## Thursday, May 01, 2014

### Part III. C.: Interlude. Dialectic of the Arithmetical Operations of the "Standard" Arithmetics. The F.E.D. Psychohistorical-Dialectical 'Meta-Equation' of Human-Social Formation(s) 'Meta-Evolution' Series.

Part III. C.:   Interlude.  The F.E.D. Psychohistorical-Dialectical 'Meta-Equation' of Human-Social Formation(s) 'Meta-Evolution' Series.

Modeling the Systematic Dialectic of ‘‘‘The Basic Operations of Arithmetic’’’ using CQ_ --
Systematically Presented via a 5-Symbol Expression.

Below is the third and final entry on the three simpler examples of systematic 'dialectical meta-models', presented as an interlude, before concluding the series on the F.E.D. Psychohistorical-Dialectical 'Meta-Equation' of Human-Social Formation(s) 'Meta-Evolution', with the problematics of the nation-state social formation, and with its 'meta-model'-predicted successor-formations.

Enjoy!

Regards,

Miguel

Introduction.  This meta-model is more “Complex” [pun intended] than the meta-models of “TV-Series”, and of ‘Modern Computerware’, presented earlier in this sub-series, in particular, because it requires some “domain-expertise” -- or, at least, some “domain familiarity” -- with respect to the domain of the so-called “Complex Numbers”, the set standardly denoted by the symbol C.

The ‘axioms-system’ of the arithmetic of the C numbers, which we denote by C, is the 6th system of arithmetic in the following standard order of standard arithmetics, with our light-spectrum ordinal color-coding added --

N, W, Z, Q, R, C,

-- for the “Natural”, “Whole”, “Integer”, “Rational”, “Real”, and “Complex” arithmetics, respectively.

About the ‘Goedelian Dialectic’ of these systems, see:  http://www.dialectics.org/dialectics/Vignettes.html, Vignette #4.

We will, in this blog-entry, use the Complex-Numbers-subsuming version of the F.E.D. ‘first dialectical algebra’ to construct, and to “solve”, a “heuristic”, ‘intensional’ [‘connotational’],‘intuitional’ meta-model of a systematic presentation of the domain of “the basic operations of arithmetic” -- encompassing both its ‘‘‘verse’’’ [e.g., addition, multiplication, exponentiation] and its inverse” [e.g., subtraction, division, root-extraction] operations, jointly, via qs with C subscripts, i.e., with Complex-number subscripts, which we also reference as Cqs.

The meta-models that we usually narrate here are constructed by interpreting the generic NQ_ version of the F.E.D. ‘first dialectical algebra’ [ see E.D. Brief # 5 and its Preface ], or, at most-advanced, by interpreting the generic WQ_ version of that algebra [ see E.D. Brief #6 and its Preface ], with the subscripts of the Nq or Wq ‘meta-numerals’ drawn from the number-space N = { 1, 2, 3, ... }, or from the number-space
W = { 0, 1, 2, 3, ... }, respectively.

This time, the subscripts of the Cqs will be drawn from the standard number-space  --

C = { R + Ri }

-- wherein R denotes the space of the standard so-called “Real” numbers, and where the i unit stands for so-called "imaginary" unity, the positive square root of -1.

FYI:  The generic Complex number is often expressed as a + bi, with a an element of R, and also with b an element of R, or as z  =  x + yi, with x an element of R, and also with y an element of R.

That is, we will be constructing our example model using the generic CQ_ version of the F.E.D. ‘first dialectical algebra’.

We use the CQ_ language this time, as it allows us to present both ‘‘‘verse’’’ & ‘‘‘reverse’’’ operations in a single model.

Herein we mean, by the word, ‘‘‘systematic’’’ in the phrase ‘‘‘systematic presentation’’’, a presentation of the major kinds of “entities” that exist in this 'Human-Phenomic', 'Meme-etic' domain, the domain of the basic operations of arithmetic -- by means of categories that classify those entities by their “kinds”, i.e., as ‘‘‘[ideo-]ontology’’’, or as “kinds of [idea-]things, or "meme-things"” -- and in strict order of rising complexity, starting from the simplest category, and moving, step-by-step, from lesser to greater, i.e., more inclusive, complexity, until we reach the most complex/inclusive extant category of this domain, or for the purposes of this example.

The model that we build will describe these categories in that strict, systematic order of rising operational complexity / inclusivity.

This will be, once again, like the previous two “interlude” models, a “snapshot” model, a “synchronic” model that takes the contemporary slice of time -- or at any rate, a recent-past slice of time -- and algorithmically generates descriptions of categories for entities that presently exist, or that might possibly presently exist, for the model’s domain, in their systematic order of inclusivity, as described above.

Our model here will not be a “chronology” model, or “diachronic” model, like the previous, major model, narrated in this series, in which the units of earlier categories are described as actually, e.g., physically, constructing, through their activity as causal agents, i.e., as subjects, the units of later categories, categories whose units did not exist until that construction took place.

That is, it will not be a model of a ‘self-advancing’ historical progression of ontology, with each historical epoch containing both old ontology, inherited from past historical epochs, plus new ontology, ontology that had never appeared before -- in past historical epochs -- until the later epoch in question, plus ‘hybrid categories’, combining / synthesizing the old with the new.

We will apply a documented, standard procedure to “solve” this ‘‘‘algebratric’’’ model -- to determine what actual category each of these generated category-descriptions refers to, and to determine which, if any, of these category-descriptions describe “empty categories”, i.e., represent ‘combinatorially’ possible entities that actually do not exist "within" this domain -- at least not presently.

To get started, we must determine the starting-point -- the point-of-departure -- for our systematic model.

This starting category will be the seed of our whole progression of generated category-descriptions, influencing every category that follows, as the “controlling source”, and as the “ever-present origin”, of all that follows from it.

The rule for getting started is to ask oneself “¿What is the least complex kind of thing, the simplest kind of thing, the least inclusive kind of thing, which inheres in this domain? -- in our case, in the domain of ‘basic arithmetic operations’ -- and to then find the answer to that question, based upon one’s prior knowledge of, or familiarity with, this domain.

The answer to this starting question that we will pursue in this example is the following:  The ‘‘‘verse’’’ operations of “Additions”, and its inverse operations”, or ‘‘‘reverse operations’’’, or “Subtractions”, are the simplest ancestors, the ultimate units, of basic arithmetical operations, ingredient in every one of the more complex operations of that domain.

A letter that the spelled names of these two kinds of operations have in common is “t”.

Therefore, we shall name/symbolize our starter category as Ct, or as Cqt, denoting the “Complex” combination of the “Additions” sub-category of elementary Real arithmetic basic operations, with the sub-category of “Subtractions”, and identifying that combination of specific sub-categories with the generic first category symbol of our generic category-arithmetic model, namely, with the symbol --

Cq[1 + 1i]

in an “identification”, an “interpretation”, or an “assignment” [ [---)’ ] that we indicate by writing:

Ct     =    Cqt    =   Cq[A + Si]    [---)  Cq[1 + i].

Our model then, will take the form of an “interpeted”, specific equation, assigned to the generic equation, like this --

C)-|-(s   =   Ct2^s   =   ( Cq[A + Si] )2^s    [---)

C|-|-|h   =   Ca2^h   =   [ Cq[1 + 1i] ]2^h

-- with the variable s indicating the step in our systematic method of presentation that the ‘accumulation of categories’, denoted by C)-|-(s, represents.

We will not, here, further recount the [Marxian] method of systematic discovery that was used to arrive at the starting category of this systematic presentation.

For more regarding that method of discovery, see Marx, Grundrisse, Penguin Books [London:  1972], pp. 100-101.

Stage 0.  Our initial step -- step s = 0 -- contains only our starting category  --

Ct     =    Cqt    =   q[A + Si] --

C)-|-(0   =   Ct2^0   =   Ct1   =   Ct    =

Cq[A + Si]    [---)  Cq[1 + 1i]

-- because 2 “raised” to the power 0 -- 20 -- is just 1

and because Ct “raised” to the power 1 is just Ct.

Stage 1.  It is when we get to the next step after step s = 0, namely, to step s = 1, that our meta-equation meta-model gives us back something initially unknown” -- and, therefore, something ‘‘‘algebraical’’’, not merely something ‘‘‘arithmetical’’’:  something to “solve-for” --

)-|-(1   =   Ct2^1   =   Ct2   =   Ct  x Ct    =

Cq[A + Si] x Cq[A + Si]   =

Cq[A + Si]     +   Cq[AA + SSi]   =   Cqt   +  Cqtt

-- because 2 “raised” to the power 1 -- 21 -- is just 2, and because our rule for multiplying a generic category, call it --

Cq[X + Yi]   =   CqZ  =  CZ

-- “by”, or “into”, itself, is, for subscripts X and Y denoting sub-category symbols, and for subscript Z denoting a category-symbol, simply --

Cq[X + Yi] x Cq[X + Yi]    =

Cq[X + Yi]  +  Cq[XX + YYi]   =   CZ  + CqZZ

-- and for x and y denoting “Real” numbers --

Cq[x + yi]  x  Cq[x + yi]     =

Cq[x + yi]   +  Cq[(x + x) + (y + y)i]     =

Cq[1x + 1yi]   +  Cq[2x + 2yi].

Note again:  Herein, Cq denotes the generic category ‘qualifier’ with “Complex” subscripts.

The subscripts that come after it are specific category descriptors.

¿But how do we discover what the resulting, initially “unknown”, or ''algebraical'', ‘category-description’, here Cqtt, means?

Well, the generic rule to “solve-for” the categorial meaning of such symbols is that, if we know what is meant by category CqZ  =  CZ, then the symbol CqZZ describes a category each of whose units is a ‘CZ OF CZs’, that is, a category for a different kind of units, called meta-CZs’, each such unit being made up out of a multiplicity of those units of which the category of the CZs is made up.

To be specific with this rule, in our example-model, CqZZ specifies a “Complex” of two sub-categories.

Each of the units of the first sub-category, the sub-category of the ‘‘‘verse’’’ operations, must be an Addition OF Additions’ that is, must be a meta-Addition’, such that each meta-Addition’ is made up out of a multiplicity of “mere” Additions.

Each of the units of the second sub-category, the sub-category of the ‘‘‘inverse’’’ operations, must be a Subtraction OF Subtractions’, that is, must be a meta-Subtraction’, such that each such meta-Subtraction’ is made up out of a multiplicity of “mere” Subtractions.

Our experiences of / "in" the domain of 'the basic operations of arithmetic' suggest that such operations do “presently” exist in the domain of “Real” arithmetic.

“Multiplication” is a basic arithmetical operation that is “made up out of multiple [repeated] additions”, viz. --

4 x 5  =  5 + 5 + 5 + 5  =  4 + 4 + 4 + 4 + 4  =  5 x 4  =  20

-- a sum of four fives, or a sum of five fours:  either order will do [a characteristic called “commutativity of addition”]!

In a partial reverse likeness, “division” is a basic arithmetical operation that is “made up out of multiple [repeated] subtractions”, viz., 5 "goes ["evenly", i.e., with 0 remainder] into 20" 4 times;  4 "goes ["evenly"] into 20" 5 times --

20 ÷ 5  =  4;

20 - 5 - 5 - 5 - 5  =  0  =  20 - 4 - 4 - 4 - 4 - 4;

20 ÷ 4  =  5

-- to see how many fours there are in twenty [not the same as how many twenties there are in four]; how many “times” four “goes ["evenly"] in to” twenty, or to see how many fives there are in twenty, [not the same as how many twenties there are in five]; how many “times” five “goes ["evenly"] in to” twenty:  but, in this case, either order will not do!

A letter that the spelled names of these two kinds of operations have in common is “n”.

Therefore, we shall name/symbolize our second category as Cn, or asCqn, denoting the “Complex” combination of the “muLtiplications” sub-category of elementary Real arithmetic basic operations, with the sub-category of “diVisions”, and identifying that combination of specific sub-categories with the generic second category symbol of our generic category-arithmetic model, namely, with the generic category-symbol

Cq[2 + 2i].

We may “assert” our solution as follows:

Cqtt  =  Cqn  =  Cn =  Cq[L + Vi]

=  Cq[AA + SSi]  [---) Cq[2 + 2i].

Again, what is dialectical about the relationship between Ct and  Ct2, or  Ct x Ct, or Ct of Ct, or Ct(Ct), the relationship of what we call meta-unit-ization, or  meta-«monad»-ization, between Ct and its already presently existing, ‘supplementary other’, Cn, is that this relationship is a synchronic double-«aufheben» relationship.

That is, each single “unit” of the “muLtiplications” sub-category of category Cn, i.e., each typical individual “multiplication” operation, is a negation, and also a preservation, by way of also being an elevation to the / forming the “higher” / more inclusive muLtiplications” sub-category / level / scale, of a whole [sub-]group of units of the “Additions” sub-category / level / scale of the Ct category.

Likewise, each single “unit” of the “diVisions” sub-category of category Cn, i.e., each typical individual “division” operation, is a negation, and also a preservation, by way of also being an elevation to the / forming the “higher” / more inclusive diVisions” sub-category / level / scale, of a whole [sub-]group of units of the “Subtractions” sub-category / level / scale of the Ct category.

So, our full solution to the step s = 1 equation of our model is --

C)-|-(1  =  Ct   + Cn   =

Additions & Subtractions  + MuLtiplications & DiVisions

[---)  Cq[1 + 1i]   +  Cq[2 + 2i].

If this meta-model is working right, Additions & Subtractions is the simplest category of the domain of ‘basic arithmetical operations’; MuLtiplications & DiVisions is the next more complex category of that domain.

Stage 2.  ¿What additional ‘category-specifications’ do we generate in our next step, step s = 2, that need “solving-for”?

Let’s find out --

C)-|-(2   =  Ct2^2  =  Ct4  =  ( Ct2 )2  =  ( Ct  + Cn )2  =

( Ct   + Cn ) x ( Ct   + Cn )   =

Ct   + Cn   +  Cqnt  +  Cqnn.

This result arises by way of two key rules of categorial algebra, plus the general rule for multiplication when one category-symbol is multiplied by a different category-symbol [we used a special case of this general rule, for the case where the same category-symbol is multiplied by itself, in step s = 1, above] --

1.  general case:  CqY x CqX   =   CqX  +  CqYX   =   CX  + CqYX;

special case:    CqX x CqX   =   CqX  +  CqXX   =   CX  + CqXX.

2.  Cqx + Cqx   =   Cqx; the same category-symbol, added to itself, does not make “two” of that category-symbol; one “copy” of each category is sufficient; two or more copies of any category would be redundant, for the purposes of this dialectical-categorial algebra.

3.  There is no Cqw such that Cqx + Cqy    =   Cqw; different category-symbols, added together [as opposed to being ‘‘‘multiplied’’’], do not reduce to a single category-symbol, just like in the proverbial case of ‘apples + oranges’, or a + o.

Well, we already know how to “solve-for” Cqnn.

It describes a category “containing” two sub-categories, the first sub-category being one of muLtiplications OF muLtiplications’, and the second sub-category being one of diVisions OF diVisions’.

The first sub-category is one each of whose units / operations is a muLtiplication OF muLtiplications’, i.e., each of which is a meta-muLtiplication’, such that each such meta-muLtiplication’ operation is made up out of a multiplicity of  muLtiplication operations.

Our experiences of / "in" the ‘basic arithmetical operations’ domain suggest that such arithmetical operations do indeed presently exist.

That sub-category-description describes the sub-category of multi-muLtiplication’ operations -- i.e., of “exPonentiations”:   “exPonentiation” is a basic arithmetical operation which is “made up out of multiple [repeated] muLtiplication operations, viz. --

23  =  2 xx 3  =  2 x 2 x 2  =   8        9   =    3 x 3  =  3 xx 3  =  32.

I.e., “two cubed”, or “two raised to the exponent three”, is “made of” a product involving three twos, that yields eight, whereas “three squared”, or “three raised to the exponent two”, yields nine:  in general, the order of “base” and “exponent” cannot be reversed without changing the result as well.

Generally, each order will return a different result.

The second sub-category should be, per our standard method, one each of whose units / operations is a diVision OF diVisions’, i.e., each of which is a meta-diVision’, such that each such meta-diVision’ operation is made up out of a multiplicity of  diVision operations.

That is, the second sub-category should be one of multi-diVision’ operations, “made up out of multiple [repeated] diVision operations.

We interpret this to be the “inverse” operation of de-exPonentiation’, or of “nth Root extraction”.

The “log” operation, which returns exponents, not bases or roots, is also a candidate for this “inverse” operation, but is not as fully this inverse operation as is the Root extraction operation.

Given our experience of / "in" the domain of ‘basic arithmetical operations’, this sub-category description may, at this point, give us pause.

Many of us may be unfamiliar with the algorithms by which the “nth” root(s) of a given number are “extracted”.

In what sense, if any, can an exponentiation be reversed, the “root” “extracted” from its “power”, by repeated division?

But let us consider the method of extracting square roots that is perhaps the oldest such method still known.

It is called “The Babylonian Method”, and also “Heron’s Method”, because the storied Heron of Ancient Alexandria is the most ancient source known to have written an explicit account of this method.

This method is, by the way, a special case of the more general “Newton’s Method”, but predates the discovery of “Newton’s Method” by many centuries.

The method involves guessing a “starting estimate” for the square root sought, followed by repeated stages of, well, diVision -- division of the square by the current best estimate of its square root -- followed by, well, diVision again -- this time division of the sum of the previous consecutive pair of estimates by two, thus averaging them -- to obtain the next better estimate of the square root, all leading to an improving estimate for the square root with each iteration of the ‘double diVision’ just described.

‘Formulaically’, the next better estimate of the square’s square root, xn+1, is derived from the previous best estimate, xn, by dividing the square, S, by the previous best estimate, xn, summing xn and S÷xn, then dividing that sum by 2:

xn+1    =    ( xn  +  S÷xn ) ÷ 2.

Let us apply this method to “extracting” the “square root” from the “square”, 9, with “starting estimate”

of x1 = 2 --

n....Current Best Estimate ( xn )...DiVide Square by That EstimateS÷xn )..DiVide Their Sum by 2 for new est.
1....2..............................9÷2 = 4.5................................(2+4.5)÷2 = 3.25
2....3.25.......................9÷3.25 2.769...................(3.25+2.769)÷2 3.01
3....3.01.......................9÷3.01 2.99......................(3.01+2.99)÷2 3.000
4....3.000....................9÷3.000 = 3.000.................(3.000+3.000)÷2 = 3.000

After n = 3, with "rounding" as shown above, the method reaches a “fixed point” / “equilibrium”
at x3+... =  3, which is the positive square root of 9

Thus we see in what sense, in this method at least, square root extraction is made up out of repeated diVisions.

We may thus “assert” our solution as follows --

Cqnn  =  Cqe  =  Ce  =  Cq[P + Ri]   =

Cq[LL + VVi]   [---)  Cq[4 + 4i].

Our step s = 2 equation-model, as we have solved it so far, thus now looks like this --

C)-|-(2   =  Ct2^2  =  Ct4   =

Ct   + Cn   +  Cqnt  + Ce

[---)

Cq[1 + 1i]   +  Cq[2 + 2i]   +  Cq[3 + 3i]  +  Cq[4 + 4i]

-- since we have not yet determined which actual category of the ‘basic arithmetic operations’ domain is described by the algorithmically-generated symbol Cqnt -- if any, i.e., if Cqnt is not an “empty category”, “inoperative” for this domain.

When, as a component of ( Ct   + Cn ) x ( Ct   + Cn ), the “higher-complexity” category, Cn, operates upon / “multiplies” the “lower-complexity” category, Ct --

Cn  x  Ct      Ct   +  Cqnt      =       Cq[A + Si]  + Cq[LA + VSi]

-- generically speaking, the categorial relationship to be called to the user’s attention by this operation, in this ‘categorial arithmetic’, is, again, a synchronic «aufheben» relationship, this time, that between Ct and Cqnt.

It calls the user to search that user’s knowledge and memory of the domain in question -- in this specific case, the domain of ‘basic arithmetical operations’ -- for a category which represents an “uplift” of category Ct entities to the level of the entities native to category Cn, thereby “canceling” the Ct-type entities concerned, at their own native level, but, by the same token, “preserving” those category Ct entities at the Cn level, combining Cn and Ct qualities, in the relationship of “elevation” of those category Ct entities within the level typical of category Cn entities.

Thus, the additional category thereby presented, Cqnt, signifies a category whose units are the operational interactions of the Ct operations with the Cn operations, as codified in the axioms, and/or theorems, and/or corollaries, and/or lemmas, and/or ‘‘‘rules’’’ of the “Real Numbers” system of arithmetic.

The first sub-category of the category Cqnt  = Cq[LA + VSi] answers to a sub-category description which connotes the way in which, or the ‘‘‘rules’’’ by which, the operation of muLtiplication “subsumes” the operation of Addition, denoted herein by L | A.

To our lights, this sub-category-description connotes the “distributive law” of “Real” arithmetic, an axiom of that system of arithmetic, which codifies the interaction of the addition operation with the multiplication operation -- the rule that the multiplication operation “distributes over” the addition operation.  This “law” involves two components, often called “left distributivity” and “right distributivity”, respectively --

·       [“left distributivity”]:    For all elements abc of R

c x (a + b)   =   (c x a) + (c x b).

·       [“right distributivity”]:  For all elements abc of R

(a + b) x c  =   (a x c) + (b x c).

The second sub-category of the category Cqnt  = Cq[LA + VSi] answers to a sub-category description which connotes the ‘‘‘rules’’’ by which the operation of diVision “subsumes” the operation of Subtraction, denoted ‘V | S’.

To our lights, this sub-category-description connotes a “non-distributive rule” of “Real” arithmetic for ‘diVision / Subtraction’, although this rule is, typically, not an explicit one in presentations and in axiomatizations of “Real” arithmetic.  It is learned informally, as a joint consequence of other rules, i.e., as [partly] already subsumed under, or included in, the “distributive law”, or is encountered as a theorem, corollary, or lemma.  First of all, note that diVision does not fully “distribute” over [ | ] Addition:

·       [‘‘‘left non-distributivity’’’, ‘V | A’]:  For some abc of R(a + b) ≠ 0

c ÷ (a + b) ≠ (c ÷ a) + (c ÷ b).

·       [‘‘‘right distributivity’’’, ‘V | A’]:       For all abc of Rc ≠ 0

(a + b) ÷ c   =   (a ÷ c) + (b ÷ c).

The (a + b) ≠ 0 and c ≠ 0 proviso’s are necessary, in these assertions about ‘V | A’, because diVisions by zero invoke a value that resides beyond the ‘‘‘number-space’’’of the set R.

But the second sub-category of Cqnt  = Cq[LA + VSi] pertains directly to the interaction of the diVision operation with the Subtraction operation, not with the Addition operation [although, given that the set R includes “signed numbers” with “negative” signs, i.e., “additive inverses”, as well as the subtraction operation-sign, additions can also express subtractions, i.e., if b  =  -d
then a + b   =   a - d], e.g.:

·       [‘‘‘left non-distributivity’’’, ‘V | S’]:  For some a, b, c of R,

(a - b) ≠ 0, c ÷ (a - b)      (c ÷ a) - (c ÷ b)

-- e.g. --

3 ÷ (1 - 2 )  =  -3    ≠   1.5  =  ( 3÷1) - (3÷2).

·       [‘‘‘right distributivity’’’, ‘V | S’]:       For all abc of R, c ≠ 0

(a - b) ÷ c   =   (a ÷ c) - (b ÷ c).

If we re-express subtractions as additions, and divisions as multiplications -- which the “Real” number arithmetic enables us to do, since it includes ratios and “multiplicative inverses”, as well as “additive inverses”, we see that the “Real” arithmetic’s '''rules''' for ‘V | S’ are partly implicit in the ‘L | A’ '''rules''', e.g., if we set b  =  -d and c  =  1÷e  =  1/e, e ≠ 0:

·       [an aspect of ‘L | A’]:  For a, -d, 1/e of R, (a + d) ≠ 0

1/e x 1/(a + d)   =   1/((e x a) + (e x b))

-- e.g. --

1/(1/3) x 1/(1 + -2 )    =

-3   =

1/( 1/3 x 1)  + ( 1/3 x -2) ).

·       [‘‘‘right distributivity’’’, ‘L | A’]:  For all a, -d, 1/e of R, e ≠ 0

(a + d) x 1/e  =  (a x 1/e) + (d x 1/e),

-- e.g. --

(1 + -2) x (1/(1/3))   =

-3   =

( 1 x (1/(1/3)) )   +   ( -2 x (1/(1/3)) ).

The subscript ‘VSi’ component of the subscript ‘[LA + VSi]’ of categopry-symbol --

Cq[LA + VSi]  = Cqnt

-- can thus be interpreted as calling attention systematically and explicitly, if somewhat redundantly, to the specific ‘V | S’ rules, which differ from the generic ‘L | A’ rules, in that the ‘V | S’ rules require the making explicit of special restrictions [e.g, 0 denominators not allowed], etc., as we have seen above.

We may therefore write our full solution for step s = 2 as --

C)-|-(2      =    Ct2^2     =     Ct4      =

Ct   + Cn   +  Cqnt  + Ce

multiplications & divisions   +

n & t interactions   +

exponentiations & de-exponentiations.

Our categorial progression so far can be summarized textually as below.

The ‘qualo-fractal’ content-structure of this psychohistorical dialectic to step 2 can be summarized as follows --

exponentiations & de-exponentiations “contain” multiplications & divisions,

which, in turn, “contain” additions & subtractions.

The “five symbolic-elements expression” for this model is thus Ct2^2 [five if we count the underscore under the t as a separate “symbolic-element”].

The systematic dialectic of the basic operations of arithmetic ‘‘‘presently’’’ and “standardly” ends here, at step s = 2, with the category named exponentiations & de-exponentiations as its ‘meta-meristemal’ category, or ‘‘‘vanguard’’’ category.

We like the compactness of the CQ_ representation of this systematic dialectic, whose “final step” can be modeled via a single equation --

C)-|-(2      =      Ct   + Cn   +  Cqnt  + Ce.

However, there is also the alternative of expressing each step of this dialectic by two separate equations, using the NQ_ dialectical algebra instead, e.g. --

verse)-|-(2      =      CA   + CL   +  CqLA  + CP

for the ‘‘‘verse’’’ operations.

inverse)-|-(2     =      CS   + CV   +  CqVS  + CR

for the ‘‘‘inverse’’’, or ‘‘‘reverse’’’, operations.

Stage 3.  To iterate our CQ_ ‘meta-equation’,

C)-|-(s   =   Ct2^s   =   ( Cq[A + Si] )2^s

for step s = 3, is to iterate the systematic presentation of the domain of basic arithmetical operations beyond the “basic”, beyond the “present”, beyond the conventional conclusion of that presentation, and beyond the “systematic reconstruction” of this domain at present, to a somewhat “preconstructive” -- somewhat “predictive” -- extrapolation of its possible future.  However, as we shall see, we have already encountered units of the “vanguard” term of step s = 3, in this very text.

Let’s see what are the additional category-descriptions that this step s = 3 ‘self-iteration’ generates:

C)-|-(3   =  Ct2^3  =  Ct8  =  ( Ct4 )2  =

(Ct   + Cn   +  Cqnt  + Ce )2   =

(Ct   + Cn   +  Cqnt  + Ce ) x (Ct   + Cn   +  Cqnt  + Ce )   =

Ct   + Cn  + Cqnt  +  Ce   +

Cqet  +  Cqen  +  Cqent  +  Cqee

[---)

Cq[1 + 1i]   + Cq[2 + 2i]   +  Cq[3 + 3i]  + Cq[4 + 4i]  +

Cq[5 + 5i]   Cq[6 + 6i]   +  Cq[7 + 7i]  + Cq[8 + 8i].

We know -- from past experience, narrated above -- how to “solve-for”category

Cqee   Cq[PP + RRi].

It describes a category “containing” two sub-categories, the first sub-category being one of Powers OF Powers’, and the second sub-category being one of Root-extractions OF Root-extractions’.

The first sub-category is one each of whose units / operations is an exPonentiation OF exPonentiations’, i.e., each of which is a meta-exPonentiation’, such that each such meta-exPonentiation’ operation is made up out of a multiplicity of  exPonentiation operations.

But that is precisely the new operation that we have encountered in this text, at the heart of the Seldon Functions in general, and at the heart of our CQ_ ‘meta-equation’ --

C)-|-(s   =   Ct2^s

-- specifically.

A unit increment in the ‘meta-exponent’ of the ‘starting-category’ symbol of that ‘meta-equation’, corresponding to a unit increment in its step-value, s, is equivalent to a two-fold exponentiation of that ‘starting-category’ symbol. e.g. --

Ct2 = Ct2^1;

(Ct2 )2  =  (Ct2^1 )2   =   (Ct2^1 )2^1 = Ct2^(1+1)  =  Ct2^2

-- because repeated exponents mutually multiply, and because 'meta-exponents' of exponents add together when those 'meta-exponents' have the same exponents as their bases, and are multiplied together.

Let’s call this sub-category Hyper-exponentiation, or H for short.

The second sub-category should be for operations which are de-exponentiations OF de-exponentiations’, i.e., which are meta-de-exponentiations’, such that each meta-de-exponentiation’ operation is made up out of a multiplicity of  de-exponentiation operations.

That is, the second sub-category should be one of multi-de-exponentiation’ operations, “made up out of multiple [repeated] de-exponentiation operations, each denoted by '', the sign of the "square-root"-extracting operation.

We can use the ‘self-example’, exemplified in this very text, to illustrate this process:

√√√Ct +Cn +Cqnt+Ce +Cqet+Cqen+Cqent+Cqee

√√√Ct2^3    =

√√Ct   + Cn  + Cqnt  Ce     =

√√Ct2^2    =

_____________
Ct   + Cn

Ct2^1    =

Ct2^(3-3)  =

Ct2^0  =

Ct
.

Let’s call this sub-category De-Hyper-exponentiation, or D for short.

We may thus “assert” our solution as follows  --

Cqee  =  Cqm  =  Cm   =   Cq[H + Di]   [---)  Cq[8 + 8i].

Our step s = 3 equation-model, as we have solved it so far, thus now looks like this --

C)-|-(3   =   Ct2^3  =  Ct8   =

Ct   + Cn  + Cqnt  + Ce   + Cqet  + Cqen  + Cqent
Cm

-- since we have not yet determined which actual categories of the ‘basic arithmetical operations’ domain are described by the algorithmically-generated ‘category-description’ symbols Cqet, Cqen, and Cqent, if any.

But we already know how to characterize the possible categories that these three category-symbols “call for”, viz. --

· Cqet   [---)  Cq[5 + 5i] “calls for” a '''hybrid''' category for the kind of ‘meta-operation’, or ‘operation of operations’,
that combines the e and t operations.
· Cqen  [---)  Cq[6 + 6i] “calls for” a '''hybrid''' category for the kind of ‘meta-operation’, or ‘operation of
operations’, that combines the e and n operations.
· Cqent [---)  Cq[7 + 7i] “calls for” a '''hybrid''' category for the kind of ‘meta-operation’, or ‘operation of
operations’, that combines the e and the Cqnt.

We may thus write our full solution for step s = 3 as --

C)-|-(3   =  Ct2^3  =  Ct8   =

Ct   + Cn  + Cqnt  + Ce   + Cqet  + Cqen  + Cqent  +
Cm   =

multiplications & divisions

n with t interactions        +

exponentiations & de-exponentiations      +

e with t interactions      +

e with n interactions

e with n & t interactions       +

meta-exponentiations & de-meta-exponentiations.

Our categorial progression so far can be summarized textually as below.

The ‘qualo-fractal’ content-structure of this psychohistorical dialectic through step 3 can be summarized as follows --

meta-exponentiations & de-meta-exponentiations “contain”

exponentiations & de-exponentiations, which “contain”

multiplications & divisions, which “contain”

The “five symbolic-elements expression” for this model, up to this step, is thus Ct2^3.

The meaning mnemonically compressed into the 5 symbolic-element expression Ct2^3 can be depicted as follows --