Hierarchy and spectrum

Hierarchy and spectrum are similar and related ideas, describing multi-level organizational structures or descriptions. They embody an ordered series of levels ranging from a lower value to a higher value in some dimension.


A hierarchy (from the Greek hierarkhia, "rule of a high priest", from hierarkhes, "president of sacred rites") is an arrangement of items (objects, names, values, categories, etc.) in which the items are represented as being "above", "below", or "at the same level as" one another.

Hierarchy is an important concept in a wide variety of fields, such as philosophy, mathematics, computer science, organizational theory, systems theory, and the social sciences (especially political philosophy).

A hierarchy can link entities either directly or indirectly, and either vertically or diagonally. The only direct links in a hierarchy, insofar as they are hierarchical, are to one's immediate superior or to one of one's subordinates, although a system that is largely hierarchical can also incorporate alternative hierarchies. Hierarchical links can extend "vertically" upwards or downwards via multiple links in the same direction, following a path. All parts of the hierarchy which are not linked vertically to one another nevertheless can be "horizontally" linked through a path by traveling up the hierarchy to find a common direct or indirect superior, and then down again. This is akin to two co-workers or colleagues; each reports to a common superior, but they have the same relative amount of authority.

Organizational forms exist that are both alternative and complementary to hierarchy. Heterarchy is one such form.


Mathematical representation

Main article: Hierarchy (mathematics)

Mathematically, in its most general form, a hierarchy is a partially ordered set or poset.[8] The system in this case is the entire poset, which is constituted of elements. Within this system, each element shares a particular unambiguous property. Objects with the same property value are grouped together, and each of those resulting levels is referred to as a class.

"Hierarchy" is particularly used to refer to a poset in which the classes are organized in terms of increasing complexity. Operations such as addition, subtraction, multiplication and division are often performed in a certain sequence or order. Usually, addition and subtraction are performed after multiplication and division has already been applied to a problem. The use of parenthesis is also a representation of hierarchy, for they show which operation is to be done prior to the following ones. For example: (2 + 5) × (7 - 4). In this problem, typically one would multiply 5 by 7 first, based on the rules of mathematical hierarchy. But when the parentheses are placed, one will know to do the operations within the parentheses first before continuing on with the problem. These rules are largely dominant in algebraic problems, ones that include several steps in order to solve. The use of hierarchy in mathematics is beneficial in order to quickly and efficiently solve a problem without having to go through the process of slowly dissecting the problem. Most of these rules are now known as the proper way into solving certain equations.


Hierarchy of Abstraction

This is a fundamental guiding structural intuition for the Weaving Unity project. Ideas are organized across hierarchical levels of abstraction

Hierarchy (mathematics)

In mathematics, a hierarchy is a set-theoretical object, consisting of a preorder defined on a set. This is often referred to as an ordered set, though that is an ambiguous term that many authors reserve for partially ordered sets or totally ordered sets. The term pre-ordered set is unambiguous, and is always synonymous with a mathematical hierarchy. The term hierarchy is used to stress a hierarchical relation among the elements.

Sometimes, a set comes equipped with a natural hierarchical structure. For example, the set of natural numbers N is equipped with a natural pre-order structure, where {\displaystyle n\leq n'} n\leq n' whenever we can find some other number {\displaystyle m} m so that {\displaystyle n+m=n'} n+m=n'. That is, {\displaystyle n'} n' is bigger than {\displaystyle n} n only because we can get to {\displaystyle n'} n' from {\displaystyle n} n using {\displaystyle m} m. This is true for any commutative monoid. On the other hand, the set of integers Z requires a more sophisticated argument for its hierarchical structure, since we can always solve the equation {\displaystyle n+m=n'} n+m=n' by writing {\displaystyle m=(n'-n)} m=(n'-n).

A mathematical hierarchy (a pre-ordered set) should not be confused with the more general concept of a hierarchy in the social realm, particularly when one is constructing computational models which are used to describe real-world social, economic or political systems. These hierarchies, or complex networks, are much too rich to be described in the category Set of sets.[1] This is not just a pedantic claim; there are also mathematical hierarchies which are not describable using set theory.[citation needed]

Another natural hierarchy arises in computer science, where the word refers to partially ordered sets whose elements are classes of objects of increasing complexity. In that case, the preorder defining the hierarchy is the class-containment relation. Containment hierarchies are thus special cases of hierarchies.


Great chain of being

The Great Chain of Being is a hierarchical structure of all matter and life, thought in medieval Christianity to have been decreed by God. The chain starts with God and progresses downward to angels, demons (fallen/renegade angels), stars, moon, kings, princes, nobles, commoners, wild animals, domesticated animals, trees, other plants, precious stones, precious metals and other minerals.

The Great Chain of Being (Latin: scala naturae, "Ladder of Being") is a concept derived from Plato, Aristotle (in his Historia Animalium), Plotinus and Proclus. Further developed during the Middle Ages, it reached full expression in early modern Neoplatonism.


Spectrum of consciousness

The Spectrum of Consciousness is a book by Ken Wilber, and general idea on the organization of mind and conceptual form.

Some of what we are doing here is intimately related to the Wilber thesis and vision, and is likely derived from the same sources. Other elements are significantly different.

Our work over time involved conceptual form and epistemological structure. In Spectrum Wilber was generally concerned with "energy levels" akin to "chakras". He was not discussing a hierarchy of concepts. Though related, he was discussing a spectrum of energy levels.