. In the 1990's, the WZ method has been the method of choice in resolving new conjectures for hypergeometric identities. The object here is to compare the WZ method with Pfaff's method. Such a comparison should (it is hoped) provide some suggestions for the further development of each method. Submitted: December 2, 1994; Accepted: November 13, 1995 1. Introduction. The first two papers in this series raise the following obvious question: Why should anyone want to resurrect a method last used in 1797 to sum hypergeometric series when it is well-known that the WZ method [13] has swept all before it. This latter state of affairs has been spelled out in delightful albeit idiosyncratic detail by Zeilberger in [16] and [17]. The reader is urged to consult these references for the complete understanding of his philosophy. Perhaps the case can be put succinctly by referring to his Meta-theorem [16] which asserts that the verification of any binomial coefficient identity via the WZ method is r...

From a philosophical viewpoint, mathematics has often and traditionally been distinguished from the natural sciences by its formal nature and emphasis on deductive reasoning. Experiments — one of the corner stones of most modern natural science — have had no role to play in mathematics. However, during the last three decades, high speed computers and sophisticated software packages such as Maple and Mathematica have entered into the domain of pure mathematics, bringing with them a new experimental flavor. They have opened up a new approach in which computer-based tools are used to experiment with the mathematical objects in a dialogue with more traditional methods of formal rigorous proof. At present, a subdiscipline of experimental mathematics is forming with its own research problems, methodology, conferences, and journals. In this paper, I first outline the role of the computer in the mathematical experiment and briefly describe the impact of high speed computing on mathematical research within the emerging sub-discipline of experimental mathematics. I then consider in more detail the epistemological claims put forward within experimental mathematics and comment on some of the discussions that experimental mathematics has provoked within the mathematical community in recent years. In the second part of the paper, I suggest the notion of exploratory experimentation as a possible framework for understanding experimental mathematics. This is illustrated by discussing the so-called PSLQ algorithm.