This small volume presents four papers containing the text of presentations given at an event that took place in 2002, a conference celebrating the 60th birthday of Bruno Buchberger. The theme of the conference was how logic is a form of glue that binds mathematics and computer science together. The papers have been modified and expanded a bit for inclusion in this book, they are:
From these papers, it is clear to all readers that the fields of mathematics and computer science have much to offer each other. Mathematics provides tools for programmers to examine their code and formally analyze it to determine if it will function correctly. Computer science provides tools to mathematicians that allow them to examine data looking for patterns and to process the symbols of mathematics to generate formal, albeit often lengthy, proofs. As Zeilberger points out, mathematics, even something as simple as 2 + 3 = 5, is the processing of symbols that are precisely defined and are manipulated using very specific rules. Those are the things that computers do very well.
The paper by Wolfram is one more instance where he is arguing for his specific “theory of everything”, namely cellular automata. Even skeptics have to admit that it is a theory that shows a lot of promise, where simple rules generate significant complexity.
While these papers are specific in their focus, the intelligent reader is convinced of how important computer science and mathematics are to each other. They form a complex and powerful positive feedback loop new progress in one leads to advancement in the other.
Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing The Journal of Recreational Mathematics. In his spare time, he reads about these things and helps his daughter in her lawn care business.
Henk Barendregt: Foundations of Mathematics from the Perspective of Computer Verification
Manfred Broy: On the Role of Logic and Algebra in Software Engineering
Stephen Wolfram: New Directions in the Foundations of Mathematics (2002)
Doron Zeilberger: Towards a Symbolic Computational Philosophy (and Methodology!) for Mathematics.