Regardless of their ultimate interests and careers, students of mathematics ought to understand something about the way in which mathematics is used in applications and the complicated interaction between mathematics and the sciences.  For many mathematicians engaged in pure research, contact with other sciences may be infrequent; they are apt to see their subject as one that is largely self-sufficient, breeding its own subdisciplines and creating its own research problems, with only occasional stimulus from outside.  Others who are more directly involved in neighboring disciplines may find it difficult to agree with this position, and indeed may lay great stress upon the role of the physical sciences as a source for mathematical ideas and techniques.

Both attitudes are of course wrong, because both are incomplete; at the same time, both are at least partially correct.  If there is any basic error of the applied mathematician, it is in an attitude that segregates certain areas of mathematics as being those which "are applied" and dismissing the rest as belonging to the nonuseful arts.  Many mathematicians share with us the belief that any branch of mathematics could become applied overnight in the right hands.

What then is the role of mathematics in the sciences?  Why is it important, and what part does it play in the development of a subject?  We think that mathematics offers the scientist a vast warehouse full of objects, each available as a model for various aspects of physical reality.  The richness and diversity of this supply are central reasons for the importance of mathematics; another is that, along with the objects, mathematics offers a system for using the models, to help raise or answer questions about physical reality, and techniques for exploring the behavior of the models themselves.
 

Physical reality	--->	Physical approximation	--->	Mathematical model	------------>	Predictions	--->	Physical reality
					Mathematical techniques

We start from physical reality (whatever that may be) and proceed from this to what we have called a physical approximation.  To think of a specific situation, suppose that the physical reality is a simple pendulum consisting of a spherical weight mounted at the end of a metal bar and swung from a pivot.  In creating the physical approximation, we may make statements such as the following: "I think we should disregard air resistance, assume the pivot has zero friction, and that the mass of the bar is negligible in comparison to the weight."

At this stage, the next step is to construct (or select) an appropriate mathematical model.  The model could be something as simple as a quadratic equation, or it could be a complicated differential equation.  At the next stage, the scientists or mathematician begins to explore the properties of the model, using mathematical techniques, and to answer specific questions that have been translated from questions about the physical reality or the physical approximation, so that they become meaningful questions about the mathematical model.  The final step, of course, is the comparison of these answers with what is known about physical reality from observation or experiment and the evaluation of the success of the model and the correctness of the scientist's intuition.

It is very important not to confuse the model with reality itself.  A model is most useful when it resemble reality closely, but there will always be aspects of reality that it does not reproduce, and it will always predict behaviors that do not in fact occur.  The skill of the scientist is in knowing how far , and which contexts, to use a particular mathematical model.  One should never expect to have only one correct mathematical model for any aspect for reality.