What is a mathematical model? | Science - OUPblog

"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?" argues Richard Brown, Teaching Professor and the Director of Undergraduate Studies in the Mathematics Department at Johns Hopkins University.

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

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.
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Recommended Reading

A Modern Introduction
to Dynamical Systems

This text is a high-level introduction to the modern theory of dynamical systems; an analysis-based, pure mathematics course textbook in the basic tools, techniques, theory and development of both the abstract and the practical notions of mathematical modelling, using both discrete and continuous concepts and examples comprising what may be called the modern theory of dynamics.

Prerequisite knowledge is restricted to calculus, linear algebra and basic differential equations, and all higher-level analysis, geometry and algebra is introduced as needed within the text...

Source: OUPblog