Does G�del Matter?
The romantic's favorite mathematician didn't prove what you think he did.
By Jordan Ellenberg
the Washington Post's SLATE/Posted Thursday, March 10, 2005, at 4:27 AM PT
The reticent and relentlessly abstract logician Kurt G�del might seem an
unlikely candidate for popular appreciation. But that's what Rebecca Goldstein
aims for in her new book _Incompleteness_, an account of G�del's most famous
theorem, which was announced 75 years ago this October. Goldstein calls G�del's
incompleteness theorem "the third leg, together with Heisenberg's uncertainty
principle and Einstein's relativity, of that tripod of theoretical cataclysms
that have been felt to force disturbances deep down in the foundations of the
'exact sciences.' "
What is this great theorem? And what difference does it really make?
Mathematicians, like other scientists, strive for simplicity; we want to boil
messy phenomena down to some short list of first principles called axioms, akin
to basic physical laws, from which everything we see can be derived. This
tendency goes back as far as Euclid, who used just five postulates to deduce
his geometrical theorems.
But plane geometry isn't all of mathematics, and other fields proved
surprisingly resistant to axiomatization; irritating paradoxes kept springing
up, to be knocked down again by more refined axiomatic systems. The so-called
"formalist program" aimed to find a master list of axioms, from which all of
mathematics could be derived by rigid logical deduction. Goldstein cleverly
compares this objective to a "Communist takeover of mathematics" in which
individuality and intuition would be subjugated, for the common good, to
logical rules. By the early 20th century, this outcome was understood to be the
condition toward which mathematics must strive.
Then G�del kicked the whole thing over.
G�del's incompleteness theorem says:
Given any system of axioms that produces no paradoxes, there exist
statements about numbers which are true, but which cannot be proved using the
given axioms.
In other words, there is no hope of reducing even mere arithmetic, the starting
point of mathematics, to axioms; any such system will miss out on some truths.
And G�del not only shows that true-but-unprovable statements exist -- he
produces one! His method is a marvel of ingenuity; he encodes the notion of
"provability" itself into arithmetic and thereby devises an arithmetic
statement P that, when decoded, reads:
P is not provable using the given axioms.
So a proof of P would imply that P was false -- in other words, the proof of P
would itself constitute a disproof of P, and we have found a paradox. So we're
forced to concede that P is not provable -- which is precisely what P claims.
So P is a true statement that cannot be proved with the given axioms. (The
dizzy-making self-reference inherent in this argument is the subject of Douglas
Hofstadter's Pulitzer Prize-winning _G�del, Escher, Bach_, a mathematical
exposition of clarity, liveliness, and scope unequalled since its publication
in 1979.)
One way to understand G�del's theorem (in combination with his 1929
"completeness theorem") is that no system of logical axioms can produce all
truths about numbers because no system of logical axioms can pin down exactly
what numbers are. My fourth-grade teacher used to ask the class to define a
peanut butter sandwich, with comic results. Whatever definition you propose
(say, "two slices of bread with peanut butter in between"), there are still
lots of non-peanut-butter-sandwiches that fall within its scope (say, two
pieces of bread laid side by side with a stripe of peanut butter spread on the
table between them). Mathematics, post-G�del, is very similar: There are many
different things we could mean by the word "number," all of which will be
perfectly compatible with our axioms. Now G�del's undecidable statement P
doesn't seem so paradoxical. Under some interpretations of the word "number,"
it is true; under others, it is false.
In his recent New York _Times_ review of _Incompleteness_, Edward Rothstein
wrote that it's "difficult to overstate the impact of G�del's theorem." But
actually, it's easy to overstate it: Goldstein does it when she likens the
impact of G�del's incompleteness theorem to that of relativity and quantum
mechanics and calls him "the most famous mathematician that you have most
likely never heard of." But what's most startling about G�del's theorem, given
its conceptual importance, is not how much it's changed mathematics, but how
little. No theoretical physicist could start a career today without a thorough
understanding of Einstein's and Heisenberg's contributions. But most pure
mathematicians can easily go through life with only a vague acquaintance with
G�del's work. So far, I've done it myself.
How can this be, when G�del cuts the very definition of "number" out from under
us? Well, don't forget that just as there are some statements that are true
under any definition of "peanut butter sandwich" -- for instance, "peanut
butter sandwiches contain peanut butter" -- there are some statements that are
true under any definition of "number" -- for instance, "2 + 2 = 4." It turns
out that, at least so far, interesting statements about number theory are much
more likely to resemble "2 + 2 = 4" than G�del's vexing "P." G�del's theorem,
for most working mathematicians, is like a sign warning us away from logical
terrain we'd never visit anyway.
What is it about G�del's theorem that so captures the imagination? Probably
that its oversimplified plain-English form -- "There are true things which
cannot be proved" -- is naturally appealing to anyone with a remotely romantic
sensibility. Call it "the curse of the slogan": Any scientific result that can
be approximated by an aphorism is ripe for misappropriation. The precise
mathematical formulation that is G�del's theorem doesn't really say "there are
true things which cannot be proved" any more than Einstein's theory means
"everything is relative, dude, it just depends on your point of view." And it
certainly doesn't say anything directly about the world outside mathematics,
though the physicist Roger Penrose does use the incompleteness theorem in
making his controversial case for the role of quantum mechanics in human
consciousness. Yet, G�del is routinely deployed by people with antirationalist
agendas as a stick to whack any offending piece of science that happens by. A
typical recent article, "Why Evolutionary Theories Are Unbelievable," claims,
"Basically, G�del's theorems prove the Doctrine of Original Sin, the need for
the sacrament of penance, and that there is a future eternity." If G�del's
theorems could prove that, he'd be even more important than Einstein and
Heisenberg!
One person who would not have been surprised about the relative inconsequence
of G�del's theorem is G�del himself. He believed that mathematical objects,
like numbers, were not human constructions but real things, as real as peanut
butter sandwiches. Goldstein, whose training is in philosophy, is at her
strongest when tracing the relation between G�del's mathematical results and
his philosophical commitments. If numbers are real things, independent of our
minds, they don't care whether or not we can define them; we apprehend them
through some intuitive faculty whose nature remains a mystery. From this point
of view, it's not at all strange that the mathematics we do today is very much
like the mathematics we'd be doing if G�del had never knocked out the
possibility of axiomatic foundations. For G�del, axiomatic foundations, however
useful, were never truly necessary in the first place. His work was
revolutionary, yes, but it was a revolution of the most unusual kind: one that
abolished the constitution while leaving the material circumstances of the
citizens more or less unchanged.
Jim Devine, e-mail: [EMAIL PROTECTED]
web: http://myweb.lmu.edu/jdevine/
