For Niklas Luhmann:* “How Recursive is Communication?” - Understanding Understanding: Essays on Cybernetics and Cognition - page 316

of taking roots belong the Eigen values 1 and also 0, since any root of 0 is
0. The essential difference between these two Eigen values is that for every
deviation from 1, recursion leads the system back to 1, while at the least
deviation from 0 the system leaves null and wanders to the stable Eigen
value “one.”
About 20 years ago there was an explosion of renewed interest in these
recursive operations, as one discovered that many functions develop not
only stable values but also a stable dynamic . One called these stabilities
“attractors,” apparently a leftover from a teleological way of thinking. Since
one can let some systems march through the most diverse Eigen behaviors
by making simple changes in the parameters, one soon stumbled onto a
most interesting behavior that is launched by certain parametric values: the
system rolls through a sequence of values without ever repeating one,
and even if one believes one has taken one of these values as the initial
value, the sequence of values cannot be reproduced: the system is chaotic.
Let me make just a couple of more remarks about stable Eigen
behaviors.
Consider next the fascinating process that recursively sifts only discrete
values out of a continuum of endless possibilities. Recall the operation of
taking roots, which lets one and only one number, namely “1,” emerge from
the endless domain of the real numbers. Can that serve as a metaphor for
the recursiveness of the natural process, sometimes also called “evolution,”
in which discrete entities are sifted out of the infinite abundance of possi-
bilities, such as a fly, an elephant, even a Luhmann? I say yes,” and hope to
contribute additional building blocks to the foundation of my assertion.
But consider also that although one can indeed make the inference from
given operations to their Eigen behaviors, one cannot make the converse
deduction from a stable behavior, an Eigen behavior, to the corresponding
generative operations. For example, “one” is the Eigen value of infinitely
many different operations. Therefore, the inference from the recursive
Eigen value “1” to the square root operation as the generator is not valid,
because the fourth, the tenth, the hundredth root, recursively applied, yield
the same Eigen value “1.” Can that serve as a metaphor for the recursive-
ness of the natural process, sometimes called the “laws of nature,” of which
there could be infinitely many versions that would explain a Milky Way, a
planetary system, indeed, even a Luhmann? I say “yes” and turn for support
to Wittgenstein's Tractatus , Point 5.1361: “The belief in the causal nexus is
the superstition.”
This result, that there emerge Eigen values, is the only thing we can rely
on. For then an opaque machine begins to behave in a predictable way, for
as soon as it has run into an Eigen state, I can of course tell you, for example,
if this Eigen state is a period, what the next value in the period is. Through
this recursive closure and only through this recursive closure do stabilities
arise that could never be discovered through input/output analysis. What is
fascinating is that while one can observe these stabilities it is in principle