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No phenomenon can be described mathematically accurately, in the sense that the mathematical model will fully correspond to the physical process. There will always be a certain “delta” discrepancy, about which it will be impossible to say anything exactly. Therefore, if you can come up with a model for any phenomenon, the only question will be the degree to which the model corresponds to the observed data, which will never be perfect. As Wintgenstein wrote, ” What cannot be said must be kept silent.”
Yes, of course, it is mathematically impossible to describe subjective sensations, which are also phenomena for an outsider.�
For example: You see a young mother with a child, and you see that the mother will experience an emotional attachment to the child. This attachment is not described mathematically.
The second class of phenomena that cannot be described mathematically is complex dynamical systems.
For example, the country's economy.
In this type of problem, the problem is as follows: in the time it takes to describe the system, changes occur in the system itself. Therefore, only a simplified approximation can be described mathematically, but not the process itself.
Since I am one of those people who studied physics very closely in my time and also consider myself a determinist, I don't think they can. In my opinion, there are no such phenomena and events in nature that do not obey any physical (natural or other) laws. And since they obey them, they can be described in mathematical terms. Another question is that this device itself was invented by people and it is limited by itself: we may not know, not see, not identify (at least not yet) any dependencies, patterns, connections, etc., but this does not mean that they are not there.
In the book” The Path to Reality”, Roger Penrose gives his philosophical triad: all reality is a special case of physics, all physics is a special case of mathematics, all mathematics is a part of reality. Penrose also says that although he considers his triad to be true in this form, it is precisely philosophical, and he ” allows for a weaker version: part of reality is described by physics and contains mathematics, and so on. So there is no very clear answer to this question, apparently, even in the philosophy of mathematics.