some sort of inner meaning inside the picture, a multidimensional aspect
trapped somehow inside those two dimensions. It is the word "meaning"
which is important here. Our minds contain interpreters which accept
two-dimensional patterns and then "pull" from them high-dimensional
notions which are so complex that we cannot consciously describe them.
The same can be said about how we respond to music, incidentally.
It feels subjectively that the pulling-out mechanism of inner meaning is
not at all akin to a decision procedure which checks for the presence or
absence of some particular quality such as well-formedness in a string.
Probably this is because inner meaning is something which reveals more of
itself over a period of time. One can never be sure, as one can about
well-formedness, that one has finished with the issue.
This suggests a distinction that could be drawn between two senses of
"form" in patterns which we analyze. First, there are qualities such as
well-formedness, which can be detected by predictably terminating tests, as in
BlooP programs. These I propose to call syntactic qualities of form. One
intuitively feels about the syntactic aspects of form that they lie close to the
surface, and therefore they do not provoke the creation of multidimen-
sional cognitive structures.
By contrast, the semantic aspects of form are those which cannot be
tested for in predictable lengths of time: they require open-ended tests. Such
an aspect is theoremhood of TNT-strings, as we have seen. You cannot just
apply some standard test to a string and find out if it is a theorem.
Somehow, the fact that its meaning is involved is crucially related to the
difficulty of telling whether or not a string is a TNT-theorem. The act of
pulling out a string's meaning involves, in essence, establishing all the
implications of its connections to all other strings, and this leads, to be sure,
down an open-ended trail. So "semantic" properties are connected to
open-ended searches because, in an important sense, an object's meaning is
not localiz.ed within the object itself. This is not to say that no understanding
of any object's meaning is possible until the end of time, for as time passes,
more and more of the meaning unfolds. However, there are always aspects
of its meaning which will remain hidden arbitrarily long.
Meaning Derives from Connections to Cognitive Structures
Let us switch from strings to pieces of music, just for variety. You may still
substitute the term "string" for every reference to a piece of music, if you
prefer. The discussion is meant to be general, but its flavor is better gotten
across, I feel, by referring to music. There is a strange duality about the
meaning of a piece of music: on the one hand, it seems to be spread
around, by virtue of its relation to many other things in the world-and yet,
on the other hand, the meaning of a piece of music is obviously derived
from the music itself, so it must be localized somewhere inside the music.
The resolution of this dilemma comes from thinking about the inter-
preter-the mechanism which does the pulling-out of meaning. (By "inter-(^582) Church, Turing, Tarski, and Others