What do all things have in common?


Knowledge Representation – Holographic Heart Torus

Holographic Heart Torus

Holographic Heart Torus by Ryan Cameron on YouTube

Knowledge Representation – Fractal Torus 1

Fractal Torus 1 by Ryan Cameron on YouTube

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^0 = 1

i^1 = i

i^2 = -1

i^3 = -i

i^4 = (i^2)^2 = (-1)^2 = 1

i^5 = i \cdot i^4 = i

i^6 = i^2 \cdot i^4 = (-1)(1) = -1

i^7 = i^2 \cdot i^5 = (-1)i = -i

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.

ComplexNumbers Example 001

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

ComplexNumbers Example 002 - Negative

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.

ComplexNumbers Example 003 - Negative Squaring

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.

ComplexNumbers Example 004 - Negative Squaring using lateral numbers01

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).

ComplexNumbers Example 004 - Negative Squaring using lateral numbers

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:

ComplexNumbers Example 004 - Negative Squaring using lateral numbers03


ComplexNumbers Example 004 - Negative Squaring using lateral numbers04

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?

Werner Heisenberg - on Language of Mathematics

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?

Equivalence Relation

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.


Limits of Category Theory and Semiotics

Category Theory 01

They are wonderful tools to explain much of our world, but lack ‘The Right Stuff’ to handle the metaphysical underpinnings of anything near a Philosophy of Mind, Philosophy of Language , or a Philosophy of Learning.

This is, because Category Theory specialises on roughly half of the Noosphere. It does a wonderful job on exteriority, but cannot sufficiently describe nor comprehensively access interiority.

Interiority Exteriority

Therefore, as is the case with Semiotics, has limited metaphysical value with respect to philosophy in general.

Semiotics 2


For example: philosophies of mind, language, or learning are not possible using only category theoretical tools and/or semiotics.

Here is an example of one attempt which fails in this regard: http://nickrossiter.org.uk/proce…

(and here: VisualizationFoundationsIEEE)

Here are two problems (of many) in the paper:

4.4.2 Knowledge is the Terminal Object of Visualisation states:

“The ultimate purpose of the visualisation process is to gain Knowledge of the original System. When this succeeds (when the diagram commutes) then the result is a ‘truth’ relationship between the Knowledge and the System. When this process breaks down and we fail to deduce correct conclusions then the diagram does not commute.”

I want to also comment on Figure 3 (which also exposes missing or false premises in the paper), but I will wait until I have discussed the assertions in the quote above which the authors of this paper reference, accept, and wish to justify/confirm.

1) The purpose of a representation is NOT to gain knowledge; rather, to express knowledge. Also, truth has nothing to do with knowledge except when that value is imposed upon it for some purpose. Truth value is a value that knowledge may or not ‘attend’ (participate in).

1a) The ‘truth value’ of the System (‘system’ is a false paradigm [later, perhaps] and a term that I also vehemently disagree with) does not always enter into the ‘dialogue’ between any knowledge that is represented and the observer interpreting that knowledge.

2) The interpretation of a representation is not to “deduce correct conclusions”; rather, to understand the meaning (semantics and epistemology) of what is represented. ‘Correct’ understanding is not exclusive to understanding nor is it necessary or sufficient for understanding a representation, because that understanding finds expression in the observer.

2a) ‘Correct’, as used in this paragraph, is coming from the outside (via the choice of which data [see Fig. 3] is represented to the observer) and may have no correspondence (hence may never ever commute) whatever to what that term means for the observer.

The authors are only talking about ontologies. That is a contrived and provincial look at the subject they are supposing to examine.

There may (and usually are) artefacts inherent in any collection and collation of data. The observer is forced to make ‘right’ (‘correct’) conclusions from that data which those who collected it have ‘seeded’ (tainted) with their own volition.

‘System’ (systematising) anything is Reductionism. This disqualifies the procedure at its outset.

They are proving essentially that manipulation leads to a ‘correct’ (their chosen version) representation of a ‘truth’ value.

I could tie my shoelaces into some kind of knot and think it were a ‘correct’ way to do so if the arrows indicate this. This is why paying too much attention to a navigation system can have one finding themselves at the bottom of a river!

The paper contains assumptions that are overlooked and terms that are never adequately defined! How can you name variables without defining their meaning? They then serve no purpose and must be removed from domain of discourse.

Categorical structures are highly portable, but they can describe/express only part of what is there. There are structure, dynamics, and resonance that ontology and functionalism completely turns a blind eye to.

The qualities of Truth, Goodness, Beauty, Clarity,… (even Falsehood, Badness, Ugliness, Obscurity,…) can be defined and identified within a knowledge representation if the representation is not restricted to ontology alone.

In order to express these qualities in semiotics and category theory, they must first be ontologised funtionally (reduced). Trying to grasp them with tools restricted to semiotics and category theory is like grasping into thin air.

That is actually the point I’m trying to make. Category Theory, and even Semiotics, each have their utility, but they are no match for the challenge of a complete representation of knowledge.

Universal Constants, Variations, and Identities #19 (Inverse Awareness)

Inverse Square
Universal Constants, Variations, and Identities
#19 The Inverse Awareness Relation

The Inverse Awareness Relation establishes a fundamental relationship in our universe:

Micro Awareness = \dfrac{1}{scope}


Macro Awareness = \dfrac{1}{depth}

\dfrac {Micro Awareness}{Macro Awareness} = \dfrac{depth}{scope}

Which essentially state:

The closer awareness is in some way to an entity, the more depth and the less scope it discerns.

The farther awareness is in some way to an entity, the more scope and the less depth it discerns.

(Be careful, this idea of closeness is not the same as distance.)