ArticlePDF Available

Digital analysis of a form

Abstract and Figures

Purpose This paper aims to show how a sociological description – a swarm analysis of the Nazi dictatorship – initially made with the means borrowed from George Spencer-Brown’s Calculus of Indications, can be transformed into a digital circuit and with which methods and tools of digital mathematics this digital circuit can be analyzed and described in its behavior. Thus, the paper also aims to contribute to a better understanding of Chapter 11 of “Laws of Form.” Design/methodology/approach The analysis uses methods of automata theory for finite, deterministic automata. Basic set operations of digital mathematics and special set operations of the Boolean Differential Calculus are used to calculate digital circuits. The software used is based on ternary logic, in which the binary Boolean logic of the elements {0, 1} is extended by the third element “Don’t care” to {0, 1, −}. Findings The paper confirms the method of transforming a form into a digital circuit derived from the comparative functional and structural analysis of the Modulator from Chapter 11 of “Laws of Form” and defines general rules for this transformation. It is shown how the indeterminacy of re-entrant forms can be resolved in the medium of time using the methods of automata theory. On this basis, a refined definition of the degree of a form is presented. Originality/value The paper shows the potential of interdisciplinary approaches between sociology and information technology and provides methods and tools of digital mathematics such as ternary logic, Boolean Differential Calculus and automata theory for application in sociology.
Content may be subject to copyright.
Digital analysis of a form
André Oksas
Research Fellow, Next Society Institute,
Kazimieras Simonavičius University, Vilnius, Lithuania
Abstract
Purpose This paper aims to show how a sociological description a swarm analysis of the Nazi
dictatorship initially made with the means borrowed from George Spencer-Browns Calculus of Indications,
can be transformed into a digital circuit and with which methods and tools of digital mathematics this digital
circuit can be analyzed and described in its behavior. Thus, the paper also aims to contribute to a better
understanding of Chapter 11 of Laws of Form.
Design/methodology/approach The analysis uses methods of automata theory for nite,
deterministic automata. Basic set operations of digital mathematics and special set operations of the Boolean
Differential Calculus are used to calculate digital circuits. The software used is based on ternary logic, in which
the binary Boolean logic ofthe elements {0, 1} is extended by the thirdelement Dontcareto {0, 1, }.
Findings The paper conrms the method of transforming a form into a digital circuit derived from the
comparative functional and structural analysis of the Modulator from Chapter 11 of Laws of Formand
denes general rules for this transformation. It is shown how the indeterminacy of re-entrant forms can be
resolved in the medium of time using the methods of automata theory. On this basis, a rened denition of the
degree of a form is presented.
Originality/value The paper shows the potential of interdisciplinary approaches between sociologyand
information technology and provides methods and tools of digital mathematics such as ternary logic, Boolean
Differential Calculus and automata theoryfor application in sociology.
Keywords Systems theory, Automata theory, Niklas Luhmann, Calculus of Indications,
Digital mathematics, George Spencer-Brown
Paper type Research paper
Introduction
With his major work, Laws of Form(Spencer-Brown, 1972), George Spencer-Brown
presented a mathematical Calculus of Indications that has found widespread application in
the processing of sociological observations. In the basic distinction-and-indication operation,
the cross operator is used to indicate the presence of a marked state and its implied
contradictory, the unmarked state. The marked state and the unmarked state are
disjunctive, mutually exclusive and jointly exhaustive. The relation of the two
distinguishable states {marked, unmarked}, which are produced by one or more distinction-
and-indication operations, is called form.Furthermore, variables are introduced, but only
as expressions, the values of which indicate, again, the marked or the unmarked state:
Let mstand for any number, greater than zero, of such expressions indicating the
marked state.
Let nstand for any number of such expressions indicating the unmarked state.
(Spencer-Brown, 1972, p. 14).
The author is grateful to Franz Hoegl (Next Society Institute, Kazimieras Simonavičius University,
Vilnius) and Leon Conrad (Next Society Institute, Kazimieras Simonavičius University, Vilnius) for
active support and inspiring conversations about these subjects. Any errors in this article, however,
remain the authors exclusive responsibility.
Digital
analysis
of a form
Received 21 October2020
Revised 30 December2020
Accepted 10 January2021
Kybernetes
© Emerald Publishing Limited
0368-492X
DOI 10.1108/K-10-2020-0682
The current issue and full text archive of this journal is available on Emerald Insight at:
https://www.emerald.com/insight/0368-492X.htm
Niklas Luhmann based his systems theory (Luhmann, 1995), which was formulated in
terms of difference theory and self-referentiality, on this Calculus of Indications. The
variables used in the differentiations of systems theory extend the Calculus of Indications,
as they no longer stand for mathematical expressions that themselves denote the marked or
unmarked state. As shown in Figure 1, the denition of a system can be formulated with the
Calculus of Indications as the difference between system and environment. System and
environment are disjunctive as variables, mutually exclusive and jointly exhaustive.
Beyond the difference-theoretical and self-referential aspects of the Calculus of Indications,
the visualization of the observers perspective implicit in the form has been widely accepted in
sociology. In the gure which symbolizes the formal denition of a system (Figure 1), the inner
distinction between system and environment implies an observer of the rst order. By contrast,
the outer distinction between the distinction of system/environment and the unmarked space
represents a second-order observer whose observation re-enters the form itself.
In Roth (2019), such distinctions between mutually exclusive and jointly exhaustive
variables are called true distinctionsand the thesis is formulated that these true
distinctionsare particularly suitable for digital processing.This thesis explores how digital
methods could be particularly suitable for observing the digitization of society. Before we
come back to this in the section Swarm analysis of the Nazi dictatorship as a form, the
method of transforming a form into a digital circuit of NOR gates will be established.
From universal NOR gate to the cross
Against the background of the assumption from Roth (2017) that George Spencer-Brown was
inspired by the Boolean NOR operation when developing the Calculus of Indications, it is worth
taking a more detailed look at the NOR operation and its electronic realization as a logic gate.
From the set of 2
4
or 16 possible Boolean logic gates with two inputs, the basic gates and the
universal gates stand out especially. Every Boolean function can be realized with a
combination of only the three basic gates: conjunction (AND), disjunction (OR) and negation
(NOT). Furthermore, according to the laws from De Morgan (1838) and using the negation
(NOT), any conjunction (AND) can be transformed into a disjunction (OR) and vice versa.
A complete algebraic logic can be represented by the combination of at least two of the
basic gates, where, in addition to the indispensable negation (NOT), either the conjunction
(AND) or the disjunction (OR) is needed. As the two universal gates, NAND (NOT AND) and
NOR (NOT OR) already contain the negation (NOT), one of these two universal gates is
sufcient to represent a complete algebraic logic.
This universality and the negation included in the operation have particularly interested
mathematicians and philosophers. In Kauffman (2001), this history of the reception is traced
from the Peirce arrow of Charles Sanders Peirce, the Sheffer stroke of Henry Maurice
Sheffer, the Nicod arrow of Jean George Pierre Nicod to the cross of George Spencer-Brown.
This series could be extended to include Edward Stamm, as can be seen from a quotation
from George Spencer-Brown published in Roth (2017) from the typewriter manuscript
Design with the NOR(Roth, 2021), which was written in 1961. Furthermore, the use of the
Sheffer stroke in Ludwig WittgensteinsTractatus Logico-Philosophicus(Wittgenstein,
1922) would also be worth mentioning in this context according to Hoegl (2021).
Figure 1.
System denition as
form, according to
Luhmann (1995)
K
How George Spencer-Browns understanding of the NOR operation and the NOR gate has
affected the development of his Calculus of Indications is of particular interest for our
investigations. The following quote by George Spencer-Brown from Design with the NOR
(Roth, 2021) deserves special attention:
So whatever, in logic, the industrial NOR unit represents, it is certainly not the Sheffer stroke.
[...] To say that the industrial NOR unit represents an operation hitherto unknown to logic
would perhaps be too strong, although if we interpret unknownas unfamiliarthe claim is
justied. For such an operation, even if known, has never been taken as standard. If we signify it
by a mark [...]calledcrosswe can compare it with the Sheffer stroke, thus.(Roth, 2017 p. 1476)
At the time of his work on Design with the NOR(Roth, 2021), George Spencer-Brown
was employed as Chief Logic Designer by the manufacturer of electronic components
Mullard Equipment Limited (Baecker, 2015). According to the quote above, the industrial
NOR gate, thus, made special demands on the logic and led directly to the notation of the
cross. In principle, although the Sheffer stroke is also suitable as a notation to represent a
universal gate such as NAND or NOR, its limitation to two variables is an obstacle to the
requirements of circuit design. Furthermore, the negation of a variable in the sense of aby
means of the Sheffer stroke would be cumbersome to represent with aja.
With the cross, George Spencer-Brown created a topologically invariant notation with any
number of disordered, but bracketed variables to represent the logical neither ... nor,which
met his requirements from the circuit design with industrial NOR gates. How, eight years later,
this special notation for the circuit design developed into the philosophically more important
Calculus of Indications remains an exciting object of research and cannot be pursued further here.
A discussion of logic gates would be incomplete without the technological perspective
afforded by the semiconductor industry. As criteria such as integration density, power
consumption and delay times play decisive roles here, a different picture than from the
philosophical-logical one results. In the realization of logic gates based on semiconductors,
the transistor is the crucial element. This leads to the fact that NAND and NOR gates can
each be realized with one transistor less than their non-negated AND and OR gates. AND
and OR gates would have to be realized as negations of NAND or NOR, respectively, which
is unacceptable, based on the above-mentioned criteria.
Thus, there a need for a direct comparison of NAND and NOR gates arises. In the 1960s,
the NOR gate was the central component of the NORBIT series YL6000 (from which it
derived its name) manufactured by Mullard Equipment Limited (Norbit, 1962). Nevertheless,
it lost almost all its importance compared to the NAND gate with the technological change
from the then common Resistor Transistor Logic (RTL) rst to Transistor Transistor Logic
(TTL) and subsequently to the CMOS technology widely used today.
In Sutherland (1999), Logical Effort was introduced as a benchmark for delay times
experienced at the different logic gates. As shown in Table 1, the delay time of a NAND gate
Table 1.
Logical effort of
different logic gates
No. of inputs
Gate type 1 2 3 4 5 n
Inverter 1
NAND 4/3 5/3 6/3 7/3 (n+ 2)/3
NOR 5/3 7/3 9/3 11/3 (2n+ 1)/3
Multiplexer 2 2 2 2 2
XOR (parity) 4 12 32
Source: Sutherland (1999,p.4)
Digital
analysis
of a form
with ninputs with a Logical Effort of (nþ2)/3 is signicantly lower compared to the
equivalent NOR gate, which has a value of (2nþ1)/3. This is the reason why the NAND gate
is the standard element in TTL and CMOS technology. However, due to its transferability,
according to De Morgans laws, this is not important for circuit technology.
For our further considerations, it remains to be noted that the cross was born from the NOR gate,
which supports the thesis of fundamental traceability of a form to a digital circuit of NOR gates.
The form as a digital circuit
Although the cross had its origin in the industrial NOR gate, it does not yet show the basic
transformability of a form into a digital circuit of NOR gates. To clarify this question and to determine
the basic rules of such a transformation it is worthwhile to have a look at Chapter 11 (Spencer-Brown,
1972, pp. 5468), where the form E4 is introduced to realize a modulator function (Figure 2).
The modulator function to be realized is dened as the conversion of a sequence of wave-
shaped signal sequences of the values 0 and 1, which are received at input a, into a signal
sequence with half the frequency at output f,i.e.a= 10101010 becomes f= 11001100. Tasks
of this kind are described in the circuit technology as modulo counters because the circuit
has to count the cycles of the input wave, to switch the output signal fafter the second cycle
in this example. A frequency halving would, therefore, correspond to a modulo-2 counter, a
frequency quartering would then correspond to a modulo-4 counter, etc.
The realization of the Modulator by means of so-called markers of the form E4 is
illustrated in Figure 3 and shows the complete function of the Modulator by means of the
wave patterns shown. According to the following quotation from George Spencer-Brown,
the Modulator only appears in a different shape than the form E4 itself:
Transgured in this way, E4 appears in a form in which it is easier to follow how the wave
structure of ais taken apart and recombined to give that of f.(Spencer-Brown, 1972,p.67).
The markers introduced in Chapter 11 (Spencer-Brown, 1972, pp. 5468) are described in
the work as follows:
Let a marker be represented by a vertical stroke, thus,
Let what is under the marker be seen to be so by lines of connexion, called leads, thus,
(Spencer-Brown, 1972,p.66).
Figure 2.
Form E4 from
Chapter 11
K
However, as the states of the connecting lines have been dened with the wave patterns
shown, a detailed description of the markers and the method used could be presented using
the methodology put forward by Kauffman (2006).
Beyond the description of the Modulator in Spencer-Brown (1972) and Kauffman (2006),
however, another perspective can be chosen. In the following, it shall be shown how, while
keeping all connections and their states described by the wave patterns, exactly the same
function can be realized only by replacing the markers with NOR gates. This interpretation
is also indicated by Spencer-Browns symbol for the marker, which shows a high level of
similarity to the transistor circuit diagram of a logic gate of the then common Resistor
Transistor Logic (RTL).
The digital circuit of NOR gates shown in Figure 4 was created by replacing all eight
markers of Figure 3 with eight NOR gates while maintaining their respective topological
positions and the connections between them. Furthermore, the wave patterns of each signal
Figure 3.
Modulator from
Chapter 11
Figure 4.
Modulator as digital
circuit of NOR gates
Digital
analysis
of a form
connection were represented as time sequences of the Boolean values 0 and 1. However, it
remains to be shown whether the structurally identical circuit of NOR gates has the same
behavior as the Modulator, i.e. whether it is functionally identical.
With the help of this chronological sequence of signal values, which was specied by
Spencer-Brown (1972), the behavior of each of the eight NOR gates can now be veried. The
rst NOR gate converts the signal sequence 1010 of the input variable atogether with the
signal sequence 1100 of the output variable finto the signal sequence NOR(a,f) of 0001. This
can be veried for all eight NOR gates and all given signal sequences. The circuit of NOR
gates, thus, realizes the same behavior as the modulator and is functionally identical to it.
If according to the above citation (Spencer-Brown, 1972,p.67),theModulatoronly
represents a different shape of form E4 and the digital circuit of NOR gates obtained
from the Modulator is structurally and functionally identical to the Modulator, the
principle of the transformability of a form into a digital circuit of NOR gates can be
derived from this.
From the transformation process from form E4 to a circuit design of NOR gates shown
above, we can now understand the basic conditions for a transformation from a form to a
digital circuit. Through a purely structural comparison of form E4 and the Modulator, the
following rules of transformation result, independent of the implementation of the switching
elements as markers or NOR gates:
Each cross corresponds to exactly one switching element.
The direct nesting of two crosses into each other is represented as a connection
between the output of the switching element belonging to the inner cross and an
input of the switching element belonging to the outer cross.
Each re-entry corresponds to a connection from the output of the respective cross to
the input of the switching element of the re-entry point.
If we now apply these rules to E4 starting with the rst, innermost cross to the eighth,
outermost cross and consider the symmetrical grouping of the form E4 into two groups of
four, we get the digital circuit shown in Figure 5.
For better assignability, the NOR gates were numbered from g1 to g8 according to the
position of their respective crosses and the signal sequences were transmitted according to
the wave patterns. With these signal sequences, the function for each individual NOR gate
Figure 5.
Digital circuit derived
from form E4 by
transformation
K
can now be traced. The circuit shown in Figure 5 is, thus, identical to the circuit shown in
Figure 4.
With this method of transforming a form into a digital circuit, derived from Spencer-
Brown (1972), we can now transform the selected sociological description of the swarm
analysis of the Nazi dictatorship from Baecker (2013) into a digital circuit.
Swarm analysis of the Nazi dictatorship as a form
In Baecker (2013), a sociological description of the Nazi dictatorship was presented by
means of swarm analysis, in two symbolic forms: rst, through extensive verbal description
and second, as a visual diagram, using Spencer-Browns Calculus of Indications (Spencer-
Brown, 1972). Figure 6 shows the second symbolic interpretation of the Nazi dictatorship, in
which the conceptual words chosen for historical-sociological observation were arranged as
variables according to rules taken from the Calculus of Indications in such a way that for an
observer familiar with the Calculus of Indications, references become visible that in classical
notation could only becommunicated with great effort and in an unclear manner.
As the digital analysis of the form presented here does not depend on the extensive
explanations of the swarm analysis method and the historical observations themselves, we refer
to Baecker (2013). However, for a better understanding of the intended semantics, the conceptual
words of the form used as variables will be described with quotations from Baecker (2013).
Hitler swarm:
A swarm is a dense and self-organizing packaging of events and values feeding on each
other. Its structure denes the differentiation by which it relates to its social and natural
environment, its culture denes how it shapes its identity in strategy and reection. The
structure is symmetrical, placing equal emphasis on all sides of differentiation (Simmel,
1989;Luhmann, 1997, pp. 595609). Culture is asymmetrical, thus biasing the interpretation
of the world (Douglas, 1982;Swidler, 1986). A swarm is considered a social system
differentiating itself from its environment and reproducing in time, using its social
dimension to produce a kind of indeterminacy, which lends it exibility in differentiation
and robustness in reproduction (Parsons, 1951;Luhmann, 1995).(Baecker, 2013,p.70).
Terror:
Terror here means the threat of violence combined with the execution of physical violence,
a form of political communication constantly calculating how much physical violence was
necessary to maintain the threat of increasing it. With this procedure, the totalitarian state
documented that there were domains just outside its sphere of inuence that were aware of
having to pay for this remaining outside in various forms.(Baecker, 2013,p.79).
Figure 6.
Hitler swarm as a
form
Digital
analysis
of a form
The doubling of terror in the calculus shows that both imagining terror and experiencing it
were important to maintain the swarm. At the same time it shows that the swarm worked
under pressure which absorbed so many resources that, of its own accord, exhaustion
became a more and more attractive option when crossing the form back into its unmarked
state.(Baecker, 2013,p.82).
n61:
The number n61means that the swarm always counted its members by adding one
member, nþ1, namely, Hitler, who at the same time was lacking in any concrete
instantiation of the swarm, n1. He was, thus, added as lacking, n61, giving the swarm
its exible and situational coherence.(Baecker, 2013 p. 79).
Loyalty:
Then, loyalty means that the trust in present fellowship was more important than
programme, principle or static morality, let alone law or economic rationality (Weber, 1978,
pp. 241246). Moreover, loyalty had the advantage that it could be tested by enforcing
immoral or illegal behavior, which was then even more binding.(Baecker, 2013,p.79).
Mass media:
This is also why the mass media became so important. They deal with actualities,
that is, with information distinguishing the new from the old. Here again, Gleichschaltung
was the rule, yet ironically it helped nd out that if the functional spheres were to be
aligned they had to develop their own professional understanding and practices in
dealing with tasks involved. The lasting effect of the suspicion of manipulationand
Kulturindustrieonly shows that, in developing their standards of information,
entertainment and advertisement, the mass media did in fact do a successful job of
differentiation (Horkheimer and Adorno, 2002;Enzensberger, 1974;Luhmann, 2000b).
(Baecker, 2013, p. 81).
Social work:
This is true for another public as well, that of social work (Volkswohlfahrt,
Arbeiterwohlfahrt,Winterhilfswerk, see Nolzen, forthcoming;Götz, 2001), which was so
important in guaranteeing support for the Nazi regime that the idea of calling the Nazi
regime an instantiation of an attempt by the functional system of social work to absorb the
whole of society seems not too farfetched. The functional system of social work is actually
the only system on the level of society that decides on questions of inclusion and exclusion
(Baecker, 1994), as otherwise only organizations can do by their usual practice of hiring and
ring. It is, thus, the social system that ensures that the promises of the French Revolution
to include the whole reading population in society are fullled at least by proxy.(Baecker,
2013, p. 81).
War economy:
For industry, for men at the front and for men and women maintaining supplies, the war
economy literally meant counting the living and the dead, conveying a clear message to all
on how to secure their survival. Then, the terror everybody heard about fed back into the
terror everybody experienced so that the motives for selecting action appropriate to the
values and distinctions captured by the form of the swarm could be both evident and self-
fullling.(Baecker, 2013,p.82).
Baeckers symbolic form, shown in Figure 6, is analyzed below using methods of digital
mathematics: the Boolean Differential Calculus, automata theory and the software
XBOOLE.
K
The Boolean Dierential Calculus
The origins of the Boolean Differential Calculus go back to applications of the work of Akers
(1959),Huffman (1958) and Reed (1954) in the synthesis and testing of integrated circuits in
the 1970s. Against the background of the constantly increasing integration density of
integrated circuits, it is easy to imagine the challenges for the generation of test vectors. For
example, if a function f(x1,...,x64) to be tested depends on 64 input variables, then all 2
64
test vectors would be needed for a complete test. To make these exponentially increasing
demands on computing time manageable, the question must be answered what the change
of a single input variable xi would mean for the change of the function result. With this kind
of dynamic consideration, the necessary amount of test vectors could be reduced
signicantly.
With Davio (1978) and Thayse (1981), comprehensive papers on the Boolean Differential
Calculus were then presented and a working group at the Chemnitz University of
Technology extended the Boolean Differential Calculus once again (Bochmann and Posthoff,
1981). At the same time, the software framework XBOOLE was developed and described in
Steinbach (1992), with which the operations of the Boolean Differential Calculus can be
calculated, even for large numbers of variables.
Analogous to the differential calculus, the Boolean Differential Calculus can be used to
analyze given functions according to their change behavior with the known operations of
Boolean algebra. The results of such derivatives of given Boolean functions are again
Boolean functions, where the results of the derivative are always simpler than the initial
functions. Boolean variables can take the values of the binary Boolean space {0, 1}.
However, these values are purely static and do not contain any information about their
change. There is no possibility of describing these values dynamically. To a certain extent,
the operations of the Boolean Differential Calculus closethis gap.
If xis a Boolean variable, then dx is the differential of this Boolean variable xwith its
values dx = 1 if the value of xchanges and dx = 0 if the value of xremains constant. The
differential dx is, thus, itself an independent variable, which as it describes the change of
xremains connected to the variable x. As the differential dx itself is also a Boolean
variable, all operations of Boolean algebra remain valid for dx.
The simple derivative operations are distinguished by the three basic operations
derivative, minimum and maximum. The derivative of a given function f(x,y) with respect to
xresults in the value 1 exactly if the function result changes due to the change of the value of
the input variable xindependent of the direction of change. As a Boolean expression, this
derivative corresponds to the antivalence (XOR) of the function result for x= 1 with the
function result for x=0:
df(x,y)/dx =f(1, y) XOR f(0, y)
For the minimum, the antivalence (XOR) is replaced by the conjunction (AND):
min
x
f(x,y)=f(1, y) AND f(0, y)
The minimum of f(x,y) with respect to xyields the value 1 exactly for the values of the input
variable x, where the function result f(x,y) is equal to 1 (and not equal to 0).
Finally, the maximumis formed with the disjunction (OR):
max
x
f(x,y)=f(1, y)ORf(0, y)
The maximum of f(x,y) with respect to xyields the value 0 exactly for the values of the input
variable x, where the function result f(x,y) is equal to 0. Or the other way round: the
Digital
analysis
of a form
maximum of f(x,y) with respect to xyields the value 1 if at least one value of xexists, where
the function result f(x,y) is equal to 1.
Using the NOR operation as an example, and thus to a certain extent also for the cross
operation itself, the following simple derivative outputs result for f(x,y) = NOR(x,y).
Derivative of NOR(x,y) with respect to x:
df(x,y)/dx = NOR(1, y) XOR NOR(0, y).
df(x,y)/dx = 0 XOR NOT y.
df(x,y)/dx = NOT y.
Minimum of NOR(x,y) with respect to x:
min
x
f(x,y) = NOR(1, y) AND NOR(0, y).
min
x
f(x,y) = 0 AND NOT y.
min
x
f(x,y)=0.
Maximum of NOR(x,y) with respect to x:
max
x
f(x,y) = NOR(1, y) OR NOR(0, y).
max
x
f(x,y) = 0 OR NOT y.
max
x
f(x,y) = NOT y.
These derivation results also correspond to the interpretation of the NOR operation with two
variables as an enableable negation operation, where an input with the value 0 enables the
negation of the remaining input to a certain extent; with only one value 1 at the inputs, the
function result keeps the value 0 constantly. This is described by the function result of
the derivation of NOR(x,y) with respect to xwith NOT y.
With a simple partial derivative, the effect of changing the value of exactly one input
variable xon the function result f(x,y) is examined. In a way, the question is answered, at
which initial assignment and a subsequent change of xthe function result f(x, y) changes as
well. If the result function is derived again after a further variable y, we speak of a double
partial derivative. The question is answered whether the function value f(x,y) changes if the
value of the input variable xis rst changed with the rst partial derivative and then
the derivative result is changed with another partial derivative of the value of the input
variable y. A further derivative of the second derivative result after another variable results
in a triple derivative and generally these multiple derivative operations are called k-fold
derivative, k-fold minimum or k-fold maximum.
While the simple derivative operations examine the dependence on an input variable x,
vectorial derivative operations extend this consideration in dependence on a set of input
variables. In contrast to the k-fold partial derivative operations, the term vectorial describes
the assumption that all input variables of the examined variable vector change
simultaneously. Together, the three k-fold derivative operations extended by three vectorial
derivative operations form the basic framework of the Boolean Differential Calculus, the use
of which is discussed in the following examples which draw on the software XBOOLE.
Circuit analysis with XBOOLE
Steinbach and Posthoff (2009) show how digital circuits can be analyzed and synthesized
with the methods of digital mathematics. Circuit analysis is used to calculate the behavior of
a given digital circuit, while circuit synthesis is used to calculate the best possible digital
K
circuit from a given behavior description. For tasks of circuit analysis and circuit synthesis,
the software XBOOLE-Monitor is a freely available tool, which can be downloaded here:
http://www.informatik.tu-freiberg.de/xboole/
The classical bivalent Boolean logic, as well as the digital circuit technology, are based
on the distinction of mutually exclusive and jointly exhaustive elements {0, 1}. Due to the
exponential dependence of the required memory capacity and computing time on the
number of Boolean variables used, complexity limits quickly arise and many problem
solutions belong to the NP-hard complexity class. This complexity can be reduced with the
methods of digital mathematics shown in Steinbach and Posthoff (2009), so that certain
Boolean problems can be solved at all or with signicantly reduced effort. The core of this
method is the extension of the bivalent Boolean logic by a third element Dont care, which
can stand for the value 0 and for the value 1. This ternary logic {0, 1, } results in a
considerable reduction of required storage capacity and computing time.
The behavior of Boolean functions can be represented as both a table and a list of all
those combinations of the values of the variables for which the Boolean function yields the
value 1 as result. All other combinations not contained in the list then inevitably result in
the value 0 as the result of the function. Each individual combination of the values of the
variables forms a variable vector (x1,...,xn) and these lists of any number of
interchangeable variable vectors are called vector lists. While the classical binary logic of
the elements {0, 1} works with binary vector lists, the ternary vector list of the elements
{0, 1, } forms the essential data type of XBOOLE. Figure 7 shows the function of the well-
known NOR gate y= NOR(x1,x2) as a ternary vector list, which unlike the classical four-
line binary vector list only consists of three lines.
Alternately, any ternary vector list in XBOOLE can be visualized as a Karnaugh plan
using the representation commonly used in digital circuit technology, as shown in Figure 8
for the function y= NOR(x1,x2).
The Hitler swarm as a digital circuit
The swarm analysis of the Nazi dictatorship from Baecker (2013), there visualized as form,
can now be transformed into a digital circuit of NOR gates according to the rules found in
the section The form as a digital circuit. The conceptual words of the form terror,n61,
loyalty,mass media,social work and war economy used as variables are taken over as
Figure 8.
Karnaugh plan
y= NOR(x1, x2)
Figure 7.
Ternary vector list
y= NOR(x1, x2)
Digital
analysis
of a form
Boolean input variables of the digital circuit terror,n1,loyalty,massmedia,socialwork and
wareconomy in computer-readable notation and each NOR gate is assigned a Boolean output
variable from y1 to y7, where y7 corresponds to the variable Hitler swarm from the form
while maintaining the order of the cross from inside to outside. Figure 9 shows the digital
circuit of the Hitler swarm as the result of this transformation.
First statements about the behavior of the circuit can already be made with heuristic
means of traditional circuit analysis and without using digital mathematics or software. A
NOR gate with two inputs describes as could be shown in the section The Boolean
Differential Calculus”–an enableable negation operation. One input of the NOR gate
enables the negation operation of the other input with the value 0. Thus, for NOR gates with
ninputs, all n1 inputs must have the value 0 to negate the remaining input signal at the
output. Direct feedback of the output signal of a negation operator to its own input generates
a signal sequence 01010... at the output as an astable circuit. From this it follows that
astable oscillation behavior can be expected under the following conditions:
The output signal of a NOR gate (at least) is fed back to the input via an odd number
of preceding NOR gates, thus generating negative feedback and
All other inputs of these NOR gates not connected in cascade have the value 0.
We will describe this astable oscillation behavior in more detail in the section Astable
oscillation behavior or imaginary state?. Further statements about the behavior of the
digital circuit cannot be made by visual inspection alone, but require analysis with the
methods of digital mathematics.
Determination of the circuit as switching circuit
In principle, digital circuits can be divided into combinatorial circuits and switching circuits,
whereby switching circuits are created by feeding back output signals to preceding inputs.
While the output function of combinatorial circuits depends only on their input variables x
according to the formula y=f(x1,...,xn), the output function of switching circuits depends
not only on the input variables xbut also on the internal state variables zresulting from the
feedbacks: y=f(x1,...,xn,z1,...,zm). Switching circuits of this kind i.e. without clock
signals, which control the synchronous switching are called asynchronous automata and
can be analyzed in their behavior using the methods of automata theory. The essential step
is the correct denition of the internal state variables z1,...,zm because the number of
Figure 9.
Digital circuit
according to the form
Hitler swarm
K
states of the automaton depends exponentially on the number of internal state variables
according to 2
m
.
The digital circuit of the Hitler swarm can now be determined as a switching circuit
according to the above classication because the output signals of the NOR gates y3,y4,y5,
y6 and y7 due to the corresponding re-entries are fed back to inputs of NOR gates which
precede these. In the following, these output functions are listed depending on their input
and output variables, with the relevant signal feedback marked in bold:
y1 =f(terror,y3,y7);
y2 =f(y1,n1,y4,y6);
y3 =f(y2,loyalty,y5);
y4 =f(y3,massmedia);
y5 =f(y4,socialwork);
y6 =f(y5,wareconomy); and
y7 =f(y6,terror).
The rst step in analyzing a switching circuit with the methods of automata theory is
the denition of the smallest set of state variables that completely describes the
behavior of the switching circuit. These state variables can be obtained by heuristic
analysis of the signal feedback that makes a purely combinatorial circuit a switching
circuit. In the present switching circuit, it can be seen that the output of the third NOR
gate y3 contains all ve signal feedbacks, and thus can be dened as state variable z.
All further output variables y4,y5,y6 and y7 are beyond y3, and thus are no longer
dependent on further state variables, but only on purely combinatorial connections
with the input variables.
With this one internal state variable z, the Hitler swarm switching circuit has exactly 2
1
or two internal states y3 = 0 and y3 = 1 and it can be concluded, based on the described
astable oscillation behavior of this kind of NOR gate cascades, that this switching circuit
will oscillate for certain input vectors (terror,n1,loyalty,massmedia,socialwork,
wareconomy) as long as these input vectors are present. Which input vectors excite this
astable oscillation behavior cannot be determined sufciently by heuristic analysis.
Answering these and other questions about the behavior of the switching circuit requires
analysis with the methods of digital mathematics and can be done using specic software
such as the XBOOLE software.
Modeling of the switching circuit with XBOOLE
In the following, the main steps of analyzing the switching circuit with the XBOOLE
software are described. With the NEW command, XBOOLE is reset and all internal
rooms and objects such as ternary vector lists are deleted; with SPACE, a new room
number 1 with a maximum of 32 Boolean variables is created. In this room, the entire
switching circuit will be calculated.
new
space 32 1
Now the AVAR command is used to dene and sort all input, output and state variables.
The question of why the variable pair zand zf are dened, instead of an internal state
variable, shall be put aside for the time being. It will be dealt with in the section entitled
Observability of re-entry signal feedbacks, below.
Digital
analysis
of a form
avar 1
zzf
terror n1 loyalty massmedia socialwork wareconomy
y1 y2 y3 y4 y5 y6 y7.
The essential step for representing the switching circuit is now done by entering the Boolean
functions of the connected NOR gates. The SBE command stands for Solve Boolean
Equation and uses /as negation, þas disjunction and =as equivalence operator. A
separate ternary vector list (TVL) TVL101 to TVL107 is calculated for each output variable.
sbe 1 101
y1=/(terrorþy7þz).
sbe 1 102
y2=/(y1þn1þy6þy4).
sbe 1 103
y3=/(y2þloyaltyþy5),
zf = y3.
sbe 1 104
y4=/(zþmassmedia).
sbe 1 105
y5=/(y4þsocialwork).
sbe 1 106
y6=/(y5þwareconomy).
sbe 1 107
y7=/(y6þterror).
With this step, all seven NOR gates are now completely described in terms of their ternary
vector lists and these can be merged into the global phase list TVL110 using the set operation
ISC. The command ISC stands for intersection and is a set operation that calculates the
average of two given ternary vector lists. This is essential for understanding the way in
which XBOOLE works because the identity of the variable names in space number 1 is used
to realize the connections between the individual NOR gates. The command OBBC reduces
the number of ternary vectors by means of the operations block formation and block
exchange while maintaining the orthogonality, which is only optimization ofcomputing time.
isc 101 102 110
isc 103 110 110
isc 104 110 110
isc 105 110 110
isc 106 110 110
isc 107 110 110
obbc 110
The result of the TVL110 shown in Figure 10 forms the global phase list of the switching circuit,
the 34 ternary vectors of which completely describe the behavior of the switching circuit.
Although this global phase list completely represents the behavior of the switching
circuit with all its output and transfer functions, this compact form of representation is only
conditionally suitable for interpretation. For example, the question of whether and under
which assignment of the input variables the switching circuit oscillates unstably can only
be answered by interpreting this global phase list, which requires considerable effort. To
answer these and other questions about the behavior of the switching circuit, suitable set
K
operations can be executed on this global phase list, the results of which allow the oscillation
behavior to be visualized more easily.
Observability of the re-entry signal feedbacks
At this point, before the oscillation behavior of the switching circuit can be analyzed,
however, it is useful to make a few comments on the observability of the signal feedback
resulting from the re-entries of the form. To observe the state transitions of a switching
circuit, the method of splitting the state variables zi into two-state variables zand zi is
applied. As only one state variable zhas already been analyzed for the present switching
circuit, only zand zf will be used in the following instead of zi and z.
Figure 10.
Global phase list of
the switching circuit
Digital
analysis
of a form
The following rule applies to the method of splitting: the state variable zis read-only,
while the state variable zf is write-accessible only. The variable z, thus, represents the
current state of the switching circuit, while zf represents the subsequent state of the
switching circuit. With this method, the transition of the switching circuit from state zto
the subsequent state zf becomes observable because the global phase list contains all four
possible combinations between zand zf:
z=0,zf = 0: switching circuit remains in state 0.
z=0,zf = 1: transition of the switching circuit from state 0 and state 1.
z=1,zf = 1: switching circuit remains in state 1.
z=1,zf = 0: transition of the switching circuit from state 1 to state 0.
In general, asynchronous switching circuits are created by signal feedbacks resulting from
re-entries. In contrast to the synchronous switching circuits, the timing of the state
transitions is not determined by a uniform clock signal but directly by the timing of the
change of the input variables. These asynchronous switching circuits are preferred to slower
synchronous switching circuits in some applications, due to their higher switching speed.
As the actual runtime behavior is determined by the physical parameters of the
semiconductor material, interconnects, etc., of the microchip, such asynchronous switching
circuits can be fully simulated in the design process as stable, deterministic automatons and
can later be realized in a microchip.
Determination of the automaton graph of the switching circuit
The behavior of a switching circuit can be visualized as a nite, deterministic automaton by
means of its automaton graph.In a state transition diagram, the states of the automaton, the
possible state transitions and their conditions are visualized. The state transitions are also
called transitions or edges of the graph. The automaton graph of our switching circuit
shown in Figure 11 with its two states, z= 0 and z= 1, has exactly four edges, which
describe all possible state transitions visualized by arrows: (z,zf) = (0, 0), (z,zf) = (0, 1),
(z,zf) = (1, 0) and (z,zf)=(1,1).
The edges of the automaton graph can now be calculated from the global phase list.
Therefore, the respective intersection of the global phase list, with all four possible state
transitions (z,zf) is calculated. The four ternary vector lists, TVL121, TVL123, TVL125 and
TVL127, contain all transition conditions of the respective state transition:
Figure 11.
Automaton graph of
the switching circuit
K
state transition (z,zf) = (0, 0) from z=0tozf =0.
tin 1 120
z zf.
00.
isc 110 120 121
state transition (z,zf) = (1, 1) from z=1tozf =1.
tin 1 122
z zf.
11.
isc 110 122 123
state transition (z,zf) = (0, 1) from z=0tozf =1.
tin 1 124
z zf.
01.
isc 110 124 125
state transition (z,zf) = (1, 0) from z=1tozf =0.
tin 1 126
z zf.
10.
isc 110 126 127
These automaton edges are only dependent on the input variables and, therefore, all state
and output variables can be hidden in the ternary vector lists by using the variable tuple (z,
zf,y1,y2,y3,y4,y5,y6,y7). This hiding of variables is realized by a set operation from the
Boolean Differential Calculus, the so-called k-fold maximum.
vtin 1 130
zzfy1y2y3y4y5y6y7.
maxk 121 130 131
maxk 123 130 133
maxk 125 130 135
maxk 127 130 137
The following four possible subgraphs can now be calculated by intersecting the respective
edges already determined. With regard to the consideration of the astable oscillation
behavior, the subgraph which describes the state transitions (z,zf) = (0, 1) from z=0tozf =1
and (z,zf) = (1, 0) from z=1 to zf = 0 is of special interest. This subgraph, along with
TVL142, thus contains all input vectors, where the automat changes from z=0tozf = 1 and
from z=1tozf = 0, thus oscillating astably between the two states:
all input vectors where the stable state 0 is reached.
isc 137 131 140
Digital
analysis
of a form
all input vectors where the stable state 1 is reached.
isc 135 133 141
all input vectors where the state astable oscillates between 01010...
isc 137 135 142
all input vectors where the stable state does not change, i.e. 0 or 1 remain.
isc 131 133 143
At this point in the complete problem program, further set operations are used to check two
things: rst, whether the four automaton edges also cover all possible combinations of input
signals and second, whether none of the combinations belongs tomore than one of these four
subgraphs. As these checks are not essential for further analysis of the forms presented
here, we refer here to the XBOOLE problem program Digital Analysis.PRPin the
Appendix.
Calculation of astable oscillation behavior
With the calculation of all edges and subgraphs of the automaton, its behavior is completely
described. Thus, for example, the astable oscillation behavior of the switching circuit can be
analyzed and visualized. The subgraph shown in Figure 12 and calculated with TVL142
already contains all input vectors for which the switching circuit or its state variable z,
shows astable oscillation.
For each combination of the input variables terror,n1,loyalty,massmedia,socialwork and
wareconomy, where thevalue 1 can be seen in the Karnaugh plan of Figure 12, the switching
circuit shows astable oscillation between z= 0 and z= 1. As we have determined the output
of the third NOR gate y3 as state variable z, we can use the TVL142 to directly infer output
variable y3.
Figure 12.
Input vectors for
which the switching
circuit oscillates
K
To determine how the oscillation behavior of the state variable zaffects output variable
y7, for example, we have to consider the purely combinatorial dependence of this output
variable. From the functions discussed in the section Determination of the circuit as
switching circuit, we can derive the dependence of input variables and state variable
for y7:
y4 =f(y3,massmedia)
y5 =f(y4,socialwork)
y6 =f(y5,wareconomy)
y7 =f(y6,terror)
!
y7 =f(massmedia,socialwork,wareconomy,terror,z)
As these combinatorial dependencies are also already included in the global phase list of the
switching circuit, all that remains is for us to calculate the intersection of TVL142 with the
global phase list.
isc 142 110 162
To visualize the oscillation behavior of output variable y7, the intersection already
calculated with TVL162 is cut with the variable tuple (y7) = (1) and displayed as TVL17 per
k-fold maximum only depending on the input variables and theinternal state variable z.
tin 1 177
y7.
1.
isc 177 162 17
maxk 17 170 17
obbc 17
As shown with the Karnaugh plan in Figure 13 for the output variable y7, the astable
oscillation behavior of the internal state variable zonly results in an astable oscillation
between value 0 and 1: 01010...for the two orange marked input vectors (terror,n1,loyalty,
massmedia,socialwork,wareconomy) = (0, 0, 0, 0, 0, 0), as well as (0, 1, 0, 0, 0, 0) due to the
Figure 13.
Oscillation behavior
of output variable y7
Digital
analysis
of a form
further combinatorial dependencies of the output variable y7 described above. For the three
other ve input vectors already visualized in Figure 12, the internal state variable zof the
switching circuit oscillates in the same way, but only between values 1 and 1: 11111....
In the complete problem program Digital Analysis.PRPoutlined in the Appendix, the
astable oscillation behavior for all output variables y1 to y7 is calculated as TVL1 to TVL7
in the way shown here as an example for the output variables y7 and displayed with TVL11
to TVL17 depending on the input variables and the state variables.
Indeterminacy and degree of re-entrant forms
With the analytical results obtained through the application of automata theory, it is worth
taking a look at the denitions of the terms indeterminacy and degree in Chapter 11
(Spencer-Brown, 1972, pp. 5468). Both the indeterminacy and the degree of an expression
are derived from a form called Echelon. First, in the ninth canon which outlines the rule that
a demonstration rests in a nite number of steps(rule of demonstration), it is shown how
the simple Echelon with a nite number of algebraic steps can be transformed into every n-
fold Echelon. Figure 14 shows the simple and the n-fold Echelon as form and the simple
Echelon as the transformed circuit of NOR gates.
According to Spencer-Brownsdenition (Spencer-Brown, 1972 p. 14), the variables m
and nare dened as expressions, which themselves indicate marked and unmarked states,
respectively:
Let mstand for any number, greater than zero, of such expressions indicating the
marked state.
Let nstand for any number of such expressions indicating the unmarked state.
(Spencer-Brown, 1972, p. 14).
By using the 2
2
or four possible variable combinations for mand n, the solutions of the
simple Echelon, as well as of each nite n-fold Echelon can be determined with a nite
number of steps. The result of the simple and the n-fold Echelon is, thus:
f(n,n)=n;
f(n,m)=n;
f(m,n)=m; and
f(m,m)=n.
The calculation of the corresponding circuits from NOR gates gives the identical result for
both the simple and the n-fold Echelon, which is shownas Karnough plan in Figure 14.
This calculability of an expression in a nite number of steps, dependent only on the
(marked or unmarked) values of the variables, is classied according to George Spencer-
Figure 14.
Transformation from
simple to n-fold
Echelon
K
Brown as an expression of the rst degree. The indeterminacy and degree of an expression
are dealt with in Spencer-Brown (1972, p. 57) as follows:
Equations of expressions with no re-entry, and thus with no unresolvable
indeterminacy, will be called equations of the rst degree, those of expressions with one re-
entry will be called of the second degree, and so on.(Spencer-Brown, 1972,p.57).
Exactly what the and so onmeans for the relation of a count of re-entries and degree,
remains unclear in this denition. Probably a linear relation in the sense of degree =
count of re-entries þ1 is intended. In any case, both, the simple and every nite n-fold
Echelon are expressions of the rst degree and correspond to purely combinatorial digital
circuits.
The introduction of re-entry in the case of the re-entrant Echelon shown in Figure 15
results in an indenite regression and the result of the expression cannot be determined with
anite number of steps. This corresponds exactly to the indeterminacy of an expression
according to Spencer-Brown (1972, p. 57). The re-entrant Echelon, having exactly one re-
entry, is, thus, of the second degree and its transformation into a digital circuit results in a
switching circuit which can be analyzed as follows.
With the rules from the section The form as a digital circuit, the re-entrant Echelon of
the second degree can be transformed into a digital circuit of NOR gates, also shown in
Figure 15. Based on this, theautomaton graph of this re-entrant Echelon can be calculated as
shown in Figure 15, using the methodology described in the section Determination of the
automaton graph of the switching circuit. Thus, the re-entrant Echelon has exactly one
state variable z. From this, the 2
1
or two states of the automaton result. At this point, it is
important to note that this automaton graph not only describes the switching circuit of NOR
gates but also the form itself. The output variable yis not only dependent on the input
variables aand bbut also on the state variable z; the Karnaugh plan of y=f(a,b,z) is also
shown in Figure 15.
Figure 15.
Re-entrant Echelon,
Karnaugh plan of y
and automaton graph
Digital
analysis
of a form
The functionality of the re-entrant Echelon can now be described completely. The output
signals of the two NOR gates result with the input vector (a,b) = (0, 0) from the negation of
the respective other input. However, as these respective other inputs depend on the value of
the internal state variable z, we have to consider both possibilities z= 0 and z= 1 for the
input vector (a,b) = (0, 0).
In the case of the internal state variable z=0, the second NOR gate would set the output
variable yto the value 1 and return this value 1 to the other input of the rst NOR gate. By
negating this 1 again by the rst NOR gate, the output state z= 0 is stabilized.
In the case of the internal state variable z= 1, the input vector (a,b) = (0, 0) results in the
output signal y= 0 and this returned value 0 stabilizes the output state z= 1 via the rst
NOR gate. Thus, if nosignal (a,b) = (0, 0) is present at the two inputs of the re-entrant
Echelon, it stores its present state with the value of its internal state variable z.
In addition to this memory function, the re-entrant Echelon can be selectively set to its
two states with the corresponding input vectors. The resetting to the state z= 0 is done with
a= 1 and this independent of the current state. With a= 1 the value 0 results for the rst
NOR gate, and thus for zand this also independently from the input variable band in
ternary logic the corresponding input vector (a,b) = (1, ) can be written. The output
variable ycorresponds in the state z= 0 to the negation of the input variable b, which is not
important for the memory function or resetting.
With the input vector (a,b) = (0, 1), the setting to the state z= 1 is carried out, likewise
independently of the current state. The value of the input variable bof 1 results in the value
0 for the output of the second NOR gate, and thus for the output variable yand this returned
value results in the value 1 for the output of the rst NOR gate, and thus for the internal
state variable z.
These observations show that George Spencer-Browns indeterminacy can be described
more precisely by means of automata theory and with the help of the medium of time. The
above-quoted analysis from Spencer-Brown (1972, p. 57) is correct in that the re-entrant
Echelon as a form of higher degree cannot be determined by a nite number of algebraic
steps from the input variables aand b. However, the re-entrant Echelon can be brought to a
dened state with a calculable and nite sequence of signals at the inputs aand band then
be completely determined in this state. Thus, the determination of a form of higher degree is
no longer done purely algebraically in the Calculus of Indications, but by a nite number of
arithmetic steps in themedium of time, which will be showna bit later in this section also for
the negative self-reference. The indeterminacy of a form of higher degree is not a total
indeterminacy if the automaton can be set to a dened state by a nite number of steps,
caused by calculable input vectors and then is completely calculable using the output
function and the transition function.
George Spencer-Brown denes the degree of an expression in direct dependence of the
number of re-entries according to the quotation above (Spencer-Brown, 1972, p. 57): degree =
number of re-entries þ1. However, although the Hitler swarm has altogether ve re-entries,
the automaton can be completely described with only one internal state variable zand 2
1
or
two states. According to the denition above, the Hitler swarm would have to be an equation
of the sixth degree, which the analysis results do not conrm. In the authors opinion, the
denition of the degree of an expression should be coupled to the number of state variables,
which are created by re-entries instead of the number of re-entries: degree = number of state
variables þ1. With this improved denition, the Hitler swarm is a form of the second degree.
Although according to this proposed denition, both the Hitler swarm and the re-entrant
Echelon are expressions of the second degree, the rst shows an astable oscillation behavior;
the other, a self-stabilizing memory behavior. This difference corresponds to the difference
K
between negative and positive self-referentiality, i.e. whether the corresponding re-entry has
been returned by a sequence of odd or even negations. Whether a form in the medium of
time now realizes memory or oscillation behavior would be a further, determinable
parameter and could play a respective role for the comparability of forms. In any case, the
algebraic indeterminacy of all forms of higher than rst degree dened in Spencer-Brown
(1972) is extended by their determinability in the medium of time.
At this point, a remark on the importance of the re-entrant Echelon for digital and
computer technology is in order. As shown in Figure 16, the re-entrant Echelon represents
the asynchronous, non-clocked RS ip-op or RS latch, where the internal state variable zis
also available as output variable Qof the rst NOR gate. The input Rlike Resetresets the
ip-op with the value 1 to the state zor Qequals 0, while the input Slike Setsets the ip-
op with the value 1 to the state zor Qequals 1.
Together with their synchronous, clocked variants, the ip-ops with their one-bit memory
function form the basic elements of digital and computer technology and also constitute the
heart of every modulo counter. The typical ip-op pattern of a crosswise interconnection of
input and output signals of a pair of NOR gates is, of course (color-coded) found in the
already discussed form E4 shown in Figure 17, as well as in the switching circuit of the
Modulator shown in Figure 18.
Although ip-ops play such a fundamental role for the digital and computer industry as
basal switching and memory elements, George Spencer-Brown did not mention these ip-
op patterns either in the E4 form or in the Modulator. However, as the ip-op was
invented as early as 1918 by two British physicists and also patented with Eccles and Jordan
(1920), he should have been familiar with the concept of ip-ops. In addition, as Chief Logic
Designer for Mullard Equipment Limited, he would also probably have been responsible for
ip-op products, as advertised in Norbit (1960) as Prefabricated Electronic Bricks(i.e.
early integrated circuits).
Figure 17.
Form E4 with color-
coded ip-ops
Figure 16.
Re-entrant Echelon as
RS ip-op
Digital
analysis
of a form
The determination of the degree of the form E4 is now almost impossible with the above-
cited denition from Spencer-Brown (1972, p. 57). E4 has seven re-entries in total and would,
thus, be of eighth degree, but of these seven re-entries only the three backward-facing ones
contribute to the emergence of internal state variables. The four forward-facing re-entries
have an only a combinatorial inuence on the values calculated with them. Based on the two
internal state variables g2z and g4z from the two RS ip-ops, automata theory provides the
automaton graph shown in Figure 19, which results in a total of 2
2
or four internal states.
Thus, the form E4 is according to our improved denition degree = number of internal state
variables þ1 of the third degree.
Astable oscillation behavior or imaginary state?
A consideration of Chapter 11 (Spencer-Brown, 1972, pp. 5468) would probably be
incomplete without a discussion of the imaginary state. In Spencer-Brown (1972, p. 58), a
function of negative self-reference is examined with form E3 shown in Figure 20, in which
the negation operation is reintroduced into itself.
This equation, leading into an innite regress or into a paradox, results neither in the
marked state nor the unmarked state as a solution and leads to the introduction of the
imaginary state. As in his consideration of indenitely extended negative self-reference
(Spencer-Brown, 1972, p. 58), the Calculus of Indications is not abandoned, this paradox is
solved with the means of time:
Figure 18.
Switching circuit of
the Modulator with
color-coded ip-ops
Figure 19.
Automaton graph of
E4 and Modulator
K
Time
As we do not wish, if we can avoid it, to leave the form, the stage we envisage is not in
space but in time.(Spencer-Brown, 1972,p.58).
With the assumption that each state transition between the two states of the Calculus of
Indications {marked, unmarked} takes a certain amount of nite time, the solution of E3 is a
resulting oscillation in the form of a rectangular wave marked-unmarked-marked-
unmarked-marked-.... But at this point, George Spencer-Brown has no further reason for
the statement of an additional imaginary state and the extension of space to {marked,
unmarked, imaginary}. Neither the effects of this new state on the operations introduced so
far are shown, nor does he describe what exactly can be done with this new state within the
Calculus of Indications. By merely naming the imaginary state, the text, at most, opens a
door for all conceivable metaphors of the imaginary, to which the explicit, graphically
produced reference to an imagined tunnel in a higher dimension also contributes. It is simply
assumed, that only in this way and without crossing the border state transitions between
the two states {marked, unmarked} are conceivable. As summarized in Fuchs and Hoegl
(2011, p. 194), however, a graphical representation of a tunnel remains only a representation
and is not a tunnel itself.
The introduction of the new imaginary state is compared in Spencer-Brown (1972, p. ix)
with the introduction of complex numbers in mathematics. However, while there are
practical applications for complex numbers, e.g. with the conversion between Cartesian
coordinates and polar coordinates which, in our estimation, have contributed signicantly
to the inclusion of complex numbers in the mathematical canon , no application of the
imaginary state is known in circuit technology and digital circuits continue to operate in the
binary space of Boolean logic {0, 1}. The following quote by George Spencer-Brown from
EE Times (1994, pp. 3135) must rather be attributed to a euphemistic self-description:
Armed with these imaginary values, digital engineers can now analyze circuits with
equations, rather than antiquated state diagrams.(EE Times, 1994, pp. 3135).
In digital circuit technology, such an astable oscillation behavior resulting in a
rectangular wave 01010... is handled according to the exception/rule scheme. With the
methods of automata theory shown, all possible states and their corresponding transition
conditions can be determined for any switching circuit. If an automaton with the same
transition conditions can change from one state to another, as well as from that state back to
the rst one, then the automaton oscillates back and forth between these states as long as
these transition conditions are present at the inputs. As it is known from the analysis, with
which transition conditions the switching mechanism can be brought into one of these two
oscillation states at all, exactly these assignments of the input variables, and thus the
oscillation behavior, can be specicallyprevented with circuit-technical measures.
Beyond the avoidance of oscillation, however, there are applications in which oscillation
behavior is specically used. In Sutherland (1999 p. 7), for example, a ring oscillator shown
in Figure 21 is described as a chain of an odd number of NOT gates, which can be used to
generate 01010 ... rectangular signals for test purposes. As already shown in the section
From universal NOR gate to the crossusing the Logical Effort approach outlined by
Sutherland (1999), the delay times of the logic gates, and thus the frequency of the
Figure 20.
Form E3 of negative
self-reference
Digital
analysis
of a form
rectangular signals, together with the odd number of NOT gates connected in the ring
oscillator, can be calculated.
But independent of the treatment as an exception or as a rule, neither an imaginary state
nor access to higher dimensions is needed to describe astable oscillation behavior in Boolean
space {0, 1}.
Observation of the results with regard to the form
The above demonstration showed how methods of digital mathematics can be used to
analyze the digital circuit of NOR gates derived from the form of the Hitler swarm with
software support. Thereby, the so-called indeterminacy of this second-order form could be
determined more precisely using state variables and the automaton graph in the medium of
time. These structural properties of the form are exclusively determined by the topological
nesting of the seven crosses with their ve re-entries and the positioning of the six input
variables within the form.
In no case are the structural properties of forms determined in a specic way by the
semantics of formalized conceptual relations. The following thought experiment will clarify
this. In the form shown in Figure 6, conceptual words chosen for the historical-sociological
swarm analysis of the Nazi dictatorship have been arranged in such a way that different
observation perspectives of relevant distinctions can be communicated. If, while retaining
the structure of the form, the conceptual words terror,n61,loyalty,mass media,social work
and war economy were replaced by infection risk,incidence,trust,mass media,intensive care
and vaccine development, the form could be interpreted as a description of an observation of
the COVID-19 pandemic. The analysis results that could be obtained with the methods
shown would be identical for (terror,n1,loyalty,massmedia,socialwork,wareconomy) and
(infectionrisk,incidence,trust,massmedia,intensivecare,vaccinedevelopment), but no
conclusions can be drawn from this about similarities or comparability on the sociological-
historical level.
To answer the question of how the results of the analysis relate to the original
sociological-historical observation of the Nazi dictatorship, we must take a closer look at the
two transformations that took place:
(1) transformation of swarm analysis from the medium of natural language into the
form.
(2) transformation of the form into the digital circuit of NOR gates.
As we have seen with the example of the Modulator from Chapter 11 (Spencer-Brown, 1972,
pp. 5468), the second transformation from the form to the digital circuit is already
established in the Calculus of Indications itself. This is further conrmed by the described
genesis of the cross from the circuit technology issues of the early industrial NOR gates.
However, one has to ask oneself what historically relevant meaning the statement can
have that the Hitler swarm gets into an astable oscillation state when terror and n61and
loyalty and mass media and social work and war economy equal 0. The same meaning would
remain for all other conceptual words in the same formal structural context of the form.
Thus, following the above thought experiment, with infection risk and incidence and trust
and mass media and intensive care and vaccine development equal to the value 0, the
pandemic would oscillate in an astable condition. These and the other statements made
about the functional result of the Hitler swarm depending on the internal state and the input
variables are mathematically correct but do not make sense in relation to the sociological
problem. As the second transformation from the form into the digital circuit is established in
K
the Calculus of Indications itself, there remains the view of the rst transformation from the
medium of natural language into the form at this point.
While the Calculus of Indications in Spencer-Brown (1972) allows the use of variables only
as expressions that themselves result in the marked or unmarked state, Baecker (2013) overcomes
this limitation and uses conceptual words designating abstract sociological categories as
variables that cannot be reduced to the marked or unmarked state. Already with this use as
variables in the form and the implied assignment of sociological categories such as terror,loyalty,
mass media,social work,war economy and abstract quantity concepts such as the number of
members n61to the marked and unmarked state or the Boolean values 0 and 1, the historical-
sociological meaning is lost. Conversely, this means that the results of the analysis can no longer
be projected back onto the initial problem formulated in the medium of natural language.
The comparison of the system denition according to Luhmann (1995) from Figure 1
with the structurally identical re-entrant Echelon gives a similar result. The memory
behavior of the re-entrant Echelon could be related to the fact that the form system/
environment quasi stores the last marked state system or environment, but this does not
seem to be very useful to us. Also worth mentioning for this interpretation is the fact, that
system and environment are disjunctive to each other, mutually exclusive and jointly
exhaustive. However, as Niklas Luhmann with his difference-theoretical and self-referential
theoretical approach only uses the unity of the operation of distinction and indication and
the re-entry and, as he does not refer to the mathematics of the Calculus of Indications, norto
indeterminacy, degree or imaginary state, questions of the traceability of the results of the
analysis to systems theory are not important.
But due to the principal algebraic indeterminacy of higher degree forms, the use of the
equals sign in the system denition from Figure 1 does not satisfy George Spencer-Browns
denition of value and equality (Spencer-Brown, 1972,p.5):
Value
Call a state indicated by an expression the value of the expression.
Equivalence
Call expressions of the same value equivalent.
Let a sign
=
of equivalence be written between equivalent expressions. (Spencer-Brown, 1972,p.5).
Figure 21.
Ring oscillator of
2nþ1 NOT gates
Figure 22.
System denition as
form according to
Luhmann (1995) with
assignment
Digital
analysis
of a form
With the use of the assignment sign :=shown in Figure 22, this algebraic contradiction to
the Calculus of Indications can be avoided. Furthermore, the assignment sign corresponds
more to the character of the denition than the equals sign. As the Calculus of Indications
itself gets along completely without deductive denitions and with Draw a distinction.
(Spencer-Brown, 1972, p. 3), there was no need in the Calculus of Indications itself to dene
the assignment operation.
Perhaps, the assumption from Roth (2019) that true distinctionsare of special importance
is a good starting point for assessing the general applicability of the presented digital methods
of analysis to sociological observations noted by means of Calculus of Indications.
Furthermore, the determinability of higher-degree forms in the medium of time, as shown
above, points to possible applications in which given sequences of input signals reproduce the
desired behavior. However, it should be noted that higher-degree forms represent nite and
deterministic automata, and thus describe non-contingent behavior, which seems rather
unsuitable for sociological applications. A further aspect of this consideration is due to the fact
that higher-degree forms describe the structure of the automaton and not its behavior, which
has to be derived from the structure by the methods shown.
Forms used in sociology such as the Hitler swarm from Figure 6 or the system denition
in Figure 1 or Figure 22, describe andcommunicate the different observation perspectives of
relevant distinctions of the respective sociological question. In doing so, they use rules
derived from the Calculus of Indications to make references visible that could be
communicated in the classical notation in the medium of natural language albeit more
obscurely and with greater effort. However, these applications of the Calculus of Indications
seem to elude purely mathematical processing by means of algebraic and arithmetic
operations. In this respect, the question of which sociological questions are at all suitable for
the mathematical processing shown by means of digital analysis opens up a wide eld for
further research.
References
Akers, S.B. (1959), On a theory of Boolean functions,Journal of the Society for Industrial and Applied
Mathematics (SIAM), Vol. 7 No. 4, pp. 487-498.
Baecker, D. (1994), Soziale Hilfe als Funktionssystem der Gesellschaft,Zeitschrift für Soziologie,
Vol. 23, pp. 93-110.
Baecker, D. (2013), The Hitler swarm, Thesis Eleven.
Baecker, D. (2015), Working the Form. George Spencer-Brown and the Mark of Distinction,in
Martínez, C. and Bischof, P. (Eds), The Future is Here, Mousse Magazine, Supplement Settimana
Basileia, pp. 42-47.
Bochmann, D. and Posthoff, C. (1981), Binäre Dynamische Systeme, Akademie-Verlag, Berlin.
Davio, M., Deschamps, J.P. and Thayse, A. (1978), Discrete and Switching Functions, McGraw-Hill
International.
De Morgan, A. (1838), Induction (mathematics), The Penny Cyclopedia.
Douglas, M. (1982), In the Active Voice, Routledge & Kegan Paul, London.
Eccles, W.H. and Jordan, F.W. (1920), Improvements in ionic relays, British patent number: GB
148582 (led: 21 June 1918; published: 5 August 1920).
EE Times (1994), Electronic Engineering Times, CMP Publications Inc., 600 Community Dr.,
Manhasset, NY 11030. Feb. 14, pp. 31-35.
Enzensberger, H.M. (1974), The Consciousness Industry: On Literature, Politics and the Media, Roloff,
M. (Ed.), Seabury Press, New York.
K
Fuchs, P. and Hoegl, F. (2011), Die Schrift der Form, in Pörksen, B. (Ed.), Schlüsselwerke des
Konstruktivismus, VS Verlag für Sozialwissenschaften.
Götz, N. (2001), Ungleiche Geschwister: Die Konstruktion von nationalsozialistischer Volksgemeinschaft
und schwedischem Volksheim, Nomos, Baden-Baden.
Horkheimer, M. and Adorno, T.W. (2002), Dialectic of Enlightenment, Schmid Noerr, G. (Ed.), Jephcott,
E. (trans.), Stanford University Press, Stanford, CA.
Huffman, D.A. (1958), Solvability criterion for simultaneous logical equations, Quarterly Progress
Report 1.56, pp. 87-88.
Hoegl, F. (2021), Meaning Negation, in Roth, S., Heidingsfelder, M., Clausen, L. and Laursen, K. (Eds),
George Spencer BrownsDesign with the NOR: With Related Essays, Emerald Publishing
Limited, Bingley, ISBN: 9781839826115, pp. 63-76.
Kauffman, L.H. (2001), The mathematics of Charles Sanders Peirce,Cybernetics and Human
Knowing, Vol. 8 No. 1/2, pp.79-110.
Kauffman, L.H. (2006), Laws of Form an Exploration in Mathematics and Foundations, Rough Draft,
available at: http://homepages.math.uic.edu/kauffman/Laws.pdf (Viewed on 30.12.2020).
Luhmann, N. (1995), Social Systems, Stanford University Press.
Luhmann, N. (1997), Die Gesellschaft der Gesellschaft, Frankfurt-am-Main, Suhrkamp.
Luhmann, N. (2000b), TheRealityoftheMassMedia,Cross,K.(trans.),StanfordUniversityPress,Stanford,CA.
Nolzen, A. (forthcoming), The Nazi Partys operational codes after 1933,inGotto,B.andSteber,M.(Eds),
A Nazi Volksgemeinschaft? German Society in the Third Reich, Oxford University Press, Oxford.
Norbit (1960), Prefabricated Electronic Bricks,Wireless World, Vol. 66 No. 8, p. 374.
Norbit (1962), Norbit Sub-Assemblies YL 6000 Series,Norbit Handbook, 2nd ed., Mullard Equipment
Limited, available at: www.electrojumble.org/DATA/Norbits.pdf (accessed 30 December 2020).
Parsons, T. (1951), The Social System, Free Press, New York.
Reed, I.S. (1954), A class of multiple-error-correcting codes and the decoding scheme,Transactions of
the IRE Professional Group on Information Theory, Vol. 4 No. 4, pp. 38-49.
Roth, S. (2017), Parsons, Luhmann, Spencer Brown. NOR design for double contingency tables,
Kybernetes, Vol. 46 No. 8, pp. 1469-1482.
Roth, S. (2019), Digital transformation of social theory. A research update,Technological Forecasting
and Social Change, Vol. 146, pp. 88-93.
Roth, S., Heidingsfelder, M., Clausen, L. and Laursen, K. (2021), George Spencer BrownsDesign with
the NOR: With Related Essays, Emerald Publishing Limited, Bingley.
Simmel, G. (1989), Über soziale Differenzierung, in Simmel, G. (Ed.), Gesamtausgabe, Aufsätze 1887
1890, Dahme H-J. Frankfurt-am-Main, Suhrkamp, Vol. 2, pp. 109-295.
Spencer-Brown, G. (1972), Laws of Form, The Julian Press, Inc. Publishers, New York, NY.
Steinbach, B. (1992), XBOOLE a Toolbox for Modelling, Simulation, and Analysis of Large Digital
systems,System Analysis and Modeling Simulation, Vol. 9 No. 4,pp. 297-312.
Steinbach, B. and Posthoff, C. (2009), Logic Functions and Equations, Examples and Exercises, Springer
Science þBusiness Media B.V.
Sutherland, I.E., Sproull, R.F. and Harris, D. (1999), Logical Effort: Designing Fast CMOS Circuits,
Morgan Kaufmann Publishers, San Francisco, CA.
Swidler, A. (1986), Culture in action: Symbols and strategies,American Sociological Review,Vol.51,pp.273-288.
Thayse, A. (1981), Boolean Calculus of Differences, Springer, Berlin, ISBN: 978-3-540-0286-1.
Weber, M. (1978), Economy and Society: An Outline of Interpretive Sociology, in Roth, G. and Wittich, G.
(Eds), Fischoff, E. et al. (trans), University of California Press, Berkeley.
Wittgenstein, L. (1922), Tractatus Logico-Philosophicus, Routledge, London.
Digital
analysis
of a form
Further reading
Spencer Brown, G. (1961), Design with the NOR, Mullard Equipment Limited, Crawley, in Roth, S.,
Heidingsfelder, M., Clausen, L. and Laursen, K. (2021), George Spencer BrownsDesign with the NOR:
With Related Essays, Bingley: Emerald Publishing Limited (8 Mar. 2021), ISBN: 9781839826115.
Appendix. XBOOLE problem program Digital Analysis.PRP
; reset XBOOLE system
new
;dene space with maximal 32 boolean variables
space 32 1
;dene all input variables
;dene all internal state variables zi as the rst yi that
; includes all feedback loops
; typically, GSB-forms should have only one internal state
; variable, but larger chains could have more than one
; zi should be used as read-onlyof the internal state variable
;zshould be used as write-only of the internal state
; variable just at the exit of thegate
;dene all output variables
avar 1
zzf
terror n1 loyalty massmedia socialwork wareconomy
y1 y2 y3 y4 y5 y6 y7.
;dene all boolean functions
sbe 1 101
y1=/(terrorþy7þz).
sbe 1 102
y2=/(y1þn1þy6þy4).
sbe 1 103
y3=/(y2þloyaltyþy5),
zf=y3.
sbe 1 104
y4=/(zþmassmedia).
sbe 1 105
y5=/(y4þsocialwork).
sbe 1 106
y6=/(y5þwareconomy).
sbe 1 107
y7=/(y6þterror).
; calculate global phase list tvl110 as the intersection of all
; boolean functions
isc 101 102 110
isc 103 110 110
isc 104 110 110
isc 105 110 110
isc 106 110 110
isc 107 110 110
obbc 110
K
; analyze the automaton graph
;dene all transitions between internal state variables
; zi/zin tvl12i
tin 1 120
z zf.
00.
tin 1 122
z zf.
11.
tin 1 124
z zf.
01.
tin 1 126
z zf.
10.
; calculate all edges of the automata as transitions in
; tvl13i based on global phase list tvl110
; show all edges of the automat as dependent on input variables in
; tvl13i, only
vtin 1 130
zzfy1y2y3y4y5y6y7.
; z/zf 0>0
isc 110 120 121
maxk 121 130 131
; z/zf 1>1
isc 110 122 123
maxk 123 130 133
; z/zf 0>1
isc 110 124 125
maxk 125 130 135
; z/zf 1>0
isc 110 126 127
maxk 127 130 137
; calculate all input vector leading to stable internal state 0
; z/zf >0
isc 137 131 140
obbc 140
; calculate all input vector leading to stable internal state 1
; z/zf >1
isc 135 133 141
obbc 141
; calculate all input vector leading to instable state oscillation
; between 0 and 1
Digital
analysis
of a form
; z/zf 0>0, 1>1
isc 137 135 142
obbc 142
; calculate all input vector leading to stable states 0 and 1
; (empty)
; z/zf 1>0, 0>1
isc 131 133 143
obbc 143
; calculate all partial automaton graphs
; check if all input vectors are covered (tvl150 should be all -)
uni 140 141 150
uni 150 142 150
uni 150 143 150
obbc 150
; calculate phase list leading to stable internal state 0
; z/zf >0
isc 140 110 160
obbc 160
; calculate phase list leading to stable internal state 1
; z/zf >1
isc 141 110 161
obbc 161
; calculate phase list leading to instable state oscillation
; between 0 and 1
; z/zf 1>0, 0>1
isc 142 110 162
obbc 162
; calculate phase list leading to stable states 0 and 1 (empty)
; z/zf 0>0, 1>1
isc 143 110 163
obbc 163
; variable tuple to show all input variables and internal
; state variable z
vtin 1 170
zf y1 y2 y3 y4 y5 y6 y7.
; variable tuple to show all output variables = 1
tin 1 171
y1.
1.
tin 1 172
y2.
1.
tin 1 173
y3.
K
1.
tin 1 174
y4.
1.
tin 1 175
y5.
1.
tin 1 176
y6.
1.
tin 1 177
y7.
1.
; calculate all output variables yi on basis of global phase list
isc 171 110 1
maxk 1 170 1
obbc 1
isc 172 110 2
maxk 2 170 2
obbc 2
isc 173 110 3
maxk 3 170 3
obbc 3
isc 174 110 4
maxk 4 170 4
obbc 4
isc 175 110 5
maxk 5 170 5
obbc 5
isc 176 110 6
maxk 6 170 6
obbc 6
isc 177 110 7
maxk 7 170 7
obbc 7
; calculate all input vector leading to instable state
; oscillation between 0 and 1
; z/zf 1>0, 0>1
isc 171 162 11
maxk 11 170 11
obbc 11
isc 172 162 12
maxk 12 170 12
Digital
analysis
of a form
obbc 12
isc 173 162 13
maxk 13 170 13
obbc 13
isc 174 162 14
maxk 14 170 14
obbc 14
isc 175 162 15
maxk 15 170 15
obbc 15
isc 176 162 16
maxk 16 170 16
obbc 16
isc 177 162 17
maxk 17 170 17
obbc 17
Corresponding author
André Oksas can be contacted at: andre.oksas@gmx.de
For instructions on how to order reprints of this article, please visit our website:
www.emeraldgrouppublishing.com/licensing/reprints.htm
Or contact us for further details: permissions@emeraldinsight.com
K
ResearchGate has not been able to resolve any citations for this publication.
Book
Full-text available
Das zwanzigste Jahrhundert ist zu Recht als ein "Jahrhundert der Extreme" bezeichnet worden. Die in der historischen Bewertung so unterschiedlichen nationalen und sozialen Integrationskonzepte der Volksgemeinschaft in Deutschland und des Volksheims in Schweden stehen exemplarisch für destruktive beziehungsweise konstruktive Versuche, den Herausforderungen dieses Jahrhunderts zu begegnen. Der empirische Hauptteil der Dissertation ist in zwei große Blöcke untergliedert. Zunächst wird die Begriffsgeschichte von Volksgemeinschaft (folkgemenskap) und Volksheim (folkhem) im Deutschen und Schwedischen verglichen. Angesichts der Vielzahl unterschiedlicher Verwender und Verwendungszusammenhänge lautet die Schlußfolgerung, daß das heute gängige Verständnis der Volksgemeinschaft als einer propagandistischen Luftblase der deutschen Nationalsozialisten und des schwedischen Volksheims als der Umschreibung für einen egalitären Wohlfahrtsstaat das Ergebnis spezifischer politischer Erfahrungen und insofern gut begründet, aber kein essentieller, vor- oder festgeschriebener Bedeutungsgehalt der Worte ist. Die in der historischen Entwicklung so unterschiedlichen Fälle Deutsches Reich und Schweden werden im ersten Teil des Vergleichs also auf Gemeinsamkeiten der Spannbreite ihrer Schlüsselbegriffe Volksgemeinschaft und Volksheim hin analysiert. Unterschiede werden dabei weniger im Ländervergleich als beim Vergleich unterschiedlicher Akteure innerhalb der beiden Länder deutlich. Im zweiten großen empirischen Block werden für den Zeitraum 1932/33 bis 1945 einzelne Politikfelder für das Deutsche Reich und Schweden untersucht, die im Zusammenhang mit Diskussionen über Volksgemeinschaft und Volksheim von besonderem Belang sind. Ausgewählt wurden für die Analyse die Bereiche Jugendpolitik, Dienstpflichtpolitik, Sozialpolitik und Bevölkerungspolitik. Mit Ausgangspunkt in der ähnlichen Begrifflichkeit im Deutschen und Schwedischen geht es in diesen Kapiteln darum, unterschiedliche Anwendungsweisen und Innovationsformen der praktischen Politik herauszuarbeiten.
Article
Full-text available
This article outlines the basic design of digitally transformed social theory. We show that any digital world is created by the drawing and cross-tabling of binary distinctions. As any theory is supposed to be concerned with truth, we introduce to and insist on the distinction between true and false distinctions. We demonstrate how flexible matrix-shaped theory architectures based on true distinctions allow for the reduction and unfolding of the entire complexity of analogue social theories. The result of our demonstrations is the idea of a theoretical Supervacuus. The social equivalent of a universal Turing machine, this supervacuous social theory is virtually empty as it is based on only one proper theoretical premise (the idea of distinction [between true and false distinctions]), and therefore able to simulate all other social theory programmes. We conclude that our digitally transformed social theory design is particularly useful for observations of a digitally transformed society.
Article
Full-text available
Purpose Cross tables are omnipresent in management, academia, and popular culture. The Matrix has us, despite all criticism, opposition, and desire for a way out. In this essay, we draw on the works of three agents of the matrix. We show that Niklas Luhmann criticised Talcott Parsons’ traditional matrix model of society and proceeded to update systems theory, the latest version of which is coded in the formal language of George Spencer Brown. As Luhmann failed to install his updates to all components of his theory platform, however, we observe regular reoccurrences of Parsonian crosstabs, particularly in the Luhmannian differentiation theory, where it results in compatibility issues and produces error messages requesting updates. In this essay, we code the missing update translating the basic matrix structure from Parsonian into Spencer Brownian formal language. Design/methodology/approach We draw on work by Boris Hennig, Louis Kauffman, and a yet unpublished manuscript by George Spencer Brown to demonstrate that the latter introduced his cross as mark to indicate NOR gates in circuit diagrams. We also show that this NOR gate marker has been taken out of and may be observed to contain the tetralemma, an ancient matrix structure from traditional Indian logics. We proceed to translate the basic structure of traditional contingency tables into a Spencer Brownian NOR equation and demonstrate the difference this translation makes in the modelling of social systems. Findings Our translation of cross tables from Parsonian into Spencer Brownian formal language results in the design a both matrix-shaped and compatible test routine that a) works as virtual window for the observation of the actually unobservable medium in which a form is drawn and b) can be used for consistency checks of expressions coded in Spencer Brownian formal language. Originality/value We quote from and discuss a so far unpublished manuscript finalised by Spencer Brown in April 1961. We translate the basic matrix structure from Parsonian into Spencer Brownian formal language. We code a Spencer Brownian NOR matrix that may be used to detect errors in expressions coded in Spencer Brownian formal language.
Chapter
Full-text available
Die Laws of Form (Spencer-Brown 1969) sind nicht voraussetzungslos in die Welt mathematischer Texte getreten. Die Probleme, auf die der Text reagiert, aber auch manche der vorgeführten Lösungen, werden einem anderen und womöglich tieferen Verständnis zugänglich, wenn man die Laws of Form im Kontext der Mathematik- und Logikgeschichte betrachtet. Die Laws of Form werden, in der hier vorgeschlagenen Perspektive, als ein spätes Werk der mathematischen „Gegenmoderne“ beobachtet, das auf eigenwillige Art begründungstheoretische Diskussionen vergangener Tage wiederbelebt, und sich hierfür den Logizismus des Frege/Russell- Programms als Kontrastfolie der eigenen Ansichten wählt; Ansichten, die im nicht-klassischen Mathematikbild des konstruktivistischen Intuitionismus eine (mathematikinterne) vorbereitete Umgebung finden – auch und gerade, was die mystisch- spirituellen Momente des Intuitionismus betrifft. Zugleich vollziehen die Laws of Form ein ästhetisches, zum Intuitionismus geradezu antithetisch aufgestelltes Formalismusprogramm (eine „Schrift der Form“), welches Ansätze aus Ludwig Wittgensteins Tractatus aufnimmt und in radikalisierender Weise weiterentwickelt. Durch diese Erweiterung des Aufmerksamkeitsbereichs mag in der anschließenden Skizze der wichtigsten Gedankenlinien der Laws of Form deutlicher hervortreten, was an den Laws of Form auf andere Weise neu ist – aber auch, was vielleicht nicht. Unser Beitrag schließt mit dem Aufnehmen der Eingangsfrage: Auf welches Problem reagiert die Systemtheorie mit ihrem (kontingenten, aber eben dadurch: spezifizierbaren) Aufgriff von Denkfiguren, die sie in den Gesetzen der Form beobachtet?
Article
Full-text available
The paper presents George Spencer-Brown's calculus of indications, published in his book Laws of Form, with respect to its proposition of a single sign, the mark of distinction, for a primary arithmetic, a primary algebra, and a possible reading of identity as memory and oscillation.
Article
Full-text available
UIC I. Introduction This paper is about G. Spencer-Brown's "Laws of Form" [LOF, SB] and its ramifications. Laws of Form is an approach to mathematics, and to epistemology, that begins and ends with the notion of a distinction. Nothing could be simpler. A distinction is seen to cleave a domain. A distinction makes a distinction. Spencer-Brown [LOF] says "We take the form of distinction for the form." There is a circularity in bringing into words what is quite clear without them. And yet it is in the bringing forth into formalisms that mathematics is articulated and universes of discourse come into being. The elusive beginning, before there was a difference, is the eye of the storm, the calm center from which these musings spring.
Article
This section presents the third volume of Max Weber's fundamental work Economy and Society which has been translated into Russian for the first time. The third volume includes two works devoted to the sociology of law. The first, 'The Economy and Laws', discusses differences between sociological and juridical approaches to studies of social processes. It describes peculiarities of normative power arenas (orders) at different levels and demonstrates how they influence the economy. The second, 'Economy and Law' ('Sociology of Law'), reviews the evolution of law orders (primarily, the three "greatest systems of law" including Roman Law, Anglo-American Law, and European Continental Law) in the context of changes in the organization of economy and structures of dominancy. Law is considered an influential factor of the rationalization of social life which in turn is affected by a rationalized economy and social management. The Journal of Economic Sociology here publishes an excerpt from the chapter 'Law, Convention and Custom' in this third volume, which shows the role of the habitual in the formation of law; explains the importance of intuition and empathy for the emergence of new orders; and discusses the changeable borders between law, convention and custom. The translation is edited by Leonid Ionin and the chapter is published with the permission of HSE Publishing House. © 2018 National Research University Higher School of Economics. All rights reserved.
Article
Culture influences action not by providing the ultimate values toward which action is oriented, but by shaping a repertoire or "tool kit" of habits, skills, and styles from which people construct "strategies of action." Two models of cultural influence are developed, for settled and unsettled cultural periods. In settled periods, culture independently influences action, but only by providing resources from which people can construct diverse lines of action. In unsettled cultural periods, explicit ideologies directly govern action, but structural opportunities for action determine which among competing ideologies survive in the long run. This alternative view of culture offers new opportunities for systematic, differentiated arguments about culture's causal role in shaping action.