Friday, September 23, 2011

How do the F.E.D. "Natural Dialectors" Emerge intuitively from the "Natural Numbers"?


How do the NQ "First Dialectical Meta-Numbers" Emerge Intuitively from the "Natural" Numbers?

Dear Readers,

How can we best understand -- intuitively and dialectically -- the emergence of the NQ_ system of the NQ "dialectical meta-numbers" as a "first contra-system" to the "<<arche'>>-system" of the "Natural" Numbers, N...=...{ 1, 2, 3, 4, . . . }?

That is, the first transition in the F.E.D. meta-systematic dialectic of the dialectical arithmetics is --

)-|-(1....=....)-|-(0^(2^1) ....= ....N_..+..NQ_

How do we "justify" this transition, from system N_ to system N_..+..system NQ_, intuitively and dialectically?

First, in what ways is system NQ_ a qualitative opposite of system N_, in the sense of a combined, "complementary opposite" and "supplementary opposite" of system N_?

Well, N_ stands for an axioms-system of arithmetic for a "space" of "Natural Numbers", such that each of these "Numbers" can be characterized as an "unqualified quantifier".

The symbol NQ_, on the contrary, stands for an axioms-system of arithmetic for a "space" of "meta-Natural meta-Numbers", such that each of these "meta-Numbers" can be characterized as an "unquantifiable [ontological] qualifier".

[Note: the phrase "three <<species>>" exemplifies a "qualified quantifier", or a "quantified qualifier" -- the "quantifier", three, is "qualified", ontologically, or in terms of "kinds" [<<gene>>] of being, by the word "<<species>>".

The phrase "three inches" also exemplifies a "qualified quantifier", or a "quantified qualifier" -- the "quantifier", three, is "qualified", metrically, or in terms of units [<<monads>>] of measurement, by "inches".

The word "three", by itself, exemplifies and "unqualified quantifier".

the word "<<species>>, by itself, exemplifies an "unquantified ontological qualifier".

the word "inches", by itself, exemplifies an "unquantified metrical qualifier".]

But, how does system NQ_, this complementary/supplementary opposite of system N_, emerge from system N_?

There is a conceptual ambiguity -- indeed, an "intra-duality", "internal duality", or "self-duality" -- in the concept of "Natural Number"; in the "first-order", "Peanic" system of the "Natural" Numbers, N_, that leaves it open to "Non-Standard Models" of the "Natural" Numbers, N, such as that of the NQ_ system.

This "intra-duality" is that of cardinality versus ordinality -- cardinal numbers versus ordinal "numbers".

The "Natural Numbers" are supposed to be, "simultaneously", "inseparately", both "finite cardinal numbers" and "finite ordinal numbers".

But note how the first four, "first order" Peano Postulates, or "Peano Axioms", of the five Peano Postulates total ["first" order because they "talk about" only individual "Natural" numbers -- not about, e.g., qualities shared by ensembles of such individual numbers, such as "odd-ness" versus "even-ness", or "composite-ness" versus "prime-ness" ["second order"], and not about qualities shared by ensembles of such ensembles ["third order"], etc.], are all about ordinality, not about cardinality:

Peano Postulate a.:  1 is a Natural Number.

Peano Postulate b.:  The successor of any Natural Number is also a Natural Number.

Peano Postulate c.:  No two distinct Natural Numbers have the same successor.

Peano Postulate d.:  1 is not the successor of any Natural Number.

Now, although, on the one hand, cardinal numbers can be reduced, by the "modern" [i.e., "capitalist"], law-of-value-permeated <<mentalite'>> of human psychohistory, to the "purely-quantitative" -- even given that cardinality is also standardly formulated as an attribute, a predicate, a qualitative determination of sets [of qualitative elements] -- on the other hand, ordinal numbers, and ordinality in general, cannot be so reduced.

There is a qualitative aspect to ordinality, to order-position in a progression, although one can certainly "count out" through, e.g., n categories, or n systems, in a categories progression, or systems progression, to determine the ordinal value of a given category, or system, in that progression, so that there is something quantitative about ordinal values as well.

"First-ness" is also a quality, though a rather abstract quality. "Second-ness" is also a quality, a qualitative determination of a term in a progression of terms/categories.

For instance --

a....b....c....d . . .

-- conveys the same sense of ordinality as --

1....2....3....4 . . .

-- although the latter have also explicit cardinal number connotations, whereas the former do not.

Each of the former represent, not cardinal numbers, which, explicitly, would have only purely-quantitative differences among them, but phonetic symbols -- "phonetic ideograms", or "phonograms" -- each one representing, albeit loosely, a distinct sound of speech, one qualitatively different, not purely quantitatively different, from the "sound of speech" value of every other such symbol in the alphabet in question.

That is, for the latter --

1 < 2 < 3 < 4 < . . .

-- but these relations do not hold for the former.

It is not true that --

a < b < c < d < . . .

-- but, on the contrary, it is true that --

a is neither less than, nor equal to, nor greater than b is neither less than, nor equal to, nor greater than c is neither less than, nor equal to, nor greater than d . . .

-- i.e., that --

a is qualitatively unequal to b is qualitatively unequal to c is qualitatively unequal to d . . ..

So, I understand the dialectical, conceptual, emergence of NQ_ from N_, intuitively, as follows:

The N_ system of arithmetic, taken to be, explicitly, that of an arithmetic of "pure, unqualified cardinalizers, or quantifiers", has a suppressed, "implicitized", inner "Janus face"; a suppressed "inner dual", which is an arithmetical axioms-system of "uncardinal ordinalizers", or ordinal qualifiers.

The "reflexion" of N_ upon N_ itself, as the "reflection" of a human subject/agent, "holding N_ in mind", and reflecting upon N_ itself, in the terms of N_ itself, from the thus "mind-embodied" point-of-view and innate criteria of N_ itself, produces a dialectical "immanent critique", or "self-critique", of N_ itself, "by" N_ itself, as "embodied" in the mind of that human subject/agent -- a complex human "mental operation", or human "mental process", which "outs", "outers", "de-suppresses", or "explicitizes", that previously hidden, previously "occulted", previously suppressed "inner Janus face".

This complex "mental process" process can be modeled "simply", in "shorthand" mnemonic form, by the [mostly] symbolic-"ideogramic" expression [wherein '---->' denotes the phrase "goes to", "becomes", or "develops into"] --

N_ ----> N_ "of" N_....=

N_( N_ ).... =

_ "times" N_....=

_ "confronts" N_....=

_ "interacts with" N_....=

_ "critiques" N_....=

_ <<aufheben>>s N_...=

_ x N_....=

_ "squared"....=....N_^2.... =



-- wherein "delta-N_" denotes a qualitative increment -- and an increment of "new" idea-ontology -- to N_, and wherein NQ_ denotes the specific, explicit, externalized form of the hidden, suppressed, implicit internal dual within N_.

Thus --

arithmetic of cardinal "pure"-quantifiers.....x.....itself........=

arithmetic of cardinal "pure"-quantifiers.....+.....arithmetic of non-cardinal ordinal qualifiers

-- whereby the arithmetic of non-cardinal ordinal qualifiers both qualitatively opposes, and also supplements, the <<arche'>> arithmetic of cardinal "pure"-quantifiers.

In the context of N_..---->..N_( N_ )........=........N_.....+.....NQ_, we can comprehend the <<arithmos>> of the NQ "meta-numbers" / "ordinal qualifiers" as arising from an <<aufheben>> operation of the <<arithmos>> N, upon the <<arithmos>> N, i.e., as a "primordial" [self-]"meta-<<monad>>-ization" of the N.

For a possible, but physical-process -- perhaps therefore easier to visualize -- one partially parallel to this "purely-conceptual"-process, consider the early, interstellar '''atomic clouds''', that were destined later to became "molecular clouds".

As the rate of stellar production -- via stellar nucleosynthesis, and via other stellar processes -- of higher atomic physio-<<species>> surged-up, driving rising rates of influx of the atom products of these stellar processes into these inter-stellar clouds, local densities -- local physical-spatial concentrations -- of atoms were achieved that were sufficient to give rise to the first relatively stable molecules ever seen in our cosmos.

The process of the emergence of molecules, as <<arithmoi>> of a new kind, or <<genos>>, of <<monads>>, or of units, from out of these self-convergences, and self-concentrations, of atoms as the <<arithmoi>> of their predecessor kind, or <<genos>>, of <<monads>>, or units, can be grasped as a mutual "self-subsumption", and as a mutual "self-internalization", of the specific kinds of sub-<<arithmoi>> of atoms that form specific kinds of molecules.

Each such molecule <<monad>>, or unit, can be seen as the result of a self-<<aufheben>> mutual internalization of a [usually] heterogeneous multiplicity of atom [sub-]<<monads>>, or [sub-]units.

The '''self-re-entry''' of atoms to form molecules; the coming together of these atoms as molecules superseded, for their locations, the former <<monad>>-ic "internity", or "in-side", and the former <<monad>>-ic "externity", or "out-side", creating a new [meta-]<<monad>>-ic "internity", or "in-side", and a new [meta-]<<monad>>-ic "externity", or "out-side", plus, in that very process, creating a new "qualo-fractal" scale, or level:  that of the new molecular <<monads>> / units, in which their predecessor, atomic <<monads>> / units were <<aufheben>>-conserved, <<aufheben>>-determinately-negated, and <<aufheben>-elevated in terms of their "qualo-fractal" scale/level.

'''Self-re-entry''' creates a new self.

"Self-internalization" of existing [ideo-]ontology creates new [ideo-]ontology.

"[Self-]Meta-<<monad>>-ization" of existing <<monads>> creates a new kind of <<arithmos>>.

"[Self-]subsumption" of existing [ideo-]ontology creates new [ideo-]ontology.

"[Self-]demotion" [cf. Hegel] of existing [ideo-]ontology immediately gives birth to/promotes
new [ideo-]ontology.

Likewise, it is the human-mental "gathering-together" of the -- in this "primordial" case, homogeneous, not heterogenous -- "unit(s)", or <<monad(s)>>, of the N "Peano Natural Numbers" -- namely, the unit/<<monad>> denoted by 1 -- into distinct multiplicities, as "ordinal <<species>> denominators";  as the <<aufheben>>-"contained"/-conserved "in-side" of the resulting new "meta-numerals", and made visible in and as these "ordinal <<species>> denominators", that creates the NQ <<genos>> qualifier for "ordinality-in-general" [given the ordinal, not cardinal, focus of the "first order" Peano Postulates], namely, q, the "qualitative numerator" of all -- of each one -- of the NQ, as the "out-side" of each NQ "meta-numeral", which could equally-well be notated as n, denoting ordinal-number-in-general, which, as a "character-ization" of the "intension" of the "extension" of all Peano "Natural" numbers, must be a qualifier, not a quantifier, and which, again, given the ordinal essence of the "first order" Peano Postulates, must stand for the generic quality of ordinal number, not for the generic quality of cardinal number.

Thus, the q/n, or the n/n, which constitute the NQ "space" of the NQ_ system of arithmetic, arise "by", or "as", a human-mind-enacted, "self", or "mutual, subsumption", and a "self", or "mutual, internalization", of the N unit(s)/monad(s), 1, and, thus, by/as a human-mind-enacted, "self", or "mutual", "meta-unit-ization" of the N unit(s), "1".

That is, "q", or "n", is a "self, or mutual, subsumption", a "self, or mutual, internalization", and a "self, or mutual, meta-unit-ization", of each specific sub-<<arithmos>> of 1s, in a way similar to, or analogous to, the way in which each given molecule is a "self, or mutual, subsumption", a "self, or mutual, internalization", and a "self, or mutual, meta-unit-ization", of a specific sub-<<arithmos>> of atoms, such that --

NQ...=...{..._______________,..________________,..________________, . . . }..=
.................<<species>> 1...<<species>> 2....<<species>> 3.................


{..____,....________,...._________, . . . }.....=.....{....____,....________,...._________, ... }..= ......1.........1.+.1........1.+.1.+.1............................1..........1.+.1.......1.+.1.+.1.

{ q1, q2, q3, . . . }......=.......{ q|1, q|2, q|3, . . . }.......=

{ q/1, q/2, q/3, . . . }.

The "division" operations invoked above -- and the "division"-signifying "division bars" used above -- should be seen as invoking a
generalized, purely-qualitative, non-amalgamative, irreducible kind of "division" that is native to "qualo-fract-al fract-tions", and to their "division bar", as signifying "division" in the sense of Platonian-dialectical <<diairesis>>, wherein their <<genos>>-"numerators", and their <<species>>-"denominators", are mutually qualitatively different, not quantitatively different, and where the "out-side" of the <<monad>>-ic entity so "depicted", as a "unit[y]", by the "numerator", is also shown, again, by showing the "in-side" of itself, via the "denominator", but, there, in the guise of a "multiplicity"; of its "division" into the [typically heterogeneous] multiplicity of its component, "internal", "implicit", <<aufheben>>-"contained"/-conserved <<species>>-[sub-]<<monads>>, or of its <<species>>-[sub-]units.

In the special, "primordial" case of the
q/n, the q/n "qualo-fract-al fract-tions" do not "amalgamate", or otherwise "reduce", not because their numerator, q, and their denominator, n in N, both represent pure arithmetical qualifiers for qualitatively distinct ontological qualities, as is the case for a whole dialectical-arithmetical-systems sub-progression of the F.E.D. meta-systematic-dialectical progression of the dialectical arithmetics, starting with --

Nq/BA [----) q/24

-- which we have exposited here in an earlier entry to this blog.

No, in the special case of the
q/n, their universal numerator, q, and the particular denominator of each, generically given by the "Natural" number variable n, do not "amalgamate", or otherwise "reduce", because their numerator is the "purely-qualitative" generic ordinality qualifier [standing for the quality of "ordinality-in-general", or of "<<genos>> ordinality"], whereas their denominators are "purely-quantitative" "Natural" number quantifiers, so that numerator and denominator do differ qualitatively, i.e., because numerator and denominator are "mutually heterogeneous", but only in this most rudimentary sense, a sense which is "primordial" with respect to the numerator / denominator qualitative differences characteristic of the rest of the progression of the F.E.D. systems of dialectical arithmetic:  numerator and denominator differ qualitatively in this limiting/extreme case because the denominator is purely-quantitative, while the numerator is purely-qualitative.

This "rudimentarity", or "primordiality", arises because NQ_ is only the second system of arithmetic -- and the first explicitly dialectical such system -- after the <<arche'>> system, N_, in the F.E.D.meta-systematic-dialectical progression of the systems of dialectical arithmetic.

It is only
q|1, or q/1, or q1 that does not have an <<arithmos>> of "two or more" N units/<<monads>> --"1s"-- as its "in-side" / denominator / subscript.

q/1 "meta-numeral" has only a "non-<<arithmos>>, a single 1unit/<<monad>>, as its
"in-side" / denominator / subscript.

Thus, as expected, the <<arche'>> differs from, and is unique in relation to, all of its
["Qualo-Peanic"] successors / descendants.

Algebras of Logic, and Arithmetics of Logic, Boolean versus "Contra-Boolean" [Dialectical].
The intuitive account of the bifurcation of
N_ into N_+ NQ_ set forth above is a special case of a general account of the human "mental operations" modeled by the arithmetical and algebraic operations of the NQ_"contra-Boolean arithmetic", and its "contra-Boolean algebra".

The F.E.D. writings often characterize, in particular, the
NQ_ arithmetic, as a "contra-Boolean arithmetic [of/for a dialectical logic]", which gives rise to a "contra-Boolean algebra".

Not surprisingly, from a dialectical point-of-view -- given the "connexion-by-opposition" between the original "Boolean arithmetic/algebra", and the "contra-Boolean arithmetic/algebra" of the
NQ_ system, there is a deep resonance, including a sharp difference, between Boole's account of the algorithmic operations of the original arithmetic/algebra that bears his name -- which he presented, beginning in his 1847 work The Mathematical Analysis of Logic, as a symbolic model of the human "mental operations" of [<<verstand>>, sub-dialectical] logical thought -- and of the generic version of the account of the algorithmic operations of the NQ_ arithmetic/algebra, as also an ideogramic-symbolic simulation of the human "mental operations" of <<vernunft>>, dialectical-logical thought.

Indeed, if we denote the axioms-system, or "rules-system", of original Boolean logic-arithmetic/logic-algebra, by
WE_, taking it to be the <<arche'>> of all arithmetics/algebras of/for logic, and per a mnemonic which we will soon explain, then, in effect, in a different -- unit-interval-restricted -- "mutually perpendicular" direction/dimension of axioms-systems progression from that of the F.E.D.meta-systematic-dialectical progression of the systems of dialectical arithmetic, we can model the bifurcation of the Boolean arithmetic/algebra of logic, WE_, into WE_+ WQ_, using the same generic symbolical-algorithmic process that we used, above, to model the bifurcation of N_ into N_+ NQ_, so that WQ_ is grasped as an opposite to WE_, in the sense of a "complementary supplement", or of a "supplementary complement", to WE_ --

WE_..---->..WE_( WE_ )........=........WE_.....+.....WQ_.......................[----).......................


Recall that
W, the "space", or "set" of the "Whole" numbers, is defined herein as that of --

 { 0, 1, 2, 3, . . .}

-- and we can also grasp the "[analytical"] geometry" of WQ_ as an <<aufheben>>
"meta-<<monad>>-ization", or "meta-unit-ization", of the unit-interval-restricted, unit-interval interior "elided", "[analytical"] geometry" of Boole's WE_, the Boolean "[analytical"] geometry" being that of an "end-points only" [explicitly, at least] single unit interval --

|<---- 0B/0B ----->|
0B..........................................1B ............<<< 1B/0B

-- wherein 1B denotes "The Universe" [the class of all individuals/<<monads>>] and wherein denotes "Nothing" [the class that is empty of all individuals / <<monads>>].

So, per Boole, in his 1847 The Mathematical Analysis of Logic, and in his 1854 Laws of Thought, Boolean-algebraic expressions/equations model human "mental operations" via symbols -- via ideograms, and via [e.g., mnemonic] phonograms converted for use as ideograms.

The generic mental operation -- represented each of by Boole's "logical ideograms" that are also "class" ideograms, or "class operators" -- is the operation that Boole calls "
Election" [or, sometimes, "selection"].

It is also for this reason that we have characterized Boole's unit-interval "arithmetic of/for logic", as "The
Elector Arithmetic", and that we have symbolized his axioms-system by WE_.

For instance, take
w as a Boolean "class ideogram" representing, by "intension", the quality of "white[-ness]", and, by "extension",  the "class" of all "white things" [but still, in Boole's algebra, always as a [members-implicit] "intension", not as an explicit "extension", or "set", of explicitly-listed "elements", or "members"], and take b as a Boolean "class ideogram" representing the "class" of "birds".

Then, with those "givens" given, the Boolean expression
w x b, or, simply, wb, symbolizes the "Election", by the "class" w, from out of the "class" b, of all of w's "likes"; of all "things", or "individuals", "within" b, that "possess" the quality denoted by w, i.e., the "Election" of all "white things" from out of the "class" of all "bird-things", and, thereby, the formation of a new "class", of "white-bird-things", as the result, or "product", of this symbolically-simulated, "logical-multiplication"-simulated, "mental operation".

That is,
wb represents the "class" of "white birds", which would be the intersection set of the corresponding sets, or "extensions", W and B.

bw represents the "Election", by the bs from out of the "class" of all of the ws, of all of the bs that are already among the ws, or are already "in" w, the result of this "logical multiplication" of b and w, again being the "class" of "bird things that are also white-things", which is equivalent to the class wb --

bw  =  wb

-- so that Boolean "logical multiplication", or his "Election" operation, is "commutative".

Now let us consider, in this light, the Boolean equation which Boole names "The Fundamental Law of [sub-dialectical, <<verstand>>] Thought", or "The Law of [exo-]Duality", namely --

x x x   =   x

-- or --

x^2  =  x^1  =  x

-- or simply --

xx  =  x

in relation to what F.E.D. names "the fundamental law of dialectical thought", or "the law of
self-duality", namely [using the underscores, as always in my blog entries here, to connote the
"contra-Boolean-ness" of the operators so underscored] --

x x x  =  x + delta-x

-- or --

x^2   =   x + delta-x

-- or, simply --

xx   =   x + delta-x

wherein delta-x symbolizes the irruption of an incremental qualitative,  categorial unit of unprecedented new ["ideo-", and/or "physi[c]o-"]ontology, as an outcome, as a result, or as a "product", of the "self-operation", of the "self-application", or "self-reflexion" of an old, already existent, already extant, or, at least, already possible, categorial unit, x.

Per F.E.D., this
delta-x "purely-qualitative categorial unit increment" of categorial ontology, will be the result of a specific "self-<<aufheben>> operation" of x upon x itself, which will, typically, if not always, be representable as a "meta-unit-ization", or as a "meta-<<monad>>-ization", of the units, or <<monads>>, held implicitly "within" the categorial-unit symbol, or "<<arithmos>>-as-a-unity" symbol, x.

In Boole's "intuitive" interpretation of the Boolean expression,
xx, for the "logical multiplication" of x by x, the expression xx connotes the "Election", from out of the [Right-Hand, or RH] "class" x, of all of the "things", or "logical individuals", implicitly "contained in" [the Left-Hand, or LH] x, i.e., that each exhibit the quality that all members of "class" x share; the quality that defines "class" x, by the LH "class operator", also x.

The LH
x "Elects" only and all xs from out of the RH x, and puts them all into the results, or "product", "class" on the RH side of the "logical equation" xx  =  x.

Thus, indeed, per Boole's "intuitive" interpretation of these symbolic operations,
xx  =  x.

"logical nonlinearity",
x^2, reduces immediately to "logical linearity", x^1  =  x:

xx   =   x^2   =   x^1   =   x.

Boole's "Fundamental Law of [linear] Thought" logical equation models a "simple reproduction of classes", i.e., a "simple reproduction" of ideas, by means of the "self-
Election" "mental operation" of the human mind --

idean x idean   =   idean

-- depicting a static universe of human knowledge, and thus implicitly denying all possibility of an "expanded reproduction of ideas"

idean x idean   =   idean + idean+n

-- modeling an irreducibly "nonlinear", dynamical, and "meta-dynamical" [i.e., a revolutionary, or [a]periodically self-revolutionizing], universe of human knowledge.

Thus, Boole's "fundamental" mental operation of "self-
Election", symbolically simulated by his algebraic "logical equation" xx = x, as mirrored by the self-multiplication of the two values of the Boolean arithmetic, 0B and 1B, i.e. --

0B x 0B   =   0B, mimicking ordinary W-arithmetic's 0 x 0  =  0

-- and --

1B x 1B   =   1B, mimicking ordinary W-arithmetic's 1 x 1  =  1

-- with the "space" of "Boolean arithmetic" defined as follows --

WE    =    { 0B, 1B }

-- arithmetically models formal logic.

Please note again that
1B connotes "Every-thing" in Boole's arithmetic of logic, and that 0B connotes "No-thing". Therefore, Boole uses the expression (1B - x), interpreted as meaning "Universe minus x", to connote "Every-thing but those things "contained in" x", or "All things in the Universe except those that are xs", i.e., (1B - x) connotes the "complement" of x, or the "complementary opposite" of x.

Indeed, Boole adopts
xx = x as his "fundamental equation of formal-logical thought" at least partly because it represents a unification, in a single expression, of two of the traditional "laws" of Aristotelian formal logic, in that the two modes of its "algebraic decomposition", yield those two, traditionally "separate", "laws", or "rules".

First, for the "multiplicative decomposition", or "factoring", of an algebraic "implicand" equation of the equation
xx = x [using '==>' to denote the word "implies"] --

x  =  xx ==>

- xx  =  xx - xx  ==>

- xx   =   0B, ==>

1B - x)   =   0B

-- which is the Boolean formulation of the traditional "law of non-contradiction", meaning "those things which exhibit both the quality
x and the quality not-x are No-things".




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