Kurt Gödel was a 5-foot-6-inch titan, a reticent introvert who ''produced the most loquacious theorems in the history of mathematics," to quote novelist Rebecca Goldstein. Close friends with Albert Einstein, the most significant logician since Aristotle, Gödel was also a man who hid from visitors and whose lifelong hypochondria ultimately deteriorated into paranoia and self-starvation.
Skeletal but frustratingly opaque, revolutionary but fearful of controversy, Gödel is an almost-paradox: a man of intense conviction with an anti-charisma, a thinker whose famous theorems have had enormous but often misappropriated implications.
A thumbnail of his personality is in many ways a thumbnail of his most famous proof. Roughly, Gödel's incompleteness theorems prove that there can be true arithmetical propositions that cannot be proved. Or, in Goldstein's words, ''A system rich enough to contain arithmetic cannot be both consistent and complete."
Paradox, conundrum, metaphor. To try to capture in prose the simple but complex nature of Gödel and his work is flat-out daunting. Of course, any attempt to systematize any person's life -- to capture between the covers of a book every contradiction, every gradation in character -- is doomed to failure, but there can be no biography in which incompleteness would weigh more heavily on a writer's mind than Gödel's.
Nonetheless, in ''Incompleteness: The Proof and Paradox of Kurt Gödel," Goldstein does a formidable job. Central to her approach is a revision: For over 70 years, rebels against objectivity (existentialists and postmodernists, in particular) have heard what they wanted to hear in Gödel's theorems. They draw from them the suggestion that formal systems, like arithmetic, are social constructs. If even our most codified systems of thought are incomplete, the reasoning follows, then all truths are obviously and essentially manufactured.
But Goldstein argues -- very convincingly -- that Gödel was in fact a Platonist who believed the universe possesses an abstract reality graspable not through the senses but through reason. Mathematical truths are not manufactured, he would argue; they are universal and objective; they exist independent of anything human. Think of them like distant planets, orbiting in the darkness; their existence has nothing to do with whether we're able to detect them or not.
The irony here is that Gödel's celebrated theorems were and continue to be hijacked by the very thinkers to whom he was diametrically opposed. Thus the Austrian exile from Nazi Germany was doubly exiled, and his isolation and ultimate descent into delusion become more poignant.
The final chapter of ''Incompleteness" deals with Gödel's late work. Enter another book about Gödel, ''A World Without Time," by Palle Yourgrau, a philosophy professor at Brandeis. Yourgrau, too, is interested in revision. In his case, he hopes to rescue Gödel's ideas about time from obscurity. There is, Yourgrau believes, a ''conspiracy of silence" shrouding the fruits of Gödel's conversations with Einstein.
In a 1949 paper Gödel used general relativity to theorize universes in which time travel is possible. Einstein declared the paper ''an important contribution," and the questions it brings up are as fascinating as they are thorny: If the past is always accessible in these hypothetical universes, then the past must coexist with the present, and if so, how can time as we understand it exist? And if time doesn't exist in any of these so-called Gödel universes, how can it in ours?
Yourgrau covers much of the same ground Goldstein does. Occasionally he fills in details Goldstein misses; occasionally the case is otherwise. Once in a while they contradict each other. ''Gödel was at heart an ironist," writes Yourgrau; ''Gödel, unlike his friend Einstein, did not have a well-developed sense of the ironic," writes Goldstein. Unfortunately, neither author answers the question I'd most hoped they would: What emotional, physical, and intellectual journey did Gödel undertake in producing his famous proof?
It's not really the biographers' fault. The problem is, a finished proof, by its very nature, no longer shows the struts and scaffolding that went into its construction. The sweat of its creator is rinsed away, and all we're left with are the clean lines of math, and the breathtaking result. Goldstein comes as close as she can get when she writes, ''It must have been an extraordinarily exhilarating experience." That's about all we get.
Maybe, though, Alan Lightman offers a glimmer of that exhilaration in his book of collected essays, ''A Sense of the Mysterious." After making a minor discovery about the theoretical properties of high-temperature gases, he recalls, ''I experienced a kaleidoscope of emotions. . . . I had found something new . . . something that no one had ever known before me, and I felt elated and powerful with knowledge." Lightman's new book is worth a look for many reasons, especially an essay entitled ''Metaphor in Science," written within the system of language about the limitations of that system, something Gödel, ironist or not, might have enjoyed. About his own small discovery, Lightman writes, ''I had shed light on a small corner of nature. Other scientists had illuminated larger corners. But there were almost certainly vast chambers and ballrooms that remained in the dark."
You can't help but wonder what kaleidoscope of emotions Gödel passed through when he arrived, at 24, at a mathematical proof that would irrevocably change human thought. I wonder if he saw how beautiful it was, if he realized, early on, how his work could creep beneath any fence you tried to put around it, exceeding mathematics, exceeding logic, radiating into the humanities, into the philosophy of the mind, the mysteries within human reach, and the ones still in shadow.
Anthony Doerr is the author of ''The Shell Collector" and ''About Grace."