What do all things have in common?

Metamathematics

Lateral Numbers – How ‘Imaginary Numbers’ May Be Understood

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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.


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


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}

and

Macro Awareness = \dfrac{1}{depth}
or

\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.)


Is Real World Knowledge More Valuable Than Fictional Knowledge?

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No.

Here an excerpt from a short summary of a paper I am writing that provides some context to answer this question:

What Knowledge is not:

Knowledge is not very well understood so I’ll briefly point out some of the reasons why we’ve been unable to precisely define what knowledge is thus far. Humanity has made numerous attempts at defining knowledge. Plato taught that justified truth and belief are required for something to be considered knowledge.

Throughout the history of the theory of knowledge (epistemology), others have done their best to add to Plato’s work or create new or more comprehensive definitions in their attempts to ‘contain’ the meaning of meaning (knowledge). All of these efforts have failed for one reason or another.

Using truth value and ‘justification’ as a basis for knowledge or introducing broader definitions or finer classifications can only fail.

I will now provide a small set of examples of why this is so.

Truth value is only a value that knowledge may attend.

Knowledge can be true or false, justified or unjustified, because

knowledge is the meaning of meaning

What about false or fictitious knowledge? [Here’s the reason why I say no.]

Their perfectly valid structure and dynamics are ignored by classifying them as something else than what they are. Differences in culture or language even make no difference, because the objects being referred to have meaning that transcends language barriers.

Another problem is that knowledge is often thought to be primarily semantics or even ontology based. Both of these cannot be true for many reasons. In the first case (semantics):

There already exists knowledge structure and dynamics for objects we cannot or will not yet know.

The same is true for objects to which meaning has not yet been assigned, such as ideas, connections and perspectives that we’re not yet aware of or have forgotten. Their meaning is never clear until we’ve become aware of or remember them.

In the second case (ontology): collations that are fed ontological framing are necessarily bound to memory, initial conditions of some kind and/or association in terms of space, time, order, context, relation,… We build whole catalogues, dictionaries and theories about them: Triads, diads, quints, ontology charts, neural networks, semiotics and even the current research in linguistics are examples.

Even if an ontology or set of them attempts to represent intrinsic meaning, it can only do so in a descriptive ‘extrinsic’ way. An ontology, no matter how sophisticated, is incapable of generating the purpose of even its own inception, not to mention the purpose of the objects to which it corresponds.

The knowledge is not coming from the data itself, it is always coming from the observer of the data, even if that observer is an algorithm.

Therefore ontology-based semantic analysis can only produce the artefacts of knowledge, such as search results, association to other objects, ‘knowledge graphs’ like Cayley,…

Real knowledge precedes, transcends and includes our conceptions, cognitive processes, perception, communication, reasoning and is more than simply related to our capacity of acknowledgement.

In fact knowledge cannot even be completely systematised; it can only be interacted with using ever increasing precision.

[For those interested, my summary is found at: A Precise Definition of Knowledge – Knowledge Representation as a Means to Define the Meaning of Meaning Precisely: http://bit.ly/2pA8Y8Y


Does Knowledge Become More Accurate Over Time?

Change lies deeper in the knowledge substrate than time.

Knowledge is not necessarily coupled with time, but it can be influenced by it. It can be influenced by change of any kind: not only time.

Knowledge may exist in a moment and vanish. The incipient perspective(s) it contains may change. Or the perspective(s) that it comprises may resist change.

Also, knowledge changes with reality and vice versa.

Time requires events to influence this relationship between knowledge and reality.

Knowledge cannot be relied upon to be a more accurate expression of reality, whether time is involved or not, because the relationship between knowledge and reality is not necessarily dependent upon time, nor is there necessarily a coupling of the relationship between knowledge and reality. The relationships of ‘more’ and ‘accurate’ are also not necessarily coupled with time.

Example: Eratosthenes calculated the circumference of the Earth long before Copernicus published. The ‘common knowledge’ of the time (Copernicus knew about Eratosthenes, but the culture did not) was that the Earth was flat.


What About Tacit Knowledge?

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A knowledge representation system is required. I’m building one right now. Mathesis Universalis.

There are other tools which are useful, such as TheBrain Mind Mapping Software, Brainstorming, GTD and Knowledgebase Software

Products and technologies like TheBrain, knowledge graphs, taxonomies, and thesauri can only manage references to and types of knowledge (ontologies).

A true knowledge representation would contain vector components which describe the answers to “Why?” and “How does one know?” or “When is ‘enough’, enough?” (epistemology).

It is only through additional epistemological representation that tacit knowledge can be stored and referenced.