|
Here is the article
February 14, 2005 Monday Late Edition - Final SECTION: Section E; Column 5; The Arts/Cultural Desk; CONNECTIONS; Pg.
1
LENGTH: 1030 words
HEADLINE: Truth, Incompleteness And the Godelian Way
BYLINE: By EDWARD ROTHSTEIN
BODY: 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
Godel'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
Godel'' (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 Godel 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 Godel, 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 Godel. 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 Godel? Most lay readers probably know of
him from Douglas R. Hofstadter's playful best-seller '' Godel, 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 Godel, ''A World Without
Time: The Forgotten Legacy of Godel 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 Godel'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. Godel 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 Godel'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, Godel 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, Godel was, Ms. Goldstein
suggests, a Platonist.
Of course, those leaps and connections could go
awry. Godel 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
Godel 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.
mailto:avromf@xxxxxxxx or mailto:af413@xxxxxxxx
----- Original Message -----
From:
mailto:cld@xxxxxxxx
To:
xywrite@xxxxxxxx
Sent: Monday, February 14, 2005 11:16 AM
Subject: Re: Countchars >I'll look for the NYT article. -- Carl Distefano mailto:cld@xxxxxxxx |