Complexity VIII. Ontology of closure in complex systems: The $C*$ hypothesis and the O° notation

Closure is a common characteristic of mathematical, natural and socio-cultural systems. Whether one is describing a graph, a molecule, a cell, a human, or a nation state, closure is implicitly understood. An objective of this paper is to continue a construction of a systematic framework for closure which is sufficient for future quantitative transdisciplinary investigations. A further objective is to extend the Birkhoff–von Neumann criterion for quantum systems to complex natural objects. The $C*$ hypothesis is being constructed to be consistent with algebraic category theory (Ehresmann and Vanbremeersch, 1987, 1997, Chandler, 1990, 1991, Chandler, Ehresmann and Vanbremeersch, 1996). Five aspects of closure will be used to construct a framework for categories of complex systems: 1. Truth functions in mathematics and the natural sciences 2. Systematic descriptions in the mks and O° notations 3. Organizational structures in hierarchical scientific languages 4. Transitive organizational pathways in the causal structures of complex behaviors 5. Composing additive, multiplicative and exponential operations in complex systems Truth functions can be formal or objective or subjective, depending on the complexity of the system and on our capability to represent the fine structure of the system symbolically, observationally or descriptively. “Complete” material representations of the fine structure of a system may allow truth functions to be created over sets of one to one correspondences. Less complete descriptions can support less stringent truth functions based on coherence or subjective judgments. The role of human values in creating and perpetuating truth functions can be placed in context of the degree of fine structure in the system's description. The organization of complex systems are hypothesized to be categorizable into degrees relative to one another, thereby creating an ordering relationship. This ordering relationship is denoted by the symbols: O°1, O°2,O°3... For example, for material systems, an ordering relation such as particles, atoms, molecules, cells, tissues, organs, individuals and social groups might be assigned to classify observations for medical purposes. The C* hypothesis asserts that any complex system can be described in terms of four enumerable concepts: closure, conformation, concatenation and cyclicity. Mappings between objects are constructed within a notation for organization. Causality is organized within C* as pathways of relationships in time. The notation of organizational degrees is used to distinguish a directionality for causality: 1. bottom-up (energy flows) 2. top-down (control processes or dominating variables), 3. outside — inward (ecoment on organism) and 4. inside — outward (organism on ecoment). Closures are asserted to emerge from evolutionary cooperation. It is asserted that truth functions emerged from the necessity of an organism to identify ecoments where life can prosper. For example, basic truth functions of mathematics (operations of addition, multiplication and exponentiation) are made operationally consistent within the biochemical operations of sustaining exponential cellular growth. These fundamental mathematical functions can provide a logical basis (in conjunction with conservation rules) for a construction of complex material categories at higher degrees of organization. It is remarked that these simple functions suggests a biochemical origin for the intuitionistic philosophy of mathematics. The emergence and success of mathematics is conjectured to result from the need to acquire a consistent basis for communication among individuals seeking to cooperate socially. This suggests a cultural closure over a collection of individual closures.