On 2/7/2020 2:36 PM, Bruce Kellett wrote:
On Sat, Feb 8, 2020 at 5:23 AM Bruno Marchal <[email protected]
<mailto:[email protected]>> wrote:
On 7 Feb 2020, at 05:59, Bruce Kellett <[email protected]
<mailto:[email protected]>> wrote:
"After N trials, the multiverse contains 2^N branches,
corresponding to all 2^N possible binary string outcomes. The
inhabitants on a string with pN zero and (1 - p)N one outcomes
will, with a degree of confidence that tends towards one as N
gets large, tend to conclude that the weight 'p' is attached to
zero outcome branches and weight (1 - p) is attached to one
outcome branches. In other words, everyone, no matter what string
they see, tends towards complete confidence in the belief that
the relative frequencies they observe represent the weights.
"Let's consider further the perspective of inhabitants on a
branch with 'pN' zero outcomes and '(1 - p)N' one outcomes. They
do not have the delusion that all observed strings have the same
relative frequency as theirs: they understand that, given the
hypothesis that they live in a multiverse, 'every' binary string,
and hence every relative frequency, will have been observed by
someone. So how do they conclude that the theory that the weights
are '(p,1 - p)' has nonetheless been confirmed?. Because they
have concluded that the weights measure the 'importance' of the
branches for theory confimation. Since they believe they have
learned that the weights are '(p,1 - p)', they conclude that a
branch with 'r' zeros and '(N - r)' ones has importance p^r(1 -
p)^{N-r}. Summing over all branches with 'pN' zeros and '(1 -
p)N' ones, or very close to those frequencies, thus gives a set
of total importance very close to 1; the remaining branches have
total importance very close to zero. So, on the set of branches
that dominate the importance measure, the theory that the weights
are (very close to) (p,1 - p) is indeed correct. All is well! By
definition, the important branches are the ones that matter for
theory confimation. The theory is inded confirmed!
"The problem, of course, is that this reasoning applies equally
well for all the inhabitants, whatever relative frequency 'p'
they see on their branch. All of them conclude that their
relative frequencies represent (to very good approximation) the
branching weights. All of them conclude that their own branches,
together with those with identical or similar relative
frequencies, are the important ones for theory confirmation. All
of them thus happily conclude that their theories have been
confirmed. And, recall, all of them are wrong: there are actually
no branching weights.”
I do not understand. If the multiverse is that sort of many
classical world, with the machine giving all outputs somewhere,
the correct weighting will be the one given by Pascal Binomial.
That comes already with the fact that we get all 2^N strings. I
might have miss something.
You certainly have. The argument that output strings that give results
inconsistent with your observations have vanishing measure overall --
an argument based on the Pascal Binomial and the law of large numbers
-- applies equally to all observers, whatever output string they
observe. So whatever data you observe, you conclude that the theory
that is consistent with that data is confirmed by the data. Which is
useless, because you reach that conclusion whatever data you observe.
The law of large numbers fails you when all possible outcomes are
observed by someone or the other.
So if the experiment is to toss a coin six times, there will be a branch
of the MW where HTHHTHHHHH is observed and this will confirm the theory
that H's are four times as probable as T's. But there will be many more
branches where it is found that P(H)=P(T) (252 vs 45). And in the limit
of large experiments almost all experimenters (in the MW) will find
P(H)~P(T). Hence almost all experimenters will conclude something close
to the presumed true value.
This however depends on the assumption that each sequence of H and T
occurs in one branch of the MW. Other probability values, like 1/pi,
are going to require very large numbers of branches to approximate.
Brent
Do you agree that in the iterated self- (WM)-duplication, the
measure is just the normal distribution?
No. As I have said before, no meaningful concept of probability can be
applied in the WM-duplication case. Since no meaningful concept of
probability applies when all outcome are guaranteed to happen, no
probability measure can be assigned.
This argument from Kent completely destroys Everett's attempt to
derive the Born rule from his many-worlds approach to quantum
mechanics. In fact, it totally undermines most attempts to derive
the Born rule from any branching theory, and undermines attempts
to justify ignoring branches on which the Born rule weights are
disconfirmed.
They normally just get relatively rare.
It is the attempted proof of this that breaks down when all outcomes
are guaranteed to occur.
In the many-worlds case, recall, all observers are aware that
other observers with other data must exist, but each is led to
construct a spurious measure of importance that favours their own
observations against the others', and this leads to an obvious
absurdity. In the one-world case, observers treat what actually
happened as important, and ignore what didn't happen: this
doesn't lead to the same difficulty.
With Mechanism (used in Darwin) I don’t see how we can evacuate
that the prediction are given by relative (even conditional)
measure, on all computations.
This has nothing to do with mechanism: it is simple an observation
about Everettian quantum mechanics. If you want to talk about some
other theory, such as mechanism, we can do that. But I think mechanism
fails at step 3 for reasons similar to those that undermine Everett.
Bruce
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