This website introduces the operator theory (or OT). Before diving in, I explain in a few sentences why this theory was developed.  

Scientists use ‘levels of complexity’ when analyzing organization in ecology and nature. What does that mean, a ‘level’? When does a ‘level’ start or end? What separates entities at one level from those at the next level? In the winter of 1993-1994 I searched for answers to such questions in textbooks, and discovered something interesting. The textbooks typically ranked ecological hierarchy in the following way:

cell → organ → organism → population

Such rankings triggered me to asking a simple question: Is the ranking ‘consistent?’ I selected two criteria for consistency: 1. A ranking includes objects of the same general kind (e.g. ‘physical units’). 2. A ranking uses one kind of rule for arriving from one level to the next (For example: ‘elements at level A are parts of elements at level B’). Surprisingly, the above ranking fails on both criteria (as I explain in the following two paragraphs). This unexpected result became the trigger for developing the OT.

2 Ranking objects

A ranking should be ‘consistent’ with respect to its entities and its relationships. In this paragraph, I discuss the entities of the above example, while addressing their ‘kind’. 

What are the kinds of the entities in the example ranking that was introduced above?

cell → organ → organism → population

Cell: On the one hand, a cell is a physical unit. On the other hand, the term cell refers to many physical objects at a time: a single cell, a cell in a eukaryotic cell, a eukaryotic cell, and a eukaryotic cell in a multicellular organism.

Organ: An organ is a multicellular unit that can function only as a part of a multicellular organism. Organs do not form from independent cells, but form inside a multicellular organisms during its development.

Organism: The organism concept refers to a broad conceptual classification of which organisms of many different complexities can be the instantiations, e.g. bacteria, protozoa, plants/fungi, and animals. Some view Lychen or the slug of a slime mold as an organism too. Currently, there is no unanimously accepted definition of the organism concept, which makes it problematic to use the concept in a hierarchical ranking.

Population: A population refers to a specific grouping of specific organisms. Basically the members of a population live in the same area (but some may leave the area, and others may enter it). Organisms of a population are normally of the same species. The ‘species’ concept has proven difficult to define in a stringent way. Generally people make use of common descent (question: how far back should common descent reach?), combine this with the capacity to mate successfully with at least one nearby member of the group (question: must we exclude organisms in a population that fail to mate successfully?), while mating outside the group must be rare (question: how infrequent is ‘rare’?). Because of such considerations, the species concept is an idealization that is not fully watertight at its edges.

Wrapping up:

the entities in the ranking belong to different logical kinds, e.g.:

1. A cell = physical object or part of an object
2. An organ = part of a multi-cellular organism
3. An organism = a broad conceptual class that includes biological units of differing complexity
4. A population = a specific grouping of specific entities (in biology: ‘organisms’).

Although the ranking does not make this explicit, the first three steps presume that objects are physical parts inside a multicellular organism. It would be more correct, therefore, to let the ranking start at the organism level and work ‘inward’:  cell ← organ ← organism. Step four in the ranking involves a conceptual grouping. Such grouping starts with organisms: organism → population. The difference in the direction of the relationship-arrows implies logical inconsistency. But there is more. Not every entity that is part of these rankings has already been defined in a ‘stringent’ way. I use the term stringent for a definition that offers criteria for including all the ‘right’ examples, and excluding all the ‘wrong’ examples. In some cases a stringent definition may not yet be available. For example, the literature offers no unanimously accepted definition of the organism concept.

3 Ranking relationships

Above I explained that in a consistent ranking, all relationships should be of a similar major kind. For example, every step towards a next level should classify as an ‘is a part of relationship’, or should describe/explain how ‘entities of level N interact to form the entity of level N+1’.

What kinds of relationships can be found in the example ranking that was introduced above?

cell → organ → organism → population

Cell → organ: The organ is viewed here as a next step in the ranking. Interestingly, there are no examples in nature where free living cells aggregate to form an organ. For this reason a ranking like ‘cell → organ’ presumes that one talks about a cell in an organ, not just any cell. In fact, the relationship presumes that the cell is ‘a part of’ an organ, and that the arrow should point in the other direction: cell ← organ.

Organ → organism: What connects an organ and an organism? If I presume we speak about a multi-cellular organism, the relationship implies that the organ ‘is a part of’ the organism. Logically it would be more correct to depicted this as: organ ← organism. The word ‘organism’ indicates a broad category, that ranges from bacteria to plants and animals. If one would choose a bacterium, any relationship between organ and organism would be unclear or even illogical. These problems arise because the words organ and organism represent different logical kinds.

Organism → population: While organisms are physical things, a populations is a conceptual grouping. Moreover, the relationship between an organisms and a population depends on the criteria that are used to define ‘population’, such as: looking similar and living in the same area, or being of the same species, etc.

Wrapping up:

The analysis of our example-ranking demonstrates that the rules for connecting entities at successive levels are not the same between levels (e.g ‘is a part of’, or ‘member of the same species’). Moreover, the relationships cannot always be defined precisely, because the entities at the different levels can’t always be defined precisely. And -because of a lack of insight in relevant relationships- it is frequently unclear at which complexity a specific level starts or ends.

Such obstacles beg the question whether or not the example ranking can serve as a foundation for theory about hierarchical organization in nature? In the most optimistic scenario the theory we currently use is not sensitive to conceptual vagueness because it possesses marked resilience on this point. In a less optimistic scenario we are stumbling around in a conceptual fog that prevents us from realizing and resolving fundamental questions about how to define much used terms in biology and ecology.

4 Three rankings

The classical picture of ecological hierarchy is that of a linear ranking. The above has shown that existing linear rankings generally mix different kinds of entities, and apply different kinds of ranking rules. As a solution, I suggest to analyze hierarchy along three dimensions: inward, outward and upward.

1: The ‘inward’ dimension.

Hierarchy on the inside of a ‘basic unit’ (I elaborate the ‘basic unit’ concept in the next paragraph)

If, for example, we look at a horse, the organization inside it, its ‘internal organization’, can be analyzed in different ways. Interestingly, the kind of relationships that one observes between the same parts of the horse depend on one’s perspective.

Here I suggest grouping all possible perspectives into four major groups: Displacement, Information, Construction, and Energy (acronym: DICE).

1. From a displacement perspective the focus is on the displacement of the horse itself, and the displacement processes that occur inside the horse, e.g. the transportation of food through the digestive system, the transportation of blood through the body, the air flows in the longs, etc.
2. From an informational perspective the focus would be more on the horse’s perception, the signals being transferred by its nerve system, the responses of the muscles.
3. From a constructional perspective, the horse has muscles, skin, bones, intestines, etc. Another informational perspective would be to focus on chemical signals based on hormones.
4. From an energetic perspective, one can rank the uptake of food, its grinding in the mouth, its digestion in the stomach, and intestines, the uptake of energy rich molecules in the intestines, transport of these molecules through the blood, etc.

Every ranking results in a proper web of relationships. If a hierarchical arrangement is discovered using one DICE perspective, this will almost certainly change if one adopts another DICE perspective. The implication is that hierarchy within any ‘basic unit’ is a relative thing, that depends on one’s perspective.

2: The ‘outward’ dimension.

 Hierarchy when ‘basic units’ produce a system of interacting ‘basic units’

Let’s assume one studies a group of basic units of some kind, for example horses. Now the interests can focus on how a stallion guards different mares. And when a foal is born, how the mother and an aunt guard the foal if the stallion is too nosy. Again such interactions can be analyzed according to the four DICE perspectives. The displacement perspective focuses on how the mother and aunt actively shield and push away the stallion of the group. The informational perspective focuses on how the horses interact by making noise, through their smell, by their posture, etc. Similar explanations hold for the construction and energy perspectives.

A system that results from interactions between ‘basic units’ is called an ‘interaction system’ in the OT. The reason for this naming is that the system consists of interacting ‘basic units’ without being a ‘basic unit’ itself. The operator theory distinguishes between two kinds of interaction systems: compound objects and groups.

3: The ‘upward’ dimension.

Hierarchy that results when low complexity ‘basic units’ produce more complex ‘basic units’

While interacting, ‘basic units’ produce all sorts of systems. A very special subset of all such systems are the systems that themselves are a ‘basic unit’. Through the pathway from ‘basic unit’ to ‘basic unit’, nature can produce increasingly complex ‘basic units’. When it comes to analyzing hierarchy, the sequence of increasingly complex ‘basic units’ has a marked theoretical relevance: it is stringent. What I mean with stringent here is that any next ‘basic unit’ needs the properties of the preceding ‘basic unit’ as a foundation. Such stringent hierarchical dependency provides the backbone for the operator theory, and is explained in the next paragraph.

5 Dual closure and unity

A special kind of organization is called a ‘loop’. A loop forms when the last element in a row interacts with the first, hereby ‘closing’ the loop. A loop has a binary logic: it is either closed or open. When a loop closes, the system attains a new configuration that offers a criterion for the ‘unity’ of the interacting elements.

A beautiful example of a loop is a group of people that each sit on another person’s lap, such that the ‘last’ person sits on the lap of the ‘first person’ (this example comes from Hofstadter’s book: “I am a strange loop”). The loop-form implies a closed series of people sitting on each-others lap, while at the same time a closed series of ‘lapping’ people implies a loop-form: the mechanism implies the form, and the form implies the mechanism. Instead of the term ‘loop’ I use the term ‘closure’. A system has closure if one can identify a loop of interactions that cause a closed process and/or form.

There is something special about closure: a system either has it, or does not have it. A loop cannot be half-closed. To be closed, a loop must have no missing links in its processes, or in the elements that construct it. If there is a single missing link, the loop is not closed. Closure is an either/or property, it is a binary variable. A binary property is of great practical value. It can be used as a stringent criterion for the identification of a new kind of organization. This is the reason why closure is used for the identification of every ‘basic unit’ in the operator theory.

In fact, the operator theory uses not one closure for every next layer, but a minimum of 3 closures. These three closures are indicated as ‘dual closure’, because the first two closures form the minimal basis:

1. Functional closure (e.g. the set of autocatalytic molecules in a bacterium)
2. Structural closure (e.g. the membrane surrounding the bacterium)
3. An obligatory interaction between the functional and the structural closure (the membrane keeps the set of autocatalytic molecules together, and the set of autocatalytic molecules produce material for the membrane)

Successions and uniformity.

Beside the above three minimum criteria for dual closure, the OT uses additional criteria for a more detailed description of the subsequent transitions in the operator hierarchy.  One of these criteria is that a dual closure must result from interactions between operators of the highest preceding level possible. Firstly, the ‘preceding level’ assures uniformity of the kind of operators involved in producing the next dual closure. Secondly, this use of ‘highest preceding level’ assures that each step in the ranking represents the shortest possible step towards any next dual closure. In general, the highest preceding level will be the level N-1 (for example: the autocatalytic set and membrane of a bacterium consist of molecules). Only in the atom does it refer to hadrons (level N-1) and electrons (level N-2).

The above criteria allow the construction/identification of a long series of successively formed operators. The name of this series is: the operator hierarchy. Interestingly, the operator hierarchy is not a linear ranking; the ranking has internal structure. This internal structure suggests some additional criteria for any next operator. While this subject is interesting, it requires substantial explanation. To do this leads beyond the goals of this website. For this reason I simply present the graph of the operator hierarchy, while suggesting some additional literature for those that want to learn more.

6 Terminology of the OT

Based on the OT, and in line with the above triplet of dimensions for ranking, one can distinguish three fundamental kinds of systems:

(A) Operator. The criterion of dual closure allows one to recognize a kind of stringently defined physical countable units named ‘operator’. The operators can be ranked hierarchically according to increasing complexity (upward dimension), where complexity is measured as the number of successive dual closures that is needed for their formation.

(B) Interaction system. The OT uses the name ‘interaction system’ for any system that consists of interacting operators without itself classifying as an operator. The set of interaction systems can be divided into two major subclasses: compound objects, and groups.

(B1) Compound object. A compound object is a system in which physical interactions between the operators in the group are stronger than interactions with entities outside the group. The interactions involved typically result from binding forces that in their reach don’t extend beyond the size of the operators involved. Due to such interactions a compound object is a single countable physical unit. It is possible that a compound object surrounds a volume that is filled with not-connected entities, such as an air bubble in water.

(B2) Group. A group ‘contains’ two or more entities that are either operator or compound object. Examples of a group are e.g. a heap of stones, a herd of cows, molecules in an air bubble in water. In many cases, it is the observer who assigns the entities to any group. In some cases, behavioral interactions of organisms can be causal to an active grouping process, e.g. bees in a hive, or people in a company. A group is a theoretical unit, also known as a ‘set’.

7 Higher order structure

There are strong indications that the operator hierarchy is not just a linear ranking of operators of different kinds, but that it has a higher order structure.

The reason for this assumption is that, for example two kinds of operators, the atoms and the molecules, are based on atoms. By analogy, four kinds of operators, the bacteria, the bacterial multicellulars (the bluegreen algae), protozoa, and plants are all based on cells. These examples demonstrate that between these two groups, the number of options seems to increase. These observations suggest a higher order structure of the operator theory, which I like to propose a hypothesis for.

The basic idea is that every dual closure introduces a new property that can be seen to recur at higher levels in the hierarchy. For example, when quarks condensed into larger particles, the hadrons (e.g. protons and neutron) these consisted of multiple quarks. The multi-particle structure was new. At higher levels it can be seen to recur in molecules (which are multi-atoms), and multicellular bacteria (the bluegreen algae) and eukaryotic multicellular organisms (such as plants). The recurrence of the multi-particle configuration can be compared with black pearls on a necklace. After some white pearls there is a black pearl. Such recurrence of the multi-particle configuration, and other properties, suggest that the operator hierarchy has a secondary structure.

The theoretical arrangement that can be deduced from the above analysis is shown in the following figure. Here symbols A, B and C indicate successively more complex dual closures. Part (a) shows how in theory the kind of closure from a low level operator recurs in a more complex from in an operator at a higher level. Part (b) shows how the logic of part (a) can be re-arranged such that similar closure kinds are arranged vertically. Par (c) illustrates how the principles of part (a) and (b) work out for real world examples.

8 The operator hierarchy

The most detailed representation of the operator hierarchy is not limited to operators, which all have dual closure, but also adds a range of systems that only have a single closure.

The reason for adding these systems is that some systems at the lower end of the operator hierarchy have a single closure only. For example hadrons consist of several quarks, but there is not proof -so far- that quarks, nor any other of the ‘fundamental particles’ in the ‘standard model’, consist of smaller parts. They can be viewed as existing on the basis of a single closure that acts as their interface. When quarks interact, this occurs on the basis of gluons. Quark gluon interactions can be viewed as a single closure that creates a hypercyclic pattern of interactions. The interface closure and the hypercycle closure together create the operators with the multi-particle property. It would lead to far to explain the dimensions of the operator hierarchy in detail (please read chapter 2 of ‘Evolution and transitions in complexity’ (Springer 2016)). Here I limit the explanation to offering a figure of the entire operator hierarchy demonstrating its full hypothetical structure.

9 A meta-periodic table

The operator hierarchy can be viewed as a periodic table for periodic tables; as a meta-periodic table. In fact, the periodicity of the operator hierarchy is not determined by a numerical analysis, as is the case for the number of protons and neutrons in an atom nucleus. Instead, every step in the operator hierarchy also offers a causal pathway for the formation of operators possessing any next kind of dual closure. In this sense the structure of the operator hierarchy reaches further than ‘just’ a periodic table.

9.1 A periodic table of the fundamental particles: The standard model

Probably the most fundamental periodic table is the ‘standard model’ used in particle physics. It categorizes the major classes of fundamental particles as either force carrying particles (bosons) or matter particles (fermions). The fermions are subsequently divided into leptons or quarks, both of which are partitioned over three groups of increasing mass.

9.2 A periodic table for the hadrons.

Another fundamental periodic table used in particle physics is the ‘eightfold way’. This table is used to organise the many ways by which quarks can combine into hadrons. Hadrons consisting of two quarks are called mesons while those made up of three quarks are called baryons (‘baryos’ means ‘heavy’), and a separate table exists for each of these types. The eightfold way was developed by Gell-Mann and Nishijima and received important contributions from Ne’eman and Zweig (Gell-Man and Neeman 1964).

9.3 The periodic table of the elements (Mendeleev 1869)

The most well-known periodic table is the periodic table of the elements. Mendeleev introduced this tabular display of the chemical elements in 1869. It organizes the reactivity of atoms and indicated a number of missing elements. Mendeleev’s discovery was so important that his table is still used as a basic tool in chemistry. Furthermore, two tables can be considered the fundaments of Mendeleev’s periodic table: the ‘nucleotide chart’ and the charts showing which sets of electron shells are to be expected in relation to a given number of protons.

9.4 A periodic table for all molecules

In fact, there are overviews of many kinds of molecules. However, the number of possible configurations exceeds any attempt at organizing them in a limited table.

9.5 A periodic table for the descend of organisms: The tree of life

Finally, and even though it may be a bit unusual to regard this arrangement as a periodic table, there are also good grounds to include the ‘‘tree of life” in this overview of tabular presentations. The only difference with the other tables is that the tree of life includes descent, a property that has no meaning in the other periodic tables discussed so far. In all other aspects, the tree of life has similar properties of creating a unique and meaningful overview of the many different forms of organisms, which enter the scheme as species.

9.6 The OT as a meta-periodic table

The operator hierarchy can be viewed as a periodic table that unites a broad range of other periodic tables. In this sense it represents a meta-periodic table.