The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems. Can We Trust in Them?

Abstract

Introduction Nowadays, mathematicians often use a computer algebra system as an aid in their mathematical research; they do the thinking and leave the tedious calculations to the computer. Everybody “knows” that computers perform this work better than people. But, of course, we must trust in the results derived via these powerful computer algebra systems. First of all, let us clarify that this paper is not, in any way, a comparison between different computer algebra systems, but a sample of the current state of the art of what mathematicians can expect when they use this kind of software. Although our example deals with a concrete system, we are sure that similar situations may occur with other programs. We are currently using Mathematica to find examples and counterexamples of some mathematical results that we are working out, with the aim of finding the correct hypotheses and eventually constructing a mathematical proof. Our goal was to improve some results of Karlin and Szegő [4] related to orthogonal polynomials on the real line. The details are not important; this is just an example of the use of a computer algebra system