What's bigger than a kazillion?

Appropriately enough for a book based on paradoxes, "Everything's" chief virtues double as drawbacks. Wallace doesn't try to hide the difficulty of his subject behind real-world examples, as many math writers do -- instead, he tackles it straight on. Unlike science, math isn't really about the real world: It's about itself, its own assumptions and the conclusions they lead to logically. The sections of "Everything" in which Wallace discusses the philosophical urges and ways of thinking that underlie math are as charming as they are insightful. Like a true mathematician, he revels in abstraction, including proofs and mathematical notation where another writer would cop out and use metaphors. Only very occasionally does he make actual errors or simplify so much that the math looks wrong.

But readers familiar with Wallace may not be surprised to learn that he fetishizes technical terms to the point of becoming irritating and inconsiderate. (Pretentious? Lui?) For example, do you know what w/r/t stands for? Well, I'll tell you, since Wallace doesn't: "with respect to," a phrase that comes up quite a bit in math classes, and that Wallace sticks into ordinary sentences from time to time in its abbreviated form. You might be able to guess its meaning from the context if you didn't already know; but then again you might not, and Wallace has already tired your brain with so many necessary abbreviations that adding gratuitous extras seems rude.

Then there's his choice to structure the book as a history of infinity, rather than a less chronological treatment. Wallace begins with paradoxes that troubled the ancient Greeks, such as Zeno's brain twisters, including the one about how you can't cross the street because you first have to go halfway across, then halfway across the remaining distance, then halfway across the remainder of that, and so on forever. (You may have encountered the version in which a hare loses a race to a slower tortoise who had a head start.) Centuries of mathematicians found infinitely large and infinitely small quantities tempting but problematical because of these and similar problems. If you know the math but not the history, it's fascinating to watch whole branches of math (analysis, set theory) grow out of attempts to avoid or justify using these dubious infinities and infinitesimals in calculations. But the chronological structure seems to make it hard for Wallace to leave out any of the background. He's like a hungry man in a grocery store, piling more and more and more into his 319-page book (or "booklet," as he wishfully calls it). His appetite is inspiring, but I could have done with a lot less of the partial differential equations -- sticky formulas from second-year calculus, sort of like algebra equations that use further, harder equations instead of the familiar x's and y's -- and a lot more of the actual theory of infinite numbers that begins with Cantor's cardinals and ordinals.

Oddly, Wallace stops with Cantor in the early 20th century. The next exciting advances in set theory come a few decades later with Gödel's Incompleteness Theorem. They don't have much to do with infinity per se, and the novelist Rebecca Goldstein will cover them in another book in the same series anyway (I can't wait -- she's terrific), so I can see why Wallace skimmed lightly over them. But infinity didn't stop with Cantor. In the 1960s, John Horton Conway, a mathematician at Princeton, invented (or as he would say, discovered) a new number system, the "surreal numbers," which takes the study of the infinite far beyond Cantor's cardinals and ordinals. You can use Cantor's numbers to measure things and, more or less, to count -- but that's about it. But with Conway's surreals, calculations involving infinitely large and small quantities make precise sense for the first time. You can add, subtract, multiply, divide and even differentiate (an operation of calculus) with the surreals -- you can use them in algebra and come up with meaningful answers. And the beauty of it is, they're relatively easy to understand and explain. Wallace does mention a precursor of the surreals, the hyperreals, but I was disappointed not to find the surreals themselves in Wallace's book.

On the whole, though, Wallace does an admirable job unwinding what he calls "the Story of Infinity's overall dynamic, whereby certain paradoxes give rise to conceptual advances that can handle those original paradoxes but in turn give rise to new paradoxes, which then generate further conceptual advances, and so on." Who needs boy-meets-girl when you can have mind-meets-mind-meets-truth?

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