Thursday, August 08, 2013

An 'Ultra-Simple' Example of Division-by-Zero 'Semantification'; of 'Full Zero' vs. 'Empty Zero'; of 'Metafinite', 'Resonance Singularity'; of 'Meta-Monadization', and of 'Onto-Dynamasis'.

 Full Title:  

An 'Ultra-Simple' Example of Division-by-Zero 'Semantification'; of 'Metafinite', 'Resonance Singularity'; of '[Ideo-]Meta-Monadization'; and of '[Ideo-]Onto-Dynamasis'.

Dear Readers,

In an earlier blog-entry here, I wrote as follows regarding the ‘metafinite singularity meta-dynamics’ of ‘ontological revolutions’, i.e., of ‘meta-evolutions’ --

"crisis" dialectic -- a 'meta-evolutionary' '''crisis''', or 'ontological revolution', forming the boundary between one epoch of '''evolution''' and the next, and involving the irruption of "new ontology" -- of new kinds of being -- is central to the F.E.D. account of [physio- and psycho]historical dialectic.

F.E.D., such ‘ontological revolutions’, or 'metafinite singularities' [not fictitious, quantitatively infinite singularities, a la 'unqualified', "purely-quantitative" nonlinear differential equations, when finite-time divisions by zero arise therein, but realistic, finite singularities, involving the irruption of qualitative change in the form of new ontology] come in two basic types, that we call 'resonance singularities', and 'conversion singularities'.
  • 'Resonance singularities' [i.e., 'self-conversion singularities', or 'auto-conversion singularities'] are exemplified, in the 'physio-dialectics' of pre-human and extra-human Nature, by "cosmological nucleosynthesis", induced when early pre-nuclear "particles" cross a certain energy-density threshold in the early cosmos -- when physical-spatially concentrated pre-nuclear "particles", e.g., "gluons and "quarks", irrupt into the formation of the new, cosmologically unprecedented ontology of the first-ever atomic nuclei, e.g., Helium ions, He++;
  • 'Resonance singularities' are exemplified, in the 'psycheo-dialectics' of human Nature, by the irruption of previously unprecedented new 'socio-ontology' in the form of the 'meme-etic', psychophysical existence of Capital-value as a new social relation of production [Marx], from a sufficient social 'densification' and concentration of the Money-value-mediated circulation of Commodity-value, which arises when a certain threshold in the growth of the social forces of production is crossed, and which begins the '''formal subsumption''', by the Capital 'socio-ontology', of its predecessor, Money and Commodity 'socio-ontology'.
  • 'Resonance singularities' characterize the 'self-«aufheben»' processes of the generation, by a natural '''eventity''', of its own supplementary "other", e.g., via a process of 'self-«aufheben» meta-unit-ization', such as I described in the summary that you requested.

  • 'Conversion singularities' [i.e., 'hetero-conversion singularities', or 'allo-conversion singularities']are exemplified, in the 'physio-dialectics' of pre-human and extra-human Nature, by the crises of "stellar nucleosynthesis".  A "star" is a "hybrid" formation -- an ontological conversion-formation -- that converts the ontology of stellar-core pre-nuclear "particles", e.g., protons ["Hydrogen ions", H+], into the new, next 'meta-ontology' of atomic nuclei, i.e., of Helium ions ["alpha particles", He++]. Helium is the "ash" -- the "entropy" -- of stellar Hydrogen fusion. When the Hydrogen in the core of a "main sequence" star is exhausted -- essentially completely converted into Helium -- then stellar nucleosynthesis in the stellar core ceases, and the star resumes catastrophic self-gravitational self-implosion, until the density-temperature of the core Helium becomes sufficient for the "Helium flash" to ignite Helium fusion in the stellar core -- a next ontological 'stellar revolution' in "stellar evolution", in which the star drives itself off of the stellar "main sequence", in a 'stellar meta-evolution', and drastically changes its dynamics: the ash, the entropy, turns into fuel, into a new free-energy resource, and the fusion-"burning" of Helium yields the new ontology of yet-higher species of atomic nuclei.
  • 'Conversion singularities' are exemplified, in the 'psycheo-dialectics' of human Nature -- of human-natural history -- by the transition from the '''formal subsumption''', by the Capital-value social-relation-of-production 'socio-ontology', of the previously-emerged 'socio-ontology' of the Money-value relation-of-production, and of the Commodity-value relation-of-production, to the '''real subsumption''', by the 'socio-ontological' conversion of more and more Money-values and Commodity-values into "Money-Capital" and "Commodity-Capital" [Marx, Capital, volume II.].
  • 'Conversion singularities' characterize the 'other-«aufheben»', «aufheben»-conversion of some of the remaining ontology / 'onto-mass' of the predecessor '''eventity''' into the ontology of the 'supplementary other' that this '''eventity''' has generated from out of itself, inside a formation which instantiates the third, 'hybrid of [it]self and its self-generated other' stage of the three stages that I described earlier, above.

My purpose in the present blog-entry is to illustrate the crucial F.E.D. concept of ‘metafinite singularity’ by an ultra-simple example, by an example from the human-cognitive domain, and by one which unifies the Seldonian concepts of ‘densification-driven meta-finite resonance singularity’, of ‘meta-monadization’, of ‘[ideo-]onto-dynamasis’, and of divisions-by-zero as signifying a 2-D, cognitive/visual version of the crisis-threshold of 3-D critical density.

The domain of discourse for this example is a system of ‘numeralic’ syntactico-semantical ‘ideo- ontology’ -- indeed, a system of written arithmetic -- that has ancient roots:  the system of the “Roman Numerals”. 

Broadly similar, systematic, presentational ‘ideo-dynamics’ and ‘ideo-meta-dynamics’ obtain for other ancient systems of numerical notation, prior to the Human-Phenomically revolutionary advent of the Hindu-Arabic system of ‘numeralic’ notation.

Suppose that the ancient ‘ideo-architect(s)’ of this system held a view that the semantic obscurity per numeral to visual ‘‘‘uptake’’’ by their users grows with juxtapositional repetition -- i.e., with 2-D textual, and cognitive/visual, ‘‘‘density’’’ -- of such numerals, and becomes ‘‘‘critical’’’ after three such repetitions, i.e., with four consecutive such repetitions.

 More specifically, such obscurity is least with a single, non-repeated occurrence of a numeral, e.g., ‘I’, increases by its single consecutive, juxtapositional repetition, e.g., ‘II’, worsens further with its double repetition, e.g., ‘III’, rises to a ‘‘‘critical’’’ level at its triple repetition, e.g., ‘IIII’, and becomes ‘‘‘super-critical’’’ at quadruple repetition or more, e.g., ‘IIIII...’.

We can form a crude metric function for this alleged human-cognitive phenomenon as follows [using the sign '=' to signify 'is equal to by definition']: 

m5( . )  =  obscurity per numeral of expression, using the parameter-value 5   =   

( 5/(5 # of occurrences of same numeral) ) x [ o°/n° ].

Note that we are ‘‘‘qualifying’’’ the quantity enclosed in ‘curve parentheses’, ( ... ), by a ratio of ‘metrical qualifiers’ -- of ‘‘‘metrical units’’’ -- enclosed in “square brackets”, [ o°/n° ], by way of ‘non-amalgamatively’ multiplying the former by the latter. 

This ‘qualification’ will be important to determining the ‘‘‘super-critical’’’ value of this metric. 

The “degree sign” superscripts, ‘°’, that form part of each of the metrical unit qualifiers, o° and  n°, are to signify that these ‘metrical unit qualifiers’ are quantifiable qualifiers’, i.e., that these units can be expressed in multiples and in fractions thereof, e.g., by the quantitative value of their ‘metrical quantifier’, or ‘‘‘coefficient’’’, ‘( 5/(5 # of occurrences of same numeral) )’.

Let us therefore see what happens when we apply this metric to various 2-D textual, cognitive/- visual repetition ‘densities’ of the Roman Numeral ‘I’ --

m5( I )  =      ( 5/(5 1) ) x [ o°/n° ]  =  5o°/4n°  =  (1 1/4)[ o°/n° ];

m5( II )  =     ( 5/(5 2) ) x [ o°/n° ]  =  5o°/3n°  =  (1 2/3)[ o°/n° ];

m5( III )  =   ( 5/(5 3) ) x [ o°/n° ]  =  5o°/2n°  =  (2 1/2)[ o°/n° ];

m5( IIII )  ( 5/(5 4) ) x [ o°/n° ]  =  5o°/1n°  =  (5)[ o°/n° ];

m5( IIIII )  = ( 5/(5 5) ) x [ o°/n° ]  =  5o°/0n°  =  5o°/. =   .

-- a monotonically mounting amount of ‘obscurity per numeral’ as numeral occurrence count mounts.

Note:  a zero-division singularity arises in our metric as of the 5 count expressed as ‘IIIII.

That final evaluation above is based upon the axiom of the Seldonian arithmetic for “dimensional analysis” -- the Seldonian arithmetic of quantifiable metrical qualifiers, of quantifiable metrical units, called ‘the Mu arithmetic’, Rm, the seventh of the dialectical arithmetics in the Seldonian dialectical progression of axioms-systems of dialectical arithmetic -- that assigns the value ‘full zero’, . -- to cases involving the multiplication of a metrical unit qualifier by that ordinary Real number which Seldon characterizes as ‘empty zero’, ‘0’, e.g., in this case, 0 x n°  =  .. 

Seldon characterizes the ‘full zero’ value . as the ‘universally dominant’ value, so that it takes over the entirety of any expression in which it occurs, and converts that entire expression into itself alone, whether . occurs in the numerator of that expression, or in its denominator, or in both. 

Seldon describes . as a 'quanto-qualitative, hybrid zero', as an ‘existential zero’, as an ‘ontology-specific zero’, or as a determinate, specific sign of ‘no-longer-existence’, or of ‘non-existence’, and characterizes its computational behavior as ‘Langolier-like’, as ‘Hridayamic’, or ‘all absorbing’, and as ‘black-hole-like’. 

The value . should not be conceived as a sign of general non-existence, or of generalized absence -- i.e., of absolute nothingness -- but only as signifying the absence of a particular, determinate [here ‘ideo-’]ontology, e.g., the absence of any intelligibility whatsoever for expressions of the form ‘IIIII...’, e.g., leading to the conversion of such expressions into a different, maximally-intelligible [‘ideo-’]ontology, hence also leading to the absence of the numerals that had formed the earlier -- ‘singularly obscure’ -- expressions.

So, per our crude metric, at five consecutive occurrences of the same numeral, cognitive ‘‘‘uptake’’’ of the meaning of the multi-numeral expression becomes non-existent, per the beliefs that we are supposing for the ancient ‘ideo-architect(s)’ of the Roman Numerals system.

In some cases within the system of the Roman Numerals, even a single repetition -- two occurrences of the same numeral -- seem to be regarded as unintelligible, so that our metric function there becomes --

m2( . )  =  obscurity per numeral of expression, using the parameter-value 2   =   

( 2/(2 # of occurrences of same numeral) ) x [ o°/n° ].

New ‘‘‘[ideo-]ontology’’’ ‘irrupts’ at the ‘‘‘singularity’’’:  ‘[ideo-]onto-dynamasis’ manifests.

I.e., ‘meta-monadization’ ‘irrupts’ at that juncture, e.g., Roman Numeral ‘V’ supersedes ‘IIIII’.

Let us, then, next, summarize the whole sequence of ‘ideo-onto-dynamical metafinite resonance singularities’ which characterizes the Roman Numerals system of ‘numeralic memes’, i.e., of ‘numeralic ideo-ontology’, as standard counting processes, expressed via Roman numerals, proceed --

Counting by I units:    .....-->  II   --> III    -->  IIII    -->  IIIII   |   V
|Metric of Obscurity|:  .5/4.............5/3................5/2.....................5/1.......................5/0....................5/4    

Counting by V units:   .....--> VV   |   X
|Metric of Obscurity|:  2/1............2/0...................2/1            

Counting by X units:   .....-->    XX  -->   XXX -->    XXXX -->    XXXXX   |   L
|Metric of Obscurity|:  5/4..................5/3.......................5/2...........................5/1...............................5/0.................................5/4  

Counting by L units:   ....L --> LL   |   C
|Metric of Obscurity|:  2/1...........2/0.................2/1            

Counting by C units:   .....C --> CC   --> CCC   --> CCC   --> CCCCC   |   D
|Metric of Obscurity|:  5/4............5/3.......................5/2.......................5/1.........................5/0................................5/4      

Counting by D units:   .....D --> DD   |   M
|Metric of Obscurity|:  2/1............2/0...................2/1            

In each case above, a ‘division-by-zero singularity’ in the “purely-quantitative” version -- the ‘quantifier-only version’ -- of our metric of ‘symbolic obscurity’, whose “purely-quantitative” value at that juncture may thus be characterized as “undefined”, as “indeterminate”, or as infinite”, is associated with a ‘metafinite [ideo-]onto-dynamasis’ at the level of the numerals, in the form of the occurrence of an ‘ideo-ontologically’ new numeral.  

That new, single numeral restores the minimal metrical reading for ‘numeralic obscurity’ by way of a ‘meta-monadization’ of the numeral used in those counting expressions that come prior to that “singularity”.

In the expressions above, the ‘-->’ horizontal arrow sign is used to represent the ‘‘‘evolutionary’’’, and merely ‘‘‘dynamical’’’ stages of the Roman-Numerals-represented counting process, whereas the ‘­|’ vertical arrow sign is used to represent the ‘‘‘revolutionary’’’, or ‘meta-evolutionary’, and also ‘meta-dynamical’, ‘‘‘irruptive’’’ transitions to a new ‘qualo-fractal’ scale / level of numeralic representation.

The ‘meta-monadizations’ implicit in the expressions above can be defined as follows --

I’ is the ‘«arché-monad» -- i.e., the ‘«arché»-unit -- of the Roman Numerals system.

V’ is a ‘meta-I monad’, made up out of a homogeneous multiplicity of exactly 5I’ monads.

X’ is a ‘meta-V monad’, made up out of a homogeneous multiplicity of exactly 2V’ monads.

L’ is a ‘meta-X monad’, made up out of a homogeneous multiplicity of exactly 5X’ monads.

C’ is a ‘meta-L monad’, made up out of a homogeneous multiplicity of exactly 2L’ monads.

D’ is a ‘meta-C monad’, made up out of a homogeneous multiplicity of exactly 5C’ monads.

M’ is a ‘meta-D monad’, made up out of a homogeneous multiplicity of exactly 2D’ monads.


  1. How will be the following?:

    The division by zero is uniquely and reasonably determined as 1/0=0/0=z/0=0 in the natural extensions of fractions. We have to change our basic ideas for our space and world:
    Mathematical Analysis, Probability and
    Applications -Plenary Lectures: Isaac 2015, Macau, China.
    (Springer Proceedings in Mathematics and Statistics, Vol. 177)
    Sep. 2016 305 pp.
    Paper:Division by Zero z/0 = 0 in Euclidean Spaces
    Dear Prof. Hiroshi Michiwaki, Hiroshi Okumura and Saburou Saitoh
    With reference to above, The Editor-in-Chief IJMC (Prof. Haydar Akca) accepted the your paper after getting positive and supporting respond from the reviewer.
    Now, we inform you that your paper is accepted for next issue of International Journal of Mathematics and Computation 9 Vol. 28; Issue 1, 2017),

  2. Please lokk the papers:

    The division by zero is uniquely and reasonably determined as 1/0=0/0=z/0=0 in the natural extensions of fractions. We have to change our basic ideas for our space and world

    Division by Zero z/0 = 0 in Euclidean Spaces
    Hiroshi Michiwaki, Hiroshi Okumura and Saburou Saitoh
    International Journal of Mathematics and Computation Vol. 28(2017); Issue 1, 2017), 1

    Relations of 0 and infinity
    Hiroshi Okumura, Saburou Saitoh and Tsutomu Matsuura:…/Camera%20ready%20manuscript_JTSS_A…

    1. Dear Dr. Saitoh,

      Many thanks for your links to the papers, of your co-authorship, on division by zero.

      Via those links, we have downloaded copies of the papers entitled --

      "Matrices and Division by Zero z/0 = 0",

      "New Meanings of the Division by Zero and Interpretations on 100/0 = 0 and on 0/0 = 0"


      "Announcement 326: The divisions by zero z/0 = 0 - its impact to human beings through education and research".

      The Foundation Encyclopedia Dialectica [F.E.D.] research collective is now scheduled to collectively read aloud and study these papers.

      As an expression of our gratitude for your calling these papers to our attention -- these papers had somehow [we will research and determine how] escaped our regular automatic scans of the internet on this topic -- we would be happy to mail to you, as a gift from us to you, a copy of volume 2 of our treatise "A Dialectical Theory of Everything", which addresses the concept of "full zero" in the context of dynamical singularities. If this gesture of gratitude is to your liking, simply e-mail me a mailing address that will get this book to you, via .

      Our most extensive on-line accessible treatment of the concept of "full zero" is accessible via the following links --,E._D._Notation_Definition,'Full_Zero'_Sign,Sheet_1_of_7,19APR2015.jpg,E._D._Notation_Definition,'Full_Zero'_Sign,Sheet_2_of_7,19APR2015.jpg,E._D._Notation_Definition,'Full_Zero'_Sign,Sheet_3_of_7,19APR2015.jpg,E._D._Notation_Definition,'Full_Zero'_Sign,Sheet_4_of_7,19APR2015.jpg,E._D._Notation_Definition,'Full_Zero'_Sign,Sheet_5_of_7,19APR2015.jpg,E._D._Notation_Definition,'Full_Zero'_Sign,Sheet_6_of_7,19APR2015.jpg,E._D._Notation_Definition,'Full_Zero'_Sign,Sheet_7_of_8,22APR2015.jpg,E._D._Notation_Definition,'Full_Zero'_Sign,Sheet_8_of_8,22APR2015.jpg

      Dialogically Yours,

      Miguel Detonacciones