"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 the N_ axioms-System





Dear Readers,

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.