Gödel, Escher, Bach: An Eternal Golden Braid (87 page)

Achilles: That was one of his droller lines, I take it.

Crab: That's just what the other character in the Dialogue thought. But that's a preposterous interpretation of the statement. (
The Crab draws deeply from his pipe,
and puffs several billows of smoke into the air.)

Achilles: Well, what does "spontaneous self-assembly" mean, then?

Crab: The idea is that when some biological units inside a cell are taken apart, they can spontaneously reassemble themselves-without being directed by any other unit. The pieces just come together, and presto!-they stick.

Achilles: That sounds like magic. Wouldn't it be wonderful if a full-sized record player could have that property? I mean, if a miniature "record player" such as a ribosome can do it, why not a big one? That would allow you to create an indestructible phonograph, right? Any time it was broken, it would just put itself together again.

Crab: Exactly my thought. I breathlessly rushed a letter off to my manufacturer explaining the concept of self-assembly, and asked him if he could build me a record player which could take itself apart and spontaneously self-assemble in another form.

Achilles: A hefty bill to fill.

Crab: True; but after several months, he wrote to me that he had succeeded, at long last-and indeed he sent me quite a hefty bill. One fine day, ho! My Grand Self-assembling Record Player arrived in the mail, and it was with great confidence that I telephoned Mr. Tortoise, and invited him over for the purpose of testing my ultimate record player.

Achilles: So this magnificent object before us must be the very machine of which you speak.

Crab: I'm afraid not, Achilles.

Achilles: Don't tell me that once again ...

Crab: What you suspect, my dear friend is unfortunately the case. I don't pretend to understand the reasons why. The whole thing is too painful to recount. To see all those springs and wires chaotically strewn about on the floor, and puffs of smoke here and there-oh, me ...

Achilles: There, there, Mr. Crab, don't take it too badly.

Crab: I'm quite all right; I just have these spells every so often. Well, to go on, after Mr.

Tortoise's initial gloating, he at last realized how sorrowful I was feeling, and took pity. He tried to comfort me by explaining that it couldn't be helped-it all had to do with somebody-or-other's "Theorem", but I couldn't follow a word of it. It sounded like "Turtle's Theorem".

Achilles: I wonder if it was that "Gödel’s Theorem" which he spoke of once before to me

... It has a rather sinister ring to it. Crab: It could be. I don't recall.

Achilles: I can assure you, Mr. Crab, that I have followed this tale with the utmost empathy for your position. It is truly sad. But, you mentioned that there was a silver lining. Pray tell, what was that?

Crab: Oh, yes—the silver lining. Well eventually, I abandoned my quest after

“Perfection” in phonographs, and decided that I might do better

to tighten up my defenses against the Tortoise's records. I concluded that a more modest aim than a record player which can play anything is simply a record player that can SURVIVE: one that will avoid getting destroyed-even if that means that it can only play a few particular records.

Achilles: So you decided you would develop sophisticated anti-Tortoise mechanisms at the sacrifice of being able to reproduce every possible sound, eh?

Crab: Well ... I wouldn't exactly say I "decided" it. More accurate would be to say that I was FORCED into that position.

Achilles: Yes, I can see what you mean.

Crab: My new idea was to prevent all "alien" records from being played on my phonograph. I knew my own records are harmless, and so if I prevented anyone else from infiltrating THEIR records, that would protect my record player, and still allow me to enjoy my recorded music.

Achilles: An excellent strategy for your new goal. Now does this giant thing before us represent your accomplishments to date along those lines?

Crab: That it does. Mr. Tortoise, of course, has realized that he must change HIS strategy, as well. His main goal is now to devise a record which can slip past my censors-a new type of challenge.

Achilles: For your part, how are you planning to keep his and other "alien" records out?

Crab: You promise you won't reveal my strategy to Mr. T, now?

Achilles: Tortoise's honor.

Crab: What!?

Achilles: Oh-it's just a phrase I've picked up from Mr. T. Don't worry-I swear your secret will remain secret with me.

Crab: All right, then. My basic plan is to use a LABELING technique. To each and every one of my records will be attached a secret label. Now the phonograph before you contains, as did its predecessors, a television camera for scanning the records, and a computer for processing the data obtained in the scan and controlling subsequent operations. My idea is simply to chomp all records which do not bear the proper label!

Achilles: Ah, sweet revenge! But it seems to me that your plan will be easy to foil. All Mr. T needs to do is to get a hold of one of your records, and copy its label!

Crab: Not so simple, Achilles. What makes you think he will be able to tell the label from the rest of the record? It may be better integrated than you suspect.

Achilles: Do you mean that it could be mixed up somehow with the actual music?

Crab: Precisely. But there is a way to disentangle the two. It requires sucking the data off the record visually and then--

Achilles: Is that what that bright green Hash was for?

Crab: That's right. That was the TV camera scanning the grooves. The groove-patterns were sent to the minicomputer, which analyzed the musical style of the piece I had put on-all in silence. Nothing had been played yet.

Achilles: Then is there a screening process, which eliminates pieces which aren't in the proper styles?

Crab: You've got it, Achilles. The only records which can pass this second test are records of pieces in my own style-and it will be hopelessly difficult for Mr. T to imitate that. So you see, I am convinced I will win this new musical battle. However, I should mention that Mr. T is equally convinced that somehow, he will manage to slip a record past my censors.

Achilles: And smash your marvelous machine to smithereens?

Crab: Oh, no-he has proved his point on that. Now he just wants to prove to me that he can slip a record-an innocuous one-by me, no matter what measures I take to prevent it. He keeps on muttering things about songs with strange titles, such as "I Can Be Played on Record Player X". But he can't scare MtE! The only thing that worries me a little is that, as before, he seems to have some murky arguments which ... which ... (He trails off into silence. Then, looking quite pensive, he takes a few puffs on his pipe.) Achilles: Hmm ... I'd say Mr. Tortoise has an impossible task on his hands. He's met his match, at long last!

Crab: Curious that you should think so ... I don't suppose that you know Henkin's Theorem forwards and backwards, do you?

Achilles: Know WHOSE Theorem forwards and backwards? I've never heard of anything that sounds like that. I'm sure it's fascinating, but I'd rather hear more about "music to infiltrate phonographs by". It's an amusing little story. Actually, I guess I can fill in the end. Obviously, Mr. T will find out that there is no point in going on, and so he will sheepishly admit defeat, and that will be that. Isn't that exactly it?

Crab: That's what I'm hoping, at least. Would you like to see a little bit of the inner workings of my defensive phonograph?

Achilles: Gladly. I've always wanted to see a working television camera.

Crab: No sooner said than done, my friend. (Reaches into the gaping"mouth" of the large phonograph, undoes a couple of snaps, and pulls out a neatly packaged instrument.) You see, the whole thing is built of independent modules, which can be detached and used independently. This TV camera, for instance, works very well by itself. Watch the screen over there, beneath the painting with the flaming tuba. (He points the camera at Achilles, whose face instantly appears on the large screen.) Achilles: Terrific! May I try it out?

Crab: Certainly.

Achilles: (pointing the camera at the Crab. There YOU are, Mt Crab, on the screen.

FIGURE 80. The Fair Captive, by Rene Magritte (1947).

Crab: So I am.

Achilles: Suppose I point the camera at the painting with the burning tuba. Now it is on the screen, too!

Crab: The camera can zoom in and out, Achilles. You ought to try it. Achilles: Fabulous!

Let me just focus down onto the tip of those flames, where they meet the picture frame

... It's such a funny feeling to be able to instantaneously "copy" anything in the room-anything I want-onto that screen. I merely need to point the camera at it, and it pops like magic onto the screen.

Crab: ANYTHING in the room, Achilles? Achilles: Anything in sight, yes. That's obvious.

Crab: What happens, then, if you point the camera at the flames on the TV screen?

(Achilles shifts the camera so that it points directly at that part of the television
screen on which the flames are-or were-displayed.)

Achilles: Hey, that's funny! That very act makes the flames DISAPPEAR from the screen! Where did they go?

Crab: You can't keep an image still on the screen and move the camera at the same time.

Achilles: So I see… But I don’t understand what’s on the screen now—not at all! It seems to be a strange long corridor. Yet I’m certainly not

FIGURE 81.
Twelve self-engulfing TV screens. I would have included one more, had 13

not been prime

pointing the camera down any corridor. I'm merely pointing it at an ordinary TV

screen.

Crab: Look more carefully, Achilles. Do you really see a corridor?

Achilles: Ahhh, now I see. It's a set of nested copies of the TV screen itself, getting smaller and smaller and smaller ... Of course! The image of the flames HAD to go away, because it came from my- pointing the camera at the PAINTING. When I point the camera at the SCREEN, then the screen itself appears, with whatever is on the screen at the time which is the screen itself, with whatever is on the screen at the time which is the screen itself, with

Crab: I believe I can fill in the rest, Achilles. Why- don't you try rotating the camera?

Achilles: Oh! I get a beautiful spiraling corridor! Each screen is rotated inside its framing screen, so that the littler they get, the more rotated they are, with respect. to the outermost screen. This idea of having a TV screen "engulf itself" is weird.