This is a good (though not new) reductio ad absurdum of the “worlds can be counted and should obviously all be treated as having the same measure” thinking that leads to Eliezer’s claims that the Born rule is mysterious, and to theories like Mangled Worlds.
I can’t really speak for them, but I somewhat believe that they believe what I believe, i.e., that the many-worlds interpretation of nonrelativistic QM models the world as a function from a (usually nondiscrete) configuration space to the complex numbers, and the norm-squared of that function is the measure that predicts the outcomes of experiments. So there’s no well-defined way to count worlds in general, although in some circumstances it may be helpful to refer to a small region of the configuration space as a “world”, and different “worlds” can have different measures.
Decoherence is implicit in quantum physics, not an extra postulate on top of it, and quantum physics is continuous. Thus, “decoherence” is not an all-or-nothing phenomenon—there’s no sharp cutoff point. Given two blobs, there’s a quantitative amount of amplitude that can flow into identical configurations between them. This quantum interference diminishes down to an exponentially tiny infinitesimal as the two blobs separate in configuration space.
I’m still confused by the Born rule. (The worlds where the Born rule makes good predictions have a lot of L^2 measure. But they have very little L^1 measure! The lion’s share of the L^1 measure is held by worlds where an L^1 version of the Born rule holds. Why does our experience accord with the Born rule?) But I have more reading to do.
What privileges L^1 measure if not world-counting? I just looked up mangled worlds again, and Hanson explicitly uses world-counting; at least, he seems to admit that worlds don’t have well-defined numbers, but he thinks this isn’t a problem for reasons I don’t really understand.
(The point of my aside about the Born probabilities is that neither the L^1 norm nor the L^2 norm are privileged. (At least, I don’t understand why our experiments favor one of them over the other.) I could just as easily have talked about the L^3 norm.)
Looking at that Mangled Worlds page, I see that you’re right — Hanson is talking about a finite number of worlds. And as far as I can tell, every world that exists is equally probable, which would correspond to an L^0 norm? I don’t really understand the proposal, though.
Sorry, I guess I got confused about what an L^1 norm meant. My non-confident recollection is that Eliezer believes, not just that we have no idea how to assign measure to worlds, but that our best idea for assigning measure to worlds actively conflicts with the Born rule, except if there exists some sort of mechanism like world-mangling. His endorsement of world-mangling as a possible solution suggests to me that he agrees with its world-counting assumption.
This is a good (though not new) reductio ad absurdum of the “worlds can be counted and should obviously all be treated as having the same measure” thinking that leads to Eliezer’s claims that the Born rule is mysterious, and to theories like Mangled Worlds.
I doubt that Yudkowsky and Hanson believe that “worlds can be counted and should obviously all be treated as having the same measure”.
I’m not confident that my memory here is correct, but what do you think they think instead?
I can’t really speak for them, but I somewhat believe that they believe what I believe, i.e., that the many-worlds interpretation of nonrelativistic QM models the world as a function from a (usually nondiscrete) configuration space to the complex numbers, and the norm-squared of that function is the measure that predicts the outcomes of experiments. So there’s no well-defined way to count worlds in general, although in some circumstances it may be helpful to refer to a small region of the configuration space as a “world”, and different “worlds” can have different measures.
From the Quantum Physics Sequence:
I’m still confused by the Born rule. (The worlds where the Born rule makes good predictions have a lot of L^2 measure. But they have very little L^1 measure! The lion’s share of the L^1 measure is held by worlds where an L^1 version of the Born rule holds. Why does our experience accord with the Born rule?) But I have more reading to do.
I don’t know much about Mangled Worlds.
What privileges L^1 measure if not world-counting? I just looked up mangled worlds again, and Hanson explicitly uses world-counting; at least, he seems to admit that worlds don’t have well-defined numbers, but he thinks this isn’t a problem for reasons I don’t really understand.
(The point of my aside about the Born probabilities is that neither the L^1 norm nor the L^2 norm are privileged. (At least, I don’t understand why our experiments favor one of them over the other.) I could just as easily have talked about the L^3 norm.)
Looking at that Mangled Worlds page, I see that you’re right — Hanson is talking about a finite number of worlds. And as far as I can tell, every world that exists is equally probable, which would correspond to an L^0 norm? I don’t really understand the proposal, though.
Sorry, I guess I got confused about what an L^1 norm meant. My non-confident recollection is that Eliezer believes, not just that we have no idea how to assign measure to worlds, but that our best idea for assigning measure to worlds actively conflicts with the Born rule, except if there exists some sort of mechanism like world-mangling. His endorsement of world-mangling as a possible solution suggests to me that he agrees with its world-counting assumption.