Numeri periodici, tree e grafi

[bertanimauro]

From APUPA created by bertanimauro: bertanimauro/APUPA#8

Quando in un grafo entri in un ciclo il numero divente uncomputable, ossia periodico. Con la logica si possono rappresentare solo tree, non grafi

_Originally posted by @bertanimauro in #6 (comment)

Bisogna ristudiare Godel e i limiti della logica del secondo ordine

[bertanimauro]

Il sab 13 feb 2021, 09:07 Alex Shkotin [email protected] ha scritto:
Frank, John, Igor et all,

We definitely need just one formula of FOL not expressed in DL and hopefully speaking about cycle.
I agree that John's formula looks like it is not fully written.

That DL is less expressive than FOL was well known from the very DL beginning (80-th, sir:-).
Let's dig. And let me cite what I have found in my archive already [1]:
"Most DLs are fragments
of first-order logic that describe a domain using concepts
(unary predicates), roles (binary predicates), and individuals
(constants)."

"Thus, DLs cannot faithfully represent objects
with nontree structures since they cannot enforce the
existence of only non-tree-like models."

Isn't that about periodic numbers?

[bertanimauro]

Curva di Peano tende a una successione di 0 o 2.
traccia1/math/butzFillingCurve.pdf

[bertanimauro]

https://www.omg.org/spec/DOL/
http://www.informatik.uni-bremen.de/cofi/index.php/CASL_documents

[bertanimauro]

You have a concept with periodic number, real number is undefined.
Gli irrazionali sono reali. I reali sono irrazionali

[bertanimauro]

traccia2/math/cicliLogica.pdf

[bertanimauro]

Il mer 17 feb 2021, 14:57 Alex Shkotin [email protected] ha scritto:
Colleagues,

As we moved to arbitrary binary relations and directed cycles, let me give this definition, to agree with.
Let R is a binary relation over a set of nodes.
A set of k>1 nodes is a di-cycled means that
There is such an ordering n1..nk of this set that R(ni ni+1) for i from 1 to k-1, and R(nk n1).

If {a} such that R(a a) is di-cycled is an open question.

R(a a) is a piece of a definition of a concept. A di-cycle is a reflexive property with a series of abductions. These series of abduction is a piece of a definition of a concept

[bertanimauro]

https://en.m.wikipedia.org/wiki/Cycle_detection

[bertanimauro]

Il sab 13 mar 2021, 20:35 [email protected] ha scritto:

Now – here’s what’s happening for me with dimensionality. I started out years ago with a little Atari ST hierarchical outline processing program (called “HippoConcept” – which I guess meant “huge concept”) – and I built a glossary of epistemological terms – just a long list of every word and term I came across in epistemology, and I started working on building a consistent interpretation of these terms so that I could “construct” them from basic primitives. I was looking for building blocks. How are these composite abstractions constructed? I went over and over this list, and the entire structure kept coming down to one idea: dimension. All these words and concepts and terms, it seemed, could all be algebraically constructed from this one basic concept: dimension. It was a compound and recursive idea. “Everything (all terms from epistemology and the theory of categories and classes) is made out of dimensions, and dimensions themselves are made out of dimensions.”

This gave me a way to define “qualitative dimensionality” in terms of “quantitative dimensionality”. Qualities have internal dimensional structure – BUT – this is only true under stipulative definitions – what Barry Smith calls “fiat” definitions. So, I can take a very abstract qualitative term like “beauty” and give it a stipulative definition – what I mean and intend by the use of that word, in some particular context – and I can probably say exactly what I mean in quantitative definitions. This process creates a “hierarchical decomposition of the abstract term” in a cascade from broad and perhaps vague or ambiguous terms (“beauty means different things to different people”) to some specific measurements and boundaries – not because these definitions are true in all cases, but because in this particular case, I stipulate these boundaries to define my intended meaning with precision. More or less, I would say, people do this all the time in normal conversations when the need for increased clarity and precision emerges.

This led me to a concept I called “synthetic dimensionality” – more or less “a dimension where the values are defined in terms of dimensions”. I discuss that idea here: http://originresearch.com/sd/sd2.cfm

This is a complex idea with a lot of moving parts – but putting it simply, what I am doing today is exploring the idea that conceptual definition and structure can not only be useful defined in synthetic dimensionality – but that this entire composite structure – capable of defining “absolutely anything, to any desired degree of accuracy (including both scientific and “vague” ideas)” – can be derived from this single underlying and all-containing ontological structure I am now calling the “Closed Loop”. This includes the definition of numbers, arithmetic definitions, sets and classes, Boolean objects, and any kind of abstraction that can be represented symbolically.

Supposedly, this entire structure – “containing all concepts” (as per my definition of concepts) can integrate the entire range of analytic thinking – from the foundations of mathematics and continuum and real number line, to every kind of logic including Boolean algebra, unfolding this entire construction from this one “simple” form of the closed loop.

Seen in the flat plane, the closed loop is a top-down hierarchy or taxonomy defined across a range of levels from higher (more abstract) to lower (more specific). The “top” is infinite, the “bottom” is infinitesimal and continuous – and “everything is contained within those bounds”. Now, subject this form to “the twist” – which does not change any of its inherent internal structure -- and now suddenly the infinite and the infinitesimal form a single line, a single boundary, “containing everything”. This is totally mind-blowing. This entire structure – with no loss of topological integrity or precision of definition – collapses into a single line – a single boundary -- maybe even a single point.

So, the strategy for proof has to be something like

Show that you can construct every categorical definition (quantitative and qualitative) from synthetic dimensions without loss of precision
Map this system of definitions into the loop

I know this is pretty wild and incredibly ambitious. It’s a rather fabulous construction in abstract synthetic objects like matrix rows with internal cellular structures. And yes, the entire idea might explode into total nonsense. But something incremental keeps pushing it. Can we really lock all conceptual form – in any language – into a single algebraic interpretation? Is this what the ancients meant by “Logos”? This entire thing does feel to me a little like the Big Bang – “you can fold absolutely everything up into a straight line of zero width….” And maybe that is a point, not a line….

Is more interesting what you say. Imagine: you have a tree that is express by n rational number between [0,1). Than you have a synthetic function f that combine the n concept in complex concept with different structure. Also this function return m, with m more and more bigger than n, rational number between [0,1). And if the n rational numbers are rapresented by 3 digit, the m rational number are rapresented by 6,9,12... digits. Now with f you combine the n+m concept and you get one rational number q of 200 digit ( for example). It is what you want but now you can combine q with each of the element of m, and so on to infinity... The knowledge don't know end.
The point is the end and the start in a close loop. It is the circle with r = 1/2*pi that start by this point that has the full knowledge.
Regards
Mauro

[bertanimauro]

Hi Mauro,

About "subspace of cartesian products". I don't know about subspace, but subset of cartesian product is relation, not concept.
This is why for me it's better to use terminology from the formal theories domain, where we have predicates, and unary predicates "represent" concepts.
A cartesian product is an algebraic operation to construct something useful, especially relations.

Nice to see BNF. Do you use https://www.gnu.org/software/bison/?

Alex

сб, 13 мар. 2021 г. в 17:45, Mauro Bertani [email protected]:
Nascondi testo citato
Hi Bruce and Alex,
--- This is also related to the ontolog list thread "twisted closed loop" that also evolved in pierce-l list ---
For me concepts are a subspace of cartesian products. Point or set of points.

ranganathan3.png

a “unit of knowledge created by a unique combination of characteristics”

From the pictures above, the concept of a professor came from two dimensions: matter and rhetoric. All the nine points are the concept of a professor. And this complex concept is based on a tree of basic concepts (for example Professor chemistry or Professor Brillant). [That a tree is a line or a number between [0,1) can be read on Dewey Classification].

At the same time we can construct a subject heading more complex starting by a hierarchical tree, that is a series of points in a line, and transform all the subjects heading in other points in a line.
I oversimplify the concept. I use a R^3 world of concept. Can be related to the firstness, secondness, and thirdness

structure.png
For example this statement in similar BNF

CompoundSubject = BasicFacet OtherFacet+

OtherFacet = RoundPersonality{0,1} RoundMatter{0,1} RoundEnergy{0,1} RoundSpace{0,1} RoundTime{0,1}

RoundPersonality = LevelPME+

RoundMatter = LevelPME+

RoundEnergy = LevelPME+

LevelPME = Personality{0,1} Matter{0,1} Energy{0,1}

RoundSpace = LevelSpace+

LevelSpace = Space | FundamentalCategoryWithOutTSP*

RoundTime = LevelTime+

LevelTime = Time | FundamentalCategoryWithOutTSP*

FundamentalCategoryWithOutTSP = Matter | Energy | Personality

Matter = Material | Property

This compound subject became a single point in R^3 but this point is related to other points that represent Space, Time, Matter, Energy and Personality that are the basic hierarchical tree.
It's only an example for understanding the theory. You postulate a hierarchical tree and you get a sort R^3 space of concepts . It is sorted by level set.
So for me concepts are a set of points and a role to combine them.
Mauro

[bertanimauro]

https://groups.io/g/lawsofform/topic/infinite/84191980
i reali tra [0,1] hanno la stessa cardinalità degli interi [0,infinity] : X_0
to prove

[bertanimauro]

https://www.jstor.org/stable/1627114?seq=1#metadata_info_tab_contents

[bertanimauro]

automa, bnf, turing, uncomputable number

Let me recommend https://en.wikipedia.org/wiki/Flex_(lexical_analyser_generator) to get all tools to create formal languages:-)