Natural systems cannot be closed to the environment. At the same time there is a necessity for closure in order to build the system. It is this quintessential tension between openness and closure that drives systems to unfold into further stages or levels of growth and development. In other words, the emergence of organization in natural systems is a result of cycles of openness and closure. There are two distinct and complementary ways by which a system will carry over closure while involved in a process of expansion across the environment. These two ways need to be expressed in any formal representation: (1) within a level this will be by means of transitive closure, which is additive; and (2) between levels (i.e., from one level to the next higher level) this requires algebraic closure, which is multiplicative. The former expresses space closure, whereas the latter expresses topological or time closure. The conjunction of these two closures generates a hierarchy of levels. Prior to, and outside of, the system lies semantic closure.

By considering an enterprise to be a system of agents that observe and construct theories about themselves immediately raises issues of closure. These in turn pose questions about the identity and evolution of that which is exhibiting such closure. We address these questions by assigning enterprises to a class of systems whose models are triply articulated. The existential articulation provides an account of the possible behaviors of the enterprise’s agents and of their interoperation. The referential articulation specifies outcomes that its agents are required to satisfy. The deontic articulation imposes constraints on the composition of the other two articulations that are sufficient to ensure that the enterprise effectively implements its specified requirements. Any of these articulations may be under-determined in that they admit more than one elaboration. The behavioral closure of an enterprise is a kind of composition (formally, a category theoretic limit construction) of its three articulations. If the enterprise is its own observer, then the articulations are its models of itself. The enterprise has many opportunities for error in constructing this model. In particular, it may find that it cannot choose among its under-determined articulations in such a way that their composition is internally consistent. Such errors necessitate changes to its model, which may be denoted as steps in an irreversible trajectory through a space of such models. This approach seems to provide a conceptual bridge across the gulf between systems theory and psychoanalysis, and has provided valuable insights into strategy formulation within large enterprises.

In this historical treatise two biological-system theories, formulated in the 1920s and 1930s, are roughly sketched. The first part discusses the concept of a thermodynamically open system, as coined by the Russian pathologist Ervin Bauer (1890–1942). Like Bertalanffy, Bauer wanted to prove the specificity of the biological sciences against physics. To achieve this, he postulated the necessity to formulate specific laws of motion which are valid for living matter alone. In the second part of the paper, the organismic-system theory of the Austrian-Canadian philosopher and biologist Ludwig von Bertalanffy (1901–1972) is outlined. The focus of this theory relied on the process dynamics that is inherent inside an organismic system. Both theories exemplify closure models for a living organism from a methodical point of view that distinguishes these earlier models from semantic closure, developed by Howard Pattee as an epistemic clue in solving the enigma of living phenomena. The objective here is to disclose the essential differences between these closure conceptions. To encourage further research on closure, the essay concludes with a few questions concerning clarification of the term.

The question of closure in biological systems is central to understanding the origins of the biological variation and complexity upon which various forms of selection act. Much of evolutionary theory, especially in the second half of the twentieth century, is concerned with the consequences of environmental selection acting on biodiversity, but neglects questions of the origin of that diversity. This has permitted us to act as if an explanation ofconsequences was the ultimate explanation in biology. However, Darwin understood that evolution was both information driven and information constrained. The link between evolutionary constraints and closure can be profitably explored by starting with Darwin’s notion of the primacy of “the nature of the organism” over “the nature of the conditions” articulated in the sixth edition of Origin of Species. Contemporary ideas of self-organization, emergence, complexity, and inherent (developmental and phylogenetic) constraints can be seen as an elaboration and refinement of Darwin’s views if we adopt the following perspective: (1) information is cheap, not costly, to produce, but may have costly consequences; and (2) information is produced by systems that are informationally closed but remain thermodynamically open.

The functional organization of the nervous system is discussed from the standpoint of organizational closure and regenerative process in order to draw parallels between life and mind. Living organization entails continual regeneration of material parts and functional relations (self-production). Similarly, dynamic stability of informational states in brains may entail coherent self-regenerating patterns of neural signals. If mind is the functional organization of the nervous system, then mental states can be seen as switchings between alternative sets of stable, self-regenerative neural signal productions. In networks of neurons, signaling resonances can be created through recurrent, reentrant neural circuits that are organized to implement a heterarchy of correlational operations. Neural representations are dynamically built-up through an interplay between externally-impressed, incoming sensory signals and internally-generated circulating signals to form pattern-resonances. Semiotic aspects of resonance states involve semantic sensori-motor linkages to and through the external environment and pragmatic linkages to evaluative mechanisms that implement internal goal states. It is hypothesized that coherent regenerative signaling may be an organizational requirement for a material system to support conscious awareness. In this view general anesthetics and seizures abolish awareness by temporarily disrupting the organizational coherence of regenerative neural signaling.

The specificity of human knowledge allow one to construct specific truths about human behavior. A structural notation and language for describing a complex hierarchically organized biological system was developed for the explicit purpose of analyzing the origins of health and disease. A specific application of these concepts to a specific patient (such as an individual suffering from the heritable disease, sickle cell anemia) requires a systematic formulation of a scientific truth. No universal law is applicable. The value of a clinical truth for the patient, as well as for the physician and society, is substantial. This value has moral, ethical, and legal weight. Both physics and chemistry use a universal external invariant reference system. Human beings and other living organisms function by an internal reference system that is neither invariant nor universal. In order to address the complexity of scientific truths within living systems, a mathematical graph is constructed from observations, descriptions, and symbolizations of the relevant human scientific activities and is placed in mutual coreference with three philosophical theories of truth. Consistency within the referencing relations of the graphic object creates an image of complex truths. When mapped over degrees of internal organization, the structural consistencies can form hierarchically transitive relations (many-to-one) creating redundancies that confirm one-another. Both structural and dynamic information can be composed within the graphic framework. The redundancies intrinsic to the degrees of organization notation, semantics, and syntax augment one another in the search for scientific truth. The degree of certitude emerging from structural implications increases in relation to the number of hierarchical degrees of organization invoked to represent the various behaviors of complex systems (as illustrated by the sickle cell anemia example). The successful synthesis of the complex image (or complex simple) of a scientific truth approaches the Heideggerian notion of identity in the sense that A = A and A is A.

Most accounts of functionality are based in etiology, either design or selection. In this paper I give an account of function as serving autonomy, which is the closure of self-maintaining processes, including those interacting with the environment. Autonomy is inherently dynamic, being based entirely on interacting processes, whose organization constitutes the integrity of the autonomous system. The etiological account focuses on external factors, either intentions in design or outcomes in selection. It ignores any but idiosyncratic organizational requirements within the biological entity and in its interactions with its environment, even though these may play a central role in the functionality of the trait in question. In particular, there is no simple relation between adaptation on the etiological account and adaptability, a higher order emergent trait that plays a central role in behavior and the evolutionary and developmental genesis of intelligence. I propose redefining adaptiveness in terms of autonomy. The definition naturally extends to adaptability and focuses on the organizational character of adaptiveness, forcing attention on this central biological characteristic, which is easily ignored in the etiological account. The result is a much richer account of both adaptation and selection.

Dynamic systems with suitable nonlinearities yield self-organizing behavior. The evolution continues until the relationship among the components becomes self-consistent; that is, until it reaches closure. Disruptions of closure that allow for continued change are also characteristic of biological evolution. Are the requisite nonlinearities add-ons that give an essentially linear world the appearance of circularity, or do they have their origin in the underlying physics of the universe? The picture developed here fits to the latter view.

Covalent bonding within chemical molecules and the internal electronic structure of atoms involve closure of phase relations in electronic wave functions, as suggested de Broglie by many years ago. The structures of crystals involving positive and negative ions can be understood in terms of replication of unit cells that may be classified in terms of symmetry. The main principle involved in crystal symmetry can be understood by examining possible patterns in decorative borders. A more widely applicable type of chemical closure occurs in oscillating reactions (dissipative structures), in which an autocatalytic process is balanced by some exit reaction. As is the case for the other types of chemical coherence, the number of distinct types of oscillating reactions is rather small. Otherwise puzzling aspects of human social and organizational behavior may be clarified by analogy with chemical oscillating reactions.

In this note epistemological problems in general theories about living systems are considered; in particular, the question of hidden connections between different areas of experience, such as folk biology and scientific biology, and hidden connections between central concepts of theoretical biology, such as function, semiosis, closure, and life.