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elativity. 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.
