What Is a Mathematical Model?

A term I thought was clear, until I realised it wasn’t

“Mathematical model” is one of those expressions I eventually just internalised.

I used it naturally in conversations to explain a result, motivate an idea, and discuss applications, especially with people outside mathematics, such as friends. Only later did I notice a pattern: I had assumed the concept was clear, but it wasn’t.

People nodded, but the nod was polite. The words landed, but the meaning didn’t.

That’s one of the reasons I decided to write this newsletter.

A model is a representation

In everyday language, a model is something that stands in for something else: a scale model, a sketch, a simplified representation.

A mathematical model works in the same way. It is a deliberate simplification of reality, expressed in mathematics, that allows us to reason about a system without carrying its full complexity at once.

It is not reality itself.

It is not “the truth”.

It is a tool.

What goes into a mathematical model?

At its core, modelling means making choices:

• choosing variables to represent quantities we care about

• specifying relationships between them (equations, functions, rules)

• deciding what to ignore.

That last point is crucial. Every model is built on omissions. A model can feel “wrong” simply because it doesn’t include everything—but including everything is impossible, and usually not even desirable.

What makes a model good is not how much it includes, but whether it accounts for the right things for the question you’re asking.

“Wrong” is not the opposite of “useful”

There’s a famous line by the statistician George Box that I think captures the mindset perfectly:

All models are wrong, but some are useful.

It sounds provocative the first time you hear it, but it’s liberating once you accept it.

A model is not meant to be a perfect copy of the world. It’s intended to be a working approximation, something you can compute with, test, refine, and learn from.

Sometimes a crude model already reveals a key mechanism. Sometimes you need layers of refinement. There is no universally “correct” model—only models that are more or less appropriate for a given purpose.

A model is a choice

This is perhaps the most non-obvious part:

A mathematical model is not found.

It is chosen.

Two people can model the same phenomenon in different ways, emphasising different aspects and addressing different questions. The mathematics comes after that conceptual decision.

Example: coffee cooling

Consider a coffee cup on a kitchen table. Denote with

\(T(t)\)

the temperature of the coffee at time t, and

\( T_{\text{room}}\)

the kitchen temperature. A straightforward model is:

\(\frac{dT(t)}{dt} = -k\,(T(t) - T_{\text{room}}),\,k>0,\)

It says that the instantaneous temperature variation is proportional to the temperature difference between the coffee cup and the room.

This is Newton’s law of cooling. It omits many details but often predicts the “cooling curve” well enough to be useful.

A final connection: models and neural networks

This way of thinking also applies to machine learning.

When we design or choose a neural network architecture, we are not just “throwing data at an algorithm”. We implicitly select a model of reality: deciding what kind of structure we believe the data has, which relationships can be captured, and what should be learned versus what should be constrained.

In that sense, a neural network is not an alternative to mathematical modelling. It is a form of modelling, one in which the structure is more flexible and the equations are learned rather than written explicitly.

If all models are wrong, but some are useful, then the real question becomes: useful for what?

And that question, whether you’re writing differential equations or choosing an architecture, is where modelling actually begins.

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