This website introduces ‘the operator theory’. The theory is based on physical entities, the ‘operators’. The abbreviation OT is used to distinguish it from mathematical operator theory. Before diving in, I will 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 led me to ask a simple question: Is the ranking ‘consistent’? I chose two criteria for consistency: 1. A ranking includes objects of the same general type (e.g. ‘physical units’). 2. A ranking uses the same rule to get from one level to the next. It came as a surprise that the above ranking fails on both criteria (more about why in the next two paragraphs). This unexpected result became the trigger for the development of the OT.
To allow for predictions, a ranking should be ‘consistent’ in terms of its entities and their relationships. In this paragraph, I will discuss the entities of the above example, addressing their ‘type’.
What are the types of entities in the example hierarchy presented 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 once: 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 only function as part of a multicellular organism. Organs do not form from independent cells, but form within a multicellular organism during its development.
Organism: The term organism refers to a broad conceptual classification. Organisms of many different complexities can be the instantiations, e.g. bacteria, protozoa, plants/fungi and animals. Some also consider lychen, or the slug of a slime mould, to be an organism. There is currently no universally accepted definition of the term organism, which makes it problematic to use the term in a hierarchical ranking.
Population: A population refers to a particular group of organisms. In principle, the members of a population live in the same area (although some may leave and others may enter). Organisms in a population belong to the same species. The concept of ‘species’ has proved difficult to define strictly. In general, people use common descent (question: how far back should common descent go?), combine this with the ability to mate successfully with at least one nearby member of the group (question: must we exclude organisms in a population that do not mate successfully?), while mating outside the group must be rare (question: how rare is ‘rare’?). Because of such considerations, the species concept is an idealisation that is not completely watertight at its edges. More importantly, the population is a concept that refers to a grouping, not to a physical thing.
The entities in the above hierarchy belong to different logical types, e.g.:
- A cell = a physical object or part of an object
- An organ = a part of a multicellular organism
- An organism = a broad conceptual class that includes biological entities of varying complexity
- A population = a specific grouping of specific entities (in biology: ‘organisms’).
Although the ranking does not make this explicit, the first three steps assume that objects are physical parts inside a multicellular organism. It would be more correct to start the ranking at the organism level and work ‘inwards’: cell <= organ <= organism. Step four in the ranking involves conceptual grouping. Such a grouping starts with organisms: organism => population. The difference in the direction of the relationship arrows implies a logical inconsistency. But there is more. Not every entity that is part of this ranking has already been defined in a stringent way. I use the term stringent for a definition that provides criteria for including all the ‘right’ examples and excluding all the ‘wrong’ examples, such that the criteria are necessary and sufficient. A stringent definition may not always be available. For example, there is no unanimously accepted definition of the term organism in the literature.
Above I explained that in a consistent ranking all relationships should be of a similar main type. For example, each step towards a next level 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 presented above (now with adapted arrows)?
cell <= organ <= organism => population
Cell <= organ: The organ is seen here as the next step in the hierarchy. Interestingly, there are no examples in nature of free-living cells aggregating to form an organ. In fact, cells become ‘part of’ an organ, and organs develop as parts of an organism (sometimes starting as a single cell inside a multicellular organism). This is why the arrow points from the organ to the cell.
Organ <= organism: What is the relationship between an organ and an organism? Assuming that we are talking about a multicellular organism, the relationship implies that the organ is ‘part of’ the organism. Logically it would be more correct to say: organ <= organism. Now there is another inconsistency: the word ‘organism’ indicates a broad category, ranging from bacteria to plants and animals. If a bacterium were chosen, any relationship between organ and organism would be unclear or even illogical. These problems arise because the words organ and organism represent different logical types.
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.
The analysis of our example ranking shows that the rules for connecting entities at successive levels are not the same between levels (e.g. ‘is part of’ or ‘belongs to the same species’). Furthermore, the relationships can’t always be precisely defined, because the entities at the different levels can’t always be precisely defined, and vice versa. And – due to a lack of insight into the relevant relationships – it is often unclear when a particular level begins or ends.
Such theoretical obstacles raise the question of whether the exemplar ranking can serve as a basis for a theory of hierarchical organisation in nature. In a very optimistic scenario, the theory we are currently using is not sensitive to conceptual fuzziness, because it is remarkably resilient in many of its aspects. In a less optimistic scenario, we are stumbling around in a conceptual fog that prevents us from recognising and resolving fundamental questions about the definitions of widely used terms in biology and ecology.
The classic picture of an ecological hierarchy is that of a linear ranking. The above has shown that existing linear rankings tend to mix different kinds of entities and apply different kinds of ranking rules. As a solution, I propose to analyse the hierarchy along three dimensions: inwards, outwards and upwards.
1: The ‘inward’ dimension.
Hierarchy within a ‘basic unit’ (I explain the concept of a ‘basic unit’ in the next paragraph)
For example, if we look at a horse, the organisation inside it, its ‘internal organisation’, can be analysed in different ways. Interestingly, the nature of the relationships you observe between the same parts of the horse depends on your perspective.
Here I propose to group all the possible perspectives into four main groups: Displacement, Information, Construction, and Energy (acronym: DICE).
- 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 transport of food through the digestive system, the transport of blood through the body, the air flows in the lungs, etc.
- From an informational point of view, the focus would be more on the horse’s perception, the signals transmitted by its nervous system, the responses of its muscles.Another information perspective would be to focus on chemical signals based on hormones.
- From a constructional perspective, the horse has muscles, skin, bones, intestines, tissues, cells, etc.
- From an energetic point of view, we can classify the ingestion of food, its grinding in the mouth, its digestion in the stomach and intestines, the absorption of energy-rich molecules in the intestines, the transport of these molecules through the blood, and so on.
Each ranking gives rise to its own web of relationships. If a hierarchical arrangement is discovered using one DICE perspective, this will almost certainly change if you adopt a different DICE perspective. The implication is that hierarchy within any ‘basic unit’ is a relative thing, depending on one’s perspective.
2: The ‘outward’ dimension.
Hierarchy as a result of ‘basic units’ beginning to interact in large complex groupings.
Let’s say you’re studying a group of basic units of some kind, such as horses. You might be interested in 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 curious. Again, such interactions can be analysed 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 information perspective focuses on how the horses interact through noise, smell, posture, etc. Similar explanations apply to the construction and energy perspectives.
A system resulting from interactions between ‘basic units’ is called an ‘interaction system’ in OT. The reason for this naming is that the system consists of interacting ‘basic units’ without being a ‘basic unit’ itself. Basically, the OT distinguishes between two kinds of interaction systems: composite objects and groups.
3: The ‘upward’ dimension.
Hierarchy that results when ‘basic units’ repeatedly give rise to more complex ‘basic units’ of the next type
When ‘basic units’ interact, they produce all sorts of systems. A very special subset of all such systems are the systems that are themselves a ‘basic unit’. By going from ‘basic unit’ to ‘basic unit’, nature can produce increasingly complex ‘basic units’. When it comes to analysing hierarchy, the sequence of increasingly complex ‘basic units’ has a distinct theoretical relevance: it has logical consistency. What I mean is that each subsequent ‘basic unit’ needs the properties of the preceding ‘basic unit’ as a basis. Such step by step hierarchical dependency is the backbone of operator theory, and is explained in the next section.
The OT is based on a special type of organisation called a ‘loop’. A loop is formed when the last element in a series interacts with the first, ‘closing’ the loop. A loop has binary logic: it is either closed or open. When a loop closes, the system reaches a new configuration that provides a criterion for the ‘unity’ of the interacting elements.
A nice example of a loop is a group of people sitting on each other’s laps, so that the ‘last’ person is sitting on the lap of the ‘first’ person (this example is from Hofstadter’s book: “I am a strange loop”). The loop form implies a closed series of people sitting on each other’s laps, 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 follow many seminal international publications by using 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 it does not. A loop cannot be half closed. To be closed, a loop must have no missing links in its processes or in the elements that make it up. If there is a single missing link, the loop is not closed. Because of this, closure is an either/or property. It is a binary property. A binary property is of great theoretical value. It can be used as a stringent criterion for identifying a (new) type of organisation. This is why closure is used to identify every ‘basic unit’ in operator theory.
In fact, operator theory does not use just a single closure for each next layer, but two closures and their interaction. The combination of these criteria is called a ‘dual closure’, because the two closures form the minimal basis:
- Process closure (functional)(e.g. the set of autocatalytic molecules in a bacterium)
- Spatial closure (structural)(e.g. the membrane surrounding the bacterium.
A bacterial cell provides an example of the obligatory interaction between process and spatial closure. Here, the membrane holds together the set of autocatalytic molecules, and the set of autocatalytic molecules produces material for the membrane.
Succession and uniformity.
In addition to the above criteria, the OT uses several criteria to describe the subsequent transitions in the operator hierarchy in more detail. One of these criteria is that a dual closure must result from interactions between operators of the highest possible preceding level. Firstly, the ‘preceding level’ ensures uniformity in the nature of the operators involved in producing the next dual closure. Secondly, this use of the ‘highest preceding level’ ensures that each step in the ranking represents the shortest possible step to any next dual closure. In general, the highest preceding level will be level N-1 (e.g. the autocatalytic set and the 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 simple ranking, because it has an internal structure. This internal structure suggests some additional criteria for each next operator. Although this is a very interesting topic, it requires a substantial explanation. This is beyond the scope of this website. For this reason, I will simply present the diagram of the entire operator hierarchy, while suggesting that those who wish to learn more should read recent publications on the OT.
Based on fundamental particles as the ‘base level’, and in line with the above triplet of dimensions for ranking hierarchy in nature, the OT distinguishes three basic types of systems:
(A) Operator. The dual closure criterion allows one to recognise a group of strictly defined physical countable units called ‘operator’. The operators can be ranked hierarchically according to increasing complexity (upward dimension), where complexity is measured as the number of successive dual closures required to form them.
(B) Interaction system. The OT uses the name ‘interaction system’ for any system that consists of interacting operators without itself being classified as an operator. The set of interaction systems can be divided into two major subclasses: composite objects and groups.
(B1) Composite object. A composite object is a system in which physical interactions between the entities, which are fundamental particles and/or operators and/or composite objects, are stronger than interactions with entities outside the group. The interactions involved typically result from binding forces acting in a space of about the size of the operators involved. Because of such interactions, a compound object is a single countable physical entity. It is possible for a compound object to surround a volume filled with unconnected entities, such as an air bubble in water.
(B2) Group. A group ‘contains’ two or more entities that are either operators or composite objects. Examples of a group are a pile 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 a group. In some cases, behavioural 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 not a physical thing. It is a theoretical entity, just like a ‘set’.
There is strong evidence to suggest that the operator hierarchy is not just a linear ranking of operators of different types, but that it has a higher-order structure.
The higher order structure is indicated by the following. For example, the operator hierarchy includes two types of operators based on atoms: atoms and molecules. By analogy, four types of operators, bacteria, multicellular bacteria (the blue-green algae), protozoa and plants, are all based on cells. These examples show that the number of possibilities seems to increase between these two groups. These observations suggest a higher order structure of operator theory, which leads to the following hypothesis.
The basic idea is that each 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 neutrons), these were made up of multiple quarks. This was the first appearance of what was called ‘multi-particle’ structure. At higher levels, this property can be seen to recur in molecules (which are multi-atoms), in multicellular bacteria (the blue-green algae) and in eukaryotic multicellular organisms (such as plants). The repeated appearance of the multi-particle configuration along the hierarchy can be compared to black pearls on a necklace. After some white pearls, there is a black pearl. Such recurrence of the multi-particle configuration at increasing distances is one of the properties that 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 the symbols A, B and C denote successively more complex dual closures. Part (a) shows how, in theory, the type of closure from a low-level operator recurs in a more complex form in a higher-level operator. Part (b) shows how the logic of part (a) can be rearranged so that similar types of closure are arranged vertically. Part (c) shows how the principles of parts (a) and (b) work for real examples.
The most detailed representation of the operator hierarchy is not limited to operators that all have dual closure, but also includes a number of primitive configurations that have only single closure.
The reason for adding these systems is that some systems at the lower end of the operator hierarchy have only a single closure. For example, hadrons are made up of several quarks, but there is no evidence – as yet – that quarks, or any of the other ‘fundamental particles’ of the ‘Standard Model’, are made up of smaller physical parts. They can be thought of as existing on the basis of a single closure that acts as their interface. When quarks interact, it is on the basis of gluons. Quark gluon interactions can be thought of as a single closure that produces a hypercyclic pattern of interactions. The interface closure and the hypercycle closure together give rise to the operators with the multiparticle property. It would go too far to explain the dimensions of the operator hierarchy in detail (see Chapter 2 of ‘Evolution and transitions in complexity’ (Springer 2016)). Here, I limit the explanation to providing a figure of the entire operator hierarchy, showing its full hypothetical structure.
The operator hierarchy can be seen as a periodic table for periodic tables; a meta-periodic table. In fact, the periodicity of the operator hierarchy is not determined by numerical analysis, as is the case with the number of protons and neutrons in an atomic nucleus. Instead, each step in the operator hierarchy also provides a causal path for the formation of operators possessing any next type of dual closure. In this sense, the structure of the operator hierarchy is more than ‘just’ a periodic table.
9.1 A periodic table of the fundamental particles: The standard model
Perhaps the most fundamental periodic table is the ‘Standard Model’ of particle physics. It categorises the main classes of fundamental particles as either force-carrying particles (bosons) or matter particles (fermions). Fermions are further divided into leptons and quarks, which are further divided into 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 in which quarks can combine to form hadrons. Hadrons made up of two quarks are called mesons, while those made up of three quarks are called baryons (‘baryos’ means ‘heavy’), and there is a separate table for each type. 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 best known periodic table is the periodic table of the elements. Of the various proposals, the table proposed by Mendeleev in 1869 has become the most famous. It organised the reactivity of atoms and identified a number of missing elements. The periodic table of the elements is still used as a basic tool in chemistry. Two other tables are directly related to Mendeleev’s periodic table: the ‘nucleotide chart’ and the charts showing which sets of electron shells can be expected in relation to a given number of protons.
9.4 A periodic table for all molecules
In fact, overviews of many types of molecules exist. However, the number of possible configurations exceeds any attempt to organise them in a limited table.
9.5 A periodic table for the descend of organisms: The tree of life
Finally, although it may be a little unusual to consider this arrangement as a periodic table, there are good reasons to include the ‘tree of life’ in this overview of tabular representations. The only difference with the other tables is that the Tree of Life includes descent, a feature that has no significance in the other periodic tables discussed so far. In all other respects, the Tree of Life provides a unique and meaningful overview of the many different forms of organisms that enter the scheme as species.
9.6 The OT as a meta-periodic table
The operator hierarchy can be thought of as a periodic table that incorporates a wide range of other periodic tables. In this sense, it is a meta-periodic table.