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It is one of the wonders of mathematics that, for every problem mathematicians solve, another awaits to perplex and galvanize them. Some of these problems are new, while others have puzzled and bewitched thinkers across the ages. Such challenges offer a tantalizing glimpse of the field’s unlimited potential, and keep mathematicians looking toward the horizons of intellectual possibility.

In Visions of Infinity, celebrated mathematician Ian Stewart provides a fascinating overview of the most formidable problems mathematicians have vanquished, and those that vex them still. He explains why these problems exist, what drives mathematicians to solve them, and why their efforts matter in the context of science as a whole. The three-century effort to prove Fermat’s last theorem—first posited in 1630, and finally solved by Andrew Wiles in 1995—led to the creation of algebraic number theory and complex analysis. The PoincarĂ© conjecture, which was cracked in 2002 by the eccentric genius Grigori Perelman, has become fundamental to mathematicians’ understanding of three-dimensional shapes. But while mathematicians have made enormous advances in recent years, some problems continue to baffle us. Indeed, the Riemann hypothesis, which Stewart refers to as the “Holy Grail of pure mathematics,” and the P/NP problem, which straddles mathematics and computer science, could easily remain unproved for another hundred years.

An approachable and illuminating history of mathematics as told through fourteen of its greatest problems, Visions of Infinity reveals how mathematicians the world over are rising to the challenges set by their predecessors—and how the enigmas of the past inevitably surrender to the powerful techniques of the present.

- Sales Rank: #234420 in eBooks
- Published on: 2013-03-05
- Released on: 2013-03-05
- Format: Kindle eBook

From Booklist

*Starred Review* Few of us share Stewart’s mathematical skills. But we relish the intellectual stimulation of joining him in exploring mathematical problems that have pushed even genius to the limit. We thrill, for instance, to the ingenuity of a great Chinese mathematician coming tantalizingly close to proving the centuries-old Goldbach Conjecture. And we feel the human meaning of mathematical achievement when a triumphant British analyst weeps before television cameras after finally proving a seventeenth-century algebraic theorem. We feel that meaning again when a brilliant Russian mathematician retreats into reclusive isolation, distressed because of initial skepticism toward his groundbreaking work on a nineteenth-century riddle. But high-level mathematics stirs deep emotions largely because it taps into the mind’s deepest impulses. Stewart repeatedly shows how a trivial mathematical curiosity can open up vital new conceptual insights. Readers learn, for example, that the apparently inconsequential four-color problem has led investigators deep into theoretical physics and has compelled fundamental rethinking of what constitutes a mathematical proof in a computerized age. Proofs incorporating computer-generated calculations may strike old-school mathematicians as unsatisfying, but Stewart assures readers that mathematics still depends on human investigators and that such investigators will not soon run out of daunting mathematical problems. A bracing mental workout for amateur mathematicians. --Bryce Christensen

Review

Science News

“As a guide to the inner workings of the mathematical jungle, Stewart provides an engaging and informative experience. If you wish to intelligently discuss the Riemann hypothesis, P/NP problems or the Hodge conjecture, you ought to read this book first.”

Choice

“A designated math popularizer, Stewart writes books that are always enlightening and enjoyable.... Again, Stewart provides another interesting read for anyone intrigued by mathematics.”

Dallas Morning News

“Anyone who has always loved math for its own sake or for the way it provides new perspectives on important real-world phenomena will find hours of brain-teasing and mind-challenging delight in the British professor’s survey of recently answered or still open mathematical questions.... Individual readers will dig deeply into certain chapters and skim others according to personal preference, but every one of them will be captivated by the technical achievements, loose ends and human insights that Stewart shares on his grand mathematical tour.”

New York Journal of Books

“Entertaining and accessible.... Ian Stewart belongs to a very small, very exclusive club of popular science and mathematics writers who are worth reading today.”

Publishers Weekly

“An entertaining history of mathematics and a fresh look at some of the most challenging problems and puzzles in the history of the field…. Stewart’s loquacious yet lucid style makes the most complex mathematics accessible, even when discussing esoteric concepts…. Once again, Stewart delivers an intriguing book that rewards random reading as much as dedicated study.”

Booklist, starred review

“Few of us share Stewart’s mathematical skills. But we relish the intellectual stimulation of joining him in exploring mathematical problems that have pushed even genius to the limit.... Stewart repeatedly shows how a trivial mathematical curiosity can open up vital new conceptual insights.... A bracing mental workout for armchair mathematicians.”

Kirkus

“Stewart’s imaginative, often-witty anecdotes, analogies and diagrams succeed in illuminating…some very difficult ideas. It will enchant math enthusiasts as well as general readers who pay close attention.”

About the Author

Ian Stewart is Emeritus Professor of Mathematics and active researcher at the University of Warwick. The author of many books on mathematics, he lives in Coventry, England.

Most helpful customer reviews

50 of 54 people found the following review helpful.

very interesting topics, but not explained very well

By Humble

I am well trained in math. I love math and tried reading two of Mr. Stewart's books: In Pursuit of the Unknown: 17 Equations That Changed the World; and Visions of Infinity. I expected the books to be amazingly interesting but was disappointed and frustrated when reading them. Mr. Stewart jumps over key steps, which are difficult to grasp and reaches for the end result!

I think less than 10% of the people with advanced degrees in the sciences could follow his explanations in any chapter and actually feel they understood the topic.

I am surprised that the book's cover jacket has endorsements from newspapers saying the book is easy and fun. My humble advice to prospective readers: borrow the book from a library and make your own judgement and then write your own review on Amazon.

27 of 34 people found the following review helpful.

Fun, Entertaining and Enlightening, GREAT READ!

By Let's Compare Options Preptorial

I've pre-read the soon to be released "golden ticket" book --ala Willy Wonka's candy bars-- The Golden Ticket: P, NP, and the Search for the Impossible-- which explores P-NP, and along with Ian's Infinity book here, these are really the only two UP TO DATE books on the millenium and other "tough" problems in math and computing, both solved and awaiting new approaches. Although P-NP is technically a computational complexity problem, knowing which problems can and can't be solved in polynomial vs. exponential time is widely believed to hold the key to "all" the other problems in computing AND math! An astonishing prospect.

Infinity is a GREAT, fun read, with just enough math to please, but not intimidate an advanced high school level reader. It is not just another of the numerous "here are the heroes --heroines-- of math" books that dwell on older, solved problems. The history and problems given lead us into both unsolved CURRENT problems, and the birth of many of the newer --over 450 at present writing-- subspecialties of analysis, math, tricky current theorems, etc. As usual, Stewart doesn't "talk down" to his readers, and you get a real "brain workout" feeling after reading this wonderful survey of the "top 14" toughest solved and unsolved, and their real-life ramifications as solved and when solved.

One of the really amazing side shows in this exciting field is the nature of humans vs. computers that is quietly raging in math. P-NP --although mostly thought to be unequal--, for example, even peeks into quantum observer and quantum computing issues about how "great" even average human brains might potentially be. It pulls back a curtain in which a variety of "modern" brute force, algorithmic, sieve and spectral methods and other algorithmic approaches to problems are finding unexpected barriers.

This is non trivial, because a modern operating system like Windows with hundreds of thousands of lines of code is "controlled" by tiny algorithms of two or three lines, as are million line fly by wire jets. Our planet is rapidly becoming algorithm driven, to the extent that most new math methods and problems are described in algorithmic terms. But P-NP is only a little slice of this wonderful book, which covers a vast number of newer and less known stumbling blocks within the top 14, including the better known millenial problems. In Systems Engineering, we know that infinite and tiny problems are much easier than the "middle" complexity problems.

Just as dynamic systems can "settle down" into chaos/fractals, strange attractors or an oscillation, the book, after taking us on a fascinating journey through the known and unknown, gives us a great, up to date feel for which problems are in which category of difficulty and likely vs. unlikely to be solved in our times. The "toughest" problems are the stuff of cryptanalysis and are "good" from the standpoint of providing security, but Ian also shows the many possible openings at the back of the tent in addition to the door, by suggesting possible "close enough" solutions and directions that are worth pursuing.

Great survey, with just enough depth in each area to tease the reader into looking further. Because of Ian's usual "analogy" explanations and diagrams, very tough problems are explained intuitively, but the actual math is sneakily grad level. You WILL hunger for more if any of these tantalizing puzzles hit home. If you're a P-NP enthusiast, you should know that Ian doesn't think it is solvable in anyone reading this books lifetimes! But don't get discouraged, he could be wrong, but does make a strong and thoughtful case.

In the golden ticket book above, Fortnow also says P/NP won't be solved in --possibly-- thousands of years, due to the kind of "proving a negative" of "showing" that EVERY potential algorithm HAS to fail. The cool thing mentioned by both Ian and Fortnow is that these REALLY tough problems tend to bring vast areas of science together that were silos in the past. If you look up Systems Science on Wiki, you'll get an idea of the missing link in a lot of these discussions. The unstated speculation is whether the human brain "perceives" higher dimensions than algorithms can reach in some fashion, and P/NP has areas as strange as the quantum observer for the possibly unexpectedly great potential of humans, as yet unknown.

Library Picks always buys the books we review, and has nothing to do with Amazon, the authors or publishers, and our comments are solely for the benefit of Amazon shoppers.

4 of 4 people found the following review helpful.

Good and well readable overview of Millennium & other math problems

By Martijn13Maart1970

As stated in the item info, " In Visions of Infinity, celebrated mathematician Ian Stewart provides a fascinating overview of the most formidable problems mathematicians have vanquished, and those that vex them still. He explains why these problems exist, what drives mathematicians to solve them, and why their efforts matter in the context of science as a whole."

After reading this book I have to say this is indeed what the book delivers, as much as I can at least imagine to be possible. So this makes for a very interesting read, also for the layman. However, some problems are so abstract, they defy a simple analogue for daily use. That is why some reviewers have commented that the book was not always clear to readers who, like me, are not having e.g. a PhD in maths. they are partially right, since there can not be a book that lists all these problems that is easily read as well. There has to be a compromise, and some feel that is why this book, as so many others, is hybrid and awkward. That is ironic, since the author tried (too?) hard to give it a try to explain all of the theorems that the current math community considers important. That does indeed lead sometimes to passages I had to skip, and I have a MSc degree in Aeronautical engineering!

The alternative would have been to omit the too abstract ones for readability, but then the title of the book would not have lived up to the promise and should have been changed in 'Visions of Infinity For Dummies'. Or, even worse, what if the author would have left out all math details and turned it into a sort of popularized Da Vinci code?

You cannot eat the cake and have it (or is it the other way around?): if you pick up a book about math, you will have to do some math reading! Unfortunately, many of the most important problems in math are not only difficult to solve (hence the million dollar prize) but, unlike for instance Fermat's Last Theorem, many problems are also difficult to state, even for people who are no layman. Math is simply so big, that many specialists do not have the faintest idea what others are doing, let alone the layman reader. Having said that, it is clear why it is difficult to write a book like this, trying also to explain the importance of these math problems to the layman. The author tries the most he can and does an excellent job in making the compromise between completeness and readability. Many fascinating subjects appear and I just suggest you skip the parts you cannot follow, since there is plenty of other interesting stuff left! For instance, the P/NP problem is explained very well, and I also liked the author's comment on Perelman's reclusive behaviour during and after his Millennium Prize.

The only thing that would have helped would have been a conclusion in the end of every subject, so it would have still been possible to grasp the main ideas without reading all the formulas in those chapters that were too difficult for some of us, but then again, maybe some math defies simple explanation with metaphors in daily life. Still, give it a try and you will be in for a good read.

Also recommended are for instance these titles

The Hilbert Challenge

The Golden Ticket: P, NP, and the Search for the Impossible

The Poincare Conjecture: In Search of the Shape of the Universe

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