The Communication of Meaning in Anticipatory Systems:

A Simulation Study of the Dynamics of Intentionality in Social Interactions

Amsterdam School of Communications Research (ASCoR),
University of Amsterdam, Kloveniersburgwal 48,

1012 CX Amsterdam, The Netherlands; http://www.leydesdorff.net ;
loet@leydesdorff.net

**Abstract. **Psychological and social systems provide us with a
natural domain for the study of anticipations because these systems are based
on and operate in terms of intentionality. Psychological systems can be
expected to contain a model of themselves and their environments; social
systems can be strongly anticipatory and therefore co-construct their
environments, for example, in techno-economic (co-)evolutions. Using Dubois’ hyper-incursive
and incursive formulations of the logistic equation, these two types of systems
and their couplings can be simulated. In addition to their structural coupling,
psychological and social systems are also coupled by providing meaning reflexively
to each other’s meaning-processing. Luhmann’s distinctions among (1)
interactions between intentions at the micro-level, (2) organization at the
meso-level, and (3) self-organization of the fluxes of meaningful communication
at the global level can be modeled and simulated using three hyper-incursive
equations. The global level of self-organizing interactions among fluxes of
communication is retained at the meso-level of organization. In a
knowledge-based economy, these two levels of anticipatory structuration can be
expected to propel each other at the supra-individual level.

Keywords: anticipation, social system, meaning, communication, incursion, double contingency

PACS: 87.23.Ge; 89.65.-s; 89.70.+c; 89.75.Fb

The hyper-incursive formulation of the logistic equation—*x*_{t}
= *ax*_{t+1}(1 – *x*_{t+1}) [1]—provides us with an
operationalization of the concept of “double contingency” which has been
central to the theory of social systems [2‑7]. According to Luhmann
[5:70], “double contingency” can be considered as the auto-catalyst of social
processes between reflexive individuals.

The concept of double contingency was first formulated by the Amercian sociologist Talcott Parsons, who defined “double contingency” in 1951 as follows:

The expectation is not defined “*Being* what I
am, alter’s treatment of me must take one of the following alternatives” but
“Depending on *which of several alternatives open to me* *I take*, I
will set *alter* a problem to which he will react in terms of the
alternative system of his own which is oriented to my action.”[2:94]

According to Parsons [3: 436], the theory of games could be considered
as a most sophisticated analysis of the implications of double contingency.
However, one can also specify the dynamics of expectations in terms of the
theory of strongly anticipatory systems [8,9] and provide the above (formal)
equation with the following (substantive) interpretation: *Ego *(at *x** _{t}*)

While Parsons based his definition mainly on American pragmatism [2,9],
the German sociologist Niklas Luhmann elaborated on double contingency in the
continental tradition. He based himself on Edmund Husserl’s transcendental
phenomenology. Husserl had concluded that “intersubjectivity” provides us with
an intentionality different from and transcendental to subjectivity [10:144].
While subjective intentionality is a natural consequence of the *cogito*
of the *cogitantes*, intersubjective intentionality remains an uncertain *cogitatum*,
that is, an inter-human construct of expectations. The two substances (or in
Husserl’s wordings “monads”) are different to the extent that they can be
considered as *analytically* independent [12,13].

Husserl noted that he had no instruments beyond a transcendental apperception of this new domain and therefore had to refrain from its empirical investigation [11:138]. Luhmann [6,14] made the communication of meaning among human beings the focus of his sociology. The communication of meaning relates to the theory of anticipatory systems since meaning is provided to information from the perspective of hindsight. At each moment of time, the communication and degree of codification of meaning can be measured, for example, by using semantic maps [15,16]. By providing meaning to the historical events from the perspective of hindsight, a reflexive system generates a model of these events or—more precisely—the expected information values contained in series of events. Such a model can be used by a reflexive mind for the weak anticipation [1,17,18].

When meanings can also be communicated between weakly anticipatory
systems, a next-order layer of *cogitata* can be expected as a result. I
shall argue that this construct can become strongly anticipatory, that is,
co-constructing its own future states. For example, the techno-economic
(co-)evolution in the social system reconstructs this system to the extent that
more recently the (sub)dynamics of a knowledge-based economy has been
constructed on top of the older—since informational—dynamics of markets [19,20].

While biological systems can already provide meaning to the information
[17,21] the *differentia specifica *of social and psychological systems is
this *processing* of meaning. Meaning-processing systems can codify
meaning into symbolic meaning and develop discursive reasoning. I follow
Luhmann’s assumption [6,12,14] that meaning is communicated in social systems,
but can be made conscious only by psychological systems.

Using the *incursive* formulation of the logistic equation [*x** _{t}*
=

Dubois’ formula for hyper-incursivity [1] captures the second
(anticipatory) dimension of the double contingency in general. However, each other
agent (e.g., *y _{t}*) can be expected to maintain a similar set of
expectations, but with potentially another parameter

The equation can be rewritten as follows [1:9]:

_{} (1)

_{}

_{}

_{}

In general, the latter equation has two solutions [24]:

*x _{t+}*

Given that 0 ≤ *x* ≤ 1, the curve *x*
= 0.5 ± 0.5 √ (1 – (4/*a*)) for *x* = 1 sets limits to the
possible values which can be reached by the hyper-incursive system. This line
is penciled into Figure 1. In addition to the two domains of possible solutions
of Equation 2 for *a* ≥ 4, Figure 1 shows the bifurcation diagram of
the logistic equation for *a *< 4, and the steady state of the
incursive equation [*x** _{t}* =

For *a* ≥ 4, two sets of expectations are
hyper-incursively generated at each time step depending on the plus or the
minus sign in the equation. After *N* time steps, 2* ^{N}*
future states would be possible. Thus, this social system of expectations
continuously needs a mechanism for making decisions between options.
Otherwise, the system would rapidly become overburdened with uncertainty [1:9].
The decisions are made in the layer of historically-rooted but weakly
anticipatory systems.

**Figure 1.** The social system as a result of hyper-incursion.

Note that the logistic equation and its hyper-incursive (but not the
incursive) formulation are symmetrical in *t*: the logistic curve refers
exclusively to (*t* – 1) and the hyper-incursive equation to (*t* +
1). Because of this symmetry in the formulas, both systems can be expected to
reach a limit value for *a* = 4.

Can *a* also have values lower than four in the strongly
anticipatory case?

**Figure 2.** Possible penetrations of the social system into the
biological variation (*a* < 4).

The term under the root in Equation 2 is positive for *x _{t}*
≤

Two more models can be derived from the logistic equation:

_{} (3)

_{} (4)

** Equation 3**
evolves into

_{} (3a)

_{} (3b)

_{} (3c)

_{} (3d)

The steady state can be found by solving *x _{t}*

_{} (5a)

_{} (5b)

_{} (5c)

In other words, the weakly anticipatory system tends to converge on a steady state which can be considered as one of the available sub-dynamics of a strongly anticipatory system.

The continuous line in Figure 1 represents the function *x *= (*a*
– 1)/*a*. This continuity allows the weakly anticipatory system to
maintain its identity for different values of *a*. Identity is based on
the expectation of continuity of the “self” in the next stage of a weakly
anticipatory system while the aperture for the reflection—the parameter *a*—may
vary. Note that institutions like individuals can be expected to develop an
identity for organizing the communication of meaning in historical time.

** Equation 4**
evolves into

**Figure 3.** Simulation of Equation 4: the value of *x* at *t*
= 1 is drawn randomly (*a* = 4).[3]

The negative value provides a mirror image of a representation at a specific moment in time, and thus allows for the reflection. Reflection enables us to bounce a communication between the subdynamics of communication systems. At the level of the social system, this subdynamic formalizes the reflexive operation of providing meaning with reference to a horizon of possible meanings [25].

In summary, the three (hyper-)incursive equations which contain the future state in the argument, specify the systemic processes of meaning-processing: identity formation (Equation 3), reflexivity (Equation 4), and the mutual expectation of expectations in the doubly contingent relations between reflexive systems (Equation 1). While identity and reflexivity are more commonly defined in social theory [26], double contingency could hitherto not be operationalized. From the perspective of the theory of anticipatory systems, the second dimension of the double contingency in intentional systems—that is, the expectation of expectations—provides us with a natural system of strong and weak anticipations.

The hyper-incursive system (Equation 1) cannot further be developed (at
*a *≥ 4) without decisions taken by reflexive agents (or
institutional agency) because of the continuous production of uncertainty. A
psychological system can be considered as the minimal unit of reflection for
making choices [27,28]. If decisions are socially organized—for example, by
using decision rules—an institutional layer can increasingly be shaped. This
institutional layer of organization provides a retention mechanism and
therefore a hold for a next round of interacting expectations at the
supra-individual level [29,30].

In other words, the social system is dually layered as a forward-moving and information-processing retention mechanism versus sets of possible expectations which flow as intentions through the networks. The fluxes contain selections on Husserl’s “horizons of meaning” [25:89f.]. These possible horizons of meaning are not given, but continuously undergoing reconstruction while weakly anticipatory systems make decisions [31]. In other words, these systems co-evolve in two layers of strong and weak anticipation, respectively. When the communication becomes more complex, the communicative competencies of the carriers of the communication are challenged to improve their capacity by learning. The communication can further be developed when layers of codification can be sustained. Knowledge, for example, can be considered as a meaning that makes a difference; the exchange of knowledge as discursive knowledge requires more codification than the communication of meaning itself.

Because the various equations are derived from the same logistic equation, the steady state of the weakly anticipatory system is equal to an available state of the strongly anticipatory system. However, the strongly anticipatory system contains more subdynamics than the weakly anticipatory one. Unlike individuals, the social system is not continuously tending towards integration, but can also operate in a distributed mode and further develop its differentiation into subsystems. When the social system is more integrated—as in a high culture—the system can be stable over long periods of time; the incursive routines prevail and tend towards a steady state. When the social system is more differentiated, the hyper-incursive equation begins to play a role because the subsystems can entertain expectations about the expectations entertained in other subsystems and thus generate a double contingency, that is, a strongly anticipatory mechanism at the level of the social system.

In summary, the social system is complex and therefore composed of
several subdynamics [21]. When, among the various subdynamics, the
hyper-incursive routine prevails, pressure to take decisions increases. This
can be experienced subjectively as time pressure; the system becomes less
predictable because it increasingly co-constructs its own future states.^{[4]} This social system
can be considered as “non-trivial” [33] and unexpected consequences of
decisions can then increasingly be expected.

** **

**Figure 4.** Simulation of the hyper-incursive equation (Equation
1) using MS Excel.

Figure 4 shows the simulation of Equation 1 for different values of *a*
(in the first row). I have highlighted cell D4. The formula bar in the Excel
sheet reads: if the random number in cell A4 (in this case, 0.317611) is smaller
than 0.5, the positive sign of the square root is used and otherwise the
negative one. The evaluation is based on the value of *a *in row one (D1 =
4.1), and the previous solution of the equation in the above cell (D3 =
0.142160957). Note that for values of *a* < 4 (as in the B-column of
Figure 3: *a *= 1.2), the number of steps is limited (as argued above).

In this simulation, the decisions were made randomly at each next time-step. This leads to simulation results as provided in Figure 5. Figure 5 shows that the system can sometimes dwell in a specific state. Next-order mechanisms like institutionalization of the rule for decision-making can be expected to stabilize these configurations because decisions are then no longer taken randomly [34].

Note that because of the structural coupling of the development of the
social system as a strongly anticipatory system to decision-making by weakly
anticipatory systems, the reduction of the uncertainty is provided by the weak
anticipation in the incursive, and not by the strong anticipation at the level
of hyper-incursive fluxes of communication as suggested by the original model
of *autopoiesis* [21,34]. The communication of meaning, therefore, is more
complex than is assumed in the *autopoiesis* model which is biologically
inspired and focuses on the self-organization of information processing. The
communication of meaning among the *models* will be addressed in a later
section; at this level, it will be shown (below) that the self-organization of
the interactions can also lead to a reduction of uncertainty.

**Figure 5.** Trajectory of a social system based on random
decisions of the decision-making units, for various values of the bifurcation
parameter *a.*

The difference between structural coupling at the level of information processing in biological systems and the structural coupling between the incursive and hyper-incursive meaning processing in anticipatory systems has been expressed in qualitative theorizing as the difference between structural coupling and interpenetration [3:437,5, 6:286 ff.,14:210 ff.]. Psychological and social systems are not only coupled structurally, but their operations in the second contingency—the intentional layer—are additionally coupled because each system can provide meaning to its own operation and reflexively to the meaning-processing in the structurally coupled one. When not only the operational systems but also their anticipatory models are structurally coupled, “interpenetration” between the systems makes it possible for the two systems to develop further by using each other’s complex dynamics reflexively as relevant selection environments [28:174].

The logistic equation can be considered as a special case of the *perturbed
recursion model* which* *Andersen [36:170 ff.] formulated as follows:

_{} (6)

*F* is a
recursive function that transforms the state *S*_{t-1}* *into a new
state *S _{t}*, using a set of parameters

Figure 6 shows the configuration when a system is continuously
perturbed. The theory of *autopoiesis* further specifies that the system *S
*itself makes a distinction between the parameters <*p _{1}…p_{n}*>
(representing the environment) and its own state <

**Figure 6.** The perturbed recursion model. Source: [36:170].

Baecker [37:86 ff.] argued that in the case of operational closure and
consequently *autopoiesis* the function (*F*)* *should be
considered as an operator of the communication system (*S*) itself.
Therefore, he reformulated Equation 6 as follows:

_{} (7)

This formulation leads to a paradox [38,39], but this paradox is a
consequence of the static framework. Using the theory and computation of
anticipatory systems, one can add time parameters to the different states and a
subscript *c* (of *coding*; see below)* *to the operation as
follows:

_{} (7a)

This rewrite of the functionality by the system itself means that the system must be able to develop and repair itself operationally in addition to reproducing itself [36:174). The repair mechanism in communication is coding; the code of the communication enables a system to distinguish reflexively between signals and noise [40-42].

Following Parsons, Luhmann [6,14] added that codified meanings can be
“condensed” in subsystems that are *functionally* differentiated. The
functionality of the differentiation can be sustained if the subsystems are
able to use the specificity of their respective codes for the reproduction of
the system as a whole. Using the various codes, events can be appreciated
differently within each subsystem; this diversity enables the system to process
more complexity and can therefore be considered as functional to the further
development of the system. The communications can then increasingly be refined.
In the social system, for example, religious and scientific communications can
be distinguished since the time of Galilei [43]. Similarly, political and
scientific communications can be distinguished as based on providing different
meanings to potentially the same events.

When functionality of the differentiation among different symbolically
generalized codes of communications prevails—that is, in modern and pluriform
societies—the external environment *E* is decomposed for each subsystem of
communication into other subsystems, with a remaining term *ε* as
representation of the residual environment. Let me use the lower-case *c*
for the coding and *s *for this level of subsystems, and then rewrite
Equation 7a as follows:

_{} (8)

In a functionally differentiated system, the windowing of the
subsystems upon each other becomes horizontal. Each subsystem (*i*) codes
(*c** _{i}*) its own previous development and the development
which it finds in its relevant environments. While the relation to the
subsystem itself proceeds historically, that is, from its state at (

_{} (9)

The state of a subsystem (*s _{i,t}*)—and, therefore, of
the system—is now dependent both on the previous state of this subsystem (

In other words, the functionally differentiated system contains an
additional Δ*t* which can be used for a local reversal of the time
axis and thus generate a transversal incursion which stands orthogonally to the
longitudinal incursions provided by weak anticipation—using *x _{t}*

The transversal incursion can be elaborated in a manner analogous to that of the incursive formulation of the logistic equation provided in Equation 5, but with an additional subscript for different subsystems:

_{} (10)

On the basis of Equation 10 one can generalize to Equation 11 for cases
where a variety of subsystems provide additional selection environments for the
development of subsystem *i*, as follows:

_{} (11)

Each subsystem (*i*) develops with reference to its own previous
state, but one can expect all other subsystems to feedback upon this
development as a relevant environment. Since each meaning-providing subsystem (*i*)
also provides meaning to its own development longitudinally, and the two types
of meaning-providing can interact, one can rewrite Equation 11 as follows:

_{} (12)

In Parsons’s structural-functionalism, the number of subsystems (*n*)
was analytically restricted—given his so-called four-function paradigm—but this
number can vary in Luhmann’s theory with the historical development of the
media of communication and the symbolic generalization of specific codes. When
the various subsystems use different frequencies for the updates, parameters
have to be added to the corresponding selection mechanisms in Equation 12 and
one obtains Equation 13:

_{} (13)

The importance of the different frequencies of selections—and the
accordingly different parameters *b*—for the further development of the
system will become manifest below when we study the interactions between these
selection terms in terms of both Equations 8 and 9, that is, recursively,
incursively, and potentially hyper-incursively.

Social order cannot be considered as a stable phenomenon because it contains an order of reproduced expectations in addition to being organized. When the expectations operate upon one another and interact, they can be expected to generate a non-linear dynamics that processes meaning. Specific meanings can be stabilized, for example, in social institutions, but all meaning arises from a horizon of possible meanings. As noted, Luhmann proposed that Husserl’s original concept of intersubjectivity needed to be refined: in addition to interactions at the micro-level, communication of meaning can also be organized at the meso-level, and the fluxes of communication can be considered as self-organizing at the macro-level of society [45].

While “double contingency” is a consequence of an internal model of
relations between intentional systems, the interaction, organization, and
self-organization of meaning can be considered as operations at the level of
the models. These models operate selectively upon the systems which are modeled
and upon each other. The interactions of the models therefore generate *selection
environments* of possible meanings [20].

In Equation 13, I specified selections upon selections as the product of the selective factors. The hyper-incursive interaction between two incursive models can accordingly be modeled as follows:

_{} (14a)

_{} (14b)

_{} (14c)

_{} (14d)

**Figure 7.** Hyper-incursive interaction*.*

Figure 7 provides the simulation of this formula after 10,000
time-steps for *b* = 4 and *b *= 8, respectively. The auxiliary line
is the linear regression line for b = 8 in this run. (The deviation from zero
in the gradient is an effect of random fluctuations in choosing the plus or the
minus signs in Equation 14d.) As expected, the interaction system swings from
one side to the other, while sometimes staying on the same side for a number of
iterations of the simulation. There is no progression in this system, nor can
it come to an end.

Two steady states can be found by equating *x*_{t+1} with *x*_{t} in
equation 14c:

_{} (15a)

_{} (15b)

_{} (15c)

While approaching to a single steady for *b* → ∞, the
two steady states are continuous for variations in this parameter (Figure 8)*.*

**Figure 8.** Two steady states of hyper-incursive interactions*.*

When a third term is added to Equation 14, we can obtain two formulas on the basis of Equations 8 and 9 above, respectively. I shall argue that this models the duality of structure which Giddens [26,46] specified in social theory [42]. Historical organizations of the interactions and potential self-organization among the fluxes of communication are modeled differently in Equations 16 and 17, respectively:

_{} (16)

_{} (17)

Equation 17 models what Strydom [47] called a “triple contingency”: not only is there interaction, but a third party, i.e., a public, can be involved additionally. A public can only arise if a different code of communications was previously stabilized.

When three selections operate upon one another, a complex dynamics is
generated [48,49]. For example, as long as the relationship is dyadic,
Equations 14 and 16 prevail as models for the selection environment of meaning.
However, when a child is added to a couple, or when the relationship is
legalized as a marriage, another social system of expectations is generated
(e.g., at the tax office). The development of this complex system is henceforth
“non-trivial” [33], whereas *two* (sub)dynamics may co-evolve along a
historical trajectory in a predictable process of mutual shaping. In addition
to mutual relations, three subsystems can be expected to develop positions in
their relations among one another [50] and thus to reduce the uncertainty in
some configurations more than in others [19,51,52].

Equation 16 modifies Equation 17 by bending the self-organization of the fluxes back to the historically present state. The self-organizing system can thus provide itself with a retention mechanism for the results of interactions among intentional meaning-processing systems. Equation 16 can be developed analogously to Equation 14 into:

_{} (16a)

_{} (16b)

_{} (16c)

**Figure 9.** Organization of interactions for different values of
the parameter *b.*

Figure 9 shows that the organization of hyper-incursive interactions in
historical time is always temporary. The light-blue curve (for *b* = 10),
for example, shows that when the organization is not sufficiently changed by
new rounds of interaction, the organization becomes unstable and fades out (in
this case at *t *= 6). Technically, this originates from the factor (1 – *x*_{t})
in the nominator under the square root of Eq. 16c: when *x*_{t}
→ 1, this nominator becomes zero, and *x*_{t+1} is no longer
defined.^{[5]}
(The computer program MS Excel provides erroneously a zero value for these
undefined states.) Unlike interactions and—as we shall see
below—self-organizations of meaning in fluxes of communications, organizations
remain historical. The other two mechanisms (interaction and self-organization)
are needed to “revive” the historical layer of organization at various moments
of time by providing variation and next-order stability, respectively.

Equation 17 can be developed as follows:

_{} (17a)

(17b)

_{} (17c)

_{} (17d)

** **

This equation has three roots of which two are complex. The real solution of 9d can (conventionally) be denoted as:

(18a)

and the complex solutions are:

(18b)

Since the further operation cannot evaluate the complex solutions in a
next time-step, the system in this case has to continue with the real solution.
This leads to a single and stable solution for each value of *b*. Are
these perhaps Luhmann’s [53:e.g., 394, 1127] “eigenvalues”?

**Figure 10.** Real and complex roots of the cubic equation in the
case of three selections operating upon one another.

Figure 10 provides the steady state values for different values of the
parameter *b. *The curve of the imaginary part goes asymptotically to
zero. Changes in *b*—because of* *the concurrent operation of
organizing meaning in historical time by using Equation 16—can move the system
along the lines in Figure 8. At each moment, however, the system quickly
converges on a single steady state.

My thesis [19] is that in a modern and pluriform society which is
functionally differentiated in terms of (symbolically generalized) codes of
communication in subsystems, the recombination and temporary reintegration of
these codes in organizations can change the system by reorganizing it (e.g., in
a techno-economic co-evolution). Therewith the value of the parameter *b*
is potentially changed in a next round, and the self-organization in the
horizon of meanings can be expected to change accordingly*.* Thus, the two
layers of the social system can auto-catalytically drive each other while
controlling the uncertainty.

The simulation of organizations showed that the organization of meaning
exchange is temporary. Interactions are continuously providing variation;
self-organization structures the system of meaning exchanges in terms of codes
of communication. The self-organization of meaning in communication provided a
single solution for all values of the parameter (*b*). However, the system
and subsystems are both concurrent and nested: each subsystem can be expected
to provide its own solution (eigenvalue *b’*) for the self-organization when observed from the
system’s level. Torres Carbonell *et al.* [23:411] provided the following
visualization for this nesting of intra- and intersystem anticipation:

**Figure 11.** Intra and Inter-Anticipation. Source: [23:411].

Change has to come from the different parameter values at the *sub*systems
level which respectively converge on different roots of the cubic equations.
These values are recombined historically by specific organization at the
meso-level of the (next-order) system. Using Luhmann’s theory, one can
associate the specificity of the various constants (that is, the converging
roots of the cubic equations) with the functionally differentiated codes of
communication in respective subsystems.

The differentiation among the codes is not only a static differentiation among eigenvectors, but also a differentiation in terms of frequencies [43]. For example, a political system can be expected to operate with an election cycle of four years, while science operates in terms of quarterly journals and annual meetings. The economy updates with prices and stock markets on a more continuous basis.

Luhmann’s social systems theory was hitherto mainly formalized using
Spencer Brown’s (1969) *Laws of Form* [54,55]*. *Spencer Brown
distinguished between two states: the ‘marked’ and the ‘unmarked.’ Luhmann
[e.g., 38]—and many others—have believed that this distinction could be used
for the development of a general calculus [39,56,57]. However, Spencer Brown’s
marks are made at specific moments of time. Their recursive elaboration leads
to a topology [39,55], but not to a calculus.

In a later discussion, Spencer Brown [58:51] added the possibility of
an evolutionary process, but he formulated this mathematically as only an
oscillation between the two states. An oscillation can, for example, be
produced by the logistic equation (3 ≤ *a *< 4). However, my algorithmic
equations were specified on the basis of substantive theorizing about the
communication of meaning in social system [cf. 56:20,59]. In summary, an
evolutionary theorist is not interested only in the momentary existence of an
observer or a second-order observer, but also in developments over time in
terms of specific operations [57,59,60].

In this study, I constructed a relation between Dubois’ incursive and hyper-incursive equations of weakly and strongly anticipatory systems [1,8,9] and Luhmann’s social systems theory [6,14]. I used both theories heuristically and perhaps eclectically because the construction of a relation requires a creative translation of the one theory into the other. Luhmann [6:605, 14:446f.], for example, formulated that “(s)elf-referential, autopoietic reproduction would not be possible without an anticipatory recursivity.” However, Luhmann did not specify this mechanism.

In his summarizing *chef-d’oeuvre* of 1997 [53], Luhmann provided two
references to Robert Rosen’s *Anticipatory Systems* [17], but both were
framed in a biological context [53: 206 and 820]. He added (at p. 821) that in
the domain of meaningful processing of information one cannot avoid a reference
to the present when defining the relation between past and future;
consequently, time has to be conceptualized as a dimension. Luhmann’s
time-dimension with the present at its origin can further be elaborated as a
degree of freedom in the theory and computation of anticipatory systems.

In other contexts, Luhmann developed a semantics for the discussion of “time” [61: 98, n.10], but for my argument other elements of social-systems theory are more important than these historical elaborations. Luhmann [53] draws two main axes of differentiation within social systems: social differentiation and systems differentiation. Social differentiation in the communication is possible because communication can be coded differently and codes of communication can be generalized symbolically. For example, one can code communication economically, scientifically or politically. Money, power, validity, affection, trust, etc., can be used to shortcut communication and thus to process more complexity [62,63]. The different function systems operate in parallel, but not synchronously. As Luhmann formulated: “Social differentiation serves as an uncoupling mechanism. It divides the time-orientations in the different systems and therefore accepts that things can be urgent in one system, while another system can take its time.”[61:128]

Systems differentiation organizes the social system at different
levels. Face-to-face communication or interaction remains the micro-operation
for hyper-incursion or strong anticipation because it is based on a double
contingency which is naturally given among reflexive agents. However,
communications can also be organized at the meso-level.^{[6]} At the macro-level of society,
Luhmann [12] proposed to use Maturana & Varela’s [36] concept of *autopoiesis*
or self-organization: interhuman coordination is possible because meaning can
also be communicated (e.g., using languages or symbolic codes of
communication). The differentiation of micro-level interaction, meso-level
organization, and macro-level self-organization in the communication of meaning
was elaborated in terms of different equations which represent the selection
environments of intentional systems.

The model of recursive perturbations allowed us to consider both
Spencer Brown’s formulae and the various formulations of the logistic map as
species of a family of equations. The spanning of the time dimension enables us
to dissolve the paradoxes in social-systems theory (as in the above case of
Equation 7). The theory and computation of anticipatory systems further
develops the time dimension into a degree of freedom which can be studied in
the forward and/or backward mode. The duality of structure in the social system
[26,46] could thus be considered as the result of a difference in the
time-subscripts of Equations 16 and 17 (above). Note that these equations are
no longer exclusively based on the model of recursive perturbations or the
logistic map. In addition to the bifurcation parameter *a*, a second
parameter *b* is involved in modeling the non-linear dynamics of
meaning-processing at the supra-individual level.

In summary: Leydesdorff & Dubois [22] showed that the social system can be considered as inherently anticipatory because of its distributedness at each moment in time. This distribution can be organized in terms of a functional differentiation among subsystems which can operate asynchronously and thus generate another ∆t. Reflexivity and intentionality add a second (longitudinal) anticipatory mechanism. The two mechanisms can be combined into a strongly anticipatory mechanism [20]. A strongly anticipatory system co-constructs its next stages as one of its subdynamics. The social system provides us with a natural example of such a system.

Marx already in 1867 expressed this expectation of a dynamics at the supra-individual level as the autonomous accumulation of capital [64]. Nowadays, we are able to refine this accumulation as the fragile progression of a knowledge-based economy.

I am grateful to Mark Burke, Sander Franse, Dirk Baecker, Daniel M. Dubois, Juan Jesús Torres Carbonell, and Daan Van den Berg for suggestions and communications in previous stages of this paper.

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[1]
For *a* = 2, the hyper-incursive system has a single (“natural”?) solution
at *x*_{t} = 0.5. In the case of *a* < 4, there exists a
basin of attraction of the for the fixed state (a – 1)/a of Equation 2 with the
plus sign, such that if *x*_{0} is in this basin, and the
hyperincursive decision rule allows the system to choose the plus equation,
then *x*_{t} will persist indefinitely. This never occurs when *a*
< 3 (Marke Burke, *personal communication*, February 1, 2008.)

[2]
Dubois (*personal communication*, July 16, 2008) noted that Eq. 4 can be
derived as the *time inverse* of his [1] incursive (Pearl-Verhulst)
equation *x** _{t}* =

_{}

is equal to the following
equation for dt = 1: _{}

The time reverse of this equation is obtaind for dt → –dt, with the negative discrete time –dt:

_{}

or, with –dt = –1: _{}

So with a time translation of t → t +1 on the whole equation, one obtains:

_{} (4)

[3]
For *a* = 4, the pulse is generated for values of *x _{t}*
> 0.8.

[4] Dubois [32: 208f.] defined “hyperincursion” as an incursion that generates multiple future states.

[5]
Daan van den Berg, *personal communication,* July 12, 2008.

[6] Organization and decision-making can also be considered as functions of the social system. Decision-making, for example, can be codified into decision rules. I elaborate this in [19: 147 ff].