Not that anyone cares but I just realized that this is not actually paradoxical and I can prove it mathematically! (I think) Bear with me since I’ve like just barely learned this stuff this week.
Proof Let S be the set of all steps needed to be taken. It can be written as S = {(distance to be traveled)(2-n): n in the Natural numbers}. Thus, S shares cardinality with the natural numbers and is countably infinite.
However, time is continuous. Thus, it has the cardinality of the continuum (real numbers) which means any time interval contains an uncountably infinite amount of moments. Let us denote an arbitrary time interval as T.
Because | T | > | S | there is no injection from T to S. Thus if each step has only 1 time value, there will be moments of time left over, and since the hand is not in two places at once we know each step must have its own time value, so this must be the case.
Therefore, when moving in steps like this, one will run out of infinite steps before they run out of moments in time to complete those steps. Hence, any finite distance can be traversed in this way over some bounded interval of time. QED.
Basically, you can traverse any distance in any time interval as long as physics allows you to move at a fast enough speed. Even if it doesn’t, there may be a limit to how fast you can traverse the distance, but it is still bounded. You can traverse any finite distance like this before existence runs out of time.
(I’m still learning. So if there’s an error in my proof please be gentle lol)
Your proof assumes a non-quantum universe. As soon as time become quantized, you actually do have a minimum finite time step, at which point any observed object by necessity must be at rest. This (well, something to this effect) has been proved by experimental data, however it’s currently unclear if this quantization is related to our methods of observation or an actual characteristic of time.
Yes, but if the universe is quantum, then there also exists a minimum finite space step. So the fractions never get infinitely small. So you either stop moving in which case of course you never reach the destination; you stopped before you did. OR you take an extra step and surpass your distance by a negligible amount which means you did move all the way.
So even in a quantized universe, the paradox is still false right?
Well, no — if anything you’ve proven it. The paradox was originally that all objects must be at rest regardless of observed movement, infinitesimality was just the quickest way for ancient Greeks to conceptualize it.
Experimentally, we’ve observed† that it is possible to “freeze” time for a quantum particle if you measure it before the wave function has the time to fully transition from “origin” to “target” when no intermediary states exist between††, i.e. a quantum object “in movement” stays at rest until one “time-step” passes, at which point it imediately exists at the next point towards the target, where it remains at rest until the next time-step. If you measure the object “between” time-steps, its position will be at the origin point, but because we’ve now collapsed the wave function, that position is manifested as reality and no other possibilities exist. As a result, a new time-step must pass before it can move — yet, if we measure again, the same observation will be repeated, so the wave function never gets to the target, even though we have declared that the wave function (and therefore the particle) is moving from origin to target.
That’s the kind of fuckery we signed up for when physicists discovered the wave-particle duality
† (potentially, there are competing hypothesis)
†† By necessity, a quantized number line follows this condition
I haven’t read the actual proof you wrote, but I think the paradox was shown to, well, not be a paradox multiple times in history. It was thought of long ago, when our understanding of mathematics and physics was more limited, and we didn’t have the tools to prove the flaws exist.
It’s pretty cool that a problem created so long ago can still be relevant, and that random people find it interesting enough to solve in Lemmy comments though! Keep learning and rock on!
I thought it was first halfway proved. And then an extra 1/4th proved. And then an additional 1/8th proved. And then the proof went an extra 1/16th. So if you can wait an infinite time, it will be proved.
I think this idea works, and it’s consistent with the way the ‘paradox’ was presented. I just need to point out that movements aren’t performed in steps, the intermediate positions are also continuous, so there are undoubtedly infinite positions to map with undoubtedly infinite instants. To me, that’s the “true” solution.
As I see it, the only reason the paradox presents halving steps is because it’s the easiest way to demonstrate that there are infinitely many steps. It doesn’t bother to show that there’s uncountably infinite steps because countable is sufficient.
But as I said, your proof (or disproof?) works for the argument as it was presented, so it’s good!
I much prefer the physics approach: disproof by experiment! “Look at my gavel. According to your theory, it has infinitely many steps to go through before it reaches the pad, so it can never hit the pad.” *bang* “Motion denied.”
I just need to point out that movements aren’t performed in steps, the intermediate positions are also continuous
That’s not what quantized time suggests. It may be that our universe does have a minimum time step, just as we know for a fact there exists a minimum distance step (the Planck length).
I don’t think Planck length is a “minimum distance step” the way you imply it is. IIRC it’s a theoretical minimum uncertainty range, or something like that. But honestly I’m not really in the know about this whole subfield so I could be wrong.
But if we take this as fact - there exists a minimum distance step - then that means there can only ever be finitely many intermediate steps and the paradox is resolved.
I don’t think Planck length is a “minimum distance step” the way you imply it is. IIRC it’s a theoretical minimum uncertainty range
They’re effectively the same thing. Subatomic theories will never represent the real universe, only the information parallel we can construct with human thought. It’s why there are countless different hypothesis for the quantum wave function, and why things could just as easily be explained by string theory instead. We can never know how the fabric of reality functions, so an insurmountable gap in our ability to ascertain a difference between two values means they may as well be exactly the same.
then that means there can only ever be finitely many intermediate steps and the paradox is resolved.
Only if the minimum scalar difference in time is equal to or greater than the minumum scalar difference in space (which we know shouldn’t be true, because of the speed of light). As I wrote in my other reply, “if you measure the object’s position between steps, its position will be at an origin point, but because we’ve now collapsed the wave function, that position is manifested as reality and no other possibilities exist. As a result, a new time-step must pass before it can move — yet, if we measure again, the same observation will be repeated, so the wave function never gets to the target, even though we have declared that the wave function (and therefore the particle) is moving from origin to target.”
Not that anyone cares but I just realized that this is not actually paradoxical and I can prove it mathematically! (I think) Bear with me since I’ve like just barely learned this stuff this week.
Proof Let S be the set of all steps needed to be taken. It can be written as S = {(distance to be traveled)(2-n): n in the Natural numbers}. Thus, S shares cardinality with the natural numbers and is countably infinite.
However, time is continuous. Thus, it has the cardinality of the continuum (real numbers) which means any time interval contains an uncountably infinite amount of moments. Let us denote an arbitrary time interval as T.
Because | T | > | S | there is no injection from T to S. Thus if each step has only 1 time value, there will be moments of time left over, and since the hand is not in two places at once we know each step must have its own time value, so this must be the case.
Therefore, when moving in steps like this, one will run out of infinite steps before they run out of moments in time to complete those steps. Hence, any finite distance can be traversed in this way over some bounded interval of time. QED.
Basically, you can traverse any distance in any time interval as long as physics allows you to move at a fast enough speed. Even if it doesn’t, there may be a limit to how fast you can traverse the distance, but it is still bounded. You can traverse any finite distance like this before existence runs out of time.
(I’m still learning. So if there’s an error in my proof please be gentle lol)
https://en.wikipedia.org/wiki/Supertask
Your proof assumes a non-quantum universe. As soon as time become quantized, you actually do have a minimum finite time step, at which point any observed object by necessity must be at rest. This (well, something to this effect) has been proved by experimental data, however it’s currently unclear if this quantization is related to our methods of observation or an actual characteristic of time.
Yes, but if the universe is quantum, then there also exists a minimum finite space step. So the fractions never get infinitely small. So you either stop moving in which case of course you never reach the destination; you stopped before you did. OR you take an extra step and surpass your distance by a negligible amount which means you did move all the way.
So even in a quantized universe, the paradox is still false right?
Well, no — if anything you’ve proven it. The paradox was originally that all objects must be at rest regardless of observed movement, infinitesimality was just the quickest way for ancient Greeks to conceptualize it.
Experimentally, we’ve observed† that it is possible to “freeze” time for a quantum particle if you measure it before the wave function has the time to fully transition from “origin” to “target” when no intermediary states exist between††, i.e. a quantum object “in movement” stays at rest until one “time-step” passes, at which point it imediately exists at the next point towards the target, where it remains at rest until the next time-step. If you measure the object “between” time-steps, its position will be at the origin point, but because we’ve now collapsed the wave function, that position is manifested as reality and no other possibilities exist. As a result, a new time-step must pass before it can move — yet, if we measure again, the same observation will be repeated, so the wave function never gets to the target, even though we have declared that the wave function (and therefore the particle) is moving from origin to target.
That’s the kind of fuckery we signed up for when physicists discovered the wave-particle duality
† (potentially, there are competing hypothesis)
†† By necessity, a quantized number line follows this condition
I haven’t read the actual proof you wrote, but I think the paradox was shown to, well, not be a paradox multiple times in history. It was thought of long ago, when our understanding of mathematics and physics was more limited, and we didn’t have the tools to prove the flaws exist.
It’s pretty cool that a problem created so long ago can still be relevant, and that random people find it interesting enough to solve in Lemmy comments though! Keep learning and rock on!
I thought it was first halfway proved. And then an extra 1/4th proved. And then an additional 1/8th proved. And then the proof went an extra 1/16th. So if you can wait an infinite time, it will be proved.
I think this idea works, and it’s consistent with the way the ‘paradox’ was presented. I just need to point out that movements aren’t performed in steps, the intermediate positions are also continuous, so there are undoubtedly infinite positions to map with undoubtedly infinite instants. To me, that’s the “true” solution.
As I see it, the only reason the paradox presents halving steps is because it’s the easiest way to demonstrate that there are infinitely many steps. It doesn’t bother to show that there’s uncountably infinite steps because countable is sufficient.
But as I said, your proof (or disproof?) works for the argument as it was presented, so it’s good!
I much prefer the physics approach: disproof by experiment! “Look at my gavel. According to your theory, it has infinitely many steps to go through before it reaches the pad, so it can never hit the pad.” *bang* “Motion denied.”
That’s not what quantized time suggests. It may be that our universe does have a minimum time step, just as we know for a fact there exists a minimum distance step (the Planck length).
I don’t think Planck length is a “minimum distance step” the way you imply it is. IIRC it’s a theoretical minimum uncertainty range, or something like that. But honestly I’m not really in the know about this whole subfield so I could be wrong.
But if we take this as fact - there exists a minimum distance step - then that means there can only ever be finitely many intermediate steps and the paradox is resolved.
They’re effectively the same thing. Subatomic theories will never represent the real universe, only the information parallel we can construct with human thought. It’s why there are countless different hypothesis for the quantum wave function, and why things could just as easily be explained by string theory instead. We can never know how the fabric of reality functions, so an insurmountable gap in our ability to ascertain a difference between two values means they may as well be exactly the same.
Only if the minimum scalar difference in time is equal to or greater than the minumum scalar difference in space (which we know shouldn’t be true, because of the speed of light). As I wrote in my other reply, “if you measure the object’s position between steps, its position will be at an origin point, but because we’ve now collapsed the wave function, that position is manifested as reality and no other possibilities exist. As a result, a new time-step must pass before it can move — yet, if we measure again, the same observation will be repeated, so the wave function never gets to the target, even though we have declared that the wave function (and therefore the particle) is moving from origin to target.”
Well now you’re just making shit up :P
I wish I was… then again, maybe I am making it up by collapsing the wave function.