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The large number of discoveries in the last few decades has caused a scientific crisis that 7 is characterised by overspecialisation and compartmentalisation. To deal with this crisis, 8 scientists look for integrating approaches, such as general laws and unifying theories. 9 Representing what can be considered a general form law, the operator hierarchy is used 10 here as a bridge between existing integrating approaches, including: a cosmic timeline, 11 hierarchy and ontology, a periodic table of periodic tables, the unification of evolutionary 12 processes, a general evolution concept, and general aspects of thermodynamics. At the 13 end of the paper an inventory of unifying concepts is presented in the form of a cross 14 table. The study ends with a discussion of major integrating principles in science.
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European Review, Vol. 22, No. S1, S113–S144 r2014 Academia Europæa. The online version of this article is
published within an Open Access environment subject to the conditions of the Creative Commons Attribution
General Laws and Centripetal Science
Alterra Wageningen UR, Alterra – Animal Ecology, PO box 47, 6700AA
Wageningen, The Netherlands. E-mail:
The large number of discoveries in the last few decades has caused a scientific crisis that
is characterised by overspecialisation and compartmentalisation. To deal with this crisis,
scientists look for integrating approaches, such as general laws and unifying theories.
Representing what can be considered a general form law, the operator hierarchy is used
here as a bridge between existing integrating approaches, including: a cosmic timeline,
hierarchy and ontology, a periodic table of periodic tables, the unification of evolutionary
processes, a general evolution concept, and general aspects of thermodynamics. At the
end of the paper an inventory of unifying concepts is presented in the form of a cross
table. The study ends with a discussion of major integrating principles in science.
The flow of scientific information is increasing daily and is causing what could be
considered a knowledge explosion crisis (e.g. Ref. 1). Dealing with this crisis requires
not only methods that handle and make accessible huge amounts of information but also
integrative theory that offers scaffolds to connect distant scientific fields. This study
focuses on the role of the operator hierarchy as a scientific integration tool in the context
of other integrative principles.
The knowledge explosion crisis is the result of science having a positive feedback
on its own development. Science depends on our capacities as humans to observe the
world, including ourselves, and to create mental representations of the observed
phenomena. In turn, better representations increase our capacities to manipulate the
world and to construct tools that improve our observations, which, in turn, accelerate
scientific development. As the result of this process, scientific ideas have simultaneously
developed extreme breadth and depth. The outcome is not only associated with
overspecialisation and compartmentalisation, but also allows a development towards
scientific integration on a cosmic scale. At any level between these extremes, a concept is
more valuable when it is efficient, pairing minimum complexity with maximum precision
and elegance. Arbitrage by these aspects has become known as ‘Ockham’s razor’, cutting
away the most complex and least elegant from any two theories explaining the same
phenomenon. If theories describe different phenomena, arbitrage is less straightforward
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because different viewpoints may highlight different aspects of natural organisation.
Considering the latter points, this paper analyses how the operator hierarchy may contri-
bute to scientific integration while focusing on major integrative ideas, for example, the
use of timelines, natural hierarchy and evolution. The paper ends with an overview
of existing integrating theories and laws and the value of scientific integration. Because
this theory is used as a common thread, this study begins with an introductive summary
of the operator theory.
The operator hierarchy, a summary
The operator hierarchy is a methodology that deals specifically with the formation of
complexity by means of emergence.
The operator hierarchy ranks discrete steps in
‘particle’ complexity in a way that also implies a temporal ranking, because the organi-
sation of complex ‘particles’ is always preceded by that of less complex ones.
To explain this approach, it is useful to start with a fundamental assumption that lies
behind the operator hierarchy, namely that nature must be analysed according to three
fundamental dimensions for organisational complexity: the upward, the inward and the
outward dimension.
The upward dimension involves the transitions from lower level ‘particles’ to higher
level ‘particles’ (Figure 1). A chemical example is the transition from atoms to molecules.
A biological example is the formation of the eukaryotic cell from two bacterial cells. The
ranking in this dimension includes fundamental particles, hadrons, atoms, molecules and
continues with bacteria, endosymbionts, multicellulars (which may be multicellular forms
Figure 1. Three independent dimensions for hierarchy in the organisation of nature:
(1) the outward dimension for hierarchy in the organization of interaction systems
(systems that consist of operators without being an operator); (2) the upward dimension
for hierarchy in the way how lower level operators create higher level operators; and
(3) the inward dimension for hierarchy in the internal organisation of operators. Only the
hierarchical ranking of the operators is strict. All other hierarchies vary according to
point of view, for example displacement, information, construction and energy.
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of bacteria or of endosymbionts) and multicellulars with neural networks. Because the
‘particles’ in this dimension include physical particles, chemical particles and organisms,
they have been given a generic name: ‘operators’.
Transitions towards higher level
operators may result from interactions between lower level operators (e.g. from unicellular
to multicellular), and may also result from internal differentiations (e.g. the engulfment
of an endosymbiont in a eukaryotic cell and the formation of a neural network in a
multicellular organism). The hierarchical ranking of the operators is called the operator
hierarchy (Figure 2) and the related theory the operator theory.
Secondly, the outward dimension involves the ways in which individual operators can
create systems that consist of interacting operators but which are not operators. As
they consist of interacting operators without showing the necessary properties to be
recognised as an operator, such systems were named ‘interaction systems’, by Jagers op
Akkerhuis and van Straalen.
Examples of interaction systems are hurricanes, waves,
ecosystems, rivers, etc.
Thirdly, the inward dimension involves the interior organisation of operators. Here the
focus is on the elements inside an operator, and if there are some, the elements in these
elements, and so on. In abiotic operators, such as atom nuclei, atoms and molecules,
the internal differentiation directly results from interactions based on condensation (from
hadrons to nuclei, from nuclei and electrons to atoms, and from atoms to molecules). For
organic operators Turchin
has formulated the law of the branching growth of the
penultimate level. This law states that ‘yafter the formation, through variation and
selection, of a control system C, controlling a number of subsystems Si, the Si will tend
to multiply and differentiate’. This law explicitly recognises that only after the formation
of a mechanism controlling the subsystems Si is there a context that allows the variety of
the subsystems to increase. Accordingly, nature has had neither a context nor the means
to develop organelles before cells or to develop organs and tissues before multicellular
organisms. For this reason, including sub-systems such as organelles, tissues, organs and
organ systems in the conventional natural hierarchy of systems is highly confusing.
The above dimensions capture independent directions for analysing natural organi-
sation: a molecule, a prokaryotic cell, a eukaryotic cell and a multicellular organism can
all be involved in ecosystem interactions and each of them shows internal organisation.
Unification Based on Timelines
After having explained the operator theory, the links are discussed with a range of
unifying approaches. The first approach is the use of timelines. Systems can be organised
by ranking them according to the moment of their first formation and the historical time
period in which they existed. When analysing phenomena in this way, timelines at
different scales are created that refer to, for example, palaeontology, particle physics,
human history, the development of the automobile. These timelines also come in
different forms, such as linear hierarchies and branching trees.
A modern timeline presenting a comprehensive overview of the organisation in nature
is Big History.
This approach ranks all systems and processes by their occurrence in
cosmic history. Big History is based on the scientific theory that the early universe was as
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small as a fundamental particle and obtained its present size following a rapid expansion;
the Big Bang. The theory that the universe has a minute origin is supported by modern
particle physics and by cosmological observations of the background radiation and the
proportionality with distance of the speed at which galaxies recede from the Earth in all
Based on these observations, it has been calculated that Big History
started about 13.7 billion years ago (Figure 3). During the universe’s first three minutes,
quarks formed and then condensed to form hadrons (such as protons and neutrons).
During the following 17 minutes, the hadrons condensed to form simple helium nuclei
(the combination of a proton and a neutron). After these initial minutes, it took about
Figure 2. This figure illustrates the evolution of the operators (the upward dimension).
The black line shows the historical pathway of subsequent first-next possible closures
and related operators. The grey columns indicate systems resulting from first-next
possible closure but are not operators. Explanation of abbreviations: Memon 5operator
showing a hypercyclic neural network with interface, SAE (‘Structural Auto
Evolution’) 5the property of an operator to autonomously evolve the structure that
carries its information, SCI (‘Structural Copying of Information’) 5the property of an
operator to autonomously copy its information (genes, learned knowledge) by simply
copying part of its structure, HMI (Hypercycle Mediating interface) 5a closure creates
an interface that mediates the functioning of the hypercycle, Multi-state 5operator
showing closure between multiple units of exactly one lower closure level,
Hypercycle 5closure based on emergent, second order recurrent interactions. Inter-
face 5closure creating an emergent limit to an operator, CALM (Categorizing And
Learning Module) 5a minimum neural memory.
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70,000 years before the dynamic balance of the transformation of matter and energy
toppled to the advantage of matter. The matter in the rapidly expanding universe now
aggregated under the influence of gravity. The aggregation process was slow because
gravity is weak at large distances. The result was a universe with a sponge-like structure
of concentrations of matter surrounding empty ‘bubbles’ of variable size that were almost
devoid of matter. After 100 million years of aggregation, the first galaxies and stars
were formed and their light started illuminating the universe. The nuclear reactions in
stars and supernovae supported the formation of elements heavier than helium. After
approximately 9.1 billion years, the Sun was formed (4.57 years ago) and then planet
Earth (4.54 billion years ago). Thereafter, it took about 1 billion years for the first life to
Figure 3. Universal timeline (modified from Wikipedia).
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emerge on Earth and another billion before cells gained the capacity of photosynthesis.
Complex Ediacaran fauna has been found in rocks of about 600 million years old.
Around 228 million years ago dinosaurs ruled the world. The first hominin fossils
originate from approximately 7 million years ago. Human history dates back to several
tens of thousands of years.
A universal timeline is a comprehensive integration tool. Its major strength is ranking
all sorts of events simultaneously. Even though every event has only a single moment of
occurrence, a timeline can flexibly adapt to variations in the moment of occurrence of
similar events by indicating a first moment at which they occur and, assuming that such
is known, a last moment. A universal timeline can thus be seen as a thickly woven cable
of many threads representing local histories and developmental rates of different parts of
the universe (Figure 3). Although all of these developmental threads unroll in different
directions, they result in similar histories. Stars are formed everywhere in the observable
universe and their formation roughly started at the same moment. Stars of the same class
also consist of similar particles and atoms, and stars the size of the sun are probably
circled by planets everywhere in the universe. One may now ask why there is so much
uniformity in the universe and whether such uniformity may be used to answer questions
about the future of the universe.
Unification Based on the Operator Hierarchy in Combination with a
Cosmic Timeline
Above, it was explained how the operator theory recognises dimensions for organisation.
It is time to return to the question of how these insights can assist in finding general laws
in cosmic development. The answer to this question rests on separating the events along
the cosmic timeline into two parallel tracks: the track of the operators and the track of the
corresponding interaction systems.
Starting with the Big Bang, the history of the universe can, in principle, be modelled
as a container full of interacting particles. Particles exert forces on each other and
interact. This interaction forms new particles and accompanying new forces. During the
universe’s initial rapid expansion, the initial quark soup condensed to also contain simple
helium nuclei. Condensation heat was radiated away into the large space of the universe.
Simultaneously, the dispersed matter started aggregating due to the force of gravity.
This created various celestial bodies, e.g. black holes, stars and planets. Nuclear reactions
in stars then allowed helium to fuse to heavier elements, which were spread by stellar
explosions. Under colder conditions, such as on planets, atoms condensed to form
molecules. Models predicting the future of this process have been based on the total
amount of matter, the gravitational constant, the expansion rate of the universe, and the
life histories of celestial bodies. Although uncertainties exist about the values of certain
parameters, such models generally predict the universe’s heat death as the consequence
of diluting matter in the vastness of an extremely large, cold space. In this formation
sequence there is no logical position for organisms.
In comparison to the latter history of the universe, the operator theory
may seem to
focus on minor details when it introduces a strict ranking of all operators, from quarks,
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through hadrons, to atoms and molecules, prokaryote cells, eukaryote cells, prokaryote
and eukaryote multicellular organisms and neural network organisms (referred to as
‘memons’) (Figure 2). Nevertheless, the use of first-next possible closure for ranking
the operators
strictly limits the sequential formation of the operators such that the
result seems to reveal a form-law at a universal scale. The idea of a form-law is sug-
gested by the observation that the sequence of first-next possible closures and related
operators is not only strict, but also follows an internal regularity (Figure 2). The operator
hierarchy thus seems to reflect a constructional form-law with three important unifying
A first unifying consequence of the operator theory, is that it implies that the limits set
by first-next possible closure apply to all operators anywhere in the universe. Accord-
ingly, the same classes of operators can be expected to exist anywhere in the universe as
long as local conditions allow for their formation. After the uniform initial conditions in
the universe ceased to exist, first-next possible closure rules offer an explanation for the
uniformity of the structural developments in unconnected local parts of the universe.
A second unifying consequence is that the timeline of Big History can now be
associated with the coming into existence of certain operators. The cosmic formation and
aggregation of matter leads to celestial bodies, and these celestial bodies show their own
life-cycles, e.g. from young stars, to supernovae, to white dwarfs and sometimes black
holes. Phases in these life cycles show a link with the existence of certain types of
operators. For example, individual quarks and hadrons only existed during the first
second of the universe. Thereafter, atomic matter and molecules are associated with the
formation and life-cycles of celestial bodies. Still later, the cell, endosymbiontic cell,
multicellular and neural network organism, were linked with different phases in the
life cycle of certain planets. If one now uses the most complex operator associated
with a celestial body as a ranking criterion, one obtains a unified ranking that offers a
structured way for organising Big History in relation to the operator theory. This ranking
is discussed in more detail at the end of the next paragraph.
A third unifying consequence is that the operator theory can be extrapolated towards
future operators, suggesting that the next operator will be a technical memon (the
generalised concept for an operator with a neural network) owing its intelligence to a
programmed neural network (see Ref. 3). This possibility for extrapolation is a unique
property of the operator hierarchy. Both cosmology and Big History focus on the
universe at large and offer no possibilities for predicting future operators.
Unification in Relation to Hierarchy and Ontology
Ontology is the study of what ‘is’ and aims at creating an organised categorisation
for describing the world. Various theories have been developed for ranking systems.
A classic example of a linear hierarchy that ranks system complexity is the ScalaNaturae
of the Greek philosopher and naturalist Aristotle. In his approach, Aristotle ranks natural
phenomena by decreasing perfection, from spiritual and divine beings to man, animals,
plants and finally rocks and formless matter. This classification is also referred to as the
Natural Ladder or the Great Chain of Being.
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A modern linear hierarchy is shown in Figure 4. With slight variations, aspects of
this ranking can be found in a broad range of textbooks and publications on natural
organisation (e.g. Refs 15–24). The linear hierarchy’s frequent occurrence in publications
shows that this integration tool has worked so well that it has become a kind of dogma.
As a consequence, people seem to accept it unquestioningly.
Ontology uses a limited number of fundamental ‘containment’ relationships, which
comply in different ways with the subsetting (set-in-set) structure of Russian dolls.
One of these is the ‘is-a-part-of’ relationship (Figure 4). Another is the ‘is-a-kind-of
relationship. In the text below we discuss various aspects of both approaches and suggest
a third fundamental ranking based on closure.
Is-A-Part-Of (Meronomy)
The ‘is-a-part-of’ relationship implies that the higher level ‘contains’ the lower. In
principle, everything ‘is-a-part-of’ the universe. This relationship is also recognized as
‘meronomy’ or ‘compositional hierarchy’. Speaking in physical terms, a specific car has
a specific chair, which has a specific handle bar, a specific screw, etc. In abstract/
conceptual language, one could say that ‘cars’, have ‘chairs’, which have ‘handle bars’,
‘screws’, etc. Scaling up to the universe, and looking in a top-down fashion at the
Figure 4. A linear hierarchy that reflects the ‘is-a-part-of’ relationships, starting with the
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‘is-a-part-of’ relationship, the largest things contained by the universe today are ‘threads’
of matter surrounding bubbles of virtually empty space. The matter in these threads has
aggregated to various forms of gas clouds, many taking the shape of galaxies. Within gas
clouds matter has condensed further to solar systems, sometimes with planets, which
may or may not have moons. And on certain planets or moons, molecules may have
formed cells. One can now say that cells may exist as parts of planets or parts of (large
enough) moons, which are parts of solar systems, which may or may not be parts of
galaxies, which are parts of the universe. Due to some ‘may be’ relationships, the ranking
includes some alternative sequences.
Above we have analysed meronomy in a top-down way, assuming that elements
always form as aggregates within pre-existing higher level systems (e.g. celestial bodies
in matter clouds). But this is not always the case. There exist many bottom-up examples
where elements first had to form the higher level organisation before they did become
parts of it. For example, atoms first have to form molecules, before the atoms involved
can be considered to be the molecule’s parts. And cells first have to form multicellulars,
before these cells can be considered parts of the just-formed multicellulars.
Due to the above differences in how an element becomes a part of a higher level
system, the ‘is-a-part-of’ relationship between a matter-cloud-that-condensed-to-a-galaxy
and a celestial body that is part of it, is very different from the ‘is-a-part-of’ relationship
that exists between an atom and the atoms-integrated-towards-a-molecule.
Meronomy is flexible with respect to the adding or skipping of levels. Using the
‘is-a-part-of’ criterion, both the ranking atom-planet and atom-molecule-planet are
correct rankings. And the ranking remains correct if one adds a few elements, e.g. atom-
molecule-stone-house-city-planet. Now a city is a part of a planet, a house is a part of a
city, a stone is part of a house, etc. This gives the impression that while the use of
meronomy makes it easy to create rankings, the freedom inherent to the methodology
puts limits on the scientific utility of the ‘levels’ in such rankings. It is of importance that
meronomy itself offers no strict rules for what determines any next level in nature, and
that for this reason, one always has to borrow information from other viewpoints for the
defining of levels.
Sometimes ‘scalar’ approaches are proposed as a basis for the identification of levels in
meronomic (‘is-a-part-of’) ranking. The scalar point of view is based on the assumption that
levels in the hierarchy have ‘average dynamical rates of different orders of magnitude’.
Assuming this viewpoint may hold true, what is the result? How about the dynamic rates of
systems such as atoms and molecules? How do these exactly differ in rates? And would
these rates show overlap if one compares a very large atom with a small molecule? And how
about a large unicellular compared with a small multicellular? And the degradation rate of a
rock of one kilogram differs by orders of magnitude from a rock of 1000 cubic metres, while
both the small and the large rocks qualify as rocks. And at low temperature, the dynamics
of bacteria may be orders of magnitude slower than those of a tornado. Such examples
compromise the rigour of scalar rankings. We therefore suggest, instead of size or dynamic
rates, focusing on the types of organisation involved, because these are scale invariant.
Another aspect of meronomy is that it can rank indiscriminatingly elements that
belong to different dimensions for organisational complexity as recognised by the
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operator theory (see also Ref. 4). For example, if one takes the following ranking: atom,
molecule, cell, organ, organism and population, this looks as a perfect ‘is-a-part-of
relationship: a given atom is a part of a molecule, which is a part of a cell, etc. Yet,
only the atom, molecule and single cell are operators. Only a cell in a multicellular
and an organ in a multicellular can be considered internal differentiations. And the organism
and population are strange elements in the ranking, because they do not exist as physical
entities, but represent abstract groupings of individual objects that take part in the global
interaction system. The organism concept groups all sorts of entities, varying from bacteria,
via endosymbionts to multicellulars and neural network organisms. Because it is a generic
abstraction, the concept of an organism does not belong in a ranking that is based on physical
‘is-a-part-of’ relationships. Furthermore, the concept of a population is a conceptual grouping
of many individual organisms that show a potential sexual relationship (as one out of many
other relationships in an ecosystem) or that have been born as the result of such a sexual
relationship. Both the concepts organism and population are logical abstractions and have no
place in physical ‘is-a-part-of’ relationships.
Is-A-Kind-Of (Taxonomy)
The ‘is-a-kind-of’ relationship implies that concepts at a higher level ‘contain’ lower
level concepts, which are more specific. This hierarchy is also recognised as a
‘taxonomy’, a ‘specification hierarchy’ and a ‘subsumption hierarchy’. A well-known
example of taxonomy is found in biology, where the group of animals has a specific
subgroup of mammals, which has a specific subgroup of primates, and a specific sub-
group of homonids. In turn, homonids are a kind of primate, which are a kind of
mammal, etc. Taxonomy shares with meronomy its flexibility to adapt to the addition or
deletion of levels (the relationship is said to be ‘transitive’ across levels). It is therefore
correct to say that homonids are a kind of primate, which are a kind of mammal, and
equally correct to, after deleting the level of the primates, say that homonids are a kind of
mammal. Just as the ‘is-a-part-of’ relationship of meronomy, the ‘is-a-kind-of’ rela-
tionship of taxonomy offers no general rules for defining exactly any next level, which
would require the formulation of the exact reason why any next level elements are a kind
of the higher level elements defining the next level in a causal/prospective way.
It is furthermore of importance that taxonomy does not map in a one to one fashion
with the evolutionary relationships in the tree of life, because species that evolved later
are sometimes more than ‘specific forms’ of earlier taxa. For example, animals evolved
from bacteria, but it is difficult to consider animals as a specific kind of bacteria. Instead,
one would generally prefer to consider for example pest bacteria as a specific kind of
bacteria. The latter implies that a different hierarchical ranking may have to be looked for
if one wants to create a containment ranking that maps with evolutionary relationships.
Operator Hierarchy and Closure
The above discussion begs the question of whether there exists an alternative way of
ranking the discussed relationships by which one can circumvent certain undesirable
consequences of meronomy or taxonomy. As has been proposed in Ref. 3, the use of
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closure may offer a new concept that adds a third perspective to hierarchical rankings
in ontology.
In relation to attempts at unification, we would like to suggest that within the
context of the operator theory the ‘is-a-part-of’ relationship can be considered as an
amalgamation of three different sets of rules, each relating to a different dimension for
organisational complexity and showing proper rules for hierarchical ranking. In this
context, the operator theory does not allow one to switch at will between these dimen-
sions and the associated rules for hierarchy. A given ranking should take place along its
proper dimension. But what exactly is gained when looking at organisation the way
the operator theory suggests? We like to explain this using the following three examples.
(1) For example the atom, the molecule, the bacterium, the endosymbiontic unicellular,
the multicellular and the neural network organisms are operators and can be ranked along
the upward dimension. The hierarchical ranking of operators is, in a bottom-up way,
determined by stepwise differences in closure configuration. A lower level operator
always show exactly one closure less than the next higher operator. (2) The organs and
cells in a multicellular organism are aspects of the internal organisation of this operator.
For this reason, their ‘is-a-part-of’ relationship strictly and only involves an internal
ranking. Here, different viewpoints can be held on hierarchy, resulting in different
options for rankings (see Ref. 4). (3) Populations and ecosystems are groupings of
elements along the outward dimension. A population is an abstraction for a specific
subset of elements of an ecosystem. Here, the ‘is-a-part-of ’ relationship of organisms
does not result in a new physical entity. What exists in nature are organisms that at
certain moments in time have sex. All the other moments they are involved in other
interactions with their environment. Accordingly, the population as a physical unity does
not exist: it is a conceptual abstraction that refers to a group of potentially mating
individuals and their offspring. Individual organisms thus represent elements that by their
interactions constitute the global ecosystem. In the same way as a population, other
elements of interaction systems, such as a tornado, form local aspects of the overall
dynamics of the larger interaction system of the earth and its atmosphere.
One could also attempt to analyse the structure of the operator hierarchy as a taxo-
nomy. In principle, taxonomy requires that any next level is a subset of the preceding
level. For example, if one starts with dogs, an element at the next level could be a
bulldog. But while bulldogs form a special subset of all dogs, molecules are not ‘just’ a
special subset of atoms. When considering subsets of the set of all atoms, one would
think primarily of ‘lanthanides’ or ‘metals’. We suggest here that the complexity ladder
of the operator hierarchy requires an altered perspective, which shows some similarity
with taxonomy, but includes emergence and the formation of supersets. For this purpose
we propose considering the use of closure both as a subsetting and as a supersetting
mechanism. As the result of closure, a subset of atoms is formed, which thereafter is
regarded as a ‘molecule’. The molecular subset represents a next level in the taxonomy
and shows new subset-plus-closure properties. The new properties of the subset have
also been referred to as emergent properties (for a historical review of the development of
the concept of emergence, see Ref. 5. What has caused much discussion is that it seems
unusual that closure simultaneously acts as a subsetting mechanism and causes a system
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that shows properties that belong to a new superset. The subsetting mechanism thus leads
to an ensemble, which is recognised under the new name of ‘molecule’, which exhibits
new (emergent) properties that did not exist in a world with only atoms. An example of
such an emergent property is the three-dimensional shape of molecules. It is because
of the supersetting properties of closure that molecules do not fit well to the classical
‘is-a-kind-of’ approach of taxonomy. For this reason the operator hierarchy suggests an
alternative kind of ranking, where closure represents a containment principle combining
a subsetting mechanism with a supersetting outcome.
Because every next level in the operator hierarchy shows exactly one additional
closure, and is more complex for this reason, the operator hierarchy shows a strict
ranking of operator complexity, where every operator has its own proper position.
Step by step, organisation in nature has to pass all the preceding levels before the higher
levels can be constructed. In fact, the closures of the operator hierarchy can be con-
sidered a third ranking, which differs from meronomy and taxonomy, because the logic is
based on closure topology. Closure offers an exact mechanism for the identification of
any next level, and is scale invariant, because it focuses on topological changes that
combine a cyclic process and an engulfing boundary. From any level, any next level
closure must represent the first-next possibility for a new process-with-boundary
topology. Because at lower levels of organisation, nature can only construct higher level
operators from preceding level operators, there is no difference between a structural
and a topological viewpoint here. It is only at higher levels that the difference
becomes apparent. For example, cellular neural networks need not physically be the
elements from which higher level neural network organisms are formed, as long as the
topology of lower level neural networks forms the basis for the topology of higher
level neural network organisms. Accordingly, it is unproblematic to shift from cell-based
neural networks to neural network architecture that is modelled in silica (computer-based
neural networks).
Using the viewpoint of closure allows an alternative way of ranking the elements of
the conventional natural hierarchy (compare Figures 4 and 5).
The most important difference of using closure in combination with the three
dimensions for organisation as proposed by the operator theory is that the new viewpoint
does not mix hierarchal dimensions and offers topological rules for the identification of
next level operators.
Unification Based on a Periodic Table of Periodic Tables
A well-known periodic table is the periodic table of the elements. Mendeleev introduced
this tabular display of the chemical elements in 1869. It organised 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.
But chemistry is not unique when it comes to periodic tables. Various periodic tables of
fundamental importance exist for other disciplines. Probably the most well-known is the
‘standard model’ used in particle physics. It categorizes the major classes of funda-
mental particles as either force-carrying particles (bosons) or matter particles (fermions).
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The fermions are subsequently divided into leptons or quarks, both of which are partitioned
over three groups of increasing mass.
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, 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.
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.
Finally, and even though it may seem 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 also
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 all basal types of operators, which enter the scheme as species.
Figure 5. Hierarchical rankings based on the three dimensions recognised by the operator
theory. Along the upward dimension, any nest level is associated with the first-next possible
new closure configuration combining a functional and a structural closure. Along the outward
dimension one finds the large systems consisting of interacting operators. The inward
dimension ranks the elements that are parts of an operator. Thick arrows indicate closure steps.
Thin arrows indicate other formative trajectories. Dashed arrows indicate the selective and or
scaffolding influences of the environment mediating the formation of next level operators.
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Every single periodic table discussed above is central to its proper field of science. But
the tables are not connected. The operator theory, however, shows that it is possible to
connect the separate tables by focusing on the types of elements in every table. If this is
done, the operator hierarchy can be used as a ‘periodic table for periodic tables’ to
organise the elements of the existing periodic tables.
As Table 1 shows, the inventory of periodic tables resulted in the identification of a
periodic table for almost every complexity level in the operator hierarchy. The inventory
furthermore indicated the following gaps for which no periodic tables were found: the
quark-gluon hypercycles, the quark confinement, the molecules, the autocatalytic sets,
the cellular membranes, the cyclic CALM networks and the sensory interfaces. With the
exception of the molecules, which may not have a periodic table because of the almost
unlimited number of combinations that can be made from the various atom species, all
the gaps involve hypercyclic sets and interfaces. One may now suggest that it is generally
impossible to create periodic tables for hypercyclic sets or for interfaces, but this
assumption is at least partially contradicted by the nucleotide chart and the classification
of potential electron shells. A reason for the absence of tables for hypercyclic sets may be
that the number of possible configurations is so large that it is impossible to classify
them, in the same way that it is hard to classify molecular configurations. Such ideas,
however, need to be worked out in more detail.
Unification Based on Organic Evolution: the Artesian Well that is
Powered by Cellular Autocatalysis
describes evolution as a ‘river that flows uphill’. Dawkins
refers to it as a
process that is ‘climbing mount improbable’. Neither of these metaphors sheds light on
the force that is needed to realise the process. To clearly indicate that a driving force is
needed to make water flow against gravity or to make evolution climb a mountain, the
metaphor of an artesian well will be used. In an artesian well, the groundwater pressure
makes the water flow naturally towards the surface allowing it to ‘defy’ gravity. But
what exactly is the pressure that makes evolution flow towards increasing complexity,
seemingly ‘against’ thermodynamic laws? As Russell
and Pross
have indicated, this
pressure is a special form of the explosive, brutal power of autocatalysis. Taking Pross’s
insight as a basis, the following text places evolution in a thermodynamic perspective
and invokes the operator hierarchy when appropriate.
Long ago, Malthus
and Verhulst
realised that population growth leads to density-
dependent stresses. Darwin
subsequently developed the idea that this stress, in combination
with reproduction and heritability of parental properties, causes reproductive disadvantage
of the least adapted individuals. However, Darwin and contemporaries had no clear idea
about what could cause the organisation of organisms. The laws of thermodynamics
that were known at that time seemingly indicated that systems could not increase their
Later, Bergson
wrote about life:
Incapable of stopping the course of material changes downward (the second law of
thermodynamics), it succeeds in retarding it yNow what do these explosions (photo-
synthetic reactions) represent, if not a storing up of the solar energy, the degradation of
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which energy is thus provisionally suspended on some of the points (the plants) where it
was being poured forth?
Later, ideas about non-equilibrium thermodynamics
and hypercyclic catalysis
offered the ingredients for a better explanation. Non-equilibrium thermodynamics solved
the problem that growth and reproduction seemed to violate the laws of thermodynamics.
What was new in open thermodynamic systems was the idea that the degradation of an
external free energy gradient could power the dynamics required for self-organisation.
For example, when a bathtub is unplugged, the self-organisation of the vortex is powered
by the degradation of the potential energy stored in the height difference between the
water in the tub and the drain at the bottom. But although non-equilibrium thermo-
dynamics offered a general solution for the powering of self-organisation, it did not
indicate what specific driving force powered evolution.
To analyse the processes that drive Darwinian evolution in more detail, evolution
will be analysed as the combination of two processes: one process explaining the
functioning of organisms, from single cells to animals, and the other process explaining
selection. The functioning of unicellular organisms requires self-organisation and a
membrane. Self-organisation is powered by transforming external energy gradients
into work. As will be discussed presently, the operator hierarchy indicates that the
organism receives the storage of heritable information for free as long as it uses
hypercyclic autocatalysis as the basis for its energetics. The membrane is required to
ensure that the information and other processes become individualised. The mechanisms
Table 1. Using the operator hierarchy for organising the periodic tables that exist for different
types of operators. Shading indicates system types that are operators
System type in operator hierarchy Organisation in specific ‘periodic table’
fundamental particles standard model
quark-gluon hypercycle ??
quark confinement ??
hadrons eightfold way
nuclear hypercycle nucleotide chart
electron shell types of shells
atoms periodic table of the elements
molecules ??
autocatalytic sets ??
cellular membrane ??
cells tree of life: prokaryotes
eukaryote cells tree of life: eukaryotes
prokaryote multicellulars tree of life: prokaryote multicellulars
eukaryote multicellulars tree of life: eukaryote multicellulars
cyclic CALM networks ??
sensors (perceptive and activating) ??
memons (hardwired) tree of life and technical hardwired memons
memons (softwired) future tree of technical life-forms
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behind selection depend on the capacity to produce offspring that receive variable
heritable information, and on selective interactions affecting the phenotypes of the
offspring differentially.
The basal self-organisation process responsible for the existence of organisms is
autocatalysis. Autocatalysis in its basic form is the process in which a certain catalytic
chemical, say A, transforms a substrate, which then leads to the production of A. Given
sufficient substrate, autocatalysis leads to the doubling of catalyst molecules with every
transformation step, from A, to 2A, 4A, etc. This process is referred to as an exponential
increase. The potential power of an exponential increase can be derived from the three
dynamic states an autocatalytic process may attain (e.g. Refs 43 and 44): (1) when the
influx of substrate is too low, the system decays; (2) when the inflow of substrate is high
enough to let the autocatalytic production of catalysts equal their decay rate, the system is
in (dynamic) balance; and (3) when there is a rich influx of substrate, the positive
feedback causes a chain reaction that will let the process grow exponentially. While
systems with decaying or balanced dynamics will go unnoticed, systems with expo-
nential growth potentially possess the brutal force of an explosion.
The explosive power of autocatalysis is not sufficient to explain Darwinian evolution
because autocatalysis lacks heritable information. The coupling of autocatalysis and
information requires an additional step. In its simplest form this second step requires the
coupling of two catalytic reaction cycles based on the molecules A and B in a second-
order cycle in which A transforms substrate to B and B transforms substrate to A. The
resulting reaction cycle is fully driven by an external free energy gradient and is a
simplified form of Eigen’s ‘catalytic hypercycle’.
Eigen, who focuses on enzymatic
reactions, has published various studies about the stability and thermodynamics of
hypercyclic catalysis. In a catalytic hypercycle, every individual catalytic molecule can
be regarded as carrying information for the overall process. The capacity of hypercycles
to carry information has recently been discussed by Silvestre and Fontanari.
The hypercycle thus combines the explosive force of autocatalysis with the information
function of the separate catalytic molecules.
Hypercyclic catalysis unleashes enormous powers while creating an informed process.
However, these properties are still insufficient to cause evolution because the process
does not yet include a spatial mediating boundary that allows the components to become
a unit of selection. Without a boundary, the catalysts of an autocatalytic hypercycle float
freely in the pre-biotic ‘soup’ and cannot be assigned to a specific group. They can dilute
or mix freely with other sets. To end up with units that selection can act on, a physical
system limit is required. This can be added quite easily as a fatty acid membrane.
Vesicles naturally form by condensation in a watery solution containing fatty acids
and the process is well understood from the aspect of thermodynamics. The combination
of a membrane with autocatalysis now defines the first primitive cell. In one of the more
recent studies on the emergence of the first cells, Martin and Russel
have discussed the
simultaneous formation of autocatalysis and membranes based on the chemical reactions
in pre-biotic submarine hydrothermal vents of volcanic origin.
Once primitive cells were produced, it was a relatively small step toward multi-
plication and heritability of information. Given a constant supply of substrate molecules,
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autocatalysis automatically increased concentrations of the catalytic molecules in a cell.
It also potentially produced fatty acids enlarging the membranous envelope. Increasing
cell volume and envelope size will destabilise the cell structure and stimulate division,
and the contents will then more or less be randomly distributed over the two ‘offspring’.
When this occurs, cell-based autocatalysis powers the motor of primitive cellular
reproduction and only selective interactions have to be added before evolution occurs.
Despite their primitive state, the above cell-based reproduction and heritability
immediately force the water in the artesian evolution well to flow upward. The reason is
that cell-based cyclic autocatalysis implies the production of numerous individuals, that
the individuals show interactions and that interactions are most detrimental for weak
performers, which Darwin referred to as the ‘less well endowed’. The latter processes
result in selective interactions that scaffold the development of increasingly complex
building plans (at least on average). Selective interactions, as used here, are not limited to
competitive interactions but also include strategies based on cooperation.
Information, first in the form of the set of autocatalytic molecules and later in the form of
RNA/DNA, plays an important role in evolution. A fundamental aspect of information
is that it is hard to avoid random changes during its use and/or reproduction. As a
consequence, the information in organisms naturally tends to change over generations. A
negative result of this uncontrolled change is that offspring may suffer a lethal accumulation
of deleterious mutations. This kind of mortality is referred to as Muller’s ratchet (the name is
derived from the random occurrence of deleterious mutations as discussed by Hermann
Joseph Muller, 1890–1967). A positive result of this change is that every once in a while a
given mutation will positively affect an organism’s fitness. As long as the production of
original types and mutants that fit equally well or better to their environment outweighs
deleterious mutations, evolution will continue. The potential for genetic evolution has
convincingly been demonstrated in experiments that investigated how the genetic material
of viruses adapted over generations when it was subjected to different chemical stresses.
The constant emergence and spread of favourable mutations unpredictably changes
the ecosystem. To maintain one’s fitness, units of selection must continuously adapt.
The continuous need for adaptation has been simulated by Schneppen and Bak
who in
a group of competing species replaced the least fit species by one that is more fit. Their
model showed that the resulting dynamics are inherently unpredictable. This was con-
cluded from the fact that when plotting the number of species involved in one extinction
event against the frequencies of such events, their model showed the fractal characteristic
of a power law distribution. Such a power law accorded well with the distribution of
species’ extinctions in the archaeological records, as van Valen
observed. After
changing the original Schneppen-Bak model to be more realistic, for example, by
including genetic adaptation and random disturbances caused by meteorite impacts, the
model was proven robust and relevant for the evolutionary process.
During evolution, selection acts not only in the direction of the capacity to evolve but
also of the capacity to evolve evolvability (e.g. Refs 55 and 56). Once evolution
has started, it becomes increasingly difficult to stop the process because selection will
favour organisms that can exploit formerly inaccessible free energy gradients. Every new
pathway implies a new kind of ‘fuel’ powering new autocatalytic processes and
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increasing the size of and/or the pressure under evolution’s artesian well. Examples of
switches towards new and larger free energy gradients are those from physico-chemical
energy to solar energy (the development of photosynthesis), from physico-chemical
energy to biochemical energy (the development of predation/herbivory), from anaerobic
pathways to the use of oxygen (yielding a twenty-fold increase in available energy), from
the exploitation of living biomass to the use of fossil biochemical energy, etc. Other
examples are (1) the switch from depending on diffusion for energy transport to active
transportation of energy-rich substrates through the cell and (2) the symbiosis with
endosymbionts, generating energy throughout the cell.
From the above switches, the switch from physico-chemical energy to biochemical
energy has especially affected the evolutionary process because the biomass of the early
organisms suddenly became a degradable free energy gradient. Exploiting this gradient,
viruses, parasites and consumers attacked the organisms. These attacks reduced the
densities, which, in turn, increased growth rates. The chisel of selection was sharpened
when indirect competition for abiotic resources was supplemented by organothrophic
interactions. Afterwards, selective forces showed diversification towards searching
for and digesting biotic resources and towards developing survival strategies to avoid
becoming a resource.
Unification Based on a General Framework for Evolution
Darwin’s theory refers to evolution as a combination of two processes: (1) the production
of numerous offspring with different combinations of heritable properties, and (2) the
selecting away of individuals that are less well endowed; that is, in comparison to nearby
organisms, their competitive and/or cooperative properties fit less well to the demands of
the abiotic and biotic environment. The focus on these processes has linked the evolu-
tionary process to heredity. In actuality, however, evolution requires nothing more than
repeating a process that combines the production of variation (a diversification step)
with selection in relation to certain criteria (a selection step) (Figure 6). As has been
indicated by Popper
and Campbell
repeated diversification and selection steps
offer a general basis for evolution of organisms. But the implications of diversification
and selection may reach further, because these concepts are not limited to organisms.
The latter realisation makes it possible to compare evolution with a recipe. A recipe
consists of two lists, one for the activities, and one for the ingredients. The search for a
generalised concept of evolution can now make use of the recipe analogy, because one
could search for the most general activities list and the most general ingredients list, and
subsequently select local activities and local ingredients belonging to specific local
evolution theories.
The production of variation is a process that may involve genes, but in a more general
interpretation of diversification it may also involve abiotic particles or computer
organisms. For example, when two fundamental particles meet, they may integrate and
split again, or they may exchange a third particle, such that, after the process, two new
particle types are formed. And when a technical memon copies its brain structure through
computer code, incidental or deliberate errors in the process may produce variation.
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The selection process, too, is not limited to organisms. In Darwinian evolution,
selection may occur at many points, including when two organisms choose each other as
mates, when semen search for an egg cell, when an embryo develops in a uterus, when
offspring are born and have to persuade their parents to feed them, and so on (Figure 6).
In particle evolution, selection depends on whether particles recombine and produce new
particles that are stable.
When examining evolution in the above way, the difference between Darwinian
evolution and the evolution of particles fades and the principle of evolution becomes
visible in its most basic and general form: a recipe based on a list of activities and a list of
ingredients. A general framework for evolution theories can now be imagined where
specific subsets of the general evolution algorithm are combined with relevant subsets
of elements. Using organisms and reproduction, variation and selection, one obtains
Darwin’s theory. Using the diversification and selection of operators, one obtains an
evolution theory for the operators. Using diversification and selection of for example a
drawing on a sheet of paper, one obtains an evolution theory for concepts.
Unification Based on Free Energy Degradation and Organisational
Degrees of Freedom
The first and second laws of thermodynamics give direction to stories about the birth of
the universe. The first law states that energy cannot be made nor destroyed. This implies
that the universe contains no net energy or that the energy of the universe must have
existed before. The second law, then, states that dynamics reduce free energy gradients
and increase the dispersal of energy, both effects being regarded as an increase in
‘entropy’. Together, these laws lead to interesting suggestions. If energy existed before
the birth of the universe, the formation of the universe must have allowed the earlier
system to reach a higher entropy state. And if the universe contains no net energy, as is
Figure 6. This figure illustrates the generalisation of the evolution concept. Both the
evolution of particles and the evolution of organisms can be regarded as consisting of
steps combining the production of variation (diversification) and selection.
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suggested, for example, by Hawking,
the autonomous splitting of positive and negative
energy and the emergence of something out of nothing must involve an entropy increase.
It falls beyond the scope of this study to discuss these possibilities for the formation of
basic gradients in more detail. Instead, the focus will be on the degradation of existing
free energy gradients after these have been formed, and on how the degradation process
causes organisational complexity.
The observation that processes in the universe degrade free energy gradients along the
fastest pathways available, has been advocated by Swenson
who used a cabin in a cold
mountain region as an example. When the door and windows of the cabin are closed, the
gradient between the warm air in the cabin and the cold air outside can only reduce
slowly by means of convection through the walls of the cabin. If a window is opened,
this represents a faster pathway, and most of the temperature equilibration will now take
place via the window.
With respect to the flow through natural ‘windows’ Bejan
has proposed the
Constructal Law, stating that if a system can change its form, it will do so in the direction
of reduced resistance to the flows through it. An example illustrating the constructal law
is a dike with a small hole in it. The water that flows through the hole will erode the
sand away, causing a bigger hole. Assuming a large enough reservoir of water behind
the dike, the hole may erode until a point where the dike gives way, hereby minimising
the resistance to the flow. The constructal law helps explaining how free energy
degradation (which implies entropy increase) is responsible for complex forms of
flow systems, such as the shape of rivers, trees, hurricanes, and so on. The observed
forms arise as configurations that offer a relatively low resistance to the flows through the
system. And because a lesser resistance to flow implies that more power can be devel-
oped, the other side of the coin is that dynamic systems can be regarded as developing
towards the least waste of power (e.g. the Carnot’s theorem, Lotka’s maximum power
and Betz law (as explained in Refs 66 and 67)). Accordingly, the maximum
entropy production principle as reflected in the reduction of resistance to flow and/or
the maximising of power, is the main formative principle for systems that belong to
the outward dimension of the operator hierarchy (galaxies, stars, planets, ecosystem,
whirlwinds, and so on).
For systems along the upward dimension, additional aspects have to be invoked. Here
we need the combination of leverages, ratchets and pawls that is proposed by Lambert.
An important aspect of Lambert’s reasoning is the following. In order to reach organi-
sational complexity, something must push a system up the complexity slide. But once the
complex state has been reached, the organisation will in principle fall apart, and go down
the complexity slide again. Interestingly, this is not what is generally observed in nature.
For example, once atoms have created a molecule, which can be considered to reside
higher on the complexity slide, the molecule will generally remain on top of the slide,
instead of immediately going down again. On top of the slide, the molecule can be
considered to have toppled over a small edge, into a potentiality well, that acts as a pawl,
and prevents deterioration of the molecule. From the lowest level up, all operators show
different ways of going up the complexity slide, each level showing its proper pawl
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For quarks, hadrons, atoms and molecules, the most general mechanism for pushing
complexity up the slide is condensation. Because the heat of formation of the higher
complexity state is radiated away into the surroundings, the now cooler system cannot
easily fall apart again. For atoms with an atom weight above that of iron, the explanation
is different. For these high-weight atoms, energy is required to make lower level nuclei
fuse. A strong leverage is required, for example the pressure in supernova explosions.
In addition, the formation of complex molecules requires a different explanation. Now
the leverage is offered by an energy-rich substrate that can be degraded and with
enzymes scaffolding the formation process.
For bacteria and higher level operators (which all qualify as organisms) a continuous
flow of energy is required to maintain their structure. For this type of organisation,
closure occurs in combination with flows through the system. The closures produce the
pawls that prevent the system from going down the complexity slide, while the main-
tenance of the organisation forms a special case of the constructal law (based on flows
through a closed organisation). The factors that are responsible for the thermodynamic
advantages of a closure, are the higher competitive value following a closure, given an
environment where individuals, or groups of individuals, compete for resources.
Unification Based on Unifying Concepts
The above paragraphs have highlighted many grand unifying concepts in science and
have shown how the operator hierarchy may contribute to these fields. The examples that
were discussed represent a limited selection of the many larger and smaller unifying
concepts that exist. To discuss all these in a single paper would detract from the major
goal of this study: to analyse relationships between unifying concepts and to discuss the
potential contribution of the operator hierarchy.
To analyse relationships between several unifying concepts while preventing endless
elaboration, it was decided to create a cross-table at a high level of abstraction. On one
axis the table shows an inventory of unifying concepts and on the other their relevance at
different levels of the operator hierarchy. It was also decided to construct not one, but
two cross-tables: one for unifying concepts relating to operators and one for unifying
concepts relating to interaction systems (the systems that consist of operators but are not
operators). In addition, it was decided to sort the unifying concepts a priori according to
the four dimensions of the DICE approach (Displacement, Information, Construction
and Energy,
such that unifying concepts dealing with similar subjects were gathered
into these four classes. The outcome of these activities is shown in Tables 2 and 3.
Undoubtedly, these two tables are not complete and the a priori assignment of a given
unifying principle to one of DICE’s four classes or the a priori assignment as being most
important for operators or interaction systems may be disputed because many principles
relate to more than one subject. It is furthermore recognised that some unifying concepts
have a narrow scope, for example, the Pauli principle, while other principles, for
example, the concept of evolution, could have been split up into a whole range of related
unifying concepts, such as the selfish gene, the moving fitness landscape, game theory,
etc. In relation to the latter remark, it has been attempted to combine a priori smaller
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concepts into overarching concepts. Concepts were separated if at least one aspect
was important enough to justify this decision. Due to these considerations, Tables 2 and
Table 3 should be considered explorative tools for identifying interesting trends.
The inventory of unifying concepts in Tables 2 and Table 3 suggests two major trends.
The first trend is that only a few concepts apply to many different levels of organisation
and, in this sense, are truly unifying. One explanation for this lies in the fact that most
theories mainly apply to either operators or interaction systems. Another explanation is
that even within the separate lists of Table 2 and of Table 3, few theories can be found
that apply to all different operators or to all different interaction systems.
Combining the information of both tables, the following unifying concepts are
relevant for all material systems:
>Gravity impacts all systems, but it acts on higher level systems in an indirect
way. From the point of view of the operator hierarchy, gravity cannot interact
with organisms at the level of their typical closure (cellularity, endosym-
bionty, multicellularity, neural network), because, even though the organism
will sense gravity, the real action of gravity is only on the fundamental
particles in the organism.
>Thermodynamic laws are obeyed by all processes in systems (although
minute disobediences due to chance effects are possible).
>The stability of all systems is limited to within a certain range of
environmental conditions.
>Self-organisation, the constructal law, the maximum power principle and the
ratchet and pawl mechanism of potentiality wells connect thermodynamics
with the formation of structural patterns and complex organisation.
>The concept of first-next possible closure allows the recognition of the
operators and the operator hierarchy, hereby offering a construction sequence
offering a specific ‘periodic table for periodic tables’.
>If systems show dynamics, these may show various patterns, such as
alternative stable states, fractal behaviour (self-organised criticality) or shifts
between stable, periodic and chaotic behaviour.
The second trend that is suggested by the inventory is the divergence between
non-dissipative or dissipative operators. Of course, there is a good reason for this:
a dissipative operator is intimately linked to properties that allow it to sustain its organi-
sation while using an external energy gradient. Examples of such properties are auto-
catalysis, a membrane, heritable information, growth, demand for food or other energy
sources, etc. More detailed subdivisions can be recognised for all levels of the operator
hierarchy because every new closure introduces new properties. For example, memic
closure introduces reflexes, learning and behaviour based on mental representations.
The operator hierarchy contributes in various ways to scientific integration. First, it
allows the structuring of a range of scientific theories by invoking the strict ranking of the
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Table 2. Cross-table shows an inventory of unifying concepts acting on or in operators ((x) 5less likely, indirect, or uncertain)
Main subject
Related subjects (it concerns states or
dynamics within operators)
particles hadrons
atom nuclei
and atoms molecules
Relativity, matter-energy x
Thermodynamic laws x x x x x x x x x x
Autocatalytic physiology, resource
dominance, energetic demands
xx x x xx
Resource allocation and trade-offs, DEB-
xx x x xx
Desire based activity and flows xx
Construction FNP closure defining the operator types x x x x x x x x x x
Functional limits irt levels of resources or
(abiotic) stressors (Eyring, von Liebig,
Paracelsus). Niche concept
xxxxxxx xxx
Selforganization and selforganized
xxxxxxx xxx
Particle-wave duality x (x) (x) (x)
Schroedinger wave functions for electron
L-systems, gnomons (internal/external
addition), fractals
x x x x x x (x)
Constructal law for flow patterns x x x x x
Growth and development, allometrics x x x x x x
Life cycle during one generation (ontogeny
and life stages), fitness, trade-offs
xx x x xx
Information Genes, genetic basis of ontology, of
bodyplan, of phenotypic plasticity
(response curves)
xx x x xx
Immune/self-other xx x x xx
Genetic basis of neural network: mental
traits, nature (instead of nurture)
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Table 2. Continued
Main subject
Related subjects (it concerns states or
dynamics within operators)
particles hadrons
atom nuclei
and atoms molecules
Neural based behavior, reflexes, memory,
intelligence, consciousness, mental
health, psychology, phychiatry,
welbeing, satisfaction and brain-body-
sensors interaction
Memes xx
Displacement/ Zwitter-ion x
dynamics Turing patterns inside operators x x x x x (x)
Transportation inside organisms x x x x x (x)
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Table 3. Interaction systems: an inventory of unifying concepts in science and the range of interaction systems to which the unifying concept apply.
The interaction systems are arranged in order of the most complex operator in the system ((x) 5less likely, indirect, or uncertain)
Main subject Related subjects
system with
system with
system with
nuclei and
system with
system with
system with
with multi-
with multi-
with memon
with memon
Temperature dependence of interactions x x x x (x) (x) (x) (x) (x) (x)
Thermodynamics of overall systems or
xxxxxxx xxx
Construction Grand Unification Theory (Theory of
Relativity, space-time x (x)
Electromagnetism x x x x
Physical chemistry x x (x) (x) (x) (x) (x)
Gravity (x) (x) (x) (x) (x) (x) (x) (x) (x) (x)
Constructal law, patterning giving high
access to flows (including interaction
chains, such as foodchain), pseudo-
fractal constrution.
xxxxxxx xxx
Selforganization and selforganized
criticality (Pareto-Zipf-Mandelbroth,
pseudo-fractal distributions)
xxxxxxx xxx
Alternative stable states/critical
xxxxxxx xxx
Biochemistry x x x x x x (x)
Biotope xx x x xx
Information Pauli principle x x
Evolution A (diversification via
production, followed by selection)
Evolution B irt moving fitness landschape
((selfish-)genes, diversification,
reproduction (sex, populations),
x x x x (x)
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Table 3. Continued
Main subject Related subjects
system with
system with
system with
nuclei and
system with
system with
system with
with multi-
with multi-
with memon
with memon
epigenetics, units of evolution, role
neutral mutations, genotype-phenotype,
defectors, game theory)
Group-against-group selection x x x x x x
Habitat xx x x xx
Stress ecology: targets,buffering and
plasticity (phenotypic, population and
community adaptation (PICT))
xx x x(x)(x)
Disease syndrome (viral, bacterial,
multicellular causal agent)
xx x x x(x)
Social behavior: nurture, altruism,
communication, moral, economy,
politics, culture, science
Lamarckian inheritance (x) x
Evolution of memons (meme-to-network) x
Displacement/ Quantum tunneling x (x)
dynamics Logistic map, Mandelbrot set, Julia set
(stability, periodicity, chaos,
(x) (x) (x) x x x x x x x
Turing patterns and waves x x x x x x (x)
Spatial occurrencce of organism
interactions, neutral theory of
xx x x xx
Transportation (moving, phoresy, goods) x x x x x x
Information Pauli principle x (x)
Evolution A (diversification via
production, followed by selection)
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Table 3. Continued
Evolution B irt moving fitness landschape
((selfish-)genes, diversification,
reproduction (sex, populations),
epigenetics, units of evolution, role
neutral mutations, genotype-phenotype,
defectors, game theory)
x x x x (x)
Group-against-group selection x x x x x x
Habitat xx x x xx
Stress ecology: targets,buffering and
plasticity (phenotypic, population and
community adaptation (PICT))
xx x x(x)(x)
Disease syndrome (viral, bacterial,
multicellular causal agent)
xx x x x(x)
Social behavior: nurture, altruism,
communication, moral, economy,
politics, culture, science
Lamarckian inheritance (x) x
Evolution of memons (meme-to-network) x
Displacement/ Quantum tunneling x (x)
dynamics Logistic map, Mandelbrot set, Julia set
(stability, periodicity, chaos,
(x) (x) (x) x x x x x x x
Turing patterns and waves x x x x x x (x)
Spatial occurrencce of organism
interactions, neutral theory of
xx x x xx
Transportation (moving, phoresy, goods) x x x x x x
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operator hierarchy. Second, it establishes connections between other unifying concepts.
The structuring capacity of the operator hierarchy results from using first-next possible
closure, which allows a strict ranking of the operators. A strict ranking means that an
operator cannot be included or excluded without disturbing the entire logic of the
operator hierarchy. If a theory possesses such strictness, this can be regarded as a special
kind of beauty. For example, Einstein said the following about his general theory of
relativity, which offered a strict framework for dealing with gravity, space-time and
matter-energy: ‘The chief attraction of this theory lies in its logical completeness. If a
single one of the conclusions drawn from it proves wrong, it must be given up; to modify
it without destroying the whole structure seems to be impossible’ (from Ref. 14). But
while the relativity theory offers an abstract quantitative framework for dealing with
matter, energy, forces and space, the operator hierarchy focuses on complementary
aspects by offering an abstract and qualitative framework for organising matter. That the
operator hierarchy deals with qualitative aspects should not be considered a flaw of the
theory, but its strength, because it addresses a blind spot in the scientific literature.
The general analysis of structural hierarchy is not a fashionable topic in science. First,
people may not think about a unifying ranking because they consider particles, such as
hadrons, atoms and molecules, as incomparable with organisms. Secondly, people may
have difficulty identifying a general ranking rule. When looking at the mechanisms,
they look differently at every level. Only the use of first-next possible closure offers a
principle that can be used across levels. Thirdly, people may consider it wrong to focus
primarily on the operators because the universe is full of interaction systems, such as
galaxies, stars, planets and at least one ecosystem. However, the operator hierarchy
cannot be created or even recognised as long as interaction systems are considered to be
part of its ranking. This aspect was already recognized by Teilhard de Chardin. Finally,
the focus of science on quantification and equations has drawn the attention away from
structural analysis, which uses completely different concepts of quantity. These and other
aspects may have contributed to the absence of the operator hierarchy in any form from
the scientific debate.
As was discussed in this study, the operator hierarchy contributes in various ways to
such fundamental topics as a cosmic timeline, a natural hierarchy, a periodic table for
periodic tables, an extension of the organic theory of evolution and an analysis of the
scope of unifying concepts. The operator hierarchy adds to these topics a unique focus on
the structural complexity of systems. This focus enables the logical integration of distant
scientific domains. These achievements support the conclusion that the operator theory
offers a practical tool for centripetal science.
This paper is an updated and elaborated version of Chapter 8, pages 174–198 of Jagers
op Akkerhuis (2010) The operator hierarchy. A chain of closures linking matter, life and
artificial intelligence. Alterra Scientific Contributions,34.
The content of this paper has not been published in a scientifically reviewed journal
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About the Author
Gerard Jagers op Akkerhuis is a system scientist with a passion for integrating theory.
He studied plant pathology at Wageningen University (cum laude). His first PhD
in ecotoxicology concerned a quantitative model of the side-effects of pesticides on
General Laws and Centripetal Science S143
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terrestrial non-target arthropods. During his second PhD he explored a topological form
law, the ‘operator hierarchy’, that seems responsible for the formation of complex
operators, from fundamental particles to neural network organisms, and beyond. He is
the author of: ‘The pursuit of complexity. The utility of biodiversity from an evolutionary
perspective’. For more information see:
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The Major Evolutionary Transitions theory of Szathmáry and Maynard Smith is famous for its contribution to the understanding of complex wholes in biology. Typical for Major Evolutionary Transitions theory is the select use of functional criteria, notably, cooperation, competition reduction and reproduction as part of a larger unit. When using such functional criteria, any group of attached cells can be viewed as multicellular, such as a plant or the slug-shaped aggregation of cells of a slime mould. In addition, one could also have used structural criteria to arrive at the conclusion that the cells in the slug of a slime mould are attached without plasma strands, while the cells of a plant are attached and connected through plasma strands. A theory which in addition to functional criteria also uses structural criteria for the identification of major transitions is the Operator Theory. Using the Operator Theory one can, for example, conclude that the slug of a slime mould represents a pluricellular organisation because its cells are not connected through plasma strands, while the cells of a plant are connected through plasma strands and for this reason represent a multicellular organism. In this chapter, the relationships between the Major Evolutionary Transitions theory and the Operator Theory are studied with a focus on transitions that lead to organisms.
The preceding chapters of this book focus on complexity, evolution and life. Relatively little attention is paid to underlying mechanisms. This is the reason why the current chapter focuses predominantly on mechanisms that can explain the organisation of complex systems, either operators or interaction systems. The main causes of organisation are sought in the intrinsic motion of fundamental particles at temperatures above absolute zero, and the capacity of bonds between particles to form and break. Such processes are analysed from a thermodynamic perspective, focusing on the degradation of free energy and the occupation of accessible microstates. Both the degradation of free energy and the occupation of accessible microstates play a role during every next step in the Operator Hierarchy. Accessible microstates are furthermore used for calculating the contributions of DNA and of the brain to complexity on earth, as well as for calculating the probability that a pattern of Darwinian evolution occurs. In Sect. 14.3 relationships are discussed with existing literature.
The concept of life is central to biology and related life sciences, but there is no convergence on a definition. With the aim to resolve this problem analogies were studied between definitions of water and life. The concept of water refers to two phenomena: material particles (the H2O molecules) and interacting water molecules (liquid water). Likewise, the concept of life can be viewed as referring to a property of special material particles (the organisms) and to the system of interacting organisms (the ecosystem). In a comparable way as chemical theory has solved the problem of defining the water molecule, one can apply the Operator Theory for solving the problem of defining the organism concept. The analogy with water subsequently offers inspiration for two ontologically distinct definitions of life: (1) a definition of life as a general indication for a property that all organisms have, and (2) a definition of life that refers to a system of interacting organisms. These two definitions refer to different ontological kinds and accordingly cannot be merged into a single definition. For this reason the concept of life can be viewed as involving two, complementary, definitions. It is discussed how findings based on the water-life analogy contribute to current discussions about the definition of life.
The Operator Theory is a new theory about the hierarchical organisation of complexity in nature. The theory is based on the idea that in the space of all possible processes, a small subset exists of highly specific processes through which small objects can integrate to form new, more complex objects. The Operator Theory focuses on this small subset of objects. The processes that the Operator Theory focuses on are referred to as uniform closure of the structural and functional kind. The combination of such closures is called a dual closure. Based on dual closures, and in a step by step way, the Operator Theory identifies a branching hierarchy of kinds of objects that have increasingly complex organisation. Any object of a kind that is included in this hierarchy is called an operator, and the branching hierarchy is called the Operator Hierarchy. Interestingly, there are strong indications that, in analogy with the primary and secondary structure of amino acids, the Operator Hierarchy has a secondary structure. The Operator Theory hypothesises that this secondary structure offers a means to one day predict the structure of future kinds of operators. By offering a stringent classification of the operators of different kinds, from quarks to multicellular animals, the Operator Theory can be used to contribute to discussions about fundamental concepts in science, e.g. individuality, organismality, hierarchy, life and (the prediction of) evolution.
In a recent publication Szathmáry has updated Major Evolutionary Transitions theory to a version 2.0. The major transition theory recognises transitions based on the select use of functional criteria, notably: cooperation, competition reduction and reproduction as part of a larger unit. These criteria apply indiscriminatingly to cells forming a bee, and bees forming a hive. The possibility of suggesting different interpretations like these has caused ambiguity about the suitability of major evolutionary transitions as hallmarks for individuality or organismality. In this chapter it is suggested to deal with such ambiguity by the additional use of structural criteria when classifying transitions and resulting kinds of system. This chapter focuses on systems of interacting organisms, for which systems a decision tree is constructed that combines three different criteria to arrive at a classification. The decision tree starts with the fraternal and egalitarian interactions sensu Queller, adds democratic and centralised coding, and transcends the limitations of functional criteria by invoking the structural classes of the Operator Theory. These classes are operators (which applies to all organisms), compound objects and behavioural groups. If these additional criteria are used, one can resolve ambiguities about the classification of different kinds of groups of interacting organisms, such as pluricellular organisations, symbioses and groups.
Full-text available
This paper compares the two known logical forms of hierarchy, both of which have been used in models of natural phenomena, including the biological. I contrast their general properties, internal formal relations, modes of growth (emergence) in applications to the natural world, criteria for applying them, the complexities that they embody, their dynamical relations in applied models, and their informational relations and semiotic aspects.
Full-text available
This paper compares the two known logical forms of hierarchy, both of which have been used in models of natural phenomena, including the biological. I contrast their general properties, internal formal relations, modes of growth (emergence) in applications to the natural world, criteria for applying them, the complexities that they embody, their dynamical relations in applied models, and their informational relations and semiotic aspects.
During the development of a multicellular organism from a zygote, a large number of epigenetic interactions take place on every level of suborganismal organization. This raises the possibility that the system of epigenetic interactions may compensate or "buffer" some of the changes that occur as mutations on its lowest levels, and thus stabilize the phenotype with respect to mutations. This hypothetical phenomenon will be called "epigenetic stability." Its potential importance stems from the fact that phenotypic variation with a genetic basis is an essential prerequisite for evolution. Thus, variation in epigenetic stability might profoundly affect attainable rates of evolution. While representing a systemic property of a developmental system, epigenetic stability might itself be genetically determined and thus be subject to evolutionary change. Whether or not this is the case should ideally be answered directly, that is, by experimentation. The time scale involved and our insufficient quantitative understanding of developmental pathways will probably preclude such an approach in the foreseeable future. Preliminary answers are sought here by using a biochemically motivated model of a small but central part of a developmental pathway. Modeled are sets of transcriptional regulators that mutually regulate each other's expression and thereby form stable gene expression patterns. Such gene-expression patterns, crucially involved in determining developmental pattern formation events, are most likely subject to strong stabilizing natural selection. After long periods of stabilizing selection, the fraction of mutations causing changes in gene-expression patterns is substantially reduced in the model. Epigenetic stability has increased. This phenomenon is found for widely varying regulatory scenarios among transcription factor genes. It is discussed that only epistatic (nonlinear) gene interactions can cause such change in epigenetic stability. Evidence from paleontology, molecular evolution, development, and genetics, consistent with the existence of variation in epigenetic stability, is discussed. The relation of epigenetic stability to developmental canalization is outlined. Experimental scenarios are suggested that may provide further evidence.
Irreversible processes are the source of order: hence 'order out of chaos.' Processes associated with randomness (openness) lead to higher levels of organisation. Under certain conditions, entropy may thus become the progenitor of order. The authors propose a vast synthesis that embraces both reversible and irreversible time, and show how they relate to one another at both macroscopic and minute levels of examination.-A.Toffler
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Systems theory differs from classic analytical science by producing statements in the context of a descriptive system rather than seeking to reduce a complex problematic situation to researchable entities. This paper analyzes the validity and applicability of systems theory as a scientific approach towards organic farming. The world-views on which organic farming and systems theory build, respectively, are discussed and the methodological consequences of these world views are clarified. The world-view inherent in organic farming, the ontological level, as reflected in stewardship towards nature, the ethic of animal husbandry, and the cycling processes in nature, is harmonious with the underlying ideas of systems theory as regards a hierarchical structure and a mainly anthropocentric stand. This world-view is not paradigmatically different from the world-view inherent in conventional agriculture but non-consistent with the world-view of deep ecology. The world-view of organic farming acknowledges the wholeness in every system with emerging phenomena we might not perceive on an epistemological level. This world-view also acknowledges that such emerging phenomena occur at the higher levels of the hierarchical structure, which is in accordance with systems theory. Originally, a system was defined by its relations to its environment. However, in order to avoid the potential reductionism that may arise when everything is reduced to the ultimate whole system, it is found necessary also to identify and describe the major mechanisms within each system. It is concluded that meeting this requirement, systems theory can be viewed as an extension of traditional methods as the problems become more complex at higher hierarchical levels. Unfortunately, biologically speaking, we still have a limited knowledge of these mechanisms or processes in organic farming, but one of the challenges of today's science in organic farming is to identify and define these mechanisms on every hierarchical level.