Some have lamented that it is difficult to distinguish randomness from probability, but I seem to be having a contrary difficulty: are they even logically dependent notions?
While introductions to Statistics abound that will take for granted a relationship between probability and the notion of randomness, I am finding this relationship to be more optional as I read into the foundations of probability theory. It seems that probability theory doesn't require a notion of randomness to satisfy the axioms of Measure Theory and Kolmogorov. Some disciplines, such as Ergodic Theory, even compute expectations (which involve probability measures) on purely deterministic systems.
Randomness itself doesn't appear to be a precise notion. Most often I see definitions similar to "the property of being unpredictable". One can notice that some systems can be extrinsically random in the sense that there are models that reduce previously-unpredictable phenomena into a pseudorandom process. According to Chaos theory, this can simply be a matter of knowing the governing equations and the relevant initial conditions with enough precision. Perhaps aspects of Quantum Mechanics are instrinsically random, but even that is not a given in my view.
The closest I have found to a formal notion of randomness is information entropy, which itself goes back to being a consequence of probability theory.
All it seems to take is complexity resulting from not knowing the state-transition rules or the state of a system and we seem to get randomness.
In summary, (1) probability theory doesn't seem to need a notion of randomness and (2) it is not clear whether randomness is anything more than limited predictability when relevant details are missing. At a glance these notions seem inseparable, but after a long gaze they seem to be at-most in a casual relationship.
Who can I read that really cuts into the meat of this problem?