vanhees71 said:
The only thing I think about probability theory, successfully applied in statistics as well as theoretical physics for about 150+/- years, is that it works. That's all I need to justify the use of any specific mathematical concept in the natural sciences.
This is almost blatantly an assumption, with little to no foundational support, especially when speaking about QT.
vanhees71 said:
where all these speculations about apparent problems, which are in fact pseudo-problems, are discussed without confusing people interested in science rather than cargo cult
The problems at the edge of knowledge are actually the only problems really worth worrying about. In this case they are distinctly physical problems, which can only become abundantly clear - i.e. to foundations skeptics such as yourself - once a new theory has actually been developed which manages to supersede the old one in some aspects. The production of any new theory which contains QM as a limit and is falsifiable is one of the major contemporary goals of theoretical physics.
The fact that such theories already exist makes the position that QT is unique untenable, regardless of how precise QT matches current experiment. If you don't believe this then you are explicitly overvaluing precision, possibly even in favor of accuracy. After the falsification of Ptolemean epicycle theory, we should all know that having a highly precise theory means exactly nothing if the theory is fundamentally conceptually inaccurate.
The only tried and true methodology to judge the accuracy of a precise theory is by comparing it with another theory; this essentially is the goal of statistical analysis. In other words, either you are assuming QT is the final theory of physics or you are assuming it isn't; if it isn't, then your stance is not scientifically justifiable beyond, being a mere plea for extending and upholding a temporary a facade of certainty about science where there should be genuine doubt and uncertainty.
vanhees71 said:
Probability is a clear defined mathematical concept with clear applications in terms of statistics. Completed by information theory and QT it provides objective assignments of probability distributions of real-world systems.
This might surprise you, but actually no, it isn't as clear cut as it seems as pointed out by many experts. There are several conceptually different notions related to probability, which in the exact sciences are usually immediately interpreted as being probability as described by the standard theory and therefore reduced to standard probability; a few of these concepts are vagueness, uncertainty, likelyhood, possibility and propensity.
Standard probability theory, axiomatized or not (NB: having/making axiomatizations is almost completely irrelevant), is a theory about a particular form of probability which by historical accident we never created an additional monikor for (such as e.g. 'Pascalian probability theory'); despite its official name it is not THE theory of probability in general in exactly the same way that Newtonian mechanics - or QM if you prefer - is not THE theory of mechanics, but merely A theory of mechanics. Statistics itself is not (a) science either, it is a methodology based on making assumptions which may or may not be provable.
Assuming otherwise, as is constantly done in the practice of exact sciences such as physics and engineering doesn't change this any bit. In fact, in many other sciences and other disciplines this fact is widely acknowledged, where the theoretical/mathematical branches of said sciences sometimes use and develop more specialized mathematical and logical tools and methods which are more adequate for their specific purposes.
Incidentally, the justification of such novel tools often end up requiring a complete rewrite of many mathematical foundations, with the developed tool usually ending up being an alternative mathematical construct which better captures the properties of the relevant analysis than standard probability theory does. In fact, we all know the most famous example of such an alternative theory: the complex amplitude calculus from QT. This is only slightly different from standard probability theory, yet is a new mathematical area ripe for research in many directions.