If I understand you correctly, we’ve got two different types of things to which we’re applying the label “probability”:
(1) A distribution on the phase space (either frequency or epistemic) for initial conditions/precise outcomes. (We can evolve this distribution forward or backward in time according to the dynamics of the system.)
(2) An “objective probability” distribution determined only the properties of the phase space.
I’m just not seeing why we should care about anything but distributions of type (1). Sure, you can put a uniform measure over points in phase space and count the proportion than ends up in a specified subset. But the only justification I can see for using the uniform measure—or any other measure—is as an approximation to a distribution of type (1).
Here’s a new toy model: the phase space is the set of real numbers in the range [0,1]. The initial state is called x_0. The time dynamics are x(t) = (x_0)^(t+1) (positive time only). The coarse outcome is round[x(t)] at some specified t. What is the “objective probability”? If it truly does depend only on the phase space, I’ve given you everything you need to answer that question.
(For macroscopic model systems like coin tosses, I go with a deterministic universe.)
Constant,
If I understand you correctly, we’ve got two different types of things to which we’re applying the label “probability”:
(1) A distribution on the phase space (either frequency or epistemic) for initial conditions/precise outcomes. (We can evolve this distribution forward or backward in time according to the dynamics of the system.) (2) An “objective probability” distribution determined only the properties of the phase space.
I’m just not seeing why we should care about anything but distributions of type (1). Sure, you can put a uniform measure over points in phase space and count the proportion than ends up in a specified subset. But the only justification I can see for using the uniform measure—or any other measure—is as an approximation to a distribution of type (1).
Here’s a new toy model: the phase space is the set of real numbers in the range [0,1]. The initial state is called x_0. The time dynamics are x(t) = (x_0)^(t+1) (positive time only). The coarse outcome is round[x(t)] at some specified t. What is the “objective probability”? If it truly does depend only on the phase space, I’ve given you everything you need to answer that question.
(For macroscopic model systems like coin tosses, I go with a deterministic universe.)