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

430 THE EMPEROR'S NEW MIND, AND OTHER FABLES
first Godel sentence in the next axiom system we choose, but it in turn will
spawn its own Godel sentence, if it is consistent, and so on forever. No single
consistent axiomatization of arithmetic can prove all the truths of arithmetic.
This might not seem to matter very much, since we seldom if ever want to
prove facts of arithmetic; we just take arithmetic for granted without proof.
But it is possible to devise Euclid-type axiom systems for arithmetic—
Peano's axioms, for instance—and prove such simple truths as "2 + 2 = 4,"
such obvious middle-level truths as "numbers evenly divisible by 10 are also
evenly divisible by 2," and such unobvious truths as "There is no largest
prime number." Before Godel devised his proof, the goal of deriving all
mathematical truth from a single set of axioms was widely regarded by
mathematicians and logicians as their great project, difficult but within reach,
the moon landing or Human Genome Project of the mathematics of the day.
But it absolutely can't be done. That is what Gödel's Theorem establishes.
Now, what does this have to do with Artificial Intelligence or evolution?
Godel proved his theorem some years before the invention of the electronic
computer, but then Alan Turing came along and extended the implications of
that abstract theorem by showing, in effect, that any formal proof procedure
of the sort covered by Godel's Theorem is equivalent to a computer program.
Godel had devised a way of putting all possible axiom systems in
"alphabetical order." In fact, they can all be lined up in the Library of Babel,
and Turing then showed that this set was a subset of another set in the
Library of Babel: the set of all possible computers. It doesn't matter what
material you make a computer out of; what matters is the algorithm it runs;
and since every algorithm is finitely specifiable, it is possible to devise a
uniform language for uniquely describing each algorithm and putting all the
specifications in "alphabetical order." Turing devised just such a system, and
in it every computer—from your laptop to the grandest parallel supercomputer
that will ever be built—has a unique description as what we now call a
Turing machine. The Turing machines can each be given a unique identification
number—its Library of Babel Number, if you like. Gödel's Theorem
can then be reinterpreted to say that each of those Turing machines that
happens to be a consistent algorithm for proving truths of arithmetic (and, not
surprisingly, these are a Vast but Vanishing subset of all the possible Turing
machines) has associated with it a Gödel sentence—a truth of arithmetic it
cannot prove. So that is what Gödel, anchored by Turing to the world of
computers, tells us: every computer that is a consistent truth-of-arithmeticprover
has an Achilles' heel, a truth it can never prove, even if it runs till
doomsday. But so what?
Gödel himself thought that the implication of his theorem was that human
beings—at least the mathematicians among us—cannot, then, be just
machines, because they can do things no machine could do. More point-
The Sword in the Stone 431
edly, at least some part of such a human being cannot be a mere machine,
even a huge collection of gadgets. If hearts are pumping machines, and lungs
are air-exchanging machines, and brains are computing machines, then
mathematicians' minds cannot be their brains, Godel thought, since
mathematicians' minds can do something that no mere computing machine
can do.
What, exactly, can they do? This is the problem of defining the feat for the
big empirical test. It is tempting to think we have already seen an example:
they can do what you used to do when you looked at the blackboard in
geometry class—using something like "intuition" or "judgment" or "pure
understanding," they can just see that certain propositions of arithmetic are
true. The idea would be that they don't need to rely on grubby algorithms to
generate their mathematical knowledge, since they have a talent for grasping
mathematical truth that transcends algorithmic processes alto-gether.
Remember that an algorithm is a recipe that can be followed by servile
dunces—or even machines; no understanding is required. Clever
mathematicians seem, in contrast, to be able to use their understanding to go
beyond what such mechanical dunces can do. But although this seems to be
what Gödel himself thought, and it certainly expresses the general popular
understanding of what Gödel's Theorem shows, it is much harder to
demonstrate than first appears. How can we distinguish a case of somebody
(or something) "grasping the truth" of a mathematical sentence from a case
of somebody (or something) just wildly guessing correctly, for instance? You
could train a parrot to utter "true" and "false" when various symbols were
written on the blackboard in front of it; how many correct guesses without an
error would the parrot have to make for us to be justified in believing that the
parrot had an immaterial mind after all (or perhaps was just a human
mathematician in a parrot costume) (Hofstadter 1979)?
This is the problem that has always given fits to those who want to use
Gödel's Theorem to prove that our minds are skyhooks, not just boring old
cranes. It won't do to say that mathematicians, unlike machines, can prove
any truth of arithmetic, for, if what we mean by "prove" is what Gödel means
by "prove" in his proof, then Gödel shows that human beings—or angels, if
such there be—cannot do it either (Dennett 1970); there is no formal proof of
a system's Gödel sentence within the system. A famous early attempt to
harness Gödel's Theorem was by the philosopher J. R. Lucas (1961; see also
1970), who decided to define the crucial feat as "producing as true" a certain
sentence—some Gödel sentence or other. This definition runs into insoluble
problems of interpretation, however, ruining the "sword-in-the stone"
definitiveness of the empirical side of the argument ( Dennett 1970, 1972;
see also Hofstadter 1979). We can see more clearly what the problem is by
considering several related feats, real and imaginary.
Rene Descartes, in 1637, asked himself how one could tell a genuine