To clarify what I am asking, it is best to make an analogy with another mathematical discipline: geometry.
In old days there was no clear separation between mathematics and real world science e.g. physics. On the other hand, euclidian geometry was usually considered as absolutely true in our physical world - Kant has believed this ...
Then came Lobachevski / Bolyai (and Gauss unpublished) ... it became more and more clear that there "exist" several geometries (incompatible with one another) - existence meaning roughly consistency ... so one could consider 1. a pure math approach and 2. the geometry of the physical world. In 1. you had first two then an infinity of geometries (Riemann manifolds) and then math. made itself free of the limitation to at most 3 dimensions, finally there could be infinte-dimensional space (Hilbert spaces are the ones most similar to classical euclidian geometry) but few thought 2. could have a question (on that side Euclide was still absolutely right for most scientists)
Then came Einstein with his general relativity: the physical world was no more exactly euclidian - near heavy objects (huge dense stars etc) even wildly non-euclidian. And Hilbert spaces came into physics via quantum mechanics ... but not replacing 3D, so now there is also more than one geometry used in physics.
So geometry is now two things: 1) the math. geometry subjects 2) several applications to the real world, using math. spaces as models AND that is the important point for my question SCIENCE about those geometries that physic theory & experience / observation describe as existing in real world.
But what are the analogues concerning probability? On one side we have a well-developed math. theory - essentially part of measure theory, the measures concerning are only restricted slightly: positive and with total mass 1 - every thing else that general measure theory doesn't treat being only there because of practical interest or due to special aspects that would not be nice where the restriction is not be made (things about stat. independance, law of large numbers, normal / the central limit theorem and the like down to Markov chains / factor analysis etc.) ... and what is on the other side? Mainly literature about avoiding fallacies and philosophy about what probabilities really MEAN alias how they have to be evaluated in real world (= the interpretations). But it seems to me: apparently NO SCIENCE at all! (This is BTW the reason why probability is often considered as math' only.) My idea is that this is largely due to the fact that all results of the theory are calculations of probabilities of complicated combinations so that no assertion is falsifiable by experiment / observation (as usual in real world science) in a strict sense - in principle even when there is a probability 0 for a not impossible event / 1 for an uncertain event (this being anyway rather abstract due to limited precision of all measurements and the impossibility to really produce infinite trial series ...)
What I am mainly wondering about is: do some people consider that there IS in fact real science there or will be in some (far?) future? In the first case (it exists already) what is it?