http://cgi.pathfinder.com/time/time100/scientist/profile/bernerslee.html
Network Designer
Tim Berners-Lee=20
=46rom the thousands of interconnected threads of the Internet, he wove=20
the World Wide Web and created a mass medium for the 21st century
------------------------------------------------------------------------
BY JOSHUA QUITTNER
Want to see how much the world has changed in the past decade? Log on=20
to the Internet, launch a search engine and type in the word enquire=20
(British spelling, please). You'll get about 30,000 hits. It turns=20
out you can "enquire" about nearly anything online these days, from=20
used Harley Davidsons for sale in Sydney, Australia ("Enquire about=20
touring bikes. Click here!"), to computer-training-by-e-mail courses=20
in India ("Where excellence is not an act but a habit"). Click once=20
to go to a site in Nairobi and enquire about booking shuttle=20
reservations there. Click again, and zip off to Singapore, to a=20
company that specializes in "pet moving." Enquire about buying=20
industrial-age nuts and bolts from "the Bolt Boys" in South Africa,=20
or teddy bears in upstate New York. Exotic cigar labels! Tantric sex=20
guides! Four-poster beds for dogs!
So what, you say? Everybody knows that with a mouse, a modem and=20
access to the Internet, these days you can point-and-click anywhere=20
on the planet, unencumbered by time or space or long-distance phone=20
tariffs.
Ah, but scroll down the list far enough, hundreds of entries deep,=20
and you'll find this hidden Rosebud of cyberspace: "Enquire Within=20
Upon Everything"--a nifty little computer program written nearly 20=20
years ago by a lowly software consultant named Tim Berners-Lee. Who=20
knew then that from this modest hack would flow the=20
civilization-altering, millionaire-spawning, information suckhole=20
known as the World Wide Web?
Unlike so many of the inventions that have moved the world, this one=20
truly was the work of one man. Thomas Edison got credit for the light=20
bulb, but he had dozens of people in his lab working on it. William=20
Shockley may have fathered the transistor, but two of his research=20
scientists actually built it. And if there ever was a thing that was=20
made by committee, the Internet--with its protocols and packet=20
switching--is it. But the World Wide Web is Berners-Lee's alone. He=20
designed it. He loosed it on the world. And he more than anyone else=20
has fought to keep it open, nonproprietary and free.
It started, of all places, in the Swiss Alps. The year was 1980.=20
Berners-Lee, doing a six-month stint as a software engineer at CERN,=20
the European Laboratory for Particle Physics, in Geneva, was noodling=20
around with a way to organize his far-flung notes. He had always been=20
interested in programs that dealt with information in a "brain-like=20
way" but that could improve upon that occasionally memory-constrained=20
organ. So he devised a piece of software that could, as he put it,=20
keep "track of all the random associations one comes across in real=20
life and brains are supposed to be so good at remembering but=20
sometimes mine wouldn't." He called it Enquire, short for Enquire=20
Within Upon Everything, a Victorian-era encyclopedia he remembered=20
from childhood.
Building on ideas that were current in software design at the time,=20
Berners-Lee fashioned a kind of "hypertext" notebook. Words in a=20
document could be "linked" to other files on Berners-Lee's computer;=20
he could follow a link by number (there was no mouse to click back=20
then) and automatically pull up its related document. It worked=20
splendidly in its solipsistic, Only-On-My-Computer way.
But what if he wanted to add stuff that resided on someone else's=20
computer? First he would need that person's permission, and then he=20
would have to do the dreary work of adding the new material to a=20
central database. An even better solution would be to open up his=20
document--and his computer--to everyone and allow them to link their=20
stuff to his. He could limit access to his colleagues at CERN, but=20
why stop there? Open it up to scientists everywhere! Let it span the=20
networks! In Berners-Lee's scheme there would be no central manager,=20
no central database and no scaling problems. The thing could grow=20
like the Internet itself, open-ended and infinite. "One had to be=20
able to jump," he later wrote, "from software documentation to a list=20
of people to a phone book to an organizational chart to whatever."
So he cobbled together a relatively easy-to-learn coding system--HTML=20
(HyperText Mark-up Language)--that has come to be the lingua franca=20
of the Web; it's the way Web-content creators put those little=20
colored, underlined links in their text, add images and so on. He=20
designed an addressing scheme that gave each Web page a unique=20
location, or url (universal resource locator). And he hacked a set of=20
rules that permitted these documents to be linked together on=20
computers across the Internet. He called that set of rules HTTP=20
(HyperText Transfer Protocol).
And on the seventh day, Berners-Lee cobbled together the World Wide=20
Web's first (but not the last) browser, which allowed users anywhere=20
to view his creation on their computer screen. In 1991 the World Wide=20
Web debuted, instantly bringing order and clarity to the chaos that=20
was cyberspace. From that moment on, the Web and the Internet grew as=20
one, often at exponential rates. Within five years, the number of=20
Internet users jumped from 600,000 to 40 million. At one point, it=20
was doubling every 53 days.
=20
Raised in London in the 1960s, Berners-Lee was the quintessential=20
child of the computer age. His parents met while working on the=20
=46erranti Mark I, the first computer sold commercially. They taught=20
him to think unconventionally; he'd play games over the breakfast=20
table with imaginary numbers (what's the square root of minus 4?). He=20
made pretend computers out of cardboard boxes and five-hole paper=20
tape and fell in love with electronics. Later, at Oxford, he built=20
his own working electronic computer out of spare parts and a TV set.=20
He also studied physics, which he thought would be a lovely=20
compromise between math and electronics. "Physics was fun," he=20
recalls. "And in fact a good preparation for creating a global=20
system."
It's hard to overstate the impact of the global system he created.=20
It's almost Gutenbergian. He took a powerful communications system=20
that only the elite could use and turned it into a mass medium. "If=20
this were a traditional science, Berners-Lee would win a Nobel=20
Prize," Eric Schmidt, CEO of Novell, once told the New York Times.=20
"What he's done is that significant."
You'd think he would have at least got rich; he had plenty of=20
opportunities. But at every juncture, Berners-Lee chose the nonprofit=20
road, both for himself and his creation. Marc Andreessen, who helped=20
write the first popular Web browser, Mosaic--which, unlike the=20
master's browser, put images and text in the same place, like pages=20
in a magazine--went on to co-found Netscape and become one of the=20
Web's first millionaires. Berners-Lee, by contrast, headed off in=20
1994 to an administrative and academic life at the Massachusetts=20
Institute of Technology. From a sparse office at M.I.T., he directs=20
the W3 Consortium, the standard-setting body that helps Netscape,=20
Microsoft and anyone else agree on openly published protocols rather=20
than hold one another back with proprietary technology. The rest of=20
the world may be trying to cash in on the Web's phenomenal growth,=20
but Berners-Lee is content to labor quietly in the background,=20
ensuring that all of us can continue, well into the next century, to=20
Enquire Within Upon Anything.
------------------------------------------------------------------------
Joshua Quittner, TIME's Personal Technology columnist, is the new=20
editor of TIME DIGITAL
QUIZ:
Tim Berners-Lee once said that one of his favorite website illustrates how t=
o:
a) Build your own computer
b) Light a barbecue
c) Build a Web site in XML
d) Build an animation in Flash
[A: the website doesn't give the link to George Goble's=20
accomplishment, though..]
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D
http://cgi.pathfinder.com/time/time100/scientist/profile/turing.html
Computer Scientist
Alan Turing=20
While addressing a problem in the arcane field of mathematical logic,=20
he imagined a machine that could mimic human reasoning. Sound=20
familiar?
------------------------------------------------------------------------
BY PAUL GRAY
If all Alan Turing had done was answer, in the negative, a vexing=20
question in the arcane realm of mathematical logic, few=20
nonspecialists today would have any reason to remember him. But the=20
method Turing used to show that certain propositions in a closed=20
logical system cannot be proved within that system--a corollary to=20
the proof that made Kurt Godel famous--had enormous consequences in=20
the world at large. For what this eccentric young Cambridge don did=20
was to dream up an imaginary machine--a fairly simple typewriter-like=20
contraption capable somehow of scanning, or reading, instructions=20
encoded on a tape of theoretically infinite length. As the scanner=20
moved from one square of the tape to the next--responding to the=20
sequential commands and modifying its mechanical response if so=20
ordered--the output of such a process, Turing demonstrated, could=20
replicate logical human thought.
The device in this inspired mind-experiment quickly acquired a name:=20
the Turing machine. And so did another of Turing's insights. Since=20
the instructions on the tape governed the behavior of the machine, by=20
changing those instructions, one could induce the machine to perform=20
the functions of all such machines. In other words, depending on the=20
tape it scanned, the same machine could calculate numbers or play=20
chess or do anything else of a comparable nature. Hence his device=20
acquired a new and even grander name: the Universal Turing Machine.
Does this concept--a fairly rudimentary assemblage of hardware=20
performing prodigious and multifaceted tasks according to the=20
dictates of the instructions fed to it--sound familiar? It certainly=20
didn't in 1937, when Turing's seminal paper, "On Computable Numbers,=20
with an Application to the Entscheidungsproblem," appeared in=20
"Proceedings of the London Mathematical Society." Turing's thoughts=20
were recognized by the few readers capable of understanding them as=20
theoretically interesting, even provocative. But no one recognized=20
that Turing's machine provided a blueprint for what would eventually=20
become the electronic digital computer.
So many ideas and technological advances converged to create the=20
modern computer that it is foolhardy to give one person the credit=20
for inventing it. But the fact remains that everyone who taps at a=20
keyboard, opening a spreadsheet or a word-processing program, is=20
working on an incarnation of a Turing machine.
Turing's 1937 paper changed the direction of his life and embroiled a=20
shy and vulnerable man ever more directly in the affairs of the world=20
outside, ultimately with tragic consequences.
Alan Mathison Turing was born in London in 1912, the second of his=20
parents' two sons. His father was a member of the British civil=20
service in India, an environment that his mother considered=20
unsuitable for her boys. So John and Alan Turing spent their=20
childhood in foster households in England, separated from their=20
parents except for occasional visits back home. Alan's loneliness=20
during this period may have inspired his lifelong interest in the=20
operations of the human mind, how it can create a world when the=20
world it is given proves barren or unsatisfactory.
=20
At 13 he enrolled at the Sherbourne School in Dorset and there showed=20
a flair for mathematics, even if his papers were criticized for being=20
"dirty," i.e., messy. Turing recognized his homosexuality while at=20
Sherbourne and fell in love, albeit undeclared, with another boy at=20
the school, who suddenly died of bovine tuberculosis. This loss=20
shattered Turing's religious faith and led him into atheism and the=20
conviction that all phenomena must have materialistic explanations.=20
There was no soul in the machine nor any mind behind a brain. But=20
how, then, did thought and consciousness arise?
After twice failing to win a fellowship at the University of=20
Cambridge's Trinity College, a lodestar at the time for=20
mathematicians from around the world, Turing received a fellowship=20
from King's College, Cambridge. King's, under the guidance of such=20
luminaries as John Maynard Keynes and E.M. Forster, provided a=20
remarkably free and tolerant environment for Turing, who thrived=20
there even though he was not considered quite elegant enough to be=20
initiated into King's inner circles. When he completed his degree=20
requirements, Turing was invited to remain at King's as a tutor. And=20
there he might happily have stayed, pottering about with problems in=20
mathematical logic, had not his invention of the Turing machine and=20
World War II intervened.
Turing, on the basis of his published work, was recruited to serve in=20
the Government Code and Cypher School, located in a Victorian mansion=20
called Bletchley Park in Buckinghamshire. The task of all those so=20
assembled--mathematicians, chess champions, Egyptologists, whoever=20
might have something to contribute about the possible permutations of=20
formal systems--was to break the Enigma codes used by the Nazis in=20
communications between headquarters and troops. Because of secrecy=20
restrictions, Turing's role in this enterprise was not acknowledged=20
until long after his death. And like the invention of the computer,=20
the work done by the Bletchley Park crew was very much a team effort.=20
But it is now known that Turing played a crucial role in designing a=20
primitive, computer-like machine that could decipher at high speed=20
Nazi codes to U-boats in the North Atlantic.
After the war, Turing returned to Cambridge, hoping to pick up the=20
quiet academic life he had intended. But the newly created=20
mathematics division of the British National Physical Laboratory=20
offered him the opportunity to create an actual Turing machine, the=20
ACE or Automatic Computing Engine, and Turing accepted. What he=20
discovered, unfortunately, was that the emergency spirit that had=20
short-circuited so many problems at Bletchley Park during the war had=20
dissipated. Bureaucracy, red tape and interminable delays once again=20
were the order of the day. Finding most of his suggestions dismissed,=20
ignored or overruled, Turing eventually left the NPL for another stay=20
at Cambridge and then accepted an offer from the University of=20
Manchester where another computer was being constructed along the=20
lines he had suggested back in 1937.
Since his original paper, Turing had considerably broadened his=20
thoughts on thinking machines. He now proposed the idea that a=20
machine could learn from and thus modify its own instructions. In a=20
famous 1950 article in the British philosophical journal Mind, Turing=20
proposed what he called an "imitation test," later called the "Turing=20
test." Imagine an interrogator in a closed room hooked up in some=20
manner with two subjects, one human and the other a computer. If the=20
questioner cannot determine by the responses to queries posed to them=20
which is the human and which the computer, then the computer can be=20
said to be "thinking" as well as the human.
Turing remains a hero to proponents of artificial intelligence in=20
part because of his blithe assumption of a rosy future: "One day=20
ladies will take their computers for walks in the park and tell each=20
other, 'My little computer said such a funny thing this morning!'"
Unfortunately, reality caught up with Turing well before his vision=20
would, if ever, be realized. In Manchester, he told police=20
investigating a robbery at his house that he was having "an affair"=20
with a man who was probably known to the burglar. Always frank about=20
his sexual orientation, Turing this time got himself into real=20
trouble. Homosexual relations were still a felony in Britain, and=20
Turing was tried and convicted of "gross indecency" in 1952. He was=20
spared prison but subjected to injections of female hormones intended=20
to dampen his lust. "I'm growing breasts!" Turing told a friend. On=20
June 7, 1954, he committed suicide by eating an apple laced with=20
cyanide. He was 41.
------------------------------------------------------------------------
TIME senior writer Paul Gray writes on a Turing machine
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D
http://cgi.pathfinder.com/time/time100/scientist/profile/godel.html
Mathematician
Kurt G=F6del=20
He turned the lens of mathematics on itself and hit upon his famous=20
"incompleteness theorem"--driving a stake through the heart of=20
formalism
------------------------------------------------------------------------
BY DOUGLAS HOFSTADTER
Kurt G=F6del was born in 1906 in Brunn, then part of the=20
Austro-Hungarian Empire and now part of the Czech Republic, to a=20
father who owned a textile factory and had a fondness for logic and=20
reason and a mother who believed in starting her son's education=20
early. By age 10, G=F6del was studying math, religion and several=20
languages. By 25 he had produced what many consider the most=20
important result of 20th century mathematics: his famous=20
"incompleteness theorem." G=F6del's astonishing and disorienting=20
discovery, published in 1931, proved that nearly a century of effort=20
by the world's greatest mathematicians was doomed to failure.
To appreciate G=F6del's theorem, it is crucial to understand how=20
mathematics was perceived at the time. After many centuries of being=20
a typically sloppy human mishmash in which vague intuitions and=20
precise logic coexisted on equal terms, mathematics at the end of the=20
19th century was finally being shaped up. So-called formal systems=20
were devised (the prime example being Russell and Whitehead's=20
Principia Mathematica) in which theorems, following strict rules of=20
inference, sprout from axioms like limbs from a tree. This process of=20
theorem sprouting had to start somewhere, and that is where the=20
axioms came in: they were the primordial seeds, the Ur-theorems from=20
which all others sprang.
The beauty of this mechanistic vision of mathematics was that it=20
eliminated all need for thought or judgment. As long as the axioms=20
were true statements and as long as the rules of inference were truth=20
preserving, mathematics could not be derailed; falsehoods simply=20
could never creep in. Truth was an automatic hereditary property of=20
theoremhood.
The set of symbols in which statements in formal systems were written=20
generally included, for the sake of clarity, standard numerals, plus=20
signs, parentheses and so forth, but they were not a necessary=20
feature; statements could equally well be built out of icons=20
representing plums, bananas, apples and oranges, or any utterly=20
arbitrary set of chicken scratches, as long as a given chicken=20
scratch always turned up in the proper places and only in such proper=20
places. Mathematical statements in such systems were, it then became=20
apparent, merely precisely structured patterns made up of arbitrary=20
symbols.
Soon it dawned on a few insightful souls, G=F6del foremost among them,=20
that this way of looking at things opened up a brand-new branch of=20
mathematics--namely, metamathematics. The familiar methods of=20
mathematical analysis could be brought to bear on the very=20
pattern-sprouting processes that formed the essence of formal=20
systems--of which mathematics itself was supposed to be the primary=20
example. Thus mathematics twists back on itself, like a self-eating=20
snake.
Bizarre consequences, G=F6del showed, come from focusing the lens of=20
mathematics on mathematics itself. One way to make this concrete is=20
to imagine that on some far planet (Mars, let's say) all the symbols=20
used to write math books happen--by some amazing coincidence--to look=20
like our numerals 0 through 9. Thus when Martians discuss in their=20
textbooks a certain famous discovery that we on Earth attribute to=20
Euclid and that we would express as follows: "There are infinitely=20
many prime numbers," what they write down turns out to look like this:
"84453298445087 87863070005766619463864545067111."
To us it looks like one big 46-digit number. To Martians, however, it=20
is not a number at all but a statement; indeed, to them it declares=20
the infinitude of primes as transparently as that set of 34 letters=20
constituting six words a few lines back does to you and me.
Now imagine that we wanted to talk about the general nature of all=20
theorems of mathematics. If we look in the Martians' textbooks, all=20
such theorems will look to our eyes like mere numbers. And so we=20
might develop an elaborate theory about which numbers could turn up=20
in Martian textbooks and which numbers would never turn up there. Of=20
course we would not really be talking about numbers, but rather about=20
strings of symbols that to us look like numbers. And yet, might it=20
not be easier for us to forget about what these strings of symbols=20
mean to the Martians and just to look at them as plain old numerals?
By such a simple shift of perspective, G=F6del wrought deep magic. The=20
G=F6delian trick is to imagine studying what might be called=20
"Martian-producible numbers" (those numbers that are in fact theorems=20
in the Martian textbooks), and to ask questions such as, "Is or is=20
not the number 8030974 Martian-producible (M.P., for short)?" This=20
question means, Will the statement '8030974' ever turn up in a=20
Martian textbook?
G=F6del, in thinking very carefully about this rather surreal scenario,=20
soon realized that the property of being M.P. was not all that=20
different from such familiar notions as "prime number," "odd number"=20
and so forth. Thus earthbound number theorists could, with their=20
standard tools, tackle such questions as, "Which numbers are M.P.=20
numbers, and which are not?" for example, or "Are there infinitely=20
many non-M.P. numbers?" Advanced math textbooks--on Earth, and in=20
principle on Mars as well--might have whole chapters about M.P.=20
numbers.
And thus, in one of the keenest insights in the history of=20
mathematics, G=F6del devised a remarkable statement that said simply,=20
"X is not an M.P. number" where X is the exact number we read when=20
the statement "X is not an M.P. number" is translated into Martian=20
math notation. Think about this for a little while until you get it.=20
Translated into Martian notation, the statement "X is not an M.P.=20
number" will look to us like just some huge string of digits--a very=20
big numeral. But that string of Martian writing is our numeral for=20
the number X (about which the statement itself talks). Talk about=20
twisty; this is really twisty! But twists were G=F6del's=20
specialty--twists in the fabric of space-time, twists in reasoning,=20
twists of all sorts.
By thinking of theorems as patterns of symbols, G=F6del discovered that=20
it is possible for a statement in a formal system not only to talk=20
about itself, but also to deny its own theoremhood. The consequences=20
of this unexpected tangle lurking inside mathematics were rich,=20
mind-boggling and--rather oddly--very sad for the Martians. Why sad?=20
Because the Martians--like Russell and Whitehead--had hoped with all=20
their hearts that their formal system would capture all true=20
statements of mathematics. If G=F6del's statement is true, it is not a=20
theorem in their textbooks and will never, ever show up--because it=20
says it won't! If it did show up in their textbooks, then what it=20
says about itself would be wrong, and who--even on Mars--wants math=20
textbooks that preach falsehoods as if they were true?
The upshot of all this is that the cherished goal of formalization is=20
revealed as chimerical. All formal systems--at least ones that are=20
powerful enough to be of interest--turn out to be incomplete because=20
they are able to express statements that say of themselves that they=20
are unprovable. And that, in a nutshell, is what is meant when it is=20
said that G=F6del in 1931 demonstrated the "incompleteness of=20
mathematics." It's not really math itself that is incomplete, but any=20
formal system that attempts to capture all the truths of mathematics=20
in its finite set of axioms and rules. To you that may not come as a=20
shock, but to mathematicians in the 1930s, it upended their entire=20
world view, and math has never been the same since.
G=F6del's 1931 article did something else: it invented the theory of=20
recursive functions, which today is the basis of a powerful theory of=20
computing. Indeed, at the heart of G=F6del's article lies what can be=20
seen as an elaborate computer program for producing M.P. numbers, and=20
this "program" is written in a formalism that strongly resembles the=20
programming language Lisp, which wasn't invented until nearly 30=20
years later.
G=F6del the man was every bit as eccentric as his theories. He and his=20
wife Adele, a dancer, fled the Nazis in 1939 and settled at the=20
Institute for Advanced Study in Princeton, where he worked with=20
Einstein. In his later years G=F6del grew paranoid about the spread of=20
germs, and he became notorious for compulsively cleaning his eating=20
utensils and wearing ski masks with eye holes wherever he went. He=20
died at age 72 in a Princeton hospital, essentially because he=20
refused to eat. Much as formal systems, thanks to their very power,=20
are doomed to incompleteness, so living beings, thanks to their=20
complexity, are doomed to perish, each in its own unique manner.
------------------------------------------------------------------------
Douglas Hofstadter is the Pulitzer-prizewinning author of "Godel, Escher, Ba=
ch"
http://vex.net/~buff/godel.html
Rudy Rucker's explanation of the Incompleteness Theorem, from=20
_Infinity and the Mind_:
The proof of G=F6del's Incompleteness Theorem is so simple, and so=20
sneaky, that it is almost embarassing to relate. His basic procedure=20
is as follows:
1. Someone introduces G=F6del to a UTM, a machine that is supposed=20
to be a Universal Truth Machine, capable of correctly answering any=20
question at all.
2. G=F6del asks for the program and the circuit design of the UTM.=20
The program may be complicated, but it can only be finitely long.=20
Call the program P(UTM) for Program of the Universal Truth Machine.
3. Smiling a little, G=F6del writes out the following sentence:=20
"The machine constructed on the basis of the program P(UTM) will=20
never say that this sentence is true." Call this sentence G for=20
G=F6del. Note that G is equivalent to: "UTM will never say G is true."
4. Now G=F6del laughs his high laugh and asks UTM whether G is true or not.
5. If UTM says G is true, then "UTM will never say G is true" is=20
false. If "UTM will never say G is true" is false, then G is false=20
(since G =3D "UTM will never say G is true"). So if UTM says G is true,=20
then G is in fact false, and UTM has made a false statement. So UTM=20
will never say that G is true, since UTM makes only true statements.
6. We have established that UTM will never say G is true. So=20
"UTM will never say G is true" is in fact a true statement. So G is=20
true (since G =3D "UTM will never say G is true").
7. "I know a truth that UTM can never utter," G=F6del says. "I=20
know that G is true. UTM is not truly universal."
Think about it - it grows on you ...
With his great mathematical and logical genius, G=F6del was able to=20
find a way (for any given P(UTM)) actually to write down a=20
complicated polynomial equation that has a solution if and only if G=20
is true. So G is not at all some vague or non-mathematical sentence.=20
G is a specific mathematical problem that we know the answer to, even=20
though UTM does not! So UTM does not, and cannot, embody a best and=20
final theory of mathematics ...
Although this theorem can be stated and proved in a rigorously=20
mathematical way, what it seems to say is that rational thought can=20
never penetrate to the final ultimate truth ... But, paradoxically,=20
to understand G=F6del's proof is to find a sort of liberation. For many=20
logic students, the final breakthrough to full understanding of the=20
Incompleteness Theorem is practically a conversion experience. This=20
is partly a by-product of the potent mystique G=F6del's name carries.=20
But, more profoundly, to understand the essentially labyrinthine=20
nature of the castle is, somehow, to be free of it.