independence

[AntonioLao]

Although independence is not a fundamental concept in the axiomatic calculus of probability it is one of its only two working principles. The other is the theory of measure which is basically axiomatic (axiom of choice). For example, it could be a choice of a minimum length. For physics, this is Planck length and its relation to Planck time, Planck energy, and all the other fundamental constants of nature (e.g. speed of light, universal constant of gravity, Boltzmann constant http://en.wikipedia.org/wiki/Boltzmann_constant, etc.). However, in the context of this post, independence is physically and mathematically related to quantitative and qualitative characteristic of equality and inequality. To say that two entities are equal is equivalent to saying that they are independent of each other influence: quantitatively, independence in space; qualitatively independence in time. The former answers questions: ‘how much?’ or ‘how many?’ the latter answers questions: ‘how long?’ or ‘how strong?’ The first represents mass property and the second represents charge property. The first gives meaning to magnitude, the second gives meaning to direction.

Magnitude is mathematically represented by positive real numbers which often include the number zero but for this post; zero is not used to represent magnitude. Direction is mathematically represented by angular measurements. In this sense, it can be represented by the entire real number line, from negative infinity, zero, and to positive infinity. Zero represents the midpoint of ‘no direction’. In a two dimensional coordinate patch, direction is described by counterclockwise and clockwise angular degrees of rotations for a given magnitude of the radius vector. A radius vector fixed at one point as the origin is described both by a magnitude and a direction. However, if each point of space-time can be considered as an independent origin for a particular radius vector then the whole of space-time contains infinite number of radius vectors with its own independent CCW or CW rotations. This independence is governed by a principle of directional invariance becoming the working principle of local infinitesimal motions (LIMs). Since the groupings of CCW and CW could form two and only two distinct topologies, further groupings form either odd or even multiples. Odds give space independence and evens give time independence. In higher dimensions where physical volumes could be defined, space independence is also equivalent to mass independence.

[AntonioLao]

The two and only two distinct topologies are shown in the image below
Attachment 244
The radii of each topology are not necessarily equal therefore the circles attract each other and formed linked loops.