**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*

_{C}**--**

__Q___

*Systematically**Presented***via a**

**5**-

**Symbol Expression**.

Dear Readers,

Below is the third and final of the three simpler examples of dialectical models, presented as an interlude, before concluding the series on the

**F**.

__.__

**E**__. Psychohistorical-Dialectical 'Meta-Equation' of Human-Social Formation(s) 'Meta-Evolution', with the problematics of the__

**D**__ation-state social formation, and with its meta-model-predicted successor-formations.__

**n**Enjoy!

Regards,

Miguel

**. This model is more “**

__Introduction__**omplex” [pun intended] than the models of “TV-Series”, and of ‘Modern Computerware’, presented earlier in this sub-series, because it requires some “domain-expertise” -- or, at least, some “domain familiarity” -- with respect to the domain of the so-called “**

__C__**omplex Numbers”, the set standardly denoted by the symbol**

__C__**C**.

The ‘axioms-system’ of the arithmetic of the

**C**numbers, which we denote by

**, is the**

__C__**6**th system of arithmetic in the following standard order of standard arithmetics, with our light-spectrum

**o**

**r**

**d**

**i**

**n**

**a**

**l**color-coding added --

**,**

__N__**,**

__W__**,**

__Z__**,**

__Q__**,**

__R__**,**

__C__-- for the “

__atural”, “__

**N**__hole”, “Integer”, “Rational”, “__

**W**__eal”, and “__

**R**__omplex” arithmetics, respectively. About the ‘Goedelian Dialectic’ of these systems, see: http://www.dialectics.org/dialectics/Vignettes.html, Vignette__

**C****#**

**4**.

We will, in this blog-entry, use
the

**omplex-Numbers-subsuming version of the**__C__**F**.**.**__E__**. ‘first dialectical algebra’ to construct, and to “solve”, a “heuristic”, ‘intuitional’ model of a**__D__*systematic***of the domain of “the basic operations of arithmetic” -- encompassing both its***presentation**‘‘‘*[e.g., addition, multiplication, exponentiation] and its__verse__’’’*“*[e.g., subtraction, division, root-extraction]__in__verse”*operations, jointly, via***s with**__q__**C**subscripts, which we also reference as_{C}**s.**__q__
The models that we usually
narrate here are constructed by interpreting the generic

_{N}**version of the**__Q___**F**.**.**__E__**. ‘first dialectical algebra’ [ see E.D. Brief # 5 and its Preface ], or, at most-advanced, by interpreting the generic**__D___{W}**version of that algebra [ see E.D. Brief #6 and its Preface ], with the subscripts of the**__Q____{N}**or**__q___{W}**‘meta-numerals’ drawn from the number-space**__q__**N**__=__**{****1**,**2**,**3**,**...****}**, or from the number-space**W**

__=__**{**

**0**,

**1**,

**2**,

**3**,

**...**

**}**, respectively.

This time, the subscripts of
the

wherein

FYI: The

_{C}**s will be drawn from the standard number-space**__q__

**C**__=__**{****R****+****R****i****}**,wherein

**R**denotes the space of the standard so-called “**R**eal” numbers, and where the**i**unit stands for so-called "**maginary" unity, the positive square root of**__i__**-****1**.FYI: The

**ric**__gene__**omplex number is often expressed as**__C__**a****+****b****i**, with**a**an element of**R**, and also with**b**an element of**R**, or as**z****=****x****+****y****i**, 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

_{C}**version of the**__Q___**F**.**.**__E__**. ‘first dialectical algebra’.**__D__
We use the

_{C}**language this time, as it allows us to present both**__Q___*‘‘‘*&__verse__’’’*‘‘‘*operations in a single model.__re__verse’’’
Herein we mean, by the word,

*‘‘‘*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**systematic***“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

That is, it will

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

**be a model of a ‘self-advancing’**__not____historical__**of ontology, with each historical epoch containing both old ontology, inherited from past historical epochs, plus new ontology, ontology that had**__progression__*never appeared***-- in past historical epochs --***before***the**__until__*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***¿***kind of thing, the**__least__complex**kind of thing, the least inclusive kind of thing, which inheres in this domain**__simplest__*?**”*-- 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 “**”, and its**__A__ddi__t__ions*“*verse operations”, or__in__*‘‘‘*verse operations’’’, or “__re__**”, are the simplest ancestors, the ultimate units, of basic arithmetical operations, ingredient in every one of the more complex operations of that domain.**__S__ubtrac__t__ions
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

in an “identification”, an “interpretation”, or an “assignment” [ ‘

_{C}**, or as**__t___{ C}__q__**, denoting the “**_{t}**omplex” combination of the “**__C__**” sub-category of elementary**__A__dditions**eal arithmetic basic operations, with the sub-category of “**__R__**”, and identifying that combination of**__S__ubtractions**sub-categories with the**__speci__fic__gene__ric**category symbol of our***first***category-arithmetic model, namely, with the symbol --**__gene__ric_{}_{C}__q___{[}_{1}_{ + }_{1}_{i}**,**_{]}in an “identification”, an “interpretation”, or an “assignment” [ ‘

**[---)**’ ] that we indicate by writing:_{}_{C}__t____=___{C}__q___{t}**=**_{C}__q___{[}_{A}_{ }_{+}_{ }_{S}_{i}_{]}**[---)**_{C}__q___{[}_{1}_{ + }_{i}**.**_{]}
Our model then, will take the
form of an “interpeted”,

**equation, assigned to the**__speci__fic**equation, like this --**__gene__ric

_{C}

__)-|-(__

_{s}

**=**

_{C}

__t__

^{2^}

^{s}

__=__

**(**

_{C}

__q___{[}

_{A}

_{ }

_{+}

_{ }

_{S}

_{i}

_{]}_{ }

**)**

^{2^}

^{s}**[---)**

_{}

_{C}

__|-|-|__

_{h}

**=**

_{C}

__h__

^{2^}

^{h}

__=__

**[**

_{C}

__q___{[}

_{1}

_{ }

_{+}

_{ 1i}

_{]}_{ }

**]**

^{2^h}

-- with the variable

We will not, here, further recount the [Marxian] method of

For more regarding that

**s**indicating the**tep in our**__s__*systematic**method of***that the ‘accumulation of categories’, denoted by***presentation*_{C}__)-|-(__**, represents.**_{s}We will not, here, further recount the [Marxian] method of

**that was used to arrive at the starting category of this***systematic discovery**systematic***.***presentation*For more regarding that

*method of***, see Marx,***discovery***, Penguin Books [London:**__Grundrisse__**1972**], pp.**100**-**101**.

__Stage__**. Our initial**

__0__**tep --**

__s__**tep**

__s__**s**

**=**

**0**-- contains only our starting category,

_{}

_{C}

__t__

__=__

_{C}

__q__

_{t}

**=**

**q**_{[}

_{A}

_{ + Si}**--**

_{]}

_{C}

__)-|-(__

_{0}

**=**

_{C}

__t__

^{2^}

^{0}

**=**

_{C}

__t__

^{1}

**=**

_{C}

__t__

**=**

**q**_{[}

_{A}

_{ }

_{+}

_{ Si}

_{]}**[---)**

_{C}

__q___{[}

_{1}

_{ }

_{+}

_{ }

_{1}

_{i}

_{]}

_{ }
-- because

and because

**2**“raised” to the power**0**--**2**-- is just^{0}**1**,and because

_{C}**“raised” to the power**__t__**1**is just_{C}**.**__t__

__Stage__**. It is when we get to the next**

__1__**tep after**

__s__**tep**

__s__**s**

**=**

**0**, namely, to

**tep**

__s__**s**

**=**

**1**, that our equation-model gives us back something initially “unknown” -- and, therefore, something ‘‘‘algebraical’’’, not merely something ‘‘‘arithmetical’’’: something to “solve-for” --

__)-|-(__

_{1}

**=**

_{C}

__t__

^{2^}

^{1}

**=**

_{C}

__t__

^{2}

**=**

_{C}

__t__**x**

_{C}

__t__

**=**

_{C}

__q___{[}

_{A}

_{ + Si}

_{]}**x**

_{C}

__q___{[}

_{A}

_{ + Si}

_{]}

**=**

_{C}

__q___{[}

_{A}

_{ + Si}

_{]}

_{ }**+**

_{C}

__q___{[}

_{AA}

_{ + SSi}

_{]}

**=**

_{C}

__q__

_{t}

_{ }**+**

_{C}

__q__

_{tt}
-- because

**2**“raised” to the power**1**--**2**-- is just^{1}**2**, and because our rule for multiplying a generic category, call it_{C}__q___{[}_{X + Yi}_{]}**=**_{C}__q___{Z}__=___{C}**, “by”, or “into”, itself, is, for subscripts**__Z__**X**and**Y**denoting*category symbols, and for subscript*__sub__-**Z**denoting a category-symbol, simply --

_{C}

__q___{[}

_{X + Yi}

_{]}**x**

_{C}

__q___{[}

_{X + Yi}

_{]}

**=**

_{C}

__q___{[}

_{X + Yi}

_{]}

_{ }**+**

_{C}

__q___{[}

_{XX + YYi}

_{]}

**=**

_{C}

__Z__

_{ }**+**

_{C}

__q__

_{ZZ}
-- and for

**x**and**y**denoting “**R**eal” numbers --

_{C}

__q___{[}_{x}

_{ + }

_{y}

_{i}

_{]}**x**

_{C}

__q___{[}_{x}

_{ + }

_{y}

_{i}

_{]}

**=**

_{C}

__q___{[}_{x}

_{ + }

_{y}

_{i}

_{]}

_{ }**+**

_{C}

__q___{[}

_{(}

_{x}

_{ + }

_{x}

_{) + (}

_{y}

_{ + }

_{y}

_{)i}

_{]}

**=**

_{}

_{C}

__q___{[}_{1x}

_{ + 1}

_{y}

_{i}

_{]}

_{ }**+**

_{C}

__q___{[}_{2x}

_{ + 2}

_{y}

_{i}**.**

_{]}__Note again__: Herein,

_{C}**denotes the**

__q__**category ‘**

__gene__ric**ualifier’ with “Complex” subscripts.**

__q__
The subscripts that come
after it are

**category descriptors.**__speci__fic**But how do we discover what the resulting, initially “unknown”, or ''algebraical'', ‘category-description’, here**

*¿*

_{C}

__q__**,**

_{tt}

__means__?
Well, the

**rule to “solve-for” the categorial**__gene__ric**of such symbols is that, if we know what is meant by category**__meaning___{C}__q___{Z}**=**_{C}**, then the symbol**__Z___{C}__q__**describes a category each of whose units is a ‘**_{ZZ}_{C}__Z____OF___{C}**s’, that is, a category for a different kind of units, called**__Z__*‘***meta**-_{C}**s’, each such unit being made up out of a**__Z__**of those units of which the category of the**__multiplicity___{C}**s is made up.**__Z__
To be

**with this rule, in our example-model,**__speci__fic_{C}__q__**specifies a “**_{ZZ}**omplex” of two sub-categories.**__C__
Each of the units of the

**sub-category, the sub-category of the***first**‘‘‘*operations, must be an**verse**’’’*‘*__A__**ddition**__OF____A__**ddition**s’ that is, must be a*‘***meta**-**’, such that each**__A__ddition*‘***meta**-**’ is made up out of a**__A__ddition**of “mere”**__multiplicity__**s.**__A__ddition
Each of the units of the

**sub-category, the sub-category of the***second**‘‘‘*operations, must be a**verse’’’**__in__*‘*__S__**ubtraction**__OF____S__**ubtraction**s’, that is, must be a*‘***meta**-**’, such that each such**__S__ubtraction*‘***meta**-**’ is made up out of a**__S__ubtraction**of “mere”**__multiplicity__**s.**__S__ubtraction
Our experiences of / "in" the domain of 'the basic operations of arithmetic' suggest that such operations do “presently” exist in the domain of “

**eal” arithmetic.**__R__
“Multiplicatio

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

**” is a basic arithmetical operation that is “made up out of multiple [repeated] subtractions”, viz.,**__n__**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 [

**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!**__not__
A letter that the spelled
names of these two kinds of operations have in common is “

**n**”.
Therefore, we shall
name/symbolize our

**category as***second*_{C}**, or as**__n___{ C}__q__**, denoting the “**_{n}**omplex” combination of the “**__C__**mu**__L__**tiplications**” sub-category of elementary**eal arithmetic basic operations, with the sub-category of “**__R__**di**__V__**isions**”, and identifying that combination of**sub-categories with the**__speci__fic__gene__ric**category symbol of our***second***category-arithmetic model, namely, with the**__gene__ric**category-symbol**__gene__ric_{C}__q___{[}_{2}_{ + }_{2}_{i}**.**_{]}
We may “assert” our solution
as follows:

_{C}

__q__

_{tt}**=**

_{C}

__q__

_{n}

__=__

_{C}

__n__

__=__

_{C}

__q___{[}

_{L}

_{ }

_{+}

_{ Vi}

_{]}

**=**

_{C}

__q___{[}

_{AA}

_{ }

_{+}

_{ SSi}

_{]}

**[---)**

_{C}

__q___{[}

_{2}

_{ + }

_{2}

_{i}**.**

_{]}
Again, what is

**about the relationship between**__dialectical___{C}**and**__t___{C}__t__**, or**^{2}_{C}__t__**x**_{C}**, or**__t___{C}__t__*of*_{C}**, or**__t___{C}__t__**(**_{C}__t__**)**, the relationship of what we call*‘*, or**meta**-**unit**-**ization**’*‘*«**meta**-**»***monad**-**ization**’*, between_{C}**and its already presently existing, ‘**__t__**supplementary**other’,_{C}**, is that this relationship is a synchronic**__n__*double**-*«**» relationship.***aufheben*
That is, each single “unit”
of the “

**mu**__L__**tiplicatio**__n__**s**” sub-category of category_{C}**, i.e., each typical individual “multiplicatio**__n__**” operation, is a**__n__**, and also a***negation***, by way of also being an***preservation**elevation**to**the*/*forming**the*“higher” / more inclusive “**mu**__L__**tiplicatio**__n__**s**” sub-category / level / scale, of a**[***whole**sub**-*]*group***of****of the “***unit*__s____A__ddi__t__**ions**” sub-category / level / scale of the_{C}**category.**__t__
Likewise, each single “unit”
of the “

**di**__V__**isio**__n__**s**” sub-category of category_{C}**, i.e., each typical individual “divisio**__n__**” operation, is a**__n__**, and also a***negation***, by way of also being an***preservation**elevation**to**the*/*forming**the*“higher” / more inclusive “**di**__V__**isio**__n__**s**” sub-category / level / scale, of a**[***whole**sub**-*]*group***of****of the “***unit*__s____S__ubtrac__t__**ions**” sub-category / level / scale of the_{C}**category.**__t__So, our full solution to the

**tep**

__s__**s**

**=**

**1**equation of our model is --

_{C}

__)-|-(__

_{1}

**=**

_{C}

__t__

_{ }**+**

_{C}

__n__

**=**

**A**

__ddi__

__t__

__ions &__

__S__

__ubtrac__

__t__

__ion__

__s__

_{ }**+**

__Mu__

__L__

__tiplicatio__

__n__

__s__

__& Di__

__V__

__isio__

__n__

__s__**[---)**

_{C}

__q___{[}

_{1}

_{ + }

_{1}

_{i}

_{]}

_{ }**+**

_{ C}

__q___{[}

_{2}

_{ + }

_{2}

_{i}**.**

_{]}
If this model is working
right,

__A____ddi____t____ion____s____&____S____ubtrac____t____ion__**is the**__s__**category of the domain of ‘basic arithmetical operations’;***simplest*__Mu____L____tiplicatio____n____s____& Di____V____isio____n__**is the**__s__**category of that domain.***next more complex*

__Stage__**.**

__2__**What additional ‘category-specifications’ do we generate in our next step,**

*¿***tep**

__s__**s**

**=**

**2**, that need “solving-for”

*?*
Let’s find out:

_{C}

__)-|-(__

_{2}

**=**

_{C}

__t__

^{2^2}

**=**

_{C}

__t__

^{4}

**=**

**(**

_{C}

__t__

^{2}^{ }

**)**

^{2}

**=**

**(**

_{ C}

__t__

_{ }**+**

_{C}

__n__^{ }

**)**

^{2}

**=**

**(**

_{ C}

__t__

_{ }**+**

_{C}

__n__

^{ }**)**

**x**

**(**

_{ C}

__t__

_{ }**+**

_{C}

__n__**)**

**=**

_{C}

__t__

_{ }**+**

_{C}

__n__

_{ }**+**

_{C}

__q__

_{n}

_{t}

**+**

_{C}

__q__**.**

_{nn}
This result arises by way of
two key rules of categorial algebra, plus the

**rule for multiplication when one category-symbol is multiplied by a different category-symbol [we used a**__gene__ral**case of this**__spec__ial**rule, for the case where the same category-symbol is multiplied by itself, in**__gene__ral**tep**__s__**s****=****1**, above] --**1**.

**case:**

__gene__ral

_{C}

__q__

_{Y}**x**

_{C}

__q__

_{X}

**=**

_{C}

__q__

_{X }**+**

_{C}

__q__

_{YX}

**=**

_{C}

__X__

_{ }**+**

_{C}

__q__**;**

_{YX}**case:**

__speci__al

_{C}

__q__

_{X}**x**

_{C}

__q__

_{X}

**=**

_{C}

__q__

_{X }**+**

_{C}

__q__

_{XX}

**=**

_{C}

__X__

_{ }**+**

_{C}

__q__**.**

_{XX}**2**.

_{C}

__q__

_{x}**+**

_{C}

__q__

_{x}

**=**

_{C}

__q__**; 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.**

_{x}**3**. There is no

_{C}

__q__**such that**

_{w}

_{C}

__q__

_{x}**+**

_{C}

__q__

_{y}

**=**

_{C}

__q__**;**

_{w}*different*category-symbols, added together [as opposed to being ‘‘‘multiplied’’’],

**to a single category-symbol, just like in the proverbial case of ‘**

*do not*__reduce__

__a__

__pples__**+**

**’, or**

__o____ranges__

__a__**+**

**.**

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

_{C}__q__**.**_{nn}
It describes a category
“containing” two sub-categories, the

**sub-category being one of***first**‘***mu**__L__**tiplication**__s____OF__**mu**__L__**tiplication**s’, and the**sub-category being one of***second**‘***di**__V__**ision**__s____OF__**di**__V__**ision**s’.
The

**sub-category is one each of whose units / operations is a***first**‘***mu**__L__**tiplication**__OF__**mu**__L__**tiplication**s’, i.e., each of which is a*‘***meta**-**mu**__L__**tiplication**’, such that each such*‘***-**__meta__**mu**__L__**tiplication**’ operation is made up out of a multiplicity of**mu**__L__**tiplication**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**-***mu**__L__**tiplication**’ operations -- i.e., of “__e____x____P__**”: “ex**__onentiations__**onentiation” is a basic arithmetical operation which is “made up out of multiple [repeated]**__P__**mu**__L__**tiplication**operations, viz. --**2**

^{3}

__=__**2 xx**

**3**

__=__**2**

**x**

**2**

**x**

**2**

**=**

**8**

**≠**

**9**

**=**

**3**

**x**

**3**

__=__**3 xx**

**3**

__=__**3**.

^{2}
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.

Generally, each order will return a different result.

The

**sub-category should be, per our standard method, one each of whose units / operations is a**

*second**‘*

**di**

__V__**ision**

__OF__**di**

__V__**ision**s’, i.e., each of which is a

*‘*

**meta**-**di**

__V__**ision**’, such that each such

*‘*

**-**__meta__**di**

__V__**ision**’ operation is made up out of a multiplicity of

**di**

__V__**ision**operations.

That is, the

**sub-category should be one of**

*second**‘*

*multi**-*

**di**

__V__**ision**’ operations, “made up out of multiple [repeated]

**di**

__V__**ision**operations.

We interpret this to be the “inverse” operation of

*‘*ex

**-**__de__**onentiation’, or of “**

__P__**n**th

**oot extraction”.**

__R__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

**oot extraction operation.**

__R__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 “

**n**th” 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

The method involves guessing a “starting estimate” for the square root sought, followed by repeated stages of, well,

‘Formulaically’, the next better estimate of the square’s square root,

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

**al case of the more**__speci__**ral “Newton’s Method”, but predates the discovery of “Newton’s Method” by many centuries.**__gene__The method involves guessing a “starting estimate” for the square root sought, followed by repeated stages of, well,

**di**__V__**ision**-- division of the square by the current best estimate of its square root -- followed by, well,**di**__V__**ision**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****di**__V__**ision**’ just described.‘Formulaically’, the next better estimate of the square’s square root,

**x****, is derived from the previous best estimate,**_{n+1}**x****, by**_{n}**dividing**the square,**S**, by the previous best estimate,**x****, summing**_{n}**x****and**_{n}**S÷x****, then**_{n}**dividing**that sum by**2**:

**x**_{n+1}**=****(****x**_{n}**+ S÷x**_{n}**)****÷****2**.Let us apply this method to “extracting” the “square root” from the “square”,

**9**, with “starting estimate” of

**x**

_{1}**=**

**2**:

**....**

__n____Current Best Estimate__(

**x**

**)...**

_{n }

__Di__

__V__

__ide____Square by That Estimate__(

**S÷x**

**)..**

_{n }

__Di__

__V__

__ide____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

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

**n****=****3**, with "rounding" as shown above, the method reaches a “fixed point” / “equilibrium” at**x**_{3+...}**=****3**, which**the positive square root of**__is__**9**.Thus we see in what sense, in this method at least, square root extraction is made up out of repeated

**di**__V__**ision**s.
We may thus “assert” our
solution as follows:

_{C}

__q__

_{nn}**=**

_{C}

__q__

_{e}

__=__

_{C}

__e__

__=__

_{C}

__q___{[}

_{P}

_{ }

_{+}

_{ Ri}

_{]}

**=**

_{C}

__q___{[}

_{LL}

_{ }

_{+}

_{ VVi}

_{] }

**[---)**

_{C}

__q___{[}

_{4}

_{ + }

_{4}

_{i}**.**

_{]}
Our

**tep**__s__**s****=****2****equation-model, as we have solved it so far, thus now looks like this --**

_{C}

__)-|-(__

_{2}

**=**

_{C}

__t__

^{2^2}

**=**

_{C}

__t__

^{4}

**=**

_{C}

__t__

_{ }**+**

_{C}

__n__

_{ }**+**

_{C}

__q__

_{n}

_{t}

**+**

_{C}

__e__**[---)**

_{C}

__q___{[}

_{1}

_{ + }

_{1}

_{i}

_{]}

_{ }**+**

_{ C}

__q___{[}

_{2}

_{ + }

_{2}

_{i}

_{]}

_{ }**+**

_{C}

__q___{[}

_{3}

_{ + }

_{3}

_{i}

_{]}

**+**

_{ C}

__q___{[}

_{4}

_{ + }

_{4}

_{i}

_{]}
-- since we have not yet
determined which actual category of the ‘basic arithmetic operations’ domain is
described by the algorithmically-generated symbol

_{C}__q___{n}**-- if any, i.e., if**_{t}_{C}__q___{n}**is not an “empty category”, “inoperative” for this domain.**_{t}
When, as a component of

**(**_{ C}__t___{ }**+**_{C}__n__^{ }**)****x****(**_{ C}__t___{ }**+**_{C}__n__**)**, the “higher-complexity” category,_{C}**, operates upon / “multiplies” the “lower-complexity” category,**__n___{C}**--**__t__

_{C}

__n__**x**

_{C}

__t__

_{ }**=**

_{ C}

__t__**+**

_{C}

__q__

_{n}

_{t}

_{ }**=**

_{ }

_{C}

__q___{[}

_{A}

_{ }

_{+}

_{ Si}

_{]}**+**

_{C}

__q___{[}

_{L}

_{A}

_{ }

_{+}

_{ VSi}

_{]}

_{}
--

**speaking, the categorial relationship to be called to the user’s attention by this operation, in this ‘categorial arithmetic’, is, again, a synchronic «**__gene__rically**» relationship, this time, that between***aufheben*_{C}**and**__t___{C}__q___{n}**.**_{t}
It calls the user to search
that user’s knowledge and memory of the domain in question -- in this

Thus, the additional category thereby presented,

**case, the domain of ‘basic arithmetical operations’ -- for a category which represents an “uplift” of category**__speci__fic_{C}**entities to the level of the entities native to category**__t___{C}**, thereby “canceling” the**__n___{C}**-type entities concerned, at their own native level, but, by the same token, “preserving” those category**__t___{C}**entities at the**__t___{C}**level, combining**__n___{C}**and**__n___{C}__t__**ualities, in the relationship of “elevation” of those category**__q___{C}**entities within the level typical of category**__t___{C}**entities.**__n__Thus, the additional category thereby presented,

_{C}__q___{n}**, signifies a category whose units are the**_{t}**of the***operational*__interactions___{C}**operations with the**__t___{C}**operations, as codified in the axioms, and/or theorems, and/or corollaries, and/or lemmas, and/or ‘‘‘rules’’’ of the “**__n__**R**eal Numbers” system of arithmetic.
The

**sub-category of the category***first*_{C}__q___{n}_{t}_{ }**=**_{C}__q___{[}_{L}_{A}_{ }_{+}_{ VSi}**answers to a sub-category description which connotes the way in which, or the ‘‘‘rules’’’ by which, the operation of**_{]}**mu**__L__**tiplicatio****“subsumes” the operation of**__n____A__ddi__t__**ion**, denoted herein by ‘**L****| A**’.
To our lights, this
sub-category-description connotes the elementary arithmetical phenomenon often named '''

**''', or the “**__D__istribution**”, e.g., of “**__D__istributive law**R**eal” 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**a**,**b**,**c**of**R**,

**c****x (a + b) = (c x a) + (c x b)**.
·
[“

**right distributivity**”]: For all elements**a**,**b**,**c**of**R**,

**(a + b) x c = (a x c) + (b x c)**.
The

**sub-category of the category***second*_{C}__q___{n}_{t}_{ }**=**_{C}__q___{[}_{L}_{A}_{ }_{+}_{ VSi}**answers to a sub-category description which connotes the ‘‘‘rules’’’ by which the operation of**_{]}**di**__V__**isio****“subsumes” the operation of**__n____S__ubtrac__t__**ion**, denoted ‘**V****| S**’.
To our lights, this
sub-category-description connotes a

*“*distributive rule” of “__-__**non****R**eal” arithmetic for ‘**di**__V__**isio****/**__n____S__ubtrac__t__**ion**’, although this rule is, typically, not an explicit one in presentations and in axiomatizations of “**R**eal” 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**di**__V__**isio****does**__n__**fully “**__not__**distribute**” over [ ‘**|**’ ]__A__ddi__t__**ion**:
·
[‘‘‘left

__non__*-***distributivity**’’’, ‘**V****| A**’]: For some**a**,**b**,**c**of**R**,**(a + b) ≠ 0**,

**c ÷ (a + b) ≠ (c ÷ a) + (c ÷ b)**.
·
[‘‘‘right

**distributivity**’’’, ‘**V****| A**’]: For all**a**,**b**,**c**of**R**,**c ≠ 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**di**__V__**isio****s by zero invoke a value that resides beyond the ‘‘‘number-space’’’of the set**__n__**R**.
But the

**sub-category of***second*_{C}__q___{n}_{t}_{ }**=**_{C}__q___{[}_{L}_{A}_{ }_{+}_{ VSi}**pertains directly to the interaction of the**_{] }**di**__V__**isio****operation with the**__n____S__ubtrac__t__**ion**operation,__with the__**not**__A__ddi__t__**ion**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**a**,**b**,**c**of**R**,**c ≠ 0**,

**(a****-****b) ÷ c = (a ÷ c)****-****(b ÷ c)**.
If we re-express subtractions
as additions, and divisions as multiplications -- which the “

**R**eal” number arithmetic enables us to do, since it includes ratios and “multiplicative inverses”, as well as “additive inverses”, we see that the “**R**eal” arithmetic’s '''rules''' for ‘**V****| S**’ are**implicit in the ‘***partly***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 ‘

_{V}

_{S}**’ component of the subscript ‘**

_{i}

_{[}

_{L}

_{A}

_{ }

_{+}

_{ VSi}**’ of category-symbol --**

_{]}

_{C}

__q___{[}

_{L}

_{A}

_{ }

_{+}

_{ VSi}

_{]}

_{ }**=**

_{C}

__q__

_{n}

_{t}-- can thus be interpreted as calling attention

**and explicitly, if**

*systematically**somewhat*redundantly, to the

**fic ‘**

__speci__**V**

**| S**’ rules, which differ from the

**ric ‘**

__gene__**L**

**| A**’ rules, in that the ‘

**V**

**| S**’ rules require the making explicit of

**al restrictions [e.g,**

__speci__**0**denominators not allowed], etc., as we have seen above.

A better

*/*

**interpretation of***the meaning of the*

**solution for**

_{C}

__q___{[}

_{VSi}

_{]}**sub-category of the**

_{ }

_{C}

__q___{[}

_{L}

_{A}

_{ }

_{+}

_{ VSi}

_{]}**category would be the [sub-]category of/for '''**

_{ }**Di**''', which can form a portal to the

__V__ided Differences__ifferential calculus, involving the Leibnizian__

**d***'*

**infinitesimal**

**d***operator,*

**ifference**'**d**-- and, which did, in part, [psycho]historically, actually serve as such, for Isaac Newton, in his pathway to that discovery:

**(Y - y) / (X - x) =**

**((y + delta(y)) - y) / ( (x + delta(x)) - x ) =**

**( (f(x + delta(x)) - f(x) ) / ( (delta(x) ) =**

**delta( f(x) ) / delta(x)**,

**such that**--

**d( f(x) ) / dx =**

**limit as delta(x) --> 0(delta( f(x) ) / delta(x)**.

We may therefore write our

*full*solution for

**tep**

__s__**s**

**=**

**2**

**as --**

_{C}

__)-|-(__

_{2}

**=**

_{C}

__t__

^{2^2}

**=**

_{C}

__t__

^{4}

**=**

_{}

_{C}

__t__

_{ }**+**

_{C}

__n__

_{ }**+**

_{C}

__q__

_{n}

_{t}

**+**

_{C}

__e__

__addi__

__t__

__ions & subtrac__

__t__

__ions__

_{ }**+**

__multiplicatio__

__n__

__s__

__& divisio__

__n__

__s__

_{ }**+**

__n__

__&__

__t__

__interactions__

_{ }**+**

__e__

__xponentiations & d__

__e____-__

__e__**.**

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

The ‘qualo-fractal’
content-structure of this psychohistorical dialectic to

**tep**__s__**2**can be summarized as follows --

__e__

__xponentiations & d__

__e____-__

__e__**“contain”**

__xponentiations__

__multiplicatio__

__ns__

__& divisio__**,**

__ns__
which, in turn, “contain”

__addi____t____ions & subtrac____t__**.**__ions__
The “five symbolic-elements
expression” for this model is thus

_{C}__t__^{2^}**[five if we count the underscore under the**^{2}**as a separate “symbolic-element”].**__t__
The systematic dialectic of
the basic operations of arithmetic ‘‘‘presently’’’ and “standardly” ends here,
at

**tep**__s__**s****=****2**, with the category named__e____xponentiations & d____e____-____e__**as its ‘meta-meristemal’ category, or ‘‘‘vanguard’’’ category.**__xponentiations__
We like the compactness of
the

_{C}**representation of this systematic dialectic, whose “final**__Q___**tep” can be modeled via a single equation --**__s__

_{C}

__)-|-(__

_{2}

**=**

_{C}

__t__

_{ }**+**

_{C}

__n__

_{ }**+**

_{C}

__q__

_{n}

_{t}

**+**

_{C}**.**

__e__
However, there is also the
alternative of expressing each

**tep of this dialectic by**__s__**, using the***two*__separate__equations_{N}**dialectical algebra instead, e.g. --**__Q___

_{verse}

__)-|-(__

_{2}

**=**

_{C}

__A__

_{ }**+**

_{C}

__L__

_{ }**+**

_{C}

__q__

_{L}

_{A}

**+**

_{C}**, for the**

__P__*‘‘‘verse’’’*operations;

_{inverse}

__)-|-(__

_{2}

**=**

_{C}

__S__

_{ }**+**

_{C}

__V__

_{ }**+**

_{C}

__q__

_{V}

_{S}

**+**

_{C}**, for the**

__R__*‘‘‘*, or

**verse’’’**__in__*‘‘‘*,

**verse’’’**__re__operations.

To isolate the

*of*

**first triads***from these paired*

**categories**

**dialectical***, we can use twin*

**categorial**-**combinatoric progressions****tep**

__s__**s**

**=**

**1**

**Triadic Seldon Function***'*

**[**

*]*

**meta**-*[*

**model***]*

**meta**-*as follows, using the modified notation also employed in the two images below --*

**equations**'

_{verse}

__)-|-(__

_{1}

**=**

_{O+}

__)-|-(___{1}**=**_{}

_{O+}

__A__

^{3^1}

**=**

**O+**

__A__

^{3}

**=**

_{O+}

__A__

_{ }**+**

_{O+}

__M__

_{ }**+**

_{O+}

__q__

_{M}

_{A}

**=**

**,**

_{O+}__A___{ }**+**_{O+}__M___{ }**+**_{O+}__D__for

*‘‘‘verse’’’*operations;

_{inverse}

__)-|-(__

_{1}

**=**_{O-}**=**

__)-|-(___{1}

_{O-}

__S__^{3^1}**=**

_{O-}

__S__^{3}**=**

_{}

_{O-}

__S__

_{ }**+**

_{O-}

__D__

_{ }**+**

_{O-}

_{}

__q__

_{D}

_{S}

**=**

_{O-}

__S___{ }**+**

_{O-}__D___{ }**+**

_{O-}

__V__**,**

for

*‘‘‘*

**verse’’’/**__in__*‘‘‘*, operations

**verse’’’**__re__-- which can then be depicted as follows --

_{}

_{C}

__)-|-(__

_{s}

**=**

_{C}

__t__

^{2^}

^{s}

__=__

**(**

_{C}

__q___{[}

_{A}

_{ }

_{+}

_{ }

_{S}

_{i}

_{]}_{ }

**)**

^{2^}**,**

^{s}for

**tep**

__s__**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

**tep**

__s__**s**

**=**

**3**, in this very text.

Let’s see what are the
additional category-descriptions that this

**tep**__s__**s****=****3**‘self-iteration’**rates:**__gene__

_{C}

__)-|-(__

_{3}

**=**

_{C}

__t__

^{2^}

^{3}

**=**

_{C}

__t__

^{8}

**=**

**(**

_{C}

__t__

^{4}^{ }

**)**

^{2}

**=**

**(**

_{C}

__t__

_{ }**+**

_{C}

__n__

_{ }**+**

_{C}

__q__

_{n}

_{t}

**+**

_{C}

__e__^{ }

**)**

^{2}

**=**

**(**

_{C}

__t__

_{ }**+**

_{C}

__n__

_{ }**+**

_{C}

__q__

_{n}

_{t}

**+**

_{C}

__e__

^{ }**)**

**x**

**(**

_{C}

__t__

_{ }**+**

_{C}

__n__

_{ }**+**

_{C}

__q__

_{n}

_{t}

**+**

_{C}

__e__**)**

**=**

_{C}

__t__

_{ }**+**

_{C}

__n__

_{ }**+**

_{C}

__q__

_{n}

_{t}

**+**

_{ C}

__e__

_{ }**+**

_{C}

__q__

_{e}

_{t}

**+**

_{C}

__q__

_{e}

_{n}

**+**

_{C}

__q__

_{e}

_{n}

_{t}

**+**

_{C}

__q__

_{ee }

**[---)**

_{C}

__q___{[}

_{1}

_{ + }

_{1}

_{i}

_{]}

_{ }**+**

_{ C}

__q___{[}

_{2}

_{ + }

_{2}

_{i}

_{]}

_{ }**+**

_{C}

__q___{[}

_{3}

_{ + }

_{3}

_{i}

_{]}

**+**

_{ C}

__q___{[}

_{4}

_{ + }

_{4}

_{i}

_{]}

**+**

_{ }

_{C}

__q___{[}

_{5}

_{ + }

_{5}

_{i}

_{]}

_{ }**+**

_{ C}

__q___{[}

_{6}

_{ + }

_{6i}

_{]}

_{ }**+**

_{C}

__q___{[}

_{7}

_{ + }

_{7}

_{i}

_{]}

**+**

_{ C}

__q___{[}

_{8}

_{ + }

_{8i}**.**

_{]}

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

_{}_{C}__q___{ee}**=**_{ C}__q___{[}_{PP}_{ }_{+}_{ RRi}**.**_{]}
It describes a category
“containing” two sub-categories, the

**sub-category being one of***first**‘*__P__**owers**__OF____P__**owers**’, and the**sub-category being one of***second**‘*__Root____-__**extractions**__OF____Root____-__’.**extractions**
The

**sub-category is one each of whose units / operations is an***first**‘***ex**__P__**onentiation**__OF__**ex**__P__**onentiation**s’, i.e., each of which is a*‘***meta**-**e****x**’, such that each such__P__onentiation*‘***-**__meta__**ex**__P__**onentiation**’ operation is made up out of a multiplicity of**ex**__P__**onentiation**operations.But that is precisely the new operation that we have encountered in this text, at the heart of the Seldon Functions in

**ral, and at the heart of our**

__gene__

_{C}**‘meta-equation’ --**

__Q___

_{}

_{C}

__)-|-(__

_{s}

**=**

_{C}

__t__

^{2^}

^{s}**--**

__speci__fically.

A unit increment in the
‘meta-exponent’ of the ‘starting-category’ symbol of that ‘meta-equation’,
corresponding to a unit increment in its

**tep-value,**__s__**s**, is equivalent to a two-fold exponentiation of that ‘starting-category’ symbol. e.g. --

_{C}

__t__

^{2}**=**

_{C}

__t__

^{2^}**;**

^{1}

**(**

_{C}

__t__

^{2}^{ }

**)**

^{2}

**=**

**(**

_{C}

__t__

^{2^}

^{1}^{ }

**)**

^{2}

**=**

**(**

_{C}

__t__

^{2^}

^{1}^{ }

**)**

^{2^}

^{1}

**=**

_{C}

__t__

^{2^(}

^{1}

^{+1)}

__=__

_{C}

__t__

^{2^}

^{2}
-- because repeated exponents
mutually

Let’s call this sub-category

The

**, and because**__multiply__*'meta-exponents' of*exponents**together when those 'meta-exponents' have the same exponents as their bases, and are multiplied together.**__add__Let’s call this sub-category

__H__**yper**-**exponentiation**, or**H**for short.The

**sub-category should be for operations which are***second**‘*__d____e____-____e__**s**__xponentiation____OF____d____e____-____e__**s’, i.e., which are**__xponentiation__*‘***meta**-__d____e____-____e__**s’, such that each**__xponentiation__*‘***meta**-__d____e____-____e__**’ operation is made up out of a multiplicity of**__xponentiation____d____e____-____e__**operations.**__xponentiation__
That is, the

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

**sub-category should be one of***second**‘*__multi__*-*__d____e____-____e__**’ operations, “made up out of multiple [repeated]**__xponentiation____d____e____-____e__**operations, each denoted by '**__xponentiation__**√**', the sign of the "square-root"-extracting operation.We can use the ‘self-example’, exemplified in this very text, to illustrate this process:

**√√√**

_{C}

__t__

_{ }**+**

_{C}

__n__

_{ }**+**

_{C}

__q__

_{n}

_{t}

**+**

_{C}

__e__

_{ }**+**

_{C}

__q__

_{e}

_{t}

**+**

_{C}

__q__

_{e}

_{n}

**+**

_{C}

__q__

_{e}

_{n}

_{t}

**+**

_{C}

__q__

_{ee }**=**

**√√√**

_{C}

__t__

^{2^}

^{3}

_{ }**=**

**√√**

_{C}

__t__

_{ }**+**

_{C}

__n__

_{ }**+**

_{C}

__q__

_{n}

_{t}

**+**

_{ C}

__e__

_{ }**=**

**√√**

_{C}

__t__

^{2^}

^{2}

_{ }**=**

**_____________**

**√**

_{C}

__t__

_{ }**+**

_{C}

__n__

_{ }**=**

**√**

_{C}

__t__

^{2^}

^{1}

_{ }**=**

_{}

_{C}

__t__

^{2^(}

^{3}^{-3)}

_{ }**=**

_{}

_{C}

__t__

^{2^}

^{0}

_{ }**=**

_{}

_{C}**.**

__t__Let’s call this sub-category

__D__**e**-

**H**

**yper**-

**exponentiation**, or

**D**

**for short.**

We may thus “assert” our solution
as follows:

_{}_{C}__q___{ee}**=**_{C}__q___{m}__=___{C}__m__**=**_{ C}__q___{[}_{H}_{ }_{+}_{ Di}_{]}_{ }**[---)**_{C}__q___{[}_{8}_{ + }_{8i}**.**_{]}Our

**tep**

__s__**s**

**=**

**3**

**equation-model, as we have solved it so far, thus now looks like this --**

_{C}

__)-|-(__

_{3}

**=**

_{C}

__t__

^{2^}

^{3}

**=**

_{C}

__t__

^{8}

**=**

_{}

_{C}

__t__

_{ }**+**

_{C}

__n__

_{ }**+**

_{C}

__q__

_{n}

_{t}

**+**

_{C}

__e__

_{ }**+**

_{C}

__q__

_{e}

_{t}

**+**

_{C}

__q__

_{e}

_{n}

**+**

_{C}

__q__

_{e}

_{n}

_{t}

**+**

_{ C}

__m__
-- since we have not yet
determined which actual categories of the ‘basic arithmetical operations’
domain are described by the algorithmically-generated ‘category-description’
symbols

_{C}__q___{e}**,**_{t}_{C}__q___{e}**, and**_{n}_{C}__q___{e}_{n}**,**_{t}**.***if any*
But we already know how to
characterize the

**categories that these three category-symbols “call for”, viz.:***possible*
·

_{C}__q___{e}_{t}_{ }**[---)**_{C}__q___{[}_{5}_{ + }_{5}_{i}**“calls for” a '''hybrid''' category for the kind of ‘meta-operation’, or ‘operation of operations’, that combines the**_{]}**and**__e__**operations.**__t__
·

_{C}__q___{e}_{n}_{ }**[---)**_{C}__q___{[}_{6}_{ + }_{6i}**“calls for” a '''hybrid''' category for the kind of ‘meta-operation’, or ‘operation of operations’, that combines the**_{]}**and**__e__**operations.**__n__
·

_{C}__q___{e}_{n}_{t}_{ }**[---)**_{C}__q___{[}_{7}_{ + }_{7}_{i}**“calls for” a '''hybrid''' category for the kind of ‘meta-operation’, or ‘operation of operations’, that combines the**_{]}**and the**__e___{C}__q___{n}**.**_{t}We may thus write our

*full*solution for step

**s**

**=**

**3**

**as --**

_{C}

__)-|-(__

_{3}

**=**

_{C}

__t__

^{2^}

^{3}

**=**

_{}

_{C}

__t__

^{8}

**=**

_{}

_{C}

__t__

_{ }**+**

_{C}

__n__

_{ }**+**

_{C}

__q__

_{n}

_{t}

**+**

_{C}

__e__

_{ }**+**

_{C}

__q__

_{e}

_{t}

**+**

_{C}

__q__

_{e}

_{n}

**+**

_{C}

__q__

_{e}

_{n}

_{t}

**+**

_{ C}

__m__

**=**

__addi__

__t__

__ions & subtrac__

__t__

__ions__

_{ }**+**

__multiplicatio__

__ns__

__& divisio__

__ns__

_{ }**+**

__n__

__with__

__t__

__interactions__

_{ }**+**

__e__

__xponentiations &__*d*

__e__

__-__

__e__

__xponentiations__

_{ }**+**

__e__

__with__

__t__

__interactions__

_{ }**+**

__e__

__with__

__n__

__interactions__

_{ }**+**

__e__

__with__

__n__

__&__

__t__

__interactions__

_{ }**+**

__m__

__eta____-__

**exponentiations**

__&__

__de__

__-__

__m__

__eta____-__.

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

The ‘qualo-fractal’ content-structure of this

*through*

**psychohistorical**__dialectic__**tep**

__s__**3**can be summarized as follows --

__m__

__eta____-__

**exponentiations**

__&__

__de____-__

__m__

__eta____-__“contain”

**exponentiations**

__e__

__xponentiations & d__

__e__

__-__

__e__**, which “contain”**

__xponentiations__

__multiplicatio__

__ns__

__& divisio__**, which “contain”**

__ns__

__addi__

__t__

__ions & subtrac__

__t__**.**

__ions__
The “five symbolic-elements
expression” for this model, up to this

**tep, is thus**__s___{C}__t__^{2^}**.**^{3}
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