Gödel, Escher, Bach: An Eternal Golden Braid (129 page)

(not Ì', nor "Ì"'). Got that?

FIGURE 140.
Pipe Dream. [Drawing by the author.]

The "Code" of Modern Art

A large number of influences, which no one could hope to pin down completely, led to further explorations of the symbol-object dualism in art. There is no doubt that John Cage, with his interest in Zen, had a profound influence on art as well as on music. His friends jasper Johns and Robert Rauschenberg both explored the distinction between objects and symbols by using objects as symbols for themselves-or, to flip the coin, by using symbols as objects in themselves. All of this was perhaps intended to break down the notion that art is one step removed from reality-that art speaks in "code", for which the viewer must act as interpreter. The idea was to eliminate the step of interpretation and let the naked object simply be, period. ("Period"-a curious case of use-mention blur.) However, if this was the intention, it was a monumental flop, and perhaps had to be.

Any time an object is exhibited in a gallery or dubbed a "work", it acquires an aura of deep inner significance-no matter how much the viewer has been warned not to look for meaning. In fact, there is a backfiring effect whereby the more that viewers are told to look at these objects without mystification, the more mystified the viewers get.

After all, if a

wooden crate on a museum floor is just a wooden crate on a museum floor, then why doesn't the janitor haul it out back and throw it in the garbage? Why is the name of an artist attached to it? Why did the artist want to demystify art? Why isn't that dirt clod out front labeled with an artist's name? Is this a hoax? Am I crazy, or are artists crazy? More and more questions flood into the viewer's mind; he can't help it. This is the "frame effect" which art-Art-automatically creates. There is no way to suppress the wonderings in the minds of the curious.

Of course, if the purpose is to instill a Zen-like sense of the world as devoid of categories and meanings, then perhaps such art is merely intended to serve-as does intellectualizing about Zen-as a catalyst to inspire the viewer to go out and become acquainted with the philosophy which rejects "inner meanings" and embraces the world as a whole. In this case, the art is self-defeating in the short run, since the viewers do ponder about its meaning, but it achieves its aim with a few people in the long run, by introducing them to its sources. But in either case, it is not true that there is no code by which ideas are conveyed to the viewer. Actually, the code is a much more complex thing, involving statements about the absence of codes and so forth-that is, it is part code, part metacode, and so on. There is a Tangled Hierarchy of messages being transmitted by the most Zen-like art objects, which is perhaps why so many find modern art so inscrutable.

Ism Once Again

Cage has led a movement to break the boundaries between art and nature. In music, the theme is that all sounds are equal-a sort of acoustical democracy. Thus silence is just as important as sound, and random sound is just as important as organized sound. Leonard B. Meyer, in his book
Music, the Arts, and Ideas
, has called this movement in music

"transcendentalism", and states:

If the distinction between art and nature is mistaken, aesthetic valuation is irrelevant. One should no more judge the value of a piano sonata than one should judge the value of a stone, a thunderstorm, or a starfish. "Categorical statements, such as right and wrong, beautiful or ugly, typical of the rationalistic thinking of tonal aesthetics," writes Luciano Berio [a contemporary composer, "are no longer useful in understanding why and how a composer today works on audible forms and musical action."

Later, Meyer continues in describing the philosophical position of transcendentalism:

... all things in all of time and space are inextricably connected with one another. Any divisions, classifications, or organizations discovered in the universe are arbitrary. The world is a complex, continuous, single event .2 [Shades of Zeno!]

I find "transcendentalism" too bulky a name for this movement. In its place, I use

"ism". Being a suffix without a prefix, it suggests an ideology

FIGURE 141.
The Human Condition I, by Rene Magritte (1933)
.

without ideas-which, however you interpret it, is probably the case. And since."ism"

embraces whatever is, its name is quite fitting. In "ism" thL- word "is" is half mentioned, half used; what could be more appropriate? Ism is the spirit of Zen in art. And just as the central problem of Zen is to unmask the self, the central problem of art in this century seems to be to figure out what art is. All these thrashings-about are part of its identity crisis.

We have seen that the use-mention dichotomy, when pushed, turns into the philosophical problem of symbol-object dualism, which links it to the mystery of mind.

Magritte wrote about his painting
The Human Condition
I (Fig. 141): I placed in front of a window, seen from a room, a painting representing exactly that part of the landscape which was hidden from view by the painting. Therefore, the tree represented in the painting hid from view the tree situated behind it, outside the room. It existed for the spectator, as it were, simultaneously in his mind, as both inside the room in the painting, and outside in the real landscape. Which is how we see the world: we see it as being outside ourselves even though it is only a mental representation of it that we

experience inside ourselves.'

Understanding the Mind

First through the pregnant images of his painting, and then in direct words, Magritte expresses the link between the two questions "How do symbols work?" and "How do our minds work?" And so he leads us back to the question posed earlier: "Can we ever hope to understand our minds! brains?"

Or does some marvelous diabolical Gödelian proposition preclude our ever unraveling our minds? Provided you do not adopt a totally unreasonable definition of

"understanding", I see no Gödelian obstacle in the way of the eventual understanding of our minds. For instance, it seems to me quite reasonable to desire to understand the working principles of brains in general, much the same way as we understand the working principles of car engines in general. It is quite different from trying to understand any single brain in every last detail-let alone trying to do this for one's own brain! I don't see how Gödel’s Theorem, even if construed in the sloppiest way, has anything to say about the feasibility of this prospect. I see no reason that Gödel’s Theorem imposes any limitations on our ability to formulate and verify the general mechanisms by which thought processes take place in the medium of nerve cells. I see no barrier imposed by Gödel’s Theorem to the implementation on computers (or their successors) of types of symbol manipulation that achieve roughly the same results as brains do. It is entirely another question to try and duplicate in a program some particular human's mind-but to produce an intelligent program at all is a more limited goal. Godel's Theorem doesn't ban our reproducing our own level of intelligence via programs any more than it bans our reproducing our own level of intelligence via transmission of hereditary information in

DNA
, followed by education. Indeed, we have seen, in Chapter XVI, how a remarkable

'Gödelian mechanism-the Strange Loop of proteins and
DNA
-is precisely what allows transmission of intelligence!

Does Gödel’s Theorem, then, have absolutely nothing to offer us in thinking about our own minds? I think it does, although not in the mystical and [imitative way which some people think it ought to. I think that the process of coming to understand Gödel’s proof, with its construction involving arbitrary codes, complex isomorphisms, high and low levels of interpretation, and the capacity for self-mirroring, may inject some rich undercurrents and flavors into one's set of images about symbols and symbol processing, which may deepen one's intuition for the relationship. between mental structures on different levels.

Accidental Inexplicability of Intelligence?

Before suggesting a philosophically intriguing "application" of Godel's proof. I would like to bring up the idea of "accidental inexplicability" of intelligence. Here is what that involves. It could be that our brains, unlike car engines, are stubborn and intractable systems which we cannot neatly decompose in any way. At present, we have no idea whether our brains will yield to repeated attempts to cleave them into clean layers, each of which can be explained in terms of lower layers-or whether our brains will foil all our attempts at decomposition.

But even if we do fail to understand ourselves, there need not be any Godelian

"twist" behind it; it could be simply an accident of fate that our brains are too weak to understand themselves. Think of the lowly giraffe, for instance, whose brain is obviously far below the level required for self-understanding-yet it is remarkably similar to our own brain. In fact, the brains of giraffes, elephants, baboons-even the brains of tortoises or unknown beings who are far smarter than we are-probably all operate on basically the same set of principles. Giraffes may lie far below the threshold of intelligence necessary to understand how those principles fit together to produce the qualities of mind; humans may lie closer to that threshold perhaps just barely below it, perhaps even above it. The point is that there may be no fundamental (i.e., Gödelian) reason why those qualities are incomprehensible; they may be completely clear to more intelligent beings.

Undecidability Is Inseparable from a High-Level Viewpoint

Barring this pessimistic notion of the accidental inexplicability of the brain, what insights might Gödel’s proof offer us about explanations of our minds/brains? Gödel’s proof offers the notion that a high-level view of a system may contain explanatory power which simply is absent on the lower levels. By this I mean the following. Suppose someone gave you G, Gödel’s undecidable string, as a string of
TNT
. Also suppose you knew nothing of' Gödel-numbering. The question you are supposed to answer is: "Why isn't this string a theorem of
TNT
?" Now you are used to such questions; for instance, if you had been asked that question about
SO=0,
you would have a ready explanation: "Its
negation
,
~S0=0
,
is a theorem
." This, together with your knowledge that
TNT
is consistent, provides an explanation of why the given string is a nontheorem. This is what I call an explanation "on the TNT-level". Notice how different it is from the explanation of why
MU
is not a theorem of the
MIU
-system: the former comes from the M-mode, the latter only from the I-mode.

Now what about
G
? The
TNT
-level explanation which worked for 50=0 does not work for
G
, because -
G
is
not
a theorem. The person who has no overview of
TNT
will be baffled as to why he can't make
G
according to the rules, because as an arithmetical proposition, it apparently has nothing wrong with it. In fact, when
G
is turned into a universally quantified string, every instance gotten from
G
by substituting numerals for the variables can be derived. The only way to explain
G
's nontheoremhood is to discover the notion of Gödel-numbering and view
TNT
on an entirely different level. It is not that it is just difficult and complicated to write out the explanation on the
TNT
-level; it is impossible. Such an explanation simply does not exist. There is, on the high level, a kind of explanatory power which simply is lacking, in principle, on the
TNT
-level.
G
's nontheoremhood is, so to speak, an
intrinsically high-level fact
. It is my suspicion that this is the case for all undecidable propositions; that is to say: every undecidable proposition is actually a Gödel sentence, asserting its own nontheoremhood in some system via some code.