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I really appreciate Andrew Gelman's way of defining 'probability' that transcends the frequentist/Bayesian schism: Probability is a mathematical concept encoded in the Kolmogorov axioms. That's it. No need to argue over a canonical interpretation. It's just math!
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This is how it’s taught in math depts afaik... sigma algebras and things. But when I audited a class set up this way, the prof listed some philosophically distinct foundational alts: one was viewing it as an empirical discipline based on experiments rather than axioms
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Math classes are taught backwards in that once a subject becomes mature enough, it redefines itself around the core definitions and cuts its mooring to its past. This is nice for actually *doing* math, but less so for understanding it.
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Off the top of my head, these subjects have done this: - group theory - algebra - category theory - topology (biggest offender) - probability - dynamical systems - algebraic topology Logic and set theory are mostly free of this just because they're meant to question foundations
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As for alternative definitions of probability, Cox came up with a longer set of axioms that Jaynes lays out in his book. They're equivalent to Kolmogorov but motivate conditional probability and the connection to logic far better. They're a pain to remember, but easy to believe.