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This means that:
Op
2
(2)
(2.4452. . .) = 2.4452 stable
Op
2
(2)
(0.4643. . .) = 0.4643 stable
A3. Consider the differential operator Op
3
:
d
dx
Op
3
=
.
The eigenfunction for this operator is the exponential function “exp:”
Op
3
(exp) = exp
i.e.,
de
dx
x
=
e
x
The generalizations of this operator are, of course, all differential equation,
integral equations, integro-differential equations, etc., which can be seen at
once when these equations are re-written in operator form, say:
F(Op
3
(n)
,Op
3
(n-1)
....,f) = 0
Of course, these operators, in turn, may be eigenvalues (eigen-operators)
of “meta-operators” and so on. This suggests that COORD, for instance,
may itself be treated as an eigen-operator, stable within bounds, and
jumping to other values whenever the boundary conditions exceed its
former stable domain:
Op(COORD
i
) = COORD
i
.
One may be tempted to extend the concept of a meta-operator to that of
a “meta-meta-operator” that computes the “eigen-meta-operators,” and so
on and up a hierarchy without end. However, there is no need to invoke
this escape as Warren S. McCulloch has demonstrated years ago in his paper
(1945): “A Heterarchy of Values Determined by the Topology of Nervous
Nets.”
It would go too far in this presentation to demonstrate the construction
of heterarchies of operators based on their composability.
A4. Consider the (self-referential) proposition:
“THIS SENTENCE HAS...LETTERS”
and complete it by writing into the appropriate space the word for the
number (or if there are more than one, the numbers) that make this propo-
sition true.
Proceeding by trial and error (comparing what this sentence says (abscissa)
with what it is (ordinate)): one finds two eigenvalues “thirty-one” and “thirty-
three.” Apply the proposition above to itself: “this sentence has thirty-one
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