13 - A computer algebra system

Summary

Symbolic computation refers to the use of machines, such as computers, to manipulate mathematical content (i.e., equations, expressions, etc.) in symbolic formand deliver a result in this form. A classical example of such a systemis one that can compute the derivative of a function expressed in some symbolic form(i.e., sin(x)+1). In this chapter we will explain why symbolic computation is interesting and what it can achieve, and then we are going to present a simple system that can differentiate functions.

Mechanical symbol manipulation

In the beginning of the twentieth century, the great German mathematician David Hilbert asked whether it would be possible to devise a method to solve mechanically any diophantine equation, that is, any polynomial equation with integer coefficients. In other words, he asked whether it is possible to solve any diophantine equation by dully following a clerical procedure. In fact, Hilbert was dreaming of a fully mechanized mathematical science. Unfortunately for Hilbert it has been shown that one cannot construct a general algorithm to solve any diophantine equation. A consequence of this proof was that the dream of a fully mechanized mathematical science was not feasible. Nevertheless, certain problems of mathematics can be solved by purely mechanical methods. For example, it has been demonstrated that certain operations like differentiation and integration can be performed mechanically. In general, such systems are known as symbolic computation or computer algebra systems.