Cantor shows that P's first derived set, P', can be "decomposed" or broken down into the union of two different subsets, Q and R, where Q is the set of all points belonging to first-species derived sets of P', and R is the set of all points that are contained in every single derived set of P', meaning R is the set of just those points that all the derived sets of P' have in common.
"Everything and More" is not for lightweights. The book has a hero: the 19th-century German mathematician Georg F.L.P. Cantor, who spent the last 20 years of his life in and out of mental hospitals. But it is less Cantor's story than the story of the problem he solved, which led him to pioneer set theory and to discover the mathematical rules that govern infinity -- which, it turns out, comes in different sizes.
The 41-year-old Wallace is probably the most important novelist of his generation, and he has fans who will follow him even into differential equations. He is the author of two story collections, a book of essays, and the 1079-page, 388-footnote novel "Infinite Jest," which is set in Greater Boston (in 1989 Wallace spent a semester as a graduate student in philosophy at Harvard), and is concerned with Alcoholics Anonymous, wheelchair-bound terrorists from Quebec, and a tennis prodigy whose late father directed a film so entertaining that it melts the minds of its viewers.
A year ago Wallace left his longtime home in central Illinois to become the Roy Edward Disney Professor in Creative Writing at Pomona College in California. Last Friday he was in New York, and he met with Ideas in his hotel room to discuss infinity, Whitey Bulger, and Platonists (people who believe that math's concepts exist independent of the mathematicians who think them). Before the interview began, Wallace, who recently quit smoking, served Ideas a glass of club soda and himself a nicotine patch.
IDEAS: A few years ago, you reviewed two novels about math for the magazine Science and took them to task for not having thought through who their audience was.
WALLACE: They were really dreadful books. One of them would take time to define very simple stuff like addition but would then throw around really high-level math terms.
IDEAS: How do you address the question of audience in your book?
WALLACE: Oh, it's so much more fun criticizing how other people have done it. There was this book by Amir Aczel, which was about Georg Cantor's mental illness, and certain supposed connections between infinity and mental illness, and infinity and the kabbalah. Obviously I wanted to do something different, and the only way I could think of was to talk about where the math actually came from. The idea is, if the book works right, you finish it with a better idea of not just how Cantorian transfinite set theory works but actually why it was a big deal and why it's beautiful and amazing.A couple of people have already said to me, "Gosh, good book but it's really hard," thinking that's a compliment, and it's not. Does it seem halfway clear to you?
IDEAS: The Uniqueness Theorem lost me.
WALLACE: It's the hardest part of the book. The idea is, if you can show that all the different series that a function expands to are equivalent, then it is just one series.
IDEAS: I pretty much followed the rest.
WALLACE: My fondest hope is that the average reader has more or less your experience. I'm hoping that even a reader who hasn't had a semester of college math will be able to follow enough to get why this stuff is a big deal and why it was beautiful, in a more substantive way than "A madman in an institution created this heavy-duty concept." My justification for the parts that are hard is that at least it's not just giving the reader pablum and abetting certain romantic but flabby ideas about madness and genius and certain mathematical concepts being so forbidden that they drive people crazy.
IDEAS: In your book, you draw a distinction between solutions that are technically correct and those that are intellectually satisfying. For example, you say that the first-year calculus answer to Zeno's paradox is correct but "semi-impoverished." [Zeno claimed that it's impossible to walk across, say, a hotel room, because you would first have to walk halfway across, then halfway across the remaining quarter-length, etc., and there would always be some space left between you and the other side of the room.]
WALLACE: I think I'm working out some of my own anger. I was very frustrated in college math classes, because they just wouldn't tell you why this stuff was important. If you ever had Zeno's dichotomy in a calc class and you saw the calc solution, you don't walk away bathed in relief at the resolution to the paradox, because they strip all the interesting context away. If you're just given "a over 1 minus r" in a math class, I'd say that's semi-impoverished. OK, I can solve that, but I still don't understand how I walked out of the room.
IDEAS: Your book convinced me that irrational numbers [such as ?] can't possibly exist. I don't know what I was thinking, accepting them as real.
WALLACE: You mean, mathematically real or physically real?
IDEAS: Physically real.
WALLACE: But are numbers physically real?
IDEAS: I had thought they were, in some way.
WALLACE: So you're a bit of a Platonist then.
IDEAS: That was my next question for you.
WALLACE: We're brainwashed into being Platonists in elementary school, because it's the easiest way to think about numbers. Nobody wants to tell a fourth-grader the metaphysics of the integer 3, so we've got this idea that 3 is this thing. These are not things. Even if you're a Platonist -- that is, even if you believe that numbers are real in some metaphysical sense, the way trees and Calebs are real, as opposed to mathematically real -- the reason you're convinced of it is we never really think about it. Well, if they are, where are they? What do they look like? What is 3? It's like the speculations that little children have, or adolescents who smoke a lot of pot at 3 o'clock in the morning.
IDEAS: A character in "Infinite Jest" describes "Marijuana Thinking" in a way that reminded me of how you talk about a certain kind of math-philosophical abyss in this book.
WALLACE: The stuff that drives us crazy is the really, really, really basic stuff. What are numbers? What exactly are these three dimensions we're sitting in? Stuff that it's embarrassing to talk about out loud because it sounds like pot-smoking "Oouhh, oouhh, oouhh," but in some sense, this is what mathematicians -- and back in the Greeks, philosophers, because there was no difference at the time -- this is what they did.
IDEAS: So, even though you've resisted the brainwashing, are you a Platonist? Do you think mathematical concepts exist?
WALLACE: Personally, yeah, I'm a Platonist. I think that God has particular languages, and one of them is music and one of them is mathematics. That's not something I can defend. It's just something I've felt in my tummy since I was a little kid, but how exactly to try to make sense of that and to fit it in any kind of a working philosophy, much less cross the street to buy a loaf of bread, is a different matter.
IDEAS: Was it hard to write about the abstract, without plots or bodies?
WALLACE: You know, in a weird way, there's really only one basic problem in all writing -- how to get some empathy with the reader. And that problem is a jewel on which there are many facets. And this is a somewhat different facet -- how to take this very, very abstract stuff, boil it down so that it fits in a pop book, and give the reader enough of the real story so that you're not lying to him, but also to make it clear enough so that it's not just understandable but halfway enjoyable for somebody who hasn't studied math for 20 years.It's really not completely different from the question, how do you get a reader to inhabit the consciousness of a character who, say, isn't a hero or isn't a very nice guy, and feel that person's humanity and something of his 3-D contours while not pretending that he's not a monster.
IDEAS: What are you teaching now at Pomona?
WALLACE: I got hired to teach writing, but they're also letting me teach some literature classes. Last spring I did eclectic modern -- Christina Stead's "The Man Who Loved Children," Joan Didion's "Play It As It Lays," and Richard Brautigan's "In Watermelon Sugar." The kind of books that are big but you know college students are never going to get them in a class.
IDEAS: Do you miss Boston?
WALLACE: I miss parts of Brighton and Allston and the Back Bay very much. But I lived there for only three years and I moved away only a decade ago. I'll go back at some point. Sure. Ride the B Line.
IDEAS: A friend of mine thinks that "Infinite Jest" should be understood as a Boston novel.
WALLACE: I think a lot of the dialect in there probably doesn't make much sense if you don't know Boston. Boston made a big impression on me, because linguistically it's very different than where I'm from.
IDEAS: Was the character Whitey Sorkin inspired by Whitey Bulger?
WALLACE: I don't think Whitey Sorkin's supposed to be an isomorphically unique mapping of Whitey Bulger, but when I was in Boston, there were rumors that Whitey had it fixed so that his people won the lottery. I mean, at least in the parts of Boston in which I was moving, Whitey was a creature of myth.
IDEAS: The Greek epigraph to "Everything and More" -- where's it from?
WALLACE: It's made up. "It is not what's inside your head, it's what your head's inside." It's a gag. I think the editor thought it was some really esoteric ancient Greek. I got a big kick out of it. It was a big deal to get him to get the diacriticals right.
IDEAS: In "Everything and More," once some of the technical questions about infinity are answered, a new abyss opens up with Godel's incompleteness theorem.
WALLACE: Infinity was the great albatross for math -- really, ever since calculus. [The 19th-century mathematicians] Karl Weierstrass, Richard Dedekind, and Cantor close all those holes, and it's beautiful, and at the same time they open what turns out to be a much worse one, as [the 20th-century mathematician and logician] Kurt Godel demonstrated.Godel is able to come up with a theorem that says, "I am not provable." And it's a theorem, which means that, by definition, math is either not consistent or it's not complete. Packed in. He is the devil, for math. After Godel, the idea that mathematics was not just a language of God but a language we could decode to understand the universe and understand everything -- that doesn't work any more. It's part of the great postmodern uncertainty that we live in.
IDEAS: What fiction are you working on now?
WALLACE: Just this morning, I delivered the post-editing draft of a book of stories. All but a couple have been in magazines, although not all under my name. I don't think any of the stories have footnotes, which I'm rather proud of. Got that monkey off my back. I think one story maybe has a couple of asterisk footnotes. You know, there are so few of them that you can use asterisks.
Caleb Crain is the author of "American Sympathy: Men, Friendship, and Literature in the New Nation."
© Copyright 2003 Globe Newspaper Company.