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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.

A theory is formal if a computer can be programmed to recognize whether or not any alleged proof is valid. Insight may be needed to devise proofs but, if they are formal, not to check them: A computer can scan a sequence of symbols and mindlessly apply mechanical rules to determine whether it represents valid logical steps starting from axioms.

Any "useful" theory can be made formal (because all mathematics can), must be consistent (not allow the deduction of contradictory results), and will incorporate at least the ability to perform elementary arithmetic (with rules for addition, multiplication, and simple algebra, such as x+y=y+x).

The First Incompleteness Theorem, slightly cleaned up, says that any such "useful" theory is incomplete; that is, there are propositions of elementary arithmetic that the theory can neither prove nor disprove. In fact, we can find such a proposition that is essentially the statement of an elaborate algebraic rule. And someone who understands what that proposition means will recognize that it is true--because, under a clever interpretation, it means that it cannot be proven.

Were we to add this proposition as a new axiom, the Incompleteness Theorem would immediately provide another truth unprovable in this beefed-up theory. There is no escape.

This is head-spinning. How, for example, can we "recognize" that some arithmetical rule is true if we cannot prove it from beloved axioms that, hitherto, seemed to say all we knew about numbers? And just how does one reason to a conclusion about the limits of reasoning?

It is also an affront. David Hilbert, one of history's greatest mathematicians, testified to an ancient faith when he said that "In mathematics, there is no ignorabimus." When we seek an answer, "we must know. We will know."

Goldstein presents a sketch of Gödel's reasoning that should be accessible to a general reader. Some small technical miscues are not seriously misleading. (For the cognoscenti: The fixed-point lemma is misstated; and the demonstration that the "Gödel sentence" is unprovable appeals unnecessarily to its meaning, and thereby relies on a stronger hypothesis than consistency.)

Gödel announced his results in one terse paragraph at the end of a three-day technical meeting--and made not a ripple. Only the great scientific polymath John von Neumann realized that something important had happened. But what?

Gödel's interpretation is startling. It follows, he says, that at least one of the following two things must be true--and he, in fact, believed both: "Either the human mind cannot be reduced to the working of the brain" or "mathematical objects and facts . . . exist objectively and independently of our mental acts and decisions." (These quotations come from Logical Dilemmas, John Dawson's biography of Gödel. Goldstein cites opinions a bit less provocative.)

Here's my guess at what he meant: If my mind is only my brain, it's some kind of computer, so all its purely deductive workings are captured within some "useful" theory. But my mind can also prove the First Incompleteness Theorem, allowing it to recognize some mathematical truths that the theory does not imply. That act of recognition can't be a deduction, so must be a perception--of an independently existing mathematical realm.

Imagine Gödel's dismay, if not despair, when many drew the opposite conclusion: that mathematics had lost its certainty and been revealed as a social practice based on arbitrary decisions about what axioms to choose. Or when, as Goldstein says, "[Ludwig] Wittgenstein never accepted that Gödel had proved what he provably did prove" since it would have contradicted Wittgenstein's deep belief that logic was necessarily empty, and could contain no surprises.

Gödel's last years became a perverse reductio ad absurdum of his sustaining belief that everything has a rational explanation. What had been intermittent episodes of paranoia and depression became chronic. He feared that his food might be poisoned, and that dangerous gases were seeping from his radiators and ice box. He starved to death.

Goldstein does not avert her eyes from this obscene ending, but does not allow it the final word. She concludes with an elegant novelistic turn. Gödel pursued philosophical questions about time in his characteristic way, by seeking results that could be established mathematically. He produced a surprising solution of Einstein's gravitational field equations, a hypothetical universe in which it is possible to travel in time. If time does loop back on itself, Goldstein says, "then a young Gödel will once again sit in a college classroom in Vienna, transfigured by the notion of the infinite eternal verities. . . . He will dream, silently and audaciously, of proving a mathematical theorem the likes of which has never before been seen, a mathematical theorem that will illuminate the nature of mathematics itself.

"And then he will do it."

David Guaspari is a mathematician and computer scientist in Ithaca, N.Y.