Probability is an application of this aforementioned math, not foundational to it.
The empirical point of view is definitely worth mentioning.
Could you elaborate on what you mean here? Im not sure I understand what you're trying to say.
Well fundamentally probability is a framework for reasoning about the future based on previous events. This framework is analogous to logic but it's separate and even though it's not as clean it's a lot more useful for dealing with the real world.