This question will be admittedly a bit vague, since I am inquiring about the existence of a theory that I am not sure exists, and if it does exists I have only a vague notion of what it might look like.
At the highest level, I am interested in the abstract study of mathematical modelling. For example, I am interested in answering questions like:
- What is the definition of a mathematical model? (e.g. does every mathematical model depend on certain assumptions or axioms? After the axioms are stated, what else is required for a mathematical model?)
- Given a model, what statements can and cannot be assigned a truth value by the model? (surely this will depend on what the definition of a model is).
- How is a statement interpreted by a model (i.e. what is the property of a model that assigns interpretation to a statement)?
Some branches of Logic (which I have no background in) seem to touch on these issues. I also thought that Model Theory was somewhat close to what I was looking for, but I am more interested in abstract study of mathematical models in physics, economics, etc. (and I am especially interested in statistical models, and the study of counterfactuals and causality within a model), and I found it hard to see how Model Theory would be practical to apply to these cases. Perhaps I am wrong about this last point.
Can anyone point me in the right direction?
A mathematical model is invariably some kind of axiomatization of some kind of structure, which is practically speaking always expressible as an explicit computable many-sorted first-order theory. I would go so far as to say that if one cannot express it in this manner, then one almost certainly does not have a precise model. So I suggest you learn about many-sorted first-order logic and how to express various axiomatizations in it, which would answer all three of your questions. But since mathematical modelling is necessarily a matter of representing real-world observations in symbols, there cannot be a purely mathematical theory of mathematical modelling. And as Noah Schweber said, model theory is a branch of mathematical logic that has very little to do with mathematical modelling (directly at least).
To make it clear, the answers to your questions based on my view:
What is the definition of a mathematical model? Something that can be expressed as a many-sorted first-order theory.
Given a model, what statements can and cannot be assigned a truth value by the model? The axiomatization that captures the model itself may not be complete (i.e. some statements may be neither provable nor disprovable), even if it may seem that every statement is either true or false under the intended interpretation of the axiomatization. Just for example, the theory of concatenation (TC) can be considered a mathematical model of finite binary strings, but it turns out that the incompleteness theorem applies to it, and so no computable extension of TC can ever be complete.
Note also that a first-order theory that expresses a mathematical model may not be strong enough to prove everything we think is true, because we may not have made a complete characterization of the intended structure. For example, if we wish to axiomatize a population model based on differential equations, we may only include some axioms concerning real numbers that we are actually certain are relevant. For example, we might only include the RCF axioms, but notice that the computable reals satisfy RCF!
Finally, in mathematics we often like to look at not just separate mathematical models but rather a single unifying foundational system in which we can express all mathematical models that we are interested in. It turns out that at least for practical real-world applications, higher-order arithmetic suffices nicely, because we can naturally express statements about naturals, reals, real sequences (functions from $\mathbb{N}$ to $\mathbb{R}$), real functions (functions from $\mathbb{R}$ to $\mathbb{R}$), and higher-order functions, and more, and easily reason about them.