[Paleopsych] NYTBR: 'Incompleteness': Waiting for Gödel

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NYTBR: 'Incompleteness': Waiting for Gödel
New York Times Book Review, 5.5.1


The Proof and Paradox of Kurt Gödel.
By Rebecca Goldstein.
Illustrated. 296 pp. Atlas Books/ W. W. Norton & Company. $22.95.

    REBECCA GOLDSTEIN, as anyone knows who has read her novels --
    particularly ''The Mind-Body Problem'' -- understands that people are
    thinking beings, and the mind's loves matter at least as much as the
    heart's. After all, she's not just a novelist, but a philosophy
    professor. She casts ''Incompleteness,'' her brief life of the
    logician Kurt Gödel (1906-78), as a touching intellectual love story.
    Though Gödel was married, his wife barely appears here; as Goldstein
    tells it, his romance was with mathematical Platonism, the idea that
    the glories of mathematics exist eternally beyond our grasp. Gödel's
    Platonism inspired him to deeds as daring as any knight's: he proved
    his famous incompleteness theorem for its sake. His Platonism also set
    him apart from his intellectual contemporaries. Only Einstein shared
    it, and could solace Gödel's loneliness, Goldstein argues. A biography
    with two focuses -- a man and an idea -- ''Incompleteness'' unfolds
    its surprisingly accessible story with dignity, tenderness and awe.

    News of Gödel's Platonism, or Einstein's, might surprise readers
    familiar with popular interpretations of their work. For centuries,
    science seemed to be tidying the mess of the real world into an
    eternal order beautiful and pure -- a heavenly file cabinet labeled
    mathematics. Then, in the early 20th century, Einstein published his
    relativity theory, Werner Heisenberg his uncertainty principle and
    Gödel his incompleteness theorem. Many thinkers -- from the logical
    positivists with whom Gödel drank coffee in the Viennese cafes of the
    1920's to existentialists, postmodernists and annoying people at
    cocktail parties -- have taken those three results as proof that
    reality is subjective and we can't see beyond our noses. You can
    hardly blame them. As Goldstein points out, the very names of the
    theories seem to mock the notion of objective truth. But she makes a
    persuasive case that Gödel and Einstein understood their work to prove
    the opposite: there is something greater than our little minds;
    reality exists, whether or not we can ever touch it.

    It's appropriate, though sad for Gödel, that his work has been
    interpreted to have simultaneously opposite meanings. The proofs of
    his famous theorems rely on just that sort of twisty thinking:
    statements like the famous Liar's Paradox, ''This statement is
    false,'' which flip their meanings back and forth. In the case of the
    Liar's Paradox, if the statement is true, then it's false -- but if
    it's false, then it's true. Like that paradox, an assertion that talks
    about itself, Gödel's theorems are meta-statements, which speak about
    themselves. Because Gödel made so much of self-reference and paradox,
    previous books about his work -- like Douglas Hofstadter's ''Gödel,
    Escher, Bach'' -- tend to emphasize the playfulness of his ideas. Not
    Goldstein's. She tells his story in a minor key, following Gödel into
    the paranoia that overtook him after Einstein's death, growing out of
    his loneliness and unrelenting rationality. After all, paranoia, like
    math, makes people dig deeper and deeper to find meaning.

    Gödel's work addresses the core of mathematics: finding proofs. Proofs
    are mathematicians' road to truth. To find them, mathematicians from
    the ancient Greeks on have set up systems consisting of three basic
    elements: axioms, true statements so intuitively obvious they are
    self-evident; rules of inference, logical principles indicating how to
    use axioms to prove new, less obviously true statements; and those new
    true statements, called theorems. (Many Americans met axioms and
    proofs for the first and last time in 10th-grade geometry.) A century
    ago, mathematicians began taking these systems to an extreme. Since
    mathematical intuition can be as unreliable as other kinds of
    intuitions -- often things that seem obvious turn out to be just plain
    wrong -- they tried to eliminate it from their axioms. They built new
    systems of arbitrary symbols and formal rules for manipulating them.
    Of course, they chose those particular symbols and rules because of
    their resemblance to mathematical systems we care about (such as
    arithmetic). But, by choosing rules and symbols that work whether or
    not there's any meaning behind them, the mathematicians kept the
    potential corruption of intuition at bay. The dream of these
    formalists was that their systems contained a proof for every true
    statement. Then all mathematics would unfurl from the arbitrary
    symbols, without any need to appeal to an external mathematical truth
    accessible only to our often faulty intuition.

    Gödel proved exactly the opposite, however. He showed that in any
    formal system complicated enough to describe the numbers and
    operations of arithmetic, as long as the axioms don't lead to
    contradictions there will always be some statement that is not
    provable -- and the contradiction of it will not be provable either.
    He also showed that there's no way to prove from within the system
    that the system itself won't give rise to contradictions. So, any
    formal system worth bothering with will either sprout contradictions
    -- which is bad news, since once you have a contradiction, you can
    prove anything at all, including 2 + 2 = 5 -- or there will be
    perfectly ordinary statements that may well be true but can never be

    You can see why this result rocked mathematics. You can also see why
    positivists, existentialists and postmodernists had a field day with
    it, particularly since, once you find one of those unprovable
    statements, you're free to add it to your system as an axiom, or else
    to add its complete opposite. Either way, you'll get a new system that
    works fine. That makes math sound pretty subjective, doesn't it?

    Well, Gödel didn't think so, and his reason grows beautifully from his
    spectacular proof itself, which Goldstein describes with lucid
    discipline. Though the proof relies on a meticulous, fiddly mechanism
    that took an entire semester to build up when I studied logic as a
    math major in college, its essence fits magically into a few pages of
    a book for laypeople. It can even, arguably, fit in a single paragraph
    of a book review -- though that may be stretching.

    To put it roughly, Gödel proved his theorem by taking the Liar's
    Paradox, that steed of mystery and contradiction, and harnessing it to
    his argument. He expressed his theorem and proof in mathematical
    formulas, of course, but the idea behind it is relatively simple. He
    built a representative system, and within it he constructed a
    proposition that essentially said, ''This statement is not provable
    within this system.'' If he could prove that that was true, he
    figured, he would have found a statement that was true but not
    provable within the system, thus proving his theorem. His trick was to
    consider the statement's exact opposite, which says, ''That first
    statement -- the one that boasted about not being provable within the
    system -- is lying; it really is provable.'' Well, is that true?
    Here's where the Liar's Paradox shows its paces. If the second
    statement is true, then the first one is provable -- and anything
    provable must be true. But remember what that statement said in the
    first place: that it can't be proved. It's true, and it's also false
    -- impossible! That's a contradiction, which means Gödel's initial
    assumption -- that the proposition was provable -- is wrong.
    Therefore, he found a true statement that can't be proved within the
    formal system.

    Thus Gödel showed not only that any consistent formal system
    complicated enough to describe the rules of grade-school arithmetic
    would have an unprovable statement, but that it would have an
    unprovable statement that was nonetheless true. Truth, he concluded,
    exists ''out yonder'' (as Einstein liked to put it), even if we can
    never put a finger on it.

    John von Neumann, the father of game theory, took up Gödel's cause in
    America; in England, Alan Turing provided an alternative proof of
    Gödel's theorem while inventing theoretical computer science. Whatever
    Gödel's work had to say about reality, it changed the course of
    mathematics forever.

    Polly Shulman is a contributing editor for Science magazine and has
    written about mathematics for many other publications.

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