What do all things have in common?

Understanding

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 \pi 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. 

Question on Quora


Knowledge Representation – Holographic Heart Torus

Holographic Heart Torus

Holographic Heart Torus by Ryan Cameron on YouTube


Which questions does Category Theory help us answer?

Category 02

Another chapter in my attempt to help break the ‘spell’ of the category theoretical ‘ontologicisation’ of our world.

This may seem to many as a purely academic question, but we all need to realise that all of what we consider a modern way of thinking rests upon ‘mental technologies’ such as Category Theory.

Academics are literally taking the ‘heart’ out of how our world is being defined!
If we don’t pay attention, humanity will continue losing its way.

Category theory is a wonderful and powerful tool; nevertheless category theory, with all of its utility, is purely ontological. It can masterfully answer questions such as ‘Who?’, ‘What?’, and ‘How?’.

Category 01

However; it is regretfully inadequate to form a comprehensive representation of knowledge, for it lacks expression of epistemological value, which are the very reasons for is use. Epistemology is about answering the questions of ‘Why?’, ‘What does it mean?’, ‘What is my purpose?’,…
Answers to questions of this kind are implicitly supplied by us during our consumption of the utility afforded by category theory. We often are so beguiled by this power of categorical expression that we don’t realise that is we ourselves who bring the ‘missing elements’ to what it offers as an expression of knowledge.
It does a wonderful job with exteriority (ontology), but cannot sufficiently describe nor comprehensively access interiority (epistemology). Therefore, it has limited metaphysical value with respect to philosophy in general.

Interiority 08

Philosophies of mind, of language, or of learning are not comprehensive using only category theoretical tools.
Categorical structures are highly portable, but they can describe/express only part of what is there. There are structures, dynamics, and resonance that the ontology and functionalism in category theory completely turns a blind eye to.
More general than category theory is knowledge representation. It includes and surpasses category theory in many areas, both in scope and depth, but in particular: knowledge representation includes not just the ontological aspects of what we know, it goes further to describe the epistemological as well.
The qualities of Truth, Goodness, Beauty, Clarity,… can be defined and identified within a knowledge representation if the representation is not restricted to ontology. When category theory is used for the purpose of defining qualia, the objects must first be ontologised and functionally reduced. Trying to grasp them with tools restricted to category theory (or even semiotics) is like grasping into thin air.

 

Category theory, although very powerful, is no match for the challenge of a complete representation of knowledge. Category theory will tell you how to tie your shoes, but it can’t tell you why you are motivated to do so.


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.


Limits of Category Theory and Semiotics

Category Theory 01

They are wonderful tools to explain much of our world, but lack ‘The Right Stuff’ to handle the metaphysical underpinnings of anything near a Philosophy of Mind, Philosophy of Language , or a Philosophy of Learning.

This is, because Category Theory specialises on roughly half of the Noosphere. It does a wonderful job on exteriority, but cannot sufficiently describe nor comprehensively access interiority.

Interiority Exteriority

Therefore, as is the case with Semiotics, has limited metaphysical value with respect to philosophy in general.

Semiotics 2

Semiotics

For example: philosophies of mind, language, or learning are not possible using only category theoretical tools and/or semiotics.

Here is an example of one attempt which fails in this regard: http://nickrossiter.org.uk/proce…

(and here: VisualizationFoundationsIEEE)

Here are two problems (of many) in the paper:

4.4.2 Knowledge is the Terminal Object of Visualisation states:

“The ultimate purpose of the visualisation process is to gain Knowledge of the original System. When this succeeds (when the diagram commutes) then the result is a ‘truth’ relationship between the Knowledge and the System. When this process breaks down and we fail to deduce correct conclusions then the diagram does not commute.”

I want to also comment on Figure 3 (which also exposes missing or false premises in the paper), but I will wait until I have discussed the assertions in the quote above which the authors of this paper reference, accept, and wish to justify/confirm.

1) The purpose of a representation is NOT to gain knowledge; rather, to express knowledge. Also, truth has nothing to do with knowledge except when that value is imposed upon it for some purpose. Truth value is a value that knowledge may or not ‘attend’ (participate in).

1a) The ‘truth value’ of the System (‘system’ is a false paradigm [later, perhaps] and a term that I also vehemently disagree with) does not always enter into the ‘dialogue’ between any knowledge that is represented and the observer interpreting that knowledge.

2) The interpretation of a representation is not to “deduce correct conclusions”; rather, to understand the meaning (semantics and epistemology) of what is represented. ‘Correct’ understanding is not exclusive to understanding nor is it necessary or sufficient for understanding a representation, because that understanding finds expression in the observer.

2a) ‘Correct’, as used in this paragraph, is coming from the outside (via the choice of which data [see Fig. 3] is represented to the observer) and may have no correspondence (hence may never ever commute) whatever to what that term means for the observer.

The authors are only talking about ontologies. That is a contrived and provincial look at the subject they are supposing to examine.

There may (and usually are) artefacts inherent in any collection and collation of data. The observer is forced to make ‘right’ (‘correct’) conclusions from that data which those who collected it have ‘seeded’ (tainted) with their own volition.

‘System’ (systematising) anything is Reductionism. This disqualifies the procedure at its outset.

They are proving essentially that manipulation leads to a ‘correct’ (their chosen version) representation of a ‘truth’ value.

I could tie my shoelaces into some kind of knot and think it were a ‘correct’ way to do so if the arrows indicate this. This is why paying too much attention to a navigation system can have one finding themselves at the bottom of a river!

The paper contains assumptions that are overlooked and terms that are never adequately defined! How can you name variables without defining their meaning? They then serve no purpose and must be removed from domain of discourse.

Categorical structures are highly portable, but they can describe/express only part of what is there. There are structure, dynamics, and resonance that ontology and functionalism completely turns a blind eye to.

The qualities of Truth, Goodness, Beauty, Clarity,… (even Falsehood, Badness, Ugliness, Obscurity,…) can be defined and identified within a knowledge representation if the representation is not restricted to ontology alone.

In order to express these qualities in semiotics and category theory, they must first be ontologised funtionally (reduced). Trying to grasp them with tools restricted to semiotics and category theory is like grasping into thin air.

That is actually the point I’m trying to make. Category Theory, and even Semiotics, each have their utility, but they are no match for the challenge of a complete representation of knowledge.


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?

hermandadblanca_universo-mente-fractal-geometria-sagrada

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