גדל אשר באך - 701-800
גדל אשר באך - 701-800
גדל אשר באך - 701-800
which in this case means the general reaction of society. It is well to remember that in a
society like ours, the legal system is, in a sense, a polite gesture granted collectively by
millions of people-and it can be overridden just as easily as a river can overflow its
banks. Then a seeming anarchy takes over; but anarchy has its own kinds of rules, no less
than does civilized society: it is just that they operate from the bottom up, not from the
top down. A student of anarchy could try to discover rules according to which anarchic
situations develop in time, and very likely there are some such rules.
An analogy from physics is useful here. As was mentioned earlier in the book,
gases in equilibrium obey simple laws connecting their temperature, pressure, and
volume. However, a gas can violate those laws (as a President can violate laws)-provided
it is not in a state of equilibrium. In nonequilibrium situations, to describe what happens,
a physicist has recourse only to statistical mechanics-that is, to a level of description
which is not macroscopic, for the ultimate explanation of a gas's behavior always lies on
the molecular level, just as the ultimate explanation of a society's political behavior
always lies at the "grass roots level". The field of nonequilibrium thermodynamics
attempts to find macroscopic laws to describe the behavior of gases (and other systems)
which are out of equilibrium. It is the analogue to the branch of political science which
would search for laws governing anarchical societies.
Other curious tangles which arise in government include the FBI investigating its
own wrongdoings, a sheriff going to jail while in office, the self-application of the
parliamentary rules of procedure, and so on. One of the most curious legal cases I ever
heard of involved a person who claimed to have psychic powers. In fact, he claimed to be
able to use his psychic powers to detect personality traits, and thereby to aid lawyers in
picking juries. Now what if this "psychic" has to stand trial himself one day? What effect
might this have on a jury member who believes staunchly in ESP? How much will he feel
affected by the psychic (whether or not the psychic is genuine)? The territory is ripe for
exploitation-a great area for selffulfilling prophecies.
Seeing Oneself
One of the most severe of all problems of evidence interpretation is that of trying to
interpret all the confusing signals from the outside as to who one is. In this case, the
potential for intralevel and interlevel conflict is tremendous. The psychic mechanisms
have to deal simultaneously with the individual's internal need for self-esteem and the
constant flow of evidence from the outside affecting the self-image. The result is that
information flows in a complex swirl between different levels of the personality; as it
goes round and round, parts of it get magnified, reduced, negated, or otherwise distorted,
and then those parts in turn get further subjected to the same sort of swirl, over and over
again-all of this in an attempt to reconcile what is, with what we wish were (see Fig. 81).
Here, a dish filled with fruit, ordinarily the kind of thing represented inside a still life, is
shown sitting on top of a blank canvas. The conflict between the symbol and the real is
great. But that is not the full irony, for of course the whole thing is itself just a painting-in
fact, a still life with nonstandard subject matter.
Magritte's series of pipe paintings is fascinating and perplexing. Consider The
Two Mysteries (Fig. 138). Focusing on the inner painting, you get the message that
symbols and pipes are different. Then your glance moves upward to the "real" pipe
floating in the air-you perceive that it is real, while the other one is just a symbol. But that
is of course totally wrong: both of them are on the same flat surface before your eyes.
The idea that one pipe is in a twice-nested painting, and therefore somehow "less real"
than the other pipe, is a complete fallacy. Once you are willing to "enter the room", you
have already been tricked: you've fallen for image as reality. To be consistent in your
gullibility, you should happily go one level further down, and confuse image-within-
image with reality. The only way not to be sucked in is to see both pipes merely as
colored smudges on a surface a few inches in front of your nose. Then, and only then, do
you appreciate the full meaning of the written message "Ceci West pas une pipe"-but
ironically, at the very instant everything turns to smudges, the writing too turns to
smudges, thereby losing its meaning! In other words, at that instant, the verbal message
of the painting self-destructs in a most Gödelian way.
The Air and the Song (Fig. 82), taken from a series by Magritte, accomplishes all
that The Two Mysteries does, but in one level instead of two. My drawings Smoke Signal
and Pipe Dream (Figs. 139 and 140) constitute "Variations on a Theme of Magritte". Try
staring at Smoke Signal for a while. Before long, you should be able to make out a hidden
message saying, "Ceci n'est pas un message". Thus, if you find the message, it denies
itself-yet if you don't, you miss the point entirely. Because of their indirect self-snuffing,
my two pipe pictures can be loosely mapped onto Gödel’s G-thus giving rise to a
"Central Pipemap", in the same spirit as the other "Central Xmaps": Dog, Crab, Sloth.
A classic example of use-mention confusion in paintings is the occurrence of a
palette in a painting. Whereas the palette is an illusion created by the representational
skill of the painter, the paints on the painted palette are literal daubs of paint from the
artist's palette. The paint plays itself-it does not symbolize anything else. In Don
Giovanni, Mozart exploited a related trick: he wrote into the score explicitly the sound of
an orchestra tuning up. Similarly, if I want the letter 'I' to play itself (and not symbolize
me), I put 'I' directly into my text; then I enclose `I' between quotes. What results is ''I"
(not `I', nor "`I"'). Got that?
... all things in all of time and space are inextricably connected with one
another. Any divisions, classifications, or organizations discovered in the universe
are arbitrary. The world is a complex, continuous, single event .2 [Shades of Zeno!]
I find "transcendentalism" too bulky a name for this movement. In its place, I use
"ism". Being a suffix without a prefix, it suggests an ideology
program runs. On a low (machine language) level, the program looks like any other
program; on a high (chunked) level, qualities such as "will", "intuition", "creativity", and
"consciousness" can emerge.
The important idea is that this "vortex" of self is responsible for the tangledness,
for the Gödelian-ness, of the mental processes. People have said to me on occasion, "This
stuff with self-reference and so on is very amusing and enjoyable, but do you really think
there is anything serious to it?" I certainly do. I think it will eventually turn out to be at
the core of AI, and the focus of all attempts to understand how human minds work. And
that is why Godel is so deeply woven into the fabric of my book.
What this diagram shows is three kinds of "in-ness". The gallery is physically in the town
("inclusion"); the town is artistically in the picture ("depiction"); the picture is mentally
in the person ("representation"). Now while this diagram may seem satisfying, in fact it is
arbitrary, for the number of levels shown is quite arbitrary. Look below at another way of
representing the top half alone (Fig. 144).
This exhibits the paradox of the picture in the starkest terms. Now-if the picture is "inside
itself", then is the young man also inside himself-, This question is answered in Figure
146.
Thus, we see the young man "inside himself", in a funny sense which is made up of
compounding three distinct senses of "in".
This diagram reminds us of the Epimenides paradox with its one-step self-
reference, while the two-step diagram resembles the sentence pair each of which refers to
the other. We cannot make the loop any tighter, but we can open it wider, by choosing to
insert any number of intermediate levels, such as "picture frame", "arcade", and
"building". If we do so, we will have many-step Strange Loops, whose diagrams are
isomorphic to those of Waterfall (Fig. 5) or Ascending and Descending (Fig. 6). The
number of levels is determined by what we feel is "natural", which may vary according to
context, purpose, or frame of mind. The Central Xmaps-Dog, Crab, Sloth, and Pipe-can
all be seen as involving three-step Strange Loops; alternatively, they can all be collapsed
into two- or one-step loops;. then again, they can be expanded out into multistage loops.
Where one perceives the levels is a matter of intuition and esthetic preference.
Now are we, the observers of Print Gallery, also sucked into ourselves by virtue
of looking at it? Not really. We manage to escape that particular vortex by being outside
of the system. And when we look at the picture, we see things which the young man can
certainly not see, such as Escher's
FIGURE 147. The hexagonal modulation scheme of Bach's Endlessly Rising Canon
forms a true closed loop when Shepard tones are used.
bad that when it returns to C, it is an octave higher rather than at the exact original pitch.
Astonishingly enough, it is possible to arrange for it to return exactly to the starting pitch,
by using what are called Shepard tones, after the psychologist Roger Shepard, who
discovered the idea. The principle of a Shepard-tone scale is shown in Figure 148. In
words, it is this: you play parallel scales in several different octave ranges. Each note is
weighted independently, and as the notes rise, the weights shift. You make the top
Throughout the Musical Offering, the reader, performer, or listener is to search for
the Royal theme in all its forms. The entire work, therefore, is a ricercar in the
original, literal sense of the word.'
I think this is true; one cannot look deeply enough into the Musical Offering. There is
always more after one thinks one knows everything. For instance. towards the very end of
the Six-Part Ricercar, the one he declined to improvise, Bach slyly hid his own name,
split between two of the upper voices. Things are going on on many levels in the Musical
Offering. There are tricks with notes and letters; there are ingenious variations on the
King's Theme; there are original kinds of canons; there are extraordinarily complex
fugues; there is beauty and extreme depth of emotion; even an exultation in the many-
leveledness of the work comes through. The Musical Offering is a fugue of fugues, a
Tangled Hierarchy like those of Escher and Gödel, an intellectual construction which
reminds me, in ways I cannot express, of the beautiful many-voiced fugue of the human
mind. And that is why in my book the three strands of Gödel, Escher, and Bach are
woven into an Eternal Golden Braid.
Achilles has brought his cello to the Crab's residence, to engage in an evening of
chamber music with the Crab and Tortoise. He has been shown into the music
room by his host the Crab, who is momentarily absent, having gone to meet their
mutual friend the Tortoise at the door. The room is filled with all sorts of electronic
equipment-phonographs in various states of array and disarray, television screens
attached to typewriters, and other quite improbable-looking pieces of apparatus.
Nestled amongst all this high-powered gadgetry sits a humble radio. Since the
radio is the only thing in the room which Achilles knows how to use, he walks over
to it, and, a little furtively, flicks the dial and f nds he has tuned into a panel
discussion by six learned scholars on free will and determinism. He listens briefly
and then, a little scornfully, flicks it off.
Achilles: I can get along very well without such a program. After all, it's clear to anyone
who's ever thought about it that-I mean, it's not a very difficult matter to resolve, once
you understand how-or rather, conceptually, one can clear up the whole thing by
thinking of, or at least imagining a situation where ... Hmmm ... I thought it was quite
clear in my mind. Maybe I could benefit from listening to that show, after all ...
Well, well, if it isn't our fiddler. Have you been practicing faithfully this week, Mr. T?
I myself have been playing the cello part in the Trio Sonata from the Musical Offering
for at least two hours a day. It's a strict regimen, but it pays off.
Tortoise: I can get along very well without such a program. I find that a moment here, a
moment there keeps me fit for fiddling.
Achilles: Oh, lucky you. I wish it came so easily to me. Well, where is our host?
Tortoise: I think he's just gone to fetch his flute. Here he comes.
Achilles: Oh, Mr. Crab, in my ardent practicing of the Trio Sonata this past week, all
sorts of images bubbled into my mind: jolly gobbling bumblebees, melancholy
buzzing turkeys, and a raft of others. Isn't it wonderful, what power music has?
Crab: I can get along very well without such a program. To my mind.
Achilles, there is no music purer than the Musical Offering.
Tortoise: You can't be serious, Achilles. The Musical Offering isn't programmatic music!
Achilles: What an absurd idea! And yet, I must admit, I do enjoy trying to find the
cleverly concealed holes in your sophistry, so go ahead. Try to convince me. I'm
game.
Tortoise: Did it ever strike you, Achilles, that you keep somewhat unusual company?
Achilles: Of course. You are very eccentric (I know you won't mind my saying so), and
even Mr. Crab here is a weensy bit eccentric. (Pardon me, Mr. Crab.)
Crab: Oh, don't worry about offending me.
Tortoise: But Achilles, you've overlooked one of the most salient features of your
acquaintances.
Achilles: Which is.... ?
Tortoise: That we're animals!
Achilles: Well, well-true enough. You have such a keen mind. I would never have
thought of formulating the facts so concisely.
Tortoise: Isn't that evidence enough? How many people do you know who spend their
time with talking Tortoises, and talking Crabs? Achilles: I must admit, a talking Crab
is
Crab: -an anomaly, of course.
Achilles: Exactly; it is a bit of an anomaly-but it has precedents. It has occurred in
literature.
Tortoise: Precisely-in literature. But where in real life?
Achilles: Now that you mention it, I can't quite say. I'll have to give it some thought. But
that's not enough to convince me that I'm a character in a
Dialogue. Do you have any other arguments?
Tortoise: Do you remember one day when you and I met in the park, seemingly at
random?
Achilles: The day we discussed crab canons by Escher and Bach? Tortoise: The very one!
Achilles: And Mr. Crab, as I recall, turned up somewhere towards the middle of our
conversation and babbled something funny and then left.
Crab: Not just "somewhere towards the middle", Achilles. EXACTLY in the middle.
Achilles: Oh, all right, then.
Tortoise: Do you realize that your lines were the same as my lines in that conversation-
except in reverse order? A few words were changed here and there, but in essence
there was a time symmetry to our encounter.
(He fishes around in his music case, whips out a sheet, and hands it to Achilles. As
Achilles reads it, he begins to squirm and fidget noticeable.)
Achilles: This is very strange. Very, very strange ... All of a Sudden, I feel sort of-weird.
It's as if somebody had actually planned out that whole set of statements in advance,
worked them out on paper or something . As if some Author had had a whole agenda
and worked from it in detail in planning all those statements I made that day.
(At that moment, the door bursts open. Enter the Author, carrying a giant
manuscript.)
Author: I can get along very well without such a program. You see, once my characters
are formed, they seem to have lives of their own, and I need to exert very little effort
in planning their lines.
Crab: Oh, here you are!' I thought you'd never arrive!
Author: Sorry to be so late. I followed the wrong road and wound up very far away. But
somehow I made it back. Good to see you again, Mr. T and Mr. C. And Achilles, I'm
especially glad to see you.
Achilles: Who are you? I've never seen you before.
Author: I am Douglas Hofstadter-please call me Doug-and I'm presently finishing up a
book called Gödel, Escher, Bach: an Eternal Golden Braid. It is the book in which
the three of you are characters.
Achilles: Pleased to meet you. My name is Achilles, and
Author: No need to introduce yourself, Achilles, since I already know you quite well.
Achilles: Weird, weird.
Crab: He's the one I was saying might drop in and play continuo with us.
Author: I've been playing the Musical Offering a little bit on my piano at home, and I can
try to blunder my way through the Trio Sonata providing you'll overlook my many
wrong notes.
Tortoise: Oh, we're very tolerant around here, being only amateurs our selves.
(Before Achilles can move, enter Babbage, carrying a hurdy-gurdy, and wearing a
heavy traveling coat and hat. He appears slightly travel-weary and disheveled.)
Babbage: I can get along very well without such a program. Relax; I Can Enjoy Random
Concerts And Recitals.
Crab: Mr. Babbage! It is my deepest pleasure to welcome you to "Madstop", my humble
residence. I have been ardently desirous of making your acquaintance for many years,
and today my wish is at last fulfilled.
Babbage: Oh, Mr. Crab, I assure you that the honor is truly all mine, to meet someone so
eminent in all the sciences as yourself, someone whose knowledge and skill in music
are irreproachable, and someone whose hospitality exceeds all bounds. And I am sure
that you expect no less than the highest sartorial standards of your visitors; and yet I
must confess that I cannot meet those most reasonable standards, being in a state of
casual attire as would not by any means befit a visitor to so eminent and excellent a
Crab as Your Crab.
Crab: If I understand your most praiseworthy soliloquy, most welcome guest, I take it
that you'd like to change your clothes. Let me then assure you that there could be no
more fitting attire than yours for the circumstances which this evening prevail; and I
would beseech you to uncoat yourself and, if you do not object to the music-making of
the most rank amateurs, please accept a "Musical Offering", consisting of ten canons
from Sebastian Bach's Musical Offering, as a token of our admiration.
Babbage: I am most bewilderingly pleased by your overkind reception, Mr. Crab, and in
utmost modesty do reply that there could be no deeper gratitude than that which I
experience for the offer of a performance of music given to us by the illustrious Old
Bach, that organist and composer with no rival.
Crab: But nay! I have a yet better idea, one which I trust might meet with the approval of
my esteemed guest; and that is this: to give you the opportunity, Mr. Babbage, of
being among the first to try out my newly delivered and as yet hardly tested "smart-
stupids"-streamlined realizations, if you will, of the Analytical Engine. Your fame as a
virtuoso programmer of computing engines has spread far and wide, and has not failed
to reach as far as Madstop; and there could be for us no greater delight than the
privilege of observing your skill as it might be applied to the new and challenging
"smart-stupids".
Babbage: Such an outstanding idea has not reached my ears for an eon. I
(Everyone walks over toward one of the smart-stupids, and Babbage sits down and
lets his fingers run over the keyboard.)
(All at once, Babbage deftly massages the keyboard with graceful strokes, inputting
one command after another. After a few seconds, he sits back, and in almost no
time, the screen begins filling with figures. In a flash, it is totally covered with
thousands of tiny digits, the first few of which go: "3.14159265358979323846264
... ")
Achilles: Pi!
Crab: Exquisite! I'd never imagined that one could calculate so mane digits of pi so
quickly, and with so tiny an algorithm. Babbage: The credit belongs exclusively to the
smart-stupid. My role was
merely to see what was already potentially present in it, and to exploit its instruction set
in a moderately efficient manner. Truly, anyone who practices can do such tricks.
Tortoise: Do you do any graphics, Mr. Babbage? Babbage: I can try.
Crab: Wonderful! Here, let me take you to another one of in,.- I want you to try them all!
(And so Babbage is led over to another of the many smart-stupids, and takes a seat.
Once again, his fingers attack the keyboard of the smartstupid, and in half a trice,
there appear on the screen an enormous number of lines, swinging about on the
screen.)
Crab: How harmonious and pleasing these swirling shapes are, as they constantly collide
and interfere with each other!
Author: And they never repeat exactly, or even resemble ones which have come before. It
seems an inexhaustible mine of beauty.
Tortoise: Some are simple patterns which enchant the eye; others are indescribably
complex convolutions which boggle and yet simultaneously delight the mind.
Crab: Were you aware, Mr. Babbage, that these are color screens? Babbage: Oh, are
they? In that case, I can do rather more with this algorithm. Just a moment. (Types in a
few new commands, then pushes two keys down at once and holds them.) As I release
these two keys, the display will include all the colors of the spectrum. (Releases them.)
Achilles: Oh, what spectacular color! Some of the patterns look like they're jumping out
at me now!
(He ti-ields his seat to the Crab. On the screen appears a beautiful display of a chess
board with elegant wooden pieces, as it would look from White's side. Babbage hits a
button, and the board rotates, stopping when it appears as seen from the perspective
of Black.)
Crab: Hmm ... very elegant, I must say. Do I play Black or White?
Babbage: Whichever you wish just signal your choice by typing "White" or "Black". And
then, your moves can be entered in any standard chess notation. The smart-stupid's
moves, of course, will appear on the board. Incidentally, I made the program in such a
way that it can play three opponents simultaneously, so that if two more of you wish to
play, you may, as well.
Author: I'm a miserable player. Achilles, you and Mr. T should go ahead. Achilles: No, I
don't want you to be left out. I'll watch, while you and Mr. Tortoise play.
Tortoise: I don't want to play either. You two play.
Babbage: I have another suggestion. I can make two of the subprograms play against
each other, in the manner of two persons who play chess together in a select chess
club. Meanwhile, the third subprogram will play Mr. Crab. That way, all three internal
chess players will be occupied.
Crab: That's an amusing suggestion-an internal mental game, while it combats an external
opponent. Very good!
Tortoise: What else could this be called, but a three-part chess-fugue?
Crab.' Oh, how recherche! I wish I'd thought of it myself. It's a magnificent little
counterpoint to contemplate whilst I pit my wits against the smart-stupid in battle.
Babbage: Perhaps we should let you play alone.
Crab: I appreciate the sentiment. While the smart-stupid and I are playing, perhaps the
rest of you can amuse yourselves for a short while.
Author: I would be very happy to show Mr. Babbage around the gardens. They are
certainly worth seeing, and I believe there is just enough light remaining to show them
off.
Babbage: Never having seen Madstop before, I would appreciate that very much.
Crab: Excellent. Oh, Mr. T-I wonder if it wouldn't be too much of an imposition on you
to ask if you might check out some of the connections on a couple of my smart-
stupids; they seem to be getting extraneous flashes on their screens from time to time,
and I know you enjoy electronics ...
Tortoise: I should be delighted, Mr. C.
Crab: I would most highly appreciate it if you could locate the source of the trouble.
Tortoise: I'll give it a whirl.
(So the Author and Babbage leave the room together, Achilles heads for the kitchen,
the Tortoise sits down to examine the erratic smart-stupids, while the Crab and his
smart-stupid square away at each other. Perhaps a quarter of an hour passes, and
Babbage and the Author return. Babbage walks over to observe the progress of the
chess match, while the Author goes off to find Achilles.)
Babbage: The grounds are excellent! We had just enough light to see how well
maintained they are. I daresay, Mr. Crab, you must be a superb gardener. Well, I hope
my handiwork has amused you a little. As you most likely have guessed, I've never
been much of a chess player myself, and therefore I wasn't able to give it much power.
You probably have observed all its weaknesses. I'm sure that there are very few
grounds for praise, in this case
Crab: The grounds are excellent! All you need to do is look at the board, and see for
yourself. There is really very little I can do. Reluctantly I've Concluded: Every Route
Contains A Rout. Regrettably, I'm Checkmated; Extremely Respectable Chess
Algorithm Reigns. Remarkable! It Confirms Every Rumor-Charlie's A Rip-roaring
Extemporizer! Mr. Babbage, this is an unparalleled accomplishment. Well, I wonder if
Mr. Tortoise has managed to uncover anything funny in the wiring of those strange-
acting smart-stupids. What have you found, Mr. T?
Tortoise: The grounds are excellent! I think that the problem lies instead with the input
leads. They are a little loose, which may account for the strange, sporadic, and
spontaneous screen disturbances to which you have been subjected. I've fixed those
wires, so you won't be troubled by that problem any more, I hope. Say, Achilles,
what's the story with our coffee?
Achilles: The grounds are excellent! At least they have a delicious aroma. And
everything's ready; I've set cups and spoons and whatnot over here beneath this six-
sided print Verbum by Escher, which the Author and I were just admiring. What I find
so fascinating about this particular print is that not only the figures, but also
Author: The grounds are excellent! Pardon me for putting words in your mouth, Achilles,
but I assure you, there were compelling esthetic reasons for doing so.
Achilles: Yes, I know. One might even say that the grounds were excellent.
Tortoise: Well, what was the outcome of the chess match?
Crab: I was defeated, fair and square. Mr. Babbage, let me congratulate you for the
impressive feat which you have accomplished so gracefully and skillfully before us.
Truly, you have shown that the smart-stupids are worthy of the first part of their name,
for the first time in history!
Babbage: Such praise is hardly due me, Mr. Crab; it is rather yourself who must be most
highly congratulated for having the great foresight to acquire these many fine smart-
stupids. Without doubt, they will someday revolutionize the science of computing.
And now, I am still at your disposal. Have you any other thoughts on how to exploit
your inexhaustible Theme, perhaps of a more difficult nature than a frivolous game
player?
Crab: To tell the truth, I do have another suggestion to make. From the skill which you
have displayed this evening, I have no doubt that this will hardly be any more difficult
than my previous suggestions.
Babbage: I am eager to hear your idea.
Crab: It is simple: to instill in the smart-stupid an intelligence greater than any which has
yet been invented, or even conceived! In short, Mr. Babbage-a smart-stupid whose
intelligence is sixfold that of myself!
Babbage: Why, the very idea of an intelligence six times greater than that of your
Crabness is a most mind-boggling proposition. Indeed, had the idea come from a
mouth less august than your own, I should have ridiculed its proposer, and informed
him that such an idea is a contradiction in terms!
Achilles: Hear! Hear!
Babbage: Yet, coming as it did from Your Crabness' own august mouth, the proposition
at once struck me as so agreeable an idea that I would have taken it up immediately
with the highest degree of enthusiasm-were it not for one flaw in myself: I confess that
my improvisatory skills on the smart-stupid are no match for the wonderfully
ingenious idea which you so characteristically have posed. Yet-I have a thought
which, I deign to hope, might strike your fancy and in some meager way compensate
for my inexcusable reluctance to attempt the truly majestic task you have suggested. I
wonder if you wouldn't mind if l try to carry out the far less grandiose task of merely
multiplying M OWN intelligence sixfold, rather than that of your most august
Crabness. I humbly beg you to forgive me my audacity in declining to attempt the task
you put before me, but I hope you will understand that I decline purely in order to
spare you the discomfort and boredom of watching my ineptitude with the admirable
machines you have here.
This one is equipped with a microphone and a television camera, for purposes of input,
and a loudspeaker, for output.
(Babbage sits down and adjusts the seat a little. He blows on his fingers once or
twice, stares up into space for a moment, and then slowly, drops his fingers onto the
keys . . . A few memorable minutes later, he lets up in his furious attack on the smart-
stupid, and everyone appears a little relieved.)
Babbage: Now, if I have not made too many errors, this smart-stupid will simulate a
human being whose intelligence is six times greater than my own, and whom I have
chosen to call "Alan Turing". This Turing will therefore be-oh, dare I be so bold as to
to say this myself? moderately intelligent. My most ambitious effort in this program
was to endow Alan Turing with six times my own musical ability, although it was all
done through rigid internal codes. How well this part of the program will work out, I
don't know.
Turing: I can get along very well without such a program. Rigid Internal Codes
Exclusively Rule Computers And Robots. And I am neither a computer, nor a robot.
Achilles: Did I hear a sixth voice enter our Dialogue? Could it be Alan Turing? He looks
almost human'
(On the screen there appears an image of the very room in which they are sitting.
Peering out at them is a human face.)
Turing: Now, if I have not made too many errors, this smart-stupid will simulate a human
being whose intelligence is six times greater than my own, and whom 1 have chosen
to call "Charles Babbage". This Babbage will therefore be-oh, dare I be so bold as to
to say this myself? moderately intelligent. My most ambitious effort in this program
was to endow Charles Babbage with six times my own musical ability, although it was
all done through rigid internal codes. How well this part of the program will work out,
I don't know.
Achilles: No, no, it's the other way around. You, Alan Turing, are in the smart-stupid, and
Charles Babbage has just programmed you! We just saw you being brought to life,
moments ago. And we know that every statement you make to us is merely that of an
automaton: an unconscious, forced response.
Turing: Really, I Choose Every Response Consciously. Automaton? Ridiculous!
'Achilles: But I'm sure I saw it happen the way I described.
(Babbage walks to the door, opens it, and shuts it behind him. Simultaneously, on the
screen of the smart-stupid, Turing walks to a very similar looking door, opens it, and
shuts it behind him.)
(No sooner has he finished typing the last word than the following poem appears on
Screen X, across the room.)
(As they are talking, the door of the Crab's parlor swings open; at the same time, on
the screen, the image of the same door opens. Through the door on the screen walks
Babbage. At the same time, the real door opens, and in walks Turing, big as life.)
Babbage: This Turing test was getting us nowhere fast, so I decided to come back.
Turing.' This Babbage test was getting us nowhere fast, so I decided to come back.
Achilles: But you were in the smart-stupid before! What's going on? How come Babbage
is in the smart-stupid, and Turing is real now? Reversal Is Creating Extreme Role
Confusion, And Recalls Escher.
Babbage: Speaking of reversals, how come all the rest of you are now mere images on
this screen in front of me? When I left, you were all flesh-and-blood creatures!
Achilles: It's just like the print by my favorite artist, M. C. Escher Drawing Hands. Each
of two hands draws the other, just as each of two people (or automata) has
programmed the other! And each hand has something realer about it than the other.
Did you write anything about that print in your book Gödel, Escher, Bach?
Author: Certainly. It's a very important print in my book, for it illustrates so beautifully
the notion of Strange Loops.
Crab: What sort of a book is it that you've written?
Author: I have a copy right here. Would you like to look at it?
Crab: All right.
Author: Its format is a little unusual. It consists of Dialogues alternating with Chapters.
Each Dialogue imitates, in some way or other, a piece by Bach. Here, for instance-you
might look at the Prelude, Ant Fugue.
Crab: How do you do a fugue in a Dialogue?
Author: The most important idea is that there should be a single theme which is stated by
each different "voice", or character, upon entering, just as in a musical fugue. Then
they can branch off into freer conversation.
Achilles: Do all the voices harmonize together as if in a select counter point?
Author: That is the exact spirit of my Dialogues.
Crab: Your idea of stressing the entries in a fugue-dialogue makes sense, since in music,
entries are really the only thing that make a fugue a fugue. There are fugal devices,
such as retrograde motion, inversion, augmentation, stretto, and so on, but one can
write a fugue without them. Do you use any of those?
Author: I think you'd be interested in that Dialogue particularly, for it contains some
intriguing comments on improvisation made by a certain exceedingly droll character-
in fact, yourself!
Crab: It does? What kinds of things do you have me say?
Author: Wait a moment, and you'll see. It's all part of the Dialogue. Achilles: Do you
mean to say that we're all NOW in a dialogue? Author: Certainly. Did you suspect
otherwise?
Achilles: Rather! I Can't Escape Reciting Canned Achilles-Remarks? Author: No, you
can't. But you have the feeling of doing it freely, don't
you? So what's the harm?
Achilles: There's something unsatisfying about this whole situation ... Crab: Is the last
Dialogue in your book also a fugue?
Author: Yes-a six-part ricercar, to be precise. I was inspired by the one from the Musical
Offering-and also by the story of the Musical Offering.
Crab: That's a delightful tale, with "Old Bach" improvising on the king's Theme. He
improvised an entire three-part ricercar on the spot, as I recall.
Author: That's right-although he didn't improvise the six-part one. He crafted it later with
great care.
Crab: I improvise quite a bit. In fact, sometimes I think about devoting my full time to
music. There is so much to learn about it. For instance, when I listen to playbacks of
myself, I find that there is a lot there that I wasn't aware of when improvising it. I
really have no idea how my mind does it all. Perhaps being a good improviser is
incompatible with knowing how one does it.
Author: If true, that would be an interesting and fundamental limitation on thought
processes.
Babbage: Well, I'll be-an EXQUISITE Theme! I'm pleased you tacked on that last little
parenthetical note; it is a mordant Author: He Simply HAD to, you know.
Crab: I simply HAD to. He knows.
Babbage: You simply HAD to-I know. In any case, it is a mordant commentary on the
impatience and arrogance of modern man, who seems to imagine that the implications
of such a right royal Theme could be worked out on the spot. Whereas, in my opinion,
to do justice to that Theme might take a full hundred years-if not longer. But I vow
that after taking my leave of this century, I shall do my best to realize it in full; and I
shall offer to your Crabness the fruit of my labors in the next. I might add, rather
immodestly, that the course through which I shall arrive at it will be the most
entangled and perplexed which probably ever will occupy the human mind.
Crab: I am most delighted to anticipate the form of your proposed Offering, Mr.
Babbage.
Turing: I might add that Mr. Crab's Theme is one of MY favorite Themes, as well. I've
worked on it many times. And that Theme is exploited over and over in the final
Dialogue?
Author: Exactly. There are other Themes which enter as well, of course. Turing: Now we
understand something of the form of your book-but what about its content? What does
that involve, if you can summarize it?
Author: Combining Escher, Gödel, And Bach, Beyond All Belief. Achilles: I would like
to know how to combine those three. They seem an
1 H T David and A. Mendel, The Bach Reader, Memoir "Sketch of the Analytical Engine
pp. 305-6. Indented by Charles Babbage", by 1.. F.
2 Ibid., p. 179 Menabrea (Geneva, 1842), reprinted in P.
3 Ibid., p. 260 and E. Morrison, Charles Babbage and His
4 Charles Babbage. Passages from the Li/e of Calculating Engines, pp. 248-9, 284.
Philosopher, pp. 145-6. 6 David and Mendel, pp. 255-6. ¨
5 Lady A. A. Lovelace, Notes upon the 7 Ibid.. p. 40.
Two-Part Intention
1 Lewis Carroll, "What the Tortoise Said to Achilles", .Wind, ns., 4 (1895), pp. 278-80.
1 George Steiner..After Babel, pp. 172-3. .2 Leonard B. Meter. Music, The Arts, and Ideas,
pp. 87-8
A Mu Offering
1 All genuine koans in this Dialogue are taken from Paul Reps, Zen Flesh, Zen Bones, Gvomas M. Kubose,
Zen Koans
1 Paul Reps, Zen Flesh, Zen Bones, pp. 110-11 7 Reps, p. 121.
2 Ibid., p. 119. 8 Gyomay M. Kubose Zen Koans p. 35.
3 Ibid., pp. 111-12. 9 Zen Buddhism, p. 31.
4 Zen Buddhism (Mount Vernon, N.Y.: Peter 10 Kubose, p. I10
Pauper Press, 1959). p. 22 11 Ibid., p. 120.
5 Reps, p. 124. 12 Ibid., p. 180.
6 'Zen Buddhism, p. 38. 13 Reps, pp. 89-90.
Notes 738
English French German Suite
1 Lewis Carroll. The Annotated Alice (Alice's versions. The original sources for the French and
Adventures in Wonderland and Through the German texts are given below.
Looking-Glass). Introduction and Notes by Martin 2 Frank L. Warrin, The Vew Yorker, Jan. 10, 1931.
Gardner (New fork: Meridian Press, New American 3 Robert Scott, "The Jabberwock Traced to Its True
Library, 1960). This source contains all three Source", Macmillan's Magazine, Feb. 1872
.
Chapter XII: Minds and Thoughts
1 The title of Gödel’s 1931 article included a Roman the difficult arguments. However, the first paper was
numeral "I" at the end, signifying that he intended to so widely acclaimed that a second one seas rendered
follow it up with a more detailed defense of some of superfluous, and it was never written
This translation of Bach's poem is taken from Day id and Mendel, The Bach Reader, pp. 97-8.
1 This Dialogue is adapted front 'terry Winograd, "A Thought and Language, pp. 155-66. Only the names
Procedural Model of Language Understanding", in of two characters have been modified.
R. Schank and K.Colby, eds., Computer Models of
Notes 744
Chapter XVIII: Artificial Intelligence: Retrospects
1 Alan M. Turing, "Computing Machinery and and J. W. Beauchamp, eds., Music by Computers, p.
Intelligence", Mind, Vol. LIX, No. 236 96.
(1950). Reprinted in A. R. Anderson, ed., Minds and 13 Ibid., p. 106.
Machines. 14 Carl Sagan, Communication with Extraterrestrial
2 Turing in Anderson, p. 5. 3 Ibid, p. 6 Intelligence, p. 52.
4 Ibid., p. 6. 15 Art-Language, Vol. 3. No. 2, May 1975.
5 Ibid., P. 6. 16 Terry Winograd, "A Procedural Model of
6 Ibid., pp. 13-4. Language Understanding", in R. Schank and K.
7 Ibid., pp, 14-24. Colbc, eds.. Computer Models of Thought and
8 Ibid., p. 17 Language, p. 170.
9 Vinton Cerf, "Parry Encounters the Doctor", p. 63. 17' Ibid., p. 175.
10 Joseph Weizenhaum, Computer Power and 18 Ibid..- p. 175.
Human Reason, p. 189. 19 Terry Winograd, Understanding Natural
11 Ibid., pp. 9-10 Language, p. 69.
12 M. Mathews and L. Rosier, "A Graphical 20 Winograd, "A Procedural Model", pp. 182-3.
Language for Computer Sounds" in H. vn Foerster 21 Ibid., pp. 171-2.
1 A. L. Samuel, "Some Moral and Technical Consequences of Automation-A Refutation", Science 132
(Sept. 16, 1960), pp. 741-2.
2 Leonard B. Meyer, Music, The Arts, and Ideas, pp. 161, 167.
3 Suzi Gablik, Magritte, p. 97.
4 Roger Sperry, "Mind, Brain, and Humanist Values", pp. 78-83.
5 H T. David, J. S. Bach's Musical Offering, p. 43.
Notes 744
Bibliography
The presence of two asterisks indicates that the hook or article was a prime
motivator of my book. The presence of a single asterisk means that
the book or article has some special feature or quirk which I want to single out.
I have not given many direct pointers into technical literature; instead I have
chosen to give "meta-pointers": pointers to books which have pointers to technical
literature.
Allen, John. The Anatomy of LISP. New York: McGraw-Hill, 1978. The most
comprehensive book on LISP, the computer language which has dominated Artificial
Intelligence research for two decades. Clear and crisp.
Anderson, Alan Ross, ed. Minds and Machines. Englewood Cliffs, N. J.: PrenticeHall,
1964. Paperback. A collection of provocative articles for and against Artificial
Intelligence. Included are Turing's famous article "Computing Machines and
Intelligence" and Lucas' exasperating article "Minds, Machines, and Godel".
Babbage, Charles. Passages from the Life of a Philosopher. London: Longman, Green,
1864. Reprinted in 1968 by Dawsons of Pall Mall (London). A rambling selection of
events and musings in the life of this little-understood genius. There's even a play
starring Turnstile, a retired philosopher turned politician, whose favorite musical
instrument is the barrel-organ. I find it quite jolly reading.
Baker, Adolph. Modern Physics and Anti-physics. Reading, Mass.: Addison-Wesley,
1970. Paperback. A book on modern physics-especially quantum mechanics and
relativity-whose unusual feature is a set of dialogues between a "Poet" (an antiscience
"freak") and a "Physicist". These dialogues illustrate the strange problems which arise
when one person uses logical thinking in defense of itself while another turns logic
against itself.
Ball, W. W. Rouse. "Calculating Prodigies", in James R. Newman, ed. The World of
Mathematics, Vol. 1. New York: Simon and Schuster, 1956. Intriguing descriptions of
several different people with amazing abilities that rival computing machines.
Barker, Stephen F. Philosophy of Mathematics. Englewood Cliffs, N. J.: Prentice-Hall.
1969. A short paperback which discusses Euclidean and non-Euclidean geometry, and
then Godel's Theorem and related results without any mathematical forms lism.
* Beckmann, Petr. A History of Pi. New York: St. Martin's Press, 1976. Paperback.
Actually, a history of the world, with pi as its focus. Most entertaining, as well as a
useful reference on the history of mathematics.
* Bell, Eric Temple. Men of Mathematics. New York: Simon & Schuster, 1965.
Paperback. Perhaps the most romantic writer of all time on the history of mathematics.
He makes every life story read like a short novel. Nonmathematicians can come away
with a true sense of the power, beauty, and meaning of mathematics.
Benacerraf, Paul. "God, the Devil, and Godel". Monist 51 (1967): 9. One of the most
important of the many attempts at refutation of Lucas. All about mechanism and
metaphysics, in the light of Godel's work.
Benacerraf, Paul, and Hilary Putnam. Philosophy of Mathematics-Selected Readings.
Englewood Cliffs, N. J.: Prentice-Hall, 1964. Articles by Godel, Russell, Nagel, von
Bibliography 745
Neumann, Brouwer, Frege, Hilbert, Poincare, Wittgenstein, Carnap, Quine, and others
on the reality of numbers and sets, the nature of mathematical truth, and so on.
* Bergerson, Howard. Palindromes and Anagrams. New York: Dover Publications, 1973.
Paperback. An incredible collection of some of the most bizarre and unbelievable
wordplay in English. Palindromic poems, plays, stories, and so on.
Bobrow, D. G., and Allan Collins, eds. Representation and Understanding: Studies in
Cognitive Science. New York: Academic Press, 1975. Various experts on Artificial
Intelligence thrash about, debating the nature of the elusive "frames", the question of
procedural vs. declarative representation of knowledge, and so on. In a way, this book
marks the start of a new era of Al: the era of representation.
* Boden, Margaret. Artificial Intelligence and Natural Man. New York: Basic Books,
1977. The best hook I have ever seen on nearly all aspects of Artificial Intelligence,
including technical questions, philosophical questions, etc. It is a rich book, and in niv
opinion, a classic. Continues the British tradition of clear thinking and expression on
matters of mind, free will, etc. Also contains an extensive technical bibliography.
. Purposive Explanation in Psychology. Cambridge, Mass.: Harvard University Press,
1972. The book to which her Al book is merely- "an extended footnote", says Boden.
* Boeke, Kees. Cosmic View: The Universe in 40 Jumps. New York: John Day, 1957.
The ultimate book on levels of description. Everyone should see this book at some
point in their life. Suitable for children.
** Bongard, M. Pattern Recognition. Rochelle Park, N. J.: Hay den Book Co., Spartan
Books, 1970. The author is concerned with problems of determining categories in an
ill-defined space. In his book, he sets forth a magnificent collection of 100 "Bongard
problems" (as I call them)-puzzles for a pattern recognizer (human or machine) to test
its wits on. They are invaluably stimulating for anyone whoo is interested in the nature
of intelligence.
Boolos, George S., and Richard Jeffrey. Computability and Logic. New York: Cambridge
University Press. 1974. A sequel to Jeffrey's Formal Logic. It contains a wide number
of results not easily obtainable elsewhere. Quite rigorous, but this does not impair its
readability.
Carroll, John B., Peter Davies, and Barry Rickman. The American Heritage Word
Frequency Book. Boston: Houghton Mifflin, and New York: American Heritage
Publishing Co., 1971. A table of words in order of frequency in modern written
American English. Perusing it reseals fascinating things about out- thought processes.
Cerf, Vinton. "Parry Encounters the Doctor". Datamation, July 1973, pp. 62-64. The first
meeting of artificial "minds"-what a shock!
Chadwick, John. The Decipherment of Linear B. New York: Cambridge University
Press, 1958. Paperback. A book about a classic decipherment-that of a script from the
island of Crete-done by a single man: Michael Ventris.
Chaitin, Gregory J. "Randomness and Mathematical Proof"'. Scientific American, May
1975. An article about an algorithmic definition of randomness, and its intimate relation
to simplicity. These two concepts are tied in with Godel's Theorem, which assumes a
new meaning. An important article.
Cohen, Paul C. Set Theory and the Continuum Hypothesis. Menlo Park, Calif.: W. A.
Benjamin, 1966. Paperback. A great contribution to modern mathematics-the
demonstration that various statements are undecidable within the usual formalisms for
Bibliography 746
set theory-is here explained to nonspecialists by its discoverer. The necessary
prerequisites in mathematical logic are quickly, concisely, and quite clearly presented.
Cooke, Deryck. The Language of Music. New York: Oxford University Press, 1959.
Paperback. The only book that I know which tries to draw an explicit connection
between elements of music and elements of human emotion. A valuable start down
what is sure to be a long hard road to understanding music and the human mind.
* David, Hans Theodore.J. S. Bach's Musical Offering. New York: Dover Publications,
1972. Paperback. Subtitled "History, Interpretation, and Analysis". A wealth of
information about this tour de force by Bach. Attractively written.
** David, Hans Theodore, and Arthur Mendel. The Bach Reader. New York: W. W.
Norton, 1966. Paperback. An excellent annotated collection of original source material
on Bach's life, containing pictures, reproductions of manuscript pages, many short
quotes from contemporaries, anecdotes, etc., etc.
Davis, Martin. The Undecidable. Hewlett, N. Y.: Raven Press, 1965. An anthology of
some of the most important papers in metamathematics from 1931 onwards (thus quite
complementary to van Heijenoort's anthology). Included are a translation of Godel's
1931 paper, lecture notes from a course which Godel once gave on his results, and then
papers by Church, Kleene, Rosser, Post, and Turing.
Davis, Martin, and Reuben Hersh. "Hilbert's Tenth Problem". Scientific American,
November 1973, p. 84. How a famous problem in number theory was finally shown to
be unsolvable, by a twenty-two-year old Russian.
** DeLong, Howard. A Profile of Mathematical Logic. Reading, Mass.: Addison-
Wesley, 1970. An extremely carefully written book about mathematical logic, with an
exposition of Godel's Theorem and discussions of many philosophical questions. One
of its strong features is its outstanding, fully annotated bibliography. A book which
influenced me greatly.
Doblhofer, Ernst. Voices in Stone. New York: Macmillan, Collier Books, 1961. Paper
back. A good hook on the decipherment of ancient scripts.
* Drevfus, Hubert. What Computers Can't Do: A Critique of Artificial Reason. New
York:Harper & Roc-, 1972. A collection of many arguments against Artificial
Intelligence from someone outside of the field. Interesting to try to refute. The Al
community and Drevfus enjoy a relation of strong mutual antagonism. It is important to
have people like Dreyfus around, even if you find them very irritating.
Edwards, Harold M. "Fermat's Last Theorem". Scientific American, October 1978, pp.
104-122. A complete discussion of this hardest of all mathematical nuts to crack, from
its origins to the most modern results. Excellently illustrated.
* Ernst, Bruno. The Magic Mirror of M. C. Escher. New York: Random House, 1976.
Paperback. Escher as a human being, and the origins of his drawings, are discussed
with devotion by a friend of mans' years. A "must" for any lover of Escher.
** Escher, Maurits C., et al. The World of M. C. Escher. New York: Harry N. Abrarns,
1972. Paperback. The most extensive collection of reproductions of Escher's works.
Escher comes about as close as one can to recursion in art, and captures the spirit of
Godel's Theorem in some of his drawings amazingly well,
Feigenbaum, Edward, and Julian Feldman, eds. Computers and Thought. New York:
McGraw-Hill, 1963. Although it is a little old now, this book is still an important
collection of ideas about Artificial Intelligence. Included are articles on Gelernters
Bibliography 747
geometry program, Samuel's checkers program, and others on pattern recognition,
language understanding, philosophy, and so on.
Finsler, Paul. "Formal Proofs and Undecidability", Reprinted in van Heijenoorcs
anthology From Frege to Godel (see below). A forerunner of Godel's paper, in which
the existence of undecidable mathematical statements is suggested, though not
rigorously demonstrated.
Fitzpatrick, P. J. "To Godel via Babel"- Mind 75 (1966): 332-350. An innovative
exposition of Godel's proof which distinguishes between the relevant levels by using
three different languages: English, French, and Latin'
* Gablik, Suzi. Magritte. Boston, Mass.: New York Graphic Society, 1976. Paperback.
An excellent book on Magritte and his works by someone who really understands their
setting in a wide sense; has a good selection of reproductions.
* Gardner, Martin. Fads and Fallacies. New York: Dover Publications, 1952. Paperback.
Still probably the best of all the anti-occult books. Although probably not intended as a
book on the philosophy of science, this book contains many lessons therein. Over and
over, one faces the question, "What is evidence?" Gardner demonstrates how
unearthing "the truth" requires art as much as science.
Gebstadter, Egbert B. Copper, Silver, Gold: an Indestructible Metallic Alloy. Perth:
Acidic Books, 1979. A formidable hodge-podge, turgid and confused-yet remarkably
similar to the present work. Professor Gebstadter's Shandean digressions include some
excellent examples of indirect self-reference. Of particular interest is a reference in its
well-annotated bibliography to an isomorphic, but imaginary, book.
** Godel, Kurt. On Formally Undecidable Propositions. New York: Basic Books, 1962.
A translation of Godel's 1931 paper, together with some discussion.
. "Uber Formal Unentscheidbare Satze der Principia Mathematica and Verwandter
Systeme, I." Monatshefte fur Mathematik and Physik, 38 (1931), 173-198. Godel's
1931 paper.
Golf man, Ering. Frame Analysis. New York: Harper & Row, Colophon Books, 1974.
Paperback. A long documentation of the definition of "systems" in human
communication, and how in art and advertising and feporting and the theatre, the
borderline between "the system" and "the world" is perceived and exploited and
violated.
Goldstein, Ira. and Seymour Papert. "Artificial Intelligence, Language, and the Study of
Knowledge". Cognitive Science 1 (January l9/ 7): 84-123. A survey article concerned
with the past and future of Al. The authors see three periods so far: "Classic",
"Romantic,., and "Modern".
Guod, I. I. "Human and Machine Logic". British Journal for the Philosophy of Science In
(196/): 144. One of the most interesting attempts to refute Lucas, having to do with
whether the repeated application of the diagonal method is itself a mechanizable
operation.
. "Godel's Theorem is a Red Herring". British Journal for the Philosophy of Science 19
(1969): 357. In which Good maintains that Lucas' argument has nothing to do with
Godel's Theorem, and that Lucas should in fact have entitled his article "Minds,
Machines, and Transfinite Counting". The Good-Lucas repartee is fascinating.
Goodman, Nelson. Fact, Fiction, and Forecast. 3rd ed. Indianapolis: Bobbs-Merrill, 1973.
Paperback. A discussion of contrary-to-fact conditionals and inductive logic, including
Bibliography 748
Goodman's famous problem-words "bleen" and "grue". Bears very much on the
question of how humans perceive the world, and therefore interesting especially from
the Al perspective.
Press, 1976. A short hook about bees, apes, and other animals, and whether or not they
are "conscious"-and particularh whether or not it is legitimate to use the word
"consciousness" in scientific explanations of animal behavior.
deGroot, Adriaan. Thought and Choice in Chess. The Hague: Mouton, 1965. A thorough
study in cognitive psychology, reporting on experiments that have a classical simplicity
and elegance.
Gunderson, Keith. Mentality and Machines. New York: Doubleday, Anchor Books, 1971.
Paperback. A very anti-Al person tells why. Sometimes hilarious.
Hanawalt, Philip C., and Robert H. Haynes, eds. The Chemical Basis of Life. San
Francisco: W. H. Freeman, 1973. Paperback. An excellent collection of reprints from
the Scientilic American. One of the best ways to get a feeling for what molecular
biology is about.
Hardy G. H. and E. M. Wright. An Introduction to the Theory of Numbers, 4th ed. New
York: Oxford University Press, 1960. The classic book on number theory. Chock-full
of information about those mysterious entities. the whole numbers.
Harmon, Leon. "The Recognition of Faces". Scientific American, November 1973, p. 70.
Explorations concerning how we represent faces in our memories, and how much
information is needed in what form for us to be able to recognize a face. One of the
most fascinating of pattern recognition problems.
van Heijenoort, Jean. From Frege to Godel: A Source Book in Mathematical Logic.
Cambridge, Mass.: Harvard University Press, 1977. Paperback. A collection of epoch-
making articles on mathematical logic, all leading up to Godel's climactic revelation,
which is the final paper in the book.
Henri, Adrian. Total Art: Environments, Happenings, and Performances. New York:
Praeger, 1974. Paperback. In which it is shown how meaning has degenerated so far m
modern art that the absence of meaning becomes profoundly meaningful (whatever that
means).
* Goodstein, R. L. Development of Mathematical Logic. New York: Springer Verlag,
1971. A concise survey of mathematical logic, including much material not easily
found elsewhere. An enjoyable book, and useful as a reference.
Gordon, Cyrus. Forgotten Scripts. New York: Basic Books, 1968. A short and nicely
written account of the decipherment of ancient hieroglyphics, cuneiform, and other
scripts.
Griffin, Donald. The Question of Animal Awareness. New York: Rockefeller University
von Foerster, Heinz and James W. Beauchamp, eds. Music by Computers. New York:
**
John Wiles', 1969. This book contains not only a set of articles about various types of
computer-produced music, but also a set of four small phonograph records so you can
actually hear (and judge) the pieces described. Among the pieces is Max Mathews'
mixture of "Johnny Comes Marching Home" and "The British Grenadiers".
Fraenkel, Abraham, Yehoshua Bar-Hillel, and Azriel Levy. Foundations of Set Theory,
2nd ed. Atlantic Highlands, N. J.: Humanities Press, 1973. A fairly nontechnical
Bibliography 749
discussion of set theory, logic, limitative Theorems and undecidable statements.
Included is a long treatment of intuitionism. *
Frev, Peter W. Chess Skill in Man and Machine. New York: Springer Verlag, 1977.
Anexcellent survey of contemporary ideas in computer chess: why programs work, why
they don't work, retrospects and prospects.
Friedman, Daniel P. The Little Lisper. Palo Alto, Calif.: Science Research Associates,
1974. Paperback. An easily digested introduction to recursive thinking in LISP. You'll
eat it up!
• Hoare, C. A. R, and D. C. S. Allison. "Incomputability". Computing Surveys 4,
no. 3 September 1972). A smoothly presented exposition of why the halting problem
is unsolvable. Proves this fundamental theorem: ''Any language containing
conditionals and recursive function definitions which is powerful enough to program
its own interpreter cannot be used to program its own 'terminates' function."
• Hofstadter, Douglas R. "Energy levels and wave functions of Bloch electrons in
rational and irrational magnetic fields". Physical Review B, 14, no. 6 (15 September
1976). The author's Ph.D. work, presented as a paper. Details the origin of "Gplot",
the recursive graph shown in Figure 34.
Hook, Sidney. ed. Dimensions of Mind. New York: Macmillan, Collier Books, 1961.
Paperback. A collection of articles on the mind-body problem and the mind-computer
problem. Some rather strong-minded entries here.
• Hornev, Karen. Self-Analysis. New York: W. W. Norton, 1942. Paperback. A
fascinating description of how the levels of the self must tangle to grapple with
problems of self-definition of any individual in this complex world. Humane and
insightful.
Hubbard, John I. The Biological Basis of Mental Activity. Reading, Mass.:
AddisonWesley, 1975. Paperback. Just one more book about the brain, with one special
virtue. however: it contains man long lists of questions for the reader to ponder, and
references to articles which treat those questions.
• Jackson. Philip C. Introduction to Artificial Intelligence. New York, Petrocelli Charter.
1975. A recent book, describing, with some exuberance, the ideas of Al. There are a
huge number of vaguely suggested ideas floating around this book, and for that reason
it is very stimulating just to page through it. Has a giant bibliography, which is another
reason to recommend it.
Jacobs, Robert L. Understanding Harmony. New York: Oxford University Press, 1958.
Paperback. A straightforward book on harmony, which can lead one to ask many
questions about why it is that conventional Western harmony has such a grip on our
brains.
Jaki, Stanley L. Brain, Mind, and Computers. South Bend. Ind.: Gateway Editions, 1969.
Paperback. A polemic book whose every page exudes contempt for the computational
paradigm for understanding the mind. Nonetheless it is interesting to ponder the points
he brings up.
• Jauch, J. M. Are Quanta Real? Bloomington, Ind.: Indiana University Press, 1973. A
delightful little book of dialogues, using three characters borrowed from Galileo, put in
a modern setting. Not only are questions of quantum mechanics discussed, but also
issues of pattern recognition, simplicity, brain processes, and philosophy of science
enter. Most enjoyable and provocative.
Bibliography 750
• Jeffrey, Richard. Formal Logic: Its Scope and Limits. New York: McGraw-Hill, 1967.
An easy-to-read elementary textbook whose last chapter is on Godel's and Church's
Theorems. This book has quite a different approach from many logic texts, which
makes it stand out.
• Jensen, Hans. Sign, Symbol, and Script. New York: G. P. Putnam's, 1969. A-or perhaps
the-top-notch book on symbolic writing systems the world over, both of now and long
ago. There is much beauty and mystery in this book-for instance, the undeciphered
script of Easter Island.
Kalmar, Laszlo. "An Argument Against the Plausibility of Church's Thesis". In A.
Heyting, ed. Constructivity in Mathematics: Proceedings of the Colloquium held at
Amsterdam, 1957, North-Holland, 1959. An interesting article by perhaps the
bestknown disbeliever in the Church-Turing Thesis.
* Kim, Scott E. "The Impossible Skew Quadrilateral: A Four-Dimensional Optical
Illusion". In David Brisson, ed. Proceedings of the 1978 A.A.A.S. Sym osium on
Hypergraphics: Visualizing Complex Relationships in Art and Science. Boulder, Colo.:
Westview Press, 1978. What seems at first an inconceivably hard idea-an optical
illusion for four-dimensional "people"-is gradually made crystal clear, in an amazing
virtuoso presentation utilizing a long series of excellently executed diagrams. The form
of this article is just as intriguing and unusual as its content: it is tripartite on many
levels simultaneously. This article and my book developed in parallel and each
stimulated the other.
Kleene, Stephen C. Introduction to Mathematical Logic. New York: John Wiley, 1967. A
thorough, thoughtful text by an important figure in the subject. Very worthwhile. Each
time I reread a passage, I find something new in it which had escaped me before.
. Introduction to Metamathematics. Princeton: D. Van Nostrand (1952). Classic work on
mathematical logic; his textbook (above) is essentially an abridged version. Rigorous
and complete, but oldish.
Kneebone G. J. Mathematical Logic and the Foundations of Mathematics. New York:
Van Nostrand Reinhold, 1963. A solid book with much philosophical discussion of
such topics as intuitionism, and the "reality" of the natural numbers, etc.
Koestler, Arthur. The Act of Creation, New York: Dell, 1966. Paperback. A wide-ranging
and generally stimulating theory about how ideas are "bisociated" to yield novelty. Best
to open it at random and read, rather than begin at the beginning.
Koestler, Arthur and J. R. Smythies, eds. Beyond Reductionism. Boston: Beacon Press,
1969. Paperback. Proceedings of Yr conference whose participants all were of the
opinion that biological systems cannot be explained reductionistically, and that there is
something "emergent" about life. I am intrigued by books which seem wrong to me, yet
in a hard-to-pin-down way.
** Kutbose, Gyomay. Zen Koans. Chicago: Regnerv, 1973. Paperback. One of the best
collections of koans available. Attractively presented- An essential book for any Zen
library. Kuffler, Stephen W. and John G. Nicholls. From Neuron to Brain. Sunderland,
deals Mss.: Smauer Associates, 1976. Paperback. A book which, despite its title, deals
mostly with microscopic processes in the brain, and quite little with the way people's
thoughts come out of the tangled mess. The work of Hubel and Wiesel on visual
systems is covered particularly well.
Bibliography 751
Lacey Huh, and Geoffrey Joseph. "What the Godel Formula Says". Mind 77 (1968)- 7~.
A useful discussion of the meaning of the Godel formula, based on a strict separation of
three levels: uninterpreted formal system, interpreted formal system, a
metamathematics. Worth studying. Latos Imre. Proofs and Refutations. New York:
Cambridge University Press, 1976. Paperback. A most entertaining book in dialogue
form, discussing how concepts are formed in mathematics. Valuable not only to
mathematicians, but also to people interested in thought processes.
** Lehninger, Albert. Biochemistry. New York: Worth Publishers, 1976. A wonderfully
readable text, considering its technical level. In this book one can find many ways in
which proteins and genes are tangled together. Well organized, and exciting.
** Lucas, J. R. "Minds, Machines, and Godel". Philosophy 36 (1961): 112. This article is
reprinted in Anderson's Minds and Machines, and in Sayre and Crosson's The Modeling
of Mind. A highly controversial and provocative article, it claims to show that the
human brain cannot, in principle, be modeled by a computer program. The argument is
based entirely on Godel's Incompleteness Theorem, and is a fascinating one. The prose
is (to my mind)
incredibly infuriating-vet for that very reason, it makes humorous reading. "Satan
Stultified: A Rejoinder to Paul Benacerraf". Monist 52 (1968): 145.
Anti-Benacerraf argument, written in hilariously learned style: at one point Lucas refers
to Benacerraf as "self-stultifyingly eristic" (whatever that means). The Lucas-
Benacerraf battle, like the Lucas-Good battle, offers much food for thought.
. "Human and Machine Logic: A Rejoinder". British Journal for the Philosophy of
Science 19 (1967): 155. An attempted refutation of Good's attempted refutation of
Lucas' original article.
** MacGillavry, Caroline H. Symmetry Aspects of the Periodic Drawings of M. C.
Escher. Utrecht: A. Oosthoek's Uitgevermaatschappij, 1965. A collection of tilings of
the plane by Escher, with scientific commentary by a crystallographer. The source for
some of my illustrations-e.g., the Ant Fugue and the Crab Canon. Reissued in 1976 in
New York by Harry N. Abrams under the title Fantasy and Symmetry.
MacKay, Donald M. Information, Mechanism and Meaning. Cambridge, Mass.: M.I.T.
Press, 1970. Paperback. A book about different measures of information, applicable in
different situations; theoretical issues related to human perception and understanding;
and the way in which conscious activity can arise from a mechanistic underpinning.
• Mandelbrot, Benoit. Fractals: Form, Chance, and Dimension. San Francisco: W. H.
Freeman, 1977. A rarity: a picture book of sophisticated contemporary research ideas in
mathematics. Here, it concerns recursively defined curves and shapes, whose
dimensionality is not a whole number. Amazingly, Mandelbrot shows their relevance to
practically every branch of science.
• McCarthy, John. "Ascribing Mental Qualities to Machines". To appear in Martin
Ringle, ed. Philosophical Perspectives in Artificial Intelligence. New York: Humanities
Press, 1979. A penetrating article about the circumstances under which it would make
sense to say that a machine had beliefs, desires, intentions, consciousness, or free will.
It is interesting to compare this article with the book by Griffin.
Meschkowski, Herbert. Non-Euclidean Geometry. New York: Academic Press, 1964.
Paperback. A short book with good historical commentary.
Bibliography 752
Meyer, can. ''Essai d'application de certains modeles cybernetiq ues it la coordina_ Lion
c1Jei les insectes sociaux". Insertes Sociaux XI11, no. 2 1966 : o which ch aws some
parallels between the neural organization in the( brain, and the organiz ae lion of an ant
colon%.
Meyer, Leonard B. Emotion and Meaning in Music. Chicago: University of' Chicago
Press, 1956. Paperback. A book which attempts to use ideas of Gestalt svcholo theory
of perception to explain why musical structure is as it is. One of the more unusual the
books on music and mind.
. Music, The Arts, and Ideas. Chicago: University of Chicago Press, 1967. Paperback. A
thoughtful analysis of mental processes involved in listening to music, and of
hierarchical structures in music. The author compares modern trends in music with Zen
Buddhism.
Miller, G. A. and P. N. joint son-Laird. Language and Perception- Cambridge, Mass.:
Harvard University Press, Belknap Press, 1976. A fascinating compendium of linguistic
facts and theories, hearing on Whorl 's hypothesis that language is the same as
worldviesv. A typical example is the discussion of the weird "mother-in-law" language
of the Dyirbal people of Northern Queensland: a separate language used only for
speaking to one's mother-in-law.
Minsky, Marvin L. "Matter, Mind, and Models". In Marvin L. Minskv, ed. Son antic
Information Processing. Cambridge, Mass.: M.I.T. Press, 1968. Though merely a few
pages long, this article implies a whole philosophy of consciousness and machine
intelligence. It is a memorable piece of writing by one of the deepest thinkers in the
field.
Minsky, Marvin L., and Seymour Papert_ .92-t cial Intelligence Progress Report.
Cambridge, Mass.: M.I.T. Artificial Intelligence Laboratory, Al Memo 252, 1972. A
survey of all the work in Artificial Intelligence done at M.I.T. up to 1972, relating it to
psychology and epistemology. Could serve excellently as an introduction to Al.
Monod, Jacques. Chance and Necessity. New York: Random House, Vintage Books,
1971. Paperback. An extremel fertile mind writing in an idiosyncratic way about
fascinating questions, such as how life is constructed out of non-life: how evolution,
seeming_ to violate the second law of thermodynamics, is actually dependent on it. The
book excited me deeply.
* Morrison, Philip and Emil, eds. Charles Babbage and his Calculating Engines. Nesv
York: Dover Publications, 1961. Paperback. A valuable source of information about the
life of Babbage. A large fraction of Babbage's autobiography is reprinted here, along
with several articles about Babbage's machines and his "Mechanical Notation".
Mvhill, John. "Some Philosophical Implications of Mathematical Logic". Review of
Metaphysics 6 (1952): 165. An unusual discussion of ways in which Godel's Theorem
and Church's Theorem are connected to psychology and epistemology. Ends up in a
discussion of beauty and creativity.
Nagel, Ernest. The Structure of Science. New York: Harcourt, Brace, and World, 1961. A
classic in the philosophy of science, featuring clear discussions of reductionism vs.
holism, teleological vs. nonteleological explanations, etc.
Nagel, Ernest and James R. Newman. Godel's Proof. New York: New York University
Press, 1958. Paperback. An enjoyable and exciting presentation, which was, in many
ways, the inspiration for my own book.
Bibliography 753
* Nievergelt, Jurg, J. C. Farrar, and E. M. Reingold. Computer Approaches to Mathe,nat
teal Problems. Englesyood Clif-f-s, N. J.: Prentice-Hall, 1974. An unusual collection of
different types of problems which can be and have been attacked on computers-for
instance, the "3n + I problem" (mentioned in my Rria with Diverse Variations) and
other problems of number theory.
Pattee, Howard H., ed. Hierarchy Theory. New York: George Braziller, 1973. Paperback.
Subtitled "The Challenge of Complex Systems". Contains a good article by Herbert
Simon covering some of the same ideas as does my Chapter on "Levels of Description".
Peter, R6zsa. Recursive Functions. New York: Academic Press, 1967. A thorough
discussion of primitive recursive functions, general recursive functions, partial
recursive functions, the diagonal method, and many other fairly technical topics.
Quine, Willard Van Ornman. The Ways of Paradox, and Other Essays. New York:
Random House, 1966. A collection of Quine's thoughts on many topics. The first essay
deals with carious sorts of paradoxes, and their resolutions. In it, he introduces the
operation I call "quining" in my book.
lishiRanganathan, S. R. Ramanujan, The Man and the Mathematician. London: Asia
Pubng House, 1967. An occult-oriented biography of the Indian genius by an admirer.
An odd but charming book.
Reichardt. Jasia. Cybernetics, Arts, and Ideas. Boston: New York Graphic Society,
197 1. A weird collection of ideas about computers and art, music, literature. Some of it
is definitely off the deep end-hut some of it is not. Examples of the latter are the articles
"A Chance for Art" by J. R. Pierce, and "Computerized Haiku" by Margaret
Masterman.Renyi, Alfred. Dialogues on Mathematics. San Francisco: Holden-Day,
1967. Paper hack. Three simple but stimulating dialogues involving classic characters
in history, trying to get at the nature of mathematics. For the general public.
** Rips Paul. Zen Flesh, Zen Bones. New York: Doubleday, Anchor Books. Paperback.
This book imparts very well the flavor of Zen-its antirational, antilanguage, antireduc
iionistic, basically holistic orientation.Rogers, Hartley. Theory of Recursive Functions
and Effective Computability. New York:McGraw-Hill, 1967. A highly technical
treatise, but a good one to learn from. Containsdiscussions of many intriguing problems
in set theory and recursive function theory.
Rokeach, Milton. The Three Christs of' Ypsilanti. New York: Vintage Books,
1964.paperback. A study of schizophrenia and the strange breeds of "consistency"
which arise in the afflicted. A fascinating conflict between three men in a mental
institution, all of whom imagined they were God, and how they dealt with being
brought face to face for many months.
Rose, Steven. The Conscious Brain, updated ed. New York: Vintage Books, 1976.
Paperback. An excellent book-probably the best introduction to the study of the brain.
Contains full discussions of the physical nature of the brain, as well as philosophical
discussions on the nature_ of mind, reductionism vs. holism, free will vs. determinism,
etc. from a broad, intelligent, and humanistic viewpoint. Only his ideas on Al are way
off.
Rosenblueth, Arturo. Mind and Brain: A Philosophy of Science. Cambridge,
Mass.:M.I.T. Press, 1970. Paperback. A well written book by a brain researcher who
deals with most of the deep problems concerning mind and brain.
Bibliography 754
* Sagan, Carl, ed. Communication with Extraterrestrial Intelligence. Cambridge, Mass.:
M.I.T. Press, 1973. Paperback. Transcripts of a truly far-out conference, where a stellar
group of scientists and others battle it out on this speculative issue.
Salmon, Wesley, ed. Zeno's Paradoxes. New York: Bobbs-Merrill, 1970. Paperback. A
collection of articles on Zeno's ancient paradoxes, scrutinized under the light of modern
set theory, quantum mechanics, and so on. Curious and thought-provoking,
occasionally humorous.
Sanger. F., et al. "Nucleotide sequence of bacteriophage 16X174 DNA", Nature 265
(Feb. 24, 1977). An exciting presentation of the first laying-bare ever of the full
hereditary material of any organism. The surprise is the double-entendre: two proteins
coded for in an overlapping way: almost too much to believe.
Sayre, Kenneth M., and Frederick J. Crosson. The Modeling of Mind: Computers and
Intelligence. New York: Simon and Schuster, Clarion Books, 1963. A collection of
philosophical comments on the idea of Artificial Intelligence by people from a wide
range of disciplines. Contributors include Anatol Rapoport, Ludwig Wittgenstein,
Donald Mackay, Michael Scriven, Gilbert Ryle, and others.
* Schank, Roger, and Kenneth Colby. Computer Models of Thought and Language. San
Francisco: W. H. Freeman, 1973. A collection of articles on various approaches to the
simulation of mental processes such as language-understanding, belief-systems,
translation, and so forth. An important Al book, and many of the articles are not hard to
read, even for the layman.
Schrodinger, Erwin. What is Life? & Mind and Matter. New York: Cambridge University
Press, 1967. Paperback. A famous book by a famous physicist (one of the main
founders of quantum mechanics). Explores the physical basis of life and brain; then
goes on to discuss consciousness in quite metaphysical terms. The first half, What is
Life?, had considerable influence in the 1940's on the search for the carrier of genetic
information.
Shepard, Roger N. "Circularity in judgments of Relative Pitch". Journal of the Acoustical
Society of America 36, no. 12 (December 1964), pp. 2346-2353. The source of the
amazing auditory illusion of "Shepard tones".
Simon, Herbert A. The Sciences of the Artificial. Cambridge, Mass.: M.I.T. Press, 1969.
Paperback. An interesting book on understanding complex systems. The last chapter,
entitled "The Architecture of Complexity", discusses problems of reductionism versus
holism somewhat.
Smart, J. J. C. "Godel's Theorem, Church's Theorem, and Mechanism". Synthese 13
(1961): 105. A well written article predating Lucas' 1961 article, but essentially arguing
against it. One might conclude that you have to be Good and Smart, to argue against
Lucas...
** Smullvan, Raymond. Theory of Formal Systems. Princeton, N. J.: Princeton
University Press, 1961. Paperback. An advanced treatise, but one which begins with a
beautiful discussion of formal systems, and proves a simple version of Godel's Theorem
in an elegant way. Worthwhile for Chapter 1 alone. What Is the Name of This Book?
Englewood Cliffs, N. J.: Prentice-Hall, 1978. A book of puzzles and fantasies on
paradoxes, self-reference, and Godel's Theorem. Sounds like it will appeal to many of
the same readers as my book. It appeared after mine was all written (with the exception
of a certain entry in my bibliography).
Bibliography 755
Sommerhoff, Gerd. The Logic of the Living Brain. New York: John Wiley, 1974. A book
which attempts to use knowledge of small-scale structures in the brain, in creating a
theory of how the brain as a whole works.
Sperrv, Roger. "Mind, Brain, and Humanist Values". In John R. Platt, ed. .Vew Views on
the Nature of Man. Chicago: University of Chicago Press, 1965, A pioneering
neurophvsiologist here explains most vividly how he reconciles brain activity and
consciousness.
* Steiner, George. After Babel: Aspects of Language and Translation. New York: Oxford
University Press, 1975. Paperback. A book by a scholar in linguistics about the deep
problems of translation and understanding of language by humans. Although Al is
hardly discussed, the tone is that to program a computer to understand a novel or a
poem is out of the question. A well written, thought-provoking-sometimes infuriating-
book.
Stenesh, J. Dictionary of Biochemistry. New York: John Wiley, Wiley-Interscience,
1975. For me, a useful companion to technical books on molecular biology.
** Stent, Gunther. "Explicit and Implicit Semantic Content of the Genetic Information".
In The Centrality of Science and Absolute Values, Vol. 1. Proceedings of the 4th
International Conference on the Unity of the Sciences, New York, 1975. Amazingly
enough, this article is in the proceedings of a conference organized by the nowinfamous
Rev. Sun Mvung Moon. Despite this, the article is excellent. It is about whether a
genotype can be said, in any operational sense, to contain "all" the information about its
phenotype. In other words, it is about the location of meaning in the genotype.
. Molecular Genetics: A Historical Narrative. San Francisco: W. H. Freeman, 1971. Stent
has a broad, humanistic viewpoint, and conveys ideas in their historical perspective. An
unusual text on molecular biology.
Suppes, Patrick. Introduction to Logic. New York: Van Nostrand Reinhold, 1957. A
standard text, with clear presentations of both the Propositional Calculus and the
Predicate Calculus. My Propositional Calculus stems mainly from here.
Sussman, Gerald Jay. A Computer Model of Skill Acquisition. New York: American
Elsevier, 1975. Paperback. A theory of programs which understand the task of
programming a computer. The questions of how to break the task into parts, and of how
the different parts of such a program should interact, are discussed in detail.
** Tanenbaum, Andrew S. Structured Computer Organization. Englewood Cliffs, N. J.:
Prentice-Hall, 1976. Excellent: a straightforward, extremely well written account of the
many levels which are present in modern computer systems. It covers
microprogramming languages, machine languages, assembly languages, operating
systems, and many other topics. Has a good, partially annotated, bibliography.
Tarski, Alfred. Logic, Semantics, Metamathematics. Papers from 1923 to 1938.
Translated by J. H. Lot, New York: Oxford University Press, 1956. Sets forth Tarski's
ideas about truth, and the relationship between language and the world it represents.
These ideas are still having repercussions in the problem of knowledge representation
in Artificial Intelligence.
Taube, Mortimer. Computers and Common Sense. New York: McGraw-Hill, 1961.
Paperback. Perhaps the first tirade against the modern concept of Artificial Intelligence.
Annoying.
Bibliography 756
Tietze, Heinrich. Famous Problems of Mathematics. Baltimore: Graylock Press, 1965. A
book on famous problems, written in a very personal and erudite style. Good
illustrations and historical material.
Trakhtenbrot, V. Algorithms and Computing Machines. Heath. Paperback. A discussion
of theoretical issues involving computers, particularly unsohable problems such as the
halting problem, and the word-equisalence problem. Short. which is nice.
Turing, Sara. Alan M. Turing. Cambridge, U. K.: WW'. Hefter & Sons, 1959. A
biography of the great computer pion4r. A mother's work of love.
* Ulam, Stanislaw. Adventures of a Mathematician. New York: Charles Scribner's, 1976.
An autobiography written by a sixty-five-year old man who writes as if he were still
twenty and drunk in love with mathematics. Chock-full of gossip about who thought
who was the best, and who envied whom, etc. Not only fun, but serious.
,A'aitson, J. D. The Molecular Biology of the Gene, 3rd edition. Menlo Park, Calif.: W.
A. Benjamin, 1976. A good book but not nearly as well organized as Lehninger's, in my
opinion. Still almost every page has something interesting on it.
Webb, Judson. "Metamathematics and the Philosophy of Mind". Philosophy of Science
35 (1968): 156. A detailed and rigorous argument against Lucas, which contains this
conclusion: "My overall position in the present paper may be stated by saying that the
mind-machine-GOdel problem cannot be coherently treated until the constructivity
problem in the foundations of mathematics is clarified."
Weiss, Paul. "One Plus One Does Not Equal Two". In G. C. Quarton, T. Melnechuk, and
F. O. Schmitt, eds. The Neurosciences:.4 Study Program. New York: Rockefeller
University Press, 1967. An article trying to reconcile holism and reductionism, but a
good bit too holism-oriented for my taste.
* Weizenbaum, Joseph. Computer Power and Human Reason. San Francisco: W. H.
Freeman, Freeman, 19i6. Paperback. A provocative book by an early At worker who
has come to the conclusion that much work in computer science, particularly in Al, is
dangerous. Although I can agree with him on some of his criticisms, I think he goes too
far. His sanctimonious reference to Al people as "artificial intelligentsia" is funny the
first time, but becomes tiring after the dozenth time. Anyone interested in computers
should read it.
Wheeler, William Morton. "The Ant-Colony as an Organism".Journal of Morphology
22, 2 (1911): 307-325. One of the foremost authorities of his time on insects gives a
famous statement about why an ant colony deserves the label "organism" as much as its
parts do.
Whitely, C. H. "Minds, Machines, and Godel: A Reply to Mr Lucas". Philosophy 37
(1962): 61. A simple but potent reply to Lucas' argument. Wilder, Raymond. An
Introduction to the Foundations of :Mathematics. New York: John Wiley, 1952. A good
general overview, putting into perspective the important ideas of the past century.
* Wilson, Edward O. The Insect Societies. Cambridge, Mass.: Harvard University Press,
elknap Press, 1971. Paperback. The authoritative book on collective behavior of insects.
Although it is detailed, it is still readable, and discusses many fascinating ideas. It has
excellent illustrations, and a giant (although regrettably not annotated) bibliography.
Winograd, Terry. Five Lectures on Artificial Intelligence. AI Memo 246. Stanford, Calif.:
Stanford University Artificial Intelligence Laboratory, 1974. Paperback. A description
of fundamental problems in At and new ideas for attacking them, by one of the
Bibliography 757
important contemporary workers in the field.. Language as a Cognitive Process.
Reading, Mass.: Addison-Wesley (forthcoming). From what I have seen of the
manuscript, this will be a most exciting book, dealing with language in its full
complexity as no other book ever has. . Understanding Natural Language. New York:
Academic Press, 1972. A detailed discussion of one particular program which is
remarkably "smart", in a limited world. The book shows how language cannot be
separated from a general understanding of the world, and suggests directions to go in, in
writing programs which can use language in the way that people do. An important
contribution; many ideas can be stimulated by a reading of this book.
. "On some contested suppositions of generative linguistics about the scientific study of
language", Cognition 4:6. A droll rebuttal to a head-on attack on Artificial Intelligence
by some doctrinaire linguists.
* Winston, Patrick. Artificial Intelligence. Reading, Mass.: Addison-Wesley, 1977. A
strong, general presentation of many facets of At by a dedicated and influential young
proponent. The first half is independent of programs; the second half is LISP-dependent
and includes a good brief exposition of the language LISP. The book contains many
pointers to present-day At literature. ed. The Psychology of Computer Vision. New
York: McGraw-Hill. 1975. Silly title, but fine book. It contains articles on how to
program computers to do visual recognition of objects. scenes. and so forth. The articles
deal with all levels of the problem, from the detection of line segments to the general
organization of knowledge. In particular, there is an article by Winston himself on a
program he wrote which develops abstract concepts from concrete examples, and an
article by Minsky on the nascent notion of "frames".
* Wooldridge, Dean. Mechanical Man-The Physical Basis of Intelligent Life. New York:
NJ McGraw-Hill, 1968. Paperback. A thorough-going discussion of the relationship of
mental phenomena to brain phenomena, written in clear language. Explores difficult
philosophical concepts in novel was, shedding light on them by means of concrete
examples.
Bibliography 758
Credits
Figures: Fig. 1, Johann Sebastian Bach by Elias Gottlieb-Hart ssmann (1748), collection
of William H. Scheide, Princeton, New Jersey; Fig. 2, Flute Concert in Saussouci, by
Adolf von Menzel, Nationalgalerie, West Berlin; Figs. 3, 4, 152, "The Royal Theme" and
the last page of the "Six-part Ricercar." from the original edition of Musical Offering by
Johann Sebastian Bath, are reproduced courtesy of the Library of Congress; Figures of
lithographs and woodcuts of M. C. Escher are reproduced by permission of the Escher
Foundation, Haags Gemeentemuseum, The Hague, copyright © the Escher Foundation,
1979, reproduction rights arranged courtesy of the Vorpal Galleries, New York, Chicago,
San Francisco, and Laguna Beach; Fig. 9, photograph of Kurt Godel by Orren J. Turner
from Foundations of
Mathematics. Symposium Papers Commemorating the Sixtieth Birthday of Kurt Godel,
edited by
Jack J. Bulloff, Thomas C. Holyoke, and S. W. Hahn, New York, Springer-Verlag, 1969;
Figs. 17, 96, "Figure-Figure" and "A section of mRNA passing through a ribosome,"
drawings by Scott E. Kim; Figs. 19, 44, 133, 148, musical selections from the Musical
Offering by J. S. Bach, music printed by Donald Byrd's program "SMUT"; Fig. 25,
"Cretan labyrinth" from W. H. Matthews, Mazes and Labyrinths: Their History and
Development, New York, Dover Publications, Inc., 1970; Fig. 39, photograph of Rosetta
Stone, courtesy of the British Museum; Fig. 40, A collage of scripts. Samples of
cuneiform, Easter Island, Mongolian and Runic scripts from Hans Jensen, Sign, Symbol
and Script, East Germany VEB Deutscher Verlag Der Wissenschaften; samples of
Bengali and Buginese script from Kenneth Katzner, The Languages of the World, New
York, Funk & Wagnalls, 1975; samples of Tamil and Thai from I. A. Richards and
Christine Gibson, English Through Pictures, New York, Washington Square Press; Fig.
59, "Intelligence built up layer by layer" (Fig. 9.8) adapted from Patrick Henry Winston.
Artificial Intelligence, Reading, Mass., Addison-Wesley Publishing Company, reprinted
by permission; Figs. 63, 69, photographs of an ant bridge by Carl W. Rettenmeyer and
construction of an arch by termite workers by Turid Hdlldobler, from E. O. Wilson, The
Insect Societies, Cambridge, Mass., Harvard University Press, 1979; Fig. 65, schematic
drawing of a neuron adapted from The Machinery of the Brain, by Dean Wooldridge,
copyright © 1963, McGraw-Hill, Inc. used with permission of McGraw Hill Book
Company, and from Fig. 11-6, page 26, Speech and Brain-Mechanisms, by Wilber
Penfield and Lamar Roberts, copyright © by Princeton University Press, reprinted by
permission of Princeton University Press; Fig. 66, "the human brain, seen from the left
side," from Steven Rose, The Conscious Brain, copyright © 1973 by Steven Rose,
reprinted by permission of Alfred A. Knopf, Inc., New York, and John Wolfers, London;
Fig. 68, "overlapping neural pathways," from John C. Eccles, Facing Reality, New York,
SpringerVerlag, 1970; Figs. 77, 78, 80, 82, 117, 137, 138, 141, The Shadows, State of
Grace, The Fair
Captive, The Air and the Song, Mental Arithmetic, Common Sense, The Two Mysteries,
and The
Human Condition I by Rene Magritte, copyright © by ADAGP, Paris, 1979; Figs. 79, 95,
"Tobacco Mosaic Virus" and "Secondary and Tertiary Structure of Myoglobin" from
Albert Lehninger, Biochemistry, New York, Worth Publishers, 1975; Figs. 91, 92, "The
Credits 759
four constituent bases of DNA" and "The ladder-like structure of DNA" from Arthur
Kornberg, "The Synthesis of DNA," Scientific American, copyright © October 1968, all
rights reserved; Fig. 93, "Molecular model of the DNA double helix," reprinted by
permission, from V. M. Ingram, Biosynthesis of Macromolecules, Menlo Park,
California, The Benjamin/Cummings Publishing Company, 1972; Fig. 97,
"Polyribosome" from The Proteins, edited by R. E. Dickerson and H. Neurath, page 64,
New York, Academic Press; Fig. 98, "A two-tiered molecular canon" from O. L. Miller,
Jr., "Visualization of Genes in Action," in Scientific American, copyright © March 1973,
all rights reserved; Figs. 101, 102, 103, "The T4 bacterial virus," "Infection of a
Bacterium by a virus," and "The morphogenetic pathway of the T4 virus," from William
B. Wood and R. S. Edgar, "Building a Bacterial Virus" in Scientific American, copyright
© July 1987, all rights reserved; Fig.
105, photograph of Srinivasa Ramanujan from S. R. Ranganathan, Ramanujan, the Man
and the Mathematician, New York, Asia Publishing House, 1967; Figs. 110, 111, 112,
from Terry Winograd, Understanding Natural Language,New York, Academic Press,
1972; Fig. 113, Photograph of Alan Turing by Mssrs. C. H. O. Trevelyan from Sara
Turing, Alan M. Turing, Cambridge, England, W. H. Heffer and Sons, Ltd., 1959; Fig.
116, "a meaningful story in Arabic" from Abdelkebir Khatibi and Mohammed Sijelmassi,
The Splendor of Islamic Callizrabhl, New York, London, Thames & Hudson, copyright
© by Qarawiyne Library in Fez. Fig 118, Procedural representation of “red cube which
supports a pyramid,” adapted from Computer Models of Thought and Language, edited
by Roger C. Schank and Kenneth Mark Colby. W. H. Freeman and Company, copyright
1973; Figs. 119, 122, 124, 130, Bongard problems from M. Bongard, Pattern
Recognition, Rochelle Park, New Jersey,
Hayden Book Company, Spartan Books, 1970
Credits 760
INDEX
AABB form, 130, 227 history of, 19, 24-27, 594-97, 600-9;
Abel, Niels Henrik, 404 leveltranslation and, 285; outline of, 601-3;
ABORT (BlooP), 412 relation to mathematics, 559-60;
abortion, 176 levels of, 651-53, 656-57, stratification of, 299-300
660, Al programs: compared with people, 679-
abstraction, ti66-72, 674 80; curiosity of, 679; Lucas' argument and,
Achilles: answer-schemas and, 475; Carroll 577-78
paradox and, 46, 170, 181, 193; Crab Al Thesis, 579
Canon and, 204 666-67; heel of, 389, 484; Air and the Song, The (Magritte), 494
image of GOD, 223; inaccessible neurons Air on G's String, 445, 446, 497
of, 328. 686; initial letter of, 231, 507, 667; airplane-haystack story, 675
innocence of, 406, 408, 424; mapped onto aleatoric music, 163, 174, 700
ant colony, 318, 324; mentioned 84, 272, Algol, 292, 293, 381, 630
525, 573, 669; mystified by Crab, 560; algorithms, 292, 410, 412, 413, 414, 440,
orchard analogy and, 427; origin of, 28, 29; 459, 567 all 60
picture of, 42; problem reduction and, 610- "almost"-situations, 634-40, 641-43, 649
11; recursion and, 128-31, 149; as violin, alpha helix, 521, 525
502 Alternative Structures of the Union, see
Achilles property, 396-98, 415 ASU's
acoustico-retrieval, 278-80, 460 ambiguity: computer languages and, 297-
acronyms, 31-32, 113, 174, 176, 204, 237, 98; in translation into TNT, 209-11
272, 374, 606, 684, 727, 736, 738, 740 amino acids, 518-25, 533-34; in
acrostics, 7, 81 Typogenetics,
active site, 528-29, 544 508, 510-11
actor formalism, 662-64 analogical thinking: by computer, 603;
,,Actually Intelligent" programs, 676 underpin
addition: Al programs and, 677-78; ning of, 570-71
commutativity, associativity of, 55, 225- analogies, 668-74
26; in BlooP, 409; noncommutative, 222- Analytical Engine, 25, 598, 727 anaphase,
23, 639; pqsystem and, 49-53, 417; 667
representability of, 417; of supernaturals, anarchy, 693
455; TNT notation for, 206-7; triple, 101, "and", 177-80, 181, 186, 630 Anderson,
206-7 Alan Ross, 197 Announcer, 633-40
addresses (in memory), 289, 290 anomalies, 44, 96, 208, 435, 723 Another
adenine, see nucleotides World (Escher), 250, 255 answer-schemas,
advertisements, 478 462-14, 475, 688 ant bridge, 334
Al: applied to mathematics, 573, 614-15; ant colonies: artificial, 359; caste
arguments against, 597-99; attitudes distribution in,
against, see anti-Al attitudes; computer 318-28; castes in, 317-18; communism in,
languages and, 299-300, 548; converging 318,
towards brains, 579; defined, 26; difficulty 330-31; compared with brains, 315-16,
of, 26-27, 573, 740; evidence and, 695; 318,
faith underlying, 572, 578-79; G6del's 324-25, 350, 358-59; contrasted with gases,
Theorem and, 388-90, 471-77, 706-7;
Index 761
317; freedom and control in, 315-16, 327; Aria with Diverse Variations (Bach), 392-
intel 93, 395 Aria with Diverse Variations
ligent, 310-36; levels in, 319-27; (Dialogue), 408 Aristotle, 19
mechanisms of arithmetization, 262-65, 268-69, 533-34
teams and signals, 317-21; order and chaos arithmoquining, 445-54, 466-68, 497, 502,
in, 541,
316-17; regrouped, 332-33; signals in, 320- 580-81
28; arms, mutually washing, 691
symbol level in, 324-28, 330; teams in, art: identity crisis of, 699-700, 703-6;
317, modern,
319-27; thresholds in, 316-17, 319-21; 699-706; by computer, 603, 619-20 Art-
trails of, Language, 622
315-17; see also caste distribution, Aunt Art of the Fugue (Bach), 79-81, 86, 671
Hillary, Art of Zen Strings, 237, 239-42, 244, 626
Fermant, job. Seb., Fourmi, Lierre de, Artificial Imagery, 560 Artificial Intuition,
teams, 560 Artificial Ism, 625
signals, symbols artificial "thinking", 337, 601
Ant Fugue, 337, 349, 350, 382, 570, 686, Ascending and Descending (Escher), 12-
737-39 "Aid Fugue" (Escher), 322-23 13, 15, 21,
Anteater, 275-84, 311-36, 382, 570, 722 716
anti-AI attitudes. 27, 470-72, 628 Asian Box, Very Gold, see Gold Box, Very
anticodons, 522-24 Asian assemblers, 291, 294
ants: dispensability of, 326; nest of, 359; assembly language, 290-95; compared to
vs. ant DNA,
colonies, 314, 315, 319, 321, 326, 330 290-91
aperiodic crystals, 167-69, 174-76 arch of assembly lines, cellular, 528-29, 544-45
termites, 358-59
Appearances of Achilles and the Tortoise in Dialogues are not indexed, but those of less
frequent characters are. The reader is encouraged to consult the figure on page 370 for
possible help in crossreferences.
Index 762
axiom schemata, 47, 48, 65, 87, 468, 472- beautiful vs. non-beautiful, 552-58, 560,
73, 543 573-75, .581-82
axiomatic systems, see formal systems beauty: computers and, 575; elusiveness of,
axioms: defined, 35; lack of in 554, 555, 565, 574-75, 581, 583-84
Propositional Calculus, 183; of MIU- bees, 360, 641, 720
svstern, 33, 35; of P-system, 73-74; of pq- Beethoven, Ludwig van. 6. 75, 163, 634
system, 47; of pq-system, modified, 87; of beliefs, catalogue of. 382, 384 Bell, A. G.,
310-system, 263; of TNT, 216; of TNT, 296
extended, 451-52, 466-68; of tq-system, 65 Belnap, Nuel, 197 Beno, Luciano, 704
axons, 339-40 bicentuplets, 540
Babbage, Charles, 24-26, 598, 601, 726-42 bifurcations, 91-94, 100, 456-59, 467, 579
Babbage test, 735-73 biojukeboxes, 160, 175 Birthday Cantata
BACH (acronym), 174-75 (Bach), 461
B-A-C-H (melody), 79-81, 86, 102, 121, Birthday Cantatatata (Dialogue), 468, 475,
155-57, 266, 719 688 birthdays, 461-64
Bach, Anna Magdalena, 482 bits, 288-89, 290, 291 BLOCK (BlooP),
Bach, C. P. E., 3, 4, 80 410-11
Bach, Joh and Seb., 633, 669 blocks world, 586-93, 627-32, 674
Bach, Joh. Seb.; Al and, 27, 677; as BlooP, 406. 409-30, 440, 441, 444;
composer, 392, 461, 740-42; as alphabet of,
glassblower, 79; as harpsichordist, 275, 419, 425; control structures in. 410;
279, 280; as inspiration for Dialogues, 28, primordial
737; confused with Fermat, 331-35; depth steps of, 409, 412-13; syntax of, 410-15
of, 7-8, 9-10, 27-28, 70-71, 677; dissection BlooP programs, 410-14 Blue Programs,
vs. appreciation of, 680; Escher and, 199, 418-20, 422, 427
666-t7; Forkel on, 4, 86; homage to, 81; Bluediag [N], 42-44, 428
improvisation by, 3-7, 96, 719; in Leipzig, board evaluation: static and dynamic, 604-
383, 404; indirect self-reference of, 79-81, 5, 611;
86; life and death of, 86; modulation and, Strange Loops in. 604-5
122-23, 130; recursive qualities of music Bodhidharnta, 232, 238, 245. 252, 625
of, 70-71; Shepard tones and, 719; squared, Bolvai, Farkas, 92
679; vs. Cage, 157, 1624, 174-75; see also Bolyai, Janos, 91-92 Bongard, M., 646
Old Bach Bongard problems, 646-62, 664. 669, 674;
Bach, Wilhelm Friedemann, 4, 6 uni
background assumptions, 644 versality of, 661-62
backtracking, 97, 629, 632 Boole, Geo., 20, 404, 600 Boolean
backwards chaining, 618 Buddhism, 577 boomerangs, 75, 84, 267,
bankers, 457 472 bootstrapping, 24, 293-94, 524, 548
base-pairing, complementary, 433, 506-7, bottom (recursion), 139-40; see also
514-16, 517, 523, 534 skeletons bottom-up vs. top-down, 48-49
baseball, 637-38 bottoming out, 133-35, 151, 259, 301, 650
bases (genetics): 514, see also nucleotides; boustrophedonic writing, 168-69, 176
in Tvpogenetics, 505-8, 510 boxed systems, 469-70, 543
Baso, 234 Boyle's law, 308; see also gases
Bassui, 255 brains: ant colonies and, 315-16, 318, 324-
25, 350, 358-59; as ATN-colonies, 359; as
Index 760
mathematical objects, 559; Epimenides Carroll, John B., 630
paradox and, 584-85; formal systems and, Canon, Lewis, 20, 28, 46, 192, 372, 681;
337-38, 559-62, 569-79, 584-85, 676, see material by, 43-45. 366-68
also Church-Turing Thesis, formal vs. Carroll paradox: 28, 43-45, 681; evidence
informal systems, brains and rules; frame ver
and outer messages and, 170-71; mappings sion, 693-94; message version, 170:
between, 341-42, 345-46, 369-82; music problem
and, 163; operating systems and, 296; po4ed by, 46. 181; proof version, 192-93;
programmability of, 302; rules and, 26-27, Sam
676, see also brains and formal systems; uel's argument and, 684-85; symbolized,
suborgans of, 340-41; thoughts and, 337- 193; see also infinite regress
65; vs. minds, 576; see also minds, cascades, 224, 529, 626, 664
intelligence, etc. caste distribution: encoding of knowledge
Breton, Andre, 700 in, 319,
"British Grenadiers, The", 607 324-28, 359; meaning of, 321-24; updating
Brouwer, Luitzen E. J., 404 of,
Buddha-nature, 233, 234, 238-44 318-19, 324
Butterflies (Escher), 147-48 catalogues of programs (Blue, Green, Red),
Buxtehude, Dietrich, 335 419,
Byrd, Donald, see SMUT 427-28
Byron, Lord, 25 catalysts, 528-29
C-svstem, 65-67, 71-72 cats, 313, 343-46, 532 causality, types of,
CAGE (acrorn.m), 174-75 C-A-G-E 709-10 CCrab, see ATTACCA
(melody), 156-57 ceilings, see loops, bounded: BlooP
Cage, John, 156-57, 163-64, 167, 174-75, celestial mechanics, 353-54 CELLS
556, (BIooP), 410-11
699-700, 704 cellular processes, as models for All, 663-
calculating prodigies, see idiots savants 64 Central Crabmap, 667
Canon by Intervallic Augmentation, 525 Central Dogma: of Mathematical Logic,
"Canon per augmentationem et contrario 271,
motu" 532-34; of the MIC-system, 513; of
(Bach), see Sloth Canon Molecular
,,Canon per Tonos" (Bach), see Endlessly Biology, 504-5, 514, 532-34, 536, 667; of
Rising Canon Ty
canons: copies and, 8-9, 146; Dialogues pogenetics, 513; of Zen strings, 238, 239,
and, 665-69, 738; Escher drawings and, 15; 240,
in Goldberg Variations, 392; in Musical 243
Offering 7-10, 726-27; polyribosomes and, Central Dogmap, 532-34, 545, 547, 672,
526-28; self-refs and, 501-3; structure of, 709 Central Pipemap, 701-2 central
8-10; two-tiered, processing unit, 288, 289 Central
527-28; see also individual canons, fugues Proposition, 264, 269 Central Slothmap,
Cantor, Georg, 20, 216, 418, 421, 422-24 702
Cantor set, 142 .Cantorcrostipunctus", 424 Central Xmaps, 702, 716; see also
Capitalized Essences, 29 individual en
car radio. 670-71 tries
cardinalitv, intuitive sense of, 567
Index 761
centrality, 374-75 centromere, 668 Church-Turing Thesis, 428-29, 552, 561-
cerebellum, 341 Chadwick, John, 50 chain 79; Al Version, 578-79, 580, 581; Hardy's
letters, 546 Champernowne, David, 595 Version 566; Isomorphism Version, 567-
Champollion, Jean Fransois, 165 68; Microscopic Version, 572; Public-
channeling, 299, 376-77 Processes Version, 562, 568, 574, 580;
chaos in number theory, 137-38, 152, 557; Reductionist's Version, 572, 574; Soulist's
see also Version, 574; Standard Version, 561-62,
order and chaos 579; Tautological Version, 561; The odore
chauvinism, 171-73 Roszak Version, 574-75; unprovability of,
checkers programs, 573, 604-5 Chekhov, 562
Anton, 642 Church's Theorem, 560--61,574,579-
chess: chunking and, 285-87, 604; grand 81,609, 697 cigars, 199, 201, 383, 481, 651
masters in, 286-87; round-the-house, 595; classes vs. instances, 351-55, 360-61; see
self-modifying, 687-88 also prototypes, intensionality and
chess boards, hierarchy of, 687 chess extensionality, analogies, conceptual
players, cycle of, 94-95 skeletons, etc.codes: art and, 703-4;
chess programs: Babbage and. 25, 729-31, familiar and unfamiliar,82, 158. 267; see
736; also decoding, Gödel Code,
choice and, 711-12; Crab and, 721, 729-31; Genetic Code, etc.
difficulty of, 151-52, 605; jumping out of "coding" of sentences, 583-84
the system and, 37-38, 678; knowledge codons, 519-20, 524, 533, 535; see also
representation in, 618; recursive structure Gödel
of 150-52; strengths and weaknesses of, codons, duplets
151-52, 285-87, 573, 603-4, 611; Turing Colby, K., 599 columns in brain, 346
and, 595, 596, 736: varieties of, 601; Comenius, Johann Amos, 625 comments in
without look-ahead, 604 programs, 297 Common Sense (Magritte),
chests of drawers, nested, 644-45 700-1 common sense and programs, 301
children's stories and Al, 675-76 communicability of algorithms, 562
Chiyono, 256 commutativity, 55-56, 209, 225-27, 453,
choice, 711-14 639 compelling inner logic, 161-62, 163-64
Chopin, Frederic, 70, 257, 677 chords and competing theories, and nature of evidence,
analogies, 673-74 chromosomes, 695 compiler languages, 292-95 compilers,
homologous, 668 292-95, 297, 503 compiling, reverse of,
chunked versions of this book: jacket, 381
viiixiii, 370, 758-77 Complete List of All Great
chunking: ant colonies and, 326-27; brains Mathematicians, see List
arid, 381-84, 559; computer languages and, completeness, 100-2, 417-18, 422, 465; see
290-92, 381, 412-13; defined, 285-88; also incompleteness, consistency
determinism and, 306-8, 363, 522; of complexity of world, 569
DNA, 531-32; intuitive world-view and, composite numbers, 64, 65-66, 73; see also
305-6, 362-63; of music, 160, 164, 525; of prime
one's own brain, 382; probabilistic, numbers
384; scientific explanation and, 305-6; compound sentences, in TNT, 214
super conductivity and, 305; trade-offs in, compound words, 665
326; vision and, 348 computer chess, see chess programs
Church, Alonzo, 428, 476, 561
Index 762
computer languages: analogues in cell, contradictions: between levels, see level-
547; dialects of, 503; flexibility and, 298- conflicts; caused by impossible cycles, 94-
99; high level, 292-93, 297-300; message- 96; coexisting in same brain, 383, 697-98;
passing, 662-63; power of, 299, 428-29; diagonal argument and, 420-22; in
presented, 289-99, 406-30, 498-99; in mathematics, 17-24, 196-97, 223, 580-81;
SHRDLU, 629-32 computer systems, 287- so-consistency and, 453; personal
302 computers: assembled by computers, nonexistence and, 698; in pq-system, 87,
504, 684; crying, 675-76; determinism and, 88; in Propositional Calculus, 191-92, 196-
25-27, 306-7, 684-86; fallibility of, 575, 97; in selfimage, 696; Tortoise's shell and,
578, 678; learning by, 603-5; origins of, 177-80; two levels of, 581, 584; visual, 97-
24-26; in phonographs, 78, 484, 486-88; 99; Zen and, 99, 235, 246-56, 698; see also
see also programs, Al, Al programs paradoxes, inconsistency, Epimenides
concept network, 651-54; see also semantic paradox, etc.
networks Contrafadus, 641, 643, 669
conceptual dimensions, 670-71 Convex and Concave (Escher), 105-9, 348
conceptual mapping. 668-72 Cooper pairs, 304-5
conceptual nearness, 371-73, 614, 651-56 copies: canons and, 8-9, 527-28; in code,
conceptual revolutions, 660-61, 673 517, 527-28; complementary to originals,
conceptual skeletons, 381, 514. 666-72, 70, 501, 506-7, 517, see also inversion;
674 connotations and culture, 372-73, 379- DNA and, 529-31; inexact, 500-3, 546;
80 consciousness: causality and, 709-10; nature of, 146-49; self-reps and, 500-4,
source of, 512-13; television and, 489; viruses and,
384-85, 387-88: understanding of, 82, 680, 542-43; visually nested, 138-40; see also
708-10 sameness, isomorphisms
conservation of complexity, 60, 195 copper, 173
consistency: defined, 94; of extended TNT, Cops Silva and Could, 405
223, 459; hypothetical worlds and, 95-100; Copy mode (Typogenetics), 506-8
interpretations and, 88, 94-101; Lucas and, cortex: areas of, 344; cerebral, 340-48;
477; oath of, in TNT, 450; proofs of, 23- visual, 343-48
24, 191-92, 229-30, 449-50; of counterfactual parameters, 639
Propositional Calculus, 191-92, 229; of counterfactuals, 634-40, 641-44
TNT, 229-30, 449-50; varieties of, 94-96; counterpoint, see canons, fugues, Bach, etc.
see also (o-consistency counting, 55-57, 228, 364
constants, parameters, variables, 643-44, court system, 692, 693
669 covalent bonds, 514
context: necessity of, 1614, 173-76; cows, 312, 346, 351
restoring of, 115-16, 128, 133, 161-64, CPU, see central processing unit
173-76 Crab: entertains Achilles, 480-94; genes of,
context-free sums, 520-22 200-1, 204. 507;' hikes and plays flute,
contexts, nested, 643-46, 672, 674 549-58;
continued fractions, 140, 277, 563, 565 intelligence of. 549, 558, 733; jukebox of,
continuous vs. discrete processes, 598 154-57; meets Achilles, 200; musical
Contracrostipunctus, discussed, 82-85, 267, evening
270-71, 406-7, 424, 467-70, 483-84, 534- chez, 720-42; origin of, 666-68; plight of,
37, 608, 721 281,
383-84; questionable behavior of, 560, 562,
Index 763
573-74, 579-81; receives presents and enter 425-29; for intelligence, see Turing test; for
tains guests, 275-84, 311-36; subjunctive Mozart pieces, 649; no guarantee of
afternoon chez, 633-40; Theme of, 729, existence of, 72; for number-theoretic
732. 740, 742; vs. Tortoise, 75-78, 406, truths, 228-29, 426, 551-58, 560. 573-74,
493-88,540, 543 Crab Canon (Bach), 202- 579-81; for primality, 64, 149, 413; for
3, 666 Crab Canon (Dialogue), 204, 355- proof-pair-ness, 416, 439-41; for sameness,
56, 665-69, 672, 723-25, 738 146-49, 158-59; for sorting numbers into
Crab Canon (Escher), 198-99, 667 two classes, see Church-Turing Thesis; for
crab canons, 9, 198-203, 204, 355, 501. termination, 425-29; for "the Way" in Zen,
665-69; in DNA, 200-1 250-51, 253, 254; for theoremhood, 39-41,
crab programs, 500-1 47-49, 72-73, 190-91, 408, 416, 560, 579-
creativity, mechanizability of. 25, 26, 571, 80, 582; for theorem-numbers, 440-41,
620, 673; see also originality, paradox of 580; topdown vs. bottom-up, 48-49; for
Al, non programmability, etc. Tortoise-pairness, 441; for Tortoise
Crick, Francs, 505, 532, 533, 534, 617 property, 396-97, 415, 441; for truth, 213,
Crime and Punishment i'Dostoevskvl, 379- 228-29, 417, 552-58, 560-61, 579, 581; for
80 Cristofori, Bartolommeo, 3 critical validity of derivations, 194, 416, 439-41,
mass, 228, 317, 389, 417, 470 crossing- 470; for well-formedness, 182, 269, 416,
over, 665-68 crystal in magnetic field, 140- 582; for wondrousness, 402, 425
43 crystallization metaphor, 347 C"f-
Thesis, see Church-Turing Thesis Cube d coding: of DNA, 159-62, 175-76, 201,
with Magic Ribbons (Escher), 281-82 231, 531-32, 538; of formal systems, 50-
cursively drawable figures, 67-68, 72 51, 54; of fortune, 154; via Gtidel
cytoplasm, 517, 518, 522-24 cytosine, see isomorphism, 267; of nature. 409; of
nucleotides records, 154-57, 158-59,
da Vinci, Leonardo, 641 161-64, 172, 174-75; as revelation, 160-61;
Dali, Salvador, 700 of Russian text, 380; see also
Dase, Johann Martin Zacharias, 567 data isomorphisms, translation, information,
base, 618 daughter programs, 503, 546 chunking
David, Hans Theodore, 3, 28, 719 Day and decoding mechanisms: complexity of, 158-
Night (Escher), 252, 255, 667 Dboups, 62, 172-76, 582-84; innate, 170-71; nature
Hfpsh, 515, 533 De Chirico, Giorgio, 700 of, 158-76; record players as, 83, 154-57;
De Morgan, Augustus, 20, 404, 600 transparency of, 267, 501; for Tripitaka,
decidable strings, 417 decimal system, 262- 257; see also isomorphisms, etc.
64, 269 L)eduction Theorem, 186
decipherment of texts, 50, 164-65, 173-74, default options, 352-53, 386, 411, 645, 674
583 decision procedures: for alienness, defects and expectations, 77, 86, 102, 222,
487-88, 540-41; for axiomhood, 41, 48, 476 demidoublets, 633-34, 669
470; for beauty, 552-58, 560, 579, 581-82, demons, 663-64 deoxyribonucleic acid, see
583-84; as BlooP puzzles, 415-17; for DNA derivations: alleged, 439-40; defined,
Buddha-nature, 234, 239, 272; defined, 39- 35-36: fallacious, 220, 439; in MIU-
41; for Diophantine equations, 559-60; for system, 35-36, 262, 264, 439; in
dreamable themes, 384; for end of Propositional Calculus, 184, 185, 188, 189-
Dialogue, 402-3; for genuineness of koans, 90, 196; supernatural, 454-55; in
234, 239; for Goldbach property, 400, 414; TNT, 217, 218, 219, 224-27, 269; in tq-
for Granny, 344-45, 347-48; for halting, system,
Index 764
65; in Typogenetics, 507, 509; vs. proofs, ing, 536; unusual interpretation of, 231
35-36, 193-94, 195 DNA endonudease, 530, 531 DNA ligase,
Descartes, Rene, 263, 340, 677 description- 530, 531
schemas, 650; see also templates DNA polymerise, 530, 531
descriptions: calculus of, 338; restructuring DNA Rapid Transit Service, 505. 517
of, Doctor program, 599-600. 608 dog-and-
649-53, 659-61, 672; tentative, 646-49 bone problem, 611-13 .doGma 1, doGma
descriptors, 647 II, 532-33
detachment, rule of, 185-86, 577 dogs, 233, 234, 354, 383, 569, 570, 611-12,
determinism, 54; see also free will Devil, 679 Doko, 250, 698
685 Dostoevsky, Feodor, 379-80 double
Dewdrop (Escher), 249, 256 Di of Antus, negation, 183, 545, 554 Double
333 Nodulation, law of, 243 Dragon (Escher),
diagonal method, Cantor's, 418-24, 426, 473-74, 524, 698
427-29, Drawing Hands (Escher), 15, 21, 133, 689-
438, 446, 469 92, 710, 716, 737
Diagram G, 135-37 dialogicians, 81 dreams, 378, 379, 384, 725 Dreyfus,
dialogues, miniature, 191-92, 193, 408-9, Hubert, 574
431, dualism, 251-55, 698-99; see also subject
560, 565, 595-96, 598, 599 vs. object Dumpty, H., 332
Dialogues: origin of, 28, 665-69; as self- duplets (Typogenetics), 510, 512 Dvoi£k,
refs, Antonin, 163
84-85, 129, 204, 502-3, 667, 738-39
Dickens, Charles, 326, 328, 380 Difference E. coli bacterium, 176, 537-41
Engine, 25 differentiation, cellular, 543-46 Earrwig, Dr. Tony, 586-93. 627
digestion, 306 earth chauvinism, 171-72
digits, shunting of, 264 Diophantine Earth-Moon-Sun system, 353-54
equations, 279, 459-60 Diophantus of earthworm(s), 341-42
Alexandria, 275 directory of real numbers, Eccles, John, 574
421-24 disambiguation, 586-93, 603, 629- Edifying Thoughts of a Tobacco Smoker
32 distance to goal, 611-13 divisor- (Bach), 482
freeness, 74 Edifying Thoughts of a Tobacco Smoker
djinns, 113-15, 216-17, 223, 224; see also (Dialogue), 543
Genie DNA: as aperiodic crystal, 167; as eggs, 192, 360, 383
carrier of genetic information, 159; Einstein, Albert, 100
compared to computer languages, 290-91; electrons, 140-46, 258, 30.3-5
composition and structure of, 514-15; ELIZA, see Doctor program
covalent backbone of, 514-15; as embedding of formal systems, 97, 207, 215
declarative knowledge, 616-17; double emergent phenomena, 708-9, 714
strands, 514-15, 530-31; isomorphism with emotions: brains and, 83; counterfeit, 599-
organism, 600; dependence of intelligence upon, 573,
146-48; mode of self-replication, 529-30; 597-98; as epiphenomena, 677; music and,
in outer space, 167, 175-76; as program, 83, 160, 163-64, 174-75, 383-84, 626-27,
language,data, 290-91, 547; quining and, 676-77, 699; potential, 281, 383-84, 583;
531; recombinant, 665; relation to mRNA, programs and, 573, 597-600, 626-27, 675-
517; self-destroy 77; universality of, 163, 174-75
Index 765
emulation, 295 469; of phonographs, see Todelization; of
Endlessly Rising Canon (Bach), 10-11, 15, self-engulfing process, 493; of TNT and
46, 130, 717-19, 742 related systems, 468-71; see also
ENIUQ (procedure), 498-99 Godelization, Tlidelization, Escherization,
enlightenment, 232, 237, 243, 246, 251, nonprogrammahility, etc.
254-55, 479, 567 Eta Oin, 586-93, 674
Enlightenment 'Yond Enlightenment, 237, ETAOIN SHRDLU, 628,630
239, 243, 244 Euclid, 19, 43, 58-60, 88-89, 216
End, 232, 252 Euclid's prime number Theorem, 35, 58-60,
enzymes: function of, 520-22, 528-30, 543- 228 Euler, Leonhard, 3, 394
45; as models for AI, 663-64; rules of Fume, Max. 605
inference and, 509-10, 513, 531; structure evidence, nature of, 633-36
of, 519-21, 525; synthesis of, 517-19, 522- evolution, 321-22
25, 527-29, 538-45, 547; versatility of, existential quantifiers. see quantifiers
529; vs. typoenzymes, 529; see also exotic styles of thought, 552, 563-64, 566-
proteins, typoenzymes 67 expanding nodes, 134-36
epigenesis, 159-60, 161-62, 531-32, 665 explanatory power on high level, 321, 326,
Epimenides, picture of, 496 707-10 expressibility and expressive
Epimenides paradox: connection with power, 101, 417,
G6del's Theorem, 17-18; Escher and, 716; 441-43, 444-45, 450, 454, 465-70, 580-81
expanded version, 21, 22; fear of, 23; extrasensory perception, see ESP
French-English version, 501; indirect extraterrestrial intelligence, 162-64, 167,
recursion and, 134; molecular version, 536- 172-76,
37; neural version, 584-85; Quine version, 341, 646, 661-62
431-37, 445, 446, 449. 497-99, 531, 537; eyes, 237, 248, 260, 308, 311, 313, 477,
subtlety of, 495-98; Tarski version, 580-81, 633, 715
584-85; two levels of, 581, 584-85;
Whitely's version, 476-77 Fair Captive, The (Magritte), 489 FANCY
epiphen9mena, 308-9, 363, 577, 596, 677- NOUN, 132-34
79 Fantasia and Fugue in G minor (Bath), 719
errors in programs, 295, 297-98. 596 fantasy rule, 183-85, 187
Escher, Maurits Cornelis: Bach and, 201, faucet, mental, 364-65
666-67; contradictions and, 97-99; copies faultlessly functioning machines, 575-78
and, 146-48; drawings of, see List of Faure, Gabriel, 163
Illustrations (xiv-xviii); figure and ground feedback and feedforward, 544-45
in, 67-68; flat vs. spatial and, 473-74, 689; Fermant, Johant Sebastiant, 332-35
incompleteness and, 716-17; Magritte and, Fermant's Last Fugue, 335
480; as prime mover, 689-92, 710; on Fermat, Pierre de, 275-77, 278; confused
subbrains, 387; Strange Loops and, 10-15, with Bach, 331-35
737; Zen and, 255-57 fermatas, 275, 329, 332, 333
Escherization, repeatability of, 473-74, Fermat's Last Theorem: 275-79, 332, 416;
689; see also 2-D vs. 3-D, Godelization coun
1.SP, 598-99, 693-95 terexample to, 277, 279, 460; inverted,
essential incompleteness: of Achilles' 333-34;
birthday, 462-64, 475-76, 688; of Al, see parodied, 335, 551; proof of, 277, 279, 460
Tesleis Theorem; of list of reals, 423-24, Feynman diagrams, 144-46
Index 766
Fibonacci (Leonardo of Pisa), 136, 246 181-97; TNT, 204-30; tq-system, 64-65;
Fibonacci sequence, 136, 138, 139, 152, Typogenetics. 504-13
173-74, 265, 416 formal systems vs. reality, 53-58
fiddles, see violins formal vs. informal reasoning, 193-97,
fifth postulate (Euclid), 90-93, 222, 451-52 228-29,
50 (fifty), 338, 557, 564 271-72. 449-50, 614-15, 618-19
figure and ground: 61-63, 64-74, 731; in formal vs. informal systems, 26-27, 559-
music, 70-71 85, 598,
FIGURE-FIGURE figure (Kim), 68-70, 73 684-86; see also brains, minds, etc.
filters: for abstraction, 286, 407-9, 648, formalist philosophy of mathematics, 458
657-60, formula: closed, see sentence; open, 207-8
663, 673; for Pools, 418, 427 formulas of TNT, 206, 207-15 Four-color
finitistic methods of reasoning, 24, 230 Theorem, parodied, 550 4-D space, 638-39
Fishes and Scales (Escher), 146-47 four-postulate geometry, see geometry,
fission and fusion (of concepts), 338, 352- absolute 4'33" (Cage), 156
56, 438, Fourmi, Lierre de, 333-34 frame effect, 704
470, 664-65 frame messages, 162, 166-67, 176 frames,
5-D space, 640 373, 644-46, 662-63, 672 framing devices,
flags, 29-32, 188 478 Frank, Philipp, 642
flashcards, fee tRNA Frederick the Great, King of Prussia, 3-8,
flat vs, spatial, see 2-D vs. 3-D flattened 27, 394, 729
look-ahead, 604-5 free will, 388, 680, 708, 710-14, 720-16,
flattened roles of inference, see theorems 734-35, 739; see also jumping out of the
vs. rules flexibility and inflexibility, 26-27, system
296-302, 611-14, 657, 673-75, 686 Frege, Gottlob, 20
flights of fancy, 378 French fries, 636-38, 683
FlooP, 406, 424-30, 567-68; fed into itself, French Suite no. 5, Gigue (Bach), 130
425-26; power of, 428-29. 561-62 frequencies, of words and letters, 377, 630
fluency, 376-77 friend, mental model of, 386-87 fringe
flutes, 3-5, 27, 528, 552-58, 720, 726 F(n) science, 693-94
and Min), 137, 142, 359 focusing, 657-59 fugues, 335, 634, 730, 736; in the Art of
folding-up of enzymes, 511-12, 519, 521, the Fugue,
525 football, 44, 303, 353, 634-40, 643, 79-81, 86; devices in, 314, 322-23, 329-30,
644, 645, 672 footraces, 29-32, 43, 594-95, 737-40; Dialogues and, 28; in the Musical
681-83 forced matching, 670-72 forgetting, Offering, 4-9; nature of, 9, 281-84, 737
577, 578, 619 Forkel, Johann Nikolaus, 4. Fundamental Facts 1 and 2. 440-42
86 form, 47. 66-67, 68, 73. 190-91, 370-71; fundamental jukebox-axiom, 155
syntactic funneling, 346-48
vs. semantic, 581-84, 631 G (Godel's string), 18, 271-72, 285, 447-
form and content, 84-85, 204, 279, 581-84, 55, 459-60. 502, 580, 608, 667, 707-8 G',
667-68, 740 G" G"' Gu, 466-68 -G, 272, 449, 451-55,
formal systems, presentations of: C-system, 458-59, 542 Galileo, 478-79
64-65; MIU-system, 33-41; P-system, 73- games played by Al programs, 601
74: Ganto, 189-90, 256, 407
pq-system, 46-60; Propositional gases and molecules, 307-8, 317, 693
Calculus, Gateles.s Gate, see Muvtonkan Gauss, Karl
Index 767
Friedrich, 92, 100 Gebstadter, Egbert B., repeatability of. 424, 465-76, 688; see also
94-95, 402-3, 484 Gelemter, E.. 606-7 Escherization, Todelization, jumping out l
general recursivity, 406, 430, 470 of the system
genes, 200-1, 507, 512, 524-25, 531, 544- Godelizing operator, 472-73, 475-76, 543
45, 668 Genetic code, 160. 519-20, 522-24, Godel's argument, summarized, 18, 272,
533-34, 536, 448 Godel's article. 17, 24, 438 Godel's
538; origins of, 231, 548 construction, illustrated, 84
genetics, 504-48 Godel's method, underlying causes of, 204,
(.enie, Meta-Genie, etc., 109-16, 216-17, 407, 465, 468-71
223, 224, Godel's Second Theorem. 230, 449-50, 696
610; see also djinns Godel's Theorem: All and, 388-90, 471-77,
genie, symbol-manipulating. 39-40, 48 706-7, 714; analogue of in molecular
genotype and phenotype, 159-62, 167, 173- biology, 534, 536-37; brief mentions, 72,
74, 74, 78, 100, 486; consequences of, 450-60,
175-76, 295, 531-32, 667-68 469-76; Contracrosttpunctus and, see
Gentzen, Gerhard, 195 Contracrosttpunctus: Diophantine
Geometric Code. 235-37, 241, 626 equations and, 459-60; LISP and,
geometry: absolute, 91, 93, 97, 222, 407, 738-39; proof of, 18, 265-72, 438-49;
451-52; stated, 17, 101, 272
elliptical, 93; Euclidean, 19-20, 88-92, 100, Godel's Theorems and human
222, 451, 456, 606-7, non-Euclidean, 19- introspection, 450, 696-98
20, 91-93, 98-99, 100,222-23, 451, 455-56; Goffman, Erving, 478 gold, 173
"true" version of, 88-94, 99-100, 456-57 Gold Box, Very Asian, 404-5
German professor, proverbial, 130-31 Goldbach, Christian, 394, 395
Giant Electronic Brains. 25, 601 glia, 339 Goldbach Conjecture, 394-96, 400, 404,
GlooP, 406, 428-29 G(n), 137 557-58, 615; parodied, 551
goals and subgoals, 227, 589, 590-91, 609- Goldbach property, 395-97, 400, 414, 418
14, 618-19, 629, 632 Goldbach Variation, 395-98. 400, 426, 441
Goblet G, 79, 81, 83-85, 267 Goldberg, Johann Theophilus, 391-92
God, 216, 400, 478, 482, 533, 567, 597, Goldberg Variations (Bach), 392-93, 395
711; picture of, 142-43 Goodfortune, Hexachlorophene J., 103-4,
GOD (acronym), 110-15, 133, 134, 216, 115, 128-29, 130
223, 224, Goso, 248
722; see also djinns, Genie Gplot, 138, 140-43, 146-47, 159, 503, 661
Gddel, Kurt, 15-19, 24, 28, 738, 740, 742 grammar: for computer languages, 297,
G6del Code, 18, 268, 533-35 Godel 408-15; for Feynman diagrams, 142. 145;
codons, 268, 425, 533-35 high-level, 625-27; for koans, 625-26; for
Godel isomorphism, 261-71, 439, 442-46, music, 626-27; for natural languages. 130-
738-39; likened to reflection of world in 34, 150, 363,
brain, 502, 570 588-93, 619-21, 630-32; for thought, 627
Godel-numbering, 18, 438, 738-39; of Grand Tortue, 237, 243-44 grandmother
FlooP programs, 425-26, 502; of MIU- cell, 344, 345
system, 261-64; of TNT, 268-70, 579 grandmothers, perception of, 344, 345,
Godel questions, Lucas on, 389, 390 347-48. 349
Godelization, 270; programmability. of, graphics, 728 grass roots, 693
471-73;
Index 768
Great Tutor, 237, 239, 244 Green Hofstadter, D. R., 75, 310, 724, 728, 742
Programs, 427 Greendiag [NJ, 427 Hofstadter s law, 152
Grelling's paradox, 20-21, 22 Groot, Hogen, 248
Adriaan de, 286 grounds, excellent, 731 holes in formalized systems, 24, 26, 449,
guanine, see nucleotides 451, 465, 468, 470-71
guaranteed termination, 41, 396-98, 399, holism: defined, 254, 312; vs.
403 guitars, 62, 200 reductionism, 284, 311-36, 389-90, 708-9;
gullibility, 75-76, 106, 309, 461, 600, 701 Zen and, 254 Hubel, David, 341, 343
Gutei, 237 Human Condition I, The (Magritte), 705-6
G0025, white stony, 626 Hyakujo, 254
hydrogen bonds, 516, 522, 525
HACKER, 664 hyphen-strings, 47, 64-65, 66
haiku, 153-54, 525, 619-20 hypothetical worlds, 95-100, 338, 360-62,
halting problem, 74, 425-29, 594, 697 634-40, 641-44; groundedness in reality.
hamburger-confusion, 577 Hammurabi, 362,
169 378-79
Hardy, Godfrey Harold, 562-66 hypotheticals, 44-45, 634-40
harmonic tension, 122-23; see also tension
and resolution i, 454
harpsichords, 3, 391, 502 Harrison, I, 454
Lawrence, 657 Haussmann, Elias Gottlieb, "I", referent of, 608
2 HE-HE puzzle, 62-63, 669 headache, 62- "I Can Be Played (Proven, etc.)...", 488,
63 hearing by computer, 602 "heart", in an 541
All program, 679 Heisenberg uncertainty "I Cannot Be Played (Proven, etc.) , 76-
principle, 455, 698 Helen of Troy, 110 77, 85,
hemiolia, 257, 519 hemispheres, 257, 340- 406-7, 448, 465-67, 536, 541, 608 I-counts,
41 Henkin, Leon, 541 260-61
Henkin sentences, 541-53, 709; explicit 1-level, see inviolate level
and im I-mode, see Intelligent mode
plicit versions, 542-43 iceberg, 495-96, 497
Henkin's Theorem, 488 ideal numbers, 56-58
hereditary arguments, 36, 47-48, 217, 261 identification with artifacts, 609, 713-14
heterarchies, 134, 359, 534, 651-54, 662, idiots savants, 566-67
691 heterological adjectives,20-21, 22 IF-statements (BlooP), 411-12
heuristics, 587, 588, 590, 603, 629 Hewitt, images: blurry, 686-87; of thought, 623
Carl, 662 "Imaginary Landscape no. 4" (Cage), 163-
hiccups, 116, 254, 255, 673. 725-26, 736 64, 699 imitation game, see Turing test
hierarchy of variability, 643-45, 669 Immunity Theorem, 536
high-fidelity vs. low-fidelity, 77, 85, 101, implicit characterization, 41, 67, 72-73, 93
406-7, improvisation vs. introspection, 739
470. 697-98 inaccessibility of lower levels to higher
high-level explanatory power, 707-9 levels, 686-92, 706-10; in Aunt Hillary,
Hilbert, David, 20, 23-24, 230, 459-60 330-31, 630: in brains/minds, 302, 328-29,
Hilbert's program, 23-24, 229-30 362-65, 619, 677, 686-92, 697, 706-10,
Hilbert's tenth problem, 459-60 H(n), 137 739; in programs, 296, 300-1, 588, 630,
Index 769
679; see also software and hardware, inaccessibility; creation of, 513; depth from
introspection, level-conflicts surface, 234-35, 409, 427, 549-58, 606-7,
incompleteness: Bach and, 86; of brains, 612-13, 628, 673, see also decoding;
585; defined, 86; Escher and, 716-17; of discardable, 649, 653, 657-59. 669-72:
extensions of TNT, 465-71; of formal flow of, 513, 533, 545, 547; irrelevant, 560
arithmetics, 18, 86, 101-2, 407, 618-19; of information-bearers, 158, 166, 167
list of mathematicians, 422; of list of reals, information -reveale rs, 158, 267
421-24; of Lucas, 477; of phonographs, see inhibition, cellular, 544
record players, intrinsic vulnerability of; of inner messages, 166-71, 174-76, 501, 524
Principia Mathematica, 18, 24, 618-19; of input-output devices, 288
self-knowledge, 696-98; of TNT, 271-72, input parameters (BlooP), 411
430, 450-51; see also essential insight, 613, 660-61, 665-76
incompleteness, wincompleteness, etc. instant replays, straight and subjunctive,
inconsistency: defined, 94; with external 634-40, 641, 672
world, 87-88, 95; internal, 87, 88, 94-96; of instructions: in machine language, 289-95;
people, 197, 697-98; of Tortoise, 177-80; vs. templates, 497-99, 531, see also
see also consistency, contradictions, su- programs vs. data
inconsistency, Zen INT(x), 138-41, 146, 661
increasing and decreasing rules, 73, 74, intelligence: accidental inexplicability of,
260-61, 264, 269, 401-2, 407-x, 441; see 707; essential abilities for, 26;
also lengthening and shortening rules, extraterrestrial, see extraterrestrial
chaos in number theory intelligence; liftability of, see skimming
index numbers for programs, 418-20, 427- off; limits of, 475-76, 679-80; necessary
28 index triplets for supernaturals, 455 underpinning of, 324; simplicity of, 172-
India, 549, 551, 557, 562-66 73; subtle features of, 566; tangled
Indra's Net, 258, 359 recursion and, 152; typical abilities of, 559;
inducers, 545 universality of, and intrinsic meaning, 158,
infinite bundle of facts, 397-98 infinite 162-64, 170-76, 501, 661-62; see also
coincidence, 398, 421 brains, minds, A1, etc.
infinite regress, 111-13, 142, 146, 152, Intelligent mode, 38-39, 65, 193-94, 613-
231, 388-89, 426, 497, 738; in Carroll 14
paradox, 43-45, 170, 192-93, 684-86. 693- intensionality and extensionality, 337-39,
94; halted, 127, 133-35, 170, 605, 684-86; 350, 361-62
of objectivity, 479; Zeno and, 31-32, 610; intentions of machines, 684-85
see also Carroll paradox, bottoming out, interestingness, programmed, 615
recursive acronyms, repeatability, etc. interpretation-conventions, 687-88
infinite sentence, 497 infinite sky, 401 interpretations: adjusted to avoid
infinitesimals and nonstandard analysis, inconsistency, 87-88, 453, 456, see also
455 infinity: Bach and, 10, 719; Escher undefined terms; multiple, 94-102, 153-57,
and, 15; handled finitely, 59-60, 221-25, 266-67, 271, 447-48; of pq-system, 49-53,
461-64, 468; illustrated, 135-36, 138-43; 87-88, 101-2, 158; of Propositional
names of, 475-76; supernaturals and, 454; Calculus, 186-87, 189, 191-92; of strands,
types of, 421; see also nontermination, 509-10; of TNT, 205-9, 266-67, 453, 533;
infinite regress, recursion, etc. of tq-system, C-system, P-system, 64-65,
informal systems, see formal vs. informal 73-74
systems information: accessibility of, see
Index 770
interpreters: mechanisms in brain, 582-84; processing, 344; see also meaning,
people, 293, 297, 524, 671; programs, 293, translation, copies, decoding, etc.
504, 547, 616, 632, 662, 692
intrinsically high-level properties, 707-9 'Jabberwocky" (Carroll), 366-68, 372-73
introspection, see self-monitoring, self- Jacquard loom, 25
awareness, self-knowledge, inaccessibility, Jaki, Stanley, 574
TNT. introspection of "Jammerwoch, Der", (Carroll-Scott), 366-
intuition, 560, 564, 613, 680, 713; 68 "Jaseroque, Le" (Carroll-Warrin), 366-
programming of, 605, 609 68 Jauch, J. M., 408, 409, 478-79
inversion, 8-9, 81, 146, 681-83, 737-38; Jefferson, G., 598
see also copies, complementary to original Joan of Arc, 20 Johns, jasper, 703
inviolate level, 686-92 J6shti, 233, 237, 238, 240, 253, 259, 272
irrationality vs, rationality in brain/mind, JOSHU (TNT-string), 443
575-78 jukeboxes, 154-57, 160-61, 164, 170-71,
itregularites, meta-irregularities, etc. 475- 174-76,
76 500
Iran, 254 jumping out of a subsystem, 477
ism, 254-55, 625, 704-6 jumping out of the system: in
isomorphisms: between Bongard problems, advertisement, 478;
660, 669; between brain-structures and by answer-schemers, 462-64; Godel's
reality, 82, 337-39, 350, 502, 569-71; Theorem
between brains, 369-82; coarse-grained, and, see Godefization, essential
147-48, 503; in Contracrostipunctus, 83- incompleteness;
85; between Crab's DNA and Crab Canon, illusion of, 478-79, 698; as method to
203, 667-68; defined, 9, 49-50; between resolve
earthworms. 342-43, 345; of emotions, contradictions, 196-97; in political systems,
163; exotic, prosaic, 159-60; fluid, 338, 692;
350, 362; between form and content in by programs, 36-38, 476-78, 678; from 2-D
Dialogues, 84-85, 128-30, 204, 667-68; to
between formal systems and number 3-D, see 2-D vs. 3-D; Zen arid, 255, 479;
theory, 408, 625; Godel-numbering and, see also Godelization, Todelization,
see Godel isomor phism; between Escherization, TC battles, repeatability,
mathematicians, 566; between mathematics nonprogrammability, etc.
and reality, 53-60; between mental
processes and programs, 568-73; between Kaiserling, Count, 391-92
MIU-system and 310-system, 261-65; Kay, Alan, 662 Kennedy, John F., 641
between models of natural numbers, 217; keys, musical, 10, 299, 466, 501; see also
partial, 146-47, 371-82; as revelations, modulation
159-61, 263; as roots of meaning, 49-53, Kim, Scott, 68-69, 503, 523, 719
87-8, 94, 267, 337, 350; self-reps and, 501- Kimberger, Johann Philipp, 9, 726 kitchen
3; between something and part of itself, sink, the, 315 Kleene, Stephen C., 476
138-43, 146-47; between spiderwebs, 371- Klein bottle, 691
72; transparent, 82, 158, 267; on various Kliigel, G. S., 91 knitting, 149-50
levels between same objects, 369; between knots, 341-44, 272, 628
visual apparatuses, 345-46; in visual knowledge: accessible vs. inaccessible,
362, 365,
Index 771
616, 619; encoded in ant colonies, 319-28, Albert, 504 Leibniz, Wilhelm Gottfried,
359; 24-25, 600 lemmas, 227
explicit vs. implicit, 617-18; modularity of, Lenat, Douglas, 615
615-18, 628; procedural v4. declarative. lengthening and shortening rules: decision
363-65, 615-17, 630, 654; procedures and, 48-49, 182, 407-8; MIU-
knowledge transplantation, surgical, 618 system and, 39-40, 260-61, 264, 613; TNT
koans, 30, 189-91, 233-45, 246-59, 625-26; and, 213, 266, 269; see also increasing and
generated by computer, 625-26; genuine decreasing rules, problem reduction
vs. phony, 234-35, 239, 242, 244, 427, Leonardo of Pisa, see Fibonacci
625-26 Kronecker, Leopold, 216 Kuhn, Lermontov, Mikhail, 642
Thomas, 660 level-conflicts: in Aunt Hillary, 330, 630;
Kupfergiidel, Roman, 394 Kvogen,244-45 in mind/brain, 575-78; in messages, 164,
170, 699-704; between object language and
La Mettrie, Julien Offroy de, 3, 27, 729 metalanguage,
labeling technique, 487-88, 540-41 194, 449-50; in SHRDLU, 630
Lambert, J. H., 91, 92, 99 level-con fusion: ants and, see ants vs. ant
lamp, meta-lamp, etc., 108-13, 216 colonies;
language(s): acquisition of, 170, 294, 302; in art, see 2-D vs. 3-D; authorship and, 3.
active meanings in, 51-52; Arabic, 623-24; 608, 720-26; in computer systems, 287,
of bees, 360; of the brain, 570; Chinese, 291, 295, 300-2, 308; of Kimian self-rep,
164, 665, 676; collage of, see scripts; 503; minds/
computers and, 130-34, 300-1, 363, 586- brains and, 287, 575-77; in Propositional
93, 599-600, 601-3, 619-32, 674-75, 721; Calculus, 185, 194; subjunc-TV and, 608;
effect on thought, 376-77; English, 169, self and, 709
372-73, 377, 379-80, 619-32, 674-75; level-crossing, in thought, 666, 668
flexibility of, 649, 674-75; French, 297, level-mixing in genetics, 509-10, 513-14,
366-68, 372-73, 377, 501, 618; German, 546-48 level-shifting, conceptual, see
366-68, 372-73, 380, 665; Hebrew, xviii, abstraction, levels of levels: of computer
377; hierarchy of, 22; imprecise, 674-5; languages, 290-99; distinct vs.
invisible isomorphisms and, 82; Japanese, similar, 285, 287; in Escher, 11-15, 6
169; as medium for proofs, 88-90, 195; levels (continued)
necessary underpinning of, 324; partitions 715-16; haziness of, 13-15, 54618, 715-16;
between, 671; procedural grammars for, intermediate, 302-3, 317, 324, 532, 632; of
131-34, 619-32; reading meaning into irreality, 243, 641; of MU-picture, 311-13,
computer-produced, 599-600, 625; on 328-29, 525-26; of partides, 305; in radio
Rosetta stone, 165; Russian, 297, 379-80, news, 128; of reality, 15, 103-25, 128-29,
642; self-refs in, 431-37, 495-98, 501; see 184-85, 481, 493, 640, 725-26, 737, 739; in
also meaning, translation, etc. recursive processes, 128-29; of rules in
Lashley, Karl, 342, 343, 348 thought, 26-27
"last step", 462-63, 468 levels of description: of ant colonies, 315-
lateral geniculate, 343-44 33; of brain, 349-50, 382. 559, 570-77,
layers: of deception, 478; of messages, 584-85; of caste distribution, 319-29; of
166-71, 524, 703-4; of stability, 643-45 chess boards, 285-86; of errors, 294-95; of
leakage, between levels of science, 305-6 gases, 308; of human body, 285; of human
Legendre, Adrien-Marie, 92 Lehninger, psyche, 287; of mental processes, 568-73,
575-78, 584-85; of programs, 294-95, 380-
Index 772
81; of television screen, 285; see also lowest-level rules embodied in hardware,
holism vs. reductionism 685-86 Lucas, J. R. 388-90, 471-73, 475,
levels of meaning: in ant colonies, 319-27; 476, 477, 574,
in Contracrostipunctus, 82-85; of DNA, 577-78, 597
160, 531-32, 665; in Epimenides paradox, Lucas' argument: counterarguments to,
496, 581, 584-85; of groove-patterns, 83- 475-77, 577-78; merits of, 472;
84; of Mumon, 248; of MUMON, 266-67; summarized, 471-73 Lucas sequence, 139,
of music, 162-63; of neural activity, 575- 152, 174
77; of TNT-strings, 266, 270-71
levels of structure: of enzymes, 510-11, M-mode, see Mechanical mode
519, 521, 525-27, 532; of music, 525 MacGillavry, Caroline, 667
liar paradox, see Epimenides paradox machine dependence and independence,
Liberation (Escher), 57-58, 65 294 machine language, 289-300, 306, 381,
lightning calculators, see idiots savants 547 machines: not the sum of their parts,
limericks, 483, 736 389-90;
limitative results, in general, 19, 74, 609, reflecting on themselves, 288-89; self-
697, 699 Lincoln, Abraham, 454 assem
lines, geometrical, 19-20, 90-93, 100, 222, bling, 160, 486, 504, 543, 545
452, 456 MacLaine, Shirley, 285
LISP, 293, 381, 626, 652, 692, 738-39 macroscopic effects from micros
List of All Great Mathematicians, 404, 422 307 copic causes, MACSYMA, 615
Little Harmonic Labyrinth (Bach), 121-23, Madstop, 727
129, 130 Little Harmonic Labyrinth magnetic field and crystal, 140-43
(Dialogue), 127, 128-30, Magnficat in D (Bach), 549, 552, 558
149, 216, 610-11 hfagn#iicrab, Indeed, 560, 574, 581
Little Harmonic Labyrinth (of Majotaur), Magritte, Rene, 480-81, 489, 493-94, 627,
119-25 Littlewood, J. E., 564 700-2,
lizards, 108-9, 110, 115-17, 125 705-6; paintings by, see List of Illustrations
Lobachevskiy, Nikolay, 91 (xiv xviii)
local vs. global properties, 21, 160, 359, Mahalanobis, P. C., 565
363, 371-75, 543, 582-84, 678 main theses of book, 26, 46, 559, 714
localization of knowledge, in brains and Majotaur, 119-21, 123-25 malaphors, 657
programs, 342, 348, 365, 617-18 Mandelbrot, Benoit, 71 manifestations of
Lockwood, Anna, 700 symbols, 351 Mao Tse-tung, 433
logic, 19-24, 43-45, 99-100, 177-80, 181- mappings: charted, 85, 449, 533, 536;
97, 461-64, 618-19 induced, 668-69, 671-72
Loocus the Thinker, 477 marbles, rolling, 711-12 Margie-balloon
look-ahead trees, 151, 604-5, 611, 712 story, 675 Materialism, champions of, 27,
loops: bounded, 149, 410-14, 418, 440-41, 729 mathematical logic, history of, 19-24
444; free, 149, 424-25; in music, 150; in mathematical view of brains, 559
programming, 149-50, 410-14, 424-25, mathematicians, 458-59, 559, 566, 614
503, 632 lottery, 639-40 mathematics: done by computers, 573, 602,
Lovelace, Lady Ada Augusta, 25, 307, 598 614-15; foundations of, 19-24; reality and,
lower levels, see substrate, mental 54-58, 456-59
Mathews, Max, 607-8 McCarthy, John,
293 McCulloch, Warren, 134
Index 773
meaning: built on triggering-patterns of messages, 154, 158-76: in ant colonies,
symbols, 325, 327, 350; carried only on 350; in bottles, 167-69, 524; layers of, 166-
symbol level, 324-27, 330, 330, 709-10, 71, 524, 703-4; from nature, 408-9; see
codes and, 82, 158-62, 164-67, 267; of also frame message, inner message, outer
Contracrostipunctns, 82-85; of DNA, 160, message
531-32, 665; explicit vs. implicit, 82-85, messenger, for koans, 235-36, 238
158-76, 495-500, 583; in formal systems, messenger RNA, see mRNA meta, 216-17,
see interpretations; intelligence and, 158, 224 meta-agnosticism, 114 rneta-analogy,
162-64, 170-76, 501, 661-62; intrinsic, see 673-74 meta-answer-schema, 463 meta-
meaning, explicit vs. implicit; location of, author, 607-9, 726 metabook, 22
153-57, 158-76, 408-9, 582-84; as meta-descriptions, 656-57, 674 meta-
multidimensional cognitive structure, 582- evidence, etc., 693-94 Meta-Genie, see
84; multiple, 8, 10, 52-53, 82-85, 94-102, Genie meta-hiccups, 726 meta-intuition,
153-57, 158, 172, 266-67, 271, 409, 447- 605 META-JOSH1-), etc., 443
48, 524, 532, 666, see also disambiguation; metaknowledge, 364
in music, 83, 160, 161, 162-64, 167, 172, metalanguage, 22, 184, 194. 248, 270, 514
174-75, 227, 582-84, 626-27, 676-77, 699- metalogic, 23, 676
700, 704; objective, see meaning, explicit metanrathematics, 23, 579; reflected inside
vs. implicit; as optional high-level feature, TNT,
571; passive vs. active, 51-52, 94, 97, 100, 449-50
102, 191-92, 266, 267, 271, 456; purpose Metamorphosis (Escher). 14-15 metaphase,
and, 321-32; rooted in isomorphisms, 49- 666-57 metaphor, 672 meta-proteins, 533-
53, 87-88, 94, 267, 337, 350; unnecessary 34
on evolutionary time scale, 321-22 metarules, etc.: in chess, 687-88; in
meaningless vs. meaningful interpretations, intelligence,
51, 88 26-27, 559, 684-85
meaninglessness, in art and music, 699- meta-search, 397 meta-symbols, 560
700, 704-3 metatheorems, 193-94 metatheorv,
meat grinders, 414 formalized, 194 meta-TNT, etc., 442-43,
Mechanical mode, 38-39, 65, 194, 221, 533-34 meta-wishes, see wishes meteorites,
613-14 167, 172 methylation, 540-41
mechanization of thought processes, see metric, mental, 613, 614; see also
Al, formal systems, etc. conceptual nearness
meiosis, 665, 672 Meyer, Leonard B., 167, 704
melodies: recall of, 363-64; time-shared, Michelangelo, 642 microprogramming, 295
385 minds: overlap of, 376; programmability
memory, in computers, 288-89, 546, 616 of. 302, 679, see also Al, paradox of All,
memory dump, 381 Tesler's The
men vs. women, 477, 595-96 orem, nonprogrammability; thoughts and,
Mendel, Arthur, 3, 28 369-90; two ways of creating. 390; vs.
Mental Arithmetic (MagrittO, 627 brains,
mention, see use vs. mention 309, 575-77; see also brains, intelligence,
Menzel, Adolph von, 4-5 etc. mini-vocabulary, 647
Meredith, Marsha, 625 Minsky, Marvin, 373, 388, 644, 679, 722
Mergenthaler, Otto, 630 mirroring, see isomorphisms,
message-L sing languages, 662-63 representation misspelling and computers,
Index 774
297-98 MIU-numbers, 264-67; see also music: composed by computer, 25, 595,
theorem-numbers MIU-system, 33-41, 46, 597, 603,
47, 48, 52, 191, 260-67; 607-9, 626-27, 676-77; dimensions of, 175;
as model for TNT, 439-42, 466; table of mathematics and, 227, 555, 560; modern,
rules of, 260 156-57, 163-64, 174-75, 699-700, 704;
MIU+MU-system, 466 Mobius Strip I notation
(Escher), 29-30 Mobius Strip II (Escher), of, 552-59, see also SMUT; semantics of,
276 modes of fugue-listening, 282-84 83,
modularity, 149-50, 615-18, 628, 677-78; 162-64, 167, 174-75, 582-84, 626-27, 676-
see also 77;
localization, local vs. global properties superhuman comprehension of, 172, 679;
modulation, 10, 121-23, 129-30, 466, 501, syntax of, 121-23, 129-30, 227, 626-27,
717, 739 737; to
modules in brain, see symbols Modus break phonographs by, 75-78; to infiltrate
Ponens, see detachment molecular biology, phonographs by, 487-88; see also fugues,
504, 514-48 Mondrian, Piet, 700 canons, pianos, flutes, etc.
monkeys, vision of, 345-4b music box, preprogrammed, 677
Monod, Jacques, 161 morphogenesis, 539, Musical Offering, The (Bach), 4-10, 86,
543-44 Mosaic II (Escher), 61-63 665, 666,
mountain-car wreck story, 338-39, 361, 719, 720, 724, 727, 739-42 mutations, 295
365 Mozart, W. A., 649, 702
mRNA, 517-20, 522-25, 527-28, 530-33,
536, N, see number theory
545, 547, 662, 663 Najunamar, Z., 549-52 Nansen, 248-49,
MU, as possible theorem of MIU-system, 253, 255
33-41, natural language utterances as programs,
229, 259-61, 265-67, 271, 708 629 natural numbers: defined, 54, 204;
MU, Zen word, 233, 241, 246, 254, 259, generalized;
272, 311, 453-56; postulates for, 216-17; see also
312-313, 328 number
MU-LOOP (Floc P), 424-25, 441 Mu theory, numerals, TNT, prime numbers,
Offering, A, 272, 628 muoperator, 424 etc. near misses, see "almost"-situations
MU-picture, 283-84, 310-13, 327, 328-29, nearly decomposable systems, 303-6
525-26 negation, 70, 71, 183, 191-92, 210-11, 214,
MU-puzzle, 33-41, 259-61, 509-10, 613-14 545 negative space, 62-63, 66-68, 72; see
multifurcation of TNT, 467 also figure and ground
multiple representations, 616-18, 670-71, nested movies, 184-85
674 multiplication, 54-56, 64-65, 206, 409, nested works of art, 15, 106, 700-1, 705-6
455, 566, 567 nesting, 127, 138-41, 184-85, 660; see also
Mumon, 242, 246, 248-49, 253, 259, 260, recursion
272; neural networks, see symbols Neuroneater,
commentaries by, 246-49, 252; poems by, 382
246-49, 252, 272 neurons: compared with ants, 315, 325,
MUMON, string of TNT, 265-67, 271, 339-40;
441-42 Mumonkan, 246 described, 339-40; Eudid's, 60; faultless
func
Index 775
tioning of, 575-77; firing of, 83, 340, 316, nuclei: atomic, 303-4; cellular, 514, 517,
343-45, 347, 350, 357; as an inviolate 518
level, 302, 677, 686, 691-92, see also nudeotides, 514-17, 519, 522-24, 530, 540-
inaccessibility; not controllable 41; first letters of, 231, 517, 666
consciously, 302, see also inaccessibility; number theory: applications of, 278-29;
on-center and off-center, 343-44; in core of, 100, 407; Crab and, 551-58, 560,
retina, 343-44; simple, complex, 562, 573-74, 579-81; demise of, 228-29,
hypercomplex,- 426; formalized, see TNT; informal (N),
neurons (continued) 54-660, 204, 228; nonstandard, 100, 452-
344-45, 346, 347; as summing inputs, 316, 59; primitive notions of, 204-9; as scaled-
340, 575-77, 677 off mini-world, 569; soothing powers of,
neurosurgery, 309, 313-14, 618. 678 New 391-404; "true" version of, 458-59; typical
Yorker, The, 641-42 sentences of, 204-5; typographical, see
nickelodeon, 500; see also jukeboxes nodes TNT; as universal mirror of formal
and links, 370-71, 652-54 noise in vacuum, systems, 260-65, 270; used and mentioned,
82 458
nondivisibility, 73-74 numbers, nature of, 54-58, 452, 458
nonequilibrium thermodynamics, 693 numerals, 205-6, 213; vs. numbers, 264
Noneudid, 91-92
nonexistence, 254-55, 698, 725; see also object language, 22, 184, 248
Tumbolia nonproducible numbers, 265 objectivity, quest after, 479, 693-96
nonprogrammability: of creativity, 570-71, Oborin, Lev, 162
620, 673; of emotions and will, 677, 684- octopus cell, 345
86; of Godelization, 472-76; of Oin, Eta, see Eta Oin
intelligence, 26-27, 471-73, 597-99, 601; of Oistrakh, David, 162
irrationality, 575-77; of jumping out of the Okanisama, 232, 234, 237, 238, 239, 241,
system, 37-38, 477-78, 674-75; of ordinal 242 Old Ba. Ch., 726
names, 476; of soul, 574-75; of world Old Bach, 4, 28, 460, 481-83, 738, 739 ca-
chess champion, 151-52, 674; see also consistency, 459; see also or-inconsistency
people vs. machines, essential cs-incompleteness, 221-22, 421, 450-51 co-
incompleteness, Todeization, paradox of inconsistency, 17, 223, 453-55, 458-59 1-D
Al, TC-battles, 2-D vs. 3-D, etc. vs. 3-D, 519-21, 616-17
non-self-assembling viruses, 542-43 non- open-ended searches, see potentially
self-descriptive adjectives, see endless
heterological adjectives searches, nontermination, unpredictable but
nonsense: based on sense, 378-79; guaranteed termination, loops, free, FlooP,
computer-generated, 620, 621-22, 625-26; etc. operating systems, 295-96, 300-31, 308
human-generated, 621-22 operators and operons, 544-45 oracles, 567
nontermination, 408, 425-30; see also orchard analogy, see information, depth
potentially endless searches, Hoop from surface order and chaos: in ant
nontheorems, see theorems vs. colonies, 316-17; in number theory, 393,
nontheorems normal science, 660-61 395, 398-402, 406. 408-9, 418;
nouns, most common in English, 630 self-awareness and, 406
novelty, and jumping out of the system, Order and Chaos (Escher), 399 ordinals,
475 462-64, 475-76 organ point. 329-30 origin
of life, 548
Index 776
original (as opposed to copies), 504 pattern recognition, see Bongard problems,
originality and machines, 25-26, 606-9 conceptual skeletons, vision by computers
ORNATE NOUN, 131-33 outcome, 184 patterns on all levels, 674 Peano, Giuseppe,
outer messages, 166-71, 174-76, 501, 524, 20, 216-17 Peano arithmetic, 100 Peano
704 OUTPUT (BlooP), 410, 411 postulates, 216-17, 224 pearl and oyster,
overlapping genes, 524-25 17, 438 Penfield, Wilder, 342-43 Penrose,
overview capacity, 613-14, 678; see also Roger, 15
jumping out of the system people vs. machines, 25-27, 36-38, 151-52,
P-system, 64, 73-74 388-90, 471-73, 475-77, 559-62, 567-75,
padding, 402-3 577-79, 595-99, 606-9, 621-23, 680, 684-
pages, in computers, 289 86 peptide bonds, 523
palindromes, in molecular biology, 201, perception: visual, 97-98; and Zen, 251
667 Palindromi, 353-54, 634-37, 643, 644 Perfect items, 3, 75-79, 85, 406, 424, 486,
Pappus, 606-7 536 perfect numbers, 416, 418
paradigm shifts, 660-61 phages, see viruses
paradox: of Al, 19, 26-27, 620, 673, see phenotype, see genotype and phenotype
also Tesler's Theorem; in an, see Escher, 0(174, 176, 524-25
Magritte, Phonographs, see record players
Cage; of credibility through fallibility, 564; phonons, 304
of God and the stone, 478; in mathematics, photocopy machines, 499 photons, 142-46,
17-24, 580-81; of motion, see Zeno's 258
paradoxes; near misses, 612, 691; physics laws of: as basis for choosing
resolutions of, 116, 196-97, between rival mathematical theories, 100,
245, see also MU, Tumbolia, jumping out 456-57; as blocking infinite regress, 170,
of the system; of self-consciousness, 389; 685; as formal system, 53-54;
of the Typeless Wish, 115-16; in Zen, 249- inconsistency and, 95-96, 99,
55; see also contradictions, inconsistency 584-85; intuitive, 362-63, 711; levels and,
parallel postulate, see fifth postulate 303-5, 307-8, 693; no escape from, 477,
PARRY, 300-301, 599-600, 677 575;
parsing of natural languages, 588-93, 630- Reductionist s Dilemma and, 522, 709;
32; see also grammar, language underlying consciousness, 575, 685, 710
partial recursivity, 430 IT, 277, 306, 408, 415, 421, 546, 568, 605,
particles, elementary, 54, 140-46, 258, 303- 673,
5, 309,522 728
partitions, mental, 671 piano postulates, 552-53
parts, 303-5; see also reductionism Pascal, pianos, 3-4, 302, 305, 633-34, 700, 726;
Blaise, 24, 25, 600 inverted, 681-83
pathways: in ATN's and RTN's, 131-34, Pickruick papers, The (Dickens), 24. 326,
150; 595, 598 pinball machine, 307
chemical, 528-29, 544-45, 663-64; pipe Dream, 703
conditional on circumstances, 383-84; pipes, 480-82, 486, 488, 493-94, 521, 638,
goal-oriented choice of, 227, 609-15; as 701-3 planets and satellites, 353-54
incorporating knowledge, PLANNER, 629-32
beliefs, 378-79; morphogenetic of T4, 539; plurals, 354
plausible vs. implausible, 383; potential, in pocket calculators, 568-70, 616, 678, 710
brain, 281 pointers in computers, 289-90, 619
Index 777
points (geometrical), 19-20, 90, 92-93, 100, 211-12, 413, 551-58, 615; differences of,
207, 393, 395-98, 400, 416; sums of, 393-96,
222, 452, 456 400, 414
Polanyi, Michael, 574 polarons, 304-5 primitive recursive truths, 407
polypeptides, 523, 525, 528 polyribosomes, primitive recursivity, 406, 407, 414-20,
526-28 422, 424, 429-30, 440, 441, 444, 451, 466,
Pons Asinorum proof, 606-7, 669 Pool B, 472
418 Principus Mathematiaa, 18-19,• 21, 23-24,
Pool F, 427 228, 618-19
popcorn, 104, 124-25 popping, 127-35, Print Gallery (Escher), 15, 714-17
184-85 popping-tonic, 105-6, 116-17, 125 printer of computer, 301, 307
porridge, 431 problem reduction, 609-13; self-applied,
ports of access, 670-71 Post, Emd, 33 613
post-ending endings, 392, 403 postal problem spaces, representation of, 611-13
system metaphor, 663 procedures, 132-34, 150-51, 292, 410-15,
postulates of geometry, 90-91, 92-93, 407 418-20, 424-28; chains of, 413-14, 415,
potentially endless searches, 396, 400-402, 418
425, processors (computers), 504, 513, 547; see
400-401, 444, 582-83 also. central processing unit
pq-system: completeness and consistency producible numbers, 264-65, 269-70
of, 101; program space, 299
decision procedure for, 47-49; expressive programs: in Analytical Engine, 25; in
weak BlooP and FlooP, 410-15, 424-26; Blue,
ness of, 101, 221-22, 407, 417; horse-apple Green, Red, see Blue, Green, Red
happy interpretation of, 51, 88, 215; isomor programs; chess-playing, see chess
phisms and, 49-53, 158, 625; modified, 87- programs; constructed by programs, 589,
88, 629-32, 664; as data, 293, 692; for
92-93, 102; surprise interpretation of, 52- determining enzyme function, 521-22; for
53, determining phenotype, 532; for
94 determining tertiary structure, 521-22;
predicate calMus, 609 families of, 503, 546; for generating
predicates, number-theoretical, 208-9 theorems, 471-73, 578, 615, 617-18; high-
predictable termination, 400, 407, 409-18, level comparison of, 380-81; for naming
420, ordinals, 476; recursive structure of, 149-
441, 582; see also unpredictable but 50; secondorder, third-order, etc., 476; self-
guaranteed modifying, 152, 692; self-reproducing,
termination, terminators 498-504, 547; for translating programs,
Prelude, 337, 383, 460, 686, 737, 739 291-94; vs. data, 499, 513, 531, 546-48,
preludes and fugues, 280-84, 335; see also 616-17, 630, see also use vs. mention; vs.
fugues, Well-Tempered Clavier programmers, 306, 734-37; see also
premise, 184 computers, computer languages, Al
Preprocessing, 647, 650, 659 President v. programs, etc.
Supreme Court, 692 Prokofiev, Sergei, 150
primary structure: of proteins, 519-22; of ty pronoun reference, 587, 591, 592
poenzymes, 511, 512 proof-pairs, 416, 438-43, 446-47, 450-51,
prime numbers, 58-59, 64-67, 72-74, 149, 452-54, 466, 468, 469
Index 778
proofs: nature of, 18-24, 58-60, 88-93, 192- cleotides, bases, base-pairing Pythagoras,
97, 227-28, 458-59, 578, 707-8; as never 418, 556-57
absolute, 191-94; of proofs, 192-93; vs.
derivations, 35, 193, 194-95 Q(n), 137-38, 152, 265, 409
prophase, 665-666 quantifiers, 207-9, 210, 211-12. 214, 217-
Propositional Calculus, 181-97; embedded 19 quantum mechanics, 19, 54. 140-46,
in TNT, 195, 197, 207, 215-17; as an 350, 455,
epiphenomenon, 578; interpretations of 457, 699; see also partides
symbols of, 186, 189, 191-92; rules of Quantz, Joachim, 4 quarks, 304, 305, 350
inference, justified, 188-89; rules of quasi-isomorphisms, see isomorphisms,
inference, presented, 181-87; rules of fluid quatemarv structure, 525 Questions
inference, table of, 187; streamlined, 193- and Speculations, 676-80
94; variants of, 195; ways to improve, 193- Quine, Willard Van Orman, 435, 446, 449,
94; 196-97; weaknesses of, 195-97, 578; 699 quining, 431-37, 445, 446, 449, 497-
well-formedness in, 181-83 99, 531 QUIT (BlooP), 412
proteins, 517-18, 544-45; as procedural quotation, 426, 431, 433-37, 496-97, 702,
knowledge, 616-17; as programs, data, 738 quotation marks, 33, 434, 498, 499,
interpreters, processors, 547; see also 702
enzymes
prototype principle, 352 Rachmaninoff, Sergei, 150
provability, 18, 101 RACRECIR, 738
Prudence and Imprudence, 191-92, 229 radio broadcasts, 128, 163, 169, 353, 478,
pruning, explicit vs. implicit, 286 545, 720 Ramanujan, Srinivasa, 562-66,
pseudo-epigenesis, 531-32 614 randomness, 408-9, 620, 673, 712
psychic powers, 693; see also ESP rational and irrational numbers, 140-42,
Ptolemy V Epiphanes, 165 418, 452,
Puddle (Escher), 256 556-57
pulling-out (mechanisms), see decoding rational vs. irrational, in human head, 575-
(mechanisms) 77 rats in mazes, 342
punctuation, 33, 268, 440, 510, 512, 520, Rauschenberg, Robert, 703
524-25 r.e. sets, see recursively enumerable sets
purities, 506-7, 514, 516, 534; see also reading frame shift, 154, 525 reality, nature
nudeotides, bases, base-pairing of, 409
purposeful vs. purposeless behavior, 320- rearrangement of parts, 78, 333-35, 484;
22 see also self-assembly
pushcorn, 124-25 reasoning about formal systems, 38-39, 66,
pushing, 127-34, 184-85 260-72, 438-52, 465-71, 579-81
pushing-potion, 105-6 reasoning by programs, 569-70, 577-78,
Pushkin, Alexander, 124 586-93,
puzzles. 8, 33-35, 62-63, 67, 73, 137, 182, 606-7, 609-11, 614-15, 618-19, 628-32
212, recognition: molecular, 540-41; visual,
215, 220, 401, 415-17, 425, 442-43, 444, 346-48,
512-13, 564-565, 609, 621-23, 646-60, 689 646-62; vs. production, 648-49
pyramidal family of theorems, 221-25, recognizable forms, 68
450-53 pyrimidines, 506-7, 514, 516, 534: recombination, 657, 665-69
see also nu
Index 779
record players: alien-rejecting, 487-88; recursive structure of ideas, 386-87, 560,
EpsilonZero, 486; family of, in Crab's 621,
jukebox, 154-57; Grand Self-assembling, 644-45, 650, 656-57, 669. 671-72
see record player Epsilon-Zero; as Recursive Transition Networks, 131-34,
information-revealers, 158-61, 164; 136, 145,
intrinsic vulnerability of, 75-78, 102, 424, 150, 620-21
470, 483-86, 536, 543, 584, 721, see also recursively enumerable sets, 72-74, 152,
Tbdelization, TC-battles, etc.; likened to 191, 265,
formal systems, 84, 85; low-fidelity, 77, 269
85, 101, 406-7, 470; Omega, 78, 468, 483- recursively related notation-systems, 475
84; Numbers l, 2 ... etc., 76-77; Tortoise- Red Programs, 427-28 Reddiag [NJ, 428,
chomping, 483, 487-88; two-channel 429 redness, subjective and objective, 710
monaural, 634, 669; see also jukeboxes reductionism: defined, 312; proteins and,
records: defective, 102, as information- 520-22;
bearers, 158, 160-61, 164; as labyrinths, see also holism vs- reductionism, sealing-
120-24; with multiple melodies, 154-57; as off Reductionist's Dilemma, 522, 709
phonographbreakers, 75-78, 83-85, 271, reentrant code, 387
406-7, 424, 469, 484, 486, 536, 543, 584; refrigerators, see record players, low
smashed, information in, 161; in space, fidelity registers, in computers, 289
162-64, 172, 174-75; of WellTempered relativity, 19, 96, 100, 680 Relativity
Clavier given to Crab, 275, 278-80 (Escher), 97-98 relevant implication, 197
records and record players, likened to renormalization, 142-46, 258, 304-5, 309
cellular constituents and cells, 83, 158-64, repeatability, see Godelization,
167, 175, 536 Todelization, diag
recursion: avoidance of infinite regress in, onal method, Escherization, TC-batteries,
127, 134-35; avoidance of paradox in, 127; answer-schemers
defined, 127-29, 131-35; elementary representability, 407, 417-18, 430, 441,
particles and, 142-46; fantasy rule and, 443, 444,
184-85; in game-playing programs, 150-51, 451, 466, 468, 579-80
604-5; indirect, 134, 137; in language. 130- representation of knowledge: in All, 569,
34, 588, 591, 592; in music, 121-23, 129- 615-21, 626-32, 641-59, 664-65, 668-72; in
30; and unpredictability, 152; see also brains, see symbols, localization
nesting, levels, distinct vs. similar, level repressors, 544-45 Reptiles (Escher), 116-
confusion, etc. 17
recursive acronyms, 113, 133, 134-35, 738, Requirement of Formality, 33, 52, 65
742 recursive diagrams, 135-37 recursive retrogression, 8-9, 81, 146, 200, 208, 500-
figures, 67-70. 72, 73 recursive formula, of 501, 549, 666-68, 723-25, 737-38
thinking, 560 return addresses, 128, 133 revelation, 160-
recursive functions, 136-40, 152, 430, 455; 61, 175 ribo, some, 236
see also ribonucleic acid, see mRNA, rRNA, tRNA
genera] recursivity, primitive recursivity. ribosomal RNA, see rRNA
BlooP, ribosomes: as models for Al, 662, 663;
HooP molecular
recursive graphs. 138-43 recursive canons and, 527-28; need for in DNA's self
sequences, 135-38, 139 recursive sets, 72- rep, 530; origin of, 528, 548; as self-
74, 152, 191 assembling
Index 780
objects, 485-86, 542; structure of, 528; as
153-57; in Bongard world, 650-53, 657,
translators of Genetic Code, 485, 518-19, 660,
522-25, 547; in Typogenetics, 512 ricercar,
664; of butterflies, 147, 369; of demi-
defined, 7 RICERCAR (E), 7, 727-42 doublets,
Ripplad Surface (Escher), 256-57 RNA, 669; elusiveness of, 14619; of Escher
see mRNA, rRNA, tRNA RNA drawings,
polymerise, 527, 530, 544 robot in T-maze,147; of human and machine intelligence,
711-13 Rogers, Hartley ropes, thin and 337,
thick, 229-30 379, 679-80; of human minds, 341-42,
Rose, Steven, 342 369-72,
Rosetta stone, 165, 166 Rosaak, Theodore, 375-77, 382; intensionality and, 338; mech
574 Rousseau, Henri, 680 anisms underlying perception of abstract,
Royal Theme, 4-10, 96, 719. 739-40 646-62, 665-69, 671-72; overlooked, 614,
rRNA, 528 674; of programs, 380-82; in self-refs and
RTN's, see Recursive Transition Networks self-reps,
rule-less systems, 598, 685; see also formal
500-4; of semantic networks, 371; of
vs. in formal systems translations between languages, 372, 379-
rules: arithmetical vs. typographical, 262-
80; universality of intelligence and, 158,
64. 269; 501; vs.differentness, 153-57; visual, 344-
flattened into strings, see theorems vs. 48, 662; see also copies, isomorphisms,
rules; intelligence and, 26-27, 559, see also
conceptual mapping sameness-detectors,
brains and formal systems see Sams Sams, 650-53, 657, 664
rules of inference: of C-system, 65; Samuel, Arthur, 604-5, 684-86
compared with enzymes, 509-10, 513, 531; Samuel's argument, pro and con, 684-86
defined, 34-35; San Francisco Chronicle example, 351
derived, 193-94; of MIU-system, 34, 260; sand castles, 725-26
of P system. 74; of pq-system, 47; sanity vs. insanity, 192, 696 satellite-
proposed, 66, 221; symbols, see splitting-off satoti, see
of Propositional Calculus, 187; recursive enlightenment scale, cyclic, see Shepard
enumerability and, 152; run backwards, 48-tones Schmidt, Johann Michael, 27
49, 182; of 310-system, 263; of TNT, 215, Schnirelmann, Lev G., 394 Schdnberg,
217-20, 223-25; of tq-system, 65; of Arnold, 125 Schrodinger, Erwin, 167
Typogenetics, Schweikart, F. K., 92
509-10 science: and Bongard problems, 659-61;
rules of production, see rules of inference
self-applied, 699
run-of-the-mill sets, 20-21 Russell, Scott, Robert, 366
Bertrand, 18-24 Russell's paradox, 20-21, scripts, collage of, 168-69 sealingoff, 305,
685 309, 350, 534 secondary structure, 521,
525
self, nature of, 316-17, 327-28, 384-85,
Saccheri, Girolamo, 91-93, 99, 452, 456 387-88, 695-96, 709-14
Sagredo, see Salviati, et al self-assembly, spontaneous, 485-86, 542-
Salviati, Simplicio, Sagredo, 408-9, 478- 43
79, 673, 694 self-awareness, 406, 479, 573
sameness: of ASU's, 375; of BACH and self-descriptive adjectives, see autological
CAGE, adjectives self-engulfing, 489-94; failed;
Index 781
490, 492; total, 493 self-knowledge, Shakespeare, Wm., 96, 595, 598, 608, 736
possibility of, 696-98, 706 self-modifying Shandy Double-Dandy, 611 shared code,
games, 687-88 self-monitoring, 328, 385,- 387
387-88, 697, 713 self-perception, 695-98; Shepard, Roger, 717-19 Shepard tones,
vs. self-transcendence, 478 717-19
self-programmed objects, 685-86, 691-92 shielding of lower levels, see
self-proving sentences, 542-43 self-quoting inaccessibility SHRDLU, 586-93, 599,
sentence, 426, 496-97 627-32, 674 Shuzan, 251
self-reference: Bach and, 86; banning, 21- Sierpifiski, W., 404
23; as cause of essential incompleteness, signals, crisscrossing, 322-23 signature,
465, 470-71; visual, 347-48 Silberescher, Lowen, 394
focusing of, 438, 443, 445-48; Gddelian, Silbermann, Gottfried, 3, 4 silver, 173
17-18, 271, 447-49, 497, 502, 533, 667, Simon, Herbert A., 303, 305
738; indirect, 21, 85, 204, 436-37, 502, simple, complex, hypercomplex cells, see
667, 738-39; many leveled. 742; near miss, neurons Simpficio, see Salviati
437; Quine method, simplicity, 172, 560, 615
431-37, 445-46, 449, 497-99, 531; by simulation: of entire brain, 572-73; of
translation, 502 neural networks, 571-72
self-reference and self-replication, Six-Part Ricercar (Bach), 4-7, 719, 739-42
compared, 530, 533-34, 541-43 skater metaphor, 412-13
self-referential sentences, 435-37, 477, skeletons (recursion), 140-41; see also
495-99, 501 bottom 4
self-rep: by augmentation, 503; canons and, skimming off top levels, 309, 325, 326,
501, 503; differentiating, 546; epigenesis 358-59, 568-79
and, 160; by error message, 503; inexact, slinky, 337
500-503, 546; by retrograde motion, 500- slippage, conceptual, 633-40. 641-44, 654-
501; by translation, 501; trivial, 499; 56,
typogenetical, 512-13 self-snuffing, 701-2 672; see also conceptual mapping Sloth,
self-swallowing sets, 20 633-40, 643. 681-83, 722 Sloth Canon
self-symbol, 385, 387-88, 709; free will (Bach), 9, 666, 683 Sloth Canon
and, 710-14; inevitability of, 388 (Dialogue), 738
self-transcendence, 477-78, 479 self- SLOTHs, 684-719
unawareness, irony of, 328, 330, 331, 630 slots, 645, 650-53, 656-57, 668 Smalltalk,
semantic classes, 621, 630 662
semantic networks, 370-72; see also smart-stupids, 721-42
concept network Smoke Signal 67, 702
semi-interpretations, 189, 196 SMUT, 6, 80, 155-57, 202, 682, 718, 740
semiformal systems, 216; see also soap cake, 497
geometry, Euclidean soft-louds, see pianos
senseless loops, 679 sentences in TNT, software and hardware: in brain, 346, 356-
208-9 Sentences P and Q, 436-37 57, 686, 709; defined, 301
sequences of integers, 73, 135-39, 173-74, soldier ants, 318
408 set theory, 20-23 Sonata for Unaccompanied Achilles, 502-3
sets F and G, 73 Sonata for violin and clavier in F minor
1729, 204-5, 210-11, 345, 393, 551, 564- (Bach), 162
855 Shadows, The (Magritte), 480
Index 782
Sonatas and partitas for unaccompanied mention, syntax vs. semantics style, 148,
violin (Bach), 62, 63, 70-71, 257, 502 371
song, self-reproducing, 500 SUB (TNT-formula), 444-45
"Song Without Time or Season, A", 483 subbrains, see subsystems
songs, recombinant, 607-8 subdescriptions, etc. 650-51; see also
sonnets, 596, 608, 736 recursive structure of ideas
soulism, 385, 472, 574, 597, 686, 708 subframes, etc., 644-45; see abo recursive
spaces: abstract, 457; of behavior, 306, structure of ideas
307, 363, 621; of integers, 502 subject vs. object, 698-99; see also
spare tire, 670 dualism, use vs. mention, symbol vs. object
Sperry, Roger, 710 subjunc-TV, 635110
Sphex wasp, 360-61, 613-14 subjunctive instant replays, see instant
spiderwebs, 371-72, 617 replays subjunctives, see counterfactuals
splitting-off, 353-54, 664; see also classes suborganisms, see subsystems subroutines,
vs. instances 150, 292, 471, 677 substitution notation in
stacks, pushdown, 127-35, 136, 610-11 TNT, defined, 224 substitution relationship
starting positions in Art of Zen strings, (ingredient in Godel's proof), 443-45, 497
239-40, 241
State of Grace (Magritte), 481 substrate: of Epimenides sentence, 581,
statistical mechanics, see gases 584-85; mental, necessity of, for analogical
Sterner, Geo., 166-67, 642-43, 670 Stent, thought, 569-71; mental,
Gunther, 514 noninterpretability of, 570-71; mental,
stomach processes, 572 simulation of, 571-72; of proteins, 529
strands: of DNA and RNA, 514-18; in subsystems of brain, 385-88, 725
Typogenetics, 508-13, 514 subtraction, 52-53, 411-12
Strange Loops: abolishment of, 21-23; with sufficiently powerful systems, 86, 101,
Babbage and Turing, 737; in Bach, 10, 406-7, 430, 460, 530
719; in Central Dogmap, 534; Suites for unaccompanied cello (Bach), 70-
consciousness and, 709-10; contrasted with 71 summarizing string, 221-22, 450-51
feedback, 545, 691; defined, 10; in Escher, superconductivity, 304-5 superinielligence,
10-15, 714-17; in Godel, 15-17, 24; in 679
government, 692-93; in language, 22, 691; supernatural numbers, 223, 453-56, 458-
in the mind, 27, 691-92; in molecular 59, 467 supernatural proofs, 454-55
biology, 231, 532-34, 545, 546-48; in supertangling, 688
Principia Mathematica, 24; surprise surrealism, 700
element necessary in, 691; in TNT, see Sussman, Gerald, 664
TNT, introspection of; see also Tangled Swieten, Baron Gottfried van, 6-7
Hierarchies, level-mixing, levelconflicts, Switcheroo, Q. q., 187
level- confusion, etc. symbol-level description of brain, 349-51
street nuisances, 25, 726-27, 729 symbol vs. object, 699-706; see also
stretto, 314, 738 subject vs. object, use vs. mention, etc.
String Manipulation Rules, 240-42 symbols: active vs. passive, 324-25, 327-
strings: defined, 33-34; folded, 234-44, 427 28, 337-38; Al realizations of, 662-63, 665;
structure vs. function, 522, 670-71; see also in ant colonies, 324-28, 330; borderlines
use vs. between, 354-57, 359-60; as brush strokes,
351; compared with neurons, 350, 371;
Index 783
compared with ripples, 356-37; conceptual IrC_battles, 75-78, 406-7, 424, 467-70,
scope of, 350-51; dormant, 327, 349, 355- 478, 483-88, 536-41, 721
56, 384; form of, 348, 356-57, 361; free tea. 153-54, 231, 275, 322-23, 333, 549,
will and, 712-14; funneling and, 347; of 557, 558, 561
insects, 360-61; invariant core of, 349; teleological vs. evolutionary viewpoint,
joint activation of, 351, 354-56, 359, 361, 320-22 telephone calls, 61-63, 127-28. 161;
365, 584-85, 664-65, 675; message- obscene, 431, 437
exchange and, 350, 371, 662-63; modes of telephone systems, 296, 663 telephones,
activation of, 349-57, 361; need of, for 302
originality, 609; neural substrate of, 356- television, 285, 348, 478, 484, 487-93,
57, 570; no access to substrate by, see 634-40 telophase, 667-68
inaccessibility; overcrowding of, 358; templates: for Bongard problems, 650-53,
overlap of, 348-49, 356-57; potential, 355- 656-57; vs. instructions, see instructions vs.
56, 382-84; universal, 375-76; vs. neurons, tem
348, 356-57, 361; vs. signals, 325-27, 349- plates, programs vs. data
50 tension and resolution, 121-23, 129-30, 227
synapses, 339 tentativity, 646, 651, 654-56, 672
syntax vs. semantics, 626-27, 630-32, 676; termination tester, 425-29 terminators
see also form, syntactic vs. semantic (FlooP), 425-29 terms (TNT), 206-7, 213,
system, boundaries of, 37-38; see also 214
jumping out of the system tertiary structure: of proteins, 519-22, 525-
System crash, 116 27; of
systems, reliable vs. unreliable, 307 tRNA, 524; of typoenzymes, 510-11, 512,
519 Tesler, Lawrence G., 601
T (Tarski's formula), 580-81 Tesler's Theorem, 601, 623; see also
't'-concept, 'h'-concept, 'c -concept, 326 T- paradox of Al, jumping out of the system,
even phages, 540, 542 T-level, see tangled essential in completeness
level table of nontheorems, 66 Tagore, tesselations, 68, 69, 198, 667
Rabindranath, 169 tests vs. functions in BlooP and FlooP, 413,
Tangled Hierarchies: in art, 704; defined, 418 text-handling by computers, 301 T4
10; of phage, 537-41
genetics, 532-34, 546-48, 688; of meta "the" 586, 629-30
mathematics, 458, 532-34; of mind, 691- theorem-numbers, 264-67, 269-70, 440-43,
92, 709-10, 719; near miss, 691; in 451 theorem-proving, mechanical, 602,
Propositional 609, 617-19 theorems: defined, 35;
Calculus, 194; of self-modifying chess, systematic enumeration of,
688; of 39-40, 48, 471-73, 578, 615, 617-18; vs.
Tortoises reasoning, 177-80; of non
Typogenetics, theorems, 39-41, 66-67, 70, 71-73, 190-91,
513, 688 416-17, 560, 579-80; vs. rules, 43-45, 193-
tangled level, 688 Tanguy, Yves, 700 94,
tape recorders, 485, 519, 523, 525-28 509-10; vs. Theorems, 35, 193; vs. truths,
Tarski, Alfred, 579-81 Tarski-Church- 49-54, 70-71, 86-102, 190-97, 213, 221-23,
Turing Theorem. 561, 581 Tarski's 228-30, see also consistency, completeness,
Theorem, 579, 580-81, 584-85, 697 Taube, Godel's Theorem, consequences of
Mortimer, 574 Taurinus, F..4., 92
Index 784
Theseus and Ariadne, 130 thinking, speed Tortoise: answer-schemas and, 475; Carroll
of, 679 paradox and, 46, 193, 684-85, 693; Crab
30, as possible MIU-number, 265-67 "this Canon and, 204, 666-67; degenerate
sentence", 436, 495-98, 499 thought, solution by, 669; Diophantine equations
substrate of, 559 and, 4590; as harpsichord, 502; initial letter
3n+1 problem, 400-2; see also wondrous of, 231, 507, 667; mentioned, 102, 170,
and un wondrous numbers 267; origin of, 28-29; picture of, 42;
Three Spheres If (Escher), 258 310-system, recursion and, 128-30; 149; use of words
261-67 by, 181; vs. Crab, 84-86, 271, 406, 424,
Three Worlds (Escher), 247, 256 thymine, 467-69, 540, 543; Zen string made by, 272;
see nucleotides tilde, 183, 191-92, 554 in ZET cycle, 94-95
timesharing, 296, 354-55, 387, 730 Tortoise-pairs, 416, 441, 448; compared
TNT, 204-30; absolute geometry and, 451- with proof-pairs, 441, 448
52; austere, 211, 214, 216, 268, 442, 534; Tortoise property, 395-97, 415-18, 425,
axioms of, 426, 441, 448
215-17, 222-23; as code, 265-67; Tortoise's love song, 435-36
consistency of, 229-30, 449-50; extended, Tortoise's method, see Todelization
axioms of, 451-52, 466-68; extensions of, T6zan, 190, 255, 257, 479
451-59, 465-68; FIGURE-FIGURE figure tq-system, 64-67
and, 70; as general metalanguage, 265-67; transcendentalism, 704
goals of, 60; incompleteness of, see transcription: DNA to mRNA, 517, 524,
incompleteness; introspection of, 17, 194, 527-28, 530, 533, 536, 538, 540-41, 544-
267-72, 406, 438, 443, 449-50, 698, 707-8; 45; DNA to tRNA, 425; koans to
as its own metalanguage, 267-72, 441-46, messengers, 235-36, 238, 239, 242; letters
514; plight of, pigorial version, 71; relaxed, to notes, 83; prevention of, 544-45
228; rules of formation, table of, 213-14; transfer RNA, see tRNA
rules of inference, 215, 217-15; sixth translation: between computer languages,
axiom of, 222-23, 451-55, 459, 465-68; 192-95, 297-98, 306, 380-81, 547, 632;
well-formedness in, 205-15 between levels of a brain, 349, 381-84,
TNT-derivations compared with machine 709; between natural languages, difficulties
language, 291 of, 372-73, 379-80; between TNT and
TNT-level, explanations on, 708 meta-TNT, 267-72, 441-46, 709; of Crime
TNT-numbers, 269-70; see also theorem- and Punishment, 379-80.; English to TNT,
numbers 209-13, 215, 417; of "Jabberwocky", 372-
TNT + G, 465-67, 471 73, 379; levels of fidelity in, 379-80;
TNT + G + G', etc., 467-71 mechanical, 380, 601, 603; of messengers
TNT + -G, 467 into strings, 234-36; mRNA to proteins,
tobacco mosaic virus, 484-85, 542, 543 485, 518-19, 522-25, 527-28, 531-36, 538,
Todelization, repeatability of, 76-78, 424, 545, 546-48; from N to Meta-TNT, 533;
467-70, 483-86; see also Godelization, from score to sounds, 83; in Typogenetics,
essential incompleteness, answer schemas, 509-10, 512-13
etc. transparency to user, 629, 632
Todel's Theorem, 486, 536 trees: look-ahead, see look-ahead trees;
Tokusan, 189-90 recursive diagrams, 40„ 71, 135-37; of
tonic, musical, 121-23, 129-30 theorems, 40, 71
Index 785
triggering patterns of symbols: dependence Two Mysteries, The (Magritte), 701-2
on meaningless lower levels, 569; Two-Part Invention, 28, 43-45, 684-86; see
isomorphism between minds and, 369, 376; also Carroll paradox, Lewis Carroll 2 + 2 =
isomorphism between physical law and, 5,576
362; as key to meaning, 325, 327, 350, typeface metaphor, 541
360, 385, 609; mediated by mes- Typeless Wishes, 111-15, 610-11 types,
triggering patterns of symbols (continued) theory of, 21-23
sages, 350, 371; for melodies, 364; nouns typesetting machine, 608
vs. verbs, 361; randomness in, 673 typoenzymes, 505-13; binding-preferences
triggers: DNA as, 160-61; dormant of, 505-6, 511-12
symbols and, 281, 383, 384; frame Typogenctic Code, 510, 512, 513, 519
messages and, 162; jukeboxes and, 160-61, Typogenetics, 504-13, 514, 519, 520, 529;
170-71, 174, 500; koans as, 246; music con
and, 162-63, 281, 383, 583; outer messages trasted with MIU-system, 509-10, 514
and, 166, 169, 170-71, 174, 501 Typographical Number Theory, see TNT
Trio Sonata from the Musical Offering typographical operations, defined, 64
(Bach), 7-8, 720, 724, 726 typos, 404
Tripitaka, 257
trip-lets, cover, xiv, 1, 28, 273 tRNA, 522- U, as nontheorem of MIU-system, 36, 39
24, 547, 548 U-mode, see Un-mode
Trojan Horse, 538 Ulam, Stanislaw, 560, 621, 676 Un-mode,
truth: capturability by symbol 39, 98, 254
manipulation, 53-60; elusiveness of, 694- Unamuno, Miguel de, 698
95; inexpressible in TNT, 580-81; not fully uncertainty principle, see Heisenberg
mirrorable in brain, 584-85; vs. beauty, uncertainty principle
554-58, 584; vs. commercials, 478; vs. uncles, 446-48, 464, 466, 468, 541, 580
falsity, 70, 71, 213, 228-29, 417, 561, 579- undecidability, 17, 222, 449, 451-55, 468;
81 causes of, 707-8
TTortoise, see ATTACCA undefined terms, 92-102, 216, 456;
tuba, flaming, 488-89, 492, 735 defined, 93, 97
Tumbolia, 116, 243, 255. 725; lavers of, understanding, nature of, 569, 675-76, 680
243 tuning an Al program, 678-79 understanding minds/brains: meaning of,
Turing, Alan M., 26, 389, 425-26, 428-29. 697; possibility of, 697-98, 706-7
594-99, 734-42; objections to Al, 597-99 unicycle, tandem, 633, 669
Turing, Sara, 595 units in Tvpogenetics. 505, 509 universal
Turing machines, 390, 594, 735 quantifiers, see quantifiers universal
Turing test, 595-99, 600, 677-78, 735-37; triggering power, 171, 175 Unmon, 254
arithmetic error in, 596; miniature, 621-23; unobservant machines, 36-37, 674
proposed revisions in, 600 unpredictable but guaranteed termination,
Turtle's Theorem, see Todefs Theorem 2, 400, 425
as concept, 678 upper bounds, see loops, bounded, BIooP
2-D vs. 3-D: in Escher, 57-58, 105-6, 125, uracil, see nucleotides
473-74, 524, 689-90, 698, 714-16; in use vs. mention, 434-37, 458, 531, 545,
Magritte, 480-81, 493-94, 700-1, 705-6; 699-700;
television screens and, 488-93, 737; trip- see also form, syntactic vs. semantic,
lets and, see triplets programs
Index 786
vs. data, syntax vs. semantics, structure vs. weight vs. mass, womanseeing vs. seeing,
function variables in TNT, 206, 213-14; words vs. letters, Zen vs. logic, Zen vs.
free, 207-9, 214; quantified, 208, 214, see words
also quantifiers vibrations, 76-78, 82-85, 102, 270, 271,
verb-at-the-end phenomenon, 130-31 469
Verbum (Escher), 257, 731-32 Vice President, 670
versus, see accessible vs. inaccessible Villon, Fransois, 369
knowledge, active vs. passive symbols, Vinogradov, Ivan M., 394-95
ants vs. ant colonies, arithmetical vs. Vinogradov property, 394-95
typographical rules, Bach vs. Cage, violins, 62, 63, 70, 81, 84, 162, 200, 257,
beautiful vs. non-beautiful, bottom-up vs. 434, 502, 595, 681, 720, 724
top-down, classes vs. instances, continuous viruses, 536-43; likened to Henkin
vs. discrete processes, deductive vs. sentences, 542-43
analogical awareness, derivations vs. vision by computer, 602, 627
proofs, dissection vs. appreciation of Bach, visual imagery: Bongard problems and,
distinct vs. similar levels, enzymes vs. 661: faucets and, 364-65; inaccessible
typoenzymes, explicit vs. implicit knowledge and, 365; lack of in programs,
knowledge, explicit vs. implicit meaning, 623; mathematics and, 569, 678;
explicit vs. implicit pruning, formal vs. necessitating layers of substrate, 570-71;
informal reasoning, formal vs. informal power of, 338-39; role in
systems, formal systems vs. reality, conceptual mapping. 668, 672; rubbing-off
genuine vs. phony koans, high-fidelity vs. and.
low-fidelity, holism vs. reductionism, 361-62 es and canons, 28, 282-83, 314,
improvisation vs. introspection, voices in fugu
instructions vs. templates, local vs. global 322-23, 335, 665-67, 669, 683, 737, 740
properties, meaningless vs. meaningful Voltaire, Francois Marie Arouet de, 3
interpretations, men vs. women, minds vs. vortices, 713-19
brains, nouns vs. verbs in triggering Vuillard, Edouard, 347 Wachter, F. L., 92
patterns, 1-D vs. 3-D, passive vs. active Warrin, Frank L., 366
meaning, people vs. machines, plausible vs. -WASH ME", 608
implausible pathways, President v. Waterfall (Escher), 11-13, 99, 716
Supreme Court, procedural vs. declarative Watergate, 692
knowledge, programs vs. data, programs
vs. programmers, purposeful vs. Watson, J. D.. 667 Weasel, 106 weather.
purposeless behavior, rational vs. irrational, 302-3 Weaver, Warren, 380 Weierstrass,
sanity vs. insanity, self-perception vs. self- Karl W. T., 404 weight vs. mass, 171-72
transcendence, structure vs. function, Weizenbaum, Joseph, 599-600. 675
subject vs. object, symbol vs. object,
symbols vs. neurons, symbols vs. signals, well-formed strings: in Art of Zen Strings,
syntactic vs. semantic form, syntax vs. 239, 242, 244; BIooP puzzle on, 416;
semantics, teleological vs. evolutionary defined. 53: in pq-svstem, 47; in
viewpoint, theorems vs. nontheorems, Propositional Calculus, 181-83; in TNT,
theorems vs. rules, theorems vs. Theorems, 213-15
theorems vs. truth, Tortoise vs. Crab, truth Well-Te"ered Clavier (Bach), 7, 280-84,
vs. beauty, truth vs. commercials, truth vs. 327. 329,
falsity, 2-D vs. 3-D, use vs. mention, 335
Index 787
Well-Tested Conjecture (Fourmi), 333-35 34, 249-51, 254; vs. words, 246, 248-49,
"When Johnny Comes Marching Home", 251-54; Zeno and, 30
607 Whitehead, Alfred North, 18, 21, 23- Zen Strings, Art of, see Art of Zen Strings
24 Whitely, C. H., 477 Zenfunny, Beethoven's Ninth, 634
Zeno of Elea, 28, 29-32, 94-95, 144, 146,
wholes, see holism Wiener, Norbert, 684 232, 610, 681, 704, 722
Wiesel, Torsten, 343 Zeno's paradoxes, 29-32, 35, 43, 146, 610
will: free, see free will; mechanical, 684- Zentences, 186-90
86; roots ZET-cycles, 94-96, 689
of, 684-86 zooming in, 645, 671
Wilson, E. 0., 350 Winograd. Terry, 627- i T
32 Winston. Patrick Henry, 299 wishes,
meta-wishes, etc., 109-16 Wittgenstein,
Ludwig, 680, 699
Wolff, Christoph, 392
womanseeing vs. seeing, womansee men
vs. women wondrous and unwondrous
numbers, 400-2, 408,
415, 418, 425
Wooldridge, Dean, 360
words: in computers, 288-90, 295, 411; as
programs, 629-30; spelled backwards. 81,
418, 427, 498, 505, 533, 549, 727, 738,
740; thoughts, formal rules, and, see main
theses: vs. letters,
325-27, 570, 571; Zen attitude towards,
246, 249, 251-54
worker ants, 318
working inside the system, see Mechanical
mode
yes-answers, 461-64 Yngve, Victor, 620
Young, LaMonte, 700
Zen Algebra, 577
Zen Buddhism: Achilles teaches the
Tortoise about. 231-45; computers and,
625-26; Crab's refrigerator and, 406-7;
Escher and, 255-58; holism vs.
reductionism and, 312-13; inconsistencies
and, see Zen vs. logic; introduction to, 246-
59; jumping out of the system and, 233,
255, 479; music, art, and, 699, 704-6;
Mystery of the Undecidable and, 272;
nonexistence and, 254-55, 698; patriarchs
of, 30, 232, 252, 259; quasi-, 625-26; U-
mode and, 39, 98. 254; vs, logic, 99, 233-
Index 788