System science has a long tradition. Accordingly, it is valuable to introduce some of the major ideas. On this page you will find a selection of thoughts of scientists that have inspired many people. In the below listing, the focus is on ideas about the identification of 'objects' and ideas about the (hierarchical) ranking of the 'complexity' of such objects. This choice is made, because this website focuses on theories about objects and their formation. We think that it is important to have access to a theory for 'objects', because such objects form the basis for the modeling -from the bottom up- of all sorts of complex dynamic phenomena, such as tipping points, self-organising criticality, fractals, chaos, evolution, etc.

For more references and information about this topic see:
The operator hierarchy. A chain of closures linking matter, life and artificial intelligence.
Jagers op Akkerhuis G.A.J.M. (2010). Alterra Scientific Contributions 34: 17-33


In his Metaphysics (Book, H, 1045: 8-10) Aristotle coined the phrase: "The whole is something over and above its parts, and not just the sum of them all". This sentence already bears in it the debate between reductionists, who view a system as the sum of its parts, and holists who emphasize the emergent properties that are typical for the whole system. In this website, both the deterministic and the holistic viewpoints are viewed as having their virtues and as supplementing each other when it comes to understanding the functioning of a system.

Leibniz (1646-1716)

Leibniz was inspired by the Aristotelian concept of entelechy ("that which realizes or makes actual what is otherwise merely potential") when proposing a ‘gestalt’ called ‘monad'. Monads can be described a follows. A monad is an indivisible ‘whole’ or ‘unit’. There is nothing to be rearranged within a monad. A monad is a metaphysical concept, of which various physical examples (‘encodings’) exist. A monad is an abstraction, and in this sense invariant. Encodings of monads, on the other hand, are dynamic space-time entities, the structure of which exist for a limited time, based on specific energy requirements. Monads have both the quality of their ‘kind’ and the qualities of their encodings that operate in the physical world. Monads are the basis of aggregates.


Whitehead (1929) presents a system of process metaphysics that defines the world as a process of becoming of actual entities. These entities are both process and outcome because they are both the subject (as the process of becoming) and the superject (as the result of the process). Whitehead reasons that processes are the basis of all entities: ‘… we are faced with the question as to whether there are not primary organisms which are incapable of further analysis. It seems very unlikely that there should be any infinite regress in nature. Accordingly, a theory of science which discards materialism must answer the question as to the character of these primary entities. There can be only one answer on this basis. We must start with the events as the ultimate unit of natural occurrence.’ (Whitehead, 1925). Whitehead refers to the events representing the ultimate units of occurrence (the lowest possible level) as actual entities or actual occasions, while “whatever things there are in any sense of ‘existence’, are derived by abstraction from actual occasions” (Whitehead, 1929). When actual occasions interact and integrate into macroscopic objects, they form a nexus (plural: nexũs). All macroscopic things we see are nexũs: a tree, a table, a human being, a star, etc. The composition of the actual entities contributing to a nexus can change. Whitehead also refers to individual actual entities as creatures because they are the product of self-creation. He uses the concept of the organism for actual entities as well as for nexũs of actual entities.


Systems hierarchy also has close relationships with the constructivist viewpoint in mathematics that originated in a 1938 paper by Gödel (Gödel, 1938).In this paper, Gödel states that only those mathematical objects that can be constructed from certain primitive objects in a finite number of steps can exist. If one applies this insight of Gödel to system science, a well organised system theory must be defined ‘from the ground up’. Gödel’s mathematical constructivism offers a useful analogy to real world systems, if it is assumed that reality emerges as a large hierarchy from primitive elements constructing more complex elements. It is of marked importance for a constructivist approach that both the ‘objects’ and the ‘levels’ are defined in a stringent way.


An early review by Feibleman (1954) shows that many general system theory rules had already been identified in the 1950s, including:
  • Each level organises the level or levels below it plus one emergent quality.
  • Complexity of levels increases upward.
  • In any organisation the higher level depends on the lower.
  • For an organisation at any given level, its mechanism lies at the level below and its purpose at the level above.
  • Time required for a change in organisation shortens as we ascend the levels.
  • The higher the level, the smaller its population of instances.
  • It is impossible to reduce the higher level to the lower.
Interestingly, the concept of a level is used in many of these rules, while it is not made clear what exactly defines the organisation of objects at the ‘next level’.

Von Bertalanffy

In the mid-20th century, Von Bertalanffy (1950, 1968) envisioned a kind of general systems theory that would one day connect a broad range of scientific disciplines. In relation to hierarchy Von Bertalanffy (1968) remarked that a "…general theory of hierarchic order obviously will be a mainstay of general systems theory". Examples of hierarchical approaches in which lower level objects are nested in higher level objects are the ‘worlds within worlds’ approach and abstract simplifications of it represented by Chinese boxes or Russian babushkas (e.g., Simon, 1962; Koestler, 1967; Lazslo, 1972, 1996).

Teilhard de Chardin

Teilhard de Chardin (1959, 1969) was one of the first scientists to develop the idea of a general particle hierarchy that included physical particles and organisms. Teilhard de Chardin called the core of his theory ‘complexification intériorisante’, or inward complexification, defining complexity as the result of relationships of elements amongst themselves. Following this reasoning, he arrived at two insights that are still relevant for particle–based reasoning: 1. "First, in the multitude of things comprising the world, an examination of their degree of complexity enables us to distinguish and separate those which may be called ‘true natural units', the ones that really matter, from the accidental pseudo-units, which are unimportant. The atom, the molecule, the cell and the living being are true units because they are both formed and centered, whereas a drop of water, a heap of sand, the earth, the sun, the stars in general, whatever their multiplicity or elaborateness of their structure, seem to possess no organisation, no ‘centricity’. However imposing their extent they are false units, aggregates arranged more or less in order of density." 2. "Secondly, the coefficient of complexity further enables us to establish, among the natural units which it has helped us to ‘identify’ and isolate, a system of classification that is no less natural and universal (ES V, p. 137; 1946)". These two points have remained relevant because they highlight some fundamental principles for identifying particles, distinguishing them from other systems and ranking them in a general hierarchy. It is remarcable that Tiilhard de Chardin seems to not have followed his own logic when he predicted that organisms will one day create a point Omega. Omega represents a group of interacting obects, and therefore can be viewed as a 'false unit'. And false units have no place in a hierarchy of 'true natural units'.


Simon (1962) focuses on the role of modularity in the evolution of complexity. He convincingly argues that it is much more efficient for nature to work with assembled modules than to start every assembly process from scratch. In a quantitative example, he introduces the watchmakers Hora and Tempus, assembling watches consisting of 100 parts. Hora and Tempus are regularly disturbed in their work. Each time they are disturbed, they lose the unit they are working on. Hora then starts to assemble units in pieces of 10 and then puts these modules together in a final assembly. When disturbed, he only loses the last 10-piece module. Tempus continues to build his watches from 100 individual elements. Every time he is disturbed, he has to start again. Simon shows that it is easy to calculate how unsuccessful Tempus will be, which -in a quantitative way- demonstrates the advantage of modularity in the evolution of system types.


Koestler (1967, p.301) introduces the holon for the elements at the nodes of hierarchical relations and states that ‘The holons which constitute an organismic or social hierarchy are Janus-faced entities: facing upward, toward the apex, they function as dependent parts of a larger whole; facing downwards, as autonomous wholes in their own right’. Important in this context is the question of what determines a 'whole'?


Turchin (1977) introduces the concept of the metasystem transition (MST) to describe structural and functional aspects of evolutionary steps from one level to the next, from system S1 to system S2, that occur when, given two or more systems S1 (S11 to S1N) that may or may not show variations, the highest control level of the individual systems S11 to S1N becomes itself controlled. Turchin also considers every metasystem transition a quantum of evolution and furthermore introduces the law of the branching growth of the penultimate level. The law states that only after the formation of a control system S will it become possible for the subsystems Si to multiply and differentiate. Accordingly, a hierarchy based on formation, will generally have to define a cell first, before that it becomes possible to identify organelles as parts of the cell.


Varela (1979) introduces the concept of autopoiesis (Greek for “self-making”) for a system in which the parts, including a boundary, support dynamics that recreate every aspect of the system. He defines autopoiesis as follows: “An autopoietic machine is a machine organised (defined as a unity) as a network of processes of production (transformation and destruction) of components which: (i) through their interactions and transformations continuously regenerate and realise the network of processes (relations) that produced them; and (ii) constitute it (the machine) as a concrete unity in space in which they (the components) exist by specifying the topological domain of its realisation as such a network.” Including the concepts of ‘regeneration’, ‘production’ and the ‘concrete unity in space’ in the definition creates a natural association with the dissipative catalytic processes and membrane as a subset of the cell’s properties but makes it difficult to use autopoiesis for describing the brain. Varela and Maturana also talk about first, second and third order autopoiesis in relation to the cell, the multicellular and the population, respectively. In these three orders, autopoiesis is linked to very different processes, because the 'self' of the units (the 'auto' part of the word autopoiesis) requires a different definition if one taks about a cell, a multicellular and a 'population'. Autopoiesis is a general, overall, functional concept. As such it does not lead to indications of exactly which structures are required for the property to emerge.


Gánti (1971, 2003a, 2003b) has developed the chemoton theory. Gánti takes the cell as the building block of life. The chemoton is constructed as a minimum complexity representation of the cell, and thus ‘minimal life’. The chemoton theory is based on networks of chemical reactions in which cyclic reactions fluidly represent the information required to produce the system. RNA or DNA are not required at this level of simplicity. The chemoton theory is supported by cycle stochiometry, a theoretical framework. Stochiometry refers to quantitative calculations for describing the number of reactants and products before and after a reaction. A chemoton consists of three compartments: (1) a chemical motor system, (2) a chemical boundary system and (3) a chemical information system. Chemotons have been suggested as an appropriate model for the cell and, as such, for the origin of life. Gánti recognises absolute and potential criteria for living organisms. The absolute criteria are (1) individuality, (2) metabolism, (3) inherent stability, (4) the presence of a subsystem that carries information for the system, and (5) the regulation and control of processes. The potential criteria are (1) growth and reproduction, (2) hereditary change, and (3) mortality. According to Gánti’s definition, any technical intelligence (e.g., an intelligent robot) cannot be regarded as life. Moreover, too narrow a focus on minimal life does not allow the definition of life, in general, because even the most excellent definition of a least complex cell has little to say about the rules that define multicellularity.

Eigen and Schuster

Eigen and Schuster (1979) have introduced the hypercycle theory. A hypercyclic process is created when catalysts that can individually perform a cyclic catalytic process interact. Their interaction results in a second-order cyclic process: a cycle of cycles, or a 'hypercycle'. Eigen and Schuster (1977, 1978a, 1978b, 1979) and Kauffman (1993) have worked out in detail aspects of the catalytic hypercycle.

Jaros and Cloete

Jaros and Cloete (1987) propose a biomatrix of interacting doublets. They regard these as comparable to Koestler’s holons. Every doublet combines an endopole, representing its internal organisation, and an exopole, representing interactions with the environment. As an example of three levels of interacting doublets, Jaros and Cloete choose the cellular level, the organism level and the societal level.


Heylighen is one of the advocates of the importance of closure in system science. Closure has a long history. Early references to closure can be found, for example, in Wilson (1969). Wilson points out that natural boundaries may result from minimum interactions or ‘some form of closure, either topological or temporal’. In its most general form, closure indicates the invariance of a set under an algebra of transformation. Thus, when an operation is performed on the elements of a set, the operation’s products are still elements of the set. Closure thus indicates “the internal invariance of a distinction (or distinction system) defining the system” (Heylighen, 1989a). As a variation on this theme, in cybernetics a system is said to be organisationally closed if its internal processes produce its own organization (Heylighen 1990). A well known example of such organisational closure is autopoiesis (Varela 1979). Heylighen proposed relational closure as a means to deal with systems that do not show a primitive level because every element is defined relative to other elements (Heylighen, 1990). Circular closure relationships break a system from any pre-existing temporal hierarchy; all relationships loose temporal order and must be considered as occurring simultaneously (Heylighen 2010). Heylighen also analyzed the role of a manager as an agent causing organisational closure in mediator evolution (Heylighen, 2006). Heylighen (1989a, 1989b) furthermore discusses how a broad range of closed structures can be defined by differently combining four basic closure types: (1) transitive closure (or recursivity), (2) cyclic closure, (3) surjective closure, and (4) inverse surjective closure. Transitivity implies that if A is lower than B and B is lower than C, then A is lower than C. Cyclicity implies the existence of inverse transformations. Surjectivity implies the many-to-one relationships, while inverse-surjectivity implies one-to-many relationships. In addition, any of the relationships in the four classes can be negated e.g., non-transitive or non-cyclic.

Alvarez de Lorenzana

Alvarez de Lorenzana (in Salthe, 1993) sketches a hierarchic application of a construction sequence to describe the development of the universe. He states that the universe (U) can be understood and explained assuming that ‘…variety and complexity can be obtained only by processes of manipulation and combination, operated on and by the initial, given, U’. Using this reasoning, he describes a general hierarchical model in which ‘the elements of each new level are made out of combinations of elements of the previous level’. Alvarez de Lorenzana further states that “Evolution of and within U takes the form of a constructible “metahierarchy” (Alvarez de Lorenzana and Ward, 1987) – a chain of evolutionary cycles with, ideally, no missing links’. That a construction sequence should not have missing links was recognised earlier by Guttman (1976).

Maynard Smith and Szathmáry

Maynard Smith and Szathmáry (1995, 1999) focus in more detail on how lower level systems construct higher level systems. They approach is based on so called major evolutionary transitions, and includes the following examples:
A. From replicating molecules to populations of molecules in protocells
B. From independent replicators to chromosomes
C. From RNA as gene-and-enzyme to DNA genes and protein enzymes
D. From bacterial cells (prokaryote) to cells with nuclei and organelles
E. From asexual clones to sexual populations
F. From single-celled organisms to animals, plants and fungi
G. From solitary individuals to colonies with non-reproductive castes (e.g., ants, bees and termites)
H. From primate societies to human societies and language
In relation to the existence of three different types classes of systems recognized by the operator theory (operators, the interiour organisation of operators, and systems which consist of interacting operators) the major evolutionary transitions can be viewed as describing three rather different types of system dynamics. For example, the transition from RNA to DNA in cells relates to a focus on the interiour of a cell. And the transition from unicellular to multicellular relates to the formation of a group of cells which interacts in such a way that the ensamble can be identified as a new, more complex type of organism. Furhtermore, both the transitions from asexual to sexual and from individuals to colonies relate to certain forms of interactions between individuals in a group.

So far, the above developments have not yet resulted in a stringent theory about how -from the bottom up- specific 'particles' can be defined as the basis for 'next level' particles. With respect to this conclusion, it is important to realise that neither the concept of a 'particle' or 'object' (at any level), nor the concept of a 'next level' have been defined in detail. For this kind of problems the operator theory suggests innovative answers.