Books on Experimental Mathematics

Bailey and Borwein have written several references works on experimental mathematics, which we feature here. As other books are published in this general area, we will feature them here also.

• Exploratory experimentation in mathematics: Selected Works. David H. Bailey and Jonathan M. Borwein have collaborated on the subject of experimental mathematics for a quarter of a century. This book collects sixteen articles written together or separately and with coauthors. These works reflect Bailey and Borwein's work on and their views about the changing face of computer-assisted "high-performance mathematics". PSIPress.
• Experimental and computational mathematics: Selected writings. A quiet revolution in mathematical computing and scientific visualization took place in the latter half of the 20th century. These developments have dramatically enhanced modes of mathematical insight and opportunities for "exploratory" computational experimentation. This volume collects the experimental and computational contributions of Jonathan and Peter Borwein over the past quarter century. PSIPress.
• Experimentelle Mathematik: Eine beispielorientierte Einführung. German translation of The Computer as Crucible: An Introduction to Experimental Mathematics. Springer.
• The Computer as Crucible: An Introduction to Experimental Mathematics. Keith Devlin and Jonathan Borwein, two well-known mathematicians with expertise in different mathematical specialties but with a common interest in experimentation in mathematics, have joined forces to create this introduction to experimental mathematics. They cover a variety of topics and examples to give the reader a good sense of the current state of play in the rapidly growing new field of experimental mathematics. The writing is clear and the explanations are enhanced by relevant historical facts and stories of mathematicians and their encounters with the field over time. AK Peters | Amazon.com
• Mathematics by Experiment: Plausible Reasoning in the 21st Century (Second Edition). "Experimental mathematics" means using advanced computing technology as a "laboratory" for mathematical research. This book presents the rationale and historical context of experimental mathematics, and includes a series of examples that best portray the experimental methodology, together with some of the numerical techniques used in this research. Numerous historical and biographical notes are also included. Amazon.com
• Experimentation in Mathematics: Computational Paths to Discovery. Following the lead of the first volume, this book gives numerous additional case studies of experimental mathematics in action, ranging from sequences, series, products, integrals, Fourier series, zeta functions, partitions, primes and polynomials. Some advanced numerical techniques are also presented. AK Peters | Amazon.com
• Experiments in Mathematics This is a CD-ROM with a hyperlinked, searchable PDF files of the above two books. This is the first time a major work of mathematics has appeared in this format. Amazon.com
• Experimental Mathematics in Action. This new book, published in May 2007 by AK Peters, is the companion volume to the MAA Short Course conducted at the San Antonio AMS-MAA meeting in Jan 2006. The authors are David Bailey, Jonathan Borwein, Neil Calkin, Roland Girgensohn, Russell Luke and Victor Moll. Each chapter presents some case studies of experimental mathematics being used in real problems of mathematical research, both pure and applied:
  1. Jonathan M. Borwein, "A Philosophical Introduction".
  2. David H. Bailey, "Algorithms I"
  3. David H. Bailey, "Algorithms II"
  4. Russell Luke, "Inverse Scattering"
  5. Roland Girgensohn, "Strange Functions"
  6. Neil Calkin, "Factoring Integers"
  7. Victor Moll, "Integrals"
  8. Jonathan M. Borwein, "A Computational Conclusion"
  9. "Exercises"
  AK Peters | Amazon.com

Other books of interest to the experimental mathematics community

• Jonathan M. Borwein, E. M. Rocha, J. F. Rodrigues (eds), Communicating Mathematics in the Digital Era. The digital era has dramatically changed the ways that researchers search, produce, publish, and disseminate their scientific work. These processes are still rapidly evolving due to improvements in information science, new achievements in computer science technologies, and initiatives such as DML and open access journals, digitization projects, scientific reference catalogs, and digital repositories. This book is a collection of contributions by key leaders in the field, offering the paradigms and mechanisms for producing, searching, and exploiting scientific and technical scholarship in mathematics in the digital era. Available at: A K Peters
• Bonnie Gold and Roger A. Simons, Proof and Other Dilemmas: Mathematics and Philosophy. During the first 75 years of the twentieth century almost all work in the philosophy of mathematics concerned foundational questions. In the last quarter of the century, philosophers of mathematics began to return to basic questions concerning the philosophy of mathematics such as, what is the nature of mathematical knowledge and of mathematical objects, and how is mathematics related to science? Two new schools of philosophy of mathematics, social constructivism and structuralism, were added to the four traditional views (formalism, intuitionalism, logicism, and platonism). The advent of the computer led to proofs and the development of mathematics assisted by computer, and to questions of the role of the computer in mathematics. This book of 16 essays, all written specifically for this volume, is the first to explore this range of new developments in a language accessible to mathematicians. Available at: Amazon
• Fernando Villegas, Experimental Number Theory. This graduate text, based on years of teaching experience, is intended for first or second year graduate students in pure mathematics. The main goal of the text is to show how the computer can be used as a tool for research in number theory through numerical experimentation. The book contains many examples of experiments in binary quadratic forms, zeta functions of varieties over finite fields, elementary class field theory, elliptic units, modular forms, along with exercises and selected solutions. Sample programs are written in GP, the scripting language for the computational package PARI, and are available for download from the author's website. Available at: Oxford University Press

Book Reviews

1. Review of Experimental Mathematics in Action by Andrew Odlyzko in American Mathematical Monthly, vol. 118 (Dec 2011), pg. 946-947: PDF
2. Review of Mathematics by Experiment: Plausible Reasoning in the 21st Century (second edition), in Mathematical Reviews, MR2473161, by Michael Otte: PDF
3. Review of The Computer as Crucible in Gazette of the Australian Mathematical Society, vol. 36, no. 9 (July 2009), pg. 208: PDF
4. Review of The Computer as Crucible in Current Reviews for Academic Libraries (CHOICE), vol. 46 no 9 (May 2009): PDF
5. Review of Experimental Mathematics in Action in Zenmtralblatt-Math 1127: PDF
   We agree with the reviewer that in some cases more historical references would have been appropriate: but the book was written as a short course and more detailed references were always available in articles cited in the book.
6. Review of Experimental Mathematics in Action in Gazette of the Australian Mathematical Society, vol. 34, no. 5 (Nov 2007), pg. 290: PDF
7. BookNews.com review of Experimental Mathematics in Action (see pg. 16): PDF
8. MAA Online book review of Experimenal Mathematics in Action: HTML
9. Experimental Math in Action is included by Science News in a listing of "new and notable" science books (Aug 2007): PDF
10. ACM Computing Reviews review of Mathematics by Experiment, Experimentation in Mathematics, and CD-ROM version (2 Mar 2007): PDF
11. Review of Mathematics by Experiment and Experimentation in Mathematics in Notes of the Canadian Mathematical Society, by Richard Crandall (vol. 38, no. 4, 2006): PDF
12. Review of Mathematics by Experiment and Experimentation in Mathematics in Notices of the American Mathematical Society, by Jeffrey Shallit (Sep 2005 issue): PDF
13. Review of Expermentation in Mathematics in Math Reviews, by Frits Beukers (May 2005): PDF
14. Article in Pour La Science (French), by Jean-Paul Delahaye (Apr 2005): PDF
15. Review of Mathematics by Experiment and Experimentation in Mathematics in Gazette of the Australian Mathematical Society, vol. 32, no. 3, pg. 199 (2005), by Andrew Rechnitzer: PDF
16. Review of Mathematics by Experiment and Experimentation in Mathematics in SIAM Review, by Reuben Hersh and Robin Frye (vol. 47 no. 4, 2005): PDF
17. MAA online book review of Mathematics by Experiment and Experimentation in Mathematics, by David P. Roberts (2005): HTML
18. Review of Mathematics by Experiment and Experimentation in Mathematics in American Scientist, by Doron Zeilberger (Mar-Apr 2005 issue): HTML
19. Review of Mathematics by Experiment in Mathematics Magazine, by Paul J. Campbell (2005): HTML
20. Zentralblatt review of Mathematics by Experiment and Experimentation in Mathematics, by Gunter M. Ziegler (2004): PDF
21. Review of Mathematics by Experiment in Math Reviews, by John H. Mason (Nov 2004): PDF
22. Review of Mathematics by Experiment and Experimentation in Mathematics in European Mathematical Society Review, by Ivan Netuka and Vladimir Soucek (Sept 2004): PDF
23. Citation in the National Academy of Science report "Getting Up to Speed: The Future of Supercomputing" (June 2004): PDF
24. Berkeley Science Review article, by Padraig Murphy (June 2004): PDF

Supplementary Information for the Books

1. A "Reader's Digest" version of "Mathematics by Experiment" and "Experimentation in Mathematics" is freely available here: PDF
2. Citation errors in Mathematics by Experiment
3. Ten Sample Problems of Experimental Math
4. Errata in Mathematics by Experiment, first edition
5. Errata in Mathematics by Experiment, second edition.
6. Errata in Experimentation in Mathematics
7. Errata in Mathematics by Experiment and Experiments in Mathematics (CD-ROM edition)
8. Errata in Experimental Mathematics in Action

Author Websites

1. Borwein's website: http://www.carma.newcastle.edu.au/~jb616/
2. Bailey's website: http://crd.lbl.gov/~dhbailey
3. Girgensohn's email: zzzzzz@mytum.de. To obtain the real address, replace "zzzzzz" by "girgen".