One of the primary roles of the applied mathematician is to create mathematical models to be used in predicting the outcomes of future events. These models take the form of equations or systems of equations, inequalities or systems of inequalities, computer simulations, or any combination of these that the mathematician finds appropriate.

Mathematics can sometimes give accurate predictions about future events as with Newtonian mechanics, whereas in other cases, such as predicting public opinion and estimating animal populations, the predictions given by mathematics are probabilistic. In cases where the underlying dynamical system is chaotic, such as the weather, mathematical models can be used to describe the long-term behavior of the system, but such systems by their very nature defy specific predictions due to their sensitivity to initial conditions.

Predictive Models and Their Validity

The predictive validity of a mathematical model can only be judged by how well it...