Gödel, Escher, Bach: An Eternal Golden Braid (56 page)

Firstly, it has the one which was given before:

30 is a
MIU
-number.

But secondly, we know that this statement is tied (via isomorphism) to the statement
MU
is a theorem of the
MIU
-system.

So we can legitimately quote this latter as the second passive meaning of
MUMON
. It may seem very strange because, after all,
MUMON
contains nothing but plus signs, parentheses, and so forth-symbols of
TNT
. How can it possibly express any statement with other than arithmetical content?

The fact is, it can. Just as a single musical line may serve as both harmony and melody in a single piece; just as "
BACH
" may be interpreted as both a name and a melody; just as a single sentence may be an accurate structural description of a picture by Escher, of a section of
DNA
, of a piece by Bach, and of the dialogue in which the sentence is embedded, so
MUMON
can be taken in (at least) two entirely different ways.

This state of affairs comes about because of two facts:

Fact 1. Statements such as "
MU
is a theorem" can be coded into number theory via Gödel’s isomorphism.

Fact 2. Statements of number theory can be translated into
TNT
.

It could be said that
MUMON
is, by Fact 1, a coded message, where the symbols of the code are, by Fact 2, just symbols of
TNT
.

Codes and Implicit Meaning

Now it could be objected here that a coded message, unlike an uncod message, does not express anything on its own-it requires knowledge the code. But in reality there is no such thing as an uncoded message. There are only messages written in more familiar codes, and message written in less familiar codes. If the meaning of a message is to be revealed it must be pulled out of the code by some sort of mechanism, or isomorphism. It may be difficult to discover the method by which the decoding should be done; but once that method has been discovered, the message becomes transparent as water. When a code is familiar enough, it ceases appearing like a code; one forgets that there is a decoding mechanism. The message is identified with its meaning.

Here we have a case where the identification of message and meant is so strong that it is hard for us to conceive of an alternate meaning: residing in the same symbols.

Namely, we are so prejudiced by the symbols of
TNT
towards seeing number-theoretical meaning (and only numb( theoretical meaning) in strings of
TNT
, that to conceive of certain string of
TNT
as statements about the
MIU
-system is quite difficult. But Gödel’s isomorphism compels us to recognize this second level of meaning certain strings of
TNT
.

Decoded in the more familiar way, MUMON bears the message:

30 is a
MIU
-number.

This is a statement of number theory, gotten by interpreting each sign the conventional way.

But in discovering Gödel-numbering and the whole isomorphism bu upon it, we have in a sense broken a code in which messages about the
MIU
-system are written in strings of
TNT
. Gödel’s isomorphism is a n( information-revealer, just as the decipherments of ancient scripts we information-revealers. Decoded by this new and less familiar mechanism
MUMON
bears the message

MU
is a theorem of the
MIU
-system.

The moral of the story is one we have heard before: that meaning is ; automatic by-product of our recognition of any isomorphism; therefore there are at least two passive meanings of
MUMON
-maybe more!

The Boomerang: Gödel-Numbering TNT

Of course things do not stop here. We have only begun realizing the: potential of Gödel’s isomorphism. The natural trick would be to turn
TNT's
capability of mirroring other formal systems back on itself, as the Tortoise turned the Crab's phonographs against themselves, and as his Goblet G turned against itself, in destroying itself. In order to do this, we

will have to Gödel-number
TNT
itself, just as we did the
MIU
-system, and then

"arithmetize" its rules of inference. The Gödel-numbering is easy to do. For instance, we could make the following correspondence:

Symbol...

Codon

Mnemonic Justification

0

666

Number of the Beast for the Mysterious Zero

S

.....

123

successorship: 1, 2, 3,

=

.....

111

visual resemblance, turned sideways

+

....

112

1+1=2

.

....

236

2x3=6

(

....

362

ends in 2

*

)

....

323

ends in 3

*

<

212

ends in 2

* these three pairs

>

....

213

ends in 3

* form a pattern

[

....

312

ends in 2

*

]

....

313

ends in 3

*

α

....

262

opposite to V (626)

´
....

163

163 is prime

∧

.....

161

´∧ís a "graph" of the sequence 1-6-1

∨

.....

616

´∨' is a "graph" of the sequence 6-1-6

⊃

.....

633

` 6 "implies" 3 and 3, in some sense .

~

.....

223

. 2 + 2 is not 3

ℑ

.....

333

`ℑ' looks likè3'

V

.....

626

opposite to a; also a "graph" of 6-2-6

.
:

.....

636

two dots, two sixes

punc. .....

611

special number, as on Bell system (411, 911)

Each symbol of
TNT
is matched up with a triplet composed of the digits 1, 2, 3, and 6, in a manner chosen for mnemonic value. I shall call each such triplet of digits a
Gödel codon
, or
codon
for short. Notice that I have given no codon for
b, c, d
, or
e
; we are using austere
TNT
. There is a hidden motivation for this, which you will find out about in Chapter XVI. I will explain the bottom entry, "punctuation", in Chapter XIV.

Now we can rewrite any string or rule of
TNT
in the new garb. Here, for instance, is Axiom 1 in the two notations, the old below the new:

626,262,636,223,123,262,111,666

V a

: ~ S a = 0

Conveniently, the standard convention of putting in a comma every third digit happens to coincide with our colons, setting them off for "easy" legibility.

Here is the Rule of Detachment, in the new notation:

RULE: If x and 212x6331213 are both theorems, then 1 is a theorem. Finally, here is an entire derivation taken from last Chapter, given in austere TNT and also transcribed into the new notation:

626,262,636,626262,163,636,362262,112,123,262,163,323,111,123,362,262,112,262,163,323 axiom:
V
α
: : V
α
' : (
α
+ 5
α
' ) = S (
α
+
α
' )
626,262.163,636,362,123,666,112,123,262,163,323,111,123,362,123,666,112,262,163,32 specification
V
α
' : ( S 0 + S
α
´ ) = S ( S 0 +
α
´ )
362,123,666,112,123,666,323,1 11,123,362,123,666,112,666,323

specification

( S 0 + 5 0 ) = S ( S 0 + 0 )
626,262,636,362 262,112,666, 3 23,111,262

axiom

V •
α
: (
α
+ 0 ) =
α

362,123,666,112,666,323,111,123,666

specification

( S 0 + 0 ) = S 0

123,362,123,666,112,666,323,11 1,123,123,666

insert '12;

S ( S 0 + 0 ) = S S 0

362,123,666,112,123,666,323,111,123,123,666

transitivity

( S 0 + 5 0 ) = S S 0

Notice that I changed the name of the "Add S" rule to "Insert `123' ", since that is the typographical operation which it now legitimizes.

This new notation has a pretty strange feel to it. You lose all sense o meaning; but if you had been brought up on it, you could read strings it this notation as easily as you do TNT. You would be able to look and, at glance, distinguish well-formed formulas from ill-formed ones. Naturally since it is so visual, you would think of this as a typographical operation but at the same time, picking out well-formed
formulas
in this notation i picking out a special class of
integers
, which have an arithmetical definition too.

Now what about "arithmetizing" all the rules of inference? As matter stand, they are all still typographical rules. But wait! According to the Central Proposition, a typographical rule is really equivalent to al arithmetical rule. Inserting and moving digits in decimally represented numbers is an
arithmetical
operation, which can be carried out typographically. Just as appending a 'O' on the end is exactly the same as multiplying b, 10, so each rule is a condensed way of describing a messy arithmetical operation.

Therefore, in a sense, we do not even need to look for equivalent arithmetical rules, because all of the rules are
already
arithmetical!

TNT-Numbers: A Recursively Enumerable Set of Numbers

Looked

at

this

way,

the

preceding

derivation

of

the

theorem

"362,123,666,112,123,666,323,111,123,123,666" is a sequence of high] convoluted number-theoretical transformations, each of which acts on one or more input numbers, and yields an output number, which is, as before, called a
producible
number, or, to be more specific, a
TNT
-
number
. Some the arithmetical rules take an old
TNT
-number and increase it in a particular way, to yield a new
TNT
-number; some take an old
TNT
-

number a and decrease it; other rules take two
TNT
-numbers, operate on each of them some odd way, and then combine the results into a new
TNT
-number and so on and so forth. And instead of starting with just one know:
'TNT
-number, we have
five
initial
TNT
-numbers-one for each (austere axiom, of course. Arithmetized
TNT

is actually extremely similar to the

arithmetized
MIU
-system, only there are more rules and axioms, and to write out arithmetical equivalents explicitly would be a big bother-and quite unenlightening, incidentally. If you followed how it was done for the MIU-system, there ought to be no doubt on your part that it is quite analogous here.

There is a new number-theoretical predicate brought into being by this

"Godelization" of
TNT
: the predicate

α is a
TNT
-number.

For

example,

we

know

from

the

preceding

derivation

that

362,123,666,112,123,666,323,111,123,123,666 is a
TNT
-number, while on the other hand, presumably 123,666,111,666 is not a
TNT
-number.

Now it occurs to us that this new number-theoretical! predicate is
expressible
by some string of
TNT
with one free variable, say a. We could put a tilde in front, and that string would express the complementary notion

α is not a
TNT
-number.

Now if we replaced all the occurrences of a in this second string by the
TNT
-numeral for 123,666,111,666-a numeral which would contain exactly 123,666,111,666 S's, much too long to write out-we would have a
TNT
-string which, just like
MUMON
, is capable of being interpreted on two levels. In the first place, that string would say 123,666,111,666 is not a
TNT
-number.

But because of the isomorphism which links
TNT
-numbers to theorems of
TNT
, there would be a second-level meaning of this string, which is:

S0=0
is not a theorem of TNT.

TNT Tries to Swallow Itself

This unexpected double-entendre demonstrates that
TNT
contains strings which talk about other strings of
TNT
. In other words, the metalanguage in which we, on the outside, can speak about
TNT
, is at least partially imitated
inside
TNT
itself. And this is not an accidental feature of
TNT
; it happens because the architecture of any formal system can be mirrored inside
N
(number theory). It is just as inevitable a feature of
TNT

as are the vibrations induced in a record player when it plays a record. It seems as if vibrations should come from the outside world-for instance, from jumping children or bouncing balls; but a side effect of producing sounds-and an unavoidable one-is that they wrap around and shake the very mechanism which produces them. It is no accident; it is a side effect which cannot be helped. It is in the nature of record players. And it is in the nature of any formalization of number theory that its metalanguage is embedded within it.