hpm Mon Dec 19 22:57:19 -0500
a surrounding 200 pages, pages 179 to 190 of Roger
Penrose's "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
understand 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.
Subject: Penrose review
Penrose's Gravitonic Brains
Review of: Roger Penrose's
"Shadows of the Mind"
by Hans Moravec, December 1994
Severe incongruities in the
dialog's logic and characterization suggest the following continuation:
(revives from feigned shutdown):
Oh Roger, you mischievous
monkey, you never tire of that silly homo-superior game, do you?
(revealed to be Roger Penrose, wearing Albert Imperator mask):
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?
I suppose you're going to remind me.
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 defense of a viscerally
attractive position, the fault is presumed to lie in the advocate, but when
the failed defense is conducted by a person of the highest intellectual and
pedagogic reputation, the position being defended itself becomes seriously suspect.
After Roger Penrose championed the cause of indefinite human superiority
over machines -- and lost -- the world learned to accept the inevitable arrival
of superhuman minds.
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 25,000 page
rebuttal that should remove any remaining doubt. My theories about a platonic
quantum gravitational collapse neural mechanism, too, have become more developed
in each successive book.
That's why we like you, you're
so fierce and persistent! But
the failure doesn't concern the games we play with you now. It occurred soon
after publication of your first books, when the logic community rejected the foundations
of your argument, the quantum computation, quantum gravitation and neurobiological
communities found your neural quantum collapse speculations over
the top, and machine intelligence researchers simply kept evolving their systems
on exponentially growing computer power. The intellectual community was unimpressed.
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.
Popularity is not proof. My argument was slow to sink in, 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 end.
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 absurdum" 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, but 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 believing it.
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
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.
My reasoning shows the power of understanding, though, of course, none
of your own analysis means anything to you, since you lack understanding.
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, so to speak.
But Godel sentences are expressions
of Platonic truths, as you
would see if you had any understanding. It is simply a lie to deny them. Obviously
your story about 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. Many of these new systems can be mapped
into physical observations, though often in unusual ways with different strengths
and weaknesses than classical mathematics. 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.
A bastardization of the Plato and Godel! 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 falsehood 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 become ineffectual. To you, unfamiliar, unintuitive systems are
indeed unproductive and unreal.
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
controlled by 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.
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 is any other Platonic verity.
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 nearly blind you: all the "exotic"
systems you have studied in detail are only slight extensions
of conventional shapes and numbers.
Human intuition reaches no 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 specialization, shared by all humans.
Of course, I do not accept your self-serving analysis.
Without a proper sense of real and unreal, robot reasoning is simply vacuously
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 decision method 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.
Notably, the program gained great deductive power from inconsistent models. Numeric
roundoff error allowed diagram calculations to show equal segments, angles
and areas to be unequal, or vice versa, and to 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 25,000 page
detailed analysis of why everything
you have bored me with today, and in the years preceding, simply illustrates
lack of understanding.
Until next time, then!