The Modeling Process

The diagram below is a schematic representation of the entire model-building process. (It is itself a mathematical model of mathematical model building!)

In the diagram:

1. Represents the natural law which actually governs the behavior of the natural system. This may be some complex combination of chemistry, biology, physics, politics, economics, etc. In general, the natural law may be quite unknowable to us.
   
   
2. Represents formalization, or the building of the mathematical model. This is where the modeler must make the difficult decisions about which parts of the natural system to model closely and which parts to ignore, and then how to assemble those pieces. Deciding how to represent a natural system in mathematical terms is often the most difficult part of the modeling process.
   
   
3. Represents mathematical deduction, where we work within the model using computations, graphing, algebra, etc. to solve a purely mathematical problem. At this stage, we are removed from the messier aspects of physical reality, working instead with mathematical representations of the essential features of the modeled system.
   
   
4. Represents interpretation of the deductions made within the model. If the model is a good one, then the results of the mathematical calculations should say something about the actual behavior of the natural system. If the model's predictions do not match reality, then it may be necessary to refine the model and cycle through the process again.