Frequentists accept all of the math behind Bayes Theorem, but are not Bayesians. The interpretation of probability is a philosophical subject, not something that is a straightforward consequence of the math.
Bayesianism is applied Bayesian probability theory.
Bayesianism is applied probability theory with additional epistemological premises. You can accept the former while rejecting the latter.
Note: I am out of my depth, and simply repeating (probably incorrectly) cached thoughts said by people smarter than me.
The interpretation of probability is a philosophical subject, not something that is a straightforward consequence of the math.
I was under the impression that there are sound decision theoretic and axiomatic justifications for the notion of subjective probability. Also, the sequences themselves provide pretty good justification. And if you want to quibble over the axioms of Cox's theorems, that seems to me squarely within the domain of ...
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