SOME GRAND THEMES

The previous chapters have illustrated how mathematical modeling can advance our understanding of behavioral and social phenomena. This and the next chapter are concerned with a slightly different topic, computational modeling. What is the difference?

In mathematical modeling, beliefs about a phenomenon are expressed as mathematical statements. These are then further developed in order to understand what the model implies about behavior in the world. Today, computers are often used as part of this enterprise. Either they can be used as tools in order to investigate a specific case, as in the ecological models of predator–prey interactions, or they can be used to connect data to theory, as they are in factor analysis and item response theory.

In computational modeling the beliefs are cast as programs to be executed by a digital computer. The program/model is then run in order to see what it implies. This is called computer simulation. Simulation is used as an alternative to mathematical analysis. Computational modeling has two advantages. It is concrete, and it is often possible to program basic beliefs (i.e., the axioms of a model) that are complex enough so that they defy mathematical analysis. On the other hand, computational modeling has important disadvantages.

Simulation reduces to analysis of specific cases. Trends can be seen but proofs cannot be obtained. Paradoxically, some people seem more disposed to trust something that “came out of the computer” than something proven by mathematical analysis, but in fact, the converse should be true.