Gödel, Escher, Bach: An Eternal Golden Braid (31 page)

Tortoise: Who said anything about a single song, Achilles?

Achilles: Every- jukebox I've ever run into obeyed the fundamental jukebox-axiom: "One record, one song".

Tortoise: This jukebox is different, Achilles. The one record sits vertically, suspended, and behind it there is a small but elaborate network of overhead rails, from which hang various record players. When push a pair of buttons, such as B-1, that selects one of the record players. This triggers an automatic mechanism that starts the record player squeakily rolling along the rusty tracks. It gets shunted alongside the record-then it clicks into playing position.

Achilles: And then the record begins spinning and music comes out -- right?

Tortoise: Not quite. The record stands still-it's the record player which rotates.

Achilles: I might have known. But how, if you have but one record to play can you get more than one song out of this crazy contraption?

Tortoise: I myself asked the Crab that question. He merely suggested I try it out. So I fished a quarter from my pocket (you get three plays for a quarter), stuffed it in the slot, and hit buttons B-1, then C-3 then B-10-all just at random.

Achilles: So phonograph B-1 came sliding down the rail, I suppose, plugged itself into the vertical record, and began spinning?

Tortoise: Exactly. The music that came out was quite agreeable, based the famous old tune
B-A-C-H
, which I believe you remember.

Achilles: Could I ever forget it?

Tortoise: This was record player B-1. Then it finished, and was s rolled back into its hanging position, so that C-3 could be slid into position.

Achilles: Now don't tell me that C-3 played another song?

Tortoise: It did just that.

Achilles: Ah, I understand. It played the flip side of the first song, or another band on the same side.

Tortoise: No, the record has grooves only on one side, and has only a single band.

Achilles: I don't understand that at all. You CAN'T pull different songs out of the same record!

Tortoise: That's what I thought until I saw Mr. Crab's jukebox. Achilles: How did the second song go?

Tortoise: That's the interesting thing ... It was a song based on the melody
C-A-G-E
.

Achilles: That's a totally different melody!

Tortoise: True.

Achilles: And isn't John Cage a composer of modern music? I seem to remember reading about him in one of my books on haiku.

Tortoise: Exactly. He has composed many celebrated pieces, such as 4'33", a three-movement piece consisting of silences of different lengths. It's wonderfully expressive-if you like that sort of thing.

Achilles: I can see where if I were in a loud and brash cafe I might gladly pay to hear Cage's 4'33" on a jukebox. It might afford some relief!

Tortoise: Right-who wants to hear the racket of clinking dishes and jangling silverware? By the way, another place where 4'33" would come in handy is the Hall of Big Cats, at feeding time.

Achilles: Are you suggesting that Cage belongs in the zoo? Well, I guess that makes some sense.

But about the Crab's jukebox ... I am baffled. How could both "
BACH
" and "
CAGE
" be coded inside a single record at once?

Tortoise: You may notice that there is some relation between the two, Achilles, if you inspect them carefully. Let me point the way. What do you get if you list the successive intervals in the melody
B-A-C-H
?

Achilles: Let me see. First it goes down one semitone, from B to A (where B is taken the German way); then it rises three semitones to C; and finally it falls one semitone, to H.

That yields the pattern:

-1, +3, -1.

Tortoise: Precisely. What about
C-A-G-E
, now?

Achilles: Well, in this case, it begins by falling three semitones, then ten semitones (nearly an octave), and finally falls three more semitones. That means the pattern is:

-3, +10, -3.

It's very much like the other one, isn't it?

Tortoise: Indeed it is. They have exactly the same "skeleton", in a certain sense. You can make
C-A-G-E
out of
B-A-C-H
by multiplying all the intervals by 31/3, and taking the nearest whole number.

Achilles: Well, blow me down and pick me up! So does that mean that only

some sort of skeletal code is present in the grooves, and that the various record players add their own interpretations to that code?

Tortoise: I don't know, for sure. The cagey Crab wouldn't fill me in on the details. But I did get to hear a third song, when record player B-1 swiveled into place.

Achilles: How did it go?

Tortoise: The melody consisted of enormously wide intervals, and we
B-C-A-H
.

The interval pattern in semitones was:

-10, +33, -10.

It can be gotten from the CAGE pattern by yet another multiplication by 3%3, and rounding to whole numbers.

Achilles: Is there a name for this kind of interval multiplication?

Tortoise: One could call it "intervallic augmentation". It is similar to tl canonic device of temporal augmentation, where all the time values notes in a melody get multiplied by some constant. There, the effect just to slow the melody down. Here, the effect is to expand the melodic range in a curious way.

Achilles: Amazing. So all three melodies you tried were intervallic augmentations of one single underlying groove-pattern in the record:

Tortoise: That's what I concluded.

Achilles: I find it curious that when you augment
BACH
you get
CAGE
and when you augment
CAGE
over again, you get BACH back, except jumbled up inside, as if
BACH
had an upset stomach after passing through the intermediate stage of
CAGE
.

Tortoise: That sounds like an insightful commentary on the new art form of Cage.

When Is One Thing Not Always the Same?

LAST CHAPTER, WE came upon the question, "When are two things the same?" In this Chapter, we will deal with the flip side of that question: "When is one thing not always the same?" The issue we are broaching is whether meaning can be said to be inherent in a message, or whether meaning is always manufactured by the interaction of a mind or a mechanism with a message-as in the preceding Dialogue. In the latter case, meaning could not said to be located in any single place, nor could it be said that a message has any universal, or objective, meaning, since each observer could bring its own meaning to each message. But in the former case, meaning would have both location and universality. In this Chapter, I want to present the case for the universality of at least some messages, without, to be sure, claiming it for all messages. The idea of an

"objective meaning" of a message will turn out to be related, in an interesting way, to the simplicity with which intelligence can be described.

Information-Bearers and Information- Revealers

I'll begin with my favorite example: the relationship between records, music, and record players. We feel quite comfortable with the idea that a record contains the same information as a piece of music, because of the existence of record players, which can

"read" records and convert the groove-patterns into sounds. In other words, there is an isomorphism between groove-patterns and sounds, and the record player is a mechanism which physically realizes that isomorphism. It is natural, then, to think of the record as an
information-bearer
, and the record-player as an
information-revealer
. A second example of these notions is given by the pq-system. There, the "information-bearers" are the theorems, and the "information-revealer" is the interpretation, which is so transparent that we don't need any electrical machine to help us extract the information from pq-theorems.

One gets the impression from these two examples that isomorphisms and decoding mechanisms (i.e., information-revealers) simply reveal information which is intrinsically inside the structures, waiting to be "pulled out". This leads to the idea that for each structure, there are certain pieces of information which
can
be pulled out of it, while there are other pieces of information which
cannot
be pulled out of it. But what does this phrase

"pull out" really mean? How hard are you allowed to pull? There are c where by investing sufficient effort, you can pull very recondite piece of information out of certain structures. In fact, the pulling-out may inv such complicated operations that it makes you feel you are putting in n information than you are pulling out.

Genotype and Phenotype

Take the case of the genetic information commonly said to reside in double helix of deoxyribonucleic acid (
DNA
). A molecule of
DNA
– a
genotype
-is converted into a physical organism-a
phenotype
-by a complex process, involving the manufacture of proteins, the replication the
DNA
, the replication of cells, the gradual differentiation of cell types and so on. Incidentally, this unrolling of phenotype from genotype
epigenesis
-

is the most tangled of tangled recursions, and in Chapter we shall devote our full attention to it. Epigenesis is guided by a se enormously complex cycles of chemical reactions and feedback loops the time the full organism has been constructed, there is not even remotest similarity between its physical characteristics and its genotype.

And yet, it is standard practice to attribute the physical structure of organism to the structure of its DNA, and to that alone. The first evidence for this point of view came from experiments conducted by Oswald A, in 1946, and overwhelming corroborative evidence has since been amassed Avery's experiments showed that, of all the biological molecules, only E transmits hereditary properties. One can modify other molecules it organism, such as proteins, but such modifications will not be transmitted to later generations. However, when DNA is modified, all successive generations inherit the modified
DNA
. Such experiments show that the only of changing the instructions for building a new organism is to change
DNA
-and this, in turn, implies that those instructions must be cc somehow in the structure of the
DNA
.

Exotic and Prosaic Isomorphisms

Therefore one seems forced into accepting the idea that the
DNA
's structure contains the information of the phenotype's structure, which is to the two are
isomorphic
. However, the isomorphism is an exotic one, by w] I mean that it is highly nontrivial to divide the phenotype and genotype into "parts" which can be mapped onto each other. Prosaic isomorphic by contrast, would be ones in which the parts of one structure are easily mappable onto the parts of the other. An example is the isomorphism between a record and a piece of music, where one knows that to any so in the piece there exists an exact

"image" in the patterns etched into grooves, and one could pinpoint it arbitrarily accurately, if the need arose Another prosaic isomorphism is that between Gplot and any of its internal butterflies.

The isomorphism between
DNA
structure and phenotype structure is anything but prosaic, and the mechanism which carries it out physically is awesomely complicated.

For instance, if you wanted to find some piece of your
DNA
which accounts for the shape of your nose or the shape of your fingerprint, you would have a very hard time. It would be a little like trying to pin down the note in a piece of music which is the carrier of the emotional meaning of the piece. Of course there is no such note, because the emotional meaning is carried on a very high level, by large "chunks" of the piece, not by single notes. Incidentally, such "chunks" are not necessarily sets of contiguous notes; there may be disconnected sections which, taken together, carry some emotional meaning.

Similarly, "genetic meaning"-that is, information about phenotype structure-is spread all through the small parts of a molecule of DNA, although nobody understands the language yet. (Warning: Understanding this "language" would not at all be the same as cracking the Genetic Code, something which took place in the early 1960's. The Genetic Code tells how to translate short portions of
DNA
into various amino acids.