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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?

möbius
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    SEP's entry on Model Theory has a section dedicated to Models and modelling with a wider scope that the Mathematical logic branch called Model Theory. – Mauro ALLEGRANZA Apr 09 '19 at 14:23
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    Model theory, despite the name, actually has essentially nothing to do with what's normally meant by "mathematical modelling" - it's much better to think of it as a generalization of abstract algebra. – Noah Schweber Apr 09 '19 at 15:17
  • @NoahSchweber Thanks for your comment. Any idea where I can go from here? Any suggestions will be helpful. – möbius Apr 09 '19 at 19:17
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    What's right and what's wrong? Do you want a practical example of mathematical modelling? Here I have a surprising one : Square Bubbles . – Han de Bruijn Apr 13 '19 at 17:54
  • @HandeBruijn No I am not looking for examples of mathematical models (although I found your example interesting). I am looking for a theory of modelling itself. For example, is there a general structure that all mathematical models follow? Admittedly the question is a tad vague, but I felt someone would know what I was referring to if there was an obvious theory out there... – möbius Apr 13 '19 at 19:16

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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.

  • How is a statement interpreted by a model? Since we are talking about capturing some kind of real-world phenomenon or conceptual structure, we would ordinarily interpret a statement according to the structure we had intended to capture. We can also consider other structures that satisfy the same theory but are not isomorphic to the intended one, but that is a matter of the study of logic, rather than a matter of modeling.

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.

user21820
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  • What does the first-order axiomatization look like for, e.g. the Lotka-Volterra predator-prey model? (https://en.wikipedia.org/wiki/Lotka–Volterra_equations) Are there multiplie sorts, or do you default to writing it in set theory with one sort being enough? I'd like to see the axiomatization, with some indication of the first-order proof that some solutions have approximate period $\sqrt{\alpha\gamma}$. –  Apr 16 '19 at 17:53
  • @MattF.: In my personal opinion, there is both an intuitive and pedagogical advantage to working in many-sorted FOL, as opposed to just a one-sorted theory such as ZFC. The reason is that axiomatizations of different basic structures combine nicely. See this post where I provide a Fitch-style system that is based on many-sorted FOL, except that it's slightly easier to use in practice since I permit using any set as a type, instead of having fixed sorts. With that in mind, ... – user21820 Apr 17 '19 at 12:00
  • ... Axiomatizing a DE-based model such as the population model you cited would involve axiomatizing the type $\mathbb{R}$ of reals, and adding constant-symbols $x,y,α,β,δ,γ$ to represent the dynamical system's variables and constants, of course with $x,y∈FN(\mathbb{R},\mathbb{R})$, and adding (first-order) axioms capturing the differential equations. Since we do not need any set theory, we just need the typing rule for function application, namely "If S,T∈set and f∈FN(S,T) and x∈S, then f(x)∈T.". Unsurprisingly, ... – user21820 Apr 17 '19 at 12:05
  • ... you can do the same axiomatization in higher-order arithmetic; in fact just third-order arithmetic will do. However, I presume that most physicists do not think of reals as Cauchy sequences of rationals, much less Dedekind cuts of rationals, or worse still subsets of $\mathbb{N}$ under some encoding. That is why I think that the cleanest way to express that population model is exactly as I stated, where the reals are axiomatized rather than constructed. Note that I did not require axiomatizing the naturals, because that is not technically part of the mathematical model. – user21820 Apr 17 '19 at 12:11
  • (In my second comment above, I should have said that we don't even need the general rule for function application, but merely "If $f∈FN(\mathbb{R},\mathbb{R})$ and $x∈\mathbb{R}$ then $f(x)∈\mathbb{R}$. So we don't even need $\mathbb{R}$ to be a "set"; we need absolutely no set theory.) This does not mean that you can prove everything about a mathematical model from its axioms alone, because you may need more axioms than just what the model says. How much more? Almost every real-world application of mathematics can in fact be proven naturally in higher-order arithmetic, so that's enough. – user21820 Apr 17 '19 at 12:18
  • I'm not sure if that's a satisfying account, but it makes sense -- I often like foundations with the strength of $ACA_0$. But it's worth clarifying in the answer that this axiomatization of the model probably can't prove the major results about the model. –  Apr 17 '19 at 12:29
  • @MattF.: Okay I will add that to my answer. Indeed, I too like to know what is the minimal assumptions needed for something, and ACA0 does suffice for plenty of stuff. Though I do think that predicative higher-order arithmetic is slightly more satisfying because we don't have to encode reals and can also handle arbitrary real functions. – user21820 Apr 17 '19 at 12:35
  • @MattF.: I decided to add more detail as well. – user21820 Apr 17 '19 at 13:10
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    @user21820 Thank you for your hard work on this answer. It answers my question and it has given me lots to think about! – möbius Apr 19 '19 at 10:52
  • @möbius: You're welcome! Feel free to ask if you need any clarification. =) – user21820 Apr 19 '19 at 12:50