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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

by 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.)
  - danbruc 4y ago
    Okay, now we are converging. I spelled this out in my first comment but was sometimes lazy later on and did not mention it every time. [...] then the resolution can not [only] be that you can not indefinitly divide space intervals in half. So your position is essentially the paradox is resolved in our universe because the unstated assumption about the divisibility of space is not satisfied, but even if it was satisfied, Xeno's reasoning would still be wrong. So there is some more fundamental flaw in the reasoning but it doesn't really matter to you because you have a resolution for our universe by denying one of the assumption. Strictly speaking you don't even know whether space and time are not indefinitely divisible in our universe, there are good reasons to think so, but it is - as far as I know - not a certainty. There are even good reasons to think that space does not exist in the same sense as temperature does not fundamentally exist and is something emerging from something more fundamental. But whether or not our space and time are indefinitely divisible, just denying this assumption seems like the cheap way out for me, sidestepping the more fundamental issue that applies even if space and time are indefinitely divisible. So let me offer you a different view that does not hinge on the divisibility of space and time. Xeno says, to reach the goal you first have to move half the way. Fine, I can do that. Then he says but in order to do that, you first have to move quarter of the way. Fine, I can do that. Then he says but in order to do that, you first have to move one eight of the way. Fine, I can do that. Then he says in order...and I get annoyed. Just tell me the first thing I have to do so we can start actually doing something. And now Xeno is in trouble, it is not that you can not do what he asks you to do, the problem is that Xeno can not make up his mind what you should do first. Xeno can pick any arbitrarily large number, subdivide that often, and give you a list of things to do and you can easily do them, no matter how large a number he picked. You could even say, hey Xeno, let me just move to the goal. Does doing so include the task of going half of the way? Yes. Does this include the task of going quarter of the way? Yes. An eight? Yes? Is there any task in your infinite list that I did not complete? No. The reasoning in the other version is more or less the same, you go half the way, then half the way of the rest, half the way of the rest, ... This time you seem stuck because Xeno can not make up his mind what the last task is you have to do, but again you can do all of them. This version has the nice property that you can actually start doing something as Xeno can at least tell you what the first task is, so you move half the way. And then another quarter. Another eight. And while it might seem as if you are stuck as Xeno keeps throwing new tasks at you, you are not. We just have this tendency to think of tasks as requiring some fixed time and doing infinitely many of them will take forever. But if you keep track of time as Xeno throws more and more tasks at you, you will notice that time grinds to a halt. You complete one task after another but as you move with constant velocity each one takes less and less time and the total time approaches some limit slower and slower. So it takes you forever when measured in number of tasks but only something finite when measured in time. And again, just walking to the goal will include any task that Xeno would ever have asked you to do, so he can do something better with his time than trying to write a complete list of tasks for you. And BTW, Newtonian mechanics are wrong, in the sense that they do not correspond to reality. (That doesn't mean they aren't useful.) Wrong is the wrong term, it is an approximation. Take special relativity, in the limit of all velocities going to zero, you recover Newtonian mechanics exactly. Take general relativity, in the limit of slow motions, small masses, and low energies, you recover Newtonian mechanics exactly. The same for quantum mechanics. So if you approximately satisfy the assumptions, you will get a result with a very small error. EDIT: I just thought of a variation of the problem, assuming indefinite divisibility of time. Xeno asks you to be in a certain place at 12:00. And 11:30. And 11:15. And 11:07:30. And so on. Admittedly being in a place is somewhat different than moving to somewhere, but would we call this a paradox? If we just stay at this place from 11:00 to 12:00 we can satisfy all requests Xeno makes, we can be there at all the times from any arbitrary finite or infinite list of times between 11:00 and 12:00. The list does not even have to be limited to countably many tasks as in the original paradox, it could contain uncountable many times, say the entire time between 11:11 and 11:12.

by drdec 4y ago

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.)
- danbruc 4y ago
  Okay, now we are converging. I spelled this out in my first comment but was sometimes lazy later on and did not mention it every time. [...] then the resolution can not [only] be that you can not indefinitly divide space intervals in half. So your position is essentially the paradox is resolved in our universe because the unstated assumption about the divisibility of space is not satisfied, but even if it was satisfied, Xeno's reasoning would still be wrong. So there is some more fundamental flaw in the reasoning but it doesn't really matter to you because you have a resolution for our universe by denying one of the assumption. Strictly speaking you don't even know whether space and time are not indefinitely divisible in our universe, there are good reasons to think so, but it is - as far as I know - not a certainty. There are even good reasons to think that space does not exist in the same sense as temperature does not fundamentally exist and is something emerging from something more fundamental. But whether or not our space and time are indefinitely divisible, just denying this assumption seems like the cheap way out for me, sidestepping the more fundamental issue that applies even if space and time are indefinitely divisible. So let me offer you a different view that does not hinge on the divisibility of space and time. Xeno says, to reach the goal you first have to move half the way. Fine, I can do that. Then he says but in order to do that, you first have to move quarter of the way. Fine, I can do that. Then he says but in order to do that, you first have to move one eight of the way. Fine, I can do that. Then he says in order...and I get annoyed. Just tell me the first thing I have to do so we can start actually doing something. And now Xeno is in trouble, it is not that you can not do what he asks you to do, the problem is that Xeno can not make up his mind what you should do first. Xeno can pick any arbitrarily large number, subdivide that often, and give you a list of things to do and you can easily do them, no matter how large a number he picked. You could even say, hey Xeno, let me just move to the goal. Does doing so include the task of going half of the way? Yes. Does this include the task of going quarter of the way? Yes. An eight? Yes? Is there any task in your infinite list that I did not complete? No. The reasoning in the other version is more or less the same, you go half the way, then half the way of the rest, half the way of the rest, ... This time you seem stuck because Xeno can not make up his mind what the last task is you have to do, but again you can do all of them. This version has the nice property that you can actually start doing something as Xeno can at least tell you what the first task is, so you move half the way. And then another quarter. Another eight. And while it might seem as if you are stuck as Xeno keeps throwing new tasks at you, you are not. We just have this tendency to think of tasks as requiring some fixed time and doing infinitely many of them will take forever. But if you keep track of time as Xeno throws more and more tasks at you, you will notice that time grinds to a halt. You complete one task after another but as you move with constant velocity each one takes less and less time and the total time approaches some limit slower and slower. So it takes you forever when measured in number of tasks but only something finite when measured in time. And again, just walking to the goal will include any task that Xeno would ever have asked you to do, so he can do something better with his time than trying to write a complete list of tasks for you. And BTW, Newtonian mechanics are wrong, in the sense that they do not correspond to reality. (That doesn't mean they aren't useful.) Wrong is the wrong term, it is an approximation. Take special relativity, in the limit of all velocities going to zero, you recover Newtonian mechanics exactly. Take general relativity, in the limit of slow motions, small masses, and low energies, you recover Newtonian mechanics exactly. The same for quantum mechanics. So if you approximately satisfy the assumptions, you will get a result with a very small error. EDIT: I just thought of a variation of the problem, assuming indefinite divisibility of time. Xeno asks you to be in a certain place at 12:00. And 11:30. And 11:15. And 11:07:30. And so on. Admittedly being in a place is somewhat different than moving to somewhere, but would we call this a paradox? If we just stay at this place from 11:00 to 12:00 we can satisfy all requests Xeno makes, we can be there at all the times from any arbitrary finite or infinite list of times between 11:00 and 12:00. The list does not even have to be limited to countably many tasks as in the original paradox, it could contain uncountable many times, say the entire time between 11:11 and 11:12.