"Section 2": On the Nature of the Opposition between the 'First Standard Arithmetic', and the Seldonian 'First Dialectical Arithmetic'.

Full Title:   

On the Nature of the Opposition between

the N_ 'First Standard Arithmetic' 

 and

the _NQ_ 'First Dialectical Arithmetic':

"Section 2" --  Fruition of the N_ ~ _NQ_ Opposition in a Dialectical Synthesis axioms-System,
  
_Nq__QN  = _Nq__U  =  _NU_, embodying a Complex Unification  


of the _NQ_ axioms-System and/with the N_ axioms-System





Dear Reader,

Below, I have reproduced, for your cognitive pleasure, the middle part, "section 2", of the forthcoming F.E.D. Vignette #21, by Aoristos Dyosphainthos, Chief Public Liaison Officer for the Foundation, which I have also added to the original blog-entry of this title, dated 14 November 2013, URL:  http://feddialectics-miguel.blogspot.com/2013/11/on-narture-of-opposition-between-first.html.


I have reproduced it here, also, as a separate blog-entry, so as to alert you to this update / addition of content to the original blog-entry of this title.

It provides a profound introduction to the Seldonian _NQ_ 'First Dialectical Arithmetic'.


Enjoy!!!



Regards,

Miguel
Member, F.E.D.,
Officer, F.E.D. Public Liaison Office

 

Fruition of the N_ ~ _NQ_ Opposition in a Dialectical Synthesis axioms-System, _Nq__QN  = _Nq__U  =  _NU_, embodying a Complex Unification of the _NQ_ axioms-System and the N_ axioms-System. 

As a generic 'category-level unit', or ' category as a unit/«monad»', each q_n stands, implicitly, for a '[meta-]unit', or for a '[super-]unit', which is, in turn, made up out of a heterogeneous multiplicity of [sub-]units.   

These [sub-]units are not "identical" to one another -- for how could they be " identical " and still be distinct and distinguishable.   

On the contrary, these [sub-]units are, instead, mutually-similar to one another, e.g., in an 'unscaled fractal' sense.   

These (1) [sub-]units are also the base-level, or most-concrete, foundational qualitative instances -- the qualitative "logical individuals", the qualitative "members", the qualitative cases, the qualitative specimens, the qualitative examples -- of some, 'relatively' more general, "kind" [cf. Plato] -- e.g., typically, (2) of a «species», but also possibly (3) of a [super⁰-]«genos», or, equivalently, 'super¹-«species»', or (4) of a 'super²-«species»', or, equivalently, of a 'super¹-«genos»', or (5) of a 'super²-«genos»', or (6) of a 'super³-«genos»', or (7) of a 'super⁴-«genos»', etc. -- a plurality of base-level [sub-]units for which that category / «arithmos» ["number of units"] / '''population''' NAME, q_n, stands, collectively, in a '[meta-]unitary', univocal way.

That is, a category denoted by / associated with / assigned to a given _NQ «aufheben»-operator / 'meta-Natural meta-number' / 'meta-Natural meta-numeral' symbol, of the general form q_n, is, in F.E.D. usage, implicitly understood to stand for an «arithmos» in something like the ancient Greek sense, i.e., an «arithmos» OF qualitative «monads», or OF qualitative units, a '''number''' OF such ultimate units ["ultimate" only relative to the universe-of-discourse in play, not in any absolute / reductionist sense].

We argue, in this section, that the mentally-perceived opposition -- in the mind(s) of the 'presentee(s) / reader(s) / thinker(s)-through of this presentation/model -- of the _NQ_ axioms-system, versus/to the N_ axioms-system, once they both come into 'co-present co-existence' within such (a) perceiving mind(s) -- is typically followed, intuitively, in such (a) mind(s), by their also mentally-constructed mutual interaction, denoted by _NQ_ x N_.   

The consequent operation / '''[re]flexion''', of the _NQ_ system-as-«aufheben»-operator, upon the N_ system-as-«aufheben»-operand, or the unique mutual '''multiplication''', mutual '''function-ing''', or mutual «aufheben»-operation / '''[de-]flexion''', specific to the _NQ_ system/operator together with the N_ system/operator, in their mutual interaction upon one another, denoted by _NQ_ x N_  =  _NQ_( N_ ), includes, at the [relative] base-level for their two, respective, universes-of-discourse, the base-level mutual opposition of the 'meta-numerals' q_n versus the numerals n.   

This, we hold, logically, intuitively, gives rise, in the perceiving human mind, to a new "kind" , i.e., to a new, third, idea / system; to a new 'ideo-ontology', of superseding, succeeding, supplementary, '''higher''' 'meta-meta-numerals' -- higher in the sense of being richer in expressive power, in capability for explicitly expressing more kinds of determinations than can either of the two previously-evoked systems / languages -- and thus 'meta-meta-numerals' which escape / transcend that [thus now former] base-level mutual opposition.

This human-mental process, whereby the mind of the 'presentor', and, if the presentation/'presentor' is successful, also the minds of the 'presentees' -- all constituting the human subjects/agents who are willingly/'will-fully' conducting these mental processes -- 'mentally-embody', and 'mentally simulate', this antithesis, this opposition of the outward meaning of _NQ_, and/with/against the outward meaning of N_, and are thereby provoked to combine these two, mutually-inadequate, opposites, by their 'dialectical multiplication' / interaction / mutual «aufheben» negation, connoted by _NQ_ x N_  =  _NQ_( N_ ), is a human, mental process which can be illustrated, pictorially, as follows, below --

 

These new, higher, 'meta-meta-numerals' are of the generic form u_n x  u^o_n, or, more simply, of the generic form u_nu^o_n.   

Therein, the explicit 'x' sign for the 'generalized / [sometimes] non-amalgamative multiplication operation', herein applied in this new, non-classical system of arithmetic, is simply understood to be indicated, implicitly, by mere juxtaposition, of u_n and/with u^o_n, alone  -- as, in classical algebra, for the classical multiplication operation -- wherein mere 'juxtapositioning' , without any intervening / mediating sign, e.g., an 'x' or "times" sign, of algebraic variables, e.g., with other(s) such, or with numerals / numeric constants, signifies their multiplication, all by itself. 

Together, u_n and u^o_n constitute the '''complex''' units, or the '''compound''' units -- compounded of both a quantifier and a qualifier; of a quantifier '''ideo-gram-matically modifying''' a qualifier -- that constitute the new, higher, 'meta-meta-number' space which we of F.E.D. denote by _NU, and of the new, higher axioms-system which we denote by _Nq__QN, or by _Nq__U, or by _NU_.   

The '^o' "degree-sign" superscript of the u^o_n qualitative-unit-qualifier -- which harks back to the sign for the explicit quantifiable 'Monad qualifier', written [approximately] as M^o, used in Dyophantus's circa 250 C.E. founding treatise on "symbolical" algebra, the «Arithmetica» -- signifies that this new unit-qualifier, 'u^o_n', is an 'addable [Plato:  «sumbeltoi»], quantifiable qualifier', unlike the old 'q_n', 'unquantifiable, unaddable [Plato: «asumbeltoi» ] qualifier' units of the _NQ_ dialectical arithmetic.

 

In these new, symbolic/ideographical 'complexes'/'compounds', the u^o_n component, or '''factor''', denotes the 'arithmetical qualifier' for the [relative] base-level qualitative units/«monads» of a given kind, of a given «arithmos», of a given ontological category, not for the category itself -- no longer for their category itself, as a unit, for which a corresponding ‘q_n’ would stand.    

The u_n component/'''factor''' denotes the 'unit-ic /«monad»-ic arithmetical quantifier', one that '''modifies'''/quantifies that qualifier as a qualifier standing for a generic [relative] base-level individual [sub-]unit / [sub-]«monad» relative to the category, q_n’, as their [meta-]unit/[meta-]«monad», by specifying the quantitative determination, the cardinal quantity determination, u_n, of the units-qualifier specifier/determination, u^o_n.  


That -- now '«monad»-ic', not 'categoric', or '''categorial''' --  quantifier, u_n, is, in its turn, '''qualified''' by that now «monad(s)»-qualifier, u^o_n, symmetrically and mutually, just as that quantifier, u_n, quantifies that qualifier, u^o_n.  


This u_n quantifier does not -- uselessly -- count the ontological category, ‘q_n’, itself, as the unit, which would always, invariably, result in a count of just 1, given the "idempotent" kind of addition of likes -- the 'super-amalgamative' kind of addition of likes -- that characterizes the 'unaddable' [Plato:  «asumbeltoi»] ontological-categorial qualifiers of the _NQ_ axioms-system of dialectical arithmetic, given it axioms.  


Instead, the value of u_n represents the count of the [relative] base-level individual units of kind q_n that are, e.g., present in the current step of the presentation, with 1u^o_n  =  u^o_n denoting a single such [relative] base-level individual unit of kind q_n.

With this new, _NU_, arithmetical / ideographical language, we can now explicitly translate -- into mathematical, arithmetical 'ideogramic' shorthand -- e.g., English, spoken or written, multi-vocal or multi-phonogramic, vocalizations / symbols-strings, or 'multi-phonetic utterances' / 'multi-symbol-writings', such as "three apples", or such as "three pounds", by means of ideogramic expressions/'compound meta-numerals' of the general form [i.e., of the algebraic form] u_nu^o_n. 

Thus, if u₁ were to be assigned the "Natural" number arithmetical quantifier value of 3, and if u^o₁ were to be assigned to the 'ontological qualifier' "apples", also denoted by a, then --  

u₁u^o₁  =  3 x u^o₁  =  3u^o₁  =  3a 

-- would stand for the English phrase "three apples". 

Or, if u₁ were to be, again, assigned to the "Natural" number arithmetical quantifier value 3, and if u^o₁ were to be assigned, instead, to the 'metrical qualifier' "pounds", also denoted by p, then --

u₁u^o₁ =  3 x u^o₁  =  3u^o₁ =  3p 

-- would stand for the English phrase "three pounds".

Note that we still cannot, yet, within the mathematical / arithmetical facilities / confines of the _NU_ language, express, e.g., English, phrases such as "three pounds [of] apples", in which both an 'ontological category qualifier', in this case, "apples", and a 'metrical unit qualifier', in this case, "pounds", as well as a 'metrical quantifier', in this case "three" -- all three components / '''factors''' -- all appear at once / all occur '''multiplied''' together, per our generalized concept of [generally non-amalgamative] '''multiplication'''.

Such triple+ conjunctions occur for "state-space" 'state-variable[ vector]s', e.g. --

(1), for the "state-variable[ vector]" --

( m₁ x dr₁(t)/dt ) x [ [ M x L ] / T ] x [ p₁ ] x [ x ]

-- for the x-axis physical-space-model's directional coordinate/component of the momentum of "particle" 1 as a function of time, t, a thus dynamical state-variable, represented by state-variable ['''ontological'''] qualifier p₁, whose 'metrical quantifier' is --

( m₁ x dr₁(t)/dt )

-- which '''modifies''' / is measured in terms of the 'metrical «monad»-qualifier' of Length, [ L ], say, measured in the unit-of-measure qualitative metrical «monad» of the "inch", divided [ / ] by Time, [ T ], say, measured in the qualitative unit-of-measure metrical «monad» of the "hour", together forming the "compound" 'metrical «monad»', or "metrical unit", of Velocity [ V ] = [ L / T ], thus measured in units of "inches per hour", thence forming the further-'''compounded''' 'metrical «monad»' of "momentum", [ P ], via Mass, [ M ], measured, say, in the unit-of-measure qualitative metrical «monad» of the "pound" --

[ P ] = [ MV ] = [ ML / T ] = [ [ M x L ] / T ]

-- thus measured, in toto, in the compound[ed] 'metrical «monad»' of "pound-inches per hour", for this classical "phase-space" type of state-space, thus simplifying this state-vector-value to --  

( m₁dr₁(t) / dt )[ Pp₁x ] 

-- or --

(2), for the "state-variable[ vector]" --

( r₁(t) ) x [ L ] x [ r₁ ] x [ x ]

-- for the x-axis physical-space-model's directional coordinate/component of the position of "particle" 1 as a function of time, t, whose 'metrical quantifier' is ( r₁(t) ), which '''modifies''' / is also measured in terms of the 'metrical «monad»-qualifier' of Length, [ L ], say, measured, again, in the unit-of-measure qualitative metrical «monad», or unit, of the "inch", and simplifying to ( r₁(t) )[ Lr₁x ] 

-- or --

(3), for a classical "phase-space"-associated "control[ parameter]-space" 'control parameter[-vector]', or, equally, for a non-classical "state-space"-associated,"control[ parameter]-space" 'control parameter[-vector]', e.g. --

( m₁ ) x [ M ] x [ c₁ ]

-- for the, typically-assumed constant, or time-non-varying, mass of "particle" 1, whose 'metrical quantifier' is ( m₁ ), which '''modifies''' / is also measured in terms of the 'metrical «monad»-qualifier' of Mass, [ M ], say, measured, again, in the unit-of-measure qualitative metrical «monad» of the "pound", and whose "control[ parameter-]space" -- in this case, 'masses-space' -- directional unit-vector for "particle" 1 is denoted by [ c₁ ].   
The 'control-parameter-vector', in this case, thus simplifies to -- 

( m₁ )[ Mc₁ ].

All three examples, above, apply to the classical, "phase-space" type of "state-space" [including to its associated "control-[parameter-]space", which we call "masses-space"], e.g., for a [nonlinear] dynamical systems theory total-differential equation [and, thus, typically also for a "singularity"-entailing, and therefore also typically meta-]dynamical mathematical model. 

Note that each of these first two, "phase space", classical "state-space", examples actually each require not three but four '''factors''', '''specifiers''', 'determinors', or '''modifiers''' -- one '[metrical] quantifier, "times" one '[metrical] qualifier' [similar to what the _NU_ system can provide], but also "times" one "state-variable" '''ontological-categorial''' qualifier, "times" one 'spatial-directional-vector' qualifier, and so exceeds the ideographical linguistic capabilities of the _NU_ arithmetical/algebraical language by not just one but by two kinds of additional 'arithmetical-ideohraphical qualifier' '''factors'''/'''specifiers'''/'determinors'/'''modifiers'''.

Capability to express quantifiers in ‘generalized-multiplicative’ combination/conjunction with both metrical qualifier units and ontological [e.g., state-variable or control-parameter 'identifier'/'specifier'] qualifier units begins not with _NU_, the third axioms-system in this axioms-systems progression presentation -- its first ‘dialectical full-synthesis’ axioms-system -- but with its seventh axioms-system -- the second ‘dialectical full-synthesis’ axioms-system -- which we of F.E.D. denote by _Nq__MQN, or by _Nq__MU, the 'Mu' axioms-system, wherein _Nq _QQ = _Nq_M  =    _NM_ denotes the fourth dialectical arithmetic axioms-system, of unquantifiable Metrical qualifiers.

All three of the symbols _Nq__QN, _Nq__U, and _NU_, stand for a new first-order axioms-system which 'complexes-together', or which '''compounds''', or which 'uni[t-i]fies' -- which constitutes a dialectical synthesis of -- the axioms-system _NQ_, and of its units, q_n, become u^o_n, and the axioms-system N_, and of its elements, n, become u_n.

As we have seen above, the u_n component of the new, ''compound unit''', stands for a «monad[s]»[-ic-level]-quantifier, or 'base-level units-quantifier', the count[or], or the cardinal  number -- the N of N number -- for the u^o_n units / «monads» present, and that this u^o_n component stands for a 'unit-qualifier', or [relative] base-level «monad[s]»[-ic]-qualifier, no longer for a 'category-as-unit-level qualifier'.   


Thus, the compound 'meta-meta-numerals' of the _NU_ explicit-dialectical arithmetic are '''complex[es of /] unities''' of the numerals of the N_ implicitly-dialectical arithmetic and of/together with the 'meta-numerals' of the _NQ_ explicitly-dialectical arithmetic.

That is, the transition from the n, and from the q_n, onward / upward to the u_nu^o_n, or to their dynamical versions, e.g., to their '''population dynamics''' versions, such as u_n(t)u^o_n, forces that which was, in _NQ_, the merely implicit presence of the base-level [sub-]units, or [sub-]«monads», of each q_n 'category-level-as-unit-level ontological qualifier', or of each  q_n '«arithmos»-qua-unit/«monad» qualifier', to the surface -- into explicit, counted -- quantified -- recognition.

It so forces because it is not meaningful or useful / 'use-valu[e-]able', in dialectical modeling, to count more than one, supposedly "identical" copy of "the same" ontological category, e.g., to have 3u^o₁ stand for the 'co-presence' of three "identical" copies of the entire category "apples", or of a, itself, 3a, rather than standing for the 'co-presence' of three [similar, but not even possibly "absolutely identical"] «monads», units, or individuals, e.g., that presently make up the entire extant "population" of that category, in this case, the "population" of three similar individual apples.

 

The 'New Old'. [Excerpt from a Recent Foundation-Internal Dialogue, for New Members of the Foundation, with Karl Seldon].

Dear Reader,

I have reproduced, below, for your convenience, an excerpt from the edited transcript of a recent, computer-assisted Foundation-internal dialogue with our co-founder, Karl H. Seldon [denoted by "KHS" in the excerpt below], which the F.E.D. Public Liaison Office has just cleared for public disclosure.

This is another one of Seldon’s highly-concentrated summaries of his ground-breaking discoveries in his advancement of Marxian dialectics.

Enjoy!

Regards,

Miguel

“... KHS:  What I discovered is that, once demystified, the concept of «karma» of the Ancient Orient, and the [e.g., Platonian] concept of «autokinesis» [i.e., of “self-movement”] of the Ancient Occident, contain the seeds of the keys to the possibility of a new, better comprehension, and to a truly present, up-to-date mathematical modeling, of our «kosmos».

“Nonlinearity” -- such as that exhibited by, and blocking the "closed-form" solution of, the differential equation formulations of the most advanced knowledge of the primary “laws” of Nature achieved by Modern humanity to-date -- is ‘karmicity’ and ‘auto-kinesicity’.

That is, this “nonlinearity” is ‘‘‘self-reflexivity’’’ and ‘self-refluxivity’. 

Dialectic is ‘self-reflexive’, ‘self-refluxive’ ‘self-«aufheben»-icity’, arising out of the ‘self-duality’, or ‘dialectical self-/internal/immanent "contradiction"’, of ‘Nature-al’ ‘‘‘eventities’’’.

This “nonlinearity” [and its inherent prone-ness to “singularity”]* is the manifestation of dialectic within a dialectic-unknowing, ‘‘‘dianoic’’’ [cf. Plato], «verstand»-imprisoned [cf. Hegel], modern mathematics.

‘Deeply-connectedly’, logical and set-theoretical “paradox” [Bertrand Russell’s diagnosis:  these paradoxes are due to “reflexiveness”] is the form in which dialectic has inescapably  manifested itself within a dialectics-denying, ideologically and polemically ‘‘‘dianoic’’’ “mathematical [‘mathematicized’] logic” and set theory.

The «kosmos» has constituted itself, in the past, and continues to constitute itself -- right now -- in the present, and into the future, as and via a process of ‘self-reflexive self-refluxion’, described, in the simplest form, via the simplest system of the ‘mathematics of dialectics’ available to us, by the following formula --

-- wherein, by the term ‘“sub-nuclear [“non-composite”] particles”’, we mean the ‘cosmo-ontological category’ that includes “non-composite” bosons, and fermions, i.e., for the fermions, the [“non-composite”] quarks, and all leptons [which, given their “Pauli Exclusion” behavior, and their lack of “strong interaction” behavior, are not expected to “compose” -- are not expected to form “composite” “particles” -- and have not been observed as doing so], and wherein also the other symbolic-elements of the formula above are defined as follows --

[For enhanced legibility, see the enlarged version of the above image at the bottom of this blog-entry.]

For descriptively richer, more complex, more concrete renditions of the dialectic «modus operandi» of ‘Nature as a Dialectical Totality’, I also discovered a progression of axioms-systems of ever more powerful ‘mathematics of dialectics’, which I later found could be organized, presented in taxonomic, ‘classificatory’, pedagogically-optimal order, and represented compactly, by the following formula, a ‘dialectical meta-equation meta-model’, formulated in the ideographical language of the first system of “mathematics of dialectics” that I discovered in 1996 --

-- which instantiates the Marxian, dialectical “method of presentation” -- that Marx wrote about in his «Grundrisse», and used in organizing and writing his world-historical, world-changing «magnum opus» -- so instantiating by way of generating, in systematic, pedagogical order, a progressive-cumulative presentation of those axioms-systems of mathematics of increasing mathematical modeling power, a ‘dialectic of the mathematics of dialectics’, if you will.

This ‘dialectic of the axioms-systems of explicitly dialectical mathematics’ starts from ³_hN__{#_}, which connotes the specification of the axioms-system of the Peano “Natural Numbers” using only the language of “first-order” symbolic logic, a specification long known, albeit “non-constructively” so, to harbor “non-standard models” of “Natural Numbers” arithmetic, hence also of its algebra.

That ³_hN__{#_} axioms-system of mathematics exhibits a ‘dialecticity’, or ‘«aufheben»-icity’, limited to a simplest, most abstract, “purely”-quantitative, or “purely”-ordinal, ‘pre-vestige’ thereof, as exhibited in the ‘‘‘ultra-simple’’’ Peano “successor function”, s, itself -- as in  s(n)  =  n + 1.

In that definition, the argument, or operand, of the s function, or operator, namely n, which can denote any individual “Natural” number, is, ‘unifiedly’, transformed in three ways.

It is (1) changed / “negated” [given that n + 1   ~=  n], (2) “elevated” in “pure”-quantitative magnitude, or in 'ordinal order-value' [in that n + 1   >  n], and also, at the “same time”, (3) “conserved” [in that n is “contained in” n + 1], i.e., in a single word, is ‘«aufheben»-ated’. ...”

“*[In simplest, most abstract form, we can illustrate differential equation ‘nonlinearity / singularity’, as it arises within the undergirding system of arithmetic of the so-called “Real” Numbers, R, as follows --

dx₂(t)/dt    =   x₂(t)( x₂(t) )    =   x₂(t)²

-- an ‘“open-form = closed-form”’-formatted solution of which is the following solution-function --

x₂(t)  =   t⁰ + t¹ + t² + t³ + t⁴ + . . .    =      

+(1 - t)^-1   =    

+1/(1 - t)⁺¹   =    

1/(1 - t)

-- so that a “singularity” arises at time t = 1 -- 

x₂(1)   =    

1/(1 - 1)  =    

1/0 

-- the problem being that the value / “number” (1/0) does not exist within -- and is "meaningless" within -- the system of the so-called “Real” Numbers, R, so that, here, suddenly, at the instant named t = 1, the solution of even this ‘‘‘simplest’’’ of nonlinear differential equations leaps beyond the the language-limitations of the numbers-system within which it was formulated, in an apparently "absolute" kind of apparently "infinitely"-sudden, "infinite"-in-magnitude dis-continuity.

What we have here is a rudimentary mathematical model of a 'non-quantitative' '''revolution''', led up to by a "purely"-quantitative "dynamical evolution" of this '''simplest''' of nonlinear "total--differential", or "ordinary"-differential, equations!   

We will release presentations of the non-“infinite”, ‘meta-finite meta-number’ value -- a value which is not quantitatively different from, but which is qualitatively different from, i.e., which is 'ideo-ontologically' different from -- any value available within the R universe of number(s), and which, beginning in the seventh axioms-system of explicitly dialectical mathematics that arises in the dialectical progression whose formula we gave above, meaningfully resolves, or ‘semantifies’, such singularities. 

We plan to do so soon, and in another venue. 

We have been, for the most part, holding back this dialectical discovery for many years.  

However, our 'psychohistorical-dialectical' calculations have indicated that the time for its public release will soon be "right"]. ...”