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# Formalizing Concept

Novel way to stay formal and free, while speaking about daily life

## Intuition

My motivation behind creating this formalisms. is so that, we don't have to waste time, for priorly specifying what we are exactly speaking about and overall, just making our lives easier when it comes to speaking about all kinds of ideas, where quantifiers like never, all cant be applied without losing formality to the system.

## Anti-concept

All concepts, have one common anti-concept. It is their non-existance. Given that every concept which doesn't exist is non-distinguishable and non-referancable(by a referencer of higher specificity), there is exactly one anti-concept, that is common for all concepts.
The only rule is that concept must exist. It can be non-referencable, or non-distinguishable. Therefore there are infinitely many concepts, that exist. Concepts, exist regardless of time, which is not obvious if we were to speak about the idea of something, because there is a point in time, from which we can say that it exists(the point in time, denoting its' creation). Concept involves formal systems, that hold property of infinity-generating.

## Axioms of concept

Purpose of this formalism, was to extend the idea of thought, into scopes with different meaning providers.
• Existence of concepts, is entirely independent from their descriptors.

## Singular concept space

My initial idea, was to guarantee singularity of all concepts, through asserting, that all of them have exactly one common anti-concept, and later by involving some kind of ideas of semi-abstract(which I describe below) metric-spaces spanned along the total set of comparators. However I provide much handier realization of the idea, by usage of singleton Hilbert space, containing all concepts, that can ever exist. In other words, if concept is referencable(by the given descriptor) it belongs to that space.
Whatever can be ever spoken off as a concept already exists in a this space, which contains, all concepts and so definition in context of a concept, instead of transforming framing object, should be thought off as accessing already existing element in that space.
Concept descriptors are functions, acting on that space and can be thought of as querying functions in common computational sense. Argument to those functions, are contextualized descriptors.
Note: I mention my initial idea here, as I believe it's important to be able to see, how this idea evolved over time, what makes it easier to understand it, but also it allows reader to get the notion, of the purpose behind it, what is of superior value itself, but also, let's others to introduce suggestions, or corrections, to a final form, what is hard accomplish, when being presented a readymade formalism.

## Meaning provider

Given the idea above, it obviously raises, the idea that same description, might represent something very different depending on who interprets the message.
I use term meaning provider instead of what's commonly associated with word interpreter because word interpreter, associates descriptor, with specified schema, which is specifying how the meaning is inferred. Nevertheless. You can treat each other as aliases.

## Contextualized descriptor

Roughly speaking, its' descriptor transformed into an object object, which I call, contextualized descriptor, because it's a primer, to the context spanned along, with relevance, by value, up to its' anti-auto-morphism.
M(x) denotes translation(can be also thought of as convolution) of descriptor with meaning of x, provided by interpreter p, into singleton concept Hilbert space, which is queryable space, common for all, meaning providers.
$M_p(x) \rightarrow mpx$
$H_c(mpx) \rightarrow res | OT:C_s$
$res | OT:C_s$
means that type of the result is a concept, pointed to by sentence s

#### Example

Let s be a concept descriptor. We can assume it's a string, in the computational sense.
Let Mx and My My denote, two meaning providers. Each of them, maps sentence s, into sentences mxs and mys, which link to concepts that are associated with given descriptor(in this case sentence s) accordingly with the associated meaning.

## Identity of concepts

As mentioned, concepts exists, in singular Hilbert Space, accessible only by creating queries, pointing, to "regions"(or other varieties) that are relevant concept representations. Therefore identity of concepts, can be inferred, by simple comparison.
• With no loss of generality, descriptors can be represented as functions on the earlier mentioned Hilbert Space, which given sentences (mxs, mys)) return, appropriate association, that is simply. Hc(mxs), Hc(mys). Where Hc represents, the singleton concept Hilbert space
Declarative definition of concepts identity
identity(meaning(x)):-(x)| meaning => [mxa, mxb] | ð = 0
To be interpreted as: Concept preserves its identity for specification x, when having null difference between contextualized descriptors mxa and mxb.
It's dual to the expression ð(Hc(ma), Hc(mb)) = 0

## Semi-Abstracts

#### Semi abstract X

Substitute X with any formal object which properties, you're interested in and here you go. You just created entirely new, formal object, that supports all those properties of X, which you need and none of those that you don't need. I call it the semi-abstract trick, it's a kind of a new way to stay formal and free
😄

### Semi-abstract distance metric.

So the only rule is that, it has to be more of a distance metric, than no-distance metric. This is very useful, for describing and comparing objects, from completely disjoint contexts/scopes and omitting tones of irrelevant bullshit, that current understanding of formal concepts requires. It's one of my favorite spells in fighting current mathematical establishment
😄
.

## Concepts, with realizers.

Concept Descriptor
Realizer
∂: Semi-abstract distance metric
A program, that provides an answer to the question: 'Whats' the meaning of life.
print(42)
∂(c, r)
We say that object r is a perfect realizer of formal concept sc, with semi-abstract distance metric ∂m, up to the degree of its formality.
An example
sc = Concept("A program, that provides an answer to the question: What's meaning of life?")
r = 'The meaning of life is: {}'.format(get_answer(sc.question))
# This example is phenomental, because it illustrates, insanely growing, complexity, when trying to clearly define all the relevant ideas
∂(sc, r)? -> val
So in the first place it raises the question, what the hell is '∂' and the whole idea is that its' whatever you want it to be and it's formal up to the degree, of formality of your specification and the meaning as long, as you can provide non contradictory meaning, everything is cool.

## Formality

Formality is just a property entirely dependent, on the knowledge of the meaning provider. This should illustrate, that it's absolutely easy for what's said to be formal object, to contain errors disqualifying them entirely, from evaluation, which conditions its' formality. While still being able to stay a perfect realizer of this concept, to another meaning provider.
Example: Syntax errors
So I thought it's pretty problematic, because the idea of formality doesn't relate anything sensible anymore. And then I reminded, myself about my black magic and casted another semi-abstract distance metric, which this time happens to produce, all the distance metrics, which perfectly describe, all properties, of the mentioned system, as if by creating a gradient and corresponding mapping, to the annotated(labeled) relevance metric which is producing interpretable state-transition matrix, from which all information, considered relevant, can be directly interpreted.
I hope it's the last time when I write such formal gibberish. Frameworks presented here, are capable of representing, all mathematical ideas, from which they derive, because they usually form a superset over, what they derive from, as composed, by non-assertive property selection, non-assertive beyond those, which itself are the desired property.
Therefore, generally speaking, they form loss-free formal system capable of describing all ideas, which, well, they are capable of describing ... [and up to its' anti-auto-morphism].