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.''

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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.   Continued...