General Systems Theory and Coherence Cybernetics

Who this chapter is for

You will learn how Coherence Cybernetics mathematically generalizes the General Systems Theory (GST) of Bertalanffy and Urmantsev. The central concepts of GST — open systems, equifinality, hierarchy — are reproduced as special cases of the $\Gamma$ formalism.

About notation

In this document:

• $\Gamma$ — coherence matrix
• $\mathcal{L}_\Omega = \mathcal{L}_0 + \mathcal{R}$ — evolution equation
• $\Phi$ — integration measure
• $R$ — reflection measure
• $P$ — purity $\mathrm{Tr}(\Gamma^2)$
• $\varphi$ — self-modelling operator
• $\mathbb{H}$ — Holon
• $V_{\mathrm{hed}}$ — hedonic valence $dP/d\tau$

Introduction: why do we need general systems theory?

In the mid-twentieth century an intellectual shift occurred that transformed the face of science: researchers in completely different fields — from cell biology to the sociology of organisations — discovered that they were describing their objects using the same differential equations. The growth of a bacterial population obeys the same laws as the spread of rumours in a social network. Heat transfer in a building is formally indistinguishable from the flow of capital in an economy. This observation raised the question: do universal laws exist that govern systems of any nature?

The answer was General Systems Theory (GST) — an interdisciplinary programme founded by Ludwig von Bertalanffy in the 1930s–1950s and developed by several schools over seventy years.

Coherence Cybernetics (CC) claims not only meta-status among theories of consciousness, but also the mathematical generalisation of GST. This is a serious claim: GST is a great intellectual tradition with proven heuristic value. A claim to generalisation obliges one to show that the CC formalism reproduces the central concepts of GST as special cases, and also adds what GST cannot.

In this section we trace the path from Bertalanffy through Urmantsev to CC, show precise correspondences, and honestly identify limitations.

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Ludwig von Bertalanffy: the birth of GST (1950–1968)

Biography and context

Ludwig von Bertalanffy (1901–1972) was an Austrian theoretical biologist who received his doctorate from the University of Vienna. In the 1930s, working on problems of organism growth, Bertalanffy discovered that the equations for cell-mass growth were formally identical to the equations of chemical kinetics. This observation became the seed of his central idea.

After World War II Bertalanffy emigrated — first to Canada (University of Alberta), then to the United States. In 1954, together with economist Kenneth Boulding, physiologist Ralph Gerard, and mathematician Anatol Rapoport, he founded the Society for General Systems Research (now the International Society for the Systems Sciences). His principal book — General System Theory: Foundations, Development, Applications (1968) — collected ideas developed since the 1930s.

Central idea

Bertalanffy argued: there exist general laws of systems, independent of the nature of the constituent elements — physical, biological, or social. These laws describe structural isomorphisms between systems of different natures.

A simple example. The Bertalanffy growth equation:

$$\frac{dW}{dt} = \eta W^{2/3} - \kappa W$$

describes the growth of an organism's mass $W$, where $\eta W^{2/3}$ is nutrient uptake (proportional to surface area) and $\kappa W$ is expenditure (proportional to mass). But exactly the same equation describes the growth of a crystal, the accumulation of capital by a firm, and the spread of infection in a population. Bertalanffy saw in this not a coincidence, but a law.

Key concepts

• Open system — a system that exchanges matter, energy, or information with its environment. This is the opposite of classical thermodynamically closed systems. Living organisms are open systems by definition: they consume food and excrete waste.

• Equifinality — the property of open systems to reach the same final state from different initial conditions. An organism reaches its adult size regardless of whether it received more or less nourishment at the start of life (within viability limits). Closed systems lack this property — their final state is uniquely determined by initial conditions.

• Isomorphisms between sciences — the same mathematical structures (systems of ODEs, feedback, hierarchy) appear in physics, biology, economics, and sociology.

Mathematical apparatus

Bertalanffy proposed an extremely general formalisation:

$$\frac{dx_i}{dt} = f_i(x_1, \ldots, x_n), \quad i = 1, \ldots, n$$

A system of ordinary differential equations (ODEs) as a universal language for describing dynamics. Any system whose dynamics can be described through the interaction of variables fits this format.

The strength and weakness of this approach are interrelated. The formalism is maximally general — it covers everything, but precisely for this reason generates no specific predictions. The statement "dynamics is described by a system of ODEs" is true for such a wide class of objects that it becomes trivial. Bertalanffy's GST is more a philosophical programme and heuristic principle than a mathematical theory with theorems and refutable predictions.

Bertalanffy's key contribution

Bertalanffy did not discover the laws of systems — he discovered the possibility of such laws. His main achievement was legitimising interdisciplinary systems thinking as a scientific programme. Before Bertalanffy, comparing a living organism with a firm was considered a metaphor; after him — a research strategy.

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Yu.A. Urmantsev: GST of objects (1978–2009)

Biography and context

Yunir Abdinovich Urmantsev (1925–2009) was a Soviet and Russian philosopher-systemologist, professor at Moscow University. Urmantsev set himself the task that Bertalanffy had not solved: to create a formal general systems theory, not a programmatic declaration. The result was General Systems Theory (1978) and subsequent works, up to Foundations of General Systems Theory (2003).

Urmantsev worked in a tradition different from the Anglo-American systems movement. Where Bertalanffy, Boulding, and Ashby were biologists and engineers, Urmantsev was a philosopher who sought logical rigour in the spirit of Soviet philosophy of science.

Central construction

Urmantsev defined a system as a quadruple:

$$\mathcal{S} = \{m, \, \mathfrak{R}, \, Z, \, S\}$$

Component	Notation	Description	Example (for a living cell)
Elements	$m$	Set of system components	Organelles: nucleus, mitochondria, ribosomes
Relations	$\mathfrak{R}$	Connections between elements	Metabolic pathways, signalling cascades
Composition laws	$Z$	Rules by which elements form the system	Genetic code, protein assembly rules
Properties	$S$	Observable characteristics of the system as a whole	Metabolic activity, capacity for division

Key results

• Law of systems transformations — Urmantsev systematically classified ways of changing a system. A system can be changed in four ways: (1) by changing elements $m$, (2) by changing relations $\mathfrak{R}$, (3) by changing laws $Z$, (4) by changing everything simultaneously. This yields a complete combinatorics of transformations.

• Polymorphism and isomorphism of systems — formal mappings between systems of different natures. Two systems are isomorphic if a bijection exists between them that preserves relations and laws.

• Algebraic approach — groupoids and polygroupoids as tools for describing systems transformations. Urmantsev was the first to attempt to give GST an algebraic form.

Mapping to CC

Urmantsev ($\mathcal{S}$)	CC formalization	Comment
Elements $m$	Dimensions $k \in \{A, S, D, L, E, O, U\}$	7 semantic roles
Relations $\mathfrak{R}$	Coherences $\gamma_{ij}$ (off-diagonal elements of $\Gamma$)	21 coherence pairs
Composition laws $Z$	Evolution operator $\mathcal{L}_\Omega$	Dynamics derived from structure $\Omega$
Properties $S$	Observables: $P$, $\Phi$, $R$, $\sigma_k$	Concrete functions of $\Gamma$

Urmantsev's advantage is the explicit attempt at algebraic formalisation. But his algebra remains descriptive: it classifies types of systems and transformations, but does not derive dynamics from structure, as L-unification does in CC.

Urmantsev and the problem of consciousness

Urmantsev never addressed the problem of consciousness. His GST is a theory of objects (systems of any nature), not a theory of subjects (systems possessing inner experience). Herein lies the fundamental limitation of his approach and, simultaneously, its honesty: he did not claim what his formalism could not deliver.

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Other GST schools

GST is not a monolithic theory but a family of approaches. Each emphasises its own aspect of "systemness". Let us consider the key schools and their connection to CC.

Mesarovic and Takahara (1975): mathematical GST

Mihajlo Mesarovic (Case Western Reserve University, USA) and Yasuhiko Takahara (Tokyo Institute of Technology) created the most rigorous mathematical GST. Their definition: a system is a mapping $S \subseteq X \times Y$ (input → output). The central theme is hierarchical multilevel systems with the task of coordinating layers.

Key concepts:

• Stratified description — one object is described at several levels of abstraction (e.g. a factory: the parts level, the workshop level, the enterprise level)
• Coordination — reconciling decisions between layers of a hierarchy

This is the closest formalism to CC in classical GST: the idea of stratification resonates with the way CC distinguishes levels of description — from $\Gamma$ (micro) through observables $P, \Phi, R$ (meso) to holon behaviour (macro). However, Mesarovic has neither quantum algebra nor a concept of consciousness.

Klir (1969, 1985): systems epistemology

George Klir (Binghamton University, USA) proposed the General Systems Problem Solver (GSPS) — an epistemological hierarchy of models. Eight levels of knowledge organisation:

1. Source (data)
2. Data → variables
3. Generative systems (rules)
4. Structured systems (compositions)
5. Metasystems (change of rules) 6–8. Meta-meta levels

The idea of systems epistemology resonates with the SAD hierarchy of CC (SAD-0: reaction, SAD-1: model of self, SAD-2: model of the model, SAD-3: reflection of the model). However, Klir has no formal thresholds for transitions between levels — no analogue of $P_{\mathrm{crit}}$ or $R_{\mathrm{th}}$.

Boulding (1956): nine levels of complexity

Kenneth Boulding (one of the co-founders of the GST society) proposed an intuitive "ladder of complexity" — nine levels of systems:

Level	Description	Analogue in CC	Comment
1	Static frameworks (crystal)	None (CC describes dynamic systems)	Structure without dynamics
2	Clockwork mechanisms	$P \ll 2/7$, deterministic dynamics	Predictable, no feedback
3	Cybernetic systems (thermostat)	Feedback, but without $\varphi$	Control without self-modelling
4	Open systems (cell)	$\mathcal{L}_\Omega = \mathcal{L}_0 + \mathcal{R}$	Exchange with environment
5	Plants (genetic society)	L0 (proto-interiority)	Growth, reproduction
6	Animals	L1 (perceptual interiority)	Sensations, movement
7	Human	L2–L3 ($P > 2/7$, SAD $\geq 1$)	Self-awareness, language
8	Social systems	Composition of holons (T-68)	Collective coherence
9	Transcendent	Open question	Unformalizable?

Boulding's ladder is intuitively correct and pedagogically valuable, but stated descriptively. CC offers formal criteria for transitions between levels: not "sufficient complexity" but concrete numbers ($P_{\mathrm{crit}} = 2/7$, $R_{\mathrm{th}} = 1/3$, $\Phi_{\mathrm{th}} = 1$).

Ackoff and Emery (1972): goal-setting

Russell Ackoff and Fred Emery placed goal-setting at the centre of systemness. A system is "purposeful" if it is capable of choosing both goals and means. Purposeful systems differ from goal-directed systems (choice of means, but not goals) and reactive systems (state-maintaining: homeostasis maintenance).

In CC the analogue of goal-setting is hedonic valence $V_{\mathrm{hed}} = dP/d\tau$ — a formally derived internal "compass" of the system that directs behaviour towards increasing coherence. Here $V_{\mathrm{hed}}$ is not a postulated "goal" but a consequence of the dynamics $\mathcal{L}_\Omega$: the system "aims" to increase $P$ because this is a mathematical property of its evolution equation.

Lem and Turchin: the metasystem transition

Stanisław Lem in Summa Technologiae (1964) discussed the metasystem transition — a qualitative leap when a control system itself becomes the object of control at the next level. Life → consciousness → mind: a chain of metasystem transitions.

Valentin Turchin (1970) formalised this idea in The Phenomenon of Science: a metasystem transition $M \to M'$ creates a new level of control. A set of systems $\{S_i\}$ is united under the control of a metasystem $S'$, which controls them as a whole.

In CC the metasystem transition corresponds to:

• Individual level: growth of SAD (self-observation observing self-observation) — each SAD level is a metasystem transition in the space of self-modelling
• Group level: composition of holons — transition from individual $\Gamma$ to group coherence

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Genealogy: from GST to CC

The connection of CC to the intellectual traditions of the twentieth century is most clearly expressed by a diagram. CC inherits ideas from three foundational programmes: general systems theory, cybernetics, and information theory.

Each arrow in the diagram is not merely "inspiration" but concrete structural inheritance. Bertalanffy gave the idea of an open system ($\mathcal{L}_\Omega$ includes exchange with the environment via $\mathcal{R}$). Wiener gave feedback ($\varphi \to \rho^* \to \mathcal{R}$). Shannon gave information measures ($S_{vN}$, $D_{KL}$). Urmantsev gave the structural quadruple (elements, relations, laws, properties). Von Foerster gave the observer ($\varphi$-operator). Tononi gave the integration measure ($\Phi$).

CC is distinguished by uniting all these elements in a single quantum-algebraic formalism, where they do not merely coexist but are derived from one another.

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How CC generalizes GST: formal justification

The key argument: CC does not add "yet another variable" to GST but derives GST concepts as projections onto a subset of dimensions.

Generalization table

GST concept	CC formalization	Status	What is added
System	Holon $\mathbb{H}$	[D]	Fixed dimension $N=7$
Open system	$\mathcal{L}_\Omega = \mathcal{L}_0 + \mathcal{R}$ (dissipation + regeneration)	[T]	Concrete dynamics, not just "exchange"
Homeostasis	$P > 2/7$ (viability region $\mathcal{V}$)	[T]	Exact numerical threshold
Feedback	$\varphi(\Gamma) \to \rho^* \to \mathcal{R}$ (self-modelling → regeneration)	[T]	Self-modelling, not just feedback
Equifinality	Primitivity $\mathcal{L}_0$ → unique attractor $I/7$ (T-39a)	[T]	Proven uniqueness of attractor
Hierarchy	L0→L4 interiority levels	[T]	Formal transition thresholds
Systems isomorphism	Substrate-independence (T-153)	[T]	Proven theorem, not just analogy
System element	Dimension $k \in \{A, S, D, L, E, O, U\}$	[D]	7 semantic roles
Connection	Coherence $\gamma_{ij}$	[T]	Quantum coherence
Integrity	$\Phi \geq 1$ — integration threshold for consciousness	[T]	Numerical threshold
Entropy	$S_{vN}(\Gamma) = -\mathrm{Tr}(\Gamma \ln \Gamma)$	[T]	Quantum (von Neumann) entropy
Goal-setting	$V_{\mathrm{hed}} = dP/d\tau$ — hedonic valence (T-103)	[T]	Not a postulated goal, but a consequence of dynamics
Metasystem transition	Composition of holons (T-68)	[C]	Quantitative threshold ($\Phi_{12} > 1$)

Status notation

[D] — definition, [T] — theorem, [C] — conditional theorem. Details: status registry.

Formal construction of the generalization

Statement. For any classical GST system $\mathcal{S} = (m, \mathfrak{R}, Z)$ one can construct a holon $\mathbb{H}$ that reproduces its structure.

Construction:

1. Elements → dimensions. Each element $m_k$ is mapped to a dimension $k$ with weight $\gamma_{kk}$. The weight reflects the "significance" of the element in the system: $\gamma_{kk} = 0$ means the element is inactive, $\gamma_{kk} = 1/7$ is the equilibrium value.

2. Relations → coherences. Each relation $r_{ij} \in \mathfrak{R}$ is mapped to a coherence $\gamma_{ij}$. If elements $m_i$ and $m_j$ are strongly connected, $|\gamma_{ij}|$ is large; if independent, $\gamma_{ij} \approx 0$.

3. Laws → evolution operator. Law $Z$ is mapped to $\mathcal{L}_\Omega$ acting on $\Gamma$. The specific form of $\mathcal{L}_\Omega$ is determined by the axioms of CC.

Two cases by number of elements:

• If the number of elements $|m| < 7$, the holon projects onto a subspace — unused dimensions have $\gamma_{kk} = 0$.
• If $|m| > 7$, elements are grouped by semantic roles. This is an inevitable compression: the 7 dimensions of CC are the minimum number covering all fundamental aspects, but not every specific element.

Thus, any GST system has a representation as a holon (with possible information loss during projection).

Limitation of the argument

The mapping $\mathcal{S} \mapsto \mathbb{H}$ is surjective but not injective: different GST systems may map to the same holon. This is the inevitable cost of 7-dimensional projection. The inverse mapping (from holon to GST system) is uniquely defined only when the interpretation of dimensions is fixed. Analogy: the projection of a three-dimensional object onto a plane loses depth information; but if the viewpoint is known, the object can be reconstructed.

Summary table: Bertalanffy — Urmantsev — CC

Aspect	Bertalanffy	Urmantsev	CC
System definition	Set of elements in interaction	$\{m, \mathfrak{R}, Z, S\}$	Holon $\mathbb{H} = (\Gamma, \varphi, E)$
Mathematics	System of ODEs	Groupoids	$\Gamma \in \mathcal{D}(\mathbb{C}^7)$, $\mathcal{L}_\Omega$
Dynamics	$\dot{x}_i = f_i(x_1, \ldots, x_n)$	Descriptive	$\dot{\Gamma} = \mathcal{L}_\Omega[\Gamma]$
Thresholds	None	None	$P_{\mathrm{crit}} = 2/7$, $R_{\mathrm{th}} = 1/3$, $\Phi_{\mathrm{th}} = 1$
Subjective experience	Not considered	Not considered	Central object (E-dimension)
Predictions	None specific	None specific	22+ falsifiable
Substrate	Abstract	Abstract	Abstract + proven independence (T-153)
Feedback	Postulated	Classified	Derived from $\varphi$
Hierarchy	Descriptive	Descriptive	L0→L4 with formal criteria
Classical/quantum	Classical	Classical	Quantum ($\Gamma$ — density matrix)

What CC adds to the GST description

Beyond generalising existing concepts, CC introduces a principally new layer absent in Bertalanffy and Urmantsev:

1. Dissipation ($\mathcal{L}_0$): Lindblad dynamics with proven primitivity (T-39a [T]) — unique attractor $I/7$ towards which the system tends without regeneration

2. Regeneration ($\mathcal{R}$): nonlinear term determined by self-model $\varphi(\Gamma)$ — the system resists decay through self-modelling

3. Observables: $P$, $\Phi$, $R$, $\sigma_k$ — concrete functions of $\Gamma$, not abstract "properties" of the system. Each observable is computable from $\Gamma$, and its value determines the qualitative state of the system

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What GST cannot do, but CC can

1. Exact thresholds instead of qualitative descriptions.

GST speaks of "sufficient complexity" for emergent properties. When is a system "sufficiently complex"? Bertalanffy does not answer. Urmantsev classifies types of complexity but specifies no numerical boundaries. CC derives concrete numbers:

• $P_{\mathrm{crit}} = 2/7$ — consciousness threshold
• $R_{\mathrm{th}} = 1/3$ — reflection threshold
• $\Phi_{\mathrm{th}} = 1$ — integration threshold

These numbers follow from the axioms and are not chosen ad hoc. They can be refuted by experiment — therein lies their strength.

2. Subjective experience.

GST is entirely silent on qualia and consciousness. Even Boulding's ladder, which includes "human" and "transcendent", does not formalise inner experience. CC formalises interiority through $\mathrm{Coh}_E$ and proves the No-Zombie theorem: any viable system necessarily possesses non-zero interiority.

3. Falsifiable predictions.

CC formulates 22+ predictions, each with a concrete numerical criterion. If even one turns out to be false, the theory will require revision. GST does not generate predictions verifiable by experiment — it is too general for that.

4. Substrate-independence with proof.

GST postulates isomorphisms between sciences — systems of different natures "resemble" one another. CC proves (T-153 [T]): any system with $\Gamma \in \mathcal{D}(\mathbb{C}^7)$ satisfying the thresholds possesses interiority — independently of physical substrate. This is not an analogy but a theorem.

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What CC cannot do (where GST excels)

Objectivity requires acknowledging areas where GST retains an advantage.

1. Decades of empirical validation.

GST has been applied in ecology (population models), biology (organism growth), management (organisational theory), engineering (systems engineering, INCOSE) — with proven heuristic value. The concepts of "feedback", "open system", and "homeostasis" have become working tools. CC is a young theory whose empirical verification still lies ahead.

2. Accessibility.

GST does not require quantum theory, category theory, or spectral geometry. It is accessible to a biologist, engineer, or manager. Bertalanffy's book can be read without specialised preparation. CC makes high demands on mathematical preparation — this limits the circle of potential users and critics.

3. Systems without consciousness.

GST naturally describes engineering, economic, and ecological systems without claiming to explain their "inner life". For a water supply system or a stock market, GST is the ideal language. CC is oriented towards systems with potential interiority; for purely mechanical systems ($P \ll 2/7$) its apparatus is excessive — why invoke $G_2$-rigidity to describe a thermostat?

4. Modularity.

GST combines easily with other approaches (control theory, operations research, synergetics). A systems engineer takes the concept of subsystem from GST, stability from control theory, and optimisation from operations research. CC represents a monolithic formalism whose integration with applied disciplines is an open task.

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Conclusion

The relationship of CC and GST can be expressed by the formula:

$$\text{CC} \;\approx\; \text{GST} \;+\; \text{quantum structure} \;+\; \text{consciousness} \;+\; \text{falsifiability}$$

CC reproduces the central concepts of GST — openness, homeostasis, equifinality, hierarchy, isomorphism — as consequences of its axioms. At the same time it adds what GST does not contain: exact thresholds, quantum-algebraic dynamics, formalisation of subjective experience, and falsifiable predictions.

However, the claim to generalisation remains programmatic until the predictions of CC have passed empirical verification. Bertalanffy's GST earned its status through decades of application; CC must earn its own — through experiment.

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