On Friday, April 8, 2022 at 3:19:08 AM UTC+3 Lawrence Crowell wrote:
> This is an appeal to some sort of imperative that demands the Born Rule
> because the counterfactual lack this certainty. This is a sort of "It must
> be true" type of argument.
Thanks for the comments! I wonder though, do you agree with my criticisms
of previous proposals for deriving the Born Rule, or are you undecided? I
will challenge you (and you all) on this matter, later in this message.
First, a correction: I have not referred to *counterfactuals* (I think that
you meant "alternatives") but now that you mention them, I may have implied
one:
"If QM were not a workable theory, *we would have no direct, experimental
clue that it is a fundamental theory in physics*".
(Not the typical use of a counterfactual, which is in an "if..." clause, as
in "*If I was a rich man*...".)
What I say is not exactly
> "It must be true"
but rather
"Although I cannot be certain, it seems to be in my interests to form this
assessment now, when I decide how to act in the present situation".
If you find this argument too loose: I have pointed out that it is the same
kind of argument that a judge uses to form a decision based on the
evidence, or an engineer uses, to trust the theory of real numbers, for her
project.
My aim has been to complete *Everett's argument,* which I outline next.
Imagine that we repeat the same trial N times, and we record the ratio
{statistical "frequency") r of one among the possible outcomes
(eigenstates). Conventional QM assigns a probability R for this outcome, so
we need an explanation why r SEEMS to approach R in the long run (though we
know that in very many worlds it will not be so!). Everett noted that, for
any positive real ε (however small), the measure of all "outlier"
sequences, that is: for which r is outside
[R-ε, R+ε],
is small, with limit zero as N increases to infinity. However, *a problem
remains:* why "small measure" or "vanishing measure" have any significance
in the interpretation of QM? *My proposal answers this question,* finding
an argument about "small measure" within the reasoned assessment that QM is
a workable theory.
*Here is my challenge to you.* I ask you if you agree with either of the
following two proposals (for deriving the Born Rule in a MWI).
First, Deutsch's (1999) proposal, here in a simplified version. Imagine a
simulated tossing of a fair coin, using a qubit instead of a coin, with
which you either win or lose one dollar. If this bet has a definite, single
value to you (presumably, by some kind of intuitive averaging over possible
futures) it will necessarily be zero, for symmetry reasons. Caveat: Deutsch
points out that we do not derive probability strictly speaking. I accept
the reasoning, but not the premise: I am uncomfortable with averaging my
future selves, and there is no direct rationale why I SHOULD do so. So, *what
do you think?*
Second (and last), proposals such as Zurek's are of the following pattern
(here I reuse the previous example): I am uncertain about the outcome, and
I expect the theory to give me some clue, which will be probability -- what
else? For symmetry reasons, the probability here must be 1/2. My objection
is that there is no randomisation in MWI (no shuffling, stirring, or God
playing dice) so that the use of probability is not rationally justified.
Again* I ask for your opinion.*
Clarification. Instead of probability proper, I derive the following. With
regard to any given application, an Everettian agent may expect "with moral
certainty" (remember the judge and the engineer!) that statistical
frequency in the long run will be as close to the Born probability as one
needs it to be (in the particular application). Some people may think
"po-tah-toes, pot-eight-os", but at some level of thinking *this* is the
crucial issue. In particular, a serious consequence for decision theory
results from failing to find any rationale for probability proper!
George K.
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