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geb.txt
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system, decision procedure, working inside/outside the system.
manipulation, and the elusive concept, "form".
recursively enumerable sets and recursive sets.
functions, physical theories, computer programs, and others.
however, that these melodies are "the same", in a peculiar sense.
intelligence to "absolute" meaning is postulated.
to preserve truth-and in particular the question
with the Dialogue by Lewis Carroll.
holism and reductionism, which is soon taken up in the Ant Fugue.
intermediate levels exist-or indeed whether any exist.
understood, in some objective sense, by an outsider?"
determining whether they are "beautiful" or not.
problem of Turing, and Tarski's Truth Theorem.
"Questions and Speculations" on Al and minds in general.
tying Godel, Escher, and Bach together once again.
Forkel, one of Bach's earliest biographers, tells the story
made a formal Dialogue between the King and the Apologist.
[Musical Offering], and dedicated it to the Inventor.'
something of the flavor of Bach's apology for his appearance. 2
Your Majesty's most august Royal grace to
Frederick, which he reported as follows:
claim that he, in turn, was even greater. The King
in five parts, and finally in eight parts.'
eight-part fugue is really beyond human capability.
because it is very melodious and sweet, almost dance-
the theme seems to shrink or to expand).
the Musical Offering is no exception. (For a simple example of
counterpoint. It is itself one large intellectual fugue, in
1972. Escher was the creator of some of the
FIGURE 5. Waterfall, by M. C. Escher (lithograph, 1961).
Tonos". The similarity of vision is
(stairs) % It is indeed true that there is an inherent
Hands. Print Gallery dates from 1956.)
which he seems to be inviting his viewers to enter.
anyway. So here is a paraphrase in more normal English:
All consistent axiomatic formulations of number theory
working out of this one beautiful spark of intuition.
but Godel realized that there was more here than meets the eye.
and also as statements about statements of number theory.
sentence G should more properly be written in English as:
the real world. How could there be many differ
clarifying what one means by "proof.
intuition with formalized, or axiomatized, reasoning systems.
and those which are not, such
sets. Intuitively, this is not the way we imagine sets.
paradox or Grelling's paradox. For people whose
mentions "this book", which should only be mentionable in a
would be the most blatant possible violation of it!
theory, you know something is wrong.
at least as far as they applied to mathematics.
whatsoever, following the methods of Russell and Whitehead?
world during the first thirty years of this century.
were not, in modern parlance, programmable.
of the pages of the Pickwick Papers,
the first to battle noise pollution.
and Babbage died a bitterly disappointed man.
computation, and the psychology of intelligence.
These same years saw the theory of computers develop by leaps and
both by the new machines, and by their unexpected limitations.
essential abilities for intelligence are certainly:
to respond to situations very flexibly;
to take advantage of fortuitous circumstances;
to make sense out of ambiguous or contradictory messages;
to recognize the relative importance of different elements of a
inflexible machines how to be flexible.
behavior, however? Certainly there must be rules on all sorts of
in the multilevel sense described above.
this book to give some perspective on the battle.
Chapters. The purpose of this structure is to allow me to
ranged from complete identity of passages on the one hand to the
resulting patterns were often symmetrical, but by no means
while in the course of study one may discover unending sub
somehow of a print by my favourite artists M.C. Escher.
TORTOISE: That is Zeno' s flag
drew? Something is wrong about the flag, I can tell.
impossible. But it is beautiful anyway, is it not?
ACHILLES: Oh, yes, there is no doubt of its beauty.
see) and the reason we are here is to have a footrace.
TORTOISE: You might give me a head start.
ACHILLES: It would have to be a huge one.
Master Zeno. As you say it is very simple indeed.
TORTOISE: Zen Koan? Zen Master? What do you mean?
it's soft fabric. And the ring cut out of it is waving, too!
(At this moment, Zeno happens by.)
Zeno: Hallo! Hulloo! What's up? What's new?
ACHILLES: This fellow must be playing the fool.
us your thoughts on this matter.
few, I promise you - in his eccentric theories.
and he mentioned especially your paradoxes.
reached the spot where the Tortoise started.
flash, Achilles covers that distance too.
and therefore Achilles can NEVER catch up with the Tortoise.
And yes, I can't quite make out what's wrong with it
Zeno: Isn't it a teaser? It's my favourite paradox.
waving down here, at the far end of the runway?
ACHILLES: The impossible one, based on an Escher print?
head start of, well, I don't know -
Tortoise, will you position yourself ten rods upwind?
( The Tortoise moves ten rods closer to the flag)
Zeno: On your mark! Get set! Go!
have worked with formal systems before.
What I mean by this is shown below, in a few examples.
the MlU-system never contain "x" "s"!
string with U in place of III.
From UMIIIMU, you could make UMUMU.
From MII11, you could make MIU (also MUI).
From IIMII, you can't get anywhere using this rule.
(The three I's have to be consecutive.)
RULE IV: If UU occurs inside one of your strings, you can drop it.
puzzle asks whether MU is a theorem of the MlU-system.
these types v appear in the book.
a proof that theorems of the MlU-system must all begin with M.
do it itself, although any person would pick up the
computer had no ability whatsoever to exit from that system.
for the M) is allowed by rule II. So the second sheet might
way of approaching thing. More about this in a few Chapters.
make no difference if you lengthen and shorten till kingdom come.
violate the spirit of the word.
possible way the genie might go about it
three new theorems: MIIU, MIUIU, MIIII.
five new theorems: MIIIIU, MIIUIIU, MIUIUIUIU, MIIIIIIII, MUI.
MIUIUIUIU MIIUIIU MIIIIU MIIIIIIII MUI MIU
Now we state the proposed "theoremhood-test":
When you have a decision procedure, then you have a very concrete
is a "litmus test" for theoremhood!
a decision procedure, but the latter may not.
was still totally unclear whether MU is, or is not, a theorem.
What the Tortoise Said to Achilles
proved that the thing couldn't be done?"
distances were constantly DIMINISHING; and so-"
several times round the world, by this time!"
"Proceed! And speak SLOWLY, please! SHORTHAND isn't invented yet!"
order to refer to them conveniently, let's call them A, B, and Z:
(A) Things that are equal to the same are equal to each other.
(Z) The two sides of this Triangle are equal to each other.
any one who accepts A and B as true, MUST accept Z as true?"
"Undoubtedly! The youngest child in a High School-as soon as High
SEQUENCE as a VALID one, I suppose?"
but I DON'T accept the Hypothetical'?"
"Certainly there might. HE, also, had better take to football."
logical necessity to accept Z as true?"
me, logically, to accept Z as true."
"A tortoise playing football would be-" Achilles was beginning.
point. Let's have Z first, and football afterwards!"
"Let's call it C," said the Tortoise.
(C) If A and B are true, Z must be true."
"That is my present position," said the Tortoise.
"Then I must ask you to accept C."
What else have you got in it?"
ALL!" (Achilles shuddered.) "Now write as I dictate:
(A) Things that are equal to the same are equal to each other.
(C) If A and B are true, Z must be true.
(Z) The two sides of this Triangle are equal to each other."
accept A and B and C, you MUST accept Z.
true. You can't dispute THAT, I imagine?"
and B and C, and STILL not accept Z, mightn't I?"
(D) If A and B and C are true, Z must be true.
Have you entered that in your note -book?"
and D, OF COURSE you accept Z."
C and D. Suppose I STILL refused to accept Z?"
Tortoise. "So enter it in your book, please. We will call it
(E) If A and B and C and D are true, Z must be true.
Turtle never made, and allow yourself to be re-named A KILL-EASE!"
rules, or do they noft That problem is the problem of this book.
they are of interest to mathematicians and logicians.
-the letters p, q, and the hyphen.
string composed of p's, q's, and hyphens.
it is called an axiom schema.
The pq-system has only one rule of production:
If ~p — q- turns out to be a theorem, then so will ~p — q~.
alone, o string such as ~p~p~p— q .
hyphen-groups should add up, in length, to the third
offspring. This shows why the addition criterion is correct.
shortening rules that gives formal systems a certain fascination..
theorems as they are generated. Here is how it is done:
(la) Throw the simplest possible axiom (-p-q~) into the bucket.
(2a) Throw the second-simplest axiom into the bucket.
(3 a) Throw the third-simplest axiom into the bucket.
This symbol-word correspondence has a name: interpretation.
q) just as much as any interpreted theorem.
language is a very important one, however. It is this:
of the pq-system. Consider the following association:
which are well-formed, but which are not theorems.
theorem whether or not we connect it with addition.
gears of a device like a cash register.
operates deterministically, which is an open question.
many of the people who give that answer have actually at
offer as proof a few marks on paper, such as are shown below:
you don't-but in the latter case, a proof won't help you a bit.
"language" and "dialect" are themselves fuzzy.
practical things to numbers as formal things. Once you have
can't be a multiple of A" (because it leaves 1 over, when you
And thus ends the demonstration of the infinitude of the primes.
label "true and what statements to label "false".
towards the formalization of number theory.
when the systems involved are very different from each other.)
the level of our thinking abilities, by using some formal system.
The telephone rings; Achilles picks it up.
Achilles: Oh, hello, Mr. T. How are you?
Achilles: How long did you hold it in that position?
twisted that way for so long?
FIGURE 14. Mosaic II, by M. C. Escher (lithograph, 1957).
Achilles: Oh, well, it's all the same to me.
Bach uses a single violin to create a piece with such interest.
Achilles: I see. Have you tried counting sheep?
reality he was just trying to think up complex word puzzles?
But I'm still in the dark on your "ADAC" puzzle.
Achilles:: Yes, t see all the black animals.
to do with the "AD AC" puzzle?
Achilles: You want to come over now? But I thought-
Achilles: I'd love to play them for you.
Achilles: You've invented a theory about them?
the harpsichord part, then, and had it published a s well.
Those sonatas seem to work very indeed as they are.
Achilles: Right. Well, I'll see you shortly.
(1) reading and recognizing any of a finite set of symbols;
(2) writing down any symbol belonging to that set;
(3) copying any of those symbols from one place to another;
(4) erasing any of those symbols;
(5) checking to see whether one symbol is the same as another;
(6) keeping and using a list of previously generated theorems.
system in which primes are distinguished from composite numbers?
AXIOM SCHEMA: xt-qx is an axiom, whenever x is a hyphen string.
x ty qz is an old theorem. Then, xty-qzx is a ne' theorem.
Below is the derivation of the theorem — t — q
the middle, you apply the rule of inference nine times in a row.
Cx, which characterize compost, numbers, as follows:
mechanically and happily, never questioning; them!
you from mixing up arithmetical facts with typographical theorems.
(by rule of inference, using line (1) as the old theorem)
(by rule of inference, using line (2) as the old theorem)
proposing a rule of the following kind:
left out of a list which is positively defined.
sudden, you will see seven letters appear in this ..
In more "technical" terminology, this becomes:
Figure", is shown in Figure 17. If you read both black and white,
FIGURE 17. FIGURE-FIGURE Figure, by Scott E. Kim (1975).
(1) as the negative space to the white regions;
(1) as the negative space to the set of all TNT-theorems;
But this hope will be dashed, because:
(1) inside the set of all nontheorems are found some truths
pictorial representation of the situation (Fig. 18).
Needless to say, Bach didn't stop at this level of complexity...
Recursively Enumerable Sets vs. Recursive Sets
role of negative space is played by strings with a
me, and continues to astonish me even today. It is a fact that:
the positive space (set of theorems) of any formal system.
There exist recursively enumerable sets which are not recursive.
It follows from the above result that:
along the way. This amounts to
time will get included you just wait a little longer.
to leave it with some intuitive fluidity.
characterize the following set of integers (or its negative space)
1 3 7 12 18 26 35 45 56 69...
How is this sequence like the FIGURE -FIGURE Figure?
Primes as Figure Rather than Ground
AXIOM SCHEMA: xy D N Dx where x and y are hyphen-strings.
RULE: If x D N Dy is a theorem, then so is x D N Dx y.
If you use the rule twice, you can generate this theorem:
What goes wrong if you try to produce it?
done, but there is a trick to it. Think about it if you want.
RULE: If -D N D z is a theorem, so is z D F-.
RULE: If z-DFz is a theorem, then Pz- is a theorem.
Oh-let's not forget that 2 is prime!
Achilles has come to visit his friend and jogging companion, the
records. What sort of music do you enjoy?
Achilles: Tell me, what kind of music is that?
Achilles: Exactly what I was thinking.
Crab-have you met him, by the way?-paid m» a visit.
much about him, I've never met him
Perhaps we could meet at random in the park on day ...
about your weird "music to smash phone graphs by", weren't you?
Achilles: Naturally, I suppose you disagreed.
called "I Cannot Be Played on Record Player 1".
Achilles: Rather unusual. Was it a present for the Crab?
record was utterly destroyed also, needless to say.
would make it vibrate and break.
salesman had told him, after all.
less than Perfect, and could not reproduce every possible sound.
coincidence that your record had those very sounds on it ...
on Record Player based on the construction of the new model.
Achilles: So what happened this time?
model up would fill the bill, and having twice the money, h e—
would destroy it. In that way, he would avoid your trick.
breaking sound, which is of coup impossible.
considerably more sophisticated than an ordinary record player.
Achilles: Let me guess how: Did it have no of cotton? Or
Omega attempt to play the record.
Incompleteness Theorem backwards and forwards, do you?
Achilles: Know WHOSE Theorem backwards and forwards? I've
more, but really I am growing quite sleepy.
Achilles: Didn't have the foggiest. What an agreeable coincidence!
Tortoise: Pray tell, who is it?
hardly a soul knows it-a: this goblet is the last piece he blew.
inestimable. But how are you sure of its maker
his name actually constitutes a melody, then
are so many clever ways of hiding things in music .. .
Poems which conceal messages that way are called "acrostics".
Achilles: A dial-a-logician? That's a new one on me.
backwards, is exactly the same as the original?
A-B-but upside down? You must be pulling my leg.
the last Contrapunctus, and here's the last theme ...
stood, only moments before. And then ... dead silence.
medium, which carries it from the objects to the ear.
explicit meaning is simply the story
(1) Isomorphism between arbitrary groove patterns and air
(2) Isomorphism between graph vibrations, arbitrary air
cause its indirect self-destruction. [Drawing by the author.
Level One: Goblets and records which backfire;
to get him in the end. Seen this
Phonograph <= =>axiomatic system for number theory
low-fidelity phonograph <= =>"weak" axiomatic system
high-fidelity phonograph <= =>"strong" axiomatic system
"Perfect" phonograph" <= => complete system for number theory'
Blueprint" of phonograph <= => axioms and rules of formal system
record <= => string of the formal system
playable record<= => theorem of the axiomatic system
unplayable record <= =>nontheorem of the axiomatic system
sound <= =>true statement of number theory
reproducible sound <= => 'interpreted theorem of the system
unreproducible sound <= => true statement which isn't a theorem:
song title <= =>implicit meaning of Godel's string:
"I Cannot Be Played "I Cannot Be Derived
on Record Player X" in Formal System X"
. until the last Contrapunctus .
A most puzzling fact about Godel's method of proof is that he uses
least that is a "romantic" way to view the situation.
Axiom SCHEMA II: If x is a hyphen-string, then xp-qx is an axiom.
very well retain their "meaning while others change.
elements, or could it have structural weaknesses?
the hopes that we will abide by that
FIGURE 21. Tower of Babel, by M. C. Escher (woodcut, 1928).
stories. The first four postulates are rather terse and elegant:
(1) A straight line segment can be drawn joining any two points.
radius and one end point as center.
(4) All right angles are congruent.
The fifth, however, did not share their grace:
title Euclid Freed of Every Flaw, and then expired.
same year, the great French mathematician
postulate, very much along the lines of Saccheri.
the manner of violets coming to light in early spring.
having found several deep results in non-Euclidean geometry.
theorems. The symbol q is especially interesting,
POINT, so two POINTS determine a LINE.
they occur, rather than explicitly, in a definition.
interpretations-it is up to the observer to look for them.
The Tortoise always beats Zeno at chess
Zeno always beats Egbert at chess.
Egbert always beats the Tortoise at chess.
merely that they come out compatible with one another.
The Tortoise was invented by Zeno.
Egbert was invented by the Tortoise.
imaginable world, instead of the one we live in. But this is
football would be an anomaly, of course.
Embedding of One Formal System In Another
Layers of Stability in Visual Perception
FIGURE 22. Relativity, by M. C. Escher (lithograph, 1953).
for the undefined terms, resulting in a com
Is Mathematics the Same in Every Conceivable World?
there would have to be exactly on parallel line ...
Is Number Theory the Same In All Conceivable Worlds?
say on it after going through Godel's Theorem in detail.
passive meanings. Where consistency is the property
which can be expressed as well-formed strings of the system, are
use our new phrase-provable within the system.
would wish them do-namely tell us everything about number theory.
How an Interpretation May Make or Break Completeness
Tortoise: This is my favorite ride. One seems to move so far, and
Achilles: You certainly are exuberant today.
Achilles: Don't tell me you believe in fortune-telling!
flying our way. In fact it's almost directly above us now.
coming so close we could practically grab it
Tortoise: Can you make out what the note says?
Achilles: All I can say is you've got a pretty fiendish laugh.
Achilles: Good grief-I wonder what he means by that!
Tortoise pie! My favorite kind of pie! (Exit.)
Achilles: Oh, boy-popcorn! I'm going to munch my head off!
think about food at a time like this?
Made Easy; Concerto for Tapdancer and Orchestra ... Hmmm.
dodecahedron and the open drawing pad?
and you could take that of Achilles.
Achilles: I'm game. Here goes nothing . . .
(They begin reading "Djinn and Tonic".)
his favorite artist, M. C. Escher.)
Tortoise: These are wonderful prints, Achilles.
consistent worlds, when juxtaposed, make a completely inconsistent
but I wouldn't want to live there.
Tortoise: I beg your pardon, but weren't we just agreeing that in
this Escher picture, an inconsistent world is portrayed?
picture. You can't visit that world.
Achilles: What on earth is pushing-potion?
drunk by someone looking at a picture, "pushes" him right into the
world of that picture. People who aren't aware of the powers of
you need only take a swallow of popping-tonic, and presto! You're
back in the rea. world, exactly where you were before you pushed
popping-tonic without having previously pushed yourself into a
one has heard from him since.
pushed right along with you into the picture you're looking at.
have already entered, and take another swig of pushing-potion?
you down inside a picture, and a pop undoes that.
not just testing the limits of my gullibility?
{Reaches into his lapel pocket, and pulls out two rather large
around.) If you're willing, we can try them. What do you say?
Achilles: Well, I guess, ahm, maybe, ahm ...
the world of Escher's Convex and Concavel
you, with your experience, manage the operation.
both he and Achilles lift their glasses to their lips.)
FIGURE 23. Convex and Concave, by M. C. Escher (lithograph, 1955).
Achilles: That's an exceedingly strange taste.
Tortoise: One gets used to it.
(The gondolier pays no attention to this request.)
Enters the sinister "Tunnel of Love"; just ahead of us.
here. Say, how do you know so much about this place, anyway? Have
They're all connected behind the frames, you know. Once you're in
one, you can get to any other one.
Oh, look at those two cute lizards!
They are the vicious guardians of that magic copper lamp hanging
Tortoise: It's a foolhardy venture, my friend. I wouldn't risk it.
Achilles: I'm going to try just once.
(He stealthily approaches the lamp, making sure not to awaken the
Achilles: Oh, Mr. T, how can I repay them?
in the town below, there's a place where they make an incomparable
cup of espresso. Invite them for a cup of espresso! Achilles: That
sip their drinks, Achilles remembers he has the lamp with him.)
magic about it? Tortoise: Oh, you know, just the usual-a genie.
Tortoise: Right. What did you expect? Pennies fry heaven?
wished this would happen to me ...
part of my Lamp, the opportunity to have any three of your wishes
Achilles: How stupefying! Don't you think so, Mr. T?
Tortoise: I surely do. Go ahead, Achilles, take the first wish.
nested) tales)-I wish that I had a HUNDRED wishes, instead of just
Tortoise: Maybe now you'll find out the answer.
Genie: I am sorry, Achilles, but I don't grant metawishes.
Achilles: I wish you'd tell me what a "meta-wish" is!
either. Achilles: Whaaat? I don't follow you at all.
Tortoise: Why don't you rephrase your last request, Achilles?
Achilles: What do you mean? Why should I?
information, why don't you just ask a question?
to grant meta-wishes. It is only within my purview to grant plain
at the Copacabana. You know-simple things like that. But meta-
wishes I cannot grant. GOD won't permit me to.
That seems like such a puny thing compared to the others you
and make your three wishes? Or at least make one of them. I don't
have all I time in the world, you know ...
I do about this. This'll just take one moment
smaller letters, so as to cover the same area.)
Genie: This is my Meta-Lamp ...
Meta-Genie: I am the Meta-Genie. You summoned me, 0 Genie? What is
Genie: This is my Meta-Meta-Lamp. . .
Meta-Meta-Genie: I am the MetaM eta-Genie. You summoned me,
0 Meta-Genie? What is your wish?
on wishes, for the duration of one Typeless Wish. Could you please
Meta-Meta-Genie: I'll have to send it through Channels, of course.
One quarter of a moment, please.
(And, twice as quickly as the Meta-Genie did, this MetaMeta-
Genie removes from the folds of his robe an object which looks
just like the gold MetaLamp, except that it is made of ...)
( ... swirls back into the MetaM eta-Meta-Lamp, which the Meta-
Meta-Genie then folds back into his robe, half as quickly as the
Your wish is granted, 0 MetaGenie.
Meta-Genie: Thank you, 0 Djinn, and GOD.
back into the Meta-Meta-Lamp, which the Meta-Genie then folds back
into her robe, half as quickly as the Meta-Meta-Genie did.)
Your wish is granted, 0 Genie.
Genie: Thank you, 0 Djinn, and GOD.
(And the Meta-Genie, as all the higher ones before her,
half as quickly as the M Genie did.)
Achilles: Thank you, 0 Djinn, and GOD.
as many "meta"'s as you wish-even infinitely many (if wish).
Before I make my wish, would you mind telling me who-or what-
The word "Djinn" is used designate Genies, Meta-Genies, Meta-Meta-
Gen etc. It is a Typeless word.
everybody knew about them. \ see, "GOD" stands for "GOD Over
Djinn"-which can be expanded as "GOD Over Djinn, O, Djinn"-and
that can, in turn, be expanded to "G( Over Djinn, Over Djinn, Over
Genie: Of course not. You can never totally expand GOD.
Meta-Genie, "I have a sped wish to make of you, 0 Djinn, and of
Djinns over her. 'I recursive acronym method accomplishes this qL
naturally. You see, when the Meta-Genie received my request, she
then had to pass it upwards to I GOD. So she forwarded a similar
why GOD is a recursive acronym. GOD is not some ultimate djinn;
GOD is the tower of djinns above any given djinn.
djinns above him or her, and no two djinns share that set.
fancy themselves somehow closer to GOD. What blasphemy!
Achilles: By gum, it must have taken genies to invent GOD.
Tortoise: I don't think I'm agnostic. Maybe I'm metaagnostic.
Achilles: Whaaat? I don't follow you at all.
not being granted. You see, the chances are infinitesimal, that a
garbling will occur at any PARTICULAR link in the chain-but when
you put an infinite number of them in a row, it becomes virtually
are very sparsely distributed in the chain.
is it? Achilles: I wish my wish would not be granted!
Achilles: What on earth does that cryptic comment mean?
Tortoise: It refers to the Typeless Wish Achilles made.
Achilles: But he hadn't yet made it.
Tortoise: Yes, he had. He said, "I wish my wish would not be
granted", and the Genie took THAT to be his wish.
Achilles: Oh, my! That sounds ominous.
(The footsteps stop; then they turn around and fade away.)
Achilles: Hey! What happened? Where is my Genie: lamp? My cup of
espresso? What happened to young friends from the Convex and
Concave worlds? What are all those little lizards doing hi
Achilles: What on earth does that cryptic comment mean?
Tortoise: I refer to the Typeless Wish you made.
Achilles: But I hadn't yet made it.
Tortoise: Yes, you had. You said, "I wish my wish would not be
granted", and the Genie took THAT to be your wish.
Achilles: Oh, my! That sounds ominous.
Tortoise; It spells PARADOX. For that Typeless wish to be
Tortoise: No. The System crashed. Achilles: What does that mean?
Tortoise: It means that you and I, Achilles, were suddenly and
instantaneously transported to Tumbolia. Achilles: To where?
bulbs. It's a sort of waiting room, where dormant software waits
for its host hardware to come back up. No telling how long the
System was down, and we were in Tumbolia. It could have been
is. But I do know that I didn't get to make my wishes! I want my
you should thank your lucky stars that we're back at all. Things
could have come out a lot worse. But I have no idea where we
an Escher picture with lizards before it went down. That's
table, next to the cycle of lizards?
The System was very kind to us, in giving us back our popping-
wonder what they are. {He picks up the smaller one, which is
open to a random page.) This looks like a moderately
Achilles: Oh, really? What is its title?
Place in Sundry Parts of the Globe. It sounds like an interesting
FIGURE 24. Reptiles, by M. C. Escher (lithograph, 1943).
take any chances with t popping-tonic-one of the lizards might
knock it off the table, so I'm going to get it right now!
rolling toward towards-the stairwell! Quick-before it falls!
which he has in his hands.) Achilles: Well, You can read it if you
want, but as for I'm not going to take any chances with t popping-
Tortoise (muttering): Eh? This story looks fascinating.
Achilles: Mr. T, Mr. T, help! Help catch the tonic-flask!
Tortoise: What's all the fuss about?
Oh no! What can we do? Mr. Tortoise-aren't you alarmed? We're
thing to do! We'll have to go down one story!
Tortoise: Go down one story? My pleasure. Won't you join me?
once, finally stays, taking the role of the Tortoise.)
of them? You can hold it out in front of you so that you don't
path is curving gently to the left as we walk? Tortoise: Very
day again. I wish I'd never listened to you, when you suggested I
swallow some of that "DRINK ME" stuff.
not a once have I ever regretted it. Relax and enjoy being small.
Achilles: Being small? What is it you've done to me, Mr. T?
Achilles: Have you made me shrink? So that this labyrinth we're
in is actually some teeny thing that someone could STEP on?
FIGURE 25. Cretan Labyrinth (Italian engraving; School of
Finiguerra). [From N Matthews, Mazes and Labyrinths: Their
History and Development (New York: Dover Publications, 1970).
Harmonic Labyrinth of the dreaded Majotaur?
it, waiting innocent victims to get lost in its fears complexity.
Then, when they wander and dazed into the center, he laughs and
laughs at them-so hard, that he laughs them to death!
Tortoise: But it's only a myth. Courage, Achilles.
(And the dauntless pair trudge on.)
But the corrugations have different sizes.
the corrugations, strange noises echo up and down the long curved
Achilles: Oh, just me, rubbing my walking stick against the wall.
Tortoise: Whew! I thought for a moment it was the bellowing of the
ferocious Majotaur! Achilles: I thought you said it was all a
Tortoise: Of course it is. Nothing to be afraid of.
walking. As he does so, some musical sounds are heard, coming from
the point where his stick is scraping the wall.)
Tortoise: Uh-oh. I have a bad feeling, Achilles.
What makes you change your mind all of a sudden? Tortoise: Do
Hey! Put that back! I want to hear the end of this piece!
(Confused, Achilles obeys, and the music resumes.)
Your stick scraping against the strange shapes in the wall acts
like a needle running down the groove, allowing us to hear the
be in such intimate contact with music before?
when I am smaller than a flea, Mr. Tortoise?
about being small. Especially not when faced with the awful
Achilles: Horrors! Are you telling me
Tortoise: I'm afraid so, Achilles. The music gave it away.
Achilles: How could it do that?
I immediately realized that the grooves we're walking through
could only be Little Harmonic Labyrinth, one of Bach's er known
organ pieces. It is so named cause of its dizzyingly frequent
tonality, as C major, which is the key of this o;
Tortoise: That's right. As the piece develops ambiguous chords and
melodies are t which lead away from the tonic. Little by little,
tension builds up-you feel at creasing desire to return home, to
had waiting my whole life to hear the ton
Tortoise: Exactly. The composer has uses knowledge of harmonic
manipulate your emotions, and to build up hopes in you to hear
Achilles: But you were going to tell me about modulations.
"modulate" partway through a piece, which means that he sets up
a temporary goal other than resolution into the tonic.
shifts the harmonic tension somehow so that I actually desire to
the back of your mind you retain the longing to hit that original
goal-in this case, C major. And when the subsidiary goal is
upward-swooping chords which mark the end of this Little
Tortoise: No, Achilles, this isn't the end. It's merely
of relief! That's some resolution! Gee!
is a disgrace to the world of music. Achilles: What do you mean?
modulated from C into G, setting up a secondary goal of hearing
G. This means that you experience two tensions at once-waiting
for resolution into G, but also keeping in mind that ultimate
desire -to resolve triumphantly into C Major.
Achilles: Why should you have to keep any
thing in mind when listening to a piece of music? Is music only an
music is not. And most of the time your ear or br the
"calculation" for you, and lets your emotions know what they
want to hear, don't have to think about it consciously in this
piece, Bach was playing tricks hoping to lead you astray. And in
Achilles: Are you telling me that I responded to a resolution in a
Achilles: It still sounded like an ending to me
It was deliberately contrived to sound like an ending but if you
follow the harmonic progression carefully, you will see that it is
in the wrong key. Apparently not just you but this miserable
record company fell for the same trick-and they truncated the
Achilles: What a dirty trick Bach played
Majotaur is in cahoots with Bach, And if you don't watch out, he
i laugh you to death-and perhaps n with you!
grooves, and escape on the outside record before the Evil
gressions which occur when time versed.
Tortoise: That's a very good question.
Suddenly t is a slight gasp, and then a "thud".)
help you out. We've got to move fast!
Achilles: Careful, Mr. T-I don't want You to fall in here, too ...
Tortoise: Don't fret, Achilles. Everything will be all —
(Suddenly, there is a slight gasp, and then a "thud".)
Achilles: Mr. T-you fell in, too! Are you all right?
Tortoise: Only my pride is hurt-otherwise I'm fine.
Achilles: Now we're in a pretty pickle, aren't we?
(Suddenly, a giant, booming laugh is heard, alarmingly close to
Tortoise: Watch out, Achilles! This is no laughing matter.
Majotaur: Hee hee hee! Ho ho! Haw haw haw!
Achilles: I'm starting to feel weak, Mr. T ...
Tortoise: Try to pay no attention to his laugh,
Achilles: I'll do my best. If only my stomach weren't empty!
popcorn around here? Achilles: I smell it, too. Where is it coming
Yes, indeed-it seems to be a bowl of popcorn!
Achilles: Oh, boy-popcorn! I'm going to munch my head off!
Tortoise: I didn't say a thing. You must be hearing things.
Achilles: Go-golly! I hope not. Well, let's dig in!
(And the two Jriends begin muncnai popcorn (or pushcorn?)-and t
once POP! I guess it was popcorn; all.)
Tortoise: What an amusing story. Did you en
pit or r Achilles-he wanted to be full-sized again
For some reason, burning. And nothing would taste bett drink of
places people very crazy over it. At the turn of the century the
Schonberg food factory stopped ma] and started making cereal
instead. You cai the uproar that caused.
Those lizards on the you see anything funny about them?
Tortoise: Umm ... not particularly. What do you see of such great
Achilles: Don't you see it? They're emerging flat picture without
drinking any pop] How are they able to do that?
Tortoise: Oh, didn't I tell you? You can ge picture by moving
perpendicularly to it you have no popping-tonic. The little li
learned to climb UP when they want to ge two-dimensional
Tortoise: Follow me, then, up this way.
house! This is YOUR house, Mr. Tortoise
Tortoise: Well, so it is-and am I glad for that! I wasn't looking
and doubt if I could have made it.
Tortoise: I'll say! This certainly is a piece of Good Fortune!
relation should be clear by the end of this Chapter.
recursion in its most precise form
■V. 1 1 1 1 1 r — htiittr
(7?Bt** and 1 f.V.vd t'f / L * ]
tension-for example a resolution into the pseudotonic —
awaiting their dire fates at the knife of Monsieur Goodfortune.
back into D, and letting us return to G once more.
listeners, we don't have very reliable deep stacks.
co construct "milk", or "big red blue green sneezes", etc.
ORNATE NOUN; RELATIVE PRONOUN; FANCY NOUN; VERB.
new one nothing were unusual. So we shall do the same.
choose the lower pat] ORNATE NOUN; PREPOSITION; FANCY NOUN. That
FANCY NOUN, with the result that the phrase
"the strange bagels that the purple cow without horns gobbled"
random, an infinite regress of that sort will not happen.
proceed into the very small RTN,
FIGURE 28. The FANCY NOUN RTN with one node recursively expanded
infinite diagram, even when an RTN calls itself.
right-hand edge is the famous sequence of Fibonacci numbers.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,
Bonacci", or "Fibonacci" for short. These numbers are best
FIGURE 29. (a) Diagram G, unexpanded. (c) Diagram H, unexpanded
(b) Diagram G, expanded once. (d) Diagram H, expanded once
FIGURE 30. Diagram G, further expanded and with numbered nodes.
defined recursively by the pair of formulas
FIBO(n) = FIBO(n- 1) + FIBO(n-2) for n > 2
We could represent this pair of formulas in an RTN (see Fig. 31).
FIGURE 31. An RTN for Fibonacci numbers.
don't need to keep track of a stack.
defined with one more nesting than G —
H(n) = n - H(H(H(n - 1 ))) for n > 0
corresponds precisely to the recursive algebraic definitions.
They are quite elegant and simple.
following recursive definition of a function:
Q(n) = Q(n - Q(n- 1)) + Q(n - Q(n-2)) for n > 2
to make the new value! The first 17 Q-numbers run as follows:
1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 8, 8, 8, 10, 9, 10, ... .
5 + 6 = 11 how far to move to the left
ephemeral to exist. Its definition sounds too circular.
completely different sequence, known as the Lucas sequence:
1, 3, 4, 7, 11, 18, 29, 47, 76, 123,...
values of x, it is continuous.
Recursion at the Lowest Level of Matter
and they are purely hypothetical creations; they don't exist.
Invention. A physicist would draw a picture like this:
may even nest them, as shown below:
would be ferociously hard to calculate.
realized in nature. For instance, the one be impossible:
electron propagates from A to B,
own virtual cloud, and so on ad infinitum.
called g-factor of the muon) to nine decimal places -- correctly!
particularly after it has practiced with INT.
FIGURE 36. Fish and Scales, by M. C. Escher (woodcut, 1959).
that even a tiny section of an
FIGURE 37. Butterflies, by M. C. Escher (wood-engraving, 1950).
When are two things the same?
"same" thing happening on several differ-
Programming and Recursion: Modularity, Loops, Procedures
knitting or crocheting-in which very small loops are
follow the alternate pathway — the non-recursive evaluation.
just one more piece of evidence for the rather recursive
probably lies at the heart of intelligence.
two, at the best Chinese restaurant in town.
Achilles: You wield a mean chopstick, Mr. T.
we just sit here and talk a little while?
Tortoise: I'd love to talk while we drink our tea. Waiter!
Could we have our bill, please, and some more tea?
Japanese poetry than you do. Have you ever read any haiku?
Tortoise: I'm afraid not. What is a haiku?
Tortoise: Hmm ... That's an evocative statement.
Thank you, waiter. Care for more tea, Achilles?
chew.) Hey! What's this funny thing inside? A piece of paper?
of your fortune. What does the rest say?
Achilles: Let's see what your fortune says, Mr. Tortoise.
messages which we run across ...
him a lot since you told me of your peculiar phonograph-battle.
silly songs, so quaint and reminiscent of bygone eras.
Tortoise: As a matter of fact, it has exactly one record.
Tortoise: No, it's just a regular jukebox-style record.
of a jukebox is it that has only a single song?
Tortoise: Who said anything about a single song, Achilles?
one song out of this crazy contraption?
buttons B-l, then C-3 then B-10-all just at random.
C-H, which I believe you remember.
Achilles: Could I ever forget it?
position, so that C-3 could be slid into position.
Achilles: Now don't tell me that C-3 played another song?
Achilles: That's a totally different melody!
him in one of my books on haiku.
4'33" on a jukebox. It might afford some relief!
coded inside a single record at once?
Tortoise: Precisely. What about C-A-G-E, now?
It's very much like the other one, isn't it?
add their own interpretations to that code?
to hear a third song, when record player B-l swiveled into place.
expand the melodic range in a curious way.
The interval pattern in semitones was:
When Is One Thing Not Always the Same?
simplicity with which intelligence can be described.
feel you are putting in n information than you are pulling out.
similarity between its physical characteristics and its genotype.
instructions must be cc somehow in the structure of the DNA.
"revelation" in this context is due to
inherent the record, for several reasons:
(the music) to an arbitrary degree of accuracy;
DNA and the Necessity of Chemical Context
has certainly lost its context. How much meaning does it carry?
Levels of Understanding of a Message
mechanism for translation of the record is a machine which
get deciphered in a way which we would consider successful.
contrary, it supplies meaning concealed inside itself.
at that time-and it would have come
-w^wsis i iaiEfl^iiuttfls»i)w*trT'3 ri .'jii:if=«tii; -
FIGURE 39. The Rosetta Stone [courtesy of the British Museum.
the extent that it acts upon intelligence in a predictable way.
decoding mechanism for the inner message.
the outer message, the inner message would be reconstructible.
his book is given by this quote:
description. See Figure 40 for some good examples.)
luflvnuixjupruj-jsu Iv In mo tun am
i-Ji,-"*/^ qa» it si_as»®. ,3*1 >,«•.,>>»
Washington Square Press, 1960), pp. 73 (Tamil), 82 (Thai).
effortless. It is as if it just gets pumped in.
and messages do get understood. How come?
Meaning Is Intrinsic If Intelligence Is Natural
the inherent, though implicit, meaning of "A-5".
could we ever realize that such beings existed?
would serve to lessen the feeling of having
second plaque, containing more dots, as follows.
1, 1, 2, 3, 5, 8, 13, 21, 34.
be inferred by intelligence from the genotype alone.
instance, the "short genotype" (1,3) would yield the phenotype
1, 3, 4, 7, 11, 18, 29, 47,...
information necessary for the revelation of the phenotype.
experiment, perhaps the DNA's meaning would finally emerge.
shaking himself dry, when who but Achilles walks by.
meadows. They're so green at this time of year.
to try to snare me in one of your wicked traps of log Mr. T.
things have nothing-nothing whatsoever-to do with logic!
Achilles: Well, didn't you just tell me Tortoise: I did.
reality don't necessarily match; that —
certainly contradicts what you said before!
Tortoise: Yes, I guess you did.
so steeped in contradiction it's impossible to argue with you!
contradictory one, and there are no two ways about it.
to point out a contradiction? Go ahead-try it out.
one thing and denies it at the same time.
Tortoise: Well, you didn't give ONE sentence. You gave TWO.
Achilles: Yes-two sentences that contradict each other!
single sentence. Then you told me that you
inconsistent it is. From the outside, however plain as day.
"Politicians lie" is true, wouldn't you?
Achilles: But you-you combined the two-in such a silly way!
Achilles: You should have used the word "and", not "in".
sounding Principles. None of that for me today, plea
only the most innocent of motivations.
Tortoise: That's what everyone believes of himself...
Achilles: Oh, I'm so sorry I started all this.
err again, in my quest for Truth.
Alphabet and First Rule of the Propositional Calculus
The first rule of this system that I will reveal is the following:
is important to define a subset of all strings, namely the
For example, all of the following are well-formed:
understood as restricted to well formed strings
call it that is quite simple.
of the fantasy, having learned from it that out.
If x were a theorem, y would be a theorem.
those lines of a derivation which take place in fantasies.
inquiring, "What if it were?") We make the following fantasy:
~~~P outcome (by double tilde rule)
If P were a theorem, so would ~~P be one.
Here is another derivation using the fantasy rule:
recursively, we present the following derivation:
P carry-over of P into inner fantasy
] pop out of inner fantasy, regain outer fantasy
] pop out of outer fantasy, reach real world!
The Converse of the Fantasy Rule
theorem, and that its first "clause" is also a theorem.
The Intended Interpretation of the Symbols
<xvy> is "either x or y-or both".
This mind is Buddha, and this mind is not Buddha.
then it is not the case that this mind is not Buddha.
If this mind is Buddha and this flax weighs three pounds,
then this flax weighs three pounds and this mind is Buddha.
then, if this flax weighs three pounds,
then this mind is Buddha and this flax weighs three pounds.
Rounding Out the List of Rules
rules is listed below, including the three new ones.
JOINING RULE: If x and y are theorems, then < xAy> is a theorem.
CONTRAPOSITIVE RULE: <xuy> and <~yzD~x> are interchangeable
DE MORGAN'S RULE: <~xA~y> and ~< xvy> are interchangeable.
SWITCHEROO RULE: <xvy> and <~xzjy> are interchangeable.
examples - which is why I only give a couple.
If you are studying it, then you are far from the Way
If you are close to the Way, then you are not studying it.
one could deny that it is a Zensible conclusion to draw.
take the theorem <Pz> — P>:
This new theorem, when interpreted, says:
Either this mind is Buddha, or this mind is not Buddha
Here is a fantasy to answer that question.
Do you see my side well? How is the Zen side?
Is There a Decision Procedure for Theorems?
would tell us that the set of theorems of the
Do We Know the System Is Consistent?
think that some rule must be wrong. Show me which one.
Prudence: Of course-in fact both MI and MIU are theorems.
you to think that '-' should be interpreted as not'?
interpretation for "a' is "and, etc.
definers all the simple words in terms of complicated ones...
reasoning forever. There comes a point where faith takes over.
proof that a proof in some system is correct. Of course,
Tortoise: Oh! You mean: ««AaB>=3Z>a<AaB»=dZ>, : don't you?
rules of inference. For instance, there is a De Morgan's Rule:
all. Therefore, there is a drawback to using such shortcuts.
make shortcuts in the top level.
language of the formal system and its metalanguage (English).
Reflections on the Strengths and Weaknesses of the System
sophisticated number-theoretical reasoning can be done.
which makes them almost impossible to grasp.
Thus, the Propositional Calculus should be thought of as part of a
Here is a fantasy with this string as its premise:
Now this theorem has a very strange semi-interpretation:
of life-and mathematics is no exception. When in times past, a
system, and that would block the derivation of the «Pa~P >z>Q>
FIGURE 42. "Crab Canon", by M. C. Escher (-1965)
Achilles and the Tortoise happen upon each other
in the park one day while strolling.
you're looking in very fine fettle these days, I must say.
Tortoise: Not at all. Here, care for one of my cigars?
markedly inferior taste, don't you think?
taste. De gustibus non est disputandum.
Achilles: To be precise, one has no frets.
Tortoise: Oh, well, it's all the same to me.
pointing to a rather prominent black eye.)
(And he disappears as suddenly as he arrived.)
Achilles: Oh, well, it's all the same to me.
Tortoise: To be precise, one has no frets.
taste. Disputandum non est de gustibus.
FIGURE 43. Here is a short section one of the Crab's
Genes, turning round and round. When the two DNA
strands are raveled and laid out side by side, they
Notice that they are the same, only one forwards while
the other goes backwards This is the defining
property of the form called "crab canon " in music. It
is reminiscent of, though a little different from
palindrome, which is a sentence that reads the same
backwards and forwards ,In molecular biology,
such segments of DNA are called "palindromes "-a
slight misnomer, since "crab canon " would be more
accurate. Not only is this DNA segment crab-
canonical-but moreover its base sequence codes for
markedly inferior taste, don't you think?
Achilles: Not at all. Here, care for one of my cigars?
Achilles: So nice to run into you.
The Crab Canon and Indirect Self-Reference
has to look at the form, as well as the content, of the Dialogue.
What We Want to Be Able to Express in TNT
Some typical sentences of N-number theory-are:
(3) 1729 is a sum of two cubes.
(4) No sum of two positive cubes is itself a cube.
(5) There are infinitely many prime numbers.
in terms of what seem to be more elementary notions.
(2) There does not exist a number b, such that b times b equals 2.
times a equals b times b times b plus c times c times c.
(6') There exists a number e such that 2 times e equals 6.
there exists a number b, such that
we did in the pq-system. Here is notation for natural numbers:
decipher complex formulas. But for now we'll be luxurious.
remarks go for the string SSSO.
simple sentences into the notation of TNT:
2 plus 3 equals 4: (SSO+SSSO)=SSSSO
2 plus 2 is not equal to 3 : ~(SSO+SSO)=SSSO
If 1 equals 0, then 0 equals 1 : <SO=0 JO=SO>
do-not have that property, such as this one
There exists a number b such that b plus 1 equals 2.
Clearly this is true. In the second instance, you get the sentence
For all numbers b, b plus 1 equals 2.
sentences are translated into TNT-notation as follows:
— 3b:(b-b)=SSO (closed; a sentence of TNT)
"is a sentence without a subject"
"improvised a six -part fugue on demand"
might not possess. One could as well stick on a "dummy
and-so" as its subject. For instance,
property which the number e might have.
the sides of the equation in the opposite order:
two existential quantifiers, one after the other, as follows:
3b:3c:SSSSSS SSSSSO=(((b • b) • b)+((c • c) • c))
((SO -SO) • SO))=(((b «b) -b)+((c • c) -c))
easiest way to do this is by negating the formula
3b:3c:SSSSSSSO=(((Sb • Sb) • Sb)+((Sc • Sc) -Sc))
b and c such that ...".) Thus we get:
-3b:3c:SSSSSSSO=(((Sb • Sb) • Sb)+((Sc -Sc) -Sc))
3b:3c:((a -a) -a)=(((Sb -Sb) • Sb)+((Sc -Sc) -Sc))
¥a:-3b:3c:((a -a) • a)=(((Sb • Sb) • Sb) +((Sc -Sc) -Sc))
An equally good translation would be this:
be written SSc. Now, our translation is extremely concise:
There exists a number which is greater than d and which is prime.
That's the translation of sentence 5 !
move leftwards again, and do the same.)
How to Distinguish True from False?
A numeral preceded by S is also a numeral.
Examples: 0 SO S50 SSSO SSSSO SSSSSO
by a prime is also a variable.
All numerals and variables are terms.
A term preceded by S is also a term.
If s and t are terms, then so are (s+ t) and (s • t).
Examples: 0 b SSa' (SO • (SSO+c)) S(Sa • (Sb • Sc))
TERMS may be divided into two categories:
(1) DEFINITE terms. These contain no variables.
(2) INDEFINITE terms. These contain variables.
rules tell how to make complete well-formed formulas.
If s and t are terms, then s = t is an atom.
four free variables in the last example.
A well-formed formula preceded by a tilde is well-formed.
free or quantified) does not change under negation.
The quantification status of a variable doesn't change here.
OPEN FORMULAS contain at least one free variable.
Examples: — c=c b=b <¥b:b=bn — c=c>
CLOSED FORMULAS (SENTENCES) contain no free variables.
the last one into an open formed formula.
All natural numbers are equal to 4.
There is no natural number which equals its own square.
Different natural numbers have different successors.
If 1 equals 0, then every number is odd.
happy" interpretations were for the pq-system's symbols.
The Five Axioms and First Rules of TNT
which can be derived in the same way as <Pv-P> was derived.
Before we give more rules, let us give the five axioms of TNT:
particular with its relation to addition.
supposed to represent. The five Peano postulates:
(2) Every djinn has a mesa (which is also a djinn).
numbers; all the rest is the work of man."
New Rules of TNT: Specification and Generalization
we wish. Here is such a rule:
wherever it occurs, by one the same term.
derived from Axiom 1 , by specification:
ways. Here is the exact statement of the rule:
free. Then ¥u:x is a theorem.
appeared free in the fantasy's premise.)
next two rules tell how to handle existential quantifiers.
For example, let us apply this rule to Axiom 1 :
good that they can be turned into each other with ease.
prime". The name of this rule is self-explanatory:
Let us apply the rule to —as usual— Axiom 1 :