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.
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.
Universal Constants, Variations and Identities – #16 (Creation/Discovery)
Universal Constants, Variations and Identities
#16 Creation and discovery compliment each other and are the means in which the Universe fundamentally unfolds and enfolds itself (Creation/Discovery)
We tend not to identify them, because there are so many variations in their harmony. Please do overestimate your thoughts… as you will see they are the beginning of your expression to and of the world.
Both Creation and Discovery will work in unison, if we allow them.
Discovery is to recognize/relate what is in your world.
Creation is to transform/synthesize it too.
Each is alone without the other.
Creation=Right ‘brain’ (right+mind)
Discovery=Left ‘brain’ (left+mind)
Their ‘magick’ (sic.) manifests not when you synchronize them; rather, when you harmonize them.
(Please take the time to watch the 4 minute video.)
Universal Constants, Variations and Identities – #15 (Change/Time)
#15 Time is a temporally ‘linear’ (directed) form of change that is not limited by dimension. (Change/Time)
Time has been arbitrarily and wrongly assigned to dimension. Change is not restricted to any dimension: therefore time is also not limited to it.
I know it’s trendy to see time as a dimension, but dimension is something completely different. Stay tuned to find out what and why.
Update: There are many reasons why time needs a proper definition. Here are a few of them:
The chemical reactions in the vessel are not really effected by some mysterious thing called time, but by the number of contacts or collisions that take place in the soup of atoms or molecules. That is what the factor ‘T’ really stands for.
1) Eternity may be a somewhat mystical overarching reality outside of the physical universe, but time is not. Nor is time a thing that anybody can do anything to. In other words: it cannot be reified.
2) The universe doesn’t exist in time, but time exists in the universe.
3) The proper definition of time is exactly: the sequence of events in the material universe.
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, Variations and Identities
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Universal Constants, Variations and Identities
#12 The ends are determined by their means. (Integrity of purpose)
(Dispelling the lie of: “The ends justify their means.” The truth is that ends are inextricably tied to the means used to achieve them. If we employ destructive means, then only destructive ends can result. Intention means nothing with regard to means… those who believe that learn: “The road to Hell is paved with good intentions.”
See http://mathesis-universalis.com for more information.
Universal Constants, Variations and Identities
Universal Constants and Variances
#11 Observation is communication between its participants (observer and observed). (Reference)
See http://mathesis-universalis.com for more information.
Universal Constants, Variations and Identities
Universal Constants and Variances
#10 All holons have at least four fundamental capacities. (Potential)
Horizontal:
1) Agency (Maintaining wholeness/identity) Pull to wholeness
2) Communion (Maintaining partness/relationship) Pull to partness
Vertical:
3) Transcendence (Becoming more wholeness) Pull upwards
4) Dissolution (Breaking down into constituent parts) Pull downwards
(see #6 for definition of holon)
See http://mathesis-universalis.com for more information.
Universal Constants, Variations and… Identities
Universal Constants, Variations and… Identities
7) pi is the identity of periodicity
8) e is the identity of change
9) phi the identity of proportion
Universal Constants and Variances
Universal Constants and Variances
#1 Awareness is primary and fundamental. (Substrate)
#2 All awareness is non-dual unless it is dual. (Duality)
#3 There is no inside without an outside nor outside without an inside. (Interiority/Exteriority)
#4 Duality is bounded, non-duality is boundless. (Boundary)
#5 Boundaries arise in a spectrum from diffuse to concise. (Crossing)
[More are coming soon in a new post…]
A few of those who have followed my posts have been asking for more information about my work. Towards that end, I’m going to start publishing my growing list of universal constants and variances. It is these constants and variances that form the foundation of my work.
There are about as many of them as there are stars in our universe (if you count the primary and derived together), so I don’t think I’ll run out of them! Most of them are self-explanatory, but if you have any questions, please don’t hesitate to ask in the appropriate thread. The numerical ordering is not yet important, as I’m still collecting and collating them as I discover them.
I have no tolerance for trolling or people who abuse others in my threads; especially on these threads about the constants and variances! So if you plan to wreak havoc here, you’ll get bumped real fast. I don’t mind criticism or skeptical opinions at all, but please be civil with everyone (including me).
See http://mathesis-universalis.com for more information.
Heurist.me (Carey G. Butler) 