## Why is it so hard to prove that e+pi or e*pi is irrational/rational?

The reason why it is so hard to prove is actually very easy to answer. **These ***constants, identities, and variations*** being referred to in this post, and others like it, all lay embedded in a far deeper substrate than current mathematics has yet explored**.

Mathematics has been, and always shall be my ‘first love’, and it has provided for me all of these years. **I am not criticising mathematics in any way.** It is my firm belief that mathematics will overcome this current situation and eventually be quite able to examine these kinds of questions in a **much more ***expansive ***and ***deeper ***way.**

We need to extend our examination of mathematical knowledge, *both in depth and in scope,* out farther and in deeper than numbers (sets and categories as well – even more below) have yet done. **I’ll introduce you to a pattern you may have already noticed in the current stage of our mathematical endeavour.**

We all know there are numbers which lay outside of *Q* which we call Irrational numbers. There are also numbers which lay outside of *R* which we call Imaginary numbers. They have both been found, **because the domain of questioning exceeded the range of answers being sought within the properties each of those numbers.** This pattern continues in other ways, as well.

**We also know there are abstractions and/or extensions of Complex numbers where the ‘air starts to get thin’ and mathematical properties start to ‘fade away’: **Quaternions, Octonians, Sedenions,…

This pattern continues in other ways: **Holors**, for example, which extend and include mathematical entities such as Complex numbers, scalars, vectors, matrices, tensors, Quaternions, and other hypercomplex numbers, yet are still capable of providing a different algebra which is consistent with real algebra.

**The framing of our answers to mathematical questions is also evolving.** Logic was, for example, limited to quite sophisticated methods that all were restricted to a boolean context. Then we found other questions which led to boundary, multi-valued, fuzzy, and fractal logics, among a few others I haven’t mentioned yet.

**Even our validity claims are evolving. **We are beginning to ask questions which require answers which transcend relationship properties such as causality, equivalence, and inference in all of their forms. Even the idea of a binary relationship is being transcended into finitary versions (which I use in my work). There are many more of these various patterns which I may write about in the future.

**They all have at least one thing in common:** *each time we extend our reach in terms of scope or depth, we find new ways of seeing things which we saw before and/or see new things which were before not seen.*

There are many ‘voices’ in this ‘mathematical fugue’ which ‘weaves’ everything together: **they are the ***constants, variations, identities,*** and the ***relationships they share with each other.*

The constants *e, π, i, ϕ, c, g, h* all denote or involve ‘special’ relationships of some kind. **Special in the sense that they are completely ***unique***.**

For example:

*e*is the(some would say proportion, but that’s not entirely correct).*identity of change**π*is the. There’s much more going on with than simply being a component of*identity of periodicity**arc*or, in a completely different context, a component of*area*…

**These relationships actually transcend mathematics.** Mathematics ‘consumes’ their utility (making use of those relationships), but they cannot be ‘corralled in’ as if they were ‘horses on the farm’ of mathematics. **Their uniqueness cannot be completely understood via equivalence classes alone.**

**They are***ubiquitous***and therefore not***algebraic***.****They are***pre-nascent***to number***,***equivalence classes, and validity claims and are therefore not***rational***.**

These are not the only reasons.

**It’s also about WHERE they are embedded in the knowledge substrate compared to the concept of number, set, category…. ***They lay more deeply embedded in that substrate.*

The reason why your question is so hard for mathematics to answer is, because our current mathematics is, as yet, unable to decide. **We need to ‘see’ these problems with a more complete set of ‘optics’ that will yield them to mathematical scrutiny. **

Apr 25, 2018 | Categories: Holors, Insight, Knowledge, Knowledge Representation, Learning, Mathematics, Mathesis Generalis, Mathesis Universalis, Metamathematics, Philosophy of Language, Philosophy of Learning, Philosophy Of Mind, Understanding, Wisdom | Tags: insight, knowledge, learning, Mathesis Generalis, Mathesis Universalis, Metamathematics, Philosophy Of Mind, understanding, Universal Constants, wisdom | Leave a comment

## Knowledge Representation – Holographic Heart Torus

Holographic Heart Torus by Ryan Cameron on YouTube

Mar 11, 2018 | Categories: Fractals, Holons, Holors, Hyperbolic Geometry, Knowledge, Knowledge Representation, Language, Learning, Linguistics, Mathematics, Mathesis Generalis, Mathesis Universalis, Metamathematics, Philosophy, Philosophy of Language, Philosophy of Learning, Philosophy Of Mind, Semantic Web, Semantics, Understanding, Wisdom | Tags: knowledge, Language, learning, Philosophy, understanding | 1 Comment

## Knowledge Representation – Fractal Torus 1

Fractal Torus 1 by Ryan Cameron on YouTube

Mar 11, 2018 | Categories: Fractals, Holons, Holors, Hyperbolic Geometry, Knowledge, Knowledge Representation, Learning, Linguistics, Mathesis Universalis, Philosophy of Language, Philosophy of Learning, Philosophy Of Mind, Semantic Web, Semantics | Tags: knowledge, Knowledge Representation, learning, Mathesis Generalis, Mathesis Universalis, Philosophia Universalis, Philosophy of Language, Philosophy Of Mind, Semantics, understanding, wisdom | Leave a comment

## 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:

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.**

Nov 14, 2017 | Categories: Constants, Holons, Holors, Hyperbolic Geometry, Knowledge, Knowledge Representation, Language, Learning, Linguistics, Mathematics, Mathesis Universalis, Metamathematics, Philosophy of Language, Philosophy of Learning, Philosophy Of Mind, Semantic Web, Semantics | Tags: Mathematica Generalis, Mathematica Universalis, Mathesis Generalis, Mathesis Universalis, Metamathematics | Leave a comment

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

**Universal Constants, Variations, and Identities**

#19 **The Inverse Awareness Relation**

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

May 15, 2017 | Categories: Discernment, Holons, Holors, Hyperbolic Geometry, Identities, Insight, Knowledge, Knowledge Representation, Language, Learning, Linguistics, Mathesis Generalis, Mathesis Universalis, Metamathematics, Metaphysics, Philosophy, Philosophy of Language, Philosophy of Learning, Philosophy Of Mind, Semantic Web, Semantics, Understanding, Universal Constants, Variations, and Identities, Variations, Wisdom | Tags: knowledge, Language, learning, Linguistics, LogicaUniversalis, Mathesis Universalis, Philosophia Universalis, Philosophy, Philosophy of Language, Philosophy of Learning, Philosophy Of Mind, understanding | Leave a comment

## Universal Constants, Variations, and Identities #18 (Dimension)

**Universal Constants, Variations, and Identities** (Dimension)

#18 **Dimension is a spectrum or domain of awareness: they essentially build an additional point of view or perspective.**

We live in a universe of potentially infinite dimension. Also, there are more spatial dimensions than three and more temporal dimensions than time (the only one science seems to recognize). Yes, I’m aware of what *temporal* means; **Temporal is a ***derived attribute*** of a much more fundamental concept: **** Change.** One important caveat: please bear in mind that my little essay here is not a complete one. The complete version will come when I publish my work.

**The idea of dimension is not at all well understood.** The fact is, science doesn’t really know what dimension is; rather, only how they may be used! Science and technology ‘consume’ their utility without understanding their richness. Otherwise they would have clarified them for us by now.

Those who may have clarified what they are get ignored and/or ridiculed, because understanding them requires a larger mental ‘vocabulary’ than Physicalism, Reductionism, and Ontology can provide.

Our present science and technology is so entrenched in dogma, collectivism, and special interest, that they no longer function as they once did. The globalist parasites running our science and technology try their best to keep us ‘on the farm’ by restricting dimension, like everything else, to the purely physical. It’s all they can imagine.

That’s why many of us feel an irritation without being able to place our finger on it when we get introduced to dimension. We seem to ‘know’ that something just doesn’t ‘rhyme’ with their version.

**Time and space may be assigned dimensionality, in a purely physical sense if necessary, but there are ***always underlying entities much deeper in meaning involved that are overlooked and/or remain unknown which provide those properties with their meaning.*** This is why the more sensitive among us sense something is wrong or that something’s missing.**

Let us temporarily divorce ourselves from the standard ‘spatial’ and ‘temporal’ kinds of ‚dimension’ for a time and observe dimension in its *essence.*

**Definitions are made from them: in fact, dimensions function for definitions just as organs do for the body. In turn, dimension has its own set of ‘organs’ as well!** I will talk about those ‘organs’ below.

Dimension may appear different to us depending upon our own state of mind, level of development, kind of reasoning we choose, orientation we prefer, expectations we may have,… but down deep…

**Everything, even attributes of all kinds, involve dimension.** We must also not forget partial dimension such as fractals over complex domains and other metaphysical entities like mind and awareness which may or may not occupy dimension. Qualia (water is ‘wet’, angry feels like ‘this’, the burden is ‘heavy’) are also dimensional.

**Dimensions are ‘compasses’ for navigating conceptual landscapes. We already think in multiple dimension without even being aware of it! Here’s is an example of how that is:
**[BTW: This is simply an

*example*to show how dimension can be ‘stacked’ or accrued. The items below were chosen

*arbitrarily*and could be replaced by any other aspects.]

♦ Imagine a point in space (we are already at 3d [x,y,z]) – actually at this level there are even more dimensions involved, but I will keep this simple for now.

♦ it moves in space and occupies a specific place in time (now 4d) 3d + 1 time dimension

♦ say it changes colour at any particular time or place (5d)

♦ let it now grow and shrink in diameter (6d)

♦ if it accelerates or slows its movement (7d)

♦ if it is rotating (8d)

♦ if it is broadcasting a frequency (9d)

♦ what if it is aware of other objects or not (10d)

♦ say it is actively seeking contact (connection) with other objects around it (11d)

♦ … (the list may go on and on)

*As you can see above, dimensions function like aspects to any object of thought.*

**Dimensionality becomes much clearer when we free ourselves from the yoke of all that Physicalism, Reductionism, and Ontology.**

Let’s now look at some of their ‘organs’ as mentioned above as well as other properties they have in common:

- They precede all entities except awareness.
- Awareness congeals into them.
- They form a first distinction.
- They have extent.
- They are integrally distributed.
- They have an axial component.
- They spin.
- They vibrate.
- They oscillate.
- They resonate.
- They may appear as scalar fields.
- Their references form fibrations.
- They are ‘aware’ of self/other.
- Their structural/dynamic/harmonic signature is unique.
- They provide reference which awareness uses to create perspective meaning.
- Holons are built from them.

http://mathesis-universalis.com

Sacred Geometry 29 by Endre @ RedBubble:

http://www.redbubble.com/people/endre/works/6920405-sacred-geometry-29?p=poster

Sep 7, 2016 | Categories: Constants, Holons, Holors, Knowledge, Knowledge Representation, Language, Learning, Linguistics, Mathematics, Mathesis Generalis, Mathesis Universalis, Meta Logic, Metamathematics, Metaphysics, Perspective, Philosophy, Scalars, Semantics, Understanding, Variations, Wisdom | Tags: BigData, First Distinction, insight, knowledge, Knowledge Representation, learning, Logica Universalis, Mathesis Universalis, Metalogic, Metaphysics, Philosophia Universalis, Scalar Field, Scalars, Scientia Universalis, Semantics, understanding, Universal Constants, Variances, wisdom | Leave a comment

## HUD Fly-by Test

Don’t take this as an actual knowledge representation; rather, simply a simulation of one. I’m working out the colour, transparent/translucent, camera movements, and other technical issues.

In any case you may find it interesting.

The real representations are coming soon.

Aug 21, 2016 | Categories: Big Data, Holons, Holors, Hyperbolic Geometry, Knowledge, Knowledge Representation, Language, Learning, Linguistics, Logic, Long Data, Mathesis Generalis, Mathesis Universalis, Meta Logic, Metamathematics, Metaphysics, Philosophy, Understanding, Wisdom | Tags: BigData, Constants, Hyperbolic Geometry, insight, knowledge, Knowledge Representation, learning, Logica Generalis, Logica Universalis, Mathesis Generalis, Mathesis Universalis, Metalogic, Metamathematics, Metaphysics, Philosophia Generalis, Philosophia Universalis, Philosophy, Philosophy of Language, Philosophy of Learning, Philosopohy of Mind, Scientia Universalis, understanding, Universal Constants, Universalis, wisdom | Leave a comment

## A New Kind of Knowledge Representation Is Coming to Be!

The project is now coming to conclusion (finally). In this video I show an example knowledge molecule being ‘examined’ by the knowledge representation.

I’ve hidden the other actors in this demonstration and have simplified the instrumentation to preserve my priority on my work.

Be patient! It won’t be long now… I have the theoretical underpinnings already behind me. Now it’s only about the representation of that work.

Aug 12, 2016 | Categories: Big Data, Holons, Holors, Knowledge, Knowledge Representation, Language, Learning, Linguistics, Long Data, Mathematics, Mathesis Generalis, Mathesis Universalis, Meta Logic, Metamathematics, Metaphysics, Philosophy, Semantics, Understanding, Wisdom | Leave a comment

## Obfuscation In A ‘Nut’ Shell

**Obfuscation In A ‘Nut’ Shell**

*Distinctions that are no differences, are incomplete, or are in discord.*

**In knowledge representation these ‘impurities’ (artificiality) and their influence are made easy to see.**

In groks you will see them as *obfuscation fields.* That means darkening and/or inversion dynamics. The term refers to the visual representation of an obfuscated field, and can also be represented as dark and/or inverted *movements* of a field or group. I concentrate more on the dark versions here and will consider the inversions (examples of lying) in a future post.

They bring dynamics that are manipulative, artificial, or non-relevant into the knowledge representation. Their dynamic signatures make them stand out out like a sore thumb.

Cymatic images reveal these dynamics too. There are multiple vortexes, each with their own semantic contribution to the overall meaning to a knowledge molecule or group.

Here is an example of a snow flake (seen below) https://www.flickr.com/photos/13084997@N03/12642300973/in/album-72157625678493236/

From Linden Gledhill.

Note that not all vortexes are continuous through the ‘bodies’ of the molecules they participate in. Also, in order to correctly visualize what I’m saying, one must realize that the cymatic images are split expressions. That means to see the relationship, you must add the missing elements which are hinted at by the image.

Every cymatic image is a cut through the dynamics it represents.

We are in effect seeing portions of something whole. Whole parts are dissected necessarily, because the surface of expression is limited to a ‘slice’ through the complete molecule.

(Only the two images marked ‘heurist.com’ are my own! The other images are only meant as approximations to aid in the understanding of my work.)

Apr 28, 2016 | Categories: Big Data, BigData, Holons, Holors, Insight, Knowledge, Knowledge Representation, Language, Learning, Linguistics, Mathematics, Mathesis Generalis, Mathesis Universalis, Metamathematics, Semantics, Wisdom | Tags: insight, knowledge, learning, Logica Universalis, Mathesis Universalis, Philosophia Universalis, understanding, wisdom | Leave a comment

## Typical Knowledge Acquisitions Node

**Knowledge Representation**

A typical knowledge acquisition node showing two layers of abstraction. Note how some of the acquisition field detection moves with the observer’s perspective. You can tell, due to the varying visual aspects of the fields and their conjunctions that it has already been primed and in use.

This node may be one of thousands/millions/billions which form when acquiring the semantics of any particular signal set.

Their purpose is to encode a waveform of meaning.

Basically it is these ‘guys’ which do the work of ‘digesting’ the knowledge contained within any given signal; sort of like what enzymes do in our cells.

The size, colour (although not here represented), orientation, quantity, sequence, and other attributes of the constituent field representations all contribute to a unique representation of those semantics the given node has encountered along its travel through any particular set of signal. The knowledge representation (not seen here) is comprised of the results of what these nodes do.

This node represents a unique cumulative ‘imprint’ or signature derived from the group of knowledge molecules it has processed during its life time in the collation similar to what a checksum does in a more or less primitive fashion for numerical values in IT applications.

I have randomized/obfuscated a bit here (in a few different ways), as usual, so that I can protect my work and release it in a prescribed and measured way over time.

In April I will be entering the 7th year of working on this phase of my work. I didn’t intentionally plan it this way, but the number 7 does seem to be a ‘number of completion’ for me as well.

The shape of the model was not intended in itself. It ‘acquired’ this shape during the course of its work. It could have just as well been of a different type (which I’m going to show here soon).

Important is the ‘complementarity’ of the two shapes as they are capable of encoding differing levels of abstraction. The inner model is more influenced by the observer than the outer one, for example. The outer shape contains a sort of ‘summary’ of what the inner shape has processed.

Jan 4, 2016 | Categories: Big Data, BigData, Consciousness, Fields, Holons, Holors, Knowledge, Knowledge Representation, Language, Learning, Linguistics, Mathesis Universalis, Semantics, Wisdom | Tags: insight, knowledge, learning, Mathesis Universalis, Metaphysics, understanding, wisdom | Leave a comment

## Knowledge Is What Awareness Does – Knowledge Representation as a Means to Define Meaning Precisely

Video is finally here!

Sep 3, 2014 | Categories: Artificial Intelligence, BigData, Holons, Holors, Insight, Knowledge, Knowledge Representation, Language, Learning, Linguistics, Logic, Mathematics, Mathesis Universalis, Metaphysics, Philosophy, Semantics, Understanding, Wisdom | Tags: ArtificialIntelligence, insight, knowledge, Language, learning, Linguistics, OrganicIntelligence, Philosopohy, Semantics, understanding, wisdom | Leave a comment

## Self-referential Paradoxum In Knowledge Representation

What do these two pictures share in common?

**They represent structure and dynamics (continuity, connectedness, and boundary [of which ***topology*** is only ONE example!]) distributed over multiple and partial dimension. That’s why they interest me and are of use. I use them to represent knowledge, because ***they are found in our knowledge!*

They always have a concealed twist (internal dynamics). You need to leave our 3D rational domain to capture their meaning though. I’ve studied these shapes for a long time now and use them in my work. Break the figures or logic apart and notice how you can tuck the parts into a cloud shape (ambiguity) and make the systems work.

Aug 29, 2014 | Categories: Holors, Knowledge, Knowledge Representation, Language, Learning, Linguistics, Logic, Mathematics, Mathesis Universalis, Metaphysics, Philosophy, Physics, Science, Semantics, Understanding, Wisdom | Tags: FractalLogic, insight, knowledge, learning, Logic, Metalogic, understanding, wisdom | Leave a comment

## Knowledge Representation – Self and Other

This knowledge representation, which I made for a presentation in Nürnberg on April 29th, 2013, depicts a partial resonance domain with respect to sentient relation and orientation. The representation is not designed to be comprehensive in any respect, rather it is intended to help those who are not accustomed to viewing knowledge in this way.

It shows how, to quote Carl Sagan: “One voice in the Cosmic fugue.” can be represented more generally. There are certainly other possibilities with respect to relation and orientation between self and other.

You may need to stop this video occasionally as some of the transitions occur very quickly.I’ve indicated what is happening in the representation to aid in the interpretation of what is being shown.

I would appreciate any feedback you may want to give. I’d be pleased to answer any questions that may arise.

Thank you for watching!

UPDATE:

A question has come in from an anonymous source asking for clarification on the use of “Resonance Domain”.

A resonance domain is comprised of resonance fields. When one or more fields are being examined, the group is called a domain.

For further clarity I’d like to add that we are looking at two separate fields (sentient beings) in terms of the relation they participate in and the orientation they share with each other.

Fields are always composed of other sub-fields and contain even partial fields from other sources!

UPDATE: “Why do you use the word ‘sentient’ in your description?”

There are basically three types of entities in our universe:

1) Having some measure of interiority (concept of self).

2) Artifacts

3) Heaps

‘Self and other’ can come in any number of combinations:

The knowledge representation above requires that at least two of the participants are of type 1 (sentience) for this knowledge representation to hold.

Examples:

Type 1) human and type 1) dog will work.

Type 1) human and type 2) book will not work.

Type 1) human and type 3) rock will not work.

This way of looking at our Kosmos (multiverse) is called Mathesis Universalis.

For those interested in more on the subject, go to http://mathesis-universalis.com.

[PS: There are also the conditions that both participants are conscious of each other and that their shared consciousness is such that differences in semantics between them are ‘reconcilable’ (coherence). (for the scientists/mathematicians among us!)] 😉

Jul 29, 2014 | Categories: Artificial Intelligence, BigData, Ethics, Holons, Holors, Knowledge Representation, Language, Linguistics, Mathematics, Mathesis Universalis, Metaphysics, Philosophy, Semantics | Tags: BigData, Holons, Holors, Knowledge Representation, organic intelligence, Semantics | Leave a comment