Many people think of mathematics as this incredibly austere and abstract subject, and mathematicians as people who are obsessed with abstraction and divorced from reality. It is true that mathematics can be abstract, and some mathematicians are quite proud of the separation between what they see as real mathematics and useful mathematics, cough Hardy cough. To some of them being able to show that an integral is finite, or that a solution exists to a system of equations, is enough. They do not actually feel the need to sit down and evaluate the integral, or solve the equations. To them the existence of the answer is the important thing, not actually constructing it.
However, in my opinion this characterisation of mathematics as purely an abstract approach is an unfair critique. There are plenty of mathematicians who evaluate integrals, construct explicit solutions to systems of equations, and who want some physical, biological, or computer science motivation behind a problem to consider it interesting and worth pursuing. And this is not just true of applied mathematics. Many pure mathematicians also love constructing explicit examples when testing out ideas, such as Dame Alison Etheridge (see here). If you are going to spend a lot of time and effort trying to tackle a problem, you want to know if you are likely to be able to solve it rather than simply banging your head against a brick wall. So we make approximations and simplifications to construct a toy model preserving the key mathematical and physical features but providing a sandbox where we can test our ideas.
Not that kind of toy model…
While there are people who study abstract foundations like quantifying the concept of oneness or, allegedly, spending hundreds of pages to establish that $1+1=2$, many mathematicians are interested in understanding problems motivated by studying the mathematics of fluids, modelling the spread of viruses, or—in my case—effective models of special magnetic materials. As a mathematical physicist, much of what I do is to study mathematical problems and questions motivated by real-world physical systems.
One such model uses a one-dimensional chain of atomic spins to understand magnetic materials. In this case we forget about the structure of an individual atom and just think of it as an arrow that can point in any direction. Next we come up with a rule for how neighbouring atoms interact, eg if they want to point in the same direction we call it a ferromagnet or in opposite directions it is called an antiferromagnet. One approximation that we make is that any given atom only cares about its nearest atoms and not the effect of all the other atoms which are further away. Finally we say how the atoms interact with a magnetic field or if they prefer to line up with a particular direction and off we go. We can understand the transition between different magnetic domains without having to study a complicated model incorporating all the details, and I can calculate this on the back of an envelope without having to touch a computer.
A spin chain of atoms modelling a domain wall in a ferromagnet.
These simplified toy models can come under criticism from two sides. Proponents of austere abstraction will attack them for lacking mathematical rigour or for involving unfounded leaps of logic, while those who are all about applications can bemoan the time spent simplifying a problem to still attack it with, potentially, complicated mathematics. Why not just study it numerically on a computer? There is definitely merit to this argument; many toy models can be and are studied numerically. However, a numerical solution does not convey the same intuition as a more analytic approach. Yes, if I solve a differential equation numerically I can play with the parameters and see how the solution changes. However, this will often not tell us why the change is happening. A good toy model, like our spin chain, can convey this.
To me, a toy model is more than just a simplified setting to play around with some interesting mathematics. In building them you need to think about what the key features are that you want the model to describe, and how they can be described mathematically. A well constructed toy model will be sophisticated enough to preserve the qualitative, and some of the quantitative, features we want, but be simple enough that we can really get our teeth into it. It will enable us to gain intuition about how changing the parameters of the model change the results that you get out. In the best cases, this intuition will help you to understand the full situation. From SIR models in epidemiology to effective field theories in physics, toy models are put to good use in many places.
What about in more abstract settings? Again, we can study a simpler object to help us gain understanding. Want to understand something about groups? Study matrix groups first: you can leverage your linear algebra experience to understand how to formulate the problem and identify potential subtleties. In a way this is what a power series or Fourier series expansion of a function is doing. You have a function with some properties that are easy to understand and others that are harder to understand, and approximating it by a power series gives you something more tractable where you can do some quick computations to check what is going on.
I guess that all I am trying to say is next time someone says that they are studying a simplified situation or playing with a toy model, do not dismiss them out of hand. Remember toy models can convey intuition and understanding beyond their humble formulation. Maybe when you are stuck on a particularly difficult problem you should think about what simplifications you could make and try one out for yourself.
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