Gödel, Escher, Bach: An Eternal Golden Braid (136 page)

Author: Oh, no! That would wreck the beauty of the scheme. The idea is to imitate Gödel’s self-referential construction, which as you know is INDIRECT, and depends on the isomorphism set up by Gödel numbering.

Crab: Oh. Well, in the programming language LISP, you can talk about your own programs directly, instead of indirectly, because programs and data have exactly the same form. Gödel should have just thought up LISP, and then

Author: But

Crab: I mean, he should have formalized quotation. With a language able to talk about itself, the proof of his Theorem would have been so much simpler!

Author: I see what you mean, but I don't agree with the spirit of your remarks. The whole point of Gödel-numbering is that it shows how even WITHOUT formalizing quotation, one can get self-reference: through a code. Whereas from hearing YOU

talk, one might get the impression that by formalizing quotation, you'd get something NEW, something that wasn't feasible through the code-which is not the case.

In any event, I find indirect self-reference a more general concept, and far more stimulating, than direct self-reference. Moreover, no reference is truly direct-every reference depends on SOME kind of coding scheme. It's just a question of how implicit it is. Therefore, no self reference is direct, not even in
LISP
.

Achilles: How come you talk so much about indirect self-reference?

Author: Quite simple-indirect self-reference is my favorite topic.

Crab: Is there any counterpart in your Dialogues to modulation between keys?

Author: Definitely. The topic of conversation may appear to change, though on a more abstract level, the Theme remains invariant. This happens repeatedly in the Prelude, Ant Fugue and other Dialogues. One can have a whole series of "modulations" which lead you from topic to topic and in the end come full circle, so that you end back in the

"tonic"-that is to say, the original topic.

Crab: I see. Your book looks quite amusing. I'd like to read it sometime.

(Flips through the manuscript, halting at the last Dialogue.)
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.

Crab: Quite Gödelian, Tell me -does your
Six-Part Ricercar
Dialogue attempt to copy in form the Bach piece it's based on?

Author: In many ways, yes. For instance, in the Bach, there's a section where the texture thins out to three voices only. I imitate that in the

Dialogue, by having only three characters interact for a while. Achilles: That's a nice touch.

Author: Thank you.

Crab: And how do you represent the King's Theme in your Dialogue?

Author: It is represented by the Crab's Theme, as I shall now demonstrate. Mr. Crab, could you sing your Theme for my readers, as well as for us assembled musicians?

Crab: Compose Ever Greater Artificial Brains (By And By).

FIGURE 151. The
Crab's Theme: C-Eb-G-Ab-B-B-A-B.

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

FIGURE152.
Last page of Six-part Ricercar, from the original edition of the
Musical Offering,
by J.S. Bach
.

unlikely threesome, at first thought. My favorite artist, Mr. T's favorite composer, and---

Crab: My favorite logician!

Tortoise: A harmonious triad, I'd say.

Babbage: A major triad, I'd say.

Turing: A minor triad, I'd say.

Author: I guess it all depends on how you look at it. But major or minor, I'd be most pleased to tell you how I braid the three together, Achilles. Of course, this project is not the kind of thing that one does in just one sitting-it might take a couple of dozen sessions. I'd begin by- telling you the story of the
Musical Offering
, stressing the Endlessly Rising Canon, and

Achilles: Oh, wonderful! I was listening with fascination to you and Mr. Crab talk about the Musical Offering and its story. From the way you two talk about it, I get the impression that the .
Musical Offering
contains a host of formal structural tricks.

Author: After describing the Endlessly Rising Canon, I'd go on to describe formal systems and recursion, getting in some comments about figures and grounds, too.

Then we'd come to self-reference and self-replication, and wind up with a discussion of hierarchical systems and the Crab's Theme.

Achilles: That sounds most promising. Can we begin tonight?

Author: Why not?

Babbage: But before we begin, wouldn't it be nice if the six of us-all of us by chance avid amateur musicians-sat down together and accomplished the original purpose of the evening: to make music?

Turing: Now we are exactly the right number to play the
Six-Part Ricercar
from the
Musical Offering
. What do you say to that?

Crab: I could get along very well with such a program.

Author: Well put, Mr. C. And as soon as we're finished, I'll begin my Braid, Achilles. I think you'll enjoy it.

Achilles: Wonderful! It sounds as if there are many levels to it, but I'm finally getting used to that kind of thing, having known Mr. T for so long. There's just one request I would like to make: could we also play the Endlessly Rising Canon? It's my favorite canon.

Tortoise: Reentering Introduction Creates Endlessly Rising Canon, After
RICERCAR
.

Notes

Introduction: A Musico-Logical Offering

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.

Chapter IV: Consistency, Completeness, and Geometry

1 Herbert Meschkovcski.
Non-Euclidean

2 Ibid., p. 33.

Geometry
. pp. 31-2

1 George Steiner..
After Babel
, pp. 172-3.

.2 Leonard B. Meter.
Music, The Arts, and Ideas
,

pp. 87-8

1 Gyomay M. Kubose,
Zen Koans
, p. 178

3 A. R. Anderson and N. D. Belnap, Jr. .

2 Ibid., p. 178.

Entailment
(Princeton, N.J.: Princeton University

Press. 1975).

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.

1 Carl Sagan, ed.
Communication with

, pp 251-2.

Extraterrestrial Intelligence
, p. 78.