Probability is a mathematical discipline whose aims are akin to those, for example, of geometry of analytical mechanics. In each field we must carefully distinguish three aspects of the theory:
(a) the formal logical content,
(b) the intuitive background,
(c) the applications.
The character, and the charm, of the whole structure cannot be appreciated without considering all three aspects in their proper relation.
All possible definitions of probability fall short of the actual practice.