minkwe wrote:Richard, are you going to answer the question or not?
How is CFD important for your derivation, if you are using the weakly objective interpretation?
The weakly objective interpretation tells me what I mean by "probability". I imagine something being repeated many many times and then I look at relative frequencies and then I imagine the number of repetitions going to infinity and the relative frequencies converging. Also known as the frequentist interpretation. But notice the word *imagine*. It's an imaginary sequence of repetitions.
Where does CFD come in? Well forget CFD for just a moment, suppose that we have a LHV model. That means that Nature chooses lambda, Alice chooses a and Bob chooses b. Alice and Bob then get to see A(a, lambda) and B(b, lambda). But since A and B are just a couple of functions I can also think of A(a', lambda) for all possible values of a' and B(b', lambda) for all possible values of b', at the same time. Within my mathematical model they are defined, too, even though they don't correspond to anything in the experiment ... well, what they correspond to, is what the outcome would have been, had Alice chosen a' instead of a, while Nature had still made the same choice lambda.
In your simulation models too, the same thing happens. You pick a lambda, also a and b get picked, there is a function AIa, lambda) which tells us Alice's outcome and so on... but I could add a line to the code where A(a', lambda) is also computed, but nothing else would change.
It is possible to go the other way round and show that CFD + locality => LHV. It's a simple little mathematical theorem. Well ... simple if you are at home with modern probability theory.
Remember what I said about the difference between models and reality?
The frequentist interpretation of probability is a bridge between part of reality and probability theory (which is part of mathematics).
LHV and CFD are both part of mathematics. They are terms which can be used within the domain of mathematical models of reality.
BTW I really appreciate that you keep putting these questions. It means that you feel that there is something going on here which needs to be sorted out and you keep on trying. This is the mark of a true scientist. A true scientist keeps on feeling that itch and keeps on trying to do something about it.
BTW there are also other interpretations of probability, for instance, the subjective (Bayesian) view, cf. Jaynes. I don't like it so much. In practice it tends to come down to the same thing, at least, in physics it does. Fortunately the "calculus" of subjective probability is identical to that of "objective" probability so the mathematical theories are identical. Every theorem of subjective probability theory is also a theorem of objective probability theory, and vice versa. The difference between the two theories is not inside the theories themselves, but in the bridges to the "real world".
We now get into self-reference paradoxes, till we realise that "the real world" is also some kind of idealisation. Hopefully it is inter-subjective. In other words, at a higher level, both Bayesian and frequentist probability concepts are themselves only "models".