Darwin's Dangerous Idea - Evolution and the Meaning of Life

The Sword in the Stone 429
CHAPTER FIFTEEN
The Emperor's New Mind,
and Other Fables
1. THE SWORD IN THE STONE
In other words then, if a machine is expected to be infallible, it cannot
also be intelligent. There are several theorems which say almost exactly
that. But these theorems say nothing about how much intelligence may
be displayed if a machine makes no pretence at infallibility.
—AUN TURING 1946, p. 124
The attempts over the years to use Godel's Theorem to prove something
important about the nature of the human mind have an elusive atmosphere of
romance. There is something strangely thrilling about the prospect of "using
science" to such an effect. I think I can put my finger on it. The key text is
not the Hans Christian Andersen tale about the Emperor's New Clothes, but
the Arthurian romance of the Sword in the Stone. Somebody (our hero, of
course) has a special, perhaps even magical, property which is quite invisible
under most circumstances, but which can be made to reveal itself quite
unmistakably in special circumstances: if you can pull the sword from the
stone, you have the property; if you can't, you don't. This is a feat or a failure
that everyone can see; it doesn't require any special interpretation or special
pleading on one's own behalf. Pull out the sword and you win, hands down.
What Godel's Theorem promises the romantically inclined is a similarly
dramatic proof of the specialness of the human mind. Godel's Theorem
defines a deed, it seems, that a genuine human mind can perform but that no
impostor, no mere algorithm-controlled robot, could perform. The technical
details of Godel's proof itself need not concern us; no mathematician
doubts its soundness. The controversy all lies in how to harness the theorem
to prove anything about the nature of the mind. The weakness in any such
argument must come at the crucial empirical step: the step where we look to
see our heroes (ourselves, our mathematicians) doing the thing that the robot
simply cannot do. Is the feat in question like pulling the sword from the
stone, a feat that has no plausible lookalikes, or is it a feat that cannot readily
(if at all) be distinguished from mere approximations of the feat? That is the
crucial question, and there has been a lot of confusion about just what the
distinguishing feat is. Some of the confusion can be blamed on Kurt Godel
himself, for he thought that he had proved that the human mind must be a
skyhook.
In 1931 Gödel, a young mathematician at the University of Vienna, published
his proof, one of the most important and surprising mathematical
results of the twentieth century, establishing an absolute limit on mathematical
proof that is really quite shocking. Recall the Euclidean geometry
you studied in high school, in which you learned to create formal proofs of
theorems of geometry, from a basic list of axioms and definitions, using a
fixed list of inference rules. You were learning your way around in an
axiomatization of plane geometry. Remember how the teacher would draw a
geometric diagram on the blackboard, showing a triangle, say, with various
straight lines intersecting its sides in various ways, meeting at various angles,
and then ask you such questions as: "Do these two lines have to intersect at a
right angle? Is this triangle over here congruent to that triangle over there?"
Often the answer was obvious: you could just see that the lines had to
intersect at a right angle, that the triangles were congruent. But it was another
matter—in fact, a considerable amount of inspired drudgery—to prove it
from the axioms, formally, according to the strict rules. Did you ever wonder,
when the teacher put a new diagram on the blackboard, whether there might
be facts about plane geometry that you could see were true but couldn't
prove, not in a million years? Or did it seem obvious to you that, if you
yourself were unable to devise a proof of some candidate geometric truth,
this would just be a sign of your own personal frailty? Perhaps you thought:
"There has to be a proof, since it's true, even if I myself can never find it!"
That's an intensely plausible opinion, but what Gödel proved, beyond any
doubt, is that when it comes to axiomatizing simple arithmetic ( not plane
geometry), there are truths that "we can see" to be true but that can never be
formally proved to be true. Actually, this claim must be carefully hedged: for
any particular axiom system that is consistent (not subtly selfcontradictory—a
disqualifying flaw), there must be a sentence of arithmetic,
now known as the Godel sentence of that system, that is not provable within
the system but is true. (In fact, there must be many such true sentences, but
one is all we need to make the point.) We can change systems, and prove that