In studying math we are, at least partially, interested in making abstraction of real world problems and solving them through rigorous techniques and methods, and then interpreting the result. Let us zoom in to one such problem - Suppose we were to create some kind of a model (perhaps a dynamical system of some sort) to govern a phenomenon we observe in real life. In constructing this model, we create equations to represent quantities and relationships in the system that are of interest to us. Once the model is in place, we go about 'solving' it for relevant quantities and then interpreting from there.
My issue lies in the detachment of the physicality of the problem during the process of solving the problem. When we solve such systems we often use techniques such as complex integration, or abstract matrix or operator manipulation which, as far as I understand, is a necessary level of abstraction that we apply in order to obtain solutions. Its as if we start the problem in the physical realm where the equation has a direct correspondence with the problem, then move to the mathematical realm where we do manipulations. In this moment we (or maybe just I?) lose sight of the actual problem and work only on the math of it, and then we convert back. What happens in this middle area and how am I to rest assured that these abstract manipulations aren't fundamentally changing things or introducing new nuances that I need to care about?
Yes I understand that the math is all quite formal, and we 'trust' it, but its almost feels like a question of scope. While working with the equations I personally often loose sight of the problem until I'm near a solution.
Also, it seems easy to look back and say, ok well our solution to some model gives valid results every time, so of course we can trust this mathematical process, but surely there's a more sound explanation. Perhaps its my personal nearsightedness that's interfering in the process, but regardless I'd like to hear some other peoples thoughts on this.
PS : I recognize that this might be borderline off topic, but its bugged me forever and this seems as good a place as any to ask, please feel free to redirect.