TABLE OF CONTENTS The mathematical scientist models natural phenomena by translating experimental results and theoretical concepts into mathematical expressions containing numbers, variables, functions, and operators. Then, using accepted methods of mathematical reasoning, these expressions are carefully manipulated or transformed into other expressions that reveal new knowledge about the phenomenon being studied. This mathematical approach to understanding the world has been an important component of the scientific method in the physical sciences since the time of Galileo and Descartes. Following in the footsteps of these scientists, Isaac Newton used this approach to formulate an axiomatic, quantitative description of the motion of objects. By using mathematical reasoning, he discovered the universal law of gravitation and derived additional laws that describe the motion of the tides and the orbits of the planets. Thus the science we call mechanics was born, and the technique of manipulating and transforming mathematical expressions was firmly established as an important tool for discovering new knowledge about the physical world.
In the past fifty years, the computer has become an indispensable experimental tool that greatly extends our ability to solve mathematical problems. Mathematical scientists routinely use computers to obtain numerical and graphical solutions to problems that are too difficult or even impossible to solve by hand. But computers are not just number crunchers. In fact, at a basic level, computers simply manipulate symbols (0s and 1s) according to well-defined rules, and it is natural to ask what other parts of the mathematical reasoning process...
