Godel, Escher, Bach: An Eternal Golden Braid by Douglas. R. Hofstadter

elena petelos

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Gödel, Escher, Bach: An Eternal Golden Braid.



Edifying Thoughts of a Tobacco Smoker
Achilles has been invited to the Crab’s home

Achilles: I see you have made a few additions since I was last here, Mr Crab. Your new paintings are especially striking.
Crab: Thank you. I am quite fond of certain painters - especially René Magritte. Most of the paintings I have are by him. He’s my favorite artist.
Achilles: They are very intriguing images. I must say. In some ways, these paintings by Magritte remind me of works by my favourite artist, M.C.Escher.
Crab: I can see that. Both Magritte and Escher use great realism in exploring the words of paradox and illusion; both have a sure sense for the evocative power of certain visual symbols, and - something which even their admirers fail to point out- both of them have a sense of the graceful line.
Achilles: Nevertheless, there is something quite different about them. I wonder how one could characterize that difference.
Crab: It would be fascinating to compare the two in detail.
Achilles: I must say, Magritte's command of realism is astonishing. For instance, I was quite taken by that painting over there of a tree with a giant pipe behind it.



Crab: You mean, a normal pipe with a tiny tree in front of it!
Achilles: Oh, is that what it is? Well, in any case, when I first spotted it, I was convinced I was smelling pipe smoke! Can you imagine how silly I felt?
Crab: I quite understand. My guests are often taken by that one.

(So saying, he reached up, removes the pipe from behind the tree in the painting, turns it over and taps it against the table, and the room begins to reek of pipe tobacco. He begins packing in a new wad of tobacco.)

This is a fine old pipe, Achilles. Believe it or not, the bowl has a copper lining, which makes it age wonderfully.
Achilles: A copper lining! You don’t say!
Crab (pulls out a box of matches, and lights his pipe):Would you care for a smoke, Achilles?
Achilles: No thank you. I only smoke cigars now and then.
Crab: No problem! I have one right here! (Reaches out towards another Magritte painting, featuring a bicycle mounted upon a lit cigar).
Achilles: Uhh - no thank you, not now.
Crab: As you will. I myself am an incurable tobacco smoker. Which reminds me – you undoubtedly know of Old Bach's predilection for pipe smoking?
Achilles: I don’t recall exactly.
Crab: Old Bach was fond of versifying, philosophizing, pipe smoking, and music making (not necessarily in that order). He combined all four into a droll poem which he set to music. It can be found in the famous musical notebook he kept for his wife, Anna Magdalena, and it is called
Edifying Thoughts of a Tobacco Smoker

Whene'er I take my pipe and stuff it
And smoke to pass the time away
My thoughts, as I sit there and puff it,
Dwell on a picture sad and grey:
It teaches me that very like
Am I myself unto my pipe.
Like me this pipe, so fragrant burning,
Is made of naught but earthen clay;
To earth I too shall be returning,
And cannot halt my slow decay.
My well used pipe, now cracked and broken,
Of mortal life is but a token.
No stain, the pipe's hue yet doth darken;
It remains white. Thus do I know
That when to death's call I must harken
My body, too, all pale will grow.
To black beneath the sod 'twill turn,
Likewise the pipe, if oft it burn.
Or when the pipe is fairly glowing,
Behold then instantaneously,
The smoke off into thin air going,
'Til naught but ash is left to see.
Man's fame likewise away will burn
And unto dust his body turn.
How oft it happens when one's smoking,
The tamper's missing from it's shelf,
And one goes with one's finger poking
Into the bowl and burns oneself.
If in the pipe such pain doth dwell
How hot must be the pains of Hell!
Thus o'er my pipe in contemplation
Of such things - I can constantly
Indulge in fruitful meditation,
And so, puffing contentedly,
On land, at sea, at home, abroad,
I smoke my pipe and worship God.


A charming philosophy, is it not?
Achilles: Indeed. Old Bach was a turner of phrases quite pleasin’.
Crab: You took the very words from my mouth. You know, in my time I have tried to write clever verses. But I fear mine don’t measure up to much. I don’t have such a way with words.
Achilles: Oh, come now, Mr. Crab. You have - how to put it? -quite a penchant for trick'ry and teasin'. I'd be honored if you'd sing me one of your songs, Mr. C.
Crab: I’m most flattered. How about if I play you a record of myself singing one of my efforts? I don’t remember when it dates from. It’s title is “A Song Without Time or Season”.
Achilles: How poetic!
(The Crab pulls a record from his selves, and walks over to a huge complex piece of apparatus. He opens it up, and inserts the record into an ominous-looking mechanical mouth. Suddenly a bright flash of greenish light sweeps over the surface of the record, and after a moment, the record is silently whisked into some hidden belly of the fantastic machine. A moment passes, and then the strains of the Crab’s voice ring out.)
 
A turner of phrases quite pleasin’
Had a penchant for tick’ry and teasin’.
       In his songs, the last line
       Migh seem sans design
What I mean is, without why or wherefore.
Achilles: Lovely! Only, I'm puzzled by one thing. It seems to me that in your song, the last line is-
Crab: Sans design?
Achilles:No… What I mean is, without rhyme or reason.
Crab: You could be right.
Achilles: Other than that, it’s a very nice song, but I must say I am even more intrigued by this monstrously complex contraption. Is it merely an oversized record player?
Crab: Oh, no, it’s much more than that. This is my Tortoise-chomping record player.
Achilles: Good grief!
Crab: Well, I don’t mean that it chomps up Tortoises. But it chomps up records produced by Mr. Tortoise.
Achilles: Whew! That’s a little milder. Is this part of that weird musical battle that evolved between you and Mr. T some time ago?
Crab: In a way. Let me explain a little more fully. You see, Mr. Tortoise’s sophistication had reached the point where he seemed to be able to destroy almost any record player I would obtain.
Achilles: But when I last heard about your rivalry, it seemed to me you had at last come into possession of an invincible phonograph – one with a built-in TV camera, minicomputer and so on, which could take itself apart and rebuild itself in such a way that it would not be destroyed.
Crab: Alack and alas! My plan was foiled. For Mr. Tortoise took advantage of one small detail which I had overlooked: the subunit which directed the disassembly and reassembly processes was itself stable during the entire process. That is, for obvious reasons, it could not take itself apart and rebuild itself, so it stayed intact.
Achilles: Yes, but what consequences did that have?
Crab: Oh, the direst ones! For you see, Mr. T focused his method down onto that subunit entirely.
Achilles: How is that?
Crab: He simply made a record which would induce fatal vibrations in the one structure he knew would never change – the disassembly-reassembly subunit.
Achilles: Oh, I see… Very sneaky.
Crab: Yes, so I though too. And his strategy worked. Not the first time, mind you. I thought I had outwitted him when my phonograph survived his first onslaught. I laughed gleefully. But the next time, he returned with a steely glint in his eye, and I knew he meant business. I placed his new record on my turntable. Then, both of us eagerly watched the computer-directed subunit carefully scan the grooves, then dismount the record, disassemble the record player, reassemble it in an astonishingly different way, remount the record – and then slowly lower the needle into the outermost groove.
Achilles: Golly!
Crab: No sooner had the first strains of sound issued forth than a loud SMASH! filled the room. The whole thing fell apart, but particularly badly destroyed was the assembler-disassembler. In that painful instant I finally realized, to my chagrin, that the Tortoise would ALWAYS be able to focus down upon -if you'll pardon the phrase- the Achille's heel of the system.
Achilles: Upon my soul! You must have felt devastated.
Crab: Yes, I felt rather forlorn for a while. But, happily, that was not the end of the story. There is a sequel to the tale, which taught me a valuable lesson, which I may pass on to you. On the Tortoise’s recommendation, I was browsing through a curious book filled with strange Dialogues about many subjects, including molecular biology, fugues, Zen Buddhism, and heaven knows what else.
Achilles: Probably some crackpot wrote it. What is the book called?
Crab: If I recall correctly, it was called Copper, Silver, Gold: an Indestructible Metallic Alloy.
Achilles: Oh, Mr. Tortoise told me about it too. It’s by a friend of his, who it appears, is quite taken with metal-logic.
Crab: I wonder which friend it is… Anyway, in one of the Dialogues, I encountered some Edifying Thoughts on the Tobacco Mosaic Virus, ribosomes, and other strange things I had never heard of.



Achilles: What is the Tobacco Mosaic Virus? What are ribosomes?
Crab: I can’t quite say, for I am a total dunce when it comes to biology. All I know is what I gathered from that Dialogue. There, it said that Tobacco Mosaic Virus are tiny cigarette-like objects that cause a disease in tobacco plants.
Achilles: Cancer?
Crab: No, not exactly, but-
Achilles: What next? A tobacco plant smoking, and getting cancer? Serves it right!
Crab: I believe you jumped to a hasty conclusion, Achilles. Tobacco plants don’t SMOKE these “cigarettes”. The nasty little things “cigarettes” just come and attack them, uninvited.
Achilles: I see. Well, now that I know all about the Tobacco Mosaic Viruses, tell me what a ribosome is.
Crab: Ribosome are apparently some sort of subcellular entities which take a message in one form and convert it into a message in another form.
Achilles: Something like a teeny tape recorder or phonograph?
Crab: Metaphorically, I suppose so. Now the thing which caught my eye was a line where this one exceedingly droll character mentions the fact that ribosomes –as well as the Tobacco Mosaic Viruses and certain other bizarre biological structures- possess “the baffling ability to spontaneously self-assemble”. Those were his exact words.
Achilles: That was one of his droller lines I take it.
Crab: That’s just what the other character in the Dialogue though. 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 pay.
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's or other’s "Theorem”, but I couldn’t follow a word of it. It sounded like “Turtle’s Theorem".
Achilles: I wonder if that was “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 I might do better to tighten my defense 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 very possible sound, eh?
Crab: Well… I couldn’t exactly say I “decided” it. More accurate would be to say 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 these 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 sear 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, sweeter revenge! But it seems to me that your plan will be easy to foil. All Mr. T needs to do is get 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 form the rest of the 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 flash 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 me 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 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 s–range titles, such as “I Can Be Played on Record Player X”. But he can’t scare ME! 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: Hmmm… 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 you know Henkin's Theorem forwards and backwards, do you?
Achilles: Know WHOSE Theorem forwards and backwards? I've never heard 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 the end. Obviously, Mr. T will find out 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 nearly 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 Crab): There YOU are, Mr. Crab, on the screen.
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 in the TV screen?
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 long strange corridor. Yet I'm certainly not 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, I see. It’s set of nested copies of the TV screen in 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-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.
Crab: What do you mean by “self-engulfing”, Achilles?
Achilles: I mean, when I point the camera at the screen –or at part of the screen. THAT’S self-engulfing.
Crab: Do you mind if I pursue this a little bit further? I’m intrigued by this new notion.
Achilles: So am I.
Crab: Very well, then. If you point the camera at a CORNER of the screen, is that still what you mean by "self-engulfing"?
Achilles: Let me try it. Hmmm-the “corridor” of screens seems to go off the edge, so there isn’t an infinite nesting any more. It's pretty but it doesn't seem to me to have the spirit of self-engulfing. It’s a “failed self-engulfing”.
Crab: If you were to swing the TV camera back towards the center of the screen, maybe you could fix it up again...
Achilles (slowly and cautiously turning the camera): Yes! The corridor is getting longer and longer… There it is! Now it’s all back. I can look down it so far that it vanishes in the distance. The corridor became infinite again precisely at the moment when the camera took in the WHOLE screen. Hmm-that reminds me of something Mr. Tortoise was saying a while back, about self-reference only occurring when a sentence talks ALL about itself…
Crab: Pardon me?
Achilles: Oh, nothing-just muttering to myself.

(As Achilles plays with the lens and other controls on the camera, a profusion of new kinds of self-engulfing images appear: swirling spirals that resemble galaxies, kaleidoscopic flower-like shapes, and other assorted patterns…)

Crab: You seem to be having a grand time.
Achilles (turns away from the camera): I’ll say! What a wealth of images this simple idea can produce! (He glances at the screen, and a look of astonishment crosses his face.) Good grief, Mr. Crab! There’s a pulsating petal-pattern on the screen! Where do the pulsations come from? The TV is still, and so is the camera.
Crab: You can occasionally set up patterns which change in time. This is because there is a slight delay in the circuitry between the moment the camera “sees” something, and the moment, it appears on the screen- around a hundredth of a second. So, if you have a nesting of depth fifty or so, roughly a half-second delay will result. If somehow a moving image gets onto the screen-then it takes a while for the more deeply nested screens to “find out” about it. This delay then reverberates through the whole system, like a visual echo. And if things are set up so the echo doesn’t die away, then you can get pulsating patterns.
Achilles: Amazing! Say-what if we tried to make it TOTAL self-engulfing?
Crab: What precisely do you mean by that?
Achilles: Well, it seems to me that this stuff with screens within screens is interesting, but I’d like to get a picture of the TV camera AND the screen, ON the screen. Only then would I really have made the system engulf itself. For the screen is only PART of the total system.
Crab: I see what you mean. Perhaps with this mirror, you can achieve the effect you want.

The Crab hands him a mirror, and Achilles maneuvers the mirror and camera in such a way that the camera and the screen are both pictured on the screen.)
Achilles: There! I've created a total self-engulfing!
Crab: It seems to me you only have the front of the mirror-what about its back? If it weren't for the back of the mirror, it wouldn't be reflective-and you wouldn't have the camera in the picture.
Achilles: You’re right. But to show both the front and the back of the mirror I need a second mirror.
Crab: But then you’ll need to show the back of that mirror, too. And what about including the back of the television, as well as its front? And then there’s the electric cord, and the inside of the television, and-
Achilles: Whoa, whoa! My head’s beginning to spin! I can see that this “total self-engulfing project” is going to pose a wee bit problem. I'm feeling a little dizzy.
Crab: I know exactly how you feel. Why don’t you sit down here and take your mind off all this self-engulfing? Relax! Look at my painting, and you’ll calm down.

Achilles lies down and sighs.
Oh-perhaps my pipe smoke is bothering you? Here, I’ll put my pipe away. (Takes the pipe from his mouth, and carefully places it above some written words in another Magritte painting.) There! Feeling any better?
Achilles: I’m still a little woozy. (Points at the Magritte.) That’s an interesting painting. I like the way it’s framed, especially the shiny inlay inside the wooden frame.




Crab: Thank you. I had it especially done-it’s a gold lining.
Achilles: A gold lining? What are those words below the pipe? They aren’t English, are they?
Crab: No, they are in French. They say, “Ceci n'est pas une pipe.”
That means, “This is not a pipe”. Which is perfectly true.
Achilles: But it is a pipe! You were just smoking it!
Crab: Oh, you misunderstand the phrase, I believe. The word ceci refers to the painting, not to the pipe. Of course the pipe is a pipe. But a painting is not a pipe.
Achilles: I wonder if that “ceci” inside the painting refers to the WHOLE painting, or just to the pipe inside the painting. Oh, my gracious! That would be ANOTHER self-engulfing! I'm not feeling at all well Mr. Crab. I think I'm going to be sick...


« Last Edit: 06 Apr, 2006, 19:36:43 by epetelos »


 

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