Mathematics: Frontiers and Perspectives [Hardcover]

V. I. Arnold (Editor), M. Atiyah (Editor), P. Lax (Editor), B. Mazur (Editor), International Mathematical Union (Corporate Author)

Book Description

ISBN-10: 0821820702 | ISBN-13: 978-0821820704 | Publication Date: December 1999

"The twentieth century has transformed mathematics from a cottage industry run by a few semi-amateurs into a worldwide industry run by an army of professionals ... "

--from the Preface by M. Atiyah

This remarkable book is a celebration of the state of mathematics at the end of the millennium. Produced under the auspices of the International Mathematical Union (IMU), the volume was born as part of the activities observing the World Mathematical Year 2000.

The volume consists of 30 articles written by some of the most influential mathematicians of our time. Authors of 15 contributions were recognized in various years by the IMU as recipients of the Fields Medal, from K. F. Roth (Fields Medalist, 1958) to W. T. Gowers (Fields Medalist, 1998). The articles offer valuable reflections about the amazing mathematical progress we have witnessed in this century and insightful speculations about the possible development of mathematics over the next century.

Some articles formulate important problems, challenging future mathematicians. Others pay explicit homage to the famous set of Hilbert Problems posed one hundred years ago, giving enlightening commentary. Yet other papers offer a deeply personal perspective, allowing singular insight into the minds and hearts of people doing mathematics today.

Mathematics: Frontiers and Perspectives is a unique volume that pertains to a broad mathematical audience of various backgrounds and levels of interest. It offers readers true and unequaled insight into the wonderful world of mathematics at this important juncture: the turn of the millennium.

The work is one of those rare volumes that can be browsed, and if you do simply browse through it, you get a wonderful sense of mathematics today. Yet it also can be intensely studied on a detailed technical level for gaining insight into some of the great problems on which mathematicians are currently working.

Editorial Reviews

Review

"This book should be in the library of every working mathematician." ---- European Mathematical Society Newsletter

"Many papers are ... broad in that they address the history of their subject and also make predictions about future developments. Many readers will be drawn to the general interest material ... Readers in search of controversy will find plenty ... especially intrigued by Arnold's elaborate schema in which the triple tetrahedron-octahedron-icosahedron corresponds to the triple reals-complexes-quaternions ... an excellent book." ---- MAA Online

"This collection demonstrates well that mathematics is alive and vital." ---- American Scientist --This text refers to the Paperback edition.

From the Publisher

Mathematics: Frontiers and Perspectives - V. Arnold, M. Atiyah, et al.

The mere names of the editors ensure that [the book] is fascinating and exciting ... Several of the 29 authors touch on the nature and/or future of mathematics ... the most interesting essays are those whose authors get out on a limb and dogmatically announce, as saving truth, propositions radically different from common opinion ... Among the authors there are many famous pure mathematicians whose contributions constitute a smorgasbord of delicacies sufficient to satisfy every taste. Do sample them!

--CMS Notes

See all Editorial Reviews

Product Details

• Hardcover: 459 pages
• Publisher: Amer Mathematical Society (December 1999)
• Language: English
• ISBN-10: 0821820702
• ISBN-13: 978-0821820704
• Product Dimensions: 10.1 x 7 x 1.1 inches
• Shipping Weight: 2.2 pounds
• Average Customer Review: 3.7 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #3,928,014 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

31 of 43 people found the following review helpful:

1.0 out of 5 stars Dissapointing., July 18, 2000

By 

Bernardo Vargas - See all my reviews

This review is from: Mathematics: Frontiers and Perspectives (Hardcover)

Judging from the publisher and the editors I expected to get a really outstanding book, but while reading it I got terribly dissapointed.

This book, with which the International Mathematical Union tried to emulate the famous list of problems proposed by David Hilbert in 1900, together with his description of the state of mathematics those days, totally fails to survey the state of this science in the year 2000 and does even less to present a comprehensive list of important problems in all branches of mathematics that should be solved in the 21st century.

The editors - all of them highly respected and renowned mathematicians - didn't make a strong effort to collect the opinion of several other high-ranked colleagues; instead they asked their buddies and pupils for contribution. I really would like to read something about the topic from Elias Stein, Steven Krantz, or Serge Lang, to name just a few. Also, the editors could have done something to reach some kind of agreement among the contributors; while some pose some problems of their own interest others claim that even trying to imitate Hilbert is nonsense. Some dissert on the interaction between mathematics and physics (there's even one who claims that mathematics is a part of physics) but almost none pays attention to the newer ineractions between mathematics and other disciplines. It seems that they haven't realized that nowadays most of the motivation for difficult and interesting problems comes from such areas as economics, communications, military and computer science. Its surprising that almost all of the contributors still think that theoretical physics is the main supplier of mathematical problems.

And the worst: they didn't cover all the branches of mathematics (true, it's a difficult task, but the AMS and the IMU should be capable of doing that). There's nothing said about operator theory, hypercomplex analysis, coding theory, commutative algebra, wavelet analysis, and many other disciplines. Their major lack is applied mathematics, I mean, probability, statistics, reliability, simulation, operations research, etc. Most of the contributors blame Bourbaki for having gone away from the actual sources of mathematical knowledge, but they remain in the Bourbaki setting themselves!

Conclsion: better spend your money in fine and focused reflections about mathematics and mathematicians, like Hardy's, Polya's, Wiener's, von Neumann's, and even Bourbaki's.

4 of 6 people found the following review helpful:

5.0 out of 5 stars an excellent book, June 16, 2002

By A Customer

This review is from: Mathematics: Frontiers and Perspectives (Paperback)

This is a wonderful collection of essays on the current state of mathematics, by some of the world's leading mathematicians. Some authors (e.g., Stanley and Sarnak) present lists of unsolved problems modelled after Hilbert's famous list. Others don't see that as useful, and instead outline the history, motivation, and current developments in their fields, or talk about neat related topics. All of the essays are fun to read, and a few (like the essay by Gowers) will become classics. Obviously, not every branch of mathematics can be represented in a single volume, but this book covers a rather broad spectrum, and does as well as anyone could reasonably expect, or even hope.

0 of 9 people found the following review helpful:

5.0 out of 5 stars That Hundred Years, and (well), The Next Hundred, August 11, 2006

By 

Peter J. Reynolds "cosmosopher" (Malden, MA USA) - See all my reviews
(REAL NAME)   

This review is from: Mathematics: Frontiers and Perspectives (Paperback)

I have perused my new book, and am delighted with it. Technically, I can't get too deeply into the 'Big Five' topics(number, geometry, topology, fractals, and probability), but I am writing in hopes that I could recommend three associated works: Steven Hawking's 'GOD CREATED THE INTEGERS', Benjamin
Yandell's 'THE HONORS CLASS', and (of course) Morris Klein's
wonderful 'MATHEMATICS'(The Loss of Certainty). I am sure that
professional mathematicians are familiar with these works, but
they might really 'get you in the mood' for Mr. Arnold's book.