[Paleopsych] NYTBR: 'Incompleteness': Waiting for Gödel
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NYTBR: 'Incompleteness': Waiting for Gödel
New York Times Book Review, 5.5.1
http://www.nytimes.com/2005/05/01/books/review/01SCHULMA.html
By POLLY SHULMAN
INCOMPLETENESS
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
proved.
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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