What is a mathematical model?

As a mathematician who focuses his attention on a field called dynamics, I am often asked when queried about my area of specialty, exactly what is a dynamical system? I usually answer something like: “I study the mathematics underlying what is means to model something mathematically.” And this seems to work as most people have a basic understanding that mathematics is used in science and engineering to model either a physical or an abstract process and to mine it for information. But thinking a little deeper, there is a better question to ask: What exactly is a mathematical model?

Every time we attempt to understand some new phenomenon or idea that may be quantifiable, our first and very natural pass at comprehension is to compare its values, behavior, and limits to something we already understand or at least have control over. Time is a common concept to measure our new phenomenon against, as in how is it changing as time passes? But we can use any known quantity with which to compare our new idea. Think of drug efficacy by dosage, say, or population growth by population size. This comparison comes in the form of a relation tying together values of our new phenomenon to values of something we already know. And when this relation between our newly quantified concept and something we already have control over is functional (meaning to each value of our known quantity, there is at most only one value of the new one), we can use our known quantity to discover, play with, and/or predict values of the new variable via studying the properties of the relationship or function.

The idea of a functional relationship tying together the values of two measurable quantities, one of which we know and the other we want to know more about, is, in essence, a mathematical model.

Sometimes, the input and output variable values can be discrete (individual real numbers with gaps between values), or continuous (like an interval of real numbers), and the properties of the functions, as mathematical models, will reflect this. In mathematics, sets of numbers (collections of valid input and output variables, the known and newly studied phenomena, respectively) and functions between them are part of the fundamental building blocks of all of our mathematical structures. We structure the vast majority of our thought processes around the functional relationships between quantifiable phenomena.

In one of these functional relationships between two quantified entities, indeed, in a mathematical model, we can vary the values of the input variable from one value to the next or to the previous one, as a means to study the function’s properties. Studying the properties of a model (a function) in this fashion is something a mathematics student begins to do at a basic level in what we call calculus, or the “calculus of functions of one independent variable,” and at a higher level in areas like analysis and topology.

We tend to also use the properties of functions (models) often without really being aware. We understand somewhat intuitively that the warmest part of a day is about two thirds of the way through the daylight hours, linking temperature to time over a day. We also know that two aspirin are more effective at pain relief than one, but intuitively understand that there is probably a maximum effective dosage that is safe before deleterious effects kick in, whether or not we choose to test the theory.

But mostly, the central power of a mathematical model, as a functional relationship between two measurable quantities, one known and one studied, is in its ability to predict, uncover, or extrapolate trends in the new quantity. And here is where my field of choice in mathematics becomes relevant: functions between quantities contain dynamical information. If we apply a function to a set, allowing its output to be reused as an input, over and over again, we can uncover properties of the function (and sometimes also of the set) by watching where individual inputs go upon repeated application of the function. This idea, iterating a function on a set (the discrete version) or using calculus to write a model as a differential equation (the continuous version), is what we call a dynamical system. In such a dynamical system, we often call the numbers that represent the iterates, or the input variable in a differential equation, the time variable, due to its common interpretation as actual time in models of science, engineering and technology. However, there is no real compelling reason why in general. But underlying both a function and its iterates (a discrete dynamical system) or a system of ordinary differential equations (the continuous one), is the idea of a function whose input and output variable values come from the same set of possibilities. So a dynamical system is the mathematical discipline that studies the structure of mathematical modeling. And a mathematical model is simply a function.

Forming and studying functional relationships to understand new things? In mathematics, this is called modeling. And in real life?

Featured image credit: ‘Koch curve’ by Fibonacci. CC BY-SA 3.0 via Wikimedia Commons.