[Paleopsych] NYT: Truth, Incompleteness and the Gödelian Way

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Arts > Connections: Truth, Incompleteness and the Gödelian Way
February 14, 2005


    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

    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

    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

    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,

    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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