Gödel, Escher, Bach: An Eternal Golden Braid (99 page)

Crab: As have all generations of Crabs.

Achilles: You don't happen to have brought your flute along, by any chance, Mr. C?

Crab: Why, certainly! I take it with me everywhere. Would you like to hear a tune or two?

Achilles: It would be delightful, in this pastoral setting. Do you play from memory?

Crab: Sad to say, that is beyond my capability. I have to read my music from a sheet. But that is no problem. I have several very pleasant pieces here in this case.

(He opens up a thin case and draws out a few pieces of paper. The topmost one has the
following symbols on it:

Va:-Sa=O

He sticks the top sheet into a little holder attached to his flute, and plays. The tune is very
short.)

Achilles: That was charming. (Peers over at the sheet on the flute, and a quizzical expression beclouds his face.) What is that statement of number theory doing, attached to your flute like that?

(The Crab looks at his flute, then his music, turns his head all around, and appears slightly
confused.)

Crab: I don't understand. What statement of number theory?

Achilles: "Zero is not the successor of any natural number." Right there, in the holder on your flute!

Crab: That's the third Piano Postulate. There are five of them, and I've arranged them all for flute. They're obvious, but catchy.

Achilles: What's not obvious to me is how a number-theoretical statement can be played as music.

Crab: But I insist, it's 'NOT a number-theoretical statement-it's a Piano Postulate! Would you like to hear another?

Achilles: I'd be enchanted.

(The Crab places another piece of paper on his flute, and this time Achilles watches more
carefully.)

Well, I watched your eyes, and they were looking at that FORMULA on the sheet. Are you sure that that is musical notation? I swear, it most amazingly resembles the notation which one might use in a formalized version of number theory.

Crab: How odd! But certainly that is music, not any kind of statement of mathematics, as far as I can tell! Of course, I am not a mathematician in any sense of the word. Would you like to hear any other tunes?

Achilles: By all means. Have you some others?

Crab: Scads.

(He takes a new sheet, and attaches it to his flute. It contains the following symbols:
3a:3b:(SSa. SSb)=SSSSSSSSSSSSSO

Achilles peers at it, while the Crab plays it.)

Isn't it lovely?

Achilles: Yes, it certainly is a tuneful little piece. But I have to say, it's looking more and more like number theory to me.

Crab: Heavens! It is just my usual music notation, nothing more. I simply don't know how you read all these extramusical connotations into a straightforward representation for sounds.

Achilles: Would you be averse to playing a piece of my own composition?

Crab: Not in the least. Have you got it with you?

Achilles: Not yet, but I have a hunch I might be able to compose some tunes all by myself.

Tortoise: I must. tell you, Achilles, that Mr. C is a harsh judge of music composed by others, so do not be disappointed if, by some chance, he is not an enthusiast for your efforts.

Achilles: That is very kind of you to forewarn me. Still, I'm willing to give it a try .

(He writes:

((SSSO . SSSO) +(SSSSO. SSSSO))=(SSSSSO. SSSSSO)

The Crab takes it, looks it over for a moment, then sets it in his music holder, and pipes.)
Crab: Why, that's quite nice, Achilles. I enjoy strange rhythms.

Achilles: What's strange about the rhythms in that piece?

Crab: Oh, naturally, to you as the composer it must seem quite bland, but to my ears, shifting from a 3/3 rhythm to 4/4 and then to 5/5 is quite exotic. If you have any other songs, I'd be glad to play them. Achilles: Thank you very much. I've never composed anything before, and I must say composing is quite different from how I had imagined it to be. Let me try my hand at another one. (
jots down a line
.)
3a:3b:(SSa - SSb) =SSSSSSSSSSSSSSO

Crab: Hmmm ... Isn't that just a copy of my earlier piece?

Achilles: Oh, no! I've added one more S. Where you had thirteen in a row, I have fourteen.

Crab: Oh, yes. Of course. (
He plays it, and looks very stern
.) Achilles: I do hope you didn't dislike my piece!

Crab: I am afraid, Achilles, that you completely failed to grasp the subtleties of my piece, upon which yours is modeled. But how could I expect you to understand it on first hearing? One does not always understand what is at the root of beauty. It is so easy to mistake the superficial aspects of a piece for its beauty, and to imitate them, when the beauty itself is locked deep inside the music, in a way which seems always to elude analysis.

Achilles: I am afraid that you have lost me a little in your erudite commentary. I understand that my piece does not measure up to your high standards, but I do not know exactly where I went astray. Could you perhaps tell me some specific way in which you find fault with my composition?

Crab: One possible way to save your composition, Achilles, would be to insert another three S's-five would do as well-into that long group of S's near the end. That would create a subtle and unusual effect.

Achilles: I see.

Crab: But there are other ways you might choose to change your piece. Personally, I would find it most appealing to put another tilde in the front. Then there would be a nice balance between the beginning and the end. Having two tildes in a row never fails to give a gay little twist to .a piece, you know.

Achilles: How about if I take both of your suggestions, and make the following piece?

-3a:3b:(SSa.SSb)=SSSSSSSSSSSSSSSSSO

Crab (a painful grimace crossing his face): Now, Achilles, it is important to learn the following lesson: never try to put too much into any single piece. There is always a point beyond which it cannot be improved,

and further attempts to improve it will in fact destroy it. Such is the case in this example.

Your idea of incorporating both of my suggestions together does not yield the desired extra amount of beauty, but on the contrary creates an imbalance which quite takes away all the charm.

Achilles: How is it that two very similar pieces, such as yours with thirteen 5's, and mine with fourteen S's, seem to you to be so different in their musical worth? Other than in that minor respect, the two are identical.

Crab: Gracious! There is a world of difference between your piece and mine. Perhaps this is a place where words fail to convey what the spirit can feel. Indeed, I would venture to say that there exists no set of rules which delineate what it is that makes a piece beautiful, nor could there ever exist such a set of rules. The sense of Beauty is the exclusive domain of Conscious Minds, minds which through the experience of living have gained a depth that transcends explanation by any mere set of rules.

Achilles: I will always remember this vivid clarification of the nature of Beauty. I suppose that something similar applies to the concept of Truth, as well?

Crab: Without doubt. Truth and Beauty are as interrelated as-as

Achilles: As interrelated as, say, mathematics and music?

Crab: Oh! You took the words right out of my mouth! How did you know that that is what I was thinking?

Tortoise: Achilles is very clever, Mr. C. Never underestimate the potency of his insight.

Achilles: Would you say that there could conceivably be any relationship between the truth or falsity of a particular statement of mathematics, and the beauty, or lack of beauty, of an associated piece of music? Or is that just a far-fetched fancy of mine, with no basis in reality?

Crab: If you are asking me, that is carrying things much too far. When I spoke of the interrelatedness of music and mathematics, I was speaking very figuratively, you know.

As for a direct connection between specific pieces of music and specific statements of mathematics, however, I harbor extremely grave doubts about its possibility. I would humbly counsel you not to give too' much time to such idle speculations.

Achilles: You are no doubt right. It would be most unprofitable. Perhaps I ought to concentrate on sharpening my musical sensitivity by composing some new pieces. Would you be willing to serve as my mentor,

Mr. C?

Crab: I would be very happy to aid you in your steps towards musical understanding.

(So Achilles takes pen in hand, and, with what appears to be a great deal of
concentration, writes
:

AOOaV'\/--nn:b+cS(33=OAD((-d)v

The Crab looks very startled.)

Y ou really want me to play that-that-that whatever-it-is?

Achilles: Oh, please do!

(So the Crab plays it, with evident difficulty.)

Tortoise: Bravo! Bravo! Is John Cage your favorite composer, Achilles? Achilles: Actually, he's my favorite anti-composer. Anyway, I'm glad you liked MY music.

Crab: The two of you may find it amusing to listen to such totally meaningless cacophony, but I assure you it is not at all pleasant for a sensitive composer to be subjected to such excruciating, empty dissonances and meaningless rhythms. Achilles, I thought you had a good feeling for music. Could it be that your previous pieces had merit merely by coincidence?

Achilles: Oh, please forgive me, Mr. Crab. I was trying to explore the limits of your musical notation. I wanted to learn directly what kinds of sound result when I write certain types of note sequences, and also how you evaluate pieces written in various styles.

Crab: Harrumph! I am not just an automatic music-machine, you know. Nor am I a garbage disposal for musical trash.

Achilles: I am very sorry. But I feel that I have learned a great deal by writing that small piece, and I am convinced that I can now write much better music than I ever could have if I hadn't tried that idea. And if you'll just play one more piece of mine, I have high hopes that you will feel better about my musical sensitivities.

Crab: Well, all right. Write it down and I'll give it a chance.

(
Achilles writes:

Ya:Vb:<(a -a) =(SSO -(b

> b))Da=0>

and the Crab plays.)

You were right, Achilles. You seem to have completely regained your musical acuity.

This is a little gem! How did you come to compose it? I have never heard anything like it.

It obeys all the rules of harmony, and yet has a certain-what shall I say?-irrational appeal to it. I can't put my finger on it, but I like it for that very reason.

Achilles: I kind of thought you might like it.

Tortoise: Have you got a name for it, Achilles? Perhaps you might call it "The Song of Pythagoras". You remember that Pythagoras and his followers were among the first to study musical sound.

Achilles: Yes, that's true. That would be a fine title.

Crab: Wasn't Pythagoras also the first to discover that the ratio of two squares can never be equal to 2? Tortoise: I believe you're right. It was considered a truly sinister discovery at the time, for never before had anyone realized that there are

numbers-such as the square root of 2-which are not ratios of integers. And thus the discovery was deeply disturbing to the Pythagoreans, who felt that it revealed an unsuspected and grotesque defect in the abstract world of numbers. But I don't know what this has to do with the price of tea in China.

Achilles: Speaking of tea, isn't that the teahouse just up there ahead of us?

Tortoise: Yes, that's it, all right. We ought to be there in a couple of minutes.

Achilles: Hmm ... That's just enough time for me to whistle for you the tune which the taxi driver this morning had on his radio. It went like this.

Crab: Hold on for a moment; I'll get some paper from my case, and jot down your tune. (
Scrounges around inside his case, and finds a blank sheet
.) Go ahead; I'm ready.

(Achilles whistles a rather long tune, and the Crab scrambles to keep up with him.)
Could you whistle the last few bars again?

Achilles: Why, certainly.

(After a couple of such repeats, the session is complete, and the Crab proudly displays
his transcription
:

<((SSSSSO.SSSSSO)+(SSSSSO.SSSSSO))=((SSSSSSSO.SSSSSSSO)+(SO.SO))n---

3b:<3c:(Sc+b)=((SSSSSSSO . SSSSSSSO)+(SO • SO))n3d:3d':3e:3e':

<---n> >>
The Crab then plays it himself.)

Tortoise: It's peculiar music, isn't it? It sounds a wee bit like music from India, to me.

Crab. Oh, I think it's too simple to be from India. But of course I know precious little about such things.

Tortoise: Well, here we are at the teahouse. Shall we sit outside here, on the verandah?

Crab: If you don't mind, I'd prefer to go inside. I've gotten perhaps enough sun for the day.

(They go inside the teahouse and are seated at a nice wooden table, and order cakes and
tea. Soon a cart of scrumptious-looking pastries is wheeled up, and each of them chooses
his favorite.)

Achilles: You know, Mr. C, I would love to know what you think of another piece which I have just composed in my head. Crab: Can you show it to me? Here, write it down on this napkin.

(Achilles writes
:

da:3b:3c:<- 3d:3e:<(SSd•SSe)=bv(SSd•SSe)=c>n(a+a)=(b+c)>
The Crab and Tortoise study it with interest
.)

Tortoise: Is it another beautiful piece, Mr. C, in your opinion?