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

hermandadblanca_universo-mente-fractal-geometria-sagrada

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.

déjà vu 01

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.

dèja vu 02

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.

dèja vu 03

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.

My Quora Answer


Which questions does Category Theory help us answer?

Category 02

Another chapter in my attempt to help break the ‘spell’ of the category theoretical ‘ontologicisation’ of our world.

This may seem to many as a purely academic question, but we all need to realise that all of what we consider a modern way of thinking rests upon ‘mental technologies’ such as Category Theory.

Academics are literally taking the ‘heart’ out of how our world is being defined!
If we don’t pay attention, humanity will continue losing its way.

Category theory is a wonderful and powerful tool; nevertheless category theory, with all of its utility, is purely ontological. It can masterfully answer questions such as ‘Who?’, ‘What?’, and ‘How?’.

Category 01

However; it is regretfully inadequate to form a comprehensive representation of knowledge, for it lacks expression of epistemological value, which are the very reasons for is use. Epistemology is about answering the questions of ‘Why?’, ‘What does it mean?’, ‘What is my purpose?’,…
Answers to questions of this kind are implicitly supplied by us during our consumption of the utility afforded by category theory. We often are so beguiled by this power of categorical expression that we don’t realise that is we ourselves who bring the ‘missing elements’ to what it offers as an expression of knowledge.
It does a wonderful job with exteriority (ontology), but cannot sufficiently describe nor comprehensively access interiority (epistemology). Therefore, it has limited metaphysical value with respect to philosophy in general.

Interiority 08

Philosophies of mind, of language, or of learning are not comprehensive using only category theoretical tools.
Categorical structures are highly portable, but they can describe/express only part of what is there. There are structures, dynamics, and resonance that the ontology and functionalism in category theory completely turns a blind eye to.
More general than category theory is knowledge representation. It includes and surpasses category theory in many areas, both in scope and depth, but in particular: knowledge representation includes not just the ontological aspects of what we know, it goes further to describe the epistemological as well.
The qualities of Truth, Goodness, Beauty, Clarity,… can be defined and identified within a knowledge representation if the representation is not restricted to ontology. When category theory is used for the purpose of defining qualia, the objects must first be ontologised and functionally reduced. Trying to grasp them with tools restricted to category theory (or even semiotics) is like grasping into thin air.

 

Category theory, although very powerful, is no match for the challenge of a complete representation of knowledge. Category theory will tell you how to tie your shoes, but it can’t tell you why you are motivated to do so.


Lateral Numbers – How ‘Imaginary Numbers’ May Be Understood

Rbi0Y

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

and,

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…)


Does Division By Zero Have Meaning?

Yes, in knowledge representation, the answer is the interior of a holon.

Ontologies go ‘out of scope’ when entering interiority. The common ontological representation via mathematical expression is 1/0.

When we ‘leave’ the exterior ontology of current mathematics by replacing number with relation, we enter the realm of interiority.

In the interior of relation, we access the epistemological aspects of any relation.

As an aide to understanding – Ontology answers questions like: ‘What?’, ‘Who?’, ‘Where?’, and ‘When?’. Epistemology answers questions like: ‘Why?’ and ‘How do we know?’

In vortex mathematics 1/0 is known as ‘entering the vortex’.

There are other connections to some new developments in mathematics involving what is called ‘inversive geometry’.

Example: (oversimplified for clarity)

If we think of say… the point [x, y, z] in space, we may assign x, y, and z any number value except where one of these coordinates gets involved in division where 0 is not allowed (up to this point in common mathematics) as a denominator. x/z is not allowed when z=0, for example.

Now, if we are dealing with interiority, numbers are replaced by relationships, such as [father, loves, son].

What if the son has died? Is the relationship still valid?

The answer to this question lies within the interior of those involved in the relation.


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.

http://bit.ly/2wPV7RN


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

Semiotics

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.