Or, put another way: asking how many worlds there are is like asking how
many experiences you had yesterday, or how many regrets a repentant criminal
has had. It makes perfect sense to say that you had many experiences or that he
had many regrets; it makes perfect sense to list the most important categories
of either; but it is a non-question to ask how many.
If this picture of the world seems unintuitive, a metaphor may help.
So, he basically concedes that the world counting is meaningless, except as a metaphor. What a letdown.
Wallace’s refusal to answer in any meaningful way reminded me of the following exchange in The Simple Truth:
Mark heaves a patient sigh. “Autrey, do you think you’re the first person to think of that question? To ask us how our own beliefs can be meaningful if all beliefs are meaningless? That’s the same thing many students say when they encounter this philosophy, which, I’ll have you know, has many adherents and an extensive literature.”
“So what’s the answer?” says Autrey.
“We named it the ‘reflexivity problem’,” explains Mark.
“But what’s the answer?” persists Autrey.
Mark smiles condescendingly. “Believe me, Autrey, you’re not the first person to think of such a simple question. There’s no point in presenting it to us as a triumphant refutation.”
I don’t understand this. Wallace does give an answer to the question “How many worlds?” His answer is something like “That’s not a question with a precise answer.” And he gives a number of reasons in support of this response. He doesn’t just say “Oh, I’ve thought about that question a lot. Believe me, it’s not that simple.” In what way is his response similar to Mark’s?
And why do you find his claim that world counting is meaningless a “letdown”? Why is giving a precise rule for world counting a desideratum for an Everettian interpretation?
My best understanding of the MWI’s take on the Born rule is that the ratio of the number of branches for each outcome to the total number of branches gives you the probability of each outcome. Both numbers must be finite for the division to make sense. If you cannot count branches, you cannot calculate probabilities, reducing the model into just a feel-good narrative (with the Born rule inserted by hand). Refusing to acknowledge this issue is similar to what Mark does in the story.
No, the probabilities in MWI are not counting discrete worlds. A world with large amplitude is not multiple identical worlds but a single world that is more real. Leaving aside the actual interpretation, your suggestion is mathematically incoherent. You seem to be demanding that the probabilities in QM are rational numbers with bounded denominator. This is an extremely radical position. It would simplify the ontology a lot, but there is no reason to believe that quantum mechanics can be approximated by a system where the amplitudes are not infinitely divisible. More precisely, a large finite subgroup of the unitary group does not look like the unitary group, but like a torus.
Sorry, I did not get your point about the group and subgroups, or at least not its relevance to the question. I would expect that to derive Born probabilities one has to assign measures to different worlds (how else would you express mathematically that “A world with large amplitude is not multiple identical worlds but a single world that is more real.”?) I agree that counting branches is not the only way to do it, just the most obvious one. Unfortunately, none of the ways of assigning “strength” to different branches seems to work any better than this naive one in deriving the Born rule (that is to say, they do not work at all).
My best understanding of the MWI’s take on the Born rule is that the ratio of the number of branches for each outcome to the total number of branches gives you the probability of each outcome.
This is not the way the Oxford Everettians understand the Born rule. See the Hilary Greaves paper I linked to for a discussion of their decision-theoretic approach to probabilities in the MWI. This approach has its problems, but they are problems that the Everettians acknowledge and attempt to address (not entirely successfully, in my opinion). That’s very different from Mark’s attitude.
Also, the Orzel post you linked to doesn’t seem to support your contention. Where do you see him committing himself to the branch counting appproach you propose? (EDIT: Actually, I see that there is discussion of the issue in the comments to that post, which is probably what you meant.)
Deutsch claimed to ‘prove’, via decision theory, that the ‘rational’ agent who believes she lives in an Everettian multiverse will nevertheless ‘make decisions as if’ the mod-squared measure gave chances for outcomes.
This must a bad wording, or something, otherwise why does a “rational” agent who does not believe “she lives in an Everettian multiverse” can still confirm the Born rule experimentally time after time?
The proof does not address rational agents who do not believe they are in an Everettian multiverse. They would have other reasons for using the Born rule.
If they are infinite, then there should at least be a well-defined way to take a limit (or one of its generalizations), which amounts to nearly the same thing, constructing a sequence of ratios of finite numbers and proving convergence.
From the Wallace paper:
So, he basically concedes that the world counting is meaningless, except as a metaphor. What a letdown.
Wallace’s refusal to answer in any meaningful way reminded me of the following exchange in The Simple Truth:
Mark heaves a patient sigh. “Autrey, do you think you’re the first person to think of that question? To ask us how our own beliefs can be meaningful if all beliefs are meaningless? That’s the same thing many students say when they encounter this philosophy, which, I’ll have you know, has many adherents and an extensive literature.”
“So what’s the answer?” says Autrey.
“We named it the ‘reflexivity problem’,” explains Mark.
“But what’s the answer?” persists Autrey.
Mark smiles condescendingly. “Believe me, Autrey, you’re not the first person to think of such a simple question. There’s no point in presenting it to us as a triumphant refutation.”
I don’t understand this. Wallace does give an answer to the question “How many worlds?” His answer is something like “That’s not a question with a precise answer.” And he gives a number of reasons in support of this response. He doesn’t just say “Oh, I’ve thought about that question a lot. Believe me, it’s not that simple.” In what way is his response similar to Mark’s?
And why do you find his claim that world counting is meaningless a “letdown”? Why is giving a precise rule for world counting a desideratum for an Everettian interpretation?
My best understanding of the MWI’s take on the Born rule is that the ratio of the number of branches for each outcome to the total number of branches gives you the probability of each outcome. Both numbers must be finite for the division to make sense. If you cannot count branches, you cannot calculate probabilities, reducing the model into just a feel-good narrative (with the Born rule inserted by hand). Refusing to acknowledge this issue is similar to what Mark does in the story.
This is a relevant discussion of the issue.
No, the probabilities in MWI are not counting discrete worlds. A world with large amplitude is not multiple identical worlds but a single world that is more real. Leaving aside the actual interpretation, your suggestion is mathematically incoherent. You seem to be demanding that the probabilities in QM are rational numbers with bounded denominator. This is an extremely radical position. It would simplify the ontology a lot, but there is no reason to believe that quantum mechanics can be approximated by a system where the amplitudes are not infinitely divisible. More precisely, a large finite subgroup of the unitary group does not look like the unitary group, but like a torus.
Sorry, I did not get your point about the group and subgroups, or at least not its relevance to the question. I would expect that to derive Born probabilities one has to assign measures to different worlds (how else would you express mathematically that “A world with large amplitude is not multiple identical worlds but a single world that is more real.”?) I agree that counting branches is not the only way to do it, just the most obvious one. Unfortunately, none of the ways of assigning “strength” to different branches seems to work any better than this naive one in deriving the Born rule (that is to say, they do not work at all).
This is not the way the Oxford Everettians understand the Born rule. See the Hilary Greaves paper I linked to for a discussion of their decision-theoretic approach to probabilities in the MWI. This approach has its problems, but they are problems that the Everettians acknowledge and attempt to address (not entirely successfully, in my opinion). That’s very different from Mark’s attitude.
Also, the Orzel post you linked to doesn’t seem to support your contention. Where do you see him committing himself to the branch counting appproach you propose? (EDIT: Actually, I see that there is discussion of the issue in the comments to that post, which is probably what you meant.)
From the paper:
This must a bad wording, or something, otherwise why does a “rational” agent who does not believe “she lives in an Everettian multiverse” can still confirm the Born rule experimentally time after time?
The proof does not address rational agents who do not believe they are in an Everettian multiverse. They would have other reasons for using the Born rule.
Is that necessarily true?
If they are infinite, then there should at least be a well-defined way to take a limit (or one of its generalizations), which amounts to nearly the same thing, constructing a sequence of ratios of finite numbers and proving convergence.