Incompleteness: The Proof and Paradox of Kurt Gödel

Incompleteness: The Proof and Paradox of Kurt Gödel

By Rebecca Goldstein
New York: W. W. Norton, 2005. Hardcover, 296 pages.

Kurt Godel is generally considered to be the pre-eminent mathematician/logician of the past century, a man whose intellectual prowess and influence is often compared to Einstein's. Godel's major theorems produced or transformed several branches of modern mathematical logic: model theory, recursion theory, set theory, proof theory, and intuitionist logic. His work has exerted a marked effect on computer science and the philosophy of consciousness, by suggesting that there are limits to what computers are capable of, and that the human mind is quite a bit more than just a computer. His incompleteness theorems were a serious blow to attempts to prove the fundamental soundness of formalized mathematical systems (systems that are entirely self-validating, depending on nothing outside of their internal rules to prove the inconsistency).

Godel's groundbreaking work resulted in the disquieting notion that within mathematical systems that are consistent, there will be propositions that can not be proven true or false. Godel further showed that proof of a mathematical system's consistency can never be ascertained by appealing to the rules of the system alone, and that, consequently, mathematical systems are not simply man-made constructs, but contain truths that are "independent of any human activities."

Godel was also a very strange man. Always intensely private, Godel, by the time of his 1978 death in Princeton, New Jersey, had disintegrated into a paranoid, anorexic recluse (his death was essentially caused by self-starvation stemming from his fears of being poisoned ), who had alienated himself from just about everyone , including Princeton's intellectual elite.

A new book, Incompleteness: The Proof and Paradox of Kurt Godel, by novelist and philosophy professor Rebecca Goldstein, is a compelling discussion of the man and his ideas. Goldstein's technical analysis of Godel's incompleteness theorems takes up about a third of the book; it is both thorough and illustrative, but it is not light reading. However, the book's biographical dimension, and Goldstein's musings on the ramifications of Godel's ideas are eminently accessible and fascinating.

Goldstein briefly touches on Godel's precocious childhood, but her focus is primarily on his adult life: Godel's connection to Wittgenstein's famous Vienna Circle (Godel, the ardent but covert Platonist, regularly attended their meetings, never revealing the metaphysical inclinations that clashed so profoundly with their radically empirical positivism); his relationship with Einstein, perhaps the deepest friendship of his life (Einstein said that, in his later years, he went to his Princeton office only "for the privilege of walking home with Godel; his global treks in search of an intellectual haven; and his gradual descent into madness.

Goldstein was a graduate student at Princeton during Godel's final years, and she sprinkles her text with memorable anecdotes: her wonder-struck pilgrimage to Godel's house (to her astonishment, a plastic pink flamingo adorned the icon's front lawn); a Godel "sighting" in a grocery store that touched off an excited discussion amongst academics about the contents of his cart; and a party, during which a daring graduate student called Godel's house, hanging up in a panic when Godel's wife called "Kurtsy" to the phone.

Goldstein tells us that, like many of Einstein's ideas, Godel's theories have been frequently misconstrued, and she postulates that both men were drawn together primarily by their shared sense of frustration.

Goldstein writes that Einstein's work has been generally seen as opening the door to a purely subjective universe that changes as often as our viewpoints. However, she declares, Einstein saw ultimate reality as unquestionably objective, though quite different from what our perceptions would lead us to believe. On the other hand, she says, Godel, because of his association with the Vienna Circle, is often thought of as a harbinger of anti-metaphysical positivism, part of the movement to squash "the old absolutist ways of thinking," when actually his work, heavily influenced by Plato's metaphysics, points to a supra-human realm of mathematical laws requiring both intuition and deduction for access (astutely, Goldstein notes that Wittgenstein was actually not a positivist, as well, believing that metaphysical concerns were essentially ineffable, but of supreme importance).

Goldstein writes that Godel's achievements range far beyond the sphere of mathematical logic, "addressing such vast and messy issues as the nature of truth and knowledge and certainty." Indeed, Goldstein declares that Godel's interpretation of his work "shows us that our minds, in knowing mathematics, are escaping the limitations of man-made systems, grasping the independent truths of abstract reality."


May/June 2006