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What's the difference between "formulas" and mathematical "models"

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A formula is a relation between variables.

A model is a set of simplifications that allow a real-life problem to be treated mathematically, together with the resulting set of formulas obtained.

A simple mathematical model may consist only of a single formula.
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As said above, simple models can consist of even just a single formula.

Given that you mentioned econometrics/stats: the (primarily linear) formulas you learn are frequently matched with a set of assumptions, specifically regarding the distributions from which the variables are drawn (often called the data generating process) in order to arrive at econometric models. This doesn’t have to be “applied” to be an econometric model, but those theoretic econometric models are frequently “applied” by simply saying what the variables and indices in the equation(s) represent.

Paper I was working on yesterday uses linear probability models (i.e. y=a +bx + e, y is a binary) to investigate how the extent of voting rights in the population influences whether business elites choose to run for office and how whether they win. The models become “applied” when I say in the paper, “in equation (4), y is a binary variable equal to 1 if a business man runs, x is the percentage of the pop eligible to vote, etc.”
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A formula is a just a combination of logical symbols in a given language while a model is a mathematical structure that abides by a given set of axioms called it's "theory". The theory typically consists of a finite list of formulas that dictates the rules of the structure.
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To put it simply, a formula is a statement that expresses how mathematical objects are related to each other and a model is a mathematical object that represents something from the real world.
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My non-precise, semi-sarcastic answer is the following.

A formula is an expression relating quantities that is always true, and may or may not be useful.

A model is an expression relating quantities that is not true, but it is useful.

Examples of formulae would be sin(x+y)=sin(x)cos(y)+sin(y)cos(x) (useful) or e=/=-7 (not particularly useful).

An example of a very useful model would be F=ma, but we know it's not actually true.

Really, the point is that formulae are relations between mathematical structures, and mathematics only accepts provably true statements (vaguely speaking, logicians don't @ me).

Models are relations between real world structures, formalised by quantifying them as mathematical structures and defining mathematical relations between them (you, as the scientist, have to decide these relations). Once you have these mathematical structures and relations, you can resort to pure mathematics to derive further properties and relations between them.
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In logic the terms "formula" and "model" have a bit of a different meaning.

A formula is a syntactic expression, an member of the set of words of a formal language. For example "∀x. x \*x >= 0" (For all x, x times x is greater than 0) is a well formulated formula (wff) in a first order language with signature (>, \*, 0)
However, without a model, a formula has no meaning and cannot be assigned a truth value, it is just a meaningless symbolic expression.  
A model is an interpretation under which a formula is true.
"∀x.x\*x >= 0" would be true in the real numbers or the natural numbers but not in the Complex Numbers.
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Most of math is basically 'build a machine that behaves like our observations or intuition' (depending on if you're doing applied or pure math, respectively)^[1].  The 'machine' is abstract, not literal, and is usually called the 'model' of the process we observe or intuit and want to simulate.

In this understanding, the nearest analog is the Lego Set. Every Lego set is made up of many bricks and parts, some bricks behave in different ways to one another (e.g., in Automata Theory you'll have a lot of production-rule 'bricks', but in Stats you'll have a lot of formulae and special variables, and in Trig you have a lot of identities -- all bricks, different flavors). As you build a new model out of bricks, you may need to borrow from other sets to help make part of your new machine work. Simple machines can be deep and complex if you use the right bricks, and big and complicated machines can be understood as a collection of simpler parts. The model, ultimately, is like the Lego model on the front of the box, it's your goal, but it's not any one part of the set, it's the sum of all the parts.



[1] Don't @ me Pure math people.
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A model is an explanation of a real-world phenomenon. A formula is the mathematical equation that allows us to use that model.
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Imo, a model is the representation of a situation whereas a formula shows the inherent similarity between 2 statements.

A simple profit model for a business will be 20% of sales. That is y=0.20x where x is sales and y is profit. Here, the lhs = rhs simply because you believe it so.

Whereas the first derivative of sin x wrt x is cos x. Here, the equality of the LHS and the rhs is an objective fact regardless of what you believe, and that's how I believe formulae vary from models.

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