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 identity of change (some would say proportion, but that’s not entirely correct).
- π is the identity of periodicity. There’s much more going on with
than simply being a component of 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.
Getting Hypertension About Hyperreals
(Links below)
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.
https://www.youtube.com/watch?v=rJWe1BunlXI (Part1)
https://www.youtube.com/watch?v=jBmJWEQTl1w (Part2)
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.
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?
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.
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:
Is Real World Knowledge More Valuable Than Fictional Knowledge?
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?
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.
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
Universal Constants, Variations, and Identities – #17 (Representation)
#17 Interiority and Exteriority arise together. (Representation)
For every interior representation there is always an exterior representation that compliments it. For every exterior representation there is always a corresponding interior one.
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.
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.
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.)
Universal Constants, Variations and Identities – #14 (Singular/Plural)
Universal Constants, Variations and Identities
#14 Singular and plural arise together. (Singular/Plural)
There is no singular without a plural representation except in the non-dual.
See http://mathesis-universalis.com for more information.
#Knowledge #Wisdom #Understanding #Learning #Insight #Constants #Variances #Philosophy #MathesisUniversalis #ScientiaUniversalis #PhilosophicaUniversalis #LogicaUniversalis #MetaMathematics #MetaLogic #MetaScience #MetaPhysics #MetaPhilosophy #Singular #Plural
Universal Constants, Variations and Identities #13 (Knowledge)
Universal Constants, Variations and Identities
#13 Knowledge is what awareness does. (Knowledge)
I’ve published this before elsewhere, but it must be restated now for what is to follow (I’m starting a new octave).
#Knowledge #Wisdom #Understanding #Learning #Insight #Constants #Variances #Metaphysics #Philosophy #MathesisUniversalis #ScientiaUniversalis #PhilosophiaUniversalis #LogicaUniversalis #MetaMathematics #MetaLogic #MetaScience #MetaPhysics #MetaPhilosophy #Awareness
Universal Constants and Variances
#6 Reality is composed of whole parts. (Holons)
Arthur Koesler coined the term ‘Holon’ that refers to entities as both wholes and parts of some other whole.
https://en.wikipedia.org/wiki/Holon_%28philosophy%29
For example: a whole atom is part of a whole molecule, which is part of a whole cell, which is part of a whole organism,… Each of these entities are neither a whole or a part; rather, a whole part or Holon.
There’s a 2000 year old philosophical squabble between atomists and holists: “Which is ultimately real – the whole or the part?” The answer is neither or both, if you prefer…
There are only ‘whole-parts’: Holons.
[More are coming soon in a new post…]
Heurist.me (Carey G. Butler) 