The subjectivist interpretation of probability is common [in] … common intuition (where people feel like they can say there’s a 30% chance of rain tomorrow without worrying about the fact that tomorrow only happens once)
Why do you say that “there’s a 30% chance of rain tomorrow” is an example of the subjective interpretation? Isn’t it just as readily interpreted as saying “on 30% of all days similar to this one [in meteorological conditions, etc.], it rains”?
Besides, “this coin flip that I am going to do right now” only happens once, too (any subsequent coin flips will be other, different, coin flips, and not that specific coin flip). Surely you don’t conclude from this that when someone says “this coin has a 50% chance of coming up heads”, it means they’re taking the subjectivist view of the coin’s behavior?
There are 0 other days “similar to” this one in Earth’s history, if “similar to” is strict enough (e.g. the exact pattern of temperature over time, cloud patterns, etc). You’d need a precise, more permissive definition of “similar to” for the statement to be meaningful.
When you say “all days similar to this one”, are you talking about all real days or all possible days? If it’s “all possible days”, then this seems like summing over the measures of all possible worlds compatible with both your experiences and the hypothesis, and dividing by the sum of the measures of all possible worlds compatible with your experiences. (Under this interpretation, jessicata’s response doesn’t make much sense; “similar to” means “observationally equivalent for observers with as much information as I have”, and doesn’t have a free variable.)
Why do you say that “there’s a 30% chance of rain tomorrow” is an example of the subjective interpretation? Isn’t it just as readily interpreted as saying “on 30% of all days similar to this one [in meteorological conditions, etc.], it rains”?
Besides, “this coin flip that I am going to do right now” only happens once, too (any subsequent coin flips will be other, different, coin flips, and not that specific coin flip). Surely you don’t conclude from this that when someone says “this coin has a 50% chance of coming up heads”, it means they’re taking the subjectivist view of the coin’s behavior?
There are 0 other days “similar to” this one in Earth’s history, if “similar to” is strict enough (e.g. the exact pattern of temperature over time, cloud patterns, etc). You’d need a precise, more permissive definition of “similar to” for the statement to be meaningful.
But the same is true of coin flips.
When you say “all days similar to this one”, are you talking about all real days or all possible days? If it’s “all possible days”, then this seems like summing over the measures of all possible worlds compatible with both your experiences and the hypothesis, and dividing by the sum of the measures of all possible worlds compatible with your experiences. (Under this interpretation, jessicata’s response doesn’t make much sense; “similar to” means “observationally equivalent for observers with as much information as I have”, and doesn’t have a free variable.)