Mathematical Treatment of the Axioms of Physics. The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which already today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics.
While Probability certainly involves some conceptually extra idea (relative to the rest of Mathematics), the issue was whether Probability required some new technical ingredient to be added to the rest of Mathematics. Kolmogorov's achievement was the realization that it didn't. Measure theory had been recently developed to resolve the technical conflict between the intuitive idea "every region in the plane has some area" and the axioms of set theory dealing with every subset of an uncountable set. This conflict has no conceptual connection with Probability, but Kolmogorov realized that the technical machinery (involved in its resolution) of measures, measurable sets, measurable functions could be reused as an axiomatic setting for Probability. In retrospect, because one special model within Probability is "pick a uniform random point from the unit square", it is clear that any general theory of Probability has to include measure theory, but (to reiterate) Kolmogorov's achievement was the realization that at the technical level it didn't require anything more.
With agreed axioms, mathematicians happily moved on with systematic development of theorem-proof Probability. The firm connection to the rest of theorem-proof Mathematics enabled researchers to use tools from other fields of Mathematics, particularly in the context of limit theorems. More prosaically, it is helpful to have coherent notation covering both discrete and continuous probability distributions and random variables.
casinos and insurance companies can and sometimes do go bankrupt, but not from conceptual inapplicability of the mathematics of Probability.But showing that a theory works in some contexts is not evidence that it will work in all contexts. Mathematicians tend to regard the axioms as self-evidently true, while admitting that in practice one might not be able to apply the resulting mathematics (see below). But this reduces to saying "it works when it works", which is rather vacuous. My own opinion is that the fundamental "philosophical" question is
in what contexts is it both possible and useful to try to assign numerical probabilities to uncertain eventsbut I don't claim to have a good answer.
Returning to the Kolmogorov axioms, in my mind there are three issues with real-world applicability. First
There has been ongoing work over many decades on genuinely different setups for thinking about uncertainty, and here are some brief comments, but in the contexts of modeling real-world phenomena (outside the quantum setting) I have never seen a convincing use of such alternatives.
The Kolmogorov axioms are technically useful in providing an agreed notion of what is a completely specified probability model within which questions have unambiguous answers. This eliminates cases like Bertrand's paradox which is simply an ambiguously defined model. But they encourage both a false sense of security (that the act of formulating a model within tbe mathematical framework somehow guarantees it is a valid representation of the real world phenomenon) and a narrowness of vision (that aspects of the real world that cannot be formulated within the framework are somehow "not probability").
Here is one illustration. Charles Dodgson (better known as Lewis Carroll) posed the "pillow problem": If an infinite number of rods be broken: find the chance that one at least is broken in the middle. Since Kolmogorov, mathematicians interpret this in a particular way which happens to give a different answer (0) than does Dodgson's interpretation (1 - 1/e). But a mathematician who thinks 0 is the "correct" answer illustrates my "false sense of security" point -- that answer depends on several particular conventions, e.g. ignoring the atomic theory of matter by modeling the rod as a continuum. Invoking usual conventions encourages you to overlook the main issue, that the problem has no more real-world meaning than a question about fairies and unicorns.