complete the argument by putting time into it, you'll find that you have an infinite number of tasks but an infinite number of slices of time in which to complete them. If you formalise that argument in terms of calculus, you just define velocity.

I assume your argument is the same with atyy's post 23. If yes then I've given my reply in my previous post 27.
 
  • #29
JK423 said:
atyy see my post 11, the problem as i phrase seems to be independent of time.

Yes but the reality is not independent of time the two are a dichotomy of motion (spacetime). The geometry implicitly includes time as a component of lengths/distance.

Read/understand the term continuum in the context of spacetime, not in the context of quantum mechanics :-p
 
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  • #30
nitsuj said:
Yes but the reality is not independent of time the two are a dichotomy of motion.
Please don't read just one sentence and just comment on that. Read the whole post and tell me your specific problem with the argument that I've given if you want to contribute.
 

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