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Eugine_Nier comments on Buridan's ass and the psychological origins of objective probability - Less Wrong
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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.
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.
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.