This page offers a comparison of things that can be ranked in hierarchies, and of mechanisms for creating a hierarchy. When constructing a hierarchy, it is practical to follow certain rules. A first rule can be to avoid mixing different types of entities in the same hierarchy. This requires an additional rule for the identification of entities of the same abstract type. A second rule can be to use a similar kind of mechanism for every step in the hierarchy. A third rule can be that every step in the hierarchy must be constructed ‘from the ground up’, as a foundation for defining a ‘higher level’. Finally, to prevent ambiguity about levels, a fourth rule is that the hierarchy must have fixed levels (a non-transitive hierarchy). To achieve these goals, we present an innovative approach to hierarchy.

Classical entities that play a role in hierarchies (particles, objects, tokens and individuals)

In the literature many concepts are used for the description of the entities in hierarchies, including: particles, objects, tokens and individuals. differences in the widths of the applicability of these concepts has led to ambiguity. For this reason we will compare the application of these concepts (Figure 1).


In philosophy, the concept of a 'token' is used to indicate a single physical entirety that shows an outer limit and exists at a certain time at a certain place. If a token has parts, these must be physically connected or linked, such that they are part of this token and not of another token. A group of objects which are not physically connected is never a token. For example an apple can be a token, or a bicycle, or an atom (with the electron shell as an outer limit). A colony of bees, or a herd of deer, is not a token, because every bee or deer is a separate token.


The concept of a 'particle', is associated with a small lump of matter (a dust particle, a piece of brick) and with the historical discoveries of increasingly small physical particles. First the atoms ('a-tomos' indicates "that what cannot be cut") were considered the smallest. Then atom nuclei were discovered. And later discoveries revealed that nuclei harbour protons and neutrons, which in turn harbour quarks and gluons.


We distinguish between two types of objects. First of all, every subset of physically connected parts of the universe can be called a physical object. This includes for example colonial organisms that grow together, such as the Portuguese ship of war. In addition, if we set our mind to an 'object oriented' attitude, we can view any token as an object, such as an apple, or a physical particle, but we can also view a group of tokens as a mental object (e.g. a galaxy). Similarly we can view a specific herd of deer, or a specific human society as a mental object. Yet, a galaxy, a herd or a human society is not a token or a particle.Individual. Next, there is the concept of the 'individual' (which means 'something that cannot be divided'). Whether an entity ‘cannot be divided’ depends on the viewpoint. For example a piece of brick, as an individual, can be broken into parts, which again are individual pieces of brick. But if an atom is divided into a nucleus and an electron shell, one does not obtain two individuals of the type atom. Moreover, people have a natural tendency to view a herd of dear, or a human society as a (mental) individual (and as a mental object). Physically speaking, however, such groups are not tokens, which is the reason that it is preferred here to consider them as groups.

Organism and operator.

These are discussed in the next paragraph.
2014 Wikispaces Tabel object particle etc.jpg
Figure 1: Comparison of concepts used for different entities. +/- indicate ambiguities.

A new way of creating hierarchy

If we want to reduce ambiguities resulting from much used concepts such as particle, object, token and individual, a new approach is required. For this new approach, we consider it a necessity that it offers fixed levels, such that every entity at every level can always be defined in a stringent way. Moreover, every level (and next type of entity at that level) must be defined from the bottom up, by using (the structure of) lower level entities as a basis for higher level entities. And there should always be a clear logic defining the transitions from entities at lower levels to entities at higher levels. A transition should lead to the next level only, in order to assure a non-transitive ranking. Finally, in order to avoid the mixing of types of entities in the ranking, all the entities in the hierarchy should be of the same abstract type. An example of a mixed hierarchy is that from bacterium, to eukaryote cell, to a plant and population of plants. The reason is that a bacterium, a eukaryote cell and a plant all comply with the criteria of an object/token, whilst the population does not, because it is a group. We have found a way to comply with all these demands by constructing a hierarchy in which any next entity is always an object/token, and in which the structure of any next entity is based on ‘shape rules’. We use the mathematical theory of ‘topology’ as the tools for such shape rules. Topology has not been used in the classical approaches to hierarchy. To allow a close link with the natural world, we use a combination of structural and functional topology for defining a next level in the hierarchy.

Structural closure

The following example can illustrate the use of topology. Suppose we start a hierarchy with a simple topological shape: a balloon. This is a three dimensionally closed shape with no hole in it. Based on its closed shape, we could call the balloon a ‘closed object’. What we propose is to construct a hierarchy of structures using balloons as the basic closed objects. As is shown in Figure 1 this can result in three different shapes that are constructed from balloons. The first is that of a balloon which is attached to another balloon. The second is that of a balloon which partially engulfs the other balloon. The third is that of a balloon which has entirely engulfed another balloon. Topology now assists in reducing the number of options, because the partially engulfed situation is identical –in a topological sense- to the attached situation. A topological analysis thus helps in reducing ambiguity and in creating clear types (attached versus engulfed). The example shows that a balloon can serve as the basis for two next structures in our hierarchy: attached balloons and engulfed balloons. Both new situations are ‘one closure step away’ from the initial balloon. And both situations are the immediate next possibilities for a new type of closure situation. Therefore, both new topologies comply with our idea of ‘first-next possible new kind of closure’, in short; first next possible closure.

Functional closure

However important shapes are, they are not the only structuring patterns in nature. Also processes may play a role in the formation and typing of higher level entities in nature. Therefore, we suggest to also apply topology when analysing processes. For example a person could –as a process- lay his left and right hand on the shoulders of two other persons. If every person does the same, this results in a topology of a branching tree. But a process can also become self-referential, and for this reason be functionally closed. For example if a person whispers a message in the ear of a next person, this may result in a process in cycle in which the last person finally whispers his message in the ear of the first person, hereby closing the process (Hoffstadter would say that we have created a ‘strange loop’).

Two examples may help explaining this. The first example is the step from atom to molecule. Atoms have their electron shell as a new property. This shell allows the exchange of electrons, which is a typical new property of atom-atom interactions. And the exchange of electrons causes covalent boding. The step from atoms to molecules is caused by 1. shared electron orbits (new structural closure) and the exchange of electrons (new functional closure).

The second example is the step from the molecule to the cell. Molecules have their three-dimensional structure as a new property. This allows (a) structures based on interacting molecules and (b) new processes in which shapes of one molecule affect the shape of another molecule. The step from molecule to cell is thus caused by: 1. the formation of a surrounding vesicle (new type of structural closure of interacting molecules), and 2. the formation of an autocatalytic set of molecules that -as a set- create each other from energy rich substrate (new type of functional closure).

Constructing a non-transitive hierarchy from the bottom up

If we now assume that fundamental particles are the smallest ‘balloons’ that exist in nature, we can start constructing closed functional and/or structural topologies from these fundamental ‘balloons’. How this works has been explained step by step in pages 36-55 of “The operator hierarchy. A chain of closures linking matter, life and artificial intelligence” (Alterra Scientific Contribution 34, 2010). A detailed explanation can furthermore be found online at These sources use closure to explain every single step in the ranking from fundamental particles (as in the standard model) to hadrons, atoms and molecules, bacterial cells, endosymbiont cells, multicellulars and multicellulars with neural networks.

Operator types and the operator hierarchy

In order to refer to all the entities in the ranking of closures and closed entities described above, both of physical and biological nature, it is time to introduce a new concept: the operator. This name has been chosen to indicate that every operator can ‘operate’ (move, be displaced, etc.) in a general way in its environment while its closures persist. Accordingly, one can always use the closures (at the respective levels) as a hallmark of whether or not an entity is of the type ‘operator’. In relation to this, the ranking of types of closures, is called the operator hierarchy. For every closure type, there exist real world entities, the ‘operators’ that form the instantiations of the type.

An important implication of the operator hierarchy: three dimensions for natural organisation

Questions and References section 1.4