On the discretization of space

[JK423]

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.

 

[phinds]

No, you have just given a restatement of Zeno's Paradox which "proves", in an invalid way, that motion is impossible

 

[JK423]

phinds thank you for the reply.

So, where is the flaw in the argument?

 

[phinds]

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

 

[JK423]

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?

 

[phinds]

If you don't think you can move, then don't. Personally, I have no trouble moving.

 

[JK423]

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?

 

[phinds]

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.

 

[Ibix]

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...

 

[JK423]

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.

 

[JK423]

Ibix, here is the problem rephrased a little bit more properly,

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.

 

[phyzguy]

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

 

[JK423]

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.

 

[phinds]

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.

 

[atyy]

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

 

[JK423]

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?

 

[JK423]

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.

 

[phyzguy]

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.

 

[phinds]

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.

 

[JK423]

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.

 

[JK423]

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!

 

[Nugatory]

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.

 

[atyy]

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.

 

[Nugatory]

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.

 

[phinds]

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

 

[Ibix]

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.

 

[JK423]

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".

 

[JK423]

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.

 

[nitsuj]

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 :tongue:

 

[JK423]

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.

 

[JK423]

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.

Ok i agree with the description that you have given, except from the sentence "Your conclusion might be valid if the points you're talking about were something physical". But what you've just said in no way implies that my argument is wrong. Why do you require from the points to be physical for my argument to be correct? What i have done, is a one-to-one correspondence between the real numbers and the points in space. If you accept that, then you have to accept the rest. No need put matter in, I'm not sure what you have in mind. Please elaborate more if you insist on your view, and in that case please formulate your opinion by defining everything that you're saying. (e.g. the sentence "...were something physical, like little tollbooths..." doesn't make sense to me)

 

[atyy]

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".

Yes, of course you can go from one point to another. At one time you are here, at the next time you are there. You don't have to move from closest point to closest point. Just move large distances in discrete time. No problem - the question is can you do so in continuous time with finite speed.

 

[JK423]

Yes, of course you can go from one point to another. At one time you are here, at the next time you are there. You don't have to move from closest point to closest point. Just move large distances in discrete time. No problem - the question is can you do so in continuous time with finite speed.

Ok, so you choose to discretize motion instead of space. That's an option as well!

 

[atyy]

Ok, so you choose to discretize motion instead of space. That's an option as well!

Yes, it's not so different from discretizing space in this sense: assume you have the real numbers, but that you can only jump from integer to integer. The integers are discrete but are embedded in the reals, so since the reals contain discrete things, you can move.

Anyway, since hopefully now you agree we can move in continuous space, why don't you take a look a look at the calculus, which answers whether you can move smoothly in continuous space.

 

[nitsuj]

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.

yea I read the whole thing. Are there an infinite number colors in a rainbow? Perhaps it becomes a matter of defining a color (coordinates), as opposed to attempting to segregate the overlapping of frequencies. Much like how space & time "overlap" continuously. So depending on our "angle of view" a meter of length for you maybe a "meter" of time for me.


You are trying to conclude something about reality, based on a false premise (spacetime concept seems fairly agreed on).

The specific problem is not considering time to be a component of motion, let alone space.