The Borwein brothers, Jonathan and Peter, are known for their work in experimental and computational mathematics. This collection of 14 mostly expository articles published in the last twenty-five years (mostly in the last ten) makes their work accessible to a wide audience. The MAA is well-represented with three Monthly papers. There are three AMS articles and two from SIAM Review. Topics range from the computation of π to the implications of experimental mathematics for the philosophy of mathematics.
Each article has a newly written short introduction placing it in context. One of these refers to Freeman Dyson’s characterization of some mathematicians as birds surveying from afar, and others as frogs living in the mud below and delighting in the details. Frogs will be at home here.
The first paper shows the relationship between the Gaussian arithmetic-geometric mean iteration and the fast computation of elementary functions. An amazingly efficient algorithm for calculating π is a by-product. Requiring only calculus, this paper (and the others) would make fun study projects for math clubs and could combine students’ interest in computers with intriguing mathematics.
Several articles thought-provokingly explore the relationship between experimental mathematics and proof. Jonathan Borwein states that his goal is insight and gives examples of how computation aids insight. He talks interestingly about what constitutes secure mathematical knowledge and in one article gives eight roles for computation illustrating each with an example. He suggests that a constructivist educational curriculum is both possible and highly desirable and includes a reference to a commercial middle-school implementation.
These articles are fun to read, recreational and educational, useful for high-school through professional levels. It is a wonderful collection.
Art Gittleman (firstname.lastname@example.org) is Professor of Computer Science at California State University Long Beach.