In the section about existing theories that could be related I was missing Luhmann’s Social Systems theory. It is not a mathematical theory but otherwise fits the desiderata. This is what o1-preview says about it (after being fed the above dialog up to the theories suggested by Claude:
Introducing Luhmann’s Social Systems Theory into our discussion is an intriguing idea. Let’s explore how it might help us address the challenge of modeling hierarchical agency with a suitable mathematical formalism.
Overview of Luhmann’s Social Systems Theory:
Niklas Luhmann’s theory conceptualizes society as a complex set of self-referential social systems composed not of individuals or actions, but of communications. In his framework:
Autopoietic Systems: Social systems are autopoietic, meaning they are self-producing and maintain their boundaries through their own operations. They continuously reproduce the elements (communications) that constitute them.
Functional Differentiation: Society is differentiated into various subsystems (e.g., economy, law, politics), each operating based on its own binary code (profit/non-profit, legal/illegal, power/no power).
Communication as the Fundamental Unit: Communications are the basic elements, and meaning is generated through the network of communications within a system.
Operative Closure and Cognitive Openness: Systems are operationally closed—they can only refer to themselves—but cognitively open, as they observe and are influenced by their environment.
Relating Luhmann’s Theory to Hierarchical Agency:
Type Consistency Across Levels:
Agents as Systems: If we consider both subagents and superagents as autopoietic systems of communications, we maintain type consistency. Both levels are constituted by the same fundamental processes.
Scale-Free Structure: Since Luhmann’s theory doesn’t privilege any particular scale, it aligns with your desideratum for a scale-free formalism.
Expressiveness for Real-World Situations:
Conflict and Integration: Luhmann’s concept of structural coupling explains how different systems interact and influence each other without losing their autonomy. This could model conflicts between layers or how a superagent gains agency at the expense of subagents.
Emergent Properties: The emergent nature of social systems in Luhmann’s theory parallels how superagents might exhibit properties not reducible to their subagents.
Representation of Intentionality:
System Intentionality: While individual intentions are abstracted away, systems exhibit purposive behavior through their operations. This could be adapted to represent the goals and beliefs of agents at different levels.
Communication of Intentions: Intentionality can be embedded in the communications that constitute the system, allowing us to model beliefs and desires.
Towards a Mathematical Formalism:
While Luhmann’s theory is primarily sociological and lacks a mathematical framework, we can attempt to formalize its concepts:
Agent-Based Modeling:
Agents as Communication Processes: Model agents (both subagents and superagents) as processes that generate and respond to communications.
Network Dynamics: Use graph theory to represent communications as edges and agents as nodes, allowing us to analyze the structure and dynamics of interactions.
Category Theory:
Objects and Morphisms: Utilize category theory to model agents as objects and communications as morphisms, preserving type consistency and composability.
Functorial Relationships: Define functors that map between categories of agents at different hierarchical levels.
Dynamic Systems Theory:
Differential Equations: Represent the evolution of communications and agent states over time using differential equations.
Attractors and Stability: Analyze how certain communication patterns lead to stable superagent formations.
Information Theory:
Entropy and Redundancy: Measure the information content of communications, which could correlate with the intentionality and decision-making processes of agents.
Mutual Information: Assess how much information is shared between subagents and superagents, indicating the degree of coupling.
Potential Benefits:
Unified Framework: By treating communications as the foundational elements, we can model both agents and superagents within the same formal structure.
Emergence and Self-Organization: The formalism could capture how complex behaviors emerge from simple interactions, which is crucial for understanding hierarchical agency.
Applicability to AI Systems: For AI alignment, this approach might help in designing AI agents that can form or integrate into higher-level systems without losing alignment with human values.
Challenges:
Abstract Nature: Luhmann’s theory is highly abstract, which might make direct mathematical translation difficult.
Intentionality Representation: Modeling intentionality purely through communications may not capture all aspects of beliefs and desires inherent in agents.
In the section about existing theories that could be related I was missing Luhmann’s Social Systems theory. It is not a mathematical theory but otherwise fits the desiderata. This is what o1-preview says about it (after being fed the above dialog up to the theories suggested by Claude: