Hofstadter's book is a classic, but it is also misunderstood. It is not just about the three guys whose names appear in the title. It is not just about math, art, and music. It is not focused on religion, the occult, new age psychology, or Zen. I'll try to outline it in a way that a reader can appreciate what the author is trying to convey.
What is It About?
The book is Hofstadter's attempt to explain how animate beings come from inanimate matter;
how a sense
of self comes from selfless components.
His conviction is that the key to understanding being or consciousness
is the kind of pattern he calls a strange loop.
This is backed up by analogies such as one in which formal systems of meaningless
symbols, when sufficiently complex isomorphisms arise, acquire
meaning, and when strange loops are present, can “perceive themselves.”
Self-awareness, according to this thesis, is a result of patterns, not
material.
In addition, there is something emergent, if not inevitable, about such patterns:
the strange loop in Gödel's proof was found in a system laboriously designed
to keep self-reference out! In fact any formal system capable of expressing
what we know of as arithmetic will necessarily have this quality.
Another way to convey this idea is to note that sometimes, it seems, content is
inseparable from form, semantics from syntax, pattern from matter.
Notes on the Text
Introduction
First line of this introduction is "Author: ". Hmmmm....
Bach's Musical Offering is awesome: it contains a 6-part fugue on a theme by
Frederick the Great.
A canon is made up of multiple transformed copies of a theme played against each
other. The copies
can be displaced in time or pitch; played at different speeds (diminution, augmentation);
or inverted. There is a notion of a retrograde copy, too.
A transformation from which the original is fully recoverable is called an
isomorphism. (This is a general notion, applicable not only to
themes BTW.)
Fugues are like canons but a bit looser. They start with one voice and as the
next voice enters (+5 or -4), the previous voices become the countersubject.
One of canons in the Musical Offering, The Endlessly Rising Canon
(Canon per Tonos) features a 6-step modulation that returns to its starting point
(Cm → Dm → Em → F#m → A♭m → B♭m → Cm)
and is an example of a strange loop.
Escher's Waterfall and Ascending and Descending contain strange loops.
Escher's Drawing Hands and Print Gallery contain a wilder kind of strange loop,
because the levels show different degrees of reality and fantasy. Think about
it!
Gödel used a strange loop by creating a mathematical statement that asserted
its own unprovability. This formed the basis for his proof that all formal systems that
attempt to capture all truth of arithmetic are incomplete. This will be shown in
Chapter 14 of GEB.
Gödel's proof was stunning in 1931 when he published it, but it is so not the drama
these days. Why was it such a big deal? Because there was the hope among many doing
mathematical logic back in the 1800s and early 1900s that a consistent, complete
formulation of mathematics was possible. Alas, it is not!
Prior to Gödel, mathematicians were already trying to deal with things like
non-Euclidean geometry, different infinities, shapes we now know of as fractals, and
Russell's Paradox (Is the set of all run-of-the mill sets run-of-the-mill or
self-swallowing?)
In Principia Mathematica (1910-1913), Russell and Whitehead banned the self-reference
of Russell's paradox by the theory of types.
But stratifying sets by type only gets rid of Russell's Paradox, but not Grelling's
(Is the word "heterological" autological or heterological?). Stratifying language goes
too far!
Hilbert issued a challenge to show PM consistent and complete. Gödel showed not
only PM, but any related system, could be complete only if it were inconsistent (First
Theorem). He also showed that a system could prove its own consistency only if it were
inconsistent (Second Theorem).
Upshot of First Theorem: In any consistent formalization of arithmetic, there MUST be true
statements that cannot be proven.
Computational analog of the First Theorem, discovered by Turing: In any sufficiently
powerful computing machine, there MUST be functions that cannot be computed.
Computers today, similar to the Analytical Engine conceived by Babbage and
described in detail by Lovelace, (1) have a control unit and a memory, and (2) hold
stored programs. They
are capable of modifying their own programs—another strange loop. We can conceive
of programs for teaching a computer to write new programs, or programs that reproduce
themselves. How about programming intelligent behavior? That's AI.
So the underlying hardware is fixed, but we can program behavior that appears
intelligent (to a degree, anyway—see page 26). Is that a contradiction?
Why should it be?
Three-Part Invention / MU-Puzzle
The dialogue Three-Part Invention introduces Achilles and the Tortoise. They discuss a
couple of Zeno's paradoxes (the Achilles and the dichotomy paradoxes), the Zen kōan
about the whether it's the flag or the wind or the mind that's moving.
Formal systems are introduced, with an up front caveat that some readers
may find it hard to always stay within the system.
The MIU-System is introduced:
Alphabet: {M, I, U}
Axiom: MI
Rule: xI → xIU
Rule: Mx → Mxx
Rule: xIIIy → xUy
Rule: xUUy → xy
Terminology: Theorem, Axiom, Rules of Inference (a.k.a.
Production Rules), Derivation (Proof).
An example derivation: MI ⇒ MII ⇒ MIIII ⇒ MIIIIU ⇒
MUIU ⇒ MUIUUIU ⇒ MUIIU.
The “MU-Puzzle” is: can you derive MU? Try it! The system has lengthening
and shortening rules, so it's kind of tough, eh?
It is possible to write a (naïve) program to derive theorems of this system, and
if you asked it to stop when it derived U, it would run forever without complaint.
But no human would be stupid enough to this!
People can not be made to be unobservant: they naturally jump out of the system.
When working on the MU-Puzzle, humans will use both M-mode and I-mode:
M-mode (mechanical mode): working entirely within the system
I-mode (intelligent mode): thinking about the system
U-mode (un-mode): to be described later
A decision procedure for a formal system is an always-terminating test for
theoremhood for a given string.
Is the following a decision procedure for the MIU-System?
Initialize a set called T to {MI}
Apply all applicable rules to each string ∈ T, adding the results to T.
If you generated the string you are looking for, announce success, else go to step 2.
NO! The procedure will eventually generate every theorem sooner or later, but it
won't ever stop when asked to decide theoremhood for a non-theorem!
Interesting. We know there are non-theorems, like U, but is there a decision
procedure? We'll have to wait for a later chapter to find out.
Two-Part Invention / Meaning and Form in Mathematics
Two-Part Invention is actually a reprint of Lewis Carroll's 1895 article
in Mind, entitled
What
the Tortoise Said to Achilles. What the Tortoise said is that there is an
infinite regress required to justify the venerable Modus Ponens.
The dialogue hints at an important philosophical problem, which
is whether human thought follows formal rules or not—which, Hofstadter
writes, "is the problem of this book".
To start attacking this problem in a simple way, Hofstadter invented a simple formal system,
the pq-system:
Alphabet: {p, q, -}
Axioms: xp-qx-, where x contains only hyphens
Rule: xpyqz → xpy-qz-, where x, y, and z contain only hyphens
There is a decision procedure to determine whether a given string is a theorem of the
pq-system! Because there are no shortening rules, all theorems can be generated in
increasing length.
The theorems of the pq-system correspond exactly to the true statements of the form
x + y = z over natural numbers. This is an isomorphism. The correspondence
works both ways.
Hofstadter's claim: it is the perception of these isomorphisms that create what we call
meaning. In fact, once we find an isomorphism, the (meaningless) symbols almost
can't avoid taking on meaning.
Of course, the interpretations have to make sense for there to be an isomorphism.
Sometimes there can be multiple ones: {p⟺plus, q⟺equals} or even
{p⟺equals, q⟺taken from}
Humans have to be careful about not working in I-mode here: --p--p-q----- is NOT a
theorem!
While the pq-system only corresponds to a tiny part of reality (natural number addition)
one might wonder whether all of reality is one giant complex formal system (i.e., is
the universe deterministic?).
Related: Let's say we try to design a formal system to capture some aspect of reality
(say, mathematics). How do we know we've captured that reality? Can we capture
all truths?
Non-obvious truths often have "simple, compelling, and beautiful" proofs attained
through reasoning. Euclid's proof on the infinitude of primes is given in the text.
Euclid's proof works by not considering an infinite number of possible cases,
but working with generalizations ("for all n...").
So we can wonder whether we can create a formal system (rules for shunting
"meaningless" symbols around to generate theorems isomorphic to truths) equal
in power to number theory.
Sonata for Unaccompanied Achilles / Figure and Ground
The dialogue gives the reader a sense of figure and ground with only Achilles speaking (figure)
and the reader filling in what the Tortoise must be saying (ground). The dialogue imitates
Bach's
Sonatas for Unaccompanied Violin.
More play in the dialogue: parts of the word "HEADACHE".
Escher's Mosaic II shows that ground can be figure,
too.
The tq-system is introduced to typographically capture the notions of multiplication
and compositeness:
Alphabet: {t, q, -}
Axioms: xt-qx, where x contains only hyphens
Rule M: xtyqz → xty-qzx, where x, y, and z contain only hyphens
Rule C: x-ty-qz → Cz, where x, y, and z contain only hyphens
As these formal systems start getting more complicated we humans have to work really
hard to constrain ourselves from thinking that --- is the number three. Hofstadter
reminds the reader not to confuse M-mode and I-mode again.
How NOT to characterize primes: A rule such as "If Cz is not a theorem, then Pz is a theorem" is
totally wrong. It is not typographical! It requires working outside the system.
Here the composites are the positively-defined figure; the primes are the "negative space",
or ground.
In music, the ground can be a figure, too, and not just in harmonies and other accompaniments.
Sometimes there are melodies in the off-beats (to name just one approach).
But it does seem that in art and in music, there are drawings and pieces in which only
the figure makes sense and the ground is not any recognizable figure at all. Is there
an analog in formal systems? Do there exist formal systems in which the negative
space is not generable by a formal system?
What does your intuition tell you about this? On the one hand, how could a set and
its complement not carry, in Hofstadter's words, "the same information?" Then again,
why shouldn't there be an asymmetry?
The answer turns out to be, for plain old natural numbers, that indeed there
are sets that can be typographically generated but whose complements can not be!
In Computer Science speak: there exist recursively enumerable sets that are not recursive.
(Recursive means both the set and its complement are typographically generatable.)
Corollary: There exist formal systems with no formal decision procedure!
Interestingly, perhaps, primes are not such a set. You can generate
primes with a formal system.
Axioms: xyDNDx where x and y are hyphen strings
Rule: xDNDy → xDNDxy
Rule: --DNDz → zDF--
Rule: zDFx , x-DNDz → zDFx-
Rule: z-DFz → Pz
Axiom: P--
Puzzle near the end of the chapter: 1 3 7 12 18 26 35 45 56 69, ...
Contracrostipunctus / Consistency, Completeness, and Geometry
This is a very important dialogue, containing paraphrases of Gödel's First
Incompleteness Theorem.
"For every record player there is a record that it cannot play" is isomorphic to
"For every sufficiently strong axiomatization of arithmetic there is a true statement
that it cannot prove." Great analogy!
"High fidelity or low fidelity, he loses either way."
The dialogue is self-referential, something that is central not only to Gödel's
proof but also, probably, to intelligence and self-awareness. (More about that
theme in later chapters.)
The dialogue is an acrostic. Did you notice? (The word "acrostic" is actually
mixed with "contrapunctus" to form the title.)
There are plenty of levels of meaning, and several isomorphisms in the dialogue.
Levels of meaning: record player grooves, air vibrations, music, phonograph vibrations....
Isomorphisms: (1) between the two halves of the dialogue, (2) between objects in the
dialogue and terms in Gödel's Proof. See Page 85.
The fact that no perfect phonograph exists is not a failure of phonographs unless you
have unrealistic expectations. Ditto for formal systems and arithmetic.
The consistency of a system is relative to its interpretation: if you change the rules,
you must change some of the interpretations because the isomorphisms may no longer hold.
You regain consistency by this reinterpretation.
This is what happened in the history of geometry when people attempted to
"prove" the 5th postulate of Euclid (they couldn't). When some smart people (Bolyai,
Лобаче́вский) denied it, they
found that they had created new, consistent geometries. And they were cool, not monstrous!
How did this happen? It is because of the undefined terms at the base of it all: point,
straight line, circle, etc. When you are able to free yourself of preconceived
notions you have a whole new way of looking at things.
The "undefined terms" should be defined in terms of just what the rules let them
do. For example, a point is "that which a straight line segment can be drawn between
two of," and so on for line and circle....
The axioms, then, hold the (implicit) meanings of the base terms. So can any term really
be said to be "meaningless"?
Certain interpretations will give consistent systems.
Consistent: all theorems (under the interpretation) come out true
Inconsistent: at least one theorem is false
Internally consistent: all theorems are "compatible" in some possible world
Sometimes you cannot "regain consistency" (e.g., Escher's Relativity). But you can
take the U-mode way out and say the picture is just a bunch of marks on a surface.
After talking about consistency, Hofstadter asks that since Geometry is a bifurcated
theory, can number theory also be bifurcated? If it is, there should also be a core.
This will be revisited later in the book.
Next comes a discussion of completeness (every true statement is a theorem). The original
pq-system is consistent and complete. The modified system is incomplete under the interpretation
of "greater or equal" but we regain completeness by trimming the interpretation to
"equals or exceeds by 1".
Systems like the pq-system are weak. When systems get stronger we see a different kind
of incompleteness. That is what Gödel's first theorem is all about.
Little Harmonic Labyrinth / Recursive Structures and Processes
The dialogue features stories within stories, illustrating how one pushes
to "go down one level" and pops to return to where one was in the previous
level.
Djinn and Tonic is a rather fabulous dialogue inside a dialogue, with great wordplay.
Pushing potion makes you feel weird but the popping tonic gives you
a deep sense of satisfaction.
Wishes, meta-wishes, and meta-meta-wishes. Lamp (L), Meta-lamp (ML), Meta-meta-lamp (MML).
Genie, Meta-Genie, etc.
Achilles' request for a meta-wish took finite time. Why?
"I wish my wish would not be granted." Awesome. System crash!
Note the echoes of the real Achilles and Tortoise's words in an inner story: shades of
the self-similarity we'll see later in the chapter.
Also there is a similarity to the overall dialogue with the piece of music (Little
Harmonic Labyrinth) never returning to the original key.
Did you notice the dialogue never got back to the outer story? Did you care?
Doesn't the outer story contain the real Achilles and Tortoise? Wait,
what? Real who?
Examples of recursion:
Stories within stories, plays within plays (e.g., Phantom of the Opera), movies
within movies, paintings inside paintings, ...
Programming
Music (modulation)
Language
Geometry, especially Fractal Geometry
Math
Recursive definitions are not circular; they have a basis.
Also usually as you descend things are a little smaller, and sometimes not quite
the same.
In music: tonic, pseudotonic. "Tension and resolution are the heart and soul
of music."
In language: RTNs. Note that real RTNs will always have a path that does not
lead to an infinite regress. Also you can expand as much as you want, there
always needs to be a way to stop the recursion.
Heterarchy: things are all tangled that there is really no "top-level." (Reminds
you of how there are no real primitive terms, which you saw in the last chapter.)
Recursive programs.
Chaos.
Fractals: self-similar shapes. Idealized mathematical fractals have infinite complexity.
Examples of fractals in nature: clouds, plants, trees, vascular systems, broccoli.
Note the recursion in Feynman diagrams.
More examples: the themes in a canon, smaller fish in Escher's Fish and Scales.
The deep question: what is the similarity here... the signature? How do we
know these things are all the same in some sense? The brain can't overanalyze. Yet
we find the invariants and the variations. This is also something that expert
programmers know how to exploit, and exploit well. The most obvious example is
writing procedures (functions).
Parameterization can also be used to supply context to RTNs, resulting in ATNs.
The idea of bottoming out really always needs to be there, no matter the context.
This is actually a common error beginning programmers make... they end up with what
is called an infinite loop.
Infinite loops only arise in programming languages that support free loops as
opposed to bounded loops only.
In game playing programs, we set a limit on the search depth.
The chapter closes by showing that even that certain systems, even though they are defined
by simple rules, seem to get very complicated and unpredictable — there's no
pattern anymore. Is this a defining property of intelligence — breaking out of such
patterns?
Canon by Intervallic Augmentation / The Location of Meaning
More wordplay with levels of meaning with the Haiku comments.
Some things to note in the dialogue: cookies as message bearers, poems containing their
own commentary....
Key idea in the dialogue is that one record sounds different when played on different
record players. (Here the semitone intervals were multiplied — intervallic
augmentation. We can illustrate this in code:
data = [-1, 3, -1] # This is "what's on the record", or is it?
def play(exponent):
for value in data:
for _ in range(exponent):
value = round(value * (10.0 / 3))
print value,
print
play(0)
play(1)
play(2)
Question of the chapter: Is meaning inherent in the message or is it required
to have a "player" (or a mind) to extract the meaning?
Record (information bearer), record player (information revealer). But does the record
player matter? You can play at different speeds!
Genotype (e.g. DNA) and Phenotype (the organism).
If the genotype is too simple, it might not mean anything....
"1 1" what comes next?
The string "come" — you need to know if it is English or Spanish or ....
Exotic vs. Prosaic isomorphisms:
Prosaic: the corresponding parts are obvious.
Exotic: way hard to see the corresponding parts.
Does all the information reside in the DNA? Two views:
The information is all there, it's intrinsic, so compelling: you only need
"intelligence" to reveal it.
The DNA is meaningless without (chemical) context — so much information is outside
— that the DNA is really more like a complex set of triggers than a true information
bearer.
"Compelling Inner Logic"... Compare Bach and Cage. Consider and think about
the following: "Bach's music is
more likely to be understood by alien intelligences than would Cage's."
Decipherment doesn't add to the meaning of a message, no matter how much work is put into it.
The Rosetta Stone would have been deciphered the same way later....
Meaning is part of an object to the extent that it acts upon intelligence
in a predictable way.
Three layers of a message:
Frame Message - says "I am a message."
Outer Message - instructions for how to decode the inner message (this is
not written in any particular language; you just have to recognize it).
Inner Message - the information that is supposed to be conveyed.
Example: Message in a bottle. Frame: Paper in a bottle. Outer: the marks on the paper,
which a reader needs to recognize as being Japanese, English, whatever.... Inner: what
the message is supposed to say.
Example: DVD. Frame: disc shape. Outer: a series of pits. Inner: the
audio or video.
It's the recognition of the outer message that's important: We have to distinguish
random marks on paper from written text or pictures; Bach from Cage; speech from static;
completely random bits from those with "structure"!
So, wait.... do we need an infinite layering of outer messages, as in the dialogue before
Chapter 2? No! Our brains are hardware. There is some universality to the
way humans respond to inputs, how babies can learn any language, etc.
Hey, maybe if these deciphering mechanisms were more universal (i.e., not just confined
to humans on Earth), then meaning would be intrinsic to messages (like mass is
intrinsic, throughout the universe).... Maybe intelligence is emergent?
But if other intelligences respond differently, would we call them intelligent? How do
we define intelligence? Like weight? Or like mass? That is, is there some
non-Earth-chauvinistic definition?
It's interesting to consider short genotypes (e.g., 1 1) vs. long genotypes
(e.g., 1 1 2 3 5 8 13 21 34 55 89 144). A long
genotype contains its frame and outer messages!
There are many
examples in Computer Science in which the outer messages are extractable from inner
messages: Lempel Ziv Compression, HTML and XML charset specifications.
Do our music and our literature and our speech contain enough compelling structure
for the outer message to be eventually revealed? Maybe not: it would seem to require
human experience, a human body, etc. for context.
Is DNA universal? If DNA ended up on a planet far away, would aliens be able to look at
it and figure out the right chemical context to give it so that epigenesis could happen?
Maybe. But if the outer message were just a string of Cs, Gs, Ts, and As, then almost
assuredly not.
Chromatic Fantasy, and Feud / The Propositional Calculus
Crab Canon / Typographical Number Theory
A Mu Offering / Mumon and Gödel
The dialogue introduces the reader to a few ideas of Zen Buddhism, relating a few of these
ideas to themes in the book, such as:
Enlightenment, satori, "no-mind": existence without thinking about existence.
Mu (無): an answer that unasks the question — "only by not asking such questions
can one know the answer to them."
One of the two kōans is genuine, but we don't know which one. This is the excluded
middle; we'll see it again with the sentence G. Funny: The Tortoise says "I can't
imagine what led him to such a belief." Awesome!!
The Tortoise mixing up Enō/Zen/Zeno, taking Okanisama's teaching to Achilles of reality
being immutable, with change and motion being illusions of the senses as "[being] Zeno" was
pretty cool.
Achilles says there's a test for genuineness of a kōan: first translate it
to a string (the yarn kind) then determine if the string has Buddha-nature.
First you transcribe the kōan into the messenger (made up of the 4 symbols).
Then after rubbing your hands in ribo (get it? - messenger RNA!) you
translate the messenger into the folded string.
Now supposedly the Great Tutor was able to just glance at the string and know if it
had Buddha-nature or not. Wait, what? How could that be?
The whole bit about triplets, the "Geometric Code," and the "Central Dogma" keeps
the analogy with biology going.
Going backwards from strings to kōans: sometimes you get nonsense, sometimes apparent
kōans, but out of these, some are genuine, some are not. Just like strings of symbols of
TNT, e.g.,
]∧∧0>eS∨[[∀∧<dd0∃
∃a:~Sa=0
∀a:<a=0∧0=0>
The "Art of Zen Strings" tells how to make the strings with Buddha-nature. That's a formal
system! It even has five "self-evident" starting positions (axioms, of course).
To make strings without Buddha-nature, you just tie a
knot (get it? knot = NOT!) at one end of a string that does.
The Tortoise asks whether there are strings with Buddha-nature that can't be made
by following the rules, to which Achilles responds that his master "said something about
somebody or other's 'Theorem.'" Pretty cool, eh?
Note also: Achilles says he was NEVER ABLE to get either of the "This mind is Buddha" and
"This mind is not Buddha" kōans by following the rules! And he doesn't have a
mechanism to determine which one is phony! The Tortoise says maybe neither one is
genuine. Do you see what he is getting at here?
The Tortoise's double knot disappears spontaneously — you better know why!
The Tortoise creates a string which represents a kōan in which the Grand Tortue creates
a string (the kōan describes the construction). When Achilles creates this string, he finds
that it is the Tortoise's string before he tied the knot in it! When the Tortoise asks
whether the string has Buddha-nature or not, Achilles was troubled and said "I would
be afraid to ask such questions."
Hofstadter says the following in posing the question "What is Zen?"
Zen resists verbal characterization or explanations.
Words don't impart enlightenment or capture truth, but (verbal) kōans may serve
as triggers that may lead to enlightenment.
Enlightenment is the transcendence of dualism, the perception
of the world as divided into categories.
Zen struggles against the reliance on words because words are inherently dualistic,
but since "what else is there but words?" the kōan all but abuses words and
verbal thinking, thereby attacking perception. Perception of an object
separates it from the universe.
Hofstadter: "Relying on words to lead you to truth is like relying on an incomplete
formal system to lead you to the truth."
What else IS there to rely on BUT formal systems? What kind
of mathematical reasoning lies outside formalization? (What else is
there to rely on but words?) The
kōan about tipping over the water vase, and the one about meeting a Zen
master on the road, may be relevant here.
Enlightenment is not the end-all.... Zen is a system and cannot be its
own meta-system; there is always something outside of Zen, which cannot be fully
understood or described with Zen.
To solve the MU-puzzle, we have to go outside the system and reason about the I-count,
which we can show can never be a multiple of three (including zero).
Gödel numbering is introduced and explained for the MIU-system. We can reason
about the MIU-system not just typographically but numerically, and in fact we
can do this for any formal system, including TNT itself! That is, any
formalization of number theory has its own metalanguage embedded within it!
We should be able to make a sentence G that says: "G is not a theorem of TNT", which
shows that TNT must be either incomplete or inconsistent. How it is made will be shown
in Chapter 14.
Prelude... / Levels of Description, and Computer Systems
The characters in the dialogue "Prelude..." are listening to preludes and fugues from Bach's
Well-Tempered Clavier;
a discussion begins which will become a lengthy treatment of holism vs. reductionism
in the next chapter.
Some things to think about when reading the dialogue:
Criticize the statement: "Who needs to see n written out
decimally? Achilles has just told us how to find it."
Number Theory — "the one branch of mathematics which has NO applications!" Haha.
The mention of the "moments of silent suspense between prelude and fugue" is of course
reflected in the dialogue itself and the next one in the book. (The end of the dialogue
is pretty cool, eh?)
The Crab speaks of thrills being replaced at a later time with familiarity, but the familiarity
has depth. Yet the thrill can be relived because it is coded for, and can be triggered from
outside (though not consciously from within).
Achilles mentions that the two modes of listening to fugues (concentrating on individual
voices — reductionism?) versus the "total effect" (holism?) cannot exist in the brain
simultaneously — one involuntarily flips back and forth. See also
Escher's Cube with Magic Ribbons
and The Spinning Dancer Illusion.
(If you can't "figure it out", see the
solution
to the dancer illusion.)
Chapter X is about how we see things on different levels. Here is a good illustration of
this; it comes from Grady Booch's Book Object Oriented Analysis and Design, in his
chapter on Abstraction:
Levels of description are everywhere:
Pixels on a screen (or paint splotches on a canvas) → images
Diodes and resistors → gates → flip-flops and memories → computer systems
Machine language → assembly language → systems language → application language
Sometimes we can only work on one level at a time; sometimes we easily navigate between
levels; sometimes lower levels are just completely shut off from observant processes
(like our consciousness).
Intelligence depends crucially on building the high-level descriptions ("chunks"). Experts
work on different levels than novices. Examples: chess, programming.
AI research deals with moving between levels.
My theory: people good at math and programming are good at moving between
levels, both (1) overcoming the barrier that keeps low levels out of our day-to-day
worries and (2) building subroutines/functions.
Key idea: programs can be written at different levels. We take this for granted, but
is it really obvious? In fact, there's an interesting question here: which
program is really running?
Translation programs: compilers and assemblers.
The isomorphism between assembly and machine languages is prosaic, while the one between a high-level
language and an assembly language is quite exotic.
Compilers are bootstrapped in a fashion similar to the way children use language
to acquire more language.
Computers as super-rigid or super-flexible: we want rigidity since we don't want
them second-guessing us, right? What if the flexibility were programmed in?
If you know what's programmed in, then you can exploit that flexibility
If you're unaware of what's programmed in, the program's not really usable
Third option: programmed-in flexibility that is so complex that you can't
really always predict how it will react. Most programmers experience this
all the time... the program did WHAT?
But actually, software is super-flexible; you can write programs to do anything.
It is the hardware, the bottommost level, which is completely fixed.
This is similar to the mind and brain.
The amazing flexibility of our minds seems nearly irreconcilable with the
notion that our brains must be made out of fixed-rule hardware.... We cannot make
our neurons fire faster or slower, we cannot rewire our brains, we cannot redesign
the interior of a neuron ... and yet we can control how we think.
One of the main goals of the book: reconciling the software of the mind with
the hardware of the brain.
Higher-level languages aren't really more powerful than lower-level ones (all you
need is machine languages) but the higher-level ones are incredibly easier to use.
Programs don't know about other programs running at the same time (let alone themselves).
Some interesting questions: do quarks really exist? They explain so much. Are pi-mesons
and gamma rays present inside a nucleus before it is split?
Two types of system:
Chaotic on the inside but regular-ish on the outside, such as a container of gas (internal
components cancel each other out).
Discrete systems, like a program: change one bit and the effect is magnified.
Epiphenomenon: a visible consequence of overall system organization (35 users, 9.3
second 100m time, ...)
The dialogue is a continuation of the previous dialogue. The dialogue itself is a fugue,
full of canons and stretti.
Holism (the whole is greater than the sum of its parts) and reductionism
(the whole can be defined completely in terms of its parts) are defined here.
"No one in his right mind could reject holism." / "No one in her left brain could
reject reductionism."
The Crab and the Anteater argue that neither holism nor reductionism can full
explain where consciousness comes from. Achilles says MU is relevant here.
The Tortoise say his MU exists on a deeper level than Achilles imagines. Do you
see why?
The Anteater's description of the Ant Colony's ability to converse (which comes from the regularity
that arises from the seemingly chaotic behavior of dumb individual ants) makes it sound to
Achilles that the "self" is a by-product of how the brain works.
English French German Suite / Minds and Thoughts
Aria with Diverse Variations / BlooP and FlooP and GlooP
Air on G's String / On Formally Undecidable Propositions of TNT and Related Systems
Birthday Cantatatata... / Jumping out of the System
Edifying Thoughts of a Tobacco Smoker / Self-Ref and Self-Rep
The Magnificrab, Indeed / Church, Turing, Tarski, and Others
SHRDLU, Toy of Man's Designing / Artificial Intelligence: Retrospects
Contrafactus / Artificial Intelligence: Prospects
Sloth Canon / Strange Loops, Or Tangled Hierarchies