Zeno's paradox

What's the thing that does happen, but apparently is impossible by logically analysing what's happening?

The light is either on or off. The paradox is: "If time is infinitely divisible, both options are bad". Atomistic maths doesn't have this difficulty (but has others).

There are other paradoxes too, which I find genuine or almost at least. For example Banach-Tarski paradox can't IMO be explained away by saying that it's impossible to cut any object so precisely.
 
Well, if you want to be picky, the arrow never reaches it's target because of the Planck distance ... it will always be never closer of 1.616252(81)×10^-35 m of the target ;) And Aquiles will never reach the turtle because of the same reason ( they will have to go sideways at best ... good ol'Pauli exclusion principle disalows 2 diferent bodies composed of matter of occupying the same space as well :D ). So neither of the original Zeno paradoxes actually happens as well ...

But the arrow DOES hit the tortoise. Achilles DOES overtake it. Despite it (apparently) being logically impossible. Hence the paradox. Which gets resolved by spotting the actual flaw in the logic.

In your example of a light, or a pigeon, what's the paradox? What's the thing that is (apparently) logically impossible, but actually does happen? Just because you can pose a question where the answer is 'don't know' or 'not enough information' doesn't make it paradoxical. In the case of asking what state is something in after infinitely many state-changes, all it makes it is a poorly thought out question.
 
No, the arrow never reaches the target. Pauli exclusion principle makes it impossible: the arrow and the target can't occupy the same space because they are both matter ( and even if we were talking about stuff that isn't covered by Pauli exclusion principle, there is the Planck lenght to consider ). Aquilles will never get to where the turtle is IRL by the same reason: it will have to go sideways at best ( but he can't because we defined a single track :D ).

So you are saying that the light example does not happen and because of that it it isn't a paradox ( fair enough, i could agree with that in principle ) but you then say that both Zeno postulates are paradoxes because they happen when they certainly can't happen as stated by Zeno....
 
I've got a tortoise lollipop that says it does happen. Trying to argue it's not really a dead tortoise because at some molecular level the arrow's atoms and the tortoise's atoms are occupying different spaces is basically rubbish. If the tortoise gets skewered, that counts as arrow hitting tortoise.

Atticus said:
The light is either on or off. The paradox is: "If time is infinitely divisible, both options are bad".

Nope, you've lost me. Why are both options bad? If you want to know the state of the light at a certain point, you look at it. If you want to know the state of the light after a finite number of state changes, you can work it out. If you want to know the state after an infinite number of changes, that is nonsensical, not paradoxical.
 
No, my friend. You can't start talking about infinite division of space when there is granularity in the same space. You can't have it both: infinite division of space and space quanta.... If you define the body of the turtle lenght as the minimum space allowed in discussion ( otherwise Aquiles never gets it due to the Pauli exclusion principle ), then there is no paradox, because all the distances below that have to discarded, making the Zeno sum a boring sum of all the steps bigger than the turtle's lenght ( a finite ammount ). For it to be a paradox, you need infinite steps, but that means that you have to put two material objects in the same place. So, by your own definition, you can only have a paradox if you run a unrealistical aproach ... so neither the lightbulb or the Zeno postulates are paradoxes using your definition.

P.S Tortoise lollipop != tortoise + arrow . you need to input energy ... and in fact the tortoise lollipop is a completely diferent object :p
 
so neither the lightbulb or the Zeno postulates are paradoxes using your definition.

And again, we can logically explain why the arrow will never hit the tortoise. But when we try and apply the logic in the real world, the arrow clearly does hit the tortoise. Therefore it's apparently a paradox, resolved when we spot the flaw in the logic. Explaining to the dead tortoise that it didn't actually get hit thanks to planck distances, Pauli exclusion principle, or whatever else is again, basically rubbish.

Zeno's paradox is an apparent paradox, resolved when you think about it a bit. But I'm yet to see the apparent (or actual) paradox wrt the light.
 
Nope, you've lost me. Why are both options bad? If you want to know the state of the light at a certain point, you look at it. If you want to know the state of the light after a finite number of state changes, you can work it out. If you want to know the state after an infinite number of changes, that is nonsensical, not paradoxical.

If the light is on, when was it last turned on? If the light is off, when was it last turned off?

You can call it nonsensical, but that's a problem too, since it's nonsense allowed by accepting infinite division of time.

Note that this is different from the situation where you'd switch the bulb on and off with minute between each switch: Then there is no time after infinite switches. Here how ever, if you have intervals 30s, 15s, 7.5s ,... There is time after all the switches, namely everything that comes after one minute has passed.
 
@sanabas

But, your definition precludes the Zeno paradox of being a paradox in the first place ;) You need to say at the same time that space is infinitely divisible ( otherwise you can't make the infinite steps ) and that the space has a minimum lenght that can't be divided, because otherwise you can't actually reach the end position in RL. Without those those two, it isn't a paradox, regardless of having a turtle pierced or not ( if you just assume infinite division of space, there is the sum of the converging series, if the space is quantified you only need to sum the terms of the series bigger than the space quanta ... and in both the turtle is dead btw :D ).

So, using your own words, that is nonsensical, not paradoxical ... or better said, incoherent because to get a paradox out of Zeno postulate you need to have 2 oposite principles in some parts of the reasoning ... atleast as long as we are talking about our world. That is a classic reductio ad absurdum :D
 
So how do we know that the turtle is fermionic anyway? The Pauli exclusion principle applies only to fermions.
 
So how do we know that the turtle is fermionic anyway? The Pauli exclusion principle applies only to fermions.
We don't, but I was responding to sanabas that was arguing that Zeno version was a paradox because it could really happen in RL ( unlike my version that in his opinion cannot happen IRL ). I don't know any bosonic turtle, so I assumed that both the turtle and the arrow were fermionic :D If both the objects are fermionic, Pauli exclusion principle does apply...

OFC that we can devise a version of the paradox with bosonic particles, but even then it is highly discussible that we should consider any distance inferior to the Planck lenght to the sum of the series terms... In the end the situation is the same: in RL and as long as we are talking of divisions of space, we can't go to infinite and still be talking inside our current understanding of physics. Time, OTOH is another piece of pie ...

P.S. My point is that Zeno's paradox as stated in the "original" ( better said, Aristotle references to him in this case ) form is not congruent with the current knowledge of physics. You might think on it as a Gedankenexperiment though or postulate a physics where space is infinitely divisible without change of properties.
 
We don't, but I was responding to sanabas that was arguing that Zeno version was a paradox because it could really happen in RL ( unlike my version that in his opinion cannot happen IRL ). I don't know any bosonic turtle, so I assumed that both the turtle and the arrow were fermionic :D If both the objects are fermionic, Pauli exclusion principle does apply...

OFC that we can devise a version of the paradox with bosonic particles, but even then it is highly discussible that we should consider any distance inferior to the Planck lenght to the sum of the series terms... In the end the situation is the same: in RL and as long as we are talking of divisions of space, we can't go to infinite and still be talking inside our current understanding of physics. Time, OTOH is another piece of pie ...

P.S. My point is that Zeno's paradox as stated in the "original" ( better said, Aristotle references to him in this case ) form is not congruent with the current knowledge of physics. You might think on it as a Gedankenexperiment, thought or postulate a physics where space is infinitely divisible without change of properties.

One could also invoke the uncertainty relation and say that because we know the momentum of the turtle with some accuracy, its position must have an uncertainty. So if the distance becomes small enough, there is no way to tell which one is ahead of the other because both are delocalized by a larger degree than the distance between them.

The same argument can be made with time: If we go to short enough times, the energy uncertainty becomes so big that there is turtles and anti-turtles all over the place and we do not know which turtle was the real one.

But Zeno would have considered anyone crazy who might have told that to him :lol:
 
One could also invoke the uncertainty relation and say that because we know the momentum of the turtle with some accuracy, its position must have an uncertainty. So if the distance becomes small enough, there is no way to tell which one is ahead of the other because both are delocalized by a larger degree than the distance between them.

The same argument can be made with time: If we go to short enough times, the energy uncertainty becomes so big that there is turtles and anti-turtles all over the place and we do not know which turtle was the real one.

But Zeno would have considered anyone crazy who might have told that to him :lol:
Dunno, Zeno was pretty crazy on his own :p Due to this thread I decided to get a library book on Vicious circles, infinity and paradoxes and I discovered in there this gem from Zeno:
If everything that exists has a place

Then every place has his place as well ... and the place of the place has it's place ( ad infinitum )

So nothing has a place :D
Or one about the non-existance of movement
A arrow can't occupy more lenght that it's original in any moment in time while in the air

This happens in every moment.

So the arrow is never on movement.
Yup, he really liked to bash the notion of infinite :p
 
One could also invoke the uncertainty relation and say that because we know the momentum of the turtle with some accuracy, its position must have an uncertainty.

I don't know if you guys are just playing now or what, but I thought that such rules only applied to quarks and such?

The same argument can be made with time: If we go to short enough times, the energy uncertainty becomes so big that there is turtles and anti-turtles all over the place and we do not know which turtle was the real one.

It's turtles all the way down. ;)
 
I don't know if you guys are just playing now or what, but I thought that such rules only applied to quarks and such?

Nope, it applies to anything you want to measure. You can only know position and velocity to a certain level of accuracy. If you want velocity to be more accurate, you'll have a greater uncertainty about position, and vice-versa. There's a formula for it, but I can't be bothered looking it up. It's probably on the wiki for Heisenberg.
 
The issue of whether the arrow could actually hit the tortoise or not is irrelevant, because you can rephrase the scenario to be about the arrow overtaking the tortoise - or if you prefer, think of Achilles racing the tortoise on a racetrack, so they are in different lanes. The paradox then is not about the arrow hitting the tortoise but about Achilles drawing level with it.
 
The issue of whether the arrow could actually hit the tortoise or not is irrelevant, because you can rephrase the scenario to be about the arrow overtaking the tortoise - or if you prefer, think of Achilles racing the tortoise on a racetrack, so they are in different lanes. The paradox then is not about the arrow hitting the tortoise but about Achilles drawing level with it.
You might interpret it that way, but the text of Aristotle point to another direction:
Aristotle (Physics said:
The second is the so-called 'Achilles', and it amounts to this, that in a race the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. This argument is the same in principle as that which depends on bisection, though it differs from it in that the spaces with which we successively have to deal are not divided into halves. The result of the argument is that the slower is not overtaken: but it proceeds along the same lines as the bisection-argument (for in both a division of the space in a certain way leads to the result that the goal is not reached, though the 'Achilles' goes further in that it affirms that even the quickest runner in legendary tradition must fail in his pursuit of the slowest), so that the solution must be the same. And the axiom that that which holds a lead is never overtaken is false: it is not overtaken, it is true, while it holds a lead: but it is overtaken nevertheless if it is granted that it traverses the finite distance prescribed
( copied from the Internet Classics Archive of MIT )


Achilles is not running in other track, he is in the same track as the turtle ( that as you can see it is never explicitely named :D ) and aiming to get to spot where the turtle is. He is not trying to overtake the turtle, just trying to reach it .

Note that all my discussion above was about the feasibility in real life of the scenario stated originally by Zeno ( or at best, the first known quoter of him, in this case Aristotle ) compared with a scenario that I posted that sanabas considered irrelevant because he considered it impossible in real life and my position on that is that both are unfeasible in our world. So in my opinion, we only have two coherent positions, or we drop all the discussion because all the Zeno paradoxes are at best Gedankenexperiment with little feasibility or we forget those little details and just look at the core of the issue ( and then we can consider both your version of the paradox or my example of the lamp, because both are the same in core, as even Aristotle recognizes above :D )

BTW congratulations on the promotion, Plotinus.
 
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