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.
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.
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.
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.
Really! Nothing Is ‘Real’
Another example of the ‘neo-snake-oil salesmen’ peddling you trendy pabulum and neo-Babylon confusion. My current project Mathesis Universalis http://mathesis-universalis.com will bring an end to this menagerie of nonsense and subtle programming.
I could write a book on this.
Don’t believe everything put forward in this… set of perspectives. This is a work in process so stay tuned… updates are coming very shortly.
I’m happy that he allows for more than 5 senses as this is a common error made by science and philosophy up to this time. I’ve taken issue with it elsewhere numerous times. Also I’m pleased that he is allowing for Neuroplasticity (Dr. Jeffrey M. Schwartz http://www.jeffreymschwartz.com/ has been leading this new model for over 10 years.)
Up to @04:27 I take issue with two important assumptions he makes:
1) That sensory information is the only way we ‘register’ reality.
2) He is a physicalist pure through. If he can’t measure and quantify it, then it doesn’t exist for him… This leads to what is known as causal ambiguity (among other things).
http://psychologydictionary.org/causal-ambiguity/
@04:57– He says that memory is stored all over the brain. This is incorrect. The effects of the phenomena of memory are manifested in various areas of the brain. There is no sufficient and necessary proof that memory is stored there! They PRESUME it to be stored there, because they can not allow or imagine anything non-physical being able to store any kind of knowledge.
@05:09– “How many memories can you fit inside your head? What is the storage capacity of the human brain?” he asks.
In addition to the presumption that memories are stored there, he then ignores the capacity of other areas of the body to imprint the effects of memory: the digestive tract, the endocrine and immune ‘systems’,… even to cell membranes (in cases of addiction, for example)!!!
@05:23– “But given the amount of neurons in the human brain involved with memory…” (the first presumption that memories are stored there) “and the number of connections a single neuron can make…” (he’s turning this whole perspective on memory into a numerical problem!) which is reductionism.
@05:27– He then refers to the work of Paul Reber, professor of psychology at Northwestern University who explained his ‘research’ into answering that question. here’s the link. I will break that further stream of presumptions down next.
http://www.scientificamerican.com/article/what-is-the-memory-capacity/
(the question is asked about middle of the 1st page of the article which contains 2 pages)
Paul Reber makes a joke and then says:
“The human brain consists of about one billion neurons. Each neuron forms about 1,000 connections to other neurons, amounting to more than a trillion connections. If each neuron could only help store a single memory, running out of space would be a problem. You might have only a few gigabytes of storage space, similar to the space in an iPod or a USB flash drive.”
“Yet neurons combine so that each one helps with many memories at a time, exponentially increasing the brain’s memory storage capacity to something closer to around 2.5 petabytes (or a million gigabytes). For comparison, if your brain worked like a digital video recorder in a television, 2.5 petabytes would be enough to hold three million hours of TV shows. You would have to leave the TV running continuously for more than 300 years to use up all that storage.”
These presumptions and observations are full of ambiguity and guesswork. Given that we are not reading a thesis on the subject, we can allow him a little slack, but even the conclusions he has arrived at are nothing substantial. More below as he reveals his lack of knowledge next.
“The brain’s exact storage capacity for memories is difficult to calculate. First, we do not know how to measure the size of a memory. Second, certain memories involve more details and thus take up more space; other memories are forgotten and thus free up space. Additionally, some information is just not worth remembering in the first place.”
He not only doesn’t know to measure memories (which he admits), he cannot even tell you what they are precisely! He offers here also no reason for us to believe that memory is reducible to information!
@05:50– “The world is real… right?” (I almost don’t want to know what’s coming next!)
And then it really gets wild…
@05:59– With his: “How do you know?” question he begins to question the existence of rocket scientists. He moves to Sun centric ideas (we’ve heard this one before) to show how wrong humanity has been in the past.
He seems to ignore or not be aware of the fact that that many pre-science explorers as far back as ancient Alexandria knew better and had documented this idea as being false. This ‘error’ of humanity reveals more about dogma of a church/religion/tradition than of humanity/reality as it truly is.
@06:29– “Do we… or will we ever know true reality?” is for him the next question to ask and then offers us to accept the possibility that we may only know what is approximately true.
@06:37 “Discovering more and more useful theories every day, but never actually reaching true objective actual reality.”
This question is based upon so much imprecision, ignorance, and arrogance that it isn’t even useful!
First of all: we cannot know “true objective actual reality” in all of its ‘essence’, because we must form a perspective around that which we observe in order to ‘see’ anything meaningful. As soon as a perspective comes into ‘being’, we lose objectivity. (ignorance, assumption)
He doesn’t define what ‘reality’ for him is. (imprecision)
He doesn’t explain what the difference between ‘true’ and ‘actual’ might be. (imprecision, assumption)
Theories are NOT discovered, rather created (implicit arrogance). They can only be discovered if they were already known/formulated at some time.
Also; theories do not stand on their own; rather, they depend upon continued affirmation by being questioned for as long as they exist. We DO NOT store knowledge in our answers; rather, in our questions.
[continued…]
A Holon’s Topology, Morphology, and Dynamics (2a)
A Holon’s Topology, Morphology, and Dynamics (2a)
This is the second video of a large series and the very first video in a mini-series about holons. In this series I will be building the vocabulary of holons which in turn will be used in my knowledge representations.
The video following this one will go into greater detail describing what you see here and will be adding more to the vocabulary.
This is the second video of a large series and the very first video in a mini-series about holons. In this series I will be building the vocabulary of holons which in turn will be used in my knowledge representations.
Ontology: Compelling and ‘Rich’
Ontologies are surfaces… even if ‘rich’. (link)
Ontology: Compelling and ‘Rich’
They are only surfaces, but they seem to provide you with depth.
This exquisite video shows how the representation of knowledge is ripe for a revolution. I’ve written about this in depth in other places so I won’t bore you with the details here unless you ask me in the comments below.
Stay tuned! I’m behind in my schedule (work load), but I’m getting very close just the same. I will publish here and elsewhere.
I’m going to use this video (and others like it) to explain why ontologies are not sufficient to represent knowledge.
Soon everyone will acknowledge this fact and claim they’ve been saying it all along! (In spite of the many thousands of papers and books obsessively claiming the opposite!!!) They do not know that how dangerous that claim is going to be. Our future will be equipped with the ability to determine if such claims are true or not. That’s some of the reason I do what I do.
#KnowledgeRepresentation #BigData #Semantics #Metaphysics #Ontology
#Knowledge #Wisdom #Understanding #Insight #Learning #MathesisUniversalis #MathesisGeneralis #PhilosophiaUniversalis #PhilosophiaGeneralis #ScientiaUniversalis #ScientiaGeneralis
Nascent Mind, Prescient Knowledge: Instinct And Envisioning
It’s at this juncture that concepts begin to coalesce. Within this ‘Holy of Holies’ concepts are born and form/generate their associated continuums. It’s like watching the blue wisping stars newly born in the constellation of Pleiades.
https://en.wikipedia.org/wiki/Pleiades
This ‘event horizon’ is so crucial to understanding and participating in mind; yet those who should know better simply ignore or overlook it.
Tesla’s statement here rings so true that it simply boggles my mind and confirms that Tesla was ‘tuned into it.’
He clearly exhibited these awarenesses on several occasions. He was able to envision many ideas to their completion before constructing them; and his instinct for somehow ‘knowing’ (flashes of insight) what to do next and where to go with an idea were so profound that it often overwhelmed and incapacitated him. His mind was so fertile that layers of creative impulses were being maintained concurrently.
Next to Socrates there are very few who inspire me. Tesla is one of those few.
Precursors Of Knowledge
Precursors Of Knowledge
Fractal fields provide a nice framework in which to think about knowledge. They are not all we need for precision, but they are helpful in a generic way. I’ll be posting more on them as the knowledge representations are published, because there are many ‘gaps to fill’ to show how these relate to knowledge.
More sources:
https://www.youtube.com/watch?v=2nTLI89vdzg
https://www.youtube.com/watch?v=1ZVNIZGw4X0
https://www.youtube.com/watch?v=Yp4ogF2w13M
https://www.youtube.com/watch?v=8UPD2_gEjvM
https://www.youtube.com/watch?v=ArZLXHVVV5I
Information Visualization Is Not Knowledge Representation
(Lynda.com – Overview of Data Visualization)
Information Visualization Is Not Knowledge Representation
This great video from Lynda.com shows how the processing language/interpreter is great for modeling information.
With such a multitude of interesting ways to model data, we find it hard to resist the temptation to call this knowledge, but it’s not!
All of the wonderful representations here still require us to interpret their meaning!
What if there were a way to present knowledge in which our own understanding is not required to interpret them? What if our understanding of what we have presented to us becomes part of the presentation itself, and in fact, influences what we take from that representation?
We obviously need knowledge representation that can provide their meaning on their own for only they can provide a true understanding of their inherent structure and dynamics.
You see real understanding is the personalization of knowledge into your own mind. If your mind cannot dialog with that knowledge, it’s not really yours and if your mind does all the work, it’s only information.
Heurist.me (Carey G. Butler) 