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.
No, you have just given a restatement of Zeno's Paradox which "proves", in an invalid way, that motion is impossible
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 x_{0} and you want to move. Motion means that you want to go from x_{0} to x_{1}, where x_{1} 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 x_{0} itself, since, if you choose any point x other than x_{0}, you can always find another point y that is closer to x_{0} 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?
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.
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...
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.
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.
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
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.
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
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.
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.
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 x_{0} to a point x_{1}, 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.