Mathematics by Experiment, 2nd Edition: Plausible Reasoning in the 21st Century [Hardcover]

Jonathan Borwein (Author), David Bailey (Author)

Book Description

ISBN-10: 1568812116 | ISBN-13: 978-1568812113 | Publication Date: December 4, 2003

This new approach to mathematics---the utilization of advanced computing technology in mathematical research---is often called experimental mathematics. The computer provides the mathematician with a "laboratory" in which she can perform experiments---analyzing examples, testing out new ideas, or searching for patterns. This book presents the rationale and historical context of experimental mathematics, and includes a series of examples that best portray the experimental methodology. For more examples and insights, the book, "Experimentation in Mathematics: Computational Paths to Discovery" is a highly recommended companion.

Editorial Reviews

Review

" Whether you take sides in the debate about the validity of experimental mathematics or are not sure what all the fuss is about, the book is a rewarding read. -Padraig Murphy, Berkeley Science Review, April 2004
Let me cut to the chase: every mathematics library requires a copy of this book. . . . Every supervisor of higher degree students requires a copy on their shelf. Welcome to the rich world of computer-supported mathematics! -Mathematiacl Reviews, December 2004
""Whether you take sides in the debate about the validity of experimental mathematics or are not sure what the fuss is all about, the book is a rewarding read."" -Padraig Murphy, Berkeley Science Review, June 2004
How large a role will computer computations play in the mathematics of tomorrow? The books under review are about many things, but it is clear that the authors are focused on this question. Their answer: very large. Their attitude: we should embrace this change. -David P. Roberts, MAA Online Read This!, January 2005
These are such fun books to read! Actually, calling them books does not do them justice. They have the liveliness and feel of great Web sites, with their bite-size fascinating factoids and their many human- and math-interest stories and other gems. But do not be fooled by the lighthearted, immensely entertaining style. You are going to learn more math (experimental or otherwise) than you ever did from any two single volumes. Not only that, you will learn by osmosis how to become an experimental mathematician. -Doron Zeilberger, American Scientist, March 2005
""It is impossible to describe the content of the whole work in detail in just a few lines."" -Ivan NetukaI, EMS, September 2004
Experimental mathematics is dedicated to expanding mathematical knowledge rapidly, occasionally with a ""temporary relaxation in rigor."" -Paul Campbell, Jr., Mathematics Magazine, April 2005
Much of the material in the book has arisen from the experiences of the authors while working on a computer based approach to different topics in mathematics. The variety obtained in this way is impressive, the authors have really touched and produced a treasure trove of lovely mathematical gems. -Fritz Beukers, AMS MathSciNet, May 2005
""Mathématiques expérimentales Certains mathématiciens défendent l'idée que les mathématiques sont une science expérimentale : l'ordinateur, dont la puissance de calcul engendre des conjectures, est pour eux une source d’inspiration."" -Jean-Paul Delahaye, Pour la Science--Logic et Calcul, April 2005
""Still, experimental mathematics is here to stay. The reader who wants to get an introduction to this exciting approach to doing mathematics can do no better than [this book]."" -Jeffrey Shallit, Notices of the AMS, September 2005
Mathematics by experiment describes the ideas behind experimental mathematics… and devotes some time to the more philosophical implications suggested by the title. -Australian Mathematical Society Gazette, August 2005
""The two books are written in an inviting, conversational, unprepossessing style. They are fascinating as a vast collection of interesting facts, anecdotes, and examples about numbers, primes, polynomials, special functions, definite integrals, series summations, and especially PI."" -Reuben Hersh, SIAM Reviews, January 2006
""The two books are written in an inviting, conversational, unprepossessing style. They are fascinating as a vast collection of interesting facts, anecdotes, and examples about numbers, primes, polynomials, special functions, definite integrals, series summations, and especially PI."" -Ruben Hersh, SIAM Reviews, January 2006"

About the Author

Jonathan M. Borwein, Dalhousie University, Canada; University of Newcastle, Australia. David H. Bailey, Chief Technologist, Computational Research Dept.Lawrence Berkeley National Laboratory. Together, Borwein and Bailey have written Mathematics by Experiment, Experimentation in Mathematics, Experimental Mathematics in Action, and Experiments in Mathematics CD. --This text refers to an alternate Hardcover edition.

Product Details

• Hardcover: 300 pages
• Publisher: A K Peters/CRC Press (December 4, 2003)
• Language: English
• ISBN-10: 1568812116
• ISBN-13: 978-1568812113
• Product Dimensions: 9.1 x 6.3 x 1 inches
• Shipping Weight: 1.2 pounds
• Average Customer Review: 5.0 out of 5 stars  See all reviews (5 customer reviews)
• Amazon Best Sellers Rank: #1,995,977 in Books (See Top 100 in Books)

More About the Author

Jonathan Michael Borwein, FRSC, FAAAS,FBAS, FAA is currently Laureate Professor in the School of Mathematical and Physical Sciences at the University of Newcastle (NSW). He directs the University's Priority Research Centre in Computer Assisted Research Mathematics and its Applications (CARMA).

A Rhodes Scholar, his research interests span pure (analysis), applied (optimization), computational (numerical and computational analysis) mathematics, and high performance computing. He has authored over a dozen books---most recently four on Experimental Mathematics (www.experimentalmath.info) and a 2010 book on Convex Functions---and over 350 refereed publications.

Customer Reviews

Most Helpful Customer Reviews

30 of 32 people found the following review helpful:

5.0 out of 5 stars A Mathematical Paradigm Shift, January 28, 2004

By 

Paul Preuss (Sausalito, CA USA) - See all my reviews
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This review is from: Mathematics by Experiment, 2nd Edition: Plausible Reasoning in the 21st Century (Hardcover)

"Mathematics by Experiment" is a ground-breaking book about a new way of doing math that generated so much excitement it was reviewed in "Scientific American" six months before it got into print. The authors are long-time collaborators David Bailey, chief technologist in the Computational Research Department of Lawrence Berkeley National Laboratory, and Jonathan Borwein, professor of science at Simon Fraser University in Vancouver, B.C.

They write that applied mathematicians and many scientists and engineers were quick to embrace computer technology, while pure mathematicians -- whose field gave rise to computers in the first place, through the work of beautiful minds like Alan Turing's -- were slower to see the possibilities. Two decades ago, when Bailey and Borwein started collaborating, "there appeared to be a widespread view in the field that 'real mathematicians don't compute.'"

Their book is testament to a paradigm shift in the making. Hardware has "skyrocketed in power and plummeted in cost," and powerful mathematical software has come on the market. Just as important, "a new generation of mathematicians is eagerly becoming skilled at using these tools" -- people comfortable with the notion that "the computer provides the mathematician with a 'laboratory' in which he or she can perform experiments: analyzing examples, testing out new ideas, or searching for patterns."

In this virtual laboratory Bailey and Borwein, with other colleagues, were among the first to discover a number of remarkable new algorithms, among them an extraordinary, simple formula for finding any hexadecimal or binary digit of pi without knowing any of the preceding digits. Further research led to proof that a wide class of fundamental constants are mathematically "normal" -- probably including pi, alhough that remains to be proved.

Their section on "proof versus truth" is an example of the gems even a mathematical tyro can find among these equations. Bailey and Borwein don't claim computers can supply rigorous proofs. Rather, the computer is a way to discover truths -- and avenues for approaching formal proofs. But often, the authors add, "computations constitute very strong evidence..., at least as compelling as some of the more complex formal proofs in the literature."

Drawing on their own work and that of others, Bailey and Borwein not only explain experimental mathematics in a lively, surprisingly accessible fashion but give many engaging examples of the "new paradigm" in action.

8 of 9 people found the following review helpful:

5.0 out of 5 stars Intuitions and experiments come first; rigorous proofs later, May 29, 2005

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Dr. Lee D. Carlson (Baltimore, Maryland USA) - See all my reviews
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This review is from: Mathematics by Experiment, 2nd Edition: Plausible Reasoning in the 21st Century (Hardcover)

If one peruses the mathematical literature for the last one hundred years one will notice that in most cases no diagrams or pictures appear. The level of rigor in all cases is impressive though, but unfortunately this makes the understanding of the results much more difficult. There seems to be an inverse relationship between rigor and understanding in mathematics, at least for those who are new to the subject at hand. In order to gain this understanding, the drawing of pictures and diagrams is useful, along with a certain amount of experimentation with the concepts at hand. Intuition, how mysterious, and however ill defined, plays a role in both the understanding of mathematical results and in their discovery. Many mathematicians do not want to acknowledge this, as a visitation to a typical conference will readily verify. The attitude has been expressed that mathematics has "always been abstract" and therefore that pictures or diagrams violate its spirit. Even a well-known geometry center whose goal was to use sophisticated computer graphics to visualize complex mathematical objects lost its funding, to the consternation of a few but with glee to most.

Thus the way to discovery of mathematics, i.e. the heavy use of intuition, the disorganized shuffling of concepts, and the experimental doodling, has been masked by the final product of this process: a superb example of logical rigor and organization called modern mathematics. The authors of this book however think otherwise, and they give the best apology for the role of experimental mathematics than anyone else in the literature. The book is packed with highly interesting examples and challenging exercises, all of which are ample proof of the need for doing experimentation in mathematics.

In addition to these considerations, the book is just plain fun to read, and even though time constraints may prohibit the working out of every exercise, the book could be used profitably in a graduate course in mathematics or even possibly in an undergraduate course at the senior level. Hopefully this approach to scholarship in mathematics will take hold in this century, and mathematicians will not only write down their final results with all their splendid rigor, but also how they got there. This would serve to educate younger generations of mathematicians in just how discovery in mathematics is done and increase their efficacy in the same. The book will also assist those who are trying to build machines capable of discovering novel results in mathematics. Machine proofs of difficult theorems and conjectures are now a reality, and in the twenty-first century we will no doubt see many more of these.

This book therefore contains a lot of hints about how to proceed in mathematics. Its acceptance will depend on how well it does its job in the creation of new mathematical results and in the teaching of them. Results in mathematics that seem plausible serve to make conjectures and motivate the construction of rigorous proofs. This book is a first step in a hopefully larger work.

2 of 4 people found the following review helpful:

5.0 out of 5 stars A Post-Modern Math Classic, May 12, 2005

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D Zeilberger (Princeton, NJ) - See all my reviews
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This review is from: Mathematics by Experiment, 2nd Edition: Plausible Reasoning in the 21st Century (Hardcover)

See my book review that appeared in American Scientist

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/mathexp.html