[Math] Modeling in pure math

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We all know that models play a major role in scientific practice. (By "model" here I mean any of various kinds of entities that function as representations or descriptions of real-world phenomena. This includes pictures, diagrams, equations, concrete physical objects, fictional or imaginary systems, etc.) Many models are valuable because they're simpler than their target systems, but they also generate useful intuition, understanding, predictions or explanations about the nature or behavior of those systems.

I'm sure that mathematicians use models in similar ways, for similar reasons. But there's virtually no academic literature (that I'm aware of) about the kinds of models found in pure mathematics, how and why they're used, how modeling practices in math compare to those in the empirical sciences, and so on. (By contrast, philosophers have written a massive amount about models in science.) As a philosopher interested in mathematical practice, this is something I'd like to understand better.

So my question is: What are some cases of mathematicians using models to better understand, predict or explain mathematical phenomena?

A few clarifications about what I'm after:

  1. I'm only asking about models in pure math. That is, the models in question should represent a mathematical object, fact or state of affairs, not an empirical one.
  2. I'm not necessarily or even primarily interested in cases involving model theory. My notion of model is broader and more informal: roughly, any thing M that can be used to give us a better handle on a system of interest S, apart from whether M satisfies some set of sentences in some formal language associated with S.
  3. The models can be (but don't have to be) mathematical objects themselves.
  4. I have no particular preference for elementary vs. sophisticated examples. Happy to see any good clear cases.
  5. It would be nice to see a published source where a mathematician explicitly describes their methods as involving a kind of modeling, but this isn't necessary.

Best Answer

This one is pretty basic: linearization, e.g. replacing a nonlinear dynamic with a linear one allows to study stability and such.

Also, linearization is behind Newton's method and Taylor polynomials give a hierarchy of models.

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