Just like the different lenses of a microscope, each of the models for the motion of a ball studies the same problem but at a different level of resolution or complexity in order to answer different questions. As the model complexity increases, the number of equations, variables, and parameters entering their mathematical formulation also increases. The equations generated by such a process increase in complexity and soon enough require a thorough numerical approach. As sketched in Figure 7, depending on the question, models can have different sizes that are roughly measured by the number of parameters and equations appearing in their formulation (a proper mathematical definition of complexity is elusive but most researchers would heartily agree that some models are more complex than others).

Model complexity showing the trade‐off between model sizes, insight, and predictive power. Of course, for some problems the Goldilocks zone will lie further to the left or further to the right.

At one end of the spectrum, a simplistic approach to a problem may lead to a small model that can be easily solved but that still offers key insights. Our model for compound interest and our first model for throwing a ball would both fall into that category. They teach us about the fundamental nature of exponential growth and ballistic trajectories, respectively. However, these simple models typically have low predictive power as they ignore key properties of the system. Our financial model would be useless to predict the market and our Newtonian model would not help the design of new footballs or solve ballistic problems with any precision.

Larger models are more difficult to solve, and typically give less insight into a phenomenon, but when properly formulated and implemented they may have great predictive power. It is in that Goldilocks zone (not too big, not too small!), that modern mathematical modelling thrives as mathematics, computer science, sciences, and engineering come together to solve technological, societal, or scientific challenges.

At the other end of the spectrum, a blind adherence to all known physical processes or an excess in the number of parameters and variables leads to models that may be valid but are so large and poorly characterized that there is no hope of extracting any meaningful information from them. These models are both useless and impossible to solve. For instance, our preposterous fifth model for the motion of a ball may contain the correct physics, but it is completely disconnected from the original problem.

The tug‐of‐war between a model’s complexity and how much information can be extracted from it is a delicate one and requires a good understanding of both the underlying scientific q