Dear Readers,

This introduction outlines steps

**0**through

**3**of the

**F**.

**.**

__E__**. dialectical model equation for its system**

__D__**-progression presentation, or**

__s__*“*atic dialectical” presentation, of its axioms-systems of dialectical arithmetic.

**meta**-**system**It aims to help our readers to understand

**F**.

**.**

__E__**.’s first psychohistorical arithmetic / algebra, the language in which the seven**

__D__**F**.

**.**

__E__**. psychohistorical equations published so far -- as well as**

__D__**F**.

**.**

__E__**.’s published equations which address other-than-human domains of the cosmos, e.g., the domain of pre-human Nature -- are all written. [For an early pictorial rendition of this application, see --**

__D__http://www.dialectics.org/dialectics/Primer_files/6_PrimerI_OCR.pdf

-- and --

http://www.dialectics.org/dialectics/Primer_files/7_PrimerII_OCR.pdf ].

It also aims to help readers to understand

**F**.

**.**

__E__**.’s second through seventh systems of psychohistorical arithmetic, each of which can express a richer version of those seven psychohistorical equations, relative to what its predecessor arithmetics can express.**

__D__This presentation of the progression of the

**F**.

**.**

__E__**. systems of psychohistorical arithmetical / algebraical language is, interestingly enough, also modeled, by the**

__D__**F**.

**.**

__E__**. psychohistorians, using this first psychohistorical language, via one of their “Seldon Function” equations, a “Seldon Equation” which I have outlined, briefly, below.**

__D__Each successive system of psychohistorical arithmetic / algebra in this systems-progression is stronger in descriptive power, and contains the arithmetical and algebraic wherewithal to describe experienced reality more richly, less abstractly, more concretely, and more specifically than its predecessor system/language.

The

**F**.

**.**

__E__**. "first psychohistorical arithmetic" / algebra is the language weakest in descriptive power in this progression of languages,**

__D__**.**

*with one exception*That weakest of all of these systems is the system of language / arithmetic / algebra of the “

**atural” Numbers, the numbers of the set**

__N__**N**

_{L}

__=__**{ 1**,

**2**,

**3**,

**. . .**,

**L }**, where

**L**denotes the effective

__finite__**imit of the “**

__L__**atural Numbers” for a given practical context of discourse, e.g., ‘‘‘the largest “**

__N__**atural” Number representable within the computer hardware/software system that we are using to facilitate our discussion’’’.**

__N__[

**: In a definition like**

__Note__**N**

_{L}

__=__**{ 1**,

**2**,

**3**,

**. . .**,

**L }**, ‘

**N**

**’ is said to be an**

_{L}*“*tension” or

__in__*“*tensional [

__in__*“*ational”] symbol”, whereas ‘

__connot__**{ 1**,

**2**,

**3**,

**. . .**,

**L }**’ is said to be an

*“*tension”, or

__ex__*“*tensional symbol”, because the latter specifies individual content of the entity defined, whereas the former merely “names” it

__ex__**].**

*as a whole*That weakest system -- the system of arithmetic that is the weakest in this progression of systems of arithmetic in terms of the ‘‘‘thought-concreteness’’’ of its descriptive power -- is the axioms-system of the “first-order” “

**atural” Numbers, which we will denote by**

__N__**, by itself, can supply only abstract,**

__N___**qualified,**

__un__**modified “quantifiers”.**

__un__Mere “quantifiers” are but one fragment of the many language-elements we need in order to describe experienced reality with any specificity or concreteness.

The second language-system in that progression-presentation, the

**F**.

**.**

__E__**. “first psychohistorical arithmetic”, is capable of far richer descriptions of reality than the first language-system in this progression-presentation,**

__D__**, even though that second system in this presentation is restricted to purely-**

__N___**itative descriptions: that second language-system cannot express**

__qual__*“*ifiers”, just as the first system,

__quant__**, cannot express**

__N___*“*ifiers” -- whether “ontological

__qual__**ifiers”, or “metrical**

__qual__**ifiers”, or “dynamical system**

__qual__**ifiers”, or other “arithmetical qualifiers”.**

__qual__Therefore, this second system presents itself as a potential “algorithmic heuristic” -- as an “intuitive-intensional”, i.e., “connotational”,

**-“extensional”, algebra, much like original Boolean algebra, but also a deep contrary to that Boolean algebra.**

__non__However, it is of a kind of contrary of that Boolean algebra that also conserves and “contains” the core “law” of Boolean algebra within itself.

In

**F**.

**.**

__E__**.’s psychohistorical theory, the [psycho]historical source of “artificial languages”, such as mathematical languages -- whether or not this is known to those who historically constructed mathematics -- is human “natural language”, first in the form of spoken language, and, later, in the form of written language as well.**

__D__They focus their presentation of their progression of psychohistorical-mathematical languages on natural language phrases of the following kind --

“two apples”

-- and --

“two pounds of apples”.

In the first phrase above, they term “two” is an “ontological quantifier” -- that is, a

*“*quantifier”, and the term “apples” an “ontological qualifier” and an “ontological

**KIND**of thing**category**[“symbolic-“]

**name**”.

In the second phrase above, the term “two” is a “metrical quantifier”, or “unit of measure quantifier”, the term “pounds” is a “metrical qualifier” and a “metrical unit category name”, and the term “apples”, is, again, an “ontological qualifier” and an “ontological

**category name**”.

Now, in ancient arithmetic -- in the arithmetic of Plato’s «

**» and of Diophantus of Alexandria’s**

*Arithmoi*__Monad__ikoi*circa*

**250**C.E. treatise «

**», which began symbolic [ideographical] algebra -- the ontological unit qualifiers, and the metrical unit qualifiers, were symbolized, at least in a generic way, in an arithmetical / proto-algebraic symbolism.**

__Arithmetiké__That is, these “qualifiers” were part of arithmetic / algebra, just as much as were the “quantifiers”.

Diophantus’s treatise uses a capital “

**M**” [the Greek letter “Mu”], with the letter “

**o**” [the Greek letter “

**o**micron”] written on top of that “

**M**”, as an abbreviation, or “syncopation”, of the ancient Greek word «

**Mo**nad», which simply means “unit” -- the fact this

**M**was a symbol for a qualitative entity, not for a quantity, notwithstanding.

^{o}Diophantus wrote, in place of our modern “

**2 + 2 = 4**”, something like “

**b**

*'***b**

*'***M**

^{o}=**d**

*'***M**”, using the “Gematria” method, defining the primed second letter “

^{o}**b**” to be the numeral for the number

*'***II**, and the primed fourth letter “

**d**” to be the numeral for the number

*'***IV**.

However, after the European Dark Ages, and during the Renaissance revival of mathematics and arithmetic, including the emergence of full “symbolic” [ideographical] algebra, the “qualifiers” dropped out of arithmetic and its algebra -- according to the

**F**.

**.**

__E__**. psychohistorians, this**

__D__*“*arose due to very profound and telling psychohistorical causes, causes which we will not go into within this blog-entry

**elision of the qualifiers**”**.**

The

**F**.

**.**

__E__**. immanent critique [“internal critique”, or “self-critique”] of the first order system of the “Natural Numbers”,**

__D__**, brings the**

__N___*“*dimension back into arithmetic.

**”**__qualifier__**0**. The initial

**tep of the**

__s__**F**.

**.**

__E__**. Seldon Function for this progression-presentation of psychohistorical arithmetics,**

__D__**tep**

__s__**s**

**=**

**0**, merely reasserts the starting point of this progression-presentation, the “

**atural” Numbers system of “pure,**

__N__**qualified quantifiers”:**

__un__**(**

__N___**)^(2^0) =**

**.**

__N___In the

*“*interpreted”

__un__**F**.

**.**

__E__**.’s “first psychohistorical arithmetic”, this maps to --**

__D__**[**

__q__**1**

**]^(2^0)**

**=**

**[**

__q__**1**

**]^1**

**=**

__q__**1**.

**1**. The next

**tep of their Seldon Function,**

__s__**tep**

__s__**s**

**=**

**1**, conserves the initial language, but also outs, and adds, its most extreme “intra-dual” [“intra” because the added system is a model of the same first order Peano Postulates that the “

**atural” Numbers,**

__N__**, are also a model of] --**

__N___**(**

__N___**)^(2^1)**

**=**

__N___**^2**

**=**

__N___**(**

**N_**

**)**

**=**

__N___

*“*

*of”*

__N___

**=**

__N___**+**

__N__

__Q___-- where

__N__**is the**

__Q___**F**.

**.**

__E__**. symbol for their “first psychohistorical arithmetic” axioms-system.**

__D__In the “uninterpreted”

**F**.

**.**

__E__**. “first psychohistorical arithmetic”, this maps to --**

__D__**[**

__q__**1**

**]^(2^1)**

**=**

**[**

__q__**1**

**]^2**

**=**

__q__**1**+

__q__**1+1**

**=**

__q__**1**+

__q__**2**.

What this equation says, per its

**F**.

**.**

__E__**. standard interpretation, is that the “**

__D__**atural” arithmetic, of “pure,**

__N__**qualified**

__un__**ifiers”, reflected upon/critiqued in its own immanent terms -- a process of self-critique connoted by**

__quant__

__N___**(**

__N___**)**, or by

**“squared” -- divulges explicitly its**

__N___**, formerly only implicit, hidden “intra-dual”, its extreme-opposite “contra-system”, which**

__in__ternal**F**.

**.**

__E__**. denotes by the symbol**

__D__

__N__**, which is the diametric**

__Q___**itative**

__qual__**of the standard “**

*opposite***atural Numbers” arithmetic: it is the “meta-**

__N__**atural” arithmetic of “pure,**

__N__**ifiable ontological-category**

__unquant__

__qual__ifiers*”*, which still follows the first four, “first order” Peano Axioms, which Axioms were intended to embrace

**the “purely**

__only__**itative” “**

__quant__**atural” Numbers, but which Axioms fail to do so**

__N__**[as explained in earlier posts].**

__exclusively__**2**. The next

**tep of their Seldon Function,**

__s__**tep**

__s__**s**

**=**

**2**, is the self-critique, or immanent critique, of the result of the previous

**tep,**

__s__**tep**

__s__**s**

**=**

**1**: it is the self-critique of

**(**

__N___

**+**

__N__

__Q___**)**--

**(**

__N___**)^(2^2)**

**=**

__N___**^4**

**=**

**(**

__N___**^2)^2**

**=**

**(**

__N___

**+**

__N__

__Q___**)((**

__N___

**+**

__N__

__Q___**))**

**=**

**(**

__N___

**+**

__N__

__Q___**) x (**

__N___

**+**

__N__

__Q___**))**

**=**

**(**

__N___

**+**

__N__

__Q___**)**

*“*

*of”*

**(**

__N___

**+**

__N__

__Q___**)**

**=**

**(**

__N___

**+**

__N__

__Q___**)**

*“squared”*

__N___

**+**

__N__

__Q___

**+**

__N__

__q__**Q**

**N**

**+**

__N__

__q__

__=__

__N___

**+**

__N__

__Q___

**+**

__N__

__U___

**+**

__N__

__M___-- where

__N__

__q__**Q**

**N**

__=__

__N__**is the**

__U___**F**.

**.**

__E__**. symbol for their “second psychohistorical arithmetic” axioms-system, an arithmetic which critiques the separation of, and the opposition between,**

__D__**and**

__N___

__N__**, by reconciling them, in an arithmetical language of explicitly**

__Q___*“*ontological-

__quant__ifiable**nit**

__U__

__qual__ifiers*”*, or of ‘ontologically-

**ifiable**

__qual__**ifiers’, and where**

__quant__

__N__

__q__

__=__

__N__**denotes their “third psychohistorical arithmetic” axioms-system, an axioms-system of an arithmetic/algebra of**

__M___*“*ifiable

__unquant__

__M__*etrical***ifiers”, which critiques the absence of**

__qual__**metrical**

*explicit***ifiers in both**

__qual__**and**

__N___

__N__**, by**

__Q___*‘‘‘*a system which

**present**-**ing**’’’**contain that kind of**

*does***ifier.**

__qual__In the “uninterpreted”

**F**.

**.**

__E__**. “first psychohistorical arithmetic”, this maps to --**

__D__**[**

__q__**1**

**]^(2^2)**

**=**

**[**

__q__**1**

**]^4**

**=**

__q__**1**

**+**

__q__**2**

**+**

__q__**2+1**

**+**

__q__**2+2**

**=**

__q__**1**

**+**

__q__**2**

**+**

__q__**3**

**+**

__q__**4**.

**3**. The next step of their Seldon Function, step

**s**

**=**

**3**, is the self-critique, or immanent critique, of the result of the previous step,

**s**

**=**

**2**.

That is, it is the self-critique of

**(**

__N___

**+**

__N__

__Q___

**+**

__N__

__U___

**+**

__N__

__M___**)**--

**(**

__N___**)^(2^3)**

**=**

__N___**^8**

**=**

**(**

__N___**^4)^2**

**=**

**(**

__N___

**+**

__N__

__Q___

**+**

__N__

__U___

**+**

__N__

__M___**) x (**

__N___

**+**

__N__

__Q___

**+**

__N__

__U___

**+**

__N__

__M___**)**

**=**

**(**

__N___

**+**

__N__

__Q___

**+**

__N__

__U___

**+**

__N__

__M___**)**

*“of”*

**(**

__N___

**+**

__N__

__Q___

**+**

__N__

__U___

**+**

__N__

__M___**)**

**=**

__N___

**+**

__N__

__Q___

**+**

__N__

__U___

**+**

__N__

__M___**+**

__N__

__q__**M**

**N**

**+**

__N__

__q__**MQ**

**+**

__N__

__q__**MU**

**+**

__N__

__q__**MM**

__=__

__N___

**+**

__N__

__Q___

**+**

__N__

__U___

**+**

__N__

__M___**+**

__N__

__q__**M**

**N**

**+**

__N__

__q__**MQ**

**+**

__N__

__q__**MQN**

**+**

__N__

__S__-- where

__N__

__q__**M**

**N**

**is**

**F**.

**.**

__E__**.’s symbol for their “fifth psychohistorical arithmetic” axioms-system, an arithmetic which critiques the absence of a**

__D__

__quant__*ifiable*metrical

**ifiers arithmetic in step**

__qual__**2**, by presenting one, where

__N__

__q__**MQ**

**denotes an axioms-system of arithmetic/algebra for explicit**

*“*

__compound__**etrical unit**

__M__**ifiers” [e.g.,**

__qual__**ML/T**, or [

**gm**.

**x cm**.]

**/sec**.’, for the ‘measuremental unit’ of “momentum”], which critiques the absence of such in step

**2**, by presenting one, where

__N__

__q__**MU**

__=__

__N__

__q__**MQN**critiques the separation of, and the opposition between, ontological

**ifiers and metrical**

__qual__**ifiers, in step**

__qual__**2**, by presenting an arithmetic which

**them [i.e., in dynamical system terms, which can explicitly, arithmetically, algorithmically, and**

__unifies__*“*itatively” express both dynamical system “state-variables” and dynamical-system “control-parameters”, including a new kind of ‘arithmetical

**o-**__quant____qual__**ifiers’ for “state-variables” and for “control-parameters”], and where**

__qual__

__N__

__q__**MM**

__=__

__N__**critiques the absence of explicit dynamical system**

__S__**ifiers in step**

__qual__**2**, by presenting a system of arithmetic of

**ifiable dynamical system**

__unquant__**ifiers.**

__qual__In the

*“*interpreted”

__un__**F**.

**.**

__E__**. “first psychohistorical arithmetic”, this maps to --**

__D__**[**

__q__**1**

**]^(2^3)**

**=**

**[**

__q__**1**

**]^8**

**=**

__q__**1**

**+**

__q__**2**

**+**

__q__**3**

**+**

__q__**4**

**+**

__q__**4+1**

**+**

__q__**4+2**

**+**

__q__**4+3**

**+**

__q__**4+4**

**=**

__q__**1**

**+**

__q__**2**

**+**

__q__**3**

**+**

__q__**4**

**+**

__q__**5**

**+**

__q__**6**

**+**

__q__**7**

**+**

__q__**8**.

The Seldon Function for the progression-presentation of the

**F**.

**.**

__E__**. psychohistorical arithmetics / algebras / languages as a whole is --**

__D__

__)-|-(__**s**

**=**

**(**

__N___**)^(2^s)**

-- wherein the symbol ‘

__)-|-(__**s**’ simply signifies the “cumulum”, “cumulator”, “accumulation”, “accumulator”, or “non-amalgamative

__sum__*”*, of the interpreted arithmetical [axioms-system] category-

**ifiers that are explicitly extant in step**

__qual__**s**of this immanent, self-critique of the “

**atural” Numbers arithmetic.**

__N__The

**F**.

**.**

__E__**. psychohistorians continue the presentation of this immanent critique of first-order “Natural” arithmetic far beyond**

__D__**s**

**=**

**3**, deriving ever richer and more powerful psychohistorical-mathematical languages.

See, for example,

**F**.

**.**

__E__**.’s early text --**

__D__http://www.dialectics.org/dialectics/Primer_files/8_Fract1-1_OCR.pdf .

To our American readers, I wish to wish them

*“*,**Safe and Happy Fourth of July Celebrations!**”
Regards,

Miguel

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