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Rebecca Goldstein (left), author of 'Incompleteness: The Proof and Paradox of Kurt Gödel,' at home in Cambridge; Gödel and Albert Einstein in Princeton, N.J., in 1954.
Rebecca Goldstein (left), author of "Incompleteness: The Proof and Paradox of Kurt Gödel," at home in Cambridge; Gödel and Albert Einstein in Princeton, N.J., in 1954. (Globe Staff Photo / Mark Wilson; Leonard McCombe Photo / Time Life Pictures, Getty Images)

The mathematician's lament

IN HER NEW BOOK, ''Incompleteness: The Proof and Paradox of Kurt Gödel,'' Rebecca Goldstein, a novelist and currently visiting professor of philosophy at Trinity College, brings all her skills to bear on a difficult man and his difficult math. As she explained in a recent interview at her Cambridge apartment, she set out to correct misinterpretations of Gödel's work, which transformed the philosophical underpinnings of mathematics, but ended up ''inhabiting his mind.''

When Gödel, born in 1906 in what is now the Czech Republic, was formulating his ideas in Vienna in the 1920s, mathematicians across the world theorized that arithmetic was a human construction. They were sure that math arose from a set of man-made rules (like a modern-day computer program), or was ''like a higher form of chess.''

That went against everything Gödel believed: For him, math was a description of an abstract reality, transcending human rules and inventions. In 1930, the 23-year-old Gödel thought he had proved that such an abstract world did exist. With his first Incompleteness theorem, he demonstrated that in a mathematical system there are things that are true that cannot be proved. He followed with a second Incompleteness theorem, which said it was impossible to prove the consistency of a mathematical system when you are working within that system.

The proofs transformed logic and branches of math, but Gödel was tragically misunderstood. Far from what he intended, many took ''incompleteness'' to mean that philosophical uncertainty had spread from the humanities and arts to the most logical human enterprise - math.

Gödel immigrated to the United States in 1940 and took up residence at the Institute for Advanced Studies in Princeton, where he found a conversation partner and confidant in Albert Einstein, but felt increasingly alienated by a world that did not understand his breakthrough. After Einstein's death, Gödel descended into ever deeper paranoia and madness.

Years ago, Goldstein immersed herself in the incompleteness proofs as a graduate student in philosophy at Princeton. After writing novels, including ''The Mind-Body Problem'' (set in Princeton), that grapple with philosophical questions and feature characters driven by some of the frenzied intellectual heroism that she saw in Gödel, she said it seemed somehow inevitable to take on the man himself.

IDEAS: You call Gödel's Incompleteness theorem one of three ''theoretical cataclysms'' that shook the world over the last century. What are they?

GOLDSTEIN: Relativity theory. Incompleteness. Uncertainty. They are often lumped together as showing the assault on objectivity and truth that was coming from the last century, even from the sciences. Really grist for the postmodern mill. And this I don't agree with. Einstein and Gödel were diametrically opposed to this point of view. Einstein [in his relativity theory] was trying to show that objective reality was very different from the way we experience it. Gödel, the way he interpreted his Incompleteness theorem was as demonstrating there was an objective, mathematical truththat we don't create our mathematical truths, because we can see there are truths that can't be proved.

IDEAS: They were great friends, but it seems an odd pairing. There is Einstein, the icon, and Gödel, the virtual unknown. What drew them together?

GOLDSTEIN: On a psychological level, this was a friendship that baffled people.... But for me, it doesn't seem so mysterious. Both men were very much at odds with the intellectual climate, even of their peers. Both men saw their work in this light of objectivity.

IDEAS: How has Gödel's theorem been misinterpreted over the years?

GOLDSTEIN: It's taken to show that math has no foundations, that math is uncertain. So just as relativity was misinterpreted - as meaning everything is just relative to points of view, that we create the truth, that it's all at heart human - [Gödel's theorems] were interpreted that way as well.

I just feel such sympathy [for Gödel]. He was somebody who hated to argue face-to-face, and he was in a [discussion] group, the Vienna Circle, which was completely at odds with everything he thought and he just sat there and he listened and meanwhile he's brewing this theorem that was going to just devastate their point of view, but he never says anything. He just stays mum. He wanted his math to do all his talking for him. And then he produced this theorem that did what he wanted - miraculously, amazingly - and everyone misinterpreted it.

IDEAS: How did his math shape his life?

GOLDSTEIN: [Gödel showed that] one of the things you can't prove is that the world you're working with is consistent - that it won't lead to a contradiction. And this is interesting because it has a very sad, poignant parallel with the man himself. Gödel did suffer from mental illness. He did have these paranoid delusions. He believed that he had enemies, that there were plots out for him, that truth would be suppressed. He really thought there was something malicious going on. The parallel I'm trying to draw is if your whole thought system is infected by persecution, how do you get outside the system to show that the system isn't leading you to untruths or, even worse, to self-destructive madness? He eventually ... starved himself to death because he thought he was being poisoned. It's an extraordinarily stunning though tragic parallel with his second Incompleteness theorem.... When you're mad, you can't get out of your system.

IDEAS: What does Gödel's theorem mean for the person on the street?

GOLDSTEIN: I think all of us want to know what kind of a world we live in, we want to know what is the ontological furniture - what exists, what's the furniture of the world? There's the existence of God, this sort of abstract entity - do objective moral values exist? And does math also present us with something other than things we can reach out and touch? Is math itself somehow descriptive of a reality other than what we can see and feel? I think the man on the street worries about what exists. God, moral values, and what else?

Carolyn Y. Johnson is a Globe staff writer. Email

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