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Robin and Eliezer, I do appreciate the significance of the proposed hypothesis, but I feel I must rethink this subject carefully. It used to be my favorite subject, a very long time ago though (pre-Deutsch). In this reply, at first I focus on your criticism of alternatives, but as a result of responding to you, a particular alternative will be shaped. (Sorry for the length.)
Ev'Hu suggests, "Well, you could have a rule that world-sides whose thickness tends toward zero, must have a degree of reality that also tends to zero. And then the rule which says that you square the thickness of a world-side, would let the probability tend toward zero as the world-thickness tended toward zero. QED."
Let me try an alternative with only one rule and without limits.
Sincere Question: Do we need a "rule" or hypothesis stating that zero transition amplitude between any two quantum states implies that either state is impossible to obtain from the other? (As opposed to somehow demonstrating or defending this proposition.)
Suppose that some general argument can be used to back the above proposition, else let us introduce it as a new axiom. (It would be in the same league as your hypothesis, but weaker.) Let us combine it with considerations of physical continuity, based on the observation that exactly zero transition amplitude is a mathematical idealization. Thus we need to form a new concept, "approximate impossibility", for actual pairs of states, without yet specifying how close to zero an amplitude must be in order for "approximate impossibility" to hold. Now note that the object "absolute value of a squared transition amplitude" shares the properties of mathematical measure and the subjective property that if it is small enough it implies approximate impossibility. It looks like a duck and it quacks like a duck! Conjecture: a full probability interpretation can be based on there two premises. At the end, if one cares, it will be possible to clarify the notion "approximate impossibility" (relative to a particular situation and particular practical concerns) but that would be mainly a consistency check rather than a useful excercise.
"That's not QED," says Po'mi. "That's a complete non-sequitur. Logical fallacy of affirming the consequent. You could have all sorts of rules that would let the reality tend toward zero as the world-thickness tended toward zero, not just the squaring rule. [...]
Conjecture: only the absolute value of the square of the transition amplitude has the properties of probability.
And in fact, all our world-sides have a thickness that 'tends toward zero' because they keep splitting.
Then let us deal with only one split at a time.
Furthermore, why would an indefinite tendency in the infinite future have any impact on what we do now?"
I think that the frequentist demons distracted you at this point. The main concern is not "what we do now"; the point is (as you hinted elsewhere) to account for the body of facts, both data and memories, which look just as if they were generated from random processes.
(Caveat: Although I did introduce "approximate impossibility", a patently subjective concept, it was only a crutch, in the way of demonstrating that the subjective interpretation of "absolute value of a squared transition amplitude" shares a common feature with the subjective interpretation of probability. That common feature was a foot in the door, to explain why this object is probability. Thus we can account for the observed frequencies being typically close to theoretical values.)
Thanks in advance for any back criticism! Also thanks for maintaining this wondrful weblogsite, which I noticed only yesterday and it has overstimulated my head. For that matter, I found a discussion somewhat related to the elusiveness of exact impossibility: <http://lesswrong.com/lw/mp/0_and_1_are_not_probabilities/>.