The Dialectic of the Dialectical Arithmetics. Introduction.

Dear Reader,


This introduction outlines steps 0 through 3 of the F.E.D. dialectical model equation for its systems-progression presentation, or “meta-systematic dialectical” presentation, of its axioms-systems of dialectical arithmetic.


It aims to help our readers to understand F.E.D.’s first psychohistorical arithmetic / algebra, the language in which the seven F.E.D. psychohistorical equations published so far -- as well as F.E.D.’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 --

http://www.dialectics.info/dialectics/Primer_files/6_PrimerI_OCR.pdf


-- and --

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


It also aims to help readers to understand F.E.D.’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.


This presentation of the progression of the F.E.D. systems of psychohistorical arithmetical / algebraical language is, interestingly enough, also modeled, by the F.E.D. psychohistorians, using this first psychohistorical language, via one of their “Seldon Function” equations, a “Seldon Equation” which I have outlined, briefly, below.


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.D. "first psychohistorical arithmetic" / algebra is the language weakest in descriptive power in this progression of languages, with one exception.


That weakest of all of these systems is the system of language / arithmetic / algebra of the “Natural” Numbers, the numbers of the set N_L  =  { 1, 2, 3, . . ., L }, where L denotes the effective finite Limit of the “Natural Numbers” for a given practical context of discourse, e.g., ‘‘‘the largest “Natural” Number representable within the computer hardware/software system that we are using to facilitate our discussion’’’.


[Note:  In a definition like N_L  =  { 1, 2, 3, . . ., L }, ‘N_L’ is said to be an “intension” or “intensional [“connotational”] symbol”, whereas ‘{ 1, 2, 3, . . ., L }’ is said to be an “extension”, or “extensional symbol”, because the latter specifies individual content of the entity defined, whereas the former merely “names” it 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” “Natural” Numbers, which we will denote by N_, by itself, can supply only abstract, unqualified, unmodified “quantifiers”.



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.D. “first psychohistorical arithmetic”, is capable of far richer descriptions of reality than the first language-system in this progression-presentation, N_, even though that second system in this presentation is restricted to purely-qualitative descriptions:  that second language-system cannot express “quantifiers”, just as the first system, N_, cannot express “qualifiers” -- whether “ontological qualifiers”, or “metrical qualifiers”, or “dynamical system qualifiers”, or other “arithmetical qualifiers”.


Therefore, this second system presents itself as a potential “algorithmic heuristic” -- as an “intuitive-intensional”, i.e., “connotational”, non-“extensional”, algebra, much like original Boolean algebra, but also a deep contrary to that Boolean algebra.


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.D.’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.
 

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 “KIND of thing quantifier”, and the term “apples” an “ontological qualifier” and an “ontological 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 «Arithmoi Monadikoi» and of Diophantus of Alexandria’s circa 250 C.E. treatise «Arithmetiké», 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.


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 “omicron”] written on top of that “M”, as an abbreviation, or “syncopation”, of the ancient Greek word «Monad», which simply means “unit” -- the fact this M^o was a symbol for a qualitative entity, not for a quantity, notwithstanding.


Diophantus wrote, in place of our modern “2 + 2  =  4”, something like “ b'b'M^o = d'M^o ”, using the “Gematria” method, defining the primed second letter “ 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.D. psychohistorians, this “elision of the qualifiers” arose due to very profound and telling psychohistorical causes, causes which we will not go into within this blog-entry.





The F.E.D. immanent critique [“internal critique”, or “self-critique”] of the first order system of the “Natural Numbers”, N_, brings the “qualifier” dimension back into arithmetic.



0.  The initial step of the F.E.D. Seldon Function for this progression-presentation of psychohistorical arithmetics, step s = 0, merely reasserts the starting point of this progression-presentation, the “Natural” Numbers system of “pure, unqualified quantifiers”:

(N_)^(2^0) = N_.


In the “uninterpreted” F.E.D.’s “first psychohistorical arithmetic”, this maps to --
[ q1 ]^(2^0)   =   [ q1 ]^1   =   q1.




1.  The next step of their Seldon Function, step 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 “Natural” Numbers, N_, are also a model of] --

(N_)^(2^1)   =   N_^2   =   N_(N_)   =   N_ “of” N_   =   N_ + NQ_

-- where NQ_ is the F.E.D. symbol for their “first psychohistorical arithmetic” axioms-system.


In the “uninterpreted” F.E.D. “first psychohistorical arithmetic”, this maps to --

[ q1 ]^(2^1)   =   [ q1 ]^2   =   q1 + q1+1   =   q1 + q2.



What this equation says, per its F.E.D. standard interpretation, is that the “Natural” arithmetic, of “pure, unqualified quantifiers”, reflected upon/critiqued in its own immanent terms -- a process of self-critique connoted by N_(N_), or by N_ “squared” -- divulges explicitly its internal, formerly only implicit, hidden “intra-dual”, its extreme-opposite “contra-system”, which F.E.D. denotes by the symbol NQ_, which is the diametric qualitative opposite of the standard “Natural Numbers” arithmetic:  it is the “meta-Natural” arithmetic of “pure, unquantifiable ontological-category qualifiers”, which still follows the first four, “first order” Peano Axioms, which Axioms were intended to embrace only the “purely quantitative” “Natural” Numbers, but which Axioms fail to do so exclusively [as explained in earlier posts].




2.  The next step of their Seldon Function, step s = 2, is the self-critique, or immanent critique, of the result of the previous step, step s = 1:  it is the self-critique of ( N_ +  NQ_ ) --

(N_)^(2^2)   =   N_^4   =   (N_^2)^2   =   ( N_ + NQ_ )(( N_ +  NQ_ ))   =

( N_ +  NQ_ ) x ( N_ +  NQ_ ))   =

( N_ +  NQ_ ) “of” ( N_ +  NQ_ )   =

( N_ +  NQ_ ) “squared”

N_ +  NQ_ +  NqQN  +  NqQQ    =    N_ +  NQ_  +   NU_  +   NM_


-- where NqQN   =   NU_ is the F.E.D. symbol for their “second psychohistorical arithmetic” axioms-system, an arithmetic which critiques the separation of, and the opposition between, N_ and NQ_, by reconciling them, in an arithmetical language of explicitly “quantifiable ontological-Unit qualifiers” , or of ‘ontologically-qualifiable quantifiers’, and where NqQQ = NM_ denotes their “third psychohistorical arithmetic” axioms-system, an axioms-system of an arithmetic/algebra of “unquantifiable Metrical qualifiers”, which critiques the absence of explicit metrical qualifiers in both N_ and NQ_, by ‘‘‘present-ing’’’ a system which does contain that kind of qualifier.


In the “uninterpreted” F.E.D. “first psychohistorical arithmetic”, this maps to --

[ q1 ]^(2^2)  =

[ q1 ]^4   =   q1 + q2 + q2+1 + q2+2   =

q1 + q2 + q3 + q4.




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_ +  NQ_  +   NU_  +   NM_ ) --


(N_)^(2^3)   =   N_^8   =   (N_^4)^2  =

( N_ +  NQ_  +   NU_  +   NM_ ) x ( N_ +  NQ_  +   NU_  +   NM_ )   =

( N_ +  NQ_  +   NU_  +   NM_ ) “of” ( N_ +  NQ_  +   NU_  +   NM_ )   =

N_ +  NQ_  +   NU_  +   NM_ + NqMN + NqMQ + NqMU + NqMM      =

N_ +  NQ_  +   NU_  +   NM_ + NqMN + NqMQ + NqMQN + NS


-- where NqMN is F.E.D.’s symbol for their “fifth psychohistorical arithmetic” axioms-system, an arithmetic which critiques the absence of a quantifiable metrical qualifiers arithmetic in step 2, by presenting one, where NqMQ denotes an axioms-system of arithmetic/algebra for explicit “compound Metrical unit qualifiers” [e.g., 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 NqMU = NqMQN critiques the separation of, and the opposition between, ontological qualifiers and metrical qualifiers, in step 2, by presenting an arithmetic which unifies them [i.e., in dynamical system terms, which can explicitly, arithmetically, algorithmically, and “quanto-qualitatively” express both dynamical system “state-variables” and dynamical-system “control-parameters”, including a new kind of ‘arithmetical qualifiers’ for “state-variables” and for “control-parameters”], and where NqMM = NS critiques the absence of explicit dynamical system qualifiers in step 2, by presenting a system of arithmetic of unquantifiable dynamical system qualifiers.



In the “uninterpreted” F.E.D. “first psychohistorical arithmetic”, this maps to --

[ q1 ]^(2^3)   =   [ q1 ]^8   =

q1 + q2 + q3 + q4 + q4+1 + q4+2 + q4+3 + q4+4   =

q1 + q2 + q3 + q4 + q5 + q6 + q7 + q8.





The Seldon Function for the progression-presentation of the F.E.D. psychohistorical arithmetics / algebras / languages as a whole is --

)-|-(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-qualifiers that are explicitly extant in step s of this immanent, self-critique of the “Natural” Numbers arithmetic.




The F.E.D. psychohistorians continue the presentation of this immanent critique of first-order “Natural” arithmetic far beyond s = 3, deriving ever richer and more powerful psychohistorical-mathematical languages.

See, for example, F.E.D.’s early text --

http://www.dialectics.info/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