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For example, how is it possible that this colloquium in honor of Luhmann
was announced, that one would hear various speakers talk, and, although
we have no idea how the brain works, that we all arrived here promptly at
9 o'clock. And what do we see? Everyone is here, everyone is listening, one
of them makes noises with his mouth, some are taking notes, etc. Indeed,
how is that possible? What's going on here?
For this I'd like now to take the next step. As I hope to show you, all of
this can only happen because these systems operate recursively!
Second Chapter: Recursors
In order to develop the following thoughts as clearly as possible, I will
increase their complexity stepwise, so that you can follow, step by step, what
it's all about.
Dimensionality 1 (Operationally Open)
I'll begin with systems of dimensionality 1. Why “dimensionality 1?”
Because here signals are linear and flow in one direction only. One could
represent this situation in its brutal simplicity by a directed line segment
(Figure 4) in which all operations that transform
x
into
y
are comprised by
the one single point “0”.
Since it is my intention here to talk later on about compositions of at
least two systems, I'll present you now with the two systems
D
and
S
, which
should help me out in the following exposition (Figure 5).
D
operates on the variable
x
and produces
y
, which is expressed by the
function
y
=
D(x)
. The same holds,
mutatis mutandis
, for machine
S
.
FIGURE 4.
FIGURE 5.
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