[FoRK] Good Gödel, Batman!
Kelley
kelley at inkworkswell.com
Sat Mar 12 15:48:43 PST 2005
For the geek squad on this lovely Saturday -- here in the USofA anyway. An
acquaintance thinks this review/book is hogwash:
From: ravi <gadfly EXITLEFT ORG>
Status: U
X-UIDL: 409562133
this whole piece is rather confusing and muddled. first the subtitle
says that godel didn't prove what you think he did. however, in the text
below, the reviewer seems to agree that most people do not even know who
godel is. no big deal. what i find unconvincing is the idea that his
result is no big deal. 1) attacks by "mainstream" mathematicians against
logicians and foundations of math folks are neither new nor opaque
(after all, why entertain/encourage someone who questions your very
programme!). 2) godel's result is indeed a metamathematical result. the
fact that this permits a run of the mill mathematician to continue his
work does little however to diminish the result, including for
mathematics. but it is the much broader impact/implication of the result
that makes it important. 3) pointing to a few misinterpretations/misuses
of godel's incompleteness theorem does not help the argument either.
my opinion: skip the review, and perhaps the book, and read either
hofstadter or penrose, or if in a lighter mood, the somewhat interesting
"uncle petros and goldbach's conjecture":
http://www.maa.org/reviews/petros.html
Additionally, "There was a delightfully intelligent and well-written review
article of this book and another one on Goedel by Jim Holt in the New
Yorker a few weeks ago. Highly recommended -- if you're in the mood for
such things:
http://www.newyorker.com/critics/atlarge/?050228crat_atlarge "
k
Does Gödel 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 Gödel might seem an
unlikely candidate for popular appreciation. But that's what Rebecca
Goldstein aims for in her new book _Incompleteness_, an account of Gödel's
most famous theorem, which was announced 75 years ago this October.
Goldstein calls Gödel'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 Gödel kicked the whole thing over.
Gödel'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 Gödel 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 _Gödel, Escher,
Bach_, a mathematical exposition of clarity, liveliness, and scope
unequalled since its publication in 1979.)
One way to understand Gödel'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-Gödel, 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
Gödel'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 Gödel's theorem." But
actually, it's easy to overstate it: Goldstein does it when she likens the
impact of Gödel'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 Gödel'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 Gödel's work. So far, I've done it myself.
How can this be, when Gödel 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
Gödel's vexing "P." Gödel'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 Gödel'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 Gödel'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, Gödel
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,
Gödel's theorems prove the Doctrine of Original Sin, the need for the
sacrament of penance, and that there is a future eternity." If Gödel'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 Gödel's theorem is Gödel 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
Gödel'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 Gödel had never knocked out the possibility of axiomatic
foundations. For Gödel, 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.
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