User:Dpleibovitz/sandbox/Equivalence

The varied formal and informal notions of equivalence are fundamental to our society. And it is because multiple notions are subjectively applied, that this article relates equivalence (even of the formal kind) to assumptions.

Equivalence spans all fields of study from mathematics, computing, engineering, cognition, and sociology. This article compares and contrasts the myriad viewpoints on equivalence, showing where they are equivalent (and how this is related to assumptions), and where they are dissimilar. In the broad scheme of things, equivalence is an informal concept, but it is best introduced in comparison to the formal approach in mathematics.

In its majestic equality, the law forbids rich and poor alike to sleep under bridges, beg in the streets and steal loaves of bread. It is a wise man who said that there is no greater inequality than the equal treatment of unequals.

Draft to Article
This draft WP:Broad-concept article is intended to replace the WP:disambiguation article Equivalence, and to become the main article for Category:Equivalence.

The old contents and history of Equivalence will be copied/moved into Equivalence (disambiguation)

Relation to Mathematics
Equivalence in general is not shrouded in mathematical terms but the conceptions targeted by this article all share some mathematical characteristics, and can deviate as well.

Mathematically, under a given metric or ordering, equivalence and similarity are binary relations that are reflexive, i.e., A ≊ A; and symmetric, i.e., if (A ≊ B) then (B ≊ A).

Deviation from Mathematics
The informal notions of equivalence deviate from the mathematical conception in two ways. Often they are notions of similarity to vague definitions or in comparison to some norm, and often they are multiply applied. This remove transitivity, and creates an informal notion of an equivalence class.

Transitivity
However, while strict equivalence is transitive, similarity is not, e.g., (1.0 ≊ 1.5) and (1.5 ≊ 2.0) but not (1.0 ≊ 2.0). Thus this category is not about the mathematical notion of equivalence, although it does subsume it.

Multiple Subjective Applications and Equivalence Classes
In mathematics, only one equivalence relation at a time can be applied, which partitions the universe into disjoint equivalence classes (of the formal kind). In the social world, ever point of view defines a different "equivalence relation", so the underlying objects, often people, belong to multiple, overlapping groupings which are not formal equivalence classes. Nevertheless, as these do define subjective classifications for which equivalence is tested against, they can be considerred as informal equavalence classes.

Thus, if (A = B) under one point of view (metric or ordering), it could be the case that (A ≠ B) under another property.

Within mathematics, there can be strict equivalence. However, different scientific theories (e.g., Einstenian mechanics vs Newtonian Mechanics), may apply different relations to the same underlying reality. What is equivalent under one theory, may be not under another, e.g., simultaneity of events. So the application of mathematics, while dividing an abstract world into strict equivalence classes, divides the real world into multiple overlapping (informal) equivalence classes, and who can say whether any two things in the world are equal or not.

Comparison
In mathematics, logic, computing science, and engineering, determining equality, i.e., testing for equality (or inequality) is considered a form of comparison. However, before comparison can begin, notions of equivalence are required to create the thing being compared. E.g., cluster learning. One does not compare apples to oranges, but how do we start off with the concept of apple or orange in the first place?


 * 1) For nominal scales
 * 2) Ternary relations internal to digital logic

Social equivalence
Social equality.

This category is for things informally related to equivalencies (and similarities) of all kinds, e.g. circuitry, functions, operators, and relations under arbitrary and possibly multiple metrics or orderings. This category is not about Equality titled articles, such as gender or social equality which are about addressing (or equalizing over) inequalities.