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> Just assume space is E³ To me, that's just begging the question. You can't assume that. > Would you want to argue that it is impossible to move in E³? I d

by drdec 4y ago

> Just assume space is E³ To me, that's just begging the question. You can't assume that. > Would you want to argue that it is impossible to move in E³? I don't know what gave you that impression. Yes, it's possible to move in Euclidean space. That has no bearing on Zeno's paradox which is not about Euclidean space.

danbruc 4y ago
You said, space in our universe is not Euclidean and that is why the paradox does not prevent us from moving or reaching the goal, correct? This implies that the paradox assumes Euclidean space, otherwise it would not matter whether or not our universe is Euclidean, correct? So finally, if space being Euclidean is what separates being able to move in our universe but not in the one portrait by the paradox, would this then not imply that either you can not move or reach the goal in Euclidean space or that there must be another reason that prevents the consequences of the paradox from applying to Euclidean space?
- drdec 4y ago
  > You said, space in our universe is not Euclidean and that is why the paradox does not prevent us from moving or reaching the goal, correct? This implies that the paradox assumes Euclidean space What I should have said is that you cannot assume that spacetime can be infinitely divided. Instead I started that spacetime is not what people usually assume it to be. The paradox (as stated) doesn't assume Euclidean space, although Euclidean space satisfies the assumptions of the paradox. Finally, if spacetime is not Euclidean then applying the paradox to a Euclidean space is just a thought experiment and everything will depend on your definitions and how you define movement in your fictional space. What Zeno was trying to get at with his paradox was a takedown of the idea of infinity applying to the real world.
  - danbruc 4y ago
    To move from 0 to 1 in R starting at time 0 and arriving at time 1 means x(t) = max(0, min(t, 1)) ignoring the problem of infinite acceleration at 0 and 1 which we could smooth out. I would find it paradoxical if that motion was not possible. All our laws of physics assume und describe motion in space build on top of the reals and tell you how it works and they can perfectly describe the motion I sketched at the top. If anything, then should expect to not be able to move in the real world if space can not be indefinitely divided because the laws showing that this motion is possible do not apply. I think the rather straight forward solution is to simply say that you can do infinitely many things in some fixed time. Where is the problem in saying after walking one meter that by doing so you managed to traverse an infinite number of intervals? This is only problematic if you think that doing an infinite number of things requires an infinite amount of time, but why would it?
    - drdec 4y ago
      Firstly, there is a little conflation between the Euclidean space and a Euclidean space with a time element going on here. It's not important to your argument but I wanted to call attention to it. Yes, one way to resolve the paradox is to allow time to be infinitely divisible like (Euclidean) space. Then you can describe where something is at any point in time. But that discussion is not interesting to me. It is NOT true that "all of our laws of physics assume and describe motion in space build on top of the reals." Both relativity and QM tell us that spacetime is NOT Euclidean.
      - danbruc 4y ago
        To be precise, relativity - special and general - do not use Euclidean space, but quantum mechanics does. I guess you where thinking of relativistic quantum mechanics or relativistic quantum field theories. But again, irrelevant. I just picked Euclidean space and an implied real time axis because I am not sure how to label the idea properly - do we need a real closed field, would rational numbers do? So do not take Euclidean too literally, understand it as three dimensions of space and one of time that all can be divided indefinitely. If we reach a point where this is not good enough, we can refine it. And I think you still did not get my point. If the resolution of the paradox hinges on the indefinite divisibility of space and time then the paradox is not resolved in a hypothetical universe where we have the ability to indefinitely divide space and time. So I argue you would then have to accept the consequences of the paradox in this hypothetical universe, would you not? And one such hypothetical universe is the universe of classical mechanics, so how would you reconcile that? Was classical mechanics wrong all the time when it described motions in a space where motions are not possible?
        drdec 4y ago
        > If the resolution of the paradox hinges on the indefinite divisibility of space and time then the paradox is not resolved in a hypothetical universe where we have the ability to indefinitely divide space and time. This does not follow logically. We have a sequence of assumptions and an argument. Just because there is an example of a system where the assumptions do not hold, that does not mean the argument is sound when the assumptions do hold. And BTW, Newtonian mechanics are wrong, in the sense that they do not correspond to reality. (That doesn't mean they aren't useful.)

by drdec 4y ago

danbruc 4y ago

You said, space in our universe is not Euclidean and that is why the paradox does not prevent us from moving or reaching the goal, correct? This implies that the paradox assumes Euclidean space, otherwise it would not matter whether or not our universe is Euclidean, correct? So finally, if space being Euclidean is what separates being able to move in our universe but not in the one portrait by the paradox, would this then not imply that either you can not move or reach the goal in Euclidean space or that there must be another reason that prevents the consequences of the paradox from applying to Euclidean space?

drdec 4y ago
> You said, space in our universe is not Euclidean and that is why the paradox does not prevent us from moving or reaching the goal, correct? This implies that the paradox assumes Euclidean space What I should have said is that you cannot assume that spacetime can be infinitely divided. Instead I started that spacetime is not what people usually assume it to be. The paradox (as stated) doesn't assume Euclidean space, although Euclidean space satisfies the assumptions of the paradox. Finally, if spacetime is not Euclidean then applying the paradox to a Euclidean space is just a thought experiment and everything will depend on your definitions and how you define movement in your fictional space. What Zeno was trying to get at with his paradox was a takedown of the idea of infinity applying to the real world.
- danbruc 4y ago
  To move from 0 to 1 in R starting at time 0 and arriving at time 1 means x(t) = max(0, min(t, 1)) ignoring the problem of infinite acceleration at 0 and 1 which we could smooth out. I would find it paradoxical if that motion was not possible. All our laws of physics assume und describe motion in space build on top of the reals and tell you how it works and they can perfectly describe the motion I sketched at the top. If anything, then should expect to not be able to move in the real world if space can not be indefinitely divided because the laws showing that this motion is possible do not apply. I think the rather straight forward solution is to simply say that you can do infinitely many things in some fixed time. Where is the problem in saying after walking one meter that by doing so you managed to traverse an infinite number of intervals? This is only problematic if you think that doing an infinite number of things requires an infinite amount of time, but why would it?
  - drdec 4y ago
    Firstly, there is a little conflation between the Euclidean space and a Euclidean space with a time element going on here. It's not important to your argument but I wanted to call attention to it. Yes, one way to resolve the paradox is to allow time to be infinitely divisible like (Euclidean) space. Then you can describe where something is at any point in time. But that discussion is not interesting to me. It is NOT true that "all of our laws of physics assume and describe motion in space build on top of the reals." Both relativity and QM tell us that spacetime is NOT Euclidean.
    - danbruc 4y ago
      To be precise, relativity - special and general - do not use Euclidean space, but quantum mechanics does. I guess you where thinking of relativistic quantum mechanics or relativistic quantum field theories. But again, irrelevant. I just picked Euclidean space and an implied real time axis because I am not sure how to label the idea properly - do we need a real closed field, would rational numbers do? So do not take Euclidean too literally, understand it as three dimensions of space and one of time that all can be divided indefinitely. If we reach a point where this is not good enough, we can refine it. And I think you still did not get my point. If the resolution of the paradox hinges on the indefinite divisibility of space and time then the paradox is not resolved in a hypothetical universe where we have the ability to indefinitely divide space and time. So I argue you would then have to accept the consequences of the paradox in this hypothetical universe, would you not? And one such hypothetical universe is the universe of classical mechanics, so how would you reconcile that? Was classical mechanics wrong all the time when it described motions in a space where motions are not possible?
      - drdec 4y ago
        > If the resolution of the paradox hinges on the indefinite divisibility of space and time then the paradox is not resolved in a hypothetical universe where we have the ability to indefinitely divide space and time. This does not follow logically. We have a sequence of assumptions and an argument. Just because there is an example of a system where the assumptions do not hold, that does not mean the argument is sound when the assumptions do hold. And BTW, Newtonian mechanics are wrong, in the sense that they do not correspond to reality. (That doesn't mean they aren't useful.)