Your butterfly formalism strikes me as a good description of what an “objective” probability is (and what ‘frequentists’ actually mean). The problem with the ‘frequentist view’ is best illustrated by your own example:
A coin has a 50% chance of landing heads because if you flip it 100 times, close to 50 of the flips will be heads. In contrast with Bayesianism, the frequentist view is perfectly objective: the limit of a ratio will be the same no matter who observes it.
In some sense, nothing is objective, there is only more and less objective. But throwing a die under carefully set up conditions (like in the casino game craps) gets you pretty close to an “objective” probability that multiple humans can agree on.
Your butterfly formalism strikes me as a good description of what an “objective” probability is (and what ‘frequentists’ actually mean). The problem with the ‘frequentist view’ is best illustrated by your own example:
Saying something is 50% likely because it happens 50% of the time is valid, but it does not actually refer to any real phenomenon. Real coins thrown by real people are not perfectly fair, because angular momentum is crucial, if you let the coin land on a flat surface.
In some sense, nothing is objective, there is only more and less objective. But throwing a die under carefully set up conditions (like in the casino game craps) gets you pretty close to an “objective” probability that multiple humans can agree on.