Mathematics is studied because it is a rich and interesting discipline, because it provides a set of ideas and tools that are effective in solving problems which arise in other fields, and because it provides concepts useful in theoretical studies in other fields. When used in problem solving, mathematics may be applied to specific problems already posed in mathematical form, or it may be used to formulate such problems. When used in theory construction, mathematics provides abstract structures which aid in understanding situations arising in other fields. Problem formulation and theory construction involve a process known as mathematical model building. Given a situation in a field other than mathematics or in everyday life, mathematical model building is the activity that begins with the situation and formulates a precise mathematical problem whose solution, or analysis in the case of theory construction, enables us to better understand the original situation.

Mathematical modeling usually begins with a situation in the real world, sometimes in the relatively controlled conditions of a laboratory and sometimes in the much less completely understood environment of meadows and forests, offices and factories, and everyday life. For example, a psychologist observes certain types of behavior in rats running in a maze, a wildlife ecologist notes the number of eggs laid by endangered sea turtles, or an economist records the volume of international trade under a specific tariff policy. Each seeks to understand the observations and to predict future behavior. These efforts may be based completely on intuition, but more often they are the result of detailed study, experience, and the recognition of similarities between the current situation and other situations which are better understood. This close study of the system, the accumulation and organization of information, is really the first step in model building. Much of this initial work must be done by a researcher who is familiar with the origin of the problem and the basic biology, economics, psychology, or whatever else is involved.

The next step (after the recognition of the problem and its initial study)
is an attempt to make the problem as precise as possible. One important
aspect of this step is to identify and select those concepts to be considered
as basic in the study and to define them carefully. This step typically
involves making certain idealizations and approximations. The purpose here
is to eliminate unnecessary information and to simplify that which is retained
as much as possible. For example, with regard to a psychologist studying
rats in a maze, the experimenter may decide that it makes no difference
that all the rats are gray or that the maze is constructed of plywood. On
the other hand, it may be significant that all the rats are siblings or
that one portion of the maze is illuminated more brightly than another.
This step of identification, idealization, and approximation will be referred
to as constructing a real model. This terminology is intended to reflect
the fact that the context is still that of real things (animals, apparatus,
etc.) but that the situation may no longer incorporate all features of the
original setting. Returning again to the maze, the psychologist may construct
a *real model* which contains rats and compartments, but with the restriction
that a rat is always in exactly one compartment. This restriction involves
the idealization that rats move instantaneously from compartment to compartment
and are never half in one compartment and half in another.

The third step (after study and formation of a real model) is usually much
less well defined and frequently involves a high degree of creativity. One
looks at the real model and attempts to identify the operative processes
at work. The goal is the expression of the entire situation in symbolic
terms. As a consequence, the real model becomes a *mathematical model*
in which the real quantities and processes are replaced by mathematical
symbols and relations (sets, functions, equations, etc.) and mathematical
operations. Usually, much of the value of the study hinges on this step
because an inappropriate identification between the real world and a mathematical
structure is unlikely to lead to useful results. It should be emphasized
that there may be several mathematical models for the same real situation.
In such circumstances, it may be the case that one model accounts especially
well for certain observation while another model accounts for others. There
may not be a "best" model; the one to be used will depend on the
precise questions to be studied.

After the problem has been transformed into symbolic terms, the resulting
mathematical system is studied using appropriate mathematical ideas and
techniques. The results of the mathematical study are theorems, from a mathematical
point of view, and predictions, from the empirical point of view. The motivation
for the mathematical study is not to produce new mathematics, i.e., new
abstract ideas or new theorems, although this may happen, but instead to
produce new information about the situation being studied. In fact, it is
likely that such information can be obtained by using well-known mathematical
concepts and techniques. The important contribution of the study may well
be the recognition of the relationship between known mathematical results
and the situation being studied.

The final step in the model-building process is the comparison of the results
predicted on the basis of the mathematical work with the real world. The
most desirable situation is that the phenomena actually observed are accounted
for in the conclusions of the mathematical study and that other predictions
are subsequently verified by experiment. In fact, in many situations an
elaborate experiment is designed to determine whether the model gives predictions
consistent with observations. Frequently, the agreement between predictions
and observations is less than desirable, at least not on the first attempt.
A much more typical situation would be that the set of conclusions of the
mathematical theory contains some which seem to agree and some which seem
to disagree with the outcomes of experiments. In such a case one has to
examine every step of the process again. Has there been a significant omission
in the step from the real world to the real model? Does the mathematical
model reflect all the important aspects of the real model, and does it avoid
introducing extraneous behavior not observed in the real world? Is the mathematical
work free from error? It usually happens that the model-building process
proceeds through several iterations, each a refinement of the preceding,
until finally an acceptable one is found. Pictorially, we can represent
this process as in the figure below. The solid lines in the figure indicate
the process of building, developing, and testing a mathematical model as
we have outlined it above. The dashed line is used to indicate an abbreviated
version of this process which is often used in practice. The shortened version
is particularly common in the social and life sciences where mathematization
of the concepts may be difficult. In either case, the steps in this process
may be complex and there may be complicated interactions between them. However,
for the purpose of studying the model-building process, such an oversimplification
is quite useful.

We also note that this distinction between real models and mathematical
models is somewhat artificial. It is a convenient way to represent a basic
part of the process, but in many cases it is very difficult to decide where
the real model ends and the mathematical model begins. In general, research
workers often do not worry about drawing such a distinction. Hence in practice
one frequently finds that predictions and conclusions are based on a sort
of hybrid model, part real and part mathematical, with no clear distinction
between the two. There is, however, some danger in this practice. While
it may well be appropriate to work with the real model in some cases and
the mathematical model in others, one should always keep in mind the setting
that is being used. At best, a failure to distinguish between a real model
and a mathematical model is confusing; at worst, it may lead directly to
incorrect conclusions. Complications may arise because problems in the social,
biological, and behavioral sciences often involve concepts, issues, and
conditions which are very difficult to quantify. Hence essential aspects
of the problem may be lost in the transition from the real model to the
mathematical model. In such cases conclusions based on the mathematical
model may well not be conclusions about the real world or the real model.
Thus there are circumstances in which it is crucial to distinguish the model
to which a conclusion refers.

© Daniel Maki & Maynard Thompson; Indiana University

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