On 4/12/2022 1:41 PM, George Kahrimanis wrote:
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.*
Physics doesn't care about "rationally justified", only about
empirically justified. Both your examples suffer from choosing the
simplest case where symmetry can be invoked. But how you know, or
assume, they are symmetric seems to already rely on the QM of qubits.
Once you've assumed the Hilbert space structure of QM, then Gleason's
theorem essentially forces the Born rule (correct me if I'm wrong, but I
think the theorem has been extended to the two-dimensional case).
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).
I think the problem is that MWI (but not Everett) assume all outcomes
are equally realized. So how does a probability become assigned to
them, what does it mean. We're told it's the probability of finding
ourself in a particular world...but that seems very much like "collapse
of the wave-function". It introduces the same problems of exactly when
and where does it happen; with only the advantage that consciousness is
not understood in detail so the mystery can be push off. Decoherence
has gone part way in solving the when/where/what basis questions, but
only part way.
Brent
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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