Visions of Infinity: The Great Mathematical Problems is Ian Stewart's latest book. Dr. Stewart is one of my heros. I interviewed him for my podcast series and I love his new book. But, I struggled to come up with the right words to write for a review. So, what I've done instead is to take snippets from a number of reviews of the book I found on the web that articulated my own impressions and turned those into a mashup review, followed by a very brief summary of my impressions.
Here's the publisher's description of the book:
For every problem mathematicians solve, another waits to perplex and galvanize them. 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 explains why these problems exist, what drives mathematicians to solve them, and why their efforts matter in the context of science as a whole. Stewart describes solved problems—like Fermat’s Last Theorem, proved three centuries after it was posited, and the Poincaré Conjecture, cracked by eccentric genius Grigori Perelman—as well as those like the P/NP problem, which 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 have risen to meet the challenges set by their predecessors—and how the enigmas of the past inevitably surrender to the powerful techniques of the present.
Below are excerpts from a half dozen reviews on the Web. Keep reading for my impression of the book as bullet points.
To be clear though, while this volume covers some of the most interesting problems in all of mathematics it is NOT a book to draw your non-mathematical friends into the math arena. Even the non-math person who wishes they could like math, and who may have enjoyed Steven Strogatz's "Joy of X," I think will find this particular book too heavy-going. But for the individual already enamored of the subject, and having some familiarity with math's deepest problems, this is a fantastic read. In fact, it's probably my favorite Stewart volume of all the ones I've read.
The four-color map problem can be understood by a bright fourth-grader (the question: whether four colors are enough to ensure that no two countries with a common border share a color). By junior high, most kids can grasp prime numbers and learn something about their properties and patterns. High school algebra students can comprehend what Fermat’s last theorem means. Yet these topics have for decades, or even centuries, occupied the world’s most sophisticated mathematicians.
Capping the discussion is a quick chapter detailing some of the problems that may give mathematicians fits and nightmares into the next century, including quaintly named perfect cuboids, Langton’s Ant, and mysterious constructs called Thrackles. Once again, Stewart delivers an intriguing book that rewards random reading as much as dedicated study.
Stewart’s imaginative, often-witty anecdotes, analogies and diagrams succeed in illuminating many but not all of some very difficult ideas. It will enchant math enthusiasts as well as general readers who pay close attention.
In the end chapters the mathematics gets more “pure” and explanations of the conjectures get more complicated. The reader has to pay more attention. CAUTION: Do not attempt these chapters without possessing full alertness. The chapter “Diophantine Dreams” addresses the search for the proof of the Birch-Swinnerton-Dyer conjecture, where the difficulty won’t just be in finding a proof. It will first be in understanding just what the conjecture is. Make this chapter your own personal Mount Everest.
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.
The reviews summed up my experience of the book.
- Ian Stewart is a great writer. He is able to weave together mathematics and story to keep the reader's attention and to provide context for why we care about solving these difficult problems.
- I'm not so sure that a motivated high school student can follow all of the ideas. I certainly couldn't. As Shecky points out in my excerpt from his MathTango review (above), "To be clear though, while this volume covers some of the most interesting problems in all of mathematics it is NOT a book to draw your non-mathematical friends into the math arena." Very well said.
- Having hammered in the last point, much of the material in the book is accessible to the motivated student who is willing to focus and persist.