Sunday, September 11, 2011

How to Calculate Purely-Qualitatively


How to Calculate Dialectically [and purely qualitatively]


Dear Readers,

I have already presented in this blog, several of the F.E.D. dialectical meta-models, formulated using the ideographical, arithmetical/algebraic language of the F.E.D. First Dialectical Arithmetic, that of the NQ dialectors.

The purpose of this post is to show, in detail, how to generate those dialectical meta-models, and others like them, from their “Dyadic Seldon Functions”, using the rules-system – the system of axioms – of the NQ, purely-qualitative dialectical language, axioms or rules that I have already presented in this thread, in an earlier blog entry here.

For instance, we presented excerpts from the “Dialectic of Nature” natural-historical-dialectical meta-model, through epoch t = 3 --

>-|-<t=3   =   <n>^(2^3) =
 
n + s + q/sn + a + q/an + q/as + q/asn + m

-- wherein, using, for n in N, q/n <----] x to assert “q/n is assigned to / interpreted as / used to model x” --

q/1 <----] n = the physio-ontological category of sub-nuclear “particles” [ <<arche’>>], e.g., quarks; gluons;

q/2 <----] s stands for the physio-ontological category of sub-atomic “particles”, e.g., protons;

q/3 <----] q/sn = physio-ontological category of processes which [expandedly re-]produce s from n, & also produce, e.g., neutrons by combination of electrons & protons;

q/4 <----] a stands for the physio-ontological category of the production of ionized, plasma atomic nuclei, e.g., by "cosmological nucleo-synthesis" ["original accumulation" of atomics];

q/5 <----] q/an = physio-ontological category of cosmos processes which [expandedly re-]produce a from n, & which also produce, e.g., electrically neutral [electron-[sub-]orbital(s)-bearing] Hydrogen atoms;

q/6 <----] q/as = physio-ontological category of cosmological processes which [expandedly re-]produce a from s, e.g., “1st generation” stars, which expandedly reproduce ionic, plasmic atoms [e.g., ionic, plasmic, normal Helium nuclei, or "alpha particles"] from plasma ionized Hydrogen atoms [e.g., from ionic normal Hydrogen nuclei, or free protons], by "stellar nucleo-synthesis";

q/7 <----] q/asn = physio-ontological category of processes which [expandedly re-]produce a from q/sn, e.g., electrically neutral, electron-rich atoms, with neutron-rich nuclei, e.g., by advanced "stellar nucleo-synthesis" ["reproductive accumulation" of atomics];

q/8 <----] m stands for the physio-ontological category “molecules”, i.e., "meta-atoms", each made up out of a [usually] heterogeneous multiplicity of atoms, e.g., H2O; CO2; CH4; O2; H2; H2S; CN, etc.


We have also presented a simplified systematic-dialectical “meta-model” of the “content-structure” of Marx’s Capital, one which captures only the circulations-process value-forms of the capitals-system, starting from the socio-ideo-ontological category of “Commodities”, or q/1 (----] C, as <<arche’>>, a systematic-dialectic which is complete, for its limited scope, as of step s = 3 --

)-|-(s=3 = (C)^(2^3) = C + M + q/MC + K + q/KC + q/KM + q/KMC + R

-- wherein --

q/1 (----] C = the Marxian human-social ideo-ontological category of “Commodities” [ <<arche’>>];

q/2 (----] M stands for the Marxian human-social ideo-ontological category of “Monies”;

q/3 (----] q/MC stands for the Marxian category “Monies Mediated Commodities Circulations”, or “MMCC”;

q/4 (----] K stands for the Marxian human-social ideo-ontological category of “<<Kapitals>>”;

q/5 (----] q/KC stands for the Marxian category of “Commodity-<<Kapitals>>”;

q/6 (----] q/KM stands for the Marxian category of “Money-<<Kapitals>>”;

q/7 (----] q/KMC stands for the Marxian category of “Subsumption of MMCC by <<Kapitals>>”, or the process of the circulations of “<<Kapitals>>” as a whole;

q/8 (----] R stands for the Marxian human-social ideo-ontological category of social “Revolutions” contra capitalism, the transition from the global-market capitalist socio-politico-economic system to its global successor socio-politico-economic system.


What both of these “<<speci>>-fic dialectical meta-models, & a host of other such dialectical meta-models besides, all have in common, is the “<<gene>>-ric dialectical meta-model

|-|-|k=3 = [ q/1 ]^(2^3)

q/1 + q/2 + q/3 + q/4 + q/5 + q/6 + q/7 + q/8

-- wherein --

q/1 [----] the generic dialectical category of <<arche’>>-thesis, or <<arche’>>-<<physis>> [ <<arche’>>];

q/2 [----] the generic dialectical category of 1st contra-thesis, or 1st meta-<<physis>>;

q/3 [----] the generic dialectical category of 1st [full] uni-thesis, or 1st [full] uni-<<physis>>;

q/4 [----] the generic dialectical category of 2nd contra-thesis, or 2nd meta-<<physis>>;

q/5 [----] the generic dialectical category of 1st partial uni-thesis, or 1st partial uni-<<physis>>;

q/6 [----] the generic dialectical category of 2nd partial uni-thesis, or 2nd partial uni-<<physis>>;

q/7 [----] the generic dialectical category of 2nd full uni-thesis, or 2nd full uni-<<physis>>;

q/8 [----] the generic dialectical category of 3rd contra-thesis, or 3rd meta-<<physis>>.

[ Note: in q/n, n is in N = {1, 2, 3, 4, . . .}, whereas, in --

|-|-|k = [ q/1 ]^(2^k),

-- k is in --

W = { 0, 1, 2, 3, 4, . . .}. ].


Therefore, we will explain the |-|-|0, |-|-|1, |-|-|2, and |-|-|3 generic Dyadic Seldon Function calculations, below, in detail, justifying each step in these calculations via explicit reference to the rules, or “axioms”, of the NQ “First Dialectical Arithmetic” –- axioms which are already gathered together, & commented upon, here, in an earlier blog-entry.

[ Note: in the title of this blog-entry, we have asserted that the dialectical calculations to be demonstrated herein, are also purely-qualitativecalculations. That assertion is true for the script level of NQ dialectical arithmetic, the level proper of the direct operations / calculations, of that arithmetic. However, the NQ arithmetic is a dialectical, or <<aufheben>>, self-subsumption, meta-unit-ization, or meta-<<monad>>-ization, of the purely-quantitative units, or <<monads>>, of the N arithmetic, and of the W arithmetic, so that the purely-quantitative calculations of the N, and of the W, arithmetics, respectively, still persist, i.e., are <<aufheben>>-conserved, at the sub-script” level, and at the super-script” level, of the NQ dialectical arithmetic, because those levels are the loci of conservation of the predecessor-system of arithmetic, that of the N Natural” numbers, with respect to which the NQ system of arithmetic is the immediate successor system, in the F.E.D. meta-system-atic dialectic of the dialectical arithmetics. The purely-quantitative calculations, at the sub-script” level of the NQ arithmetic, arise only in the context of its “multiplication” operation, as defined via Axiom # 9.]

(0) For the k = 0, “0th self-iteration” from q/1, we need only know, from “Standard Arithmetic”, that n^0 = 1 for all n in N, e.g., that 2^(+1) = 2, and that 2^(-1) = 1/2, and therefore that --

2^(0) = 2/2 = 2/1 x 1/2 = 2^(+1) x 2^(-1) = 2^(+1-1) = 2^(0) = 1

-- and also that --

x^(+1) = x , e.g., [q/1]^(+1) = q/1,

so that we calculate as follows:

|-|-|0 = [ q/1 ]^(2^0) = [ q/1 ]^(1) = [ q/1 ] = q/1 = [ q/1 ]^1

[ Note: Confining the super-script” level arithmetical operations for the arithmetical system of the NQ to those of N = { 1, 2, 3, 4, . . .}, as we strictly should, rather than to those of the immediate successor system of N, namely, to those of the "Whole" numbers, W = { 0, 1, 2, 3, 4, . . .}, this calculation doesn't even arise. It is included here to illustrate the fuller scope of the Dyadic Seldon Function, for it would explicitly appear in these calculations for the arithmetical system of the WQ, as also for its successors, e.g., for those of the ZQ, of the QQ, of the RQ, of the CQ, and of the HQ, etc., wherein Z denotes the “integers”, Q the “rational” numbers, R the “Real” numbers, C the “Complex” numbers, H the Hamilton Quaternions, etc.]



(1) For the k = 1, “1st self-iteration” from q/1, we need know only, in addition, that, as in “Standard Arithmetic” --

x^(2) = x x x = x-“squared”

-- for a straightforward application of NQ rule # 9 --

Axiom # 9: For every j & k in N,

[q/k] x [q/j] = [q/j] + [q/(k+j)]

-- &, thus, per that rule # 9, with, in this case, j = k = 1, we calculate as follows --

|-|-|1 = [ q/1 ]^(2^1) = [ q/1 ]^(2) = [ q/1 ] x [ q/1 ] = q/1 + q/(1+1) =

|-|-|0 x |-|-|0 = |-|-|0^2 =

|-|-|1 = [ q/1 ]^(2) = q/1 + q/2,

which, by rule #8 --

Axiom # 8: For every j & k in N:

If j is quantitatively unequal to k, then

q/j + q/k is qualitatively unequal to q/x

for every x in N

-- cannot be further “amalgamated”/cannot be “reduced”.



(2) For the k = 2, “2nd [cumulative] self-iteration” from q/1, we have 2 options.

We can (A.) carry out the full, “bi-nomial” self-multiplication, or “[re-]squaring”, of –-

[ q/1 ]^(2) = q/1 + q/2

-- calculating as follows --

|-|-|2 = [ q/1 ]^(2^2) = [ q/1 ]^(4) = [[ q/1 ]^(2)] x [[ q/1 ]^(2)] =

[q/1 + q/2] x [q/1 + q/2] =

|-|-|1 x |-|-|1 = |-|-|1^2,

then, multiplying each term in the left-hand sum onto each term in the right-hand sum, using Axiom # 9 4 times, =

[ [q/1 x q/1] + [q/1 x q/2] ] + [ [q/2 x q/1] + [q/2 x q/2] ] =

[q/1 + q/(1+1)] + [q/2 + q/(1+2)] + [q/1 + q/(2+1)] + [q/2 + q/(2+2)] =

[q/1 + q/2] + [q/2 + q/3] + [q/1 + q/3] + [q/2 + q/4],

then, applying rule # 10 --

Axiom # 10: For every j & k in N,

q/k + q/j = q/j + q/k
[commutative “law” of addition for NQ “dialectical meta-numbers”]

-- six times, & rule # 11 --

Axiom # 11: For every i, j, & k in N,

[q/i + q/j] + q/k = q/i + [q/j + q/k]
[associative “law” of addition for NQ “meta-numbers”]

-- four times, =

[q/1 + q/1] + [q/2 + q/2 + q/2] + [q/3 + q/3] + q/4,

then, applying rule # 7 --

Axiom # 7: For every n in N,

q/n + q/n = q/n
[idempotency “law” of addition for the NQ; the unquantifiability rule”]

-- three times, =

[q/1] + [q/2 + q/2] + [q/3] + q/4,

then, applying rule # 7 one more time, =

[q/1] + [q/2] + [q/3] + q/4 =

|-|-|2 = [ q/1 ]^(4) = q/1 + q/2 + q/3 + q/4,

which, by rule # 8, i.e., by Axiom # 8, cannot be “reduced”.

Alternatively, we can (B.) carry out an abbreviated calculation for |-|-|2, by applying the F.E.D. meristemal principle, which is, in general, a rule of non-distributive multiplication for NQ “dialectical meta-numbers” –-

Meristemal Principle. If i, j, . . ., k are all in N, & were just listed in “progressive order”, i.e., if i < j < . . . < k, and if a, b, . . ., c are all in N, & were just listed in “progressive order”, then the calculation of product of 2 generic poly-qualinomials can be abbreviated as follows --

[ q/i + q/j + . . . + q/k ] x [ q/a + q/b + . . . + q/c ] =

[ q/a + q/b + . . . + q/c ] + [q/k x q/a] + [q/k x q/b] + . . . + [q/k x q/c]

-- i.e., after <<aufheben>>-conserving the "multiplicand"/"operand" poly-qualinomial --

[ q/a + q/b + . . . + q/c ],

-- it is necessary only to apply the “meristemal”, or “most advanced”/“most progressed” ontological categorogram of the "multiplier"/"operator" categories-sum, "cumulum", or poly-qualinomial, here q/k, to each of the terms contained in that "multiplicand"/"operand" poly-qualinomial, then calculating as follows --

|-|-|2 = [ q/1 ]^(2^2) = [ q/1 ]^(4) = [[ q/1 ]^(2)] x [[ q/1 ]^(2)] =

[q/1 + q/2] x [q/1 + q/2] =

|-|-|1 x |-|-|1 = |-|-|1^2 =

[q/1 + q/2] + [q/2 x q/1] + [q/2 x q/2] =

[q/1 + q/2] + [q/1 + q/3] + [q/2 + q/4],

then, applying rule # 10 –-

Axiom # 10: For every j & k in N,

q/k + q/j = q/j + q/k
[commutative “law” of addition for NQ “meta-numbers”]

-- 5 times, & rule # 11 --

Axiom # 11: For every i, j, & k in N,

[q/i + q/j] + q/k = q/i + [q/j + q/k]
[associative “law” of addition for NQ “meta-numbers”]

-- 4 times, =

[q/1 + q/1] + [q/2 + q/2] + q/3 + q/4,

then, applying rule # 7 --

Axiom # 7: For every n in N,

q/n + q/n = q/n
[idempotency “law” of addition for the NQ; the unquantifiability rule”]

-- 2 times, we obtain:

|-|-|2 = [ q/1 ]^(4) = q/1 + q/2 + q/3 + q/4,

which, again, by rule #8 –- Axiom # 8 -- cannot be “reduced”.



(3) For the k = 3, “3rd [cumulative] self-iteration” from q/1, we have a straightforward application of the F.E.D. non-distributivemeristemal principle” shorthand rule, once we have applied our learnings from the 1st & 2nd “self-iterations” from q/1, namely, from |-|-|1 and |-|-|2 –-

|-|-|3 = [ q/1 ]^(2^3) = [ q/1 ]^(8) = [[ q/1 ]^(4)] x [[ q/1 ]^(4)] =

|-|-|2 x |-|-|2 = |-|-|2^2 =

[q/1 + q/2 + q/3 + q/4] x [q/1 + q/2 + q/3 + q/4],

then, applying the meristemal rule of poly-qualinomial multiplication, for the above tetra-qualinomials, =

[q/1 + q/2 + q/3 + q/4] +

[q/4 x q/1] + [q/4 x q/2] + [q/4 x q/3] + [q/4 x q/4 ] =

[q/1 + q/2 + q/3 + q/4] +

[q/1 + q/5] + [q/2 + q/6] + [q/3 + q/7] + [q/4 + q/8],

then, applying rule # 10

Axiom # 10: For every j & k in N,

q/k + q/j = q/j + q/k
[commutative “law” of addition for NQ “meta-numbers”]

-- multiple times, & applying rule # 11 --

Axiom # 11: For every i, j, & k in N,

[q/i + q/j] + q/k = q/i + [q/j + q/k]
[associative “law” of addition for NQ “meta-numbers”]

-- also multiple times, =

[q/1 + q/1] + [q/2 + q/2] + [q/3 + q/3] + [q/4 + q/4] +

q/5 + q/6 + q/7 + q/8,

then, applying rule # 7 –-

Axiom # 7: For every n in N,

q/n + q/n = q/n
[idempotency “law” of addition for the NQ; the unquantifiability rule”]

-- 4 times, we obtain:

|-|-|3 = [ q/1 ]^(8) = q/1 + q/2 + q/3 + q/4 + q/5 + q/6 + q/7 + q/8,

which, again, by rule #8, i.e., by Axiom #8, cannot be “reduced”.



The further Seldon Function “self-iterations” from q/1 follow by “recursive induction”, e.g. --

(4) For the k = 4, “4th [cumulative] self-iteration” from the q/1 <<arche’>>, we obtain --

|-|-|4 = [ q/1 ]^(2^4) = [ q/1 ]^(16) = [[ q/1 ]^(8)] x [[ q/1 ]^(8)] =

|-|-|3 x |-|-|3 = |-|-|3^2 =

[q/1 + q/2 + q/3 + q/4 + q/5 + q/6 + q/7 + q/8]^2 =

[ q/1 ]^(16) =

[q/1 + q/2 + q/3 + q/4 + q/5 + q/6 + q/7 + q/8 + q/9 + q/10 + q/11 + q/12 + q/13 + q/14 + q/15 + q/16].


Of course, such further cumulative self-iteration steps are immediately meaningful only to the extent that the new “physio-ontology” categories, or new “ideo-ontology” categories, thus generated, have been largely or wholly solved, or semantified, for the universe[-of-discourse] being delineated by such a dialectical meta-model”.

The number of meaningful steps –- and, especially, the number of ontological categorogram terms -- meaningfully generated, can be quite large.

For example, the [12-symbol version of the] Theory of Everything Dialectical  Equation “meta-model” of the cosmos as a totality --

>-|-<t = <n>^(2^t)

-- has been mostly solved, i.e., semantified, or meaningfully self-iterated, by the F.E.D. dialectical-encyclopedists, to t = 9, or for 512 terms, whose physical ontology has been at least generically characterized, and, in some cases, identified with specific cosmological processes, already fully extant and recognized by Standard Science, or known to be at least partially extant, in the Terran-humanity-known cosmos, if not identified with previously unrecognized, but already recorded, cosmological processes, and to t = 10, or for 1,024 terms, for the predicted ontology of that epoch of the Dialectic of Nature immediately subsequent to the present [local] epoch, again, as presently known to Terran humanity.



So far, we have been demonstrating the qualitative calculations for the successive “self-iterations” of the Dyadic Seldon Function, for generic , or minimally-interpreted”, minimally-assigned”, “dialectograms”, q/1, q/2, q/3, etc.

The algorithms for these qualitative calculations for specifically-interpreted, specifically-assigned “dialectograms” are less abstract, and more directly intuitive, than those for generically-interpreted, generically-assigned “dialectograms”.

For example, if we have, as our <<arche’>>-thesis, or <<arche’>>-<<physis>>, for a given dialectical progression that we are modeling, q/X, or X, where “X” is a mnemonic abbreviation for the name of the ontological category for which X, or q/X, stands, e.g., such that “X” is the first letter of that name, and if q/XX has been solved by “Y”, or by q/Y, e.g., because “Y” is the first letter of the name of the ontological category identified as fitting q/XX, [e.g., because “Y” is the first letter of the name of the ontological category identified as fitting q/XX, i.e., if “Y” abbreviates the name of the category/<<arithmos>> of units/<<monads>> which has, as its own units/<<monads>>, meta-units, or meta-<<monads>>”, of the units, or <<monads>>, of the category/<<arithmos>> denoted by X or q/X], then –-

(0) s = 0, )-|-(0 = (X)^(2^0) = X^1 = X;

(1) s = 1, )-|-(1 = (X)^(2^1) = X^2 = X + q/XX = X + Y

(2) s = 2, )-|-(2 = (X)^(2^2) = X^4 = (X)^(2^2) =

(X
+
Y)^2 = (X + Y) x (X + Y) =

(X + Y) + YxX + YxY =

(X
+
Y) + q/YX + q/YY = X + Y + q/YX + Z

-- if q/YY has been solved by “Z”, or q/Z, [e.g., because “Z” is the 1st letter of the name of the ontological category identified as fitting q/YY, i.e., if “Z” abbreviates the name of the category / <<arithmos>> of units/<<monads>> which has, as its own units/<<monads>>, meta-units, or meta-<<monads>>”, of the units, or <<monads>>, of the category/<<arithmos>> denoted by Y or q/Y].

In the above equation, the term “q/YX” stands for a new ontological category, for a category wherein category Y is ontologically hybridized, or dialectically synthesized, with category X, forming a complex unity of Y and X, or stands for a category of ontological process wherein category “Y” expandedly-reproduces itself / its population, or <<arithmos>>, of <<monads>>, or units, by converting <<monads>>, or units, of category X into those of category Y.

For example, q/MC stands for the socio-ontological category of the “socio-ontological hybridization”, or “dialectical synthesis”, or “real subsumption”, of the “Commodities” socio-ontological category &/by the “Monies” socio-ontological category, in the Capital-dialectic Equation, connoting the socio-ontological process-category of “Monies Mediated Commodities Circulations”.

Likewise, q/KC stands for the socio-ontological category of “Commodity-Capital”, the result of the “socio-ontological hybridization”, or “dialectical synthesis”, or “real subsumption”, of the “Commodities” socio-ontological category &/by the “Kapitals” socio-ontological category, in the Capital-dialectic Equation.

Likewise, q/KM stands for the socio-ontological category of “Money-Capital”, the result of the “socio-ontological hybridization”, or “dialectical synthesis”, or “real subsumption”, of the “Monies” socio-ontological category &/by the “Kapitals” socio-ontological category, in the Capital-dialectic Equation.

Likewise, q/KMC stands for the socio-ontological process-category of “Monies-and-Commodities-Mediated Kapitals-Circulations”, which connotes, in historical-dialectical context, the result of the “socio-ontological hybridization”, or “dialectical synthesis”, or “real subsumption”, of the “Monies Mediated Commodities Circulations” [q/MC or MMCC] socio-ontological category &/by the “Kapitals” socio-ontological category, in the Capital-dialectic Equation; the “appropriation” of MMCC by K, or, in systematic-dialectical context, the “de-abstraction” of K from q/MC, or the “re-concretization”/”re-determination” of q/MC with/by q/K, yielding q/KMC.


Similarly, q/as stands for the "uni-<<physis>>" of a & s, the physio-ontological category of cosmological processes which [expandedly re-]produce, or reproductively accumulate, a, atoms, by transforming/consuming/depleting s, sub-atomic “particles” populations, or <<arithmoi>>, e.g., protons, converting them, together with electrons and neutrons, into atoms populations, or <<arithmoi>>, e.g., Helium atoms [e.g., of stellar-plasma/ionized Helium atoms: Helium nuclei or "alpha particles"].

What can we identify as cosmological units, or <<monads>>, known to conduct such cosmological processes?

Such are the units, or <<monads>>, known as “first generation” stars, in which the units, or <<monads>>, of atoms [e.g., Helium atoms], & the units, or <<monads>>, of sub-atomic “particles” [e.g., protons], inter-mingle, and whose very “life-process” is thequalo-quantitatively”-expanding self-reproduction of atoms [e.g., of plasma/ionized Helium nuclei] by these stars’ conversion of their interiorized “sub-atomic particles” called protons [e.g., of protons as stellar-plasma/ionized Hydrogen atoms: “Hydrogen nuclei”], plus electrons, and neutrons, into their interiorized atoms of “higher atomic <<species>>” [e.g., into Helium atoms, i.e., of stellar-plasma/ionized Helium atoms: Helium nuclei or "alpha particles"].


Regards,

Miguel

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