1.5 Mathematical Modeling
Mathematical Ideas
Remember the following proverb: All mathematical models are wrong, but
some mathematical models are useful.
Mathematical Modeling
Mathematical modeling involves two equally important activities:
- Building a mathematical structure, a model, based on hypotheses about
relations among the quantities that describe the real world situation,
and then deriving new relations, - Evaluating the model, comparing the new relations with the real world
and making predictions from the model.
Good mathematical modeling explains the hypotheses, the development of
the model and its solutions, and then supports the findings by comparing
them mathematically with the actual circumstances. Successful modeling
requires a balance between so much complexity that making predictions from
the model may be intractable and so little complexity that the predictions
are unrealistic and useless. A successful model must allow a user to consider
the effects of different policies.
At a more detailed level, mathematical modeling involves successive steps
in the cycle of modeling:
- Factors and observations,
- Mathematical structure,
- Testing and sensitivity analysis,
- Effects and observations.
Consider the diagram in Figure 1.4 which illustrates the cycle of modeling.
Steps 1 and 2 in the more detailed cycle are the first activity described above
and steps 3 and 4 are the second activity.