[Paleopsych] NYT: Truth, Incompleteness and the Gödelian Way
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Arts > Connections: Truth, Incompleteness and the Gödelian Way
http://www.nytimes.com/2005/02/14/arts/14conn.html
February 14, 2005
CONNECTIONS
By EDWARD ROTHSTEIN
Relativity. Incompleteness. Uncertainty.
Is there a more powerful modern Trinity? These reigning deities
proclaim humanity's inability to thoroughly explain the world. They
have been the touchstones of modernity, their presence an unwelcome
burden at first, and later, in the name of postmodernism, welcome
company.
Their rule has also been affirmed by their once-sworn enemy: science.
Three major discoveries in the 20th century even took on their names.
Albert Einstein's famous Theory (Relativity), Kurt Gödel's famous
Theorem (Incompleteness) and Werner Heisenberg's famous Principle
(Uncertainty) declared that, henceforth, even science would be
postmodern.
Or so it has seemed. But as Rebecca Goldstein points out in her
elegant new book, "Incompleteness: The Proof and Paradox of Kurt
Gödel" (Atlas Books; Norton), of these three figures, only Heisenberg
might have agreed with this characterization.
His uncertainty principle specified the inability to be too exact
about small particles. "The idea of an objective real world whose
smallest parts exist objectively," he wrote, "is impossible." Oddly,
his allegiance to an absolute state, Nazi Germany, remained
unquestioned even as his belief in absolute knowledge was quashed.
Einstein and Gödel had precisely the opposite perspective. Both fled
the Nazis, both ended up in Princeton, N.J., at the Institute for
Advanced Study, and both objected to notions of relativism and
incompleteness outside their work. They fled the politically absolute,
but believed in its scientific possibility.
And therein lies Ms. Goldstein's tale. From the late 1930's until
Einstein's death in 1955, Einstein and Gödel, the physicist and the
mathematician, would take long walks, finding companionship in each
other's ideas. Late in his life, in fact, Einstein said he would go to
his office just to have the "privilege" of walking with Gödel. What
was their common ground? In Ms. Goldstein's interpretation, they both
felt marginalized, "disaffected and dismissed in profoundly similar
ways." Both thought that their work was being invoked to support
unacceptable positions.
Einstein's convictions are fairly well known. He objected to quantum
physics and its probabilistic clouds. God, he famously asserted, does
not play dice. Also, he believed, not everything depends on the
perspective of the observer. Relativity doesn't imply relativism.
The conservative beliefs of an aging revolutionary? Perhaps, but
Einstein really was a kind of Platonist: He paid tribute to science's
liberating ability to understand what he called the "extra-personal
world."
And Gödel? Most lay readers probably know of him from Douglas R.
Hofstadter's playful best-seller "Gödel, Escher, Bach," a book that is
more about the powers of self-referentiality than about the limits of
knowledge. But the latter is the more standard association. "If you
have heard of him," Ms. Goldstein writes, perhaps too cautiously,
"then there is a good chance that, through no fault of your own, you
associate him with the sorts of ideas - subversively hostile to the
enterprises of rationality, objectivity, truth - that he not only
vehemently rejected but thought he had conclusively, mathematically,
discredited."
Ms. Goldstein's interpretation differs in some respects from that of
another recent book about Gödel, "A World Without Time: The Forgotten
Legacy of Gödel and Einstein" by Palle Yourgrau (Basic), which sees
him as more of an iconoclastic visionary. But in both he is portrayed
as someone widely misunderstood, with good reason perhaps, given his
work's difficulty.
Before Gödel's incompleteness theorem was published in 1931, it was
believed that not only was everything proven by mathematics true, but
also that within its conceptual universe everything true could be
proven. Mathematics is thus complete: nothing true is beyond its
reach. Gödel shattered that dream. He showed that there were true
statements in certain mathematical systems that could not be proven.
And he did this with astonishing sleight of hand, producing a
mathematical assertion that was both true and unprovable.
It is difficult to overstate the impact of his theorem and the
possibilities that opened up from Gödel's extraordinary methods, in
which he discovered a way for mathematics to talk about itself. (Ms.
Goldstein compares it to a painting that could also explain the
principles of aesthetics.)
The theorem has generally been understood negatively because it
asserts that there are limits to mathematics' powers. It shows that
certain formal systems cannot accomplish what their creators hoped.
But what if the theorem is interpreted to reveal something positive:
not proving a limitation but disclosing a possibility? Instead of "You
can't prove everything," it would say: "This is what can be done: you
can discover other kinds of truths. They may be beyond your
mathematical formalisms, but they are nevertheless indubitable."
In this, Gödel was elevating the nature of the world, rather than
celebrating powers of the mind. There were indeed timeless truths. The
mind would discover them not by following the futile methodologies of
formal systems, but by taking astonishing leaps, making unusual
connections, revealing hidden meanings.
Like Einstein, Gödel was, Ms. Goldstein suggests, a Platonist.
Of course, those leaps and connections could go awry. Gödel was an
intermittent paranoiac, whose twisted visions often left his
colleagues in dismay. He spent his later years working on a proof of
the existence of God. He even died in the grip of a perverse
esotericism. He feared eating, imagined elaborate plots, and literally
wasted away. At his death in 1978, he weighed 65 pounds.
But he was no postmodernist. Late in his life Gödel said of
mathematics: "It is given to us in its entirety and does not change,
unlike the Milky Way. That part of it of which we have a perfect view
seems beautiful, suggesting harmony." That beauty, he proposed, would
be mirrored by the world itself. These are not exactly the views of an
acolyte devoted to Relativity, Incompleteness and Uncertainty. And
Einstein was his fellow dissenter.
The Connections column will appear every other Monday.
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