Philosophical Equality

Equality is both a descriptive concept and a normative concept. As a descriptive concept, equality is, by definition, an adjectival relation between entities that are identical in some specific respect.
No two entities can be identical in all respects, for then they would not be two entities but the same entity. The equality may be one of quantity or quality. Equality may be predicated of things, persons, or social entities such as institutions, groups, and so on. Equality is also a normative concept. As a normative concept, equality is the notion that there is some special respect in which all
human beings are in fact equal (descriptive) but that this factual equality requires that we treat them in a special way. Special treatment may mean ensuring identical treatment, or it may mean differential treatment to restore them to or to aid them in reaching or realizing the specific factual state. We often use the term equivalent to acknowledge that there may be tiny differences in a given comparison. As a logical function implemented by humans, equality is an ideal and makes considerable presumptions.

Knot Equivalence

A knot is created by beginning with a one-dimensional line segment, wrapping it around itself arbitrarily, and then fusing its two free ends together to form a closed loop (Adams 2004)(Sossinsky 2002). When topologists consider knots and other entanglements such as links and braids, they consider the space surrounding the knot as a viscous fluid. If the knot can be pushed about smoothly in the fluid, without intersecting itself, to coincide with another knot, the two knots are considered equivalent. The idea of knot equivalence is to give a precise definition of when two knots should be considered the same even when positioned quite differently in space. A formal mathematical definition is that two knots are equivalent if one can be transformed into the other via a type of deformation of R3 upon itself, known as an ambient isotopy. The basic problem of knot theory, the recognition problem, is determining the equivalence of two knots. Algorithms exist to solve this problem, with the first given by Wolfgang Haken in the late 1960s (Hass 1998). Nonetheless, these algorithms can be extremely time-consuming, and a major issue in the theory is to understand how hard this problem really is (Hass 1998). The special case of recognizing the unknot, called the unknotting problem, is of particular interest (Hoste 2005).

There is here an equivalence in the structure of the knot, and then there is the equivalence of interactions relative to all other knots in the full context of The Emergence Model. There is a massive amount of R&D needed to understand Architectures of Mass and how any given configuration manifests physical properties. Logical requirements (e.g. M5) are always a bit more precise than might be real requirements (e.g. M6) if only because there is usually more than one way to accomplish something.

Supervenience

Strategically at issue here is context and that is compartmentalized by Encapsulated Interpretative Model (EIM). What we mean by compartmentalization here is that context is completely made manifest within the 2D Articulation Layer of Translation Matrices, Paradigms Of Interest/Nature (POI/N) to POI/N for any particular EIM. Consequently philosophically implemented supervience must also be mode shifted several times for any given investigation. Once inside those cells EIM by EIM relative to and respective of any given POI/N. Then again within each subsequent analytical layer of Translation Matrices as appropriate for that particular layer. The unified Universe is always the final arbitor under Elegant Reasonism. Supervenient equivalence is contextual EIM to EIM and always prioritized philosophically as a predicate priority consideration relative to and respective of the unified Universe.