Usage of the word "model" In science we usually say some concept on paper is a "model" of some real-life concept. For example the Bohr model of the atom describes how real-world atoms behave in a particular context or situation and we use that model when we find it useful. So the Bohr model is a "model" of the atom.
But in mathematics it seems to be the other way around? A real life-concept being a "model" of some mathematical concept on paper? For example the natural numbers being a model for the Peano axioms.
Why is it "backwards" like this? Why can't we say the Peano axioms are a model of the natural numbers, or ZFC is a model of some real-life concept we call a "set" and so on?
 A: Personally, I don't think it's backwards at all.
The intuition that worked for me is that a model of a theory is an example of how that theory can work. In real life, a model of some object is usually a simpler thing satisfying the same basic properties as that object. The mathematical analogue is the idea of a model of the theory of a structure: each structure $\mathcal{A}$ has an associated set of sentences $Th(\mathcal{A})$, its theory, and we could (we don't, but we could) say that a "model of $\mathcal{A}$" is a model of $Th(\mathcal{A})$ in the usual sense.
This is not a trivial notion: for almost every $\mathcal{A}$ (specifically: exactly when $\mathcal{A}$ is infinite) there are lots of structures not isomorphic to $\mathcal{A}$ but which are models of $Th(\mathcal{A})$. And indeed we often have situations where $\mathcal{A}$ is complicated but $Th(\mathcal{A})$ has simple models.
So I'd say that the real conceptual shift is realizing that when we speak of (say) "the Bohr model of the atom," we have in the back of our minds some "basic properties" of atoms which we're trying to capture. Making these explicit we see a two step process:


*

*Decide what the basic properties of atoms that we care about are.

*Build stuff satisfying those properties.
We then think of the actual model-building process as that second step, with the first step being something much more fundamental and complicated.
