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

438 THE EMPEROR'S NEW MIND, AND OTHER FABLES The Library of Toshiba 439
Toshiba, but they are Vanishingly hard to "find"—that's why software companies
make quite a few millionaires along with their software.
Now, every megabyte-length bit string is an algorithm in one sense—the
sense that matters to us: it is a recipe, stupid or wise, that can be followed by
a mechanism, my Toshiba. If we try bit strings at random, most of the time
the Toshiba will just sit there emitting a faint hum (it won't even flash an
amber light); there are Vastly more ways of being a dead program than a live
one, to echo Dawkins. Only a Vanishing subset of these algorithms are
interesting in any way at all, and only a Vanishing subset of them have
anything at all to do with truths of arithmetic, and only a Vanishing subset of
these attempt to generate formal proofs of arithmetical truths, and only a
Vanishing subset of those are consistent. Godel shows us that not a single
one of the algorithms in that subset ( and there are still Vastly many of them,
even for my little Toshiba) can generate proofs of all the truths of arithmetic.
But Godel's Theorem tells us nothing at all about any other algorithm in
the Library of Toshiba. It does not tell us whether there are any algorithms
that can play decent chess. There are in fact Vastly many, and a few actual
ones reside on my actual Toshiba, and I've never beaten any of them! It does
not tell us whether there are any algorithms that are pretty darn good at
playing the Turing Test or imitation game. In fact, there is one actual one on
my Toshiba, a stripped-down version of Joseph Weizenbaum's famous ELIZA
program, and I have seen it fool uninitiated people into concluding, like
Edgar Allan Poe, that there must be a human being issuing the answers. At
first I was baffled by how any sane human being could think there was a tiny
guy in my laptop Toshiba, sitting, unattached to anything, on a card table, but
I had forgotten how resourceful a persuaded mind can be—there must be,
these wily skeptics concluded, a cellular phone in my Toshiba!
Godel's Theorem in particular has nothing at all to tell us about whether
there might be algorithms in the Library of Toshiba that could do an impressive
job of "producing as true" or "detecting as true or false" candidate
sentences of arithmetic. If human mathematicians can do an impressive job
of "just seeing" with "mathematical intuition" that certain propositions are
true, perhaps a computer can imitate this talent, the same way it can imitate
chess-playing and conversation-holding: imperfectly, but impressively. That
is exactly what people in AI believe: that there are risky, heuristic algorithms
for human intelligence in general, just as there are for playing good checkers
and good chess and a thousand other tasks. And here is where Penrose made
his big mistake: he ignored this set of possible algorithms— the only set of
algorithms that AI has ever concerned itself with—and concentrated on the
set of algorithms that Godel's Theorem actually tells us something about.
Mathematicians, Penrose says, use "mathematical insight" to see that a
certain proposition follows from the soundness of a certain system. He then
goes to some length to argue that there could be no algorithm, or at any rate
no practical algorithm, "for" mathematical insight. But, in going to all this
trouble, he overlooks the possibility that some algorithm—many different
algorithms, in fact—might yield mathematical insight even though that was
not just what it was "for." We can see the mistake clearly in a parallel
argument.
Chess is a finite game (since there are rules for terminating go-nowhere
games as draws). That means that there is, in principle, an algorithm for
determining either checkmate or a draw—I have no idea which. In fact, I can
specify the algorithm for you quite simply: (1) Draw the entire decision tree
of all possible chess games (a Vast but finite number). (2) Go to the end node
of each game; it will be either a win for white or black, or a draw. (3) "Color"
the node black, white, or gray, depending on the outcome. (4) Work
backwards, one whole step ( one white move plus one black move ) at a time;
if on the previous move all the paths from any one of white's moves lead
through all black's responses to a white-colored node, color that node white
and move back again, and so forth. (5 ) Do the same for any guaranteed
winning paths for black. (6) Color all other nodes gray. At the end of this
procedure (way past the universe's bedtime), you will have colored in every
node of the tree of all possible chess games, leading back to white's opening
move. Now it is time to play. If any one of the twenty legal moves is colored
white, take it! There is a guaranteed checkmate ahead that can be reached just
by always staying on the white nodes. Shun any black move, of course, since
that opens up a guaranteed win for your opponent. If there are no white
moves at the outset, choose a gray move, and hope that sometime later in the
game you'll be offered a white move. The worst you can do is a draw. (If all
the opening moves for white are colored black, most improbably, your only
hope is to choose one at random and hope that your opponent, playing black,
goofs at some later stage of play and lets you escape by getting on a gray or a
white path.)
That is an algorithm, clearly. No step in the recipe requires any insight, and
I have specified it unambiguously in a finite form. The trouble is that it is not
remotely feasible or practical, because the tree it exhaustively searches is
Vast. But I suppose it is nice to know that in principle there is an algorithm
for playing perfect chess, however useless. There might be a feasible
algorithm for playing perfect chess. No one has ever found one, thank
goodness, since it would turn chess into a game of scarcely more interest than
tic-tac-toe. No one knows whether there is such a feasible algorithm, but the
general consensus is that it is very unlikely. Not knowing for sure, let's choose
the supposition that makes for the worst case for AI. Let's suppose that there
is no feasible algorithm for checkmate or a guaranteed draw—none at all.