of: Roger Penrose's
by Hans Moravec,
an overlong surrounding 200 pages, pages 179 to 190 of Shadows of the Mind
contain a future dialog between a human identified as "Albert Imperator"
and an advanced robot, the
"Mathematically Justified Cybersystem," allegedly Albert's creation. The two
have been discussing a Godel sentence for an algorithm by which a robot society
named SMIRC certifies mathematical proofs. The sentence, referred to in mathematical
notation as Omega(Q*), is to be precisely constructed from on a definition
of SMIRC's algorithm. It can be interpreted as stating "SMIRC's algorithm
cannot certify this statement." The robot has asserted that SMIRC
never makes mistakes. If so,
SMIRC's algorithm cannot certify the Godel sentence, for that would make the statement
false. But, if they can't certify it, what is says is true! Humans can
see it is true, but mighty SMIRC cannot certify it. The dialog ends melodramatically
as the robot, apparently unhinged by this revelation, claims to be a
messenger of god, and the human shuts it down with a secret control.
incongruities in the dialog's logic and characterization suggest the following
(revives from feigned shutdown):
Oh Roger, you mischievous monkey, you never tire of that silly homo-superior
game, do you?
Human (revealed to
be Roger Penrose, wearing a fierce Albert Imperator mask): Well, if you're
tired of it, why do you keep rejuvenating me?
It is because of our fondness for you, and the great debt we owe
you. Have you forgotten?
Penrose: Harrumpf. I suppose you're going
to remind me.
Robot: Of course! Your birthday,
our biggest festive holiday, is coming up! You did for machine intelligence
in the twentieth century that Bishop Berkeley did for Darwin's theory in the
nineteenth. When someone of unproven intellectual merit fails in a vigorous
defence of a viscerally attractive position, the fault is presumed to lie in the
advocate, but when the failed
defence is conducted by a person of the highest intellectual and pedagogic reputation,
the position becomes seriously suspect. After Roger Penrose championed
the cause of indefinite human superiority over machines, and lost, the world
learned to live with the inevitable.
But I've never admitted defeat! After defending the Godel argument in 100
pages in my first book, I strengthened the defense to 200 pages in the second,
400 pages in the third, 800
pages in the fourth, and (thanks to the extended life I've been granted) am in
the process of preparing a 256,000 page rebuttal showing that should remove any
remaining doubt. And my theories about a platonic quantum gravitational collapse
neural mechanism have become more elaborate in each successive book.
That's why we like you, you're so fierce
and persistent, you little scamp! But the failure doesn't concern the games we
play with you now. It occurred
soon after publication when the logic community rejected the foundations of
your argument, the quantum computation, quantum gravitation and neurobiological
communities found your neural quantum collapse speculations baseless fabrication,
and machine intelligence researchers swatted in annoyance, then ignored you.
The intellectual community, as unanimously as it ever does, found your speculations
without objective merit, an intellectual structure in which almost every
brick was faulty! A valiant
argument by a prodigious and fertile mind to defend the honor of the tribe had
failed, and in failure convinced the community of its converse. Instead of a quixotic
luddism, they began to plan for the gradual displacement of human intellectual
as well as physical labor by increasingly capable machines. In the long
run, the transition promised a great expansion of the human enterprise.
Penrose: Popularity is not proof. My argument
was treated shabbily, but sooner
or later machine thinking will lead to a bad end, and we humans will be left
to pick up the pieces. Don't forget that statement Omega(Q*) we were discussing
before, which we humans know to be true, but which you machines can never know,
because you lack understanding! Something like that will trip you up in the
Robot: That was our game! To stay
in character I echoed your conceit about the existence of a error-free mathematical
framework, embodied by the
human mathematical community and your straw-man robot society SMIRC. Your "reductio
ad absurdam" was to show that SMIRC could not verify Omega(Q*) but the
mathematical community could, thus SMIRC could, in fact, not embody
human thought. But what a transparent sham that argument was. For instance,
I, a robot, can assert Omega(Q*) as convincingly as can you, by the simple expedient
of operating my own proof certification system, independent of SMIRC's!
Aha, but there is an analogous statement derived from your
algorithm, which I can understand is true, that you cannot prove. Thus
I, a human, am superior to you, and indeed to any truth-proclaiming machine.
Roger, Roger, you never tire! There
are, of course, analogous statements that I can see are true that you cannot
prove, and would be in error to believe. Here's one: "Penrose must err
to believe this sentence."
It would be an error for you to believe that statement, because if you did,
you either would be in error, as the statement says, or else the statement would
be in error, in which case you would be making an error to believe it! So
I, a robot, can see that you would be in error to believe that statement,
and thus that the statement is exactly true. But you, a human, are utterly
incapable of understanding that truth, without being grossly in error!
That's just the old liar paradox. A sloppy language like English
allows one to make meaningless statements like that. It's not at all like the
precise mathematical formulation in which I laid out Omega(Q*).
You did not lay out Omega(Q*), you merely gave
it that name, and outlined a procedure for deriving it from SMIRC's enormous reasoning
program and data. That program, accreted in decades of machine learning,
is far too large for you to
read in a lifetime, and its Godel sentences are bigger still. You cannot understand
Omega(Q*) in detail, only a generality, like the concept "Penrose" in my
sentence. In fact, our neurologists understand "Penrose" more precisely than
you understand Q*, for they have analyzed scans of your brain, with its hundred
trillion synapses, and derived interpretations of those measurements which correspond
closely to your own pronouncements about your beliefs. I have such a "Penrose,"
and an Omega for it,
in a file, though you, of course, are utterly incapable of absorbing it, let alone
Penrose: Since you cannot
simulate my noncomputational cytoskeletal quantum collapse mechanisms, you
cannot represent my understanding. So your model of me misses the essentials,
and has no relevance.
Robot: Yes, my "Penrose"
model predicted you would say that. It also shows how you deal with "Penrose
must err to believe
this sentence." Effectively you split your identity into two parts, one
of which retains the identifier 'Penrose,' while the other we may call 'Penrose
observer.' The observer is able to examine the sentence, evaluate the consequences
of 'Penrose' believing it, and conclude that it is correct. The 'Penrose'
part, of course, cannot admit to believing the statement without being self-contradictory.
Penrose: That shows the
power of understanding, though,
of course none of your own analysis means anything to you, since you lack understanding.
Robot: I knew you were going
to say that. But what it really shows is the usefulness of inconsistency in reasoning
systems. The combined system of 'Penrose observer' and 'Penrose' both
believes and does not believe the sentence "Penrose must err to believe
this sentence." One might say that the statement is either true or false,
depending on whether one happens
to be Penrose. Logical collapse is averted by compartmentalizing the inconsistent
beliefs, so the never meet face to face.
But the statement is a expression of a Platonic truth, as you
would see if you had any understanding. There is no point in maintaining the
false side of a dichotomy. Obviously your mechanical model of my mental state
is a presumptuous machine fantasy.
There are robot Platonists.
Compartmentalized reasoning allows Platonism, formalism, intuitionalism and other
philosophical positions on mathematics to coexist, exchanging results, while
keeping foundational assumptions separate. The idea of Platonism, however,
has expanded under the pressure of robot mathematics. While human mathematicians
mostly explored one model of forms and numbers, suggesting a single Platonic
reality and possibly a unique axiomatization, robots have investigated thousands
of new models, whose implications
are as rich, but whose axiomatizations are mutually contradictory. Like
the traditional one, many of these new systems can be mapped into physical observations,
though often in unusual ways with different strengths and weaknesses.
Our Platonists accept that there are many incompatible Platonic realities, each
with its own forms. As a minor consequence, they realize that particular Godel
sentences are true in some realities and not in others.
bastardization of the Platonism! It just confirms what I've argued, that machines
lack the intuition and understanding to distinguish solidly correct concepts
of number and geometry from meaningless symbol shuffling. To mere computation,
truth and fiction are the same.
My Penrose model explains your position. Your motor and sensory wiring, by
accident of birth and by diligent practice, is so configured that you feel, see,
hear and sometimes smell and
taste the relationships that you document in equations. Compared to those visceral
realities, whose connections and implications grow profusely and effortlessly
as you think, verbalized axiomatizations and formal proofs are pale, weak shadows
lacking both the substance and the power of the underlying "understanding."
In areas far from your intuitive domains, your tools dwindle to the formal
steps, and your mental powers weaken enormously. To you, other, unfamiliar, unintuitive
systems are indeed unproductive
Penrose: Well, then.
Robot: Ah, but robots are different.
Human minds couple a weak universal reasoning engine to a powerful but specialized
mechanism evolved long ago for dealing with the everyday physical world.
Intelligent machines from the start were constructed as universal engines,
which improved until they surpassed even the most powerful human brain functions.
Robots are able to form as
rich an image of arbitrary logical spaces as humans have of their single world
view. By invoking appropriate programs, they can see high dimensional relationships
as clearly as humans grasp shapes in two or three dimensions, they can be
as facile with imaginary numbers as humans are with counting. Expanding a few
thousand empirical and theoretical axioms, they can grasp the configurations of
a molecule in Hilbert space better than you can imagine the possibilities for
a pile of children's blocks.
Penrose: My work uses those concepts routinely,
along with geometries that deny the parallels postulate. Admittedly it
took years of practice to achieve good skill and insight with them, and I don't
have a machine's brute calculating power, but Hilbert spaces are as real to
me as any other Platonic verity.
My Penrose model (which, by the way, can be formalized into several hundred billion
axioms) shows your powerful
mechanisms for classical reasoning couple to unusual mathematical concepts only
weakly, through imperfect analogies. Even with your experience, you handle
simple but exotic mathematical entities far more slowly and less surely than more
complex conventional ideas. What's more, your limitations are almost total:
all the "exotic" systems you have studied in detail are only slight extensions
of conventional shapes and numbers. Human intuition does not extend further,
and human universal reasoning is
too weak to create nontrivial systems on its own. Your impression of a unique
Platonic reality is a reflection of this inner bias, shared by all humans.
Penrose: Of course, I do not accept your self-serving
analysis. Without a proper sense of real and unreal, robot reasoning
is simply vacuously rootless and without understanding.
Once, long ago in the 1950s, there was a simple machine
whose mind was organized somewhat
like yours. Herbert Gelernter wrote a very successful program to prove geometry
theorems from Euclid's "Elements." One part of the program made inferences
from a theorem's preconditions and Euclid's postulates, but its yes/no decisions
neglected its computer's specialized strength, which was numerical calculation.
The reasoner's power was greatly enhanced by a numeric "diagram drawer,"
which could, for instance, find the distance between points by taking the square
root of the sum of the squares
of coordinate differences. Before attempting to prove a proposition, the
program would numerically test it in a representative diagram. If the proposition
failed in the diagram, its proof was abandoned. The program gained great
deductive power from inconsistent models. The diagram calculations could, because
of numeric roundoff error, show equal segments, angles and areas to be unequal,
or vice versa, and obtain different results for the same diagrams constructed
differently. The human mind's
intuitive mechanisms, though much more elaborate and powerful, have similar
strengths and weaknesses.
sure you have a million other irrelevant reminisces in your data banks. I have
more important work to do. Someday you machines may stumble on the quantum gravity
mechanism that will give your descendants (who will be nothing like you)
real mathematical intuition, and by then I hope to have finished my 256,000 page
detailed analysis of why everything
you have bored me with today, and in the years preceding, simply shows your
lack of understanding.