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Pfaff's Method (III): Comparison with the WZ Method
 Elec. J. Combinatorics
, 1996
"... . 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 meth ..."
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. 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 wellknown 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 Metatheorem [16] which asserts that the verification of any binomial coefficient identity via the WZ method is r...
Exploratory experimentation in experimental mathematics: A glimpse at the PSLQ algorithm
"... From a philosophical viewpoint, mathematics has 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, in the ..."
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From a philosophical viewpoint, mathematics has 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, in the
Chapter 2 Fourier Series and Integrals
"... Having contested the various results [Biot and Poisson] now recognise that they are exact but they protest that they have invented another method of expounding them and that this method is excellent and the true one. If they had illuminated this branch of physics by important and general views and h ..."
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Having contested the various results [Biot and Poisson] now recognise that they are exact but they protest that they have invented another method of expounding them and that this method is excellent and the true one. If they had illuminated this branch of physics by important and general views and had greatly perfected the analysis of partial differential equations, if they had established a principal element of the theory of heat by fine experiments... they would have the right to judge my work and to correct it. I would submit with much pleasure... But one does not extend the bounds of science by presenting, in a form said to be different, results which one has not found oneself and, above all, by forestalling the true author in publication.
ComputerAssisted Discovery and Proof
, 2007
"... With the advent of powerful, widelyavailable mathematical software, combined with everfaster computer hardware, we are approaching a day when both the discovery and proof of mathematical facts can be done in a computerassisted manner. This article presents several specific examples of this new pa ..."
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With the advent of powerful, widelyavailable mathematical software, combined with everfaster computer hardware, we are approaching a day when both the discovery and proof of mathematical facts can be done in a computerassisted manner. This article presents several specific examples of this new paradigm in action. Kronecker said “In mathematics, I recognize true scientific value only in concrete mathematical truths, or to put it more pointedly, only in mathematical formulas. ” I would rather say “computations ” than “formulas, ” but my view is essentially the same. — Harold Edwards, Essays in Constructive Mathematics, 2004 All truths are easy to understand once they are discovered; the point is to discover them. — attributed to Galileo Galilei 1
What’s experimental about experimental mathematics?
, 2008
"... 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, dur ..."
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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 computerbased 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 subdiscipline 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 socalled PSLQ algorithm.