Waiting for Gödel

Positivism started as a kind of intellectual housecleaning following World War I. In Vienna, Goldstein writes, "the overall topic was the moral and intellectual death and decay of all that had come before, and the need to construct entirely new methodologies, forms and foundations." The Vienna Circle celebrated empirical science as the basis for those new foundations, but as the ideas of positivism evolved and filtered into such disciplines as literary theory, anthropology and linguistics, science itself became a target of skepticism. Enter, postmodernism. Every form of knowledge came to be analyzed as a set of rules created by flawed human beings whose biases inevitably infected those rules. The relativity, uncertainty and incompleteness of Einstein, Heisenberg and Gvdel became metaphors for the unreliability of what we once took to be truth.

Einstein, whose famous theory is so often misrepresented in the maxim "Everything is relative," might not have been so ardent a Platonist as Gvdel, but he was no subjectivist. According to Goldstein, "Einstein interpreted his theory as representing the objective nature of space-time, so very different from our human, subjective point of view of space and time." Both men believed that an objective, abstract reality existed and that the human mind could behold and grasp this reality. As Gvdel saw it, we achieve this through a faculty called mathematical intuition.

Gvdel's faith in mathematical intuition put him in opposition to the mathematical equivalent of positivism, a movement called formalism. Formalism, led by the mathematician David Hilbert, believed that mathematics was, in Hilbert's words, "a game played according to certain simple rules with meaningless marks on paper." Or, as Goldstein describes it, "mathematicians, according to formalism, are not in the business of discovering descriptive truths, whether of the real world of things in physical space or the trans-empirical world of numbers and sets ... They are simply in the business of manipulating the mechanical rules of self-enclosed formal systems."

If formalism were correct, then it followed that mathematics could also be overhauled so that every part of it was "consistent" and the entire system was "complete." It could be boiled down to a set of rules or axioms and procedures so basic and ironclad that a machine -- the computers that were just beyond the historical horizon -- could perform it. It could be finally purged of the paradoxes that had been plaguing the field for hundreds of years. Mathematical intuition, the source of ideas that can't be formally proven but possessing what Goldstein calls, "the urgent cogency that compels belief," has no place in such a system.

Gvdel's theorem used the rules of formalism itself to demonstrate that the formalist project could never be achieved. In what Goldstein calls "one of the most astounding pieces of mathematical reasoning ever produced," he demonstrated that in the kind of system that the formalists aspired to, it was possible to make a statement that was both unprovable and yet also true. This works a little like the famous "liar's paradox," in which the statement "This statement is false," can only be true if it is also false and vice versa. But Gvdel's theorem was not a paradox, precisely because it pointed to the difference between what could be proven and what was true.

It's easy to see why the distinction confuses people. While Gvdel might seem to be challenging reason itself, he was, to put a finer point on it, actually demonstrating the inherent problem in a particularly limited form of reasoning. He never abandoned his own faith in another, broader form of reason that includes intuition. Goldstein writes that Gvdel's adherence to his unfashionable Platonic beliefs made him a kind of intellectual exile, and that this, as well as genius, sealed the bond he shared with the equally alienated Einstein.

The two men famously walked together through the streets of Princeton to the Institute for Advanced Studies every day, and Einstein once said that he kept his position there only for the opportunity it gave him to walk with Gvdel. One of the many topics they discussed was the theory of relativity, and on Einstein's 70th birthday, Gvdel presented the older man with a paper that used Einstein's field observations to argue that time itself does not exist. (Given a sufficiently fast rocket ship traveling in a sufficiently wide curve, it's possible, Gvdel claimed, to make a round trip that would carry you into the past.) There is some difference of opinion on whether Einstein accepted this interpretation, but the philosophy professor Palle Yourgrau says he did and champions Gvdel's theory in the somewhat more technical new book "A World Without Time: The Forgotten Legacy of Gvdel and Einstein."

After Einstein's death in 1955, Gvdel became increasingly tormented by fears of persecution, some of which were projected onto the great 17th century German mathematician Gottfried Wilhelm Leibniz. Leibniz's work, Gvdel insisted, had been villainously suppressed in order to "make men stupid." He ate less and less, ostensibly for fear of being poisoned, until finally he died of malnutrition, weighing a mere 65 pounds.

Goldstein calls Gvdel's mental illness "that rationality run amuck which is paranoia." If Gvdel's gifts finally turned on him, it could hardly have been more cruel than the realization that his work and that of his closest friend had been so commonly misunderstood. Goldstein begins her book with several quotations from books on philosophy and history -- one published during Gvdel's lifetime and assigned to her in college -- that characterize Gvdel's work as undermining the very possibility of discovering truth. Einstein is depicted in Michael Frayn's 1998 play "Copenhagen" as proving that "the universe exists only as a series of approximations. Only within the limits determined by our relationship with it. Only through the understanding lodged within the human head." Given what Einstein and Gvdel really did believe, you can hardly blame Gvdel for suspecting a conspiracy to make men stupid.

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