### Semantics

This system is quite interesting if we allow ourselves to talk about the qualities of infinite sets as if we can know their character completely. The problem is, any discussion of an infinite set includes their definition which MAY NOT be the same as any characterisation which they may actually have.

Also, and more importantly, interiority as well as exteriority are accessible without the use of this system. These ‘Hyperreals’ are an ontological approach to epistemology via characteristics/properties we cannot really know. There can be no both true and verifiable validity claim in this system.

## Knowledge Representation – Holographic Heart Torus

Holographic Heart Torus by Ryan Cameron on YouTube

## Knowledge Representation – Fractal Torus 1

Fractal Torus 1 by Ryan Cameron on YouTube

## “How much knowledge does the understanding in words contain?”

Words are symbolic indications and/or conveyors of meaning and are not that meaning in themselves.

Meaning is found, stored, and manipulated in our minds. This is why different languages are capable, in varying degrees of usefulness, to convey meaning which is very similar to that found via the symbols of any other.

It It is also the reason why there are words indicating meaning that are not found in other languages; or, if found in a different language, the other language requires more of its own structure, dynamics, and resonance to convey the same meaning.

For example: the words ‘déjà vu’ in French are found in German ‘schon gesehen’ and in English ‘already seen’, but these phrases do not convey the full meaning found in the French version. To counter this deficit, their meaning in other languages must be ‘constructed’ out of or ‘fortified’ by the careful use of longer strings of symbols. This additional construction and/or fortification may even fail at times. This is often where the word phrase from a different language is simply added to the language in which the concept is missing.

This same situation is found in the literature of many languages. The words used to convey meaning are condensed and may contain more meaning than is usually the case. In this regard, even the person reading/hearing the words may not possess the competence necessary to catch this condensed meaning in its fullness.

Mathematical expressions, albeit more precise, are also indications of meaning. They are more robust in their formulation, but at ever-increasing depth or scope, even they may fail to reliably or conveniently convey meaning.

Our understanding of what words mean is not always accurate, but where our mutual understanding of the meaning of words overlaps, and the degree to which they overlap, is where their meaning can be shared.

Our own personal understanding of words is measured by our ability to apply their meaning in our lives.

There is also a false meme, which I would like to clarify.

“Knowledge is Power!”

It is wrongly said that ‘Knowledge is power’. The truth is another: Knowledge is the measure of usefulness of what we understand and is the only true expression of its ‘power’.

The value of Knowledge is found in its usefulness and not in its possession.

## Lateral Numbers – How ‘Imaginary Numbers’ May Be Understood

First, allow me to rename theses numbers during the remainder of this post to lateral numbers, in accordance to the naming convention as was recommended by Gauss. I have a special reason for using this naming convention. It will later become apparent why I’ve done this.

If we examine lateral numbers algebraically, a pattern emerges:

### $i^8 = i^4 \cdot i^4 = (1)(1) = 1$

When we raise lateral numbers to higher powers, the answers do not get higher and higher in value like other numbers do. Instead, a pattern emerges after every 4th multiplication. This pattern never ceases.

All other numbers, besides laterals, have a place on what currently is called the ‘Real number line’.

I qualify the naming of the Real Numbers, because even their conceptualisation has come into question by some very incisive modern mathematicians. That is a very ‘volatile’ subject for conventional mathematicians and would take us off on a different tangent, so I’ll leave that idea for a different post.

If we look for laterals on any conventional Real number line, we will never ‘locate’ them. They are found there, but we need to look at numbers differently in order to ‘see’ them.

Lateral numbers solve one problem in particular: to find a number, which when multiplied by itself, yields another negative number.
Lateral numbers unify the number line with the algebraic pattern shown above.

2 is positive and, when multiplied by itself, yields a positive number. It maintains direction on the number line.

When one of the numbers (leaving squaring briefly) being multiplied is negative, the multiplication yields a negative number. The direction ‘flips’ 180° into the opposite direction.

Multiplying -2 by -2 brings us back to the positive direction, because of the change resulting in multiplying by a negative number, which always flips our direction on the number line.

So, it appears as if there’s no way of landing on a negative number, right? We need a number that only rotates 90°, instead of the 180° when using negative numbers. This is where lateral numbers come into play.

If we place another lateral axis perpendicular to our ‘Real’ number line, we obtain the desired fit of geometry with our algebra.

When we multiply our ‘Real’ number 1 by i, we get i algebraically, which geometrically corresponds to a 90° rotation from 1 to i.

Now, multiplying by i again results in i squared, which is -1. This additional 90° rotation equals the customary 180° rotation when multiplying by -1 (above).

We may even look at this point as if we were viewing it down a perpendicular axis of the origin itself (moving in towards the origin from our vantage point, through the origin, and then out the back of our screen).

###### [If we allow this interpretation, we can identify the ‘spin’ of a point around the axis of its own origin! The amount of spin is determined by how much the point moves laterally in terms of i. We may even determine in which direction the rotation is made. I’ll add how this is done to this post soon.]

Each time we increase our rotation by multiplying by a factor of i, we increase our rotation another 90°, as seen here:

and,

The cycle repeats itself on every 4th power of i.

We could even add additional lateral numbers to any arbitrary point. This is what I do in my knowledge representations of holons. For example a point at say 5 may be expressed as any number of laterals i, j, k,… simply by adding or subtracting some amount of i, j, k,…:

5 + i + j +k +…

Or better as:

[5, i, j, k,…]

Seeing numbers in this fashion makes a point n-dimensional.

## Strictly Speaking Can’t! Natural Language Won’t?

Physics is only complex, because it’s in someone’s interest to have it that way. The way to understanding, even if you don’t understand science, was paved with words. Even if those words led only to a symbolic form of understanding.

Common ordinary language is quite capable of explaining physics. Mathematics is simply more precise than common language. Modern Mathematics pays the price for that precision by being overly complex and subservient to causal and compositional relations. These are limitations that metaphysics and philosophy do not have.

Words in language have a structure that mathematics alone will never see as it looks for their structure and dynamics in the wrong places and in the wrong ways. Modern pure mathematics lacks an underlying expression of inherent purpose in its ‘tool set’.

With natural language we are even able to cross the ‘event horizon’ into interiority (where unity makes its journey through the non-dual into the causal realm). It is a place where mathematics may also ‘visit’ and investigate, but only with some metaphysical foundation to navigate with. The ‘landscape’ is very different there… where even time and space ‘behave’ (manifest) differently. Yet common language can take us there! Why? It’s made of the ‘right stuff’!

The mono-logical gaze with its incipient ontological foundation, as found in (modern) pure mathematics, is too myopic. That’s why languages such as Category Theory, although subtle and general in nature, even lose their way. They can tell us how we got there, but none can tell us why we wanted to get there in the first place!

It’s easy to expose modern corporate science’s (mainstream) limitations with this limited tool set – you need simply ask questions like: “What in my methodology inherently expresses why am I looking in here?” (what purpose) or “What assumptions am I making that I’m not even aware of?” or “Why does it choose to do that? and you’re already there where ontology falls flat on its face.

Even questions like these are met with disdain, intolerance and ridicule (the shadow knows it can’t see them and wills to banish what it cannot)! And that’s where science begins to resemble religion (psyence).

Those are also some of the reasons why philosophers and philosophy have almost disappeared from the mainstream. I’ll give you a few philosophical hints to pique your interest.

Why do they call it Chaos Theory and not Cosmos Theory?
Why coincidence and not synchronicity?
Why entropy and not centropy?

Why particle and not field?
(many more examples…)

## Are sets, in an abstract sense, one of the most fundamental objects in contemporary mathematics?

Yes and no.

The equivalence relation lies deeper within the knowledge representation and it’s foundation.

There are other knowledge prerequisites which lie even deeper within the knowledge substrate than the equivalence relation.

The concepts of a boundary, of quantity, membership, reflexivity, symmetry, transitivity, and relation are some examples.

http://bit.ly/2wPV7RN