For Niklas Luhmann:* “How Recursive is Communication?” - Understanding Understanding: Essays on Cybernetics and Cognition - page 317

impossible to find out what generates these stabilities. One cannot analyt-
ically determine how this system operates, although we see that it does
operate in a way that permits us to make predictions.
Third Chapter: Compositions
I have up till now spoken of systems as entities, spoken about their behav-
ior, their synthesis, analysis, and taxonomy. But here I am in the company
of scholars of sociology, i.e., the science of the “ socius ,” of the companion
and comrade, of the “ secundus ,” the follower, the second. So I must concern
myself with at least two systems, with their behavior, their synthesis and
analysis. Indeed a society usually consists of more than two members, but
if the process of integration, the “composition” of two systems has been
established, one can use stepwise recursion to apply the established com-
position rule to an arbitrary number of new arrivals.
How does such a composition come about?
Here, I believe, is perhaps the essential step in my exposition, for through
the composition of two systems of dimensionality 2, the recursors, there
emerge systems that are irreducibly of dimensionality 3.
But how is one to proceed?
Dimensionality 3 (Calculus of Recursive Functors)
The systems in Figure 8 should help me out here. I'll go back to the two
machines, the recursors D and S from Figure 7.
In step one (orientation) I rotate recursor S 90°, so that the variables and
parameters in D and S are aligned with one another; in step two (compo-
sition) I push the two together, so that out of the two separate systems D
and S a new machine now arises, a DS -composition. This new machine is
distinguished by its double closure, first a closure on u , that previously, as a
parameter, controlled D , and then the closure on x , that previously, as a
parameter, controlled S . So now both systems control one another recip-
rocally; the operational functions of the one system become functions of
the other: two recursive functors.
Extensions of the Second Order
0.5 Functors: functions of functions (functions of the second order)
From your middle school years your can surely recall the differential and
integral calculus. One wrote dy/dx and spoke of the “derivative of y with
respect to x ,” whereby y is a function of x : y = f(x) . That is, the derivative,
or differential operator Di , as I'd like to call it, is a functor, for it operates