The Magazine

Visions of Infinity

A mathematical voyage brought Kurt Gödel to the shores of madness.

Jun 19, 2006, Vol. 11, No. 38 • By DAVID GUASPARI
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Incompleteness

The Proof and Paradox of Kurt Gödel

by Rebecca Goldstein

Norton, 296 pp., $22.95

REBECCA GOLDSTEIN'S FINE BOOK makes Kurt Gödel the protagonist of a tragic love story. Enamored, at age 20, with the Platonic vision of a realm of objective truth, he made his life a quest for it. His work in mathematics and logic created new fields of research and provoked voluminous commentary. Arguably, it shed light on the nature of the mind and, therefore, on what it means to be human. But the story ends in bitter irony. Gödel's work was routinely enlisted in a war against the very possibility of objective truth, and his final years were consumed by what is, at least figuratively, a disease of reason gone wild: clinical paranoia.

Goldstein, a philosopher and novelist, presents a moving picture of a passionate life devoted to "abstruse" concerns and invokes, appropriately, the Platonic theme that a genuine philosopher hungers for truth with an intensity that is erotic. Incompleteness is much more rewarding than the garrulous publishing phenomenon Gödel, Escher, Bach--which labors under the handicap that Gödel, Escher, and Bach have no nontrivial connections with one another.

What can be plausibly said about the inner life of someone as guarded and opaque as Kurt Gödel? Someone who limited his public statements to propositions that he could rigorously prove, and was given in private to show-stopping gnomic utterance: e.g., "I don't believe in natural science."

Goldstein finds a key in Gödel's deep friendship with Albert Einstein, who said that he went to his office every day "for the privilege of walking home with Gödel." They were intellectual peers. (Gödel, though unknown to the general public, was venerated by the world's best mathematicians.) But their personalities could not have been more different--one furtive and enigmatic, the other with the public face of an affable secular saint.

What bound them together? Both believed deeply that science is a search for objective truth, and worked only on problems they believed to be scientifically and philosophically important. Most significant, Goldstein suggests, is that these very commitments made them "intellectual exiles," because the 20th century saw (and we are still seeing) an "intellectual revolt against objectivity and rationality." Goldstein reads Gödel's work as a direct response. (I should note that the eminent logician Solomon Feferman, one of the editors of the exemplary edition of Gödel's collected works, has sharply and convincingly criticized Goldstein for reading Gödel's later philosophical convictions into his younger self.)

To understand why a mathematical result might have philosophical implications requires attention to the surprisingly central role that the philosophy of mathematics has played in the history of Western thought. Anyone who wonders about what we know and how we know it will be struck by the peculiar nature of mathematics. It seems to offer truths that are certain but not based on any actual experience of the world. Immanuel Kant summed up this peculiarity in a famous question that he placed at the center of the Critique of Pure Reason: How is pure mathematics possible?

We'll consider two broad answers that predate (and postdate) Kant, which I'll call Platonist and anti-Platonist. Plato regarded mathematics as a model of true knowledge, of apprehending the world's underlying reality undistorted by its appearance. The anti-Platonists, a heterogeneous coalition of the unwilling, hold that any supposed "knowledge" making no appeal to experience can be certain only insofar as it says nothing at all about the world. "All unmarried men are mortal" is an awfully good bet but, like any summary of experience, is open to revision. "All unmarried men are bachelors" is certain precisely because it relies on no facts about men or marriage, but only on the definitions of the words it contains.

The anti-Platonists could be refuted by proving mathematical results with philosophical significance. The results would be certain, because mathematical; yet the very fact that they provoked--better yet, required--a philosophical response would show that they were most assuredly about something. The heart of Goldstein's book is an account of such a result, Gödel's First Incompleteness Theorem.

I will state it anachronistically, by reference to computing. Euclid presents geometry as a theory: a body of axioms, taken as self-evident, from which further geometrical truths are deduced by proofs, which are sequences of logical steps. By the end of the 19th century it had become clear that any known field of mathematics (and anything likely ever to be accepted as mathematics) could, in principle, be expressed in a formal theory.