I'm reading through Peter Turchin's 1998 book, Quantitative Analysis of Movement, for a project I'm working on. I found that his second chapter gives a very nice introduction to the reasons why biology depends on formal mathematical models. This is a topic I often review in my courses, so I'll quote some of his discussion.
He lists six objectives for model-building on pp. 33-35, each with some explanatory text. This amounts to a paragraph or so for each reason; I'm only giving one or two sentences of each, with much omitted.
Formal statement of the problem ...The necessity of stating the assumptions of the model is another benefit. A mathematical description of a problem forces one to be very clear about what the different variables and parameters in the model are, and how they are interrelated.
Identifying knowledge gaps ...It may turn out that good quantitative data are available to estimate some functions and parameters but not others, immediately suggesting a focus for the empirical program. When there are many gaps, one has to decide which parameters need to be estimated precisely, and for which parameters ``guesstimates'' will do....
Gaining theoretical insights There is a large class of models that are never intended to be directly confronted with data.... The purpose of such models is to gain insights into possible causal interconnections between various factors and, in general, extend our intuition...
Quantitative tests of theory ...A qualitative prediction allows one to test the theory that generated it, but it does not provide a very strong test. Because there are only a few possible outcomes in a qualitative situation (e.g., factor X will either increase, stay the same, or decrease), the probability that the ``correct'' outcome will happen by chance is correspondingly high. A quantitative prediction, on the other hand, can be a much stronger test of the theory, because it will not only say that X will increase, but how much...
Interpreting the data Sometimes an investigator is motivated not by a desire to test general theory, but by the necessity of measuring some specific quantity [that would be impossible to measure directly]...
Forecasting and prediction ...Forecasting is weaker than prediction, and uses the knowledge of the past behavior of the system to forecast its future state. Forecasting does not necessarily require an in-depth understanding of the system's dynamics, and can be done at the phenomenological level. However, forecasting will most likely fail if the system's dynamics change. I use prediction in its strongest sense: that is, to predict a situation that was not encountered in the past. For example, it may be necessary to generate predictions about how a system's behavior will change as a result of a certain human intervention. Prediction, in general, requires a mechanistic understanding of the system....
I especially appreciate the point about quantitative tests --- one that has eluded many paleontologists who are content with categorical statements that are essentially untestable, because they only assert that something should happen ``regularly'' or ``more often'' than something else.
Also, the final point, about forecasting and prediction, is valuable -- although perhaps idiosyncratic, as I have not seen that distinction made elsewhere. Still, it applies far beyond theoretical biology and into historical science generally. If we consider our state of knowledge about climate change in response to human activity, clearly this is an example where the distinction between forecasting and prediction is relevant. We can have confidence in a prediction only if it entails a suitable understanding of the mechanisms of change in the system, whereas forecasting is accurate only to the extent that we can depend on a uniformitarian assumption -- that the conditions observed in the past followed the same mechanistic relations that will be relevant to the future.
I tend to lecture about genetic models, for which there is a great value in simplicity (point 3), but which may require quite complicated extensions to handle reasonable biological populations (point 2). In that connection, some reasonable people go to extremes of interpretation -- sometimes claiming that the data necessitate some assumption on the basis of a very simplified model, and in other cases claiming that no model can apply to the complex history of the population. It is our task (my task) to determine which factors are important and conceivably affect results, and which will always be too weak to influence the interpretation of the data (point 1). And the end will often be to discover evidence for values in past human populations for which we have no direct means of estimating aside from genetic variation (point 5).
References:
Turchin P. 1998. Quantitative analysis of movement. Sinauer Associates, Sunderland MA.
