Saturday, December 14, 2013

"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' 
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:

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'.



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 unitmonad»', each qn 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, qn, 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 qn, 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' qn 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 un uon, or, more simply, of the generic form unuon.   

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 un and/with uon, 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, un and uon 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 uon qualitative-unit-qualifier -- which harks back to the sign for the explicit quantifiable 'Monad qualifier', written [approximately] as Mo, used in Dyophantus's circa 250 C.E. founding treatise on "symbolical" algebra, the «Arithmetica» -- signifies that this new unit-qualifier, 'uon', is an 'addable [Plato:  «sumbeltoi»], quantifiable qualifier', unlike the old 'qn', 'unquantifiable, unaddable [Plato: «asumbeltoi» ] qualifier' units of the NQ_ dialectical arithmetic.
In these new, symbolic/ideographical 'complexes'/'compounds', the uon component, or '''factor''', denotes the 'arithmetical qualifier' for the [relative] base-level qualitative unitsmonads» 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 ‘qn’ would stand.    

The un 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, qn’, as their [meta-]unit/[meta-]«monad», by specifying the quantitative determination, the cardinal quantity determination, un, of the units-qualifier specifier/determination, uon.  

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

This un quantifier does not -- uselessly -- count the ontological category, ‘qn’, 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 un represents the count of the [relative] base-level individual units of kind qn that are, e.g., present in the current step of the presentation, with 1uon  =  uon denoting a single such [relative] base-level individual unit of kind qn.

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] unuon. 

Thus, if u1 were to be assigned the "Natural" number arithmetical quantifier value of 3, and if uo1 were to be assigned to the 'ontological qualifier' "apples", also denoted by a, then --  

u1uo1  3 x uo1  3uo1  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 uo1 were to be assigned, instead, to the 'metrical qualifier' "pounds", also denoted by p, then --

u1uo1 =  3 x uo1  3uo1  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 = NqM  =    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, qn, become uon, and the axioms-system N_, and of its elements, n, become un.

As we have seen above, the un 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 uon units / «monads» present, and that this uon 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 qn, onward / upward to the unuon, or to their dynamical versions, e.g., to their '''population dynamics''' versions, such as un(t)uon, forces that which was, in NQ_, the merely implicit presence of the base-level [sub-]units, or [sub-monads», of each qn 'category-level-as-unit-level ontological qualifier', or of each  qn '«arithmos»-qua-unitmonad» 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 3uo1 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.

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