Proliferating Paradoxes! Infinity Lost in Translation

Everything and More: A Compact History of Infinity , by David Foster Wallace. W.W. Norton, 319 pages, $23.95.

In college, I developed a crush on a fellow student in a course on Gerard Manley Hopkins. She was thin, silent and smoked a corncob pipe during class. One day she said, “Read this,” and handed me a small book. It was the Kama Sutra . I was excited. My excitement evaporated, however, when I actually read the book. I was stimulated by the subject matter, but I couldn’t do any of it.

I have a similar reaction to math books. I thus welcomed a new history of infinity from the hip literary novelist David Foster Wallace. Certainly, he would have sympathy for those of us who lacked flexibility. There were some good signs: The jacket portrait shows the author sitting in front of a bookcase holding a large black dog in his lap-his, presumably, but possibly a rental. By covering two of the four author-photo clichés (the others being The Tree and The Body of Water), the publisher assures us that we are dealing with a real writer and not some mathematician. Also, the author looks like one of the lesser-known and more dissolute Baldwin brothers, so I was prepared for math aimed at the Baldwin level.

Everything and More , despite being called a “booklet” of “pop technical writing” by Mr. Wallace, is 300-plus pages long. Don’t worry: The pages are small and filled mostly with equations, symbols, abbreviations and footnotes, so there are very few real words to read. The footnotes, conveniently set at the bottom of the page, are quite nice. Sometimes humorous, often irrelevant and always casual in a controlled fashion, they will entertain you for a long time while you avoid the principle text.

The book is about infinity, I think, the mathematical kind (though infinity isn’t a number), principally from the time of the ancient Greek Zeno to the triumphs of Georg Cantor (1845-1918), the German mathematician. Cantor was obsessed with infinity and spent much of his life in mental hospitals, about which much has been made. To Mr. Wallace’s credit, he doesn’t dwell on the infinity/crazy connection.

Mr. Wallace doesn’t spell out where he’s going, so I read other books about infinity and threw myself on the mercy of a real mathematician to talk me through the topic, though he also wasn’t sure what Mr. Wallace was writing about. We begin with Zeno and his various paradoxes, which claim, for instance, that a runner cannot finish a race: Before the runner can cross the finish line, he must run half the distance, and before that, half of that half, and before that, a half of a half of that half, and so on. There are an infinite number of these fractions, and so the runner never gets going. Zeno introduced the idea-to the West, at least-that infinities are a peck of trouble (and perhaps delayed the Olympic games for centuries).

Galileo’s Paradox notes that the naturals (counting numbers) are equaled by the squares of those numbers: That is, if you counted to infinity (1, 2, 3, 4, 5 …), you could write the square of each of those counting numbers next to it: 1 next to 1, 4 next to 2, 9 next to 3, and so on. This is a paradox because the squares would seem to be fewer and, when spread out over the natural numbers, would seem to get farther and farther apart as you count up-and yet you never run out of them; you can always find one to pair up with a natural, a one-to-one correlation. These two infinities appear to be equal. Maddening.

Cantor is given credit for making sense of all this. He found equal infinite sets beyond that of Galileo’s example, figuring out ways of pairing all the naturals with an apparently larger set, that of all the naturals plus their negatives and zero. And then he proved that the infinite set of naturals was also equal to all of the rationals (natural numbers plus their fractions). Cantor then did the unthinkable, finding different “sizes” of infinity-showing, for example, that there are more real numbers than naturals, as the reals contain irrationals. Because of the irrationals, on a number line the reals would be denser than the plain-vanilla counting numbers. Cantor devised a fiendishly clever proof of this, which I cannot adequately explain (and neither, to my untrained eye, can Mr. Wallace).

Along the way, Mr. Wallace teaches us mathematics. He lectures us like schoolchildren-a treatment we perhaps deserve, but it’s not pleasant. He covers 2,500 years of mathematics, but it feels like much longer. Mr. Wallace rambles on in an incoherent and haphazard fashion. It’s like learning trig from Lieutenant Columbo: “Oh, just one more thing ….” Concepts I used to understand, I no longer do after reading Everything and More . For example, the proof that the square root of two is irrational, by Hippasus of Metapontum, was always clear to me; in fact, I revisited the proof last year in an refresher course. Somehow, Mr. Wallace rewrites this elegant proof so that it’s totally mystifying.

My mathematician friend insists the book is “a parody of math books.” He couldn’t make head nor tail of it, and said that Mr. Wallace is either an “obscurantist” or just showing off. The text is a thicket of symbols and equations. Mr. Wallace lists 37 abbreviations on pages 3 and 4, supposedly for convenience. It is not convenient. When I come across “ZFS” on page 305, I have to flip back to page 4 to find out this stands for the “Zermelo-Fraenkel-Skolem system of axioms for set theory.” To which I say to the author, “FU.” (My glossary available upon request.) Not all of the confusion is Mr. Wallace’s fault. Inexplicably, the publisher hasn’t provided an index.

Mr. Wallace’s math history is thin. He relies in part on Morris Kline, whom he calls a “towering figure.” Kline is to mathematical history what Kenny G. is to the saxophone: wildly popular but not necessarily respected by his peers. Mr. Wallace parrots Kline in insisting that the Greeks invented math and “were the first people to treat numbers … as abstractions.” He says the Babylonians, Egyptians and Indians who came before the Greeks could conceive of “five oranges” but not just “5.” This is nuts. Mr. Wallace buys into the argument that non-Europeans can’t count, and certainly not to infinity.

The objection isn’t that he’s racist, but that he misses out on a rich history of infinity. Long before Zeno, the Indian atomists knew infinity was a problem when they decided that elementary particles had to be finite, not infinitesimal, as the latter would imply that mole hills had the same number of particles as mountains. (Infinities continue to trouble particle physics to this day.) Jain mathematicians (800-200 B.C.) postulated five different kinds of infinity and, like Cantor, discarded the notion that all were the same or equal. In the 10th century A.D., the physicist Abu Ali al-Hasan ibn al-Haitham (Alhazen, in Latin) figured out that we can see a mountain because a tiny replica of the mountain is formed in light on our retina. The multitudinous points of light in the big mountain are all replicated point for point in the tiny mountain-a concept Cantor would confirm centuries later.

But the real problem here is math envy. David Foster Wallace is not a mathematician: He took some math courses in college and, damn it, we’re going to hear about it. Everything and More is a bit like the Kama Sutra , had the latter been written by a very arrogant virgin: clumsy, painful and ultimately unsatisfying.

Dick Teresi is the author of Lost Discoveries: The Ancient Roots of Modern Science-from the Babylonians to the Maya (Simon and Schuster).