It turns out that these intellectual romances can be disastrous for science, too, as Rebecca Goldstein shows in her masterful new book, "Incompleteness: The Proof and Paradox of Kurt Gödel." The book is part of the Great Discoveries series, in which each volume is dedicated to a single important scientific breakthrough. The designated breakthrough in this volume is the revolutionary demonstration, by the Viennese logician Kurt Gödel, that in any formal system complex enough to handle numbers, there inevitably exists at least one formula that is both true and unprovable, and that, by extension, no such formal system can prove itself to be consistent or complete.

A professor of philosophy as well as the author of five interesting, if highly cerebral novels, Goldstein makes the ideal guide to both Gödel's theorem and the way it's been represented -- or, more often, grievously misrepresented -- in the humanities. She even met Gödel once, at a garden party at Princeton, where she was a graduate student and young logician in the early 1970s, though she reports that she was, like the rest of the students present, "awed into stupidity" by the very presence of the great man. (Gödel, in residence at Princeton's fabled Institute for Advanced Studies since World War II, died in 1978.)

	Read our interview with Andrew Sean Greer and save 30% on The Confessions of Max Tivoli
	"Incompleteness: The Proof and Paradox of Kurt Godel" By Rebecca Goldstein Norton/Atlas Books 296 pages Nonfiction Buy this book
	"A World Without Time: The Forgotten Legacy of Godel and Einstein" By Palle Yourgrau Basic Books 210 pages Nonfiction Buy this book

Gödel's theorem often gets grouped with two better-known and equally epochal discoveries in physics, Einstein's Theory of Relativity and Heisenberg's Uncertainty Principle. Some thinkers see all three as, in Goldstein's words, "high-grade grist for the postmodern mill, pulverizing the old absolutist ways of thinking about truth and certainty, objectivity and rationality." For those who wish to argue that "the very notion of the objectively true is a socially constructed myth," these three towering minds make for impressive allies.

That's all well and good for Werner Heisenberg who, Goldstein says, found such notions congenial. But Einstein and Gödel neither intended nor saw their work as undermining the idea of objective truth; in fact, quite the opposite. Goldstein's "Incompleteness" is both an eminently lucid explanation of Gödel's theorem and its implications, and a full-blooded defense of it from those who would claim it for the cause of pomo relativism.

Gödel used arithmetic in proving his theorem, but since Goldstein is most interested in the philosophical implications of his work, "Incompleteness" doesn't use many numbers or equations. It's a challenging book, nevertheless, because of its focus on abstract concepts, but it doesn't require specialized knowledge and it's never gratuitously murky. (There are only about three pages, concerning a technique Gödel developed for turning the very syntax of mathematics into numbers -- the "Gödel numbering" -- that remain stubbornly opaque to this nonmathematician.) It's more concise and less cosmic than another popular exploration of some of the same ideas, Douglas Hofstadter's "Gödel, Escher, Bach."

According to Goldstein, Gödel and his close friend Einstein stood in opposition to a philosophical movement that came to dominate 20th century thought, positivism. The epicenter of positivism was Vienna, Austria, between the two world wars and specifically within an elite group known as the Vienna Circle. The demigod of the positivists was the philosopher Ludwig Wittgenstein; Wittgenstein himself insisted that he was not a positivist at all and became so disgusted with the group's misunderstanding of his ideas that during some of their meetings he'd sit facing the wall and recite verse by a mystically inclined Indian poet. (The choice was deliberately provocative, since the Vienna Circle disdained anything smacking of metaphysics.) Gödel also attended the Circle's meetings, though he kept his objections to himself.

Very roughly, the positivists declared that only statements that can in principle be factually verified by the evidence provided by our senses have any real meaning. Other statements, dealing with abstractions or the definitions of words (i.e., "All bilingual people speak two languages") they dubbed "trivial" or "meaningless" because their meanings exist only within the system of language. (Wittgenstein call such statements "language games.")

Goldstein uses the very helpful metaphor of chess to illuminate this distinction. The game of chess has a complex and unified set of rules, but it isn't about anything but itself. As magnificently as they function, the rules of chess are a set of entirely man-made ideas and don't describe anything in the real, material world. Likewise, according to the positivists (and, in this instance, Wittgenstein), much of what philosophy had traditionally concerned itself with -- abstract questions about the nature of being, reality and God -- were "meaningless" outside the closed, man-made system of language (including the language of mathematics).

Gödel, on the other hand, was a Platonist. Like the ancient Greek philosopher, he believed that there exists, in objective reality external to the human mind, an ideal realm. Plato called the objects in this realm "the forms," and described the physical world as we experience it as consisting of mere shadows of the forms of this ideal realm. Where the positivists would argue that mathematics is merely a system of symbols created entirely by human beings, Gödel considered it an apprehension or understanding of some transcendent objective (if not necessarily material) reality. For the positivists, the number "2" was an idea and a symbol; for Gödel it was an actual, and very beautiful, thing.