The Millennium Problems 1 Hardcover – Oct 16 2002
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From Publishers Weekly
The noble idea that advanced mathematics can be made comprehensible to laypeople is tested in this sometimes engaging but ultimately unsatisfying effort. Mathematician and NPR commentator Devlin (The Math Gene) bravely asserts that only "a good high-school knowledge of mathematics" is needed to understand these seven unsolved problems (each with a million-dollar price on its head from the Clay Mathematics Institute), but in truth a Ph.D. would find these thickets of equations daunting. Devlin does a good job with introductory material; his treatment of topology, elementary calculus and simple theorems about prime numbers, for example, are lucid and often fun. But when he works his way up to the eponymous problems he confronts the fact that they are too abstract, too encrusted with jargon, and just too hard. He finally throws in the towel on the Birch and Sinnerton-Dyer Conjecture ("Don't feel bad if you find yourself getting lost... the level of abstraction is simply too great for the nonexpert"), while the chapter on the Hodge Conjecture is so baffling that the second page finds him morosely conceding that "the wise strategy might be to give up." Nor does Devlin make a compelling case for the real-world importance of many of these problems, rarely going beyond vague assurances that solving them "would almost certainly involve new ideas that will... have other uses." Sadly, this quixotic book ends up proving that high-level mathematics is beyond the reach of all but the experts.
Copyright 2002 Reed Business Information, Inc.
From School Library Journal
Adult/High School-In May, 2000, the Clay Mathematics Institute posted a million-dollar prize to anyone able to solve any of what it considered the seven most important mathematical problems of the 21st century. They were chosen not for theoretical beauty alone, but because many of them deal with concepts in fields like physics, computer science, and engineering, and exist because practitioners in those fields are already using theoretical or practical design solutions that have not been mathematically proven. Devlin, "The Math Guy" from NPR's Weekend Edition, does a good job explaining the background of the problems and why theoretical mathematics as a discipline should matter to a general audience. Each problem has a chapter of its own and is given a treatment that, where applicable, extends back to the ancient Greeks. A passing knowledge of mathematics is important for taking in Devlin's work but a major in the subject is not, and this book should satisfy anyone looking for a layman's guide to modern theoretical mathematics. Or hoping to win a million dollars.
Sheryl Fowler, Chantilly Regional Library, VA
Copyright 2003 Reed Business Information, Inc.
Top Customer Reviews
Now to the math. A reader having an extensive math background (say college major or above) will find little real math of interest here. Be prepared to face a page and a half explanation of the amazing growth in magnitude of factorials.
There is astoundingly some algebra done wrong here (see page 192).
The research in some of the chapters appears to have been done hastily and the exposition is not clear (P vs NP Problem for example).
There are factual errors, from the minor (the year of Isaac Newton's birth) to Mr Devlin having Daniel Bernoulli and Euler working in the 19th century (p. 132; it was the eighteenth).
This gives the feel of a book which has been written hastily and one wonders if it was reviewed by another mathematician before publication.
In fairness, the description of most of the problems is deftly handled and will draw the interest of most readers.
Please give the climbing the mountain to view the math landscape analogy a rest.
The book is ridiculously overpriced at US $16.00 and Canadian $25.00.
The going isn't too hard thanks to Devlin's expository ability, but alas, I think this will be true only for aficionados of mathematics and physics. In his columns for the Mathematical Association of America, Keith has always had in mind a varied audience of readers. But how can he hope to communicate to the non-mathematician when so much meaning resides in the equations that appear throughout the book? Still, his pedagogy prevents this from becoming "The Idiot's Guide to the Millennium Problems". (I suppose it'll appear real soon.)
Devlin hints at a disturbing idea. Will cutting edge problems become so abstruse some day that it will take the best minds all the fruitful years of their lives just to arrive at a position of comprehension? What then, mathematical AI?
There are some silly mistakes, perhaps caused by a looming deadline. One involves a mix-up between the relativistic precession of Mercury's orbit and the relativistic bending of light rays. A logical error appears in a footnote on pg.54, where the word "a" should replace "no". Another one appears in the caption of Fig. 5.5, where "Example" should replace "Proof". Would it be too much to ask that copy editors who are assigned technical books have a dim awareness of mathematical argumentation?
One can of course think of many other problems that fit the stature of the millennium problems, such as the invariant subspace conjecture, or developing a complete mathematical model of the cell, but these seven will no doubt spark the curiosity of a few young persons as they further their studies in mathematics. Some of the millennium problems, such as the Riemann hypothesis, the NP problem, the Poincare conjecture, and the Navier-Stokes equations, require only an undergraduate education. The others definitely require more background, just to understand even the statement of the problem. All of the them are fascinating, and will no doubt stimulate some incredibly interesting mathematical constructions.
Personal note for anyone interested (from someone who has worked on one of these problems for several years): For those readers who are thinking about attacking one of these problems, it is important to be really interested in solving it, for your own satisfaction, and not to be concerned about the financial reward or what the solution will bring you in terms of professional advancement.Read more ›
Make no mistake, these problems are very hard. Even with all his mathematical expertise. Devlin readily admits that he really does not understand them all and had a very difficult time writing about them at a level so that a general audience could understand the basics of the problems. The seven problems are
· The Riemann hypothesis
· Yang-Mills Theory and the Mass Gap Hypothesis
· The P vs. NP Problem
· The Navier-Stokes Equations
· The Poincare Conjecture
· The Birch and Swinnerton-Dyer Conjecture
· The Hodge Conjecture
and the Riemann hypothesis is distinguished in that it is the only one that was also on Hilbert's list at the turn of the previous century. In his descriptions of the last two problems, it is clear that Devlin is struggling to understand the fundamentals of the problems.
Nevertheless, he does manage to inform the reader about what the problems are about, as well as a taste of how difficult they are. Like the problems David Hilbert stated in 1900, this collection of problems forms a marker by which the mathematical progress of this century will be measured. For that reason, all mathematicians should learn something about them, and this book is an ideal initial step.
Published in Recreational Mathematics e-mail newsletter, reprinted with permission.
Most recent customer reviews
This book is more pedantic than I thought it would be. Being a smallish book and a smaller audience, it is understandable that the mathematical details are trimmed down (almost... Read morePublished on Jan. 11 2004 by Randy Given
Great book. Although pushing the limits of "accessible", those who understand it will be intrigued by Devlin's discussion of the Millenium Problems, a set of puzzling... Read morePublished on July 8 2003 by D. Wang
Reviewer Ted Sung pointed out a sloppy remark of physics history made by the book. There is another serious mistake appeared on Page 91: The Yang-Mills gauge theory has never been... Read morePublished on June 15 2003 by Kwong Chung Ping
I've read several of Devlin's books and have loved them. So I was quite excited to see this book published. Read morePublished on Jan. 22 2003 by love-physics-and-math
... but keep it in mind for that teenage nerd in your life.
To help you evaluate my evaluation, let me note up front that I have three long-ago years of graduate math courses... Read more
Would you like to win a million dollars? Would you like to win it by solving a math problem? You have entered the right millennium to do so. Read morePublished on Dec 10 2002 by Rob Hardy