Overview of Ian Stewart's "

**Visions of Infinity**"

*"Mathematics is newer, and more diverse, than most of us imagine. At a rough estimate, the world's research mathematicians number about a hundred thousand, and they produce more than two million pages of new mathematics every year...*

*"When we think of mathematics, what springs to mind is endless pages of dense symbols and formulas. However, those two million pages generally contain more words than symbols... As the great Carl Friedrich Gauss remarked around 1800, the essence of mathematics is 'notions, not notations'. Ideas, not symbols."*

-- from Ian Stewart's Preface

Ian Stewart (or one of his clones… because I refuse to be duped into believing that

*one*person can write all these books!) has a new volume out, "

**Visions of Infinity**." There are several popular math books available that review a variety of the most famous, intriguing often-unsolved problems in mathematics (the exact selection can vary slightly from volume to volume, though certain standards almost always show up). Anyway, this is Stewart's contribution to the genre… and… as one might expect, it is excellent. If only it had been written first, some of the other attempts might even have been unnecessary.

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.

Stewart gives enough background and depth to each discussion to hold the attention, and exercise the brain, of most math buffs, without getting so technical as to lose them in the dust (though parts, toward the end especially, are definitely tough-going). I have no idea if he was able to produce this work largely based on his past writing and knowledge, or whether it required lots of research on his part to fill in all the detail, history, and richness that are on display here... either way the book is a great accomplishment.

Fermat's Last Theorem, the Riemann Hypothesis, the four-color theorem, the Goldbach conjecture, the Poincare' conjecture, and P vs. NP, are among the classics Stewart rolls out for scrutiny (the title may be a bit misleading in so much as "infinity" is not really the central topic). He does a great job of not only explaining these mathematical conundrums in an accessible way, but also of detailing their histories, context, and relevance (when it exists) to other matters. The first few chapters tend to warm the reader up. From there the book moves on to more complex and difficult problems as it goes along. The last couple of conjectures dealt with, the Birch--Swinnerton-Dyer conjecture and the Hodge conjecture, are the densest to follow (...but Stewart notes of the Hodge conjecture that it "

*is arguably more representative of real mathematics, as done by mathematicians of the twentieth and twenty-first centuries, than any other topic in this book*"). Soon after those chapters Stewart follows up with a lighter chapter titled "Twelve For the Future," briefly outlining 12 more unsolved problems, some fairly well-known. This section ends with the ABC conjecture, and was apparently written before last year's announcement from Shinichi Mochizuki that he had proven the conjecture, as there is no mention of the claimed proof (which has yet to be confirmed, and indeed some say may not be comprehensible to anyone

*except*Mochizuki!).

I suppose my favorite chapter (pardon the bias) is on the Riemann Hypothesis, where Stewart seems to capture in a chapter what others have written whole books about. Another chapter that I found especially good was on the Mass Gap Hypothesis, and more generally on particle physics, but your own interests will determine which chapters you most enjoy. They are all good. This will likely be the initial 'go-to' volume on my shelf whenever I wish to check on something regarding one of these classic problems.

The book ends with a handy 11-page glossary, an interesting set of succinct "notes," and a brief bibliography for "further reading" -- on a side note, I was pleased to see that the brief bibliography cites Matthew Watkins' "

**The Mystery of the Prime Numbers**," a book I've been touting for quite awhile, but which I'd not yet seen referenced by any prominent author (Watkins gets a very brief mention in the body of the book as well).

This is simply a fabulous book for most mathematics enthusiasts... no matter which clone of Stewart's wrote it!

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