Gödel, Escher, Bach: An Eternal Golden Braid (14 page)

Achilles: Oh, well, it's all the same to me.

Achilles: You're right; I wonder why I never noticed that difference between fiddles and guitars before. Speaking of fiddling, how would you like to come over and listen to one of the sonatas for unaccompanied violin by your favorite composer, J. S.

Bach? I just bought a marvelous recording of them. I still can't get over the way Bach uses a single violin to create a piece with such interest.

Achilles: A headache too? That's a shame. Perhaps you should just go to bed.

Achilles: I see. Have you tried counting sheep?

Achilles: Oh, oh, I see. Yes, I fully know what you mean. Well, if it's THAT distracting, perhaps you'd better tell it to me, and let me try to work on it, too.

Achilles: A word with the letters `
A', `D', À', `C'
consecutively inside it ... Hmm ...

What about "abracadabra"?

Achilles: True, "
ADAC
" occurs backwards, not forwards, in that word. Achilles: Hours and hours? It sounds like I'm in for a long puzzle, then. Where did you hear this infernal riddle?

Achilles: You mean he looked like he was meditating on esoteric Buddhist matters, but in reality he was just trying to think up complex word puzzles?

Achilles: Aha!-the snail knew what this fellow was up to. But how did you come to talk to the snail?

Achilles: Say, I once heard a word puzzle a little bit like this one. Do you want to hear it?

Or would it just drive you further into distraction? Achilles: I agree-can't do any harm. Here it is: What's a word that begins with the letters "
HE
" and also ends with "
HE
"?

Achilles: Very ingenious-but that's almost cheating. It's certainly not what I meant!

Achilles: Of course you're right-it fulfills the conditions, but it's a sort of "degenerate"

solution. There's another solution which I had in mind. Achilles: That's exactly it!

How did you come up with it so fast? Achilles: So here's a case where having a headache actually might have helped you, rather than hindering you. Excellent!

But I'm still in the dark on your "
ADAC
" puzzle.

Achilles: Congratulations! Now maybe you'll be able to get to sleep! So tell me, what is the solution?

Achilles: Well, normally I don't like hints, but all right. What's your hint? Achilles: I don't know what you mean by "figure" and "ground" in this case.

Achilles: Certainly I know Mosaic II! I know ALL of Escher's works. After all, he's my favorite artist. In any case, I've got a print of Mosaic II hanging on my wall, in plain view from here.

Achilles:: Yes, t see all the black animals.

Achilles: Yes, I also see how their "negative space" -- what's left out-- defines the white animals.

Achilles: So THAT'S what you mean by "figure" and "ground". But what does that have to do with the "
ADAC
" puzzle?

Achilles: Oh, this is too tricky for me. I think I'M starting to get a headache Achilles: You want to come over now? But I thought--

Achilles: Very well. Perhaps by then I'll have thought of the right answer to YOUR

puzzle, using your figure-ground hint, relating it to MY puzzle

Achilles: I'd love to play them for you.

Achilles: You've invented a theory about them?

Achilles: Accompanied by what instrument?

Achilles: Well, if that's the case, it seems a little strange that he would have written out the harpsichord part, then, and had it published a s well.

Achilles: I see -- sort of an optional feature. One could listen to them either way -- with or without accompaniment. But how would one know what the accompaniment is supposed to sound like?

Achilles: Ah, yes, I guess that it is best, after all, to leave it to the listener’s imagination.

And perhaps, as you said, Bach never even had accompaniment in mind at all.

Those sonatas seem to work very indeed as they are.

Achilles: Right. Well, I'll see you shortly.

Achilles: Good-bye, Mr. T.

Primes
vs.
Composites

THERE IS A strangeness to the idea that concepts can be captured by simple typographical manipulations. The one concept so far captured is that of addition, and it may not have appeared very strange. But suppose the goal were to create a formal system with theorems of the form
P
x, the letter `x' standing for a hyphen-string, and where the only such theorems would be ones in which the hyphen-string contained exactly a prime number of hyphens. Thus
, P---
would be a theorem, but
P----
would not. How could this be done typographically? First, it is important to specify clearly what is meant by
typographical
operations. The complete repertoire has been presented in the
MIU
-system and the pq-system, so we really only need to make a list of the kinds of things we have permitted:

(1) reading and recognizing any of a finite set of symbols;

(2) writing down any symbol belonging to that set;

(3) copying any of those symbols from one place to another;

(4) erasing any of those symbols;

(5) checking to see whether one symbol is the same as another;

(6) keeping and using a list of previously generated theorems.

The list is a little redundant, but no matter. What is important is that it clearly involves only trivial abilities, each of them far less than the ability to distinguish primes from nonprimes. How, then, could we compound some of these operations to make a formal system in which primes are distinguished from composite numbers?

The tq-System

A first step might be to try to solve a simpler, but related, problem. We could try to make a system similar to the pq-system, except that it represents multiplication, instead of addition. Let's call it the
tq-system
, `t' for times'. More specifically, suppose X, Y, and Z

are, respectively, the numbers of hyphens in the hyphen-strings x, y, and z. (Notice I am taking special pains to distinguish between a string and the number of hyphens it contains.) Then we wish the string x ty q z to be a theorem if and only if X times Y

equals Z. For instance,
--t---q-----
should be a theorem because 2 times 3 equals 6, but --

t--q--- should not be a theorem. The tq-system can be characterized just about as easily as the pq-system namely, by using just one axiom schema and one rule of inference: AXIOM SCHEMA: x
t
-
q
x is an axiom, whenever x is a hyphen string.

.

RULE OF INFERENCE: Suppose that x, y, and z are all hyphen-strings. An suppose that x
t
y
q
z is an old theorem. Then, xty-qzx is a ne' theorem.

Below is the derivation of the theorem
--t---q-----

:

(1)

--t-q--

(axiom)

(2)

--t—q----

(by rule of inference, using line (1) as the old theorem)

(3)

--t---q ------- (by rule of inference, using line (2) as the old theorem) Notice how the middle hyphen-string grows by one hyphen each time the rule of inference is applied; so it is predictable that if you want a theorem with ten hyphens in the middle, you apply the rule of inference nine times in a row.

Capturing Compositeness

Multiplication, a slightly trickier concept than addition, has now bee] "captured"

typographically, like the birds in Escher's Liberation. What about primeness? Here's a plan that might seem smart: using the tq-system define a new set of theorems of the form
C
x, which characterize compost. numbers, as follows:

RULE: Suppose x, y, and z are hyphen-strings. If x-ty-qz is a theorem then
C
z is a theorem.

This works by saying that Z (the number of hyphens in z) is composite a long as it is the product of two numbers greater than 1-namely, X + (the number of hyphens in x-), and Y

+ 1 (the number of hyphens in y I am defending this new rule by giving you some

"Intelligent mode justifications for it. That is because you are a human being, and want t, know
why
there is such a rule. If you were operating exclusively in the "Mechanical mode", you would not need any justification, since M-mod. workers just follow the rules mechanically and happily, never questioning; them!

Because you work in the I-mode, you will tend to blur in your mind the distinction between strings and their interpretations. You see, things Cal become quite confusing as soon as you perceive "meaning" in the symbol which you are manipulating.

You have to fight your own self to keep from thinking that the
string'
---' is the
number
3.

The Requirement of Formality, which in Chapter I probably seemed puzzling (because it seemed so obvious), here becomes tricky, and crucial. It is the essential thing which keeps you from mixing up the I-mode with the M-mode; or said another way, it keeps you from mixing up arithmetical facts with typographical theorems.

Illegally Characterizing Primes

It is very tempting to jump from the C-type theorems directly to P-type theorems, by proposing a rule of the following kind:

PROPOSED RULE: Suppose x is a hyphen-string. If
C
x is not a theorem, then
P
x is a theorem.

The fatal flaw here is that checking whether
C
x is not a theorem is not an explicitly typographical operation. To know for sure that
MU
is not a theorem of the MIU-system, you have to go
outside
of the system ... and so it is with this Proposed Rule. It is a rule which violates the whole idea of formal systems, in that it asks you to operate informally-that is, outside the system. Typographical operation (6) allows you to look into the stockpile of previously found theorems, but this Proposed Rule is asking you to look into a hypothetical "Table of Nontheorems". But in order to generate such a table, you would have to do some reasoning
outside the system
-reasoning which shows why various strings cannot be generated inside the system. Now it may well be that there is another formal system which can generate the "Table of Nontheorems", by purely typographical means.

In fact, our aim is to find just such a system. But the Proposed Rule is not a typographical rule, and must be dropped.

This is such an important point that we might dwell on it a bit more. In our
C
-

system (which includes the tq-system and the rule which defines C-type theorems), we have theorems of the form
C
x, with `x' standing, as usual, for a hyphen-string. There are also nontheorems of the form
C
x. (These are what I mean when I refer to "nontheorems", although of course
tt-Cqq
and other ill-formed messes are also nontheorems.) The difference is that theorems have a composite number of hyphens, nontheorems have a prime number of hyphens. Now the theorems all have a common "form", that is, originate from a common set of typographical rules. Do all nontheorems also have a common

"form", in the same sense? Below is a list of C-type theorems, shown without their derivations. The parenthesized numbers following them simply count the hyphens in them.

C---- (4)

C-------- (6)

C----------------(8)

C-----------------(9)

C--------------------(10)

C -------------------- (12)

C------------------------ (14)

C ------------------------ (15)

C----------------------------(16)

C----------------------------(18)

.

.

.

I he "holes" in this list are the nontheorems. I o repeat the earlier quest Do the holes also have some "form" in common? Would it be reasonable say that merely by virtue of being the holes in this list, they share a common form? Yes and no. That they share
some
typographical quality is and able, but whether we want to call it "form" is unclear. The reason hesitating is that the holes are only
negatively
defined-they are the things that are left out of a list which is
positively
defined.

Figure and Ground

This recalls the famous artistic distinction between
figure
and
ground
. When a figure or

"positive space" (e.g., a human form, or a letter, or a still life is drawn inside a frame, an unavoidable consequence is that its complementary shape-also called the "ground", or

"background", or "negative space"-has also been drawn. In most drawings, however, this fig ground relationship plays little role. The artist is much less interested in ground than in the figure. But sometimes, an artist will take interest in ground as well.