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To Infinity and Beyond by Eli Maor

Jacket: Paperback
Pages: 304 pages
Publisher: Princeton University Press
Genre: Science
ISBN: 0691025118


Review: Maor has written a book for both mathematicians and poets. Since he is a mathematician himself there is, to be sure, plenty of math in Maor's book. But the book should also appeal to the aesthetic side of many readers (me included) by exploring human perspectives of infinity, such as how we try to relate to the concept at a personal level, and how different people have tried to capture the notion in art and prose.

The book is arranged in four parts, dealing with the mathematical concept of infinity (how it shows up in algebra, etc.), geometrical infinity, aesthetic infinity (both art and poetry) and cosmological infinity.

The section on mathematical infinity has the typical assortment of historical examples, beginning with examples like the runner's paradox made famous by Zeno. There are also examples of infinite series that converge, including examples of how ancient mathematicians invented infinite series for transcendental numbers like pi. There's a plethora of little tidbits found throughout this section in little mini chapters that are short essays, only a few pages long, that give surprisingly succinct, tantalizing, and often delicious examples of mathematical infinity. Reading this book I was struck by what good reading it makes for any student preparing to take a class in calculus.

Some of the author's most interesting material is the author's discussions about infinite series. I particularly enjoyed his  examples how the associative property doesn't hold for infinite series (a non-intuitive fact that often comes as a surprise to many new students). Ordinarily, if you have a string of numbers that are connected by addition (x1+x2+x3+..+xn) for example, you can rearrange their order and get the same result. One of the strange things about infinity, though, is that rearranging the terms in an infinite series can result in the limit of the series changing from one number to another.

Of course no discussion about infinity would be complete  without mentioning Cantor, which Maor does with particular  clarity for first-time readers. Indeed, this is one of the things I  like about Maor best - he's written a book that is fun to read, even if you already know most of the stuff. It's engaging and entertaining, and full of "ahh" and "ohhh" even when you find yourself reading about something you studied many years ago. At the same time this is a good introductory text for anyone (I'm thinking youngsters in high school) who wants to start exploring some of these mathematical concepts, and need a friendly introductory text. If you can manage first-year algebra you have the tools you need to follow what Maor is talking about, though be advised that he doesn't shirk when producing equations, though most of the math is relegated to the appendices.

The section on geometric infinity is punctuated by nice illustrations and those geometrical shapes that you may have heard about - the ones with things like finite volume but infinite surface area. This was one of those rare occasions where I found myself wishing Maor had gone a little further. Instead of simply showing how such objects exist in mathematics, he really should have explained the apparent "paradox" (it's not hard). Instead, he makes the example more of a "paradox" than it really is by mixing metaphors in talking about "painting" the surface. Of course mathematicians have one idea about painting a surface (mathematical paint has no thickness), but the beginning reader is likely to be mostly confused - too bad, since Maor clearly has the skill to explain the trick.

Maor's exploration of the infinite is (almost) infinite. He has a wonderful section on tiling, and some brilliant plates representing some of the best mathematical art that attempts to depict the nfinite. The section on cosmology and the infinite is a nice summary of the history of astronomy and how astronomers and cosmologists have vacillated over the years between a cosmos that is infinite, then finite and bounded. I thoroughly enjoyed reading this book. It is well written and both easy and fun to read. My only complaints are rather minor. Several times Maor treats infinity as a "big number" (it's not a number at all, and he makes that clear, but his terminology on this score isn't as  consistent as it should be). And, he refers to mathematics as a science. Well, I suppose he's entitled to his opinion on that one,
though I imagine it will continue to be debated. Count me as one of those who puts mathematics in the "tools" category, separate from science.

The fact these inconsequential gripes are all I can find to complain about tells you what a really fine book this is. If you  love mathematics, this book really needs to be in your library.

— Reviewed by:
Duwayne Anderson Duwayne Anderson

duwaynea@hotmail.com

 
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