The Problem only assumes the universe is continuous. If you move a particle by a sufficiently small amount, you can guarantee an arbitrarily small change any finite distance in the future. Thanks to the butterfly effect, it has to be an absurdly tiny amount, but it’s only necessary that it exists.
Also, it assumes that the Ass will eventually die, but that’s really more for effect. The point is that it can’t make the decision in bounded time.
If you could do an interrupt, you could just make it go to the left if it takes too long to decide.
You can make it so that it gets more left-biased as it gets hungrier, but this just means that the equilibrium has it slowly moving to the right thereby increasing the pull to the right enough to counter out the increased pull to the left from hunger.
As it stands the ass will (after an indefinite amount of time) wind up in one of three positions: a) eating from the left bale, b) eating from the right bale, or c) dead.
I’m trying to arrange it so that it always winds up in one of (a) or (b).
If it can pick (a) or (b) then it can also pick something somewhere in between. The only way to get around this is to somehow define (a) and (b) so that they border on each other.
For example, if you talk about which bale it eats first, and it only needs to eat some of it, then you could have something where it walks to the right bale, is about to take a bite, but then changes its mind and goes to the left bale. If you change it by epsilon, it takes an epsilon-sized bite, and eats from the right bale first instead of the left bale.
If it can pick (a) or (b) then it can also pick something somewhere in between. The only way to get around this is to somehow define (a) and (b) so that they border on each other.
Only if you assume bounded time. A ball unstably balanced on a one-dimensional hill will after an indefinite amount of time fall to one side or the other, even though the two equilibria aren’t next to each other.
Just because you can think of a possibility does not make it possible. In the absence of classical mechanics, finite temperature will cause it to fall in very finite time. With quantum mechanics, quantum zero point fluctuations will cause it to fall in finite time even if it was at zero temperature.
Finite temperature will cause it to fall in a finite time if you start with it balanced perfectly. You just need to tilt it a little to counter that. This is an argument by continuity, not an argument by symmetry.
There’s some set of starting positions that result in it falling to the left, and another set that result in it falling to the right. If you start it on a boundary point and it falls right after time t, that means that you can get a point arbitrarily close to it that will eventually fall left, so is clearly nowhere near that at time t. That means that physics isn’t being continuous.
Finite temperature will cause it to fall in a finite time if you start with it balanced perfectly. You just need to tilt it a little to counter that.
You cannot counter it by tilting it because the thermal perturbation is random. At one moment is being pushed to the left, at another to the right.
f you start it on a boundary point and it falls right after time t, that means that you can get a point arbitrarily close to it that will eventually fall left, so is clearly nowhere near that at time t. That means that physics isn’t being continuous.
To be clear, I don’t want to argue against the hypothesis. As long as you are NOT talking about the real world, which is what I take “physics” to mean, you can talk about continuous and balance point, and arbitrarily long times to fall one way or the other. The point is that in the real world, any real world experiment analyzed in detail will have “noise” which causes it to fall sometimes one way and sometimes the other in finite time when placed at its balance point. That noise is usually dominated by thermal fluctuations, but even in the absence of thermal fluctuations, there are still quantum fluctuations which behave very much like thermal noise, which behave in very many ways as though you can’t get the temperature below some limit.
In principle, you can build a system where the thing is cooled enough, and where it is designed so that the quantum fluctuations are small enough, that you can have a relatively long time for it to fall one way or the other off its balance point. That is, the amount of time it takes to be pushed by quantum noise is large on human or intuitive scales. However, you can’t make it arbitrarily large without building an arbitariily large system. So if you are concerned about the difference between “really big” and “infinite,” you don’t get to claim a physical system balanced on an unstable equilibrium point can have an infinitely long time to fall.
Of course this doesn’t mean that in a non-physical “world” with no quantum and rigid objects (another approximation that can’t be realized in the real world) you couldn’t do it. So the math is safe, its just not physics.
You cannot counter it by tilting it because the thermal perturbation is random.
Randomness is an attribute of the map, not the territory. You do not know how much to tilt it, but there is still a correct position.
This isn’t something you can feasibly do in real life. It’s not hard to make it absurdly difficult to find the point at which you have to position the needle. It’s just that there is a position.
You do not know how much to tilt it, but there is still a correct position.
The amount to tilt it takes is changing in time. Thermal and/or quantum fluctuations continue to start the thing falling in one direction or another, and you have to keep seeing how it’s going and move whatever your balancing it on to catch it and stop it from falling.
You have created a dynamic equilibrium instead of a static one by using active feedback based on watching what the random and thermal noise are doing. You have not created a situation where an unstable equilibrium takes an arbitrarily long time to be lost. You have invented active feedback to modify the overall system into a stable equilibrium.
If you tilt it the correct way, it will not just stand there. It will fall almost perfectly into every breeze that comes its way. Almost, because it has to be off a little so that it will fall almost perfectly into the next breeze.
You can’t keep it from ever tilting, but that doesn’t mean that you can’t keep it from falling completely.
Falling to the left with t==1 second also has probability zero. Remaining balanced for a period of time between a google and 3^^^3 times the current age of the universe, then falling left, has positive probability.
There is no upper bound to the amount of time that the ball can remain balanced in a continuous deterministic universe.
Sorry, I’m not sure I understand what you mean. What particle should we move to change the fact that the ass will eventually get hungry and choose to walk forward towards one of the piles at semi-random? It seems to me like you can move a particle to guarantee some arbitrarily small change, but you can’t necessarily move one to guarantee the change you want (unless the particle in question happens to be in the brain of the ass).
If you slowly move the particles one at a time from one bale to the other, you know that once you’ve moved the entire bale the Ass will change its decision. At some point before that it won’t be sure.
There might not actually be a choice where the Ass stands there until it starves. It might walk forward, or split in half down the middle and have half of it take one bale of hay and half take the other, or any number of other things. It’s really more that there’s a point where the Ass will eventually take a third option, even if you make sure all third options are worse than the first two.
Thanks (and I actually read the other new comments on the post before responding this time!) I still have two objections.
The first one (which is probably just a failure of my imagination and is in some way incorrect) is that I still don’t see how some simple algorithms would fail. For example, the ass stares at the bales for 15 seconds, then it moves towards whichever one it estimates is larger (ignoring variance in estimates). If it turns out that they are exactly equal, it instead picks one at random. For simplicity, let’s say it takes the first letter of the word under consideration (h), plugs the corresponding number (8) as a seed into a pseudorandom integer generator, and then picks option 1 if the result is even, option 2 if it’s odd. It does seem like this might induce a discontinuity in decisions, but I don’t see where it would fail (so I’d like someone to tell me =)).
The second objection is that our world is, in fact, not continuous (with the Planck length and whatnot). My very mediocre grasp of QM suggests to me that if you try to use continuity to break the ass’s algorithm (and it’s a sufficiently good algorithm), you’ll just find the point where its decisions are dominated by quantum uncertainty and get it to make true random choices. Or something along those lines.
For example, the ass stares at the bales for 15 seconds, then it moves towards whichever one it estimates is larger (ignoring variance in estimates). If it turns out that they are exactly equal, it instead picks one at random.
Your problem is that you’re using an algorithm that can only be approximated on an analog computer. You can’t do flow control like that. If you want it to do A if it has 0 as an input and B if it has 1 as an input, you can make it do A+(B-A)x where x is the input, but you can’t just make it do A under one condition and B under another. If continuity is your only problem, you can make it do A+(B-A)f(x), where f(x)=0 for 0<=x<=0.49 and f(x)=1 for 0.51<=x<=1, but f(x) still has to come out to 1⁄2 when x is somewhere between 0.49<x<0.51.
If you tried to do your algorithm, after 15 seconds, there’d have to be some certainty level where the Ass will end up doing some combination of going left and choosing at random, which will keep it in the same spot if “random” was right. If “random” is instead left, then it stops if it’s half way between that and right.
The second objection is that our world is, in fact, not continuous (with the Planck length and whatnot).
I’m not really sure where that idea came from. Quantum physics is continuous. In fact, derivatives are vital to it, and you need continuity to have them. The position of an object is spread out over a waveform instead of being at a specific spot like a billiard ball, but the waveform is a continuous function of position. The waveform has a center of mass that can be specified however much you want. Also, the Planck length seems kind of arbitrary. It means something if you have an object with size one Planck mass (about the size of a small flea), but a smaller object would have a more spread out waveform, and a larger object would have a tighter one.
get it to make true random choices.
That would make it so you can’t purposely fool the Ass, but it won’t keep that from happening on accident. For example, if you try to balance a needle on the tip outside when there’s a little wind, you’re (probably) not going to be able to do it by making it stand up perfectly straight. It’s going to have to tilt a little so it leans into every gust of wind. But there’s still some way to get it to balance indefinitely.
The second objection is that our world is, in fact, not continuous (with the Planck length and whatnot).
I’m not really sure where that idea came from. Quantum physics is continuous. In fact, derivatives are vital to it, and you need continuity to have them. The position of an object is spread out over a waveform instead of being at a specific spot like a billiard ball, but the waveform is a continuous function of position. The waveform has a center of mass that can be specified however much you want. Also, the Planck length seems kind of arbitrary. It means something if you have an object with size one Planck mass (about the size of a small flea), but a smaller object would have a more spread out waveform, and a larger object would have a tighter one.
The Plank length is irrelevant but quantization isn’t. Specifically, with with quantum mechanics it’s possible to get the ass to be in a superposition of eating from one or the other (but not in the middle) in bounded time.
Okay, thanks for the explanation. It does seem that you’re right*, and I especially like the needle example.
*Well, assuming you’re allowed to move the hay around to keep the donkey confused (to prevent algorithms where he tilts more and more left or whatever from working). Not sure that was part of the original problem, but it’s a good steelman.
You don’t have to move the hay during the experiment. The donkey is the one that moves.
If he goes left as he gets hungry, you move the bale to his right a tad closer, and he’ll slowly inch towards it. He’ll slow down instead of speed up as he approaches it because he’s also getting hungrier.
Does that really work for all (continuous? differentiable?) functions. For example, if his preference for the bigger/closer one is linear with size/closeness, but his preference for the left one increases quadratically with time, I’m not sure there’s a stable solution where he doesn’t move. I feel like if there’s a strong time factor, either a) the ass will start walking right away and get to the size-preferred hay, or b) he’ll start walking once enough time has past and get to the time-preferred hay. I could write down an equation for precision if I figure out what it’s supposed to be in terms of, exactly...
I’m not sure there’s a stable solution where he doesn’t move.
Like I said, the hay doesn’t move, but the donkey does. He starts walking right away to the bigger pile, but he’ll slow down as time passes and he starts wanting the other one.
Interestingly, that trick does get the ass to walk to at least one bale in finite time, but it’s still possible to get it to do silly things, like walk right up to one bale of hay, then ignore it and eat the other.
I’m not sure there’s a stable solution
The solutions are almost certainly unstable. That is, once you find some ratio of bale sizes that will keep the donkey from eating, an arbitrarily small change can get it to eat eventually.
Interestingly, that trick does get the ass to walk to at least one bale in finite time, but it’s still possible to get it to do silly things, like walk right up to one bale of hay, then ignore it and eat the other.
Okay, sure, but that seems like the problem is “solved” (i.e. the donkey ends up eating hay instead of starving).
The Problem only assumes the universe is continuous. If you move a particle by a sufficiently small amount, you can guarantee an arbitrarily small change any finite distance in the future. Thanks to the butterfly effect, it has to be an absurdly tiny amount, but it’s only necessary that it exists.
Also, it assumes that the Ass will eventually die, but that’s really more for effect. The point is that it can’t make the decision in bounded time.
I’m not convinced this is actually true for the same reason of continuity.
Its possible survival is not guaranteed by continuity. It is possible in real life, but it takes more than continuity to prove that.
I know. I was thinking that it might be possible for the ass to guarantee it won’t die by having an interrupt based on how hungry it is.
If you could do an interrupt, you could just make it go to the left if it takes too long to decide.
You can make it so that it gets more left-biased as it gets hungrier, but this just means that the equilibrium has it slowly moving to the right thereby increasing the pull to the right enough to counter out the increased pull to the left from hunger.
My idea is the following:
As it stands the ass will (after an indefinite amount of time) wind up in one of three positions:
a) eating from the left bale,
b) eating from the right bale, or
c) dead.
I’m trying to arrange it so that it always winds up in one of (a) or (b).
If it can pick (a) or (b) then it can also pick something somewhere in between. The only way to get around this is to somehow define (a) and (b) so that they border on each other.
For example, if you talk about which bale it eats first, and it only needs to eat some of it, then you could have something where it walks to the right bale, is about to take a bite, but then changes its mind and goes to the left bale. If you change it by epsilon, it takes an epsilon-sized bite, and eats from the right bale first instead of the left bale.
Only if you assume bounded time. A ball unstably balanced on a one-dimensional hill will after an indefinite amount of time fall to one side or the other, even though the two equilibria aren’t next to each other.
No. It almost certainly will fall eventually, but there is at least one possibility where it never does.
Just because you can think of a possibility does not make it possible. In the absence of classical mechanics, finite temperature will cause it to fall in very finite time. With quantum mechanics, quantum zero point fluctuations will cause it to fall in finite time even if it was at zero temperature.
Finite temperature will cause it to fall in a finite time if you start with it balanced perfectly. You just need to tilt it a little to counter that. This is an argument by continuity, not an argument by symmetry.
There’s some set of starting positions that result in it falling to the left, and another set that result in it falling to the right. If you start it on a boundary point and it falls right after time t, that means that you can get a point arbitrarily close to it that will eventually fall left, so is clearly nowhere near that at time t. That means that physics isn’t being continuous.
You cannot counter it by tilting it because the thermal perturbation is random. At one moment is being pushed to the left, at another to the right.
To be clear, I don’t want to argue against the hypothesis. As long as you are NOT talking about the real world, which is what I take “physics” to mean, you can talk about continuous and balance point, and arbitrarily long times to fall one way or the other. The point is that in the real world, any real world experiment analyzed in detail will have “noise” which causes it to fall sometimes one way and sometimes the other in finite time when placed at its balance point. That noise is usually dominated by thermal fluctuations, but even in the absence of thermal fluctuations, there are still quantum fluctuations which behave very much like thermal noise, which behave in very many ways as though you can’t get the temperature below some limit.
In principle, you can build a system where the thing is cooled enough, and where it is designed so that the quantum fluctuations are small enough, that you can have a relatively long time for it to fall one way or the other off its balance point. That is, the amount of time it takes to be pushed by quantum noise is large on human or intuitive scales. However, you can’t make it arbitrarily large without building an arbitariily large system. So if you are concerned about the difference between “really big” and “infinite,” you don’t get to claim a physical system balanced on an unstable equilibrium point can have an infinitely long time to fall.
Of course this doesn’t mean that in a non-physical “world” with no quantum and rigid objects (another approximation that can’t be realized in the real world) you couldn’t do it. So the math is safe, its just not physics.
Randomness is an attribute of the map, not the territory. You do not know how much to tilt it, but there is still a correct position.
This isn’t something you can feasibly do in real life. It’s not hard to make it absurdly difficult to find the point at which you have to position the needle. It’s just that there is a position.
The amount to tilt it takes is changing in time. Thermal and/or quantum fluctuations continue to start the thing falling in one direction or another, and you have to keep seeing how it’s going and move whatever your balancing it on to catch it and stop it from falling.
You have created a dynamic equilibrium instead of a static one by using active feedback based on watching what the random and thermal noise are doing. You have not created a situation where an unstable equilibrium takes an arbitrarily long time to be lost. You have invented active feedback to modify the overall system into a stable equilibrium.
If you tilt it the correct way, it will not just stand there. It will fall almost perfectly into every breeze that comes its way. Almost, because it has to be off a little so that it will fall almost perfectly into the next breeze.
You can’t keep it from ever tilting, but that doesn’t mean that you can’t keep it from falling completely.
Your argument now boils down to “the physical world is not both continuous and deterministic”.
With probability zero.
Falling to the left with t==1 second also has probability zero. Remaining balanced for a period of time between a google and 3^^^3 times the current age of the universe, then falling left, has positive probability.
There is no upper bound to the amount of time that the ball can remain balanced in a continuous deterministic universe.
Sorry, I’m not sure I understand what you mean. What particle should we move to change the fact that the ass will eventually get hungry and choose to walk forward towards one of the piles at semi-random? It seems to me like you can move a particle to guarantee some arbitrarily small change, but you can’t necessarily move one to guarantee the change you want (unless the particle in question happens to be in the brain of the ass).
If you slowly move the particles one at a time from one bale to the other, you know that once you’ve moved the entire bale the Ass will change its decision. At some point before that it won’t be sure.
There might not actually be a choice where the Ass stands there until it starves. It might walk forward, or split in half down the middle and have half of it take one bale of hay and half take the other, or any number of other things. It’s really more that there’s a point where the Ass will eventually take a third option, even if you make sure all third options are worse than the first two.
Thanks (and I actually read the other new comments on the post before responding this time!) I still have two objections.
The first one (which is probably just a failure of my imagination and is in some way incorrect) is that I still don’t see how some simple algorithms would fail. For example, the ass stares at the bales for 15 seconds, then it moves towards whichever one it estimates is larger (ignoring variance in estimates). If it turns out that they are exactly equal, it instead picks one at random. For simplicity, let’s say it takes the first letter of the word under consideration (h), plugs the corresponding number (8) as a seed into a pseudorandom integer generator, and then picks option 1 if the result is even, option 2 if it’s odd. It does seem like this might induce a discontinuity in decisions, but I don’t see where it would fail (so I’d like someone to tell me =)).
The second objection is that our world is, in fact, not continuous (with the Planck length and whatnot). My very mediocre grasp of QM suggests to me that if you try to use continuity to break the ass’s algorithm (and it’s a sufficiently good algorithm), you’ll just find the point where its decisions are dominated by quantum uncertainty and get it to make true random choices. Or something along those lines.
Your problem is that you’re using an algorithm that can only be approximated on an analog computer. You can’t do flow control like that. If you want it to do A if it has 0 as an input and B if it has 1 as an input, you can make it do A+(B-A)x where x is the input, but you can’t just make it do A under one condition and B under another. If continuity is your only problem, you can make it do A+(B-A)f(x), where f(x)=0 for 0<=x<=0.49 and f(x)=1 for 0.51<=x<=1, but f(x) still has to come out to 1⁄2 when x is somewhere between 0.49<x<0.51.
If you tried to do your algorithm, after 15 seconds, there’d have to be some certainty level where the Ass will end up doing some combination of going left and choosing at random, which will keep it in the same spot if “random” was right. If “random” is instead left, then it stops if it’s half way between that and right.
I’m not really sure where that idea came from. Quantum physics is continuous. In fact, derivatives are vital to it, and you need continuity to have them. The position of an object is spread out over a waveform instead of being at a specific spot like a billiard ball, but the waveform is a continuous function of position. The waveform has a center of mass that can be specified however much you want. Also, the Planck length seems kind of arbitrary. It means something if you have an object with size one Planck mass (about the size of a small flea), but a smaller object would have a more spread out waveform, and a larger object would have a tighter one.
That would make it so you can’t purposely fool the Ass, but it won’t keep that from happening on accident. For example, if you try to balance a needle on the tip outside when there’s a little wind, you’re (probably) not going to be able to do it by making it stand up perfectly straight. It’s going to have to tilt a little so it leans into every gust of wind. But there’s still some way to get it to balance indefinitely.
The Plank length is irrelevant but quantization isn’t. Specifically, with with quantum mechanics it’s possible to get the ass to be in a superposition of eating from one or the other (but not in the middle) in bounded time.
Okay, thanks for the explanation. It does seem that you’re right*, and I especially like the needle example.
*Well, assuming you’re allowed to move the hay around to keep the donkey confused (to prevent algorithms where he tilts more and more left or whatever from working). Not sure that was part of the original problem, but it’s a good steelman.
You don’t have to move the hay during the experiment. The donkey is the one that moves.
If he goes left as he gets hungry, you move the bale to his right a tad closer, and he’ll slowly inch towards it. He’ll slow down instead of speed up as he approaches it because he’s also getting hungrier.
Does that really work for all (continuous? differentiable?) functions. For example, if his preference for the bigger/closer one is linear with size/closeness, but his preference for the left one increases quadratically with time, I’m not sure there’s a stable solution where he doesn’t move. I feel like if there’s a strong time factor, either a) the ass will start walking right away and get to the size-preferred hay, or b) he’ll start walking once enough time has past and get to the time-preferred hay. I could write down an equation for precision if I figure out what it’s supposed to be in terms of, exactly...
Like I said, the hay doesn’t move, but the donkey does. He starts walking right away to the bigger pile, but he’ll slow down as time passes and he starts wanting the other one.
Interestingly, that trick does get the ass to walk to at least one bale in finite time, but it’s still possible to get it to do silly things, like walk right up to one bale of hay, then ignore it and eat the other.
The solutions are almost certainly unstable. That is, once you find some ratio of bale sizes that will keep the donkey from eating, an arbitrarily small change can get it to eat eventually.
Okay, sure, but that seems like the problem is “solved” (i.e. the donkey ends up eating hay instead of starving).
It can also use the “always eat the left bale first” strategy, although that gets kind of odd if it does it with a bale of size zero.
There is a problem if you want to make it make an actual binary decision, like go to one bale and stay.
See Daniel’s comment here.