Saturday, October 29, 2011

Comparison: The Dialectic of Algebraic Formal Logic versus The Dialectic of '''Whole numbers''' Arithmetic

Comparison:

The Dialectic of Algebraic Formal Logic

versus

The
Dialectic
of '''
Whole numbers''' Arithmetic





Dear Readers,

The foci of this post are two F.E.D. "dialectical equations", both of which are expressed in the "ideo-graph-ical language" of the algebra of their NQ_ "First Dialectical Arithmetic".

The first of the two dialectical equations expresses the dialectical elaboration of the "Boolean" algebra of formal logic as <<arche'>>.

The second of the two dialectical equations expresses the dialectical elaboration of the "Peanoian" "natural" axiomatic arithmetic of "unqualified quantifiers", the "number-space" of the Whole numbers --

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

-- as <<arche'>>.

Per the F.E.D. "solution" -- or "semantification" -- for the two post-<<arche'>> terms of both of these two equations, the categorial-progressions, or axioms-system progressions, that these equation generate both converge and intersect in an in-common second, "contra-system" term, which connotes the axioms-system of the W-based arithmetic of pure, unquantifiable ontological/categorial qualifiers --

WQ_

-- whose space of "meta-numbers" is --

WQ...=...{q/0, q/1, q/2, q/3, . . . }.


When a "meta-numeral" -- generically, call it x -- is "underscored", or underlined, as are the qs in the q/n above, except for q/0, then, per the F.E.D. Encyclopedia Dialectica notation, this signifies that the "meta-number" thus notated exhibits the "contra-Boolean" characteristic", i.e., that this "meta-number", as "self-function", i.e., when it operates upon itself, e.g., when it is self-multiplied, or "squared", produces, as its "self-product", a value which is qualitatively unequal to its "unsquared" value, viz. --

x^2...=...xx...=...x(x) is not greater than & is not equal to & is not less than x

-- whereas, for a non-underscored [meta-]number, i.e., for a "Boolean" [meta-]number, generically denoted by x --

x^2...=...xx...=...x(x) is greater than OR is equal to OR is less than x.

For F.E.D.'s exposition of the centrality of this "not-not-not" -- or "neti-neti-neti" -- relation that characterizes their dialectical "meta-numbers" -- this relationship of "non-quantitative inequality", or of "qualitative inequality" -- and its ubiquity, and the relation-sign that they "coined" for it, and the revolution in mathematics implied by its transcendence of the mathematical "trichotomy principle", and its supercession by their dialectical-mathematical "tetrachotomy principle", see --

http://www.dialectics.org/dialectics/Primer_files/3_F.E.D.%20Intro.%20Letter,%20Supplement%20A-1_OCR.pdf


Unlike F.E.D., which uses the Dyadic Seldon Function equations to express both of these dialectical axioms-systems progressions, I will use the Triadic Seldon Function equations, for step s..=..1, to express these dialectical axioms-systems progressions, in order to focus attention on the first three terms/systems of each systems progression only.

Specifically, the two equations in question are the following --

(WE_)^(3^1)......=.....?

-- and --

(W_)^(3^1)......=......?

-- wherein WE_ connotes the "Elector" algebra of the "Boolean" axiomatic arithmetic system for algebraic formal logic, whose space is WE...=...{0, 1}, and wherein W_ connotes the "Peanoian" "Whole Number" axiomatic system of arithmetic, whose space is --

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

The general, s-as-variable form of the first of our two "dialectical equations" --

(WE_)^(3^s)......=.....?

-- asks the question "What new axioms-systems follow, dialectically, from the premise/<<arche'>> of the "Boolean" axioms-system of "quasi-quanto-qualitative" algebraic formal logic?".


The general, s-as-variable form of the second of our two "dialectical equations" --

(W_)^(3^s)......=......?

-- asks the question "What new axioms-systems follow, dialectically, from the premise/<<arche'>> of the "Whole Number" axioms-system of "purely-quantitative", "Peanoian/Natural" arithmetic?".


For step s..=..1, F.E.D.'s answers to these question/solutions to these equations -- their "semantification" of, or "meaning-discovery" for, the two new terms generated in each case -- is as follows.

Exercising the generic "self-combination" rules for the expansion of these Triadic Seldon Function equations, to obtain their still specifically unsolved generic expressions, we have --

(WE_)^(3^1)......=

WE_..+..Wq/EE_..+..Wq/EEE_

-- and --

(W_)^(3^1)......= ......

W_..+..q/WW_..+..q/WWW_.




The F.E.D. solution, or "semantification", for the first equation's second and third terms is based upon the following overall intuition --

(Whole-Numbers-based, unit-interval Formal-Certainty Logic System)^(3^1)......

=

(Whole-Numbers-based, unit-interval Formal-Certainty Logic System)..+..

(Whole-Numbers-based, unit-intervals Ontological Possibility Logic System)..+.

(Whole-Numbers-based, unit-intervals Ontological Actualization Logic System)

-- and they abbreviate/assign ['(---)'] the two "ideo-ontologically" new terms/axioms-systems symbols thus generated essentially as follows --

Wq/EE_......(---)......WQ_



-- whose space is --


WQ...=...{q/0, q/1, q/2, q/3, . . . }

-- and --

Wq/EEE_......(---)......Wq/QE_

-- whose space is --

Wq/QE...=...

{b
./0, b1(t)b./1, b2(t)b./2, b3(t)b./3, ...}.



The '.' "subscript" signifies unit-interval-restricted, "quasi-quantifiablility" of the qualifier "meta-numeral" so "subscripted".



The F.E.D. solution, or "semantification", for the second equation's second and third terms is based upon the following overall intuition --

(Whole-Numbers, Pure, Unqualified Quantifiers Arithmetic System)^(3^1)......=


(Whole-Numbers, Pure, Unqualified Quantifiers Arithmetic System)..+..

(Whole-Numbers-based, Pure, Unquantifiable Qualifiers Arithmetic System)..+.

(Whole-Numbers-based, Quantifiable Qualifiers Arithmetic System)

-- and they abbreviate/assign ['(---)'] the two "ideo-ontologically" new terms/axioms-systems symbols thus generated essentially as follows --

q/WW_......(---)......WQ_


-- whose space is --


WQ...=...{q/0, q/1, q/2, q/3, . . . }

-- and --

q/WWW_......(---)......Wq/QW_......(---)......WU_

-- whose space is --


WU_...=...

{uo/0, u1(t)uo/1, u2(t)uo/2, u3(t)uo/3, . . . }.


The 'o' "subscript" signifies "full-multiplicity quantifiablility" of the qualifier "meta-numeral" so "subscripted" [reflecting the "syncopated" notation for the generic <<Monad>>, or generic "qualitative unit", in Diophantus of Alexandria's circa 250 C.E. proto-ideographic-algebraical manuscript, The <<Arithmetike'>>].



Note that the second axioms-system -- the "contra-system" -- in each of the two progressions per the above "semantification"/solution is the same:  the WQ_ axioms-system of arithmetic.

This implies that the WQ_ axioms-system is both a system of "dialectical ARITHMETIC" and a system of "[arithmetic for an algebra of] dialectical LOGIC".

Note that these two solutions, which converge and intersect in the WQ_ system for their second terms, diverge again thereafter, for the third terms of each.

The third system, or "uni-system", in each case, is qualitatively different from -- qualitatively unequal to -- the third system of the other case --

Wq/QE_......is qualitatively, "ideo-ontologically" unequal to......WU_.


For example, the bn(t) "quasi-quantifiers" of the Wq/QE_ system's "ontological qualifier" ontological category symbols, b./n, are restricted to the "Boolean" unit-interval. That is, they are "formal-logical quantifiers" -- "all (1), or none (0)" quasi-quantifiers -- whose only admissible values are confined to the "Boolean", "unit-interval end-points" set WE...=...{0, 1}, inherited from the WE_ component of this third, "uni-system" axioms-system as a "complex unity".  The b./n "quasi-quantifiable ontological qualifiers" still connote whole ontological categories.



If, in a given dialectical progression being modeled using the Wq/QE_ system, the ontological category assigned to the ontological qualifier b./k actualizes during stage t, then the "quasi-quantifier" function for that qualifier, bk(t), in a well-fitting "meta-model", should generate the value 1 for stage t, so that the combined value of bk(t) and/"times" b./k will be 1b./k, or, equivalently, just b./k, indicating that the ontology assigned to category b./k is extant/manifest in stage t.


If, in a given dialectical progression being modeled using the Wq/QE_ system, the ontological category assigned to the ontological qualifier b./k does not actualize during stage t, then the "quasi-quantifier" function for that qualifier, bk(t), in a well-fitting "meta-model", should generate the value 0 for stage t, so that the combined value of bk(t) and/"times" b./k will be 0b./k, or, equivalently, just b./0, a "quanto-qualitative zero/absence" value, indicating that the ontology assigned to category b./k is unmanifest throughout stage t.


On the contrary, the un(t) full-multiplicity quantifiers [in the Whole-numbers sense] of the WU_ system's qualifiers, uo/n, range over the entire W space.

The uo/n "quantifiable qualifiers" do not make sense as connoting whole ontological categories, as do the q/n, and the b./n.

They do make sense as connoting the units, or <<monads>>, of the given ontological category, or <<arithmos>>.

If uo/k denotes a single unit, or <<monad>> of the ontological category, or <<arithmos>>, q/k, then the uk(t) "quantifier" function for that uo/k <<monad>>-ic qualifier, in a well-fitting "meta-model" of a given dialectical ontological progression, should generate the "population census count" -- the "number" of <<monads>> of kind, or, e.g., of <<genos>>, q/k -- that were/will-be extant as of stage t, e.g., the average number of type uo/k units / <<monads>> / logical-individuals that were/will-be in existence during epoch/stage t.

If uk(t)...=...0, then uk(t)uo/k...=...0uo/k...=...uo/0, which signifies either the extinction of all <<monads>> of the q/k <<arithmos>>, or the not-yet-manifest, not-yet-irrupted state of the q/k <<arithmos>>/ontology.



Thus, per the F.E.D. solution, the Wq/QE_ "uni-system" "inherits" the "unit-interval-endpoints" restrictedness that characterize its "quasi-quantifier functions", the bn(t), from its "<<arche'>>-system", the "Boolean" system of a formal-logic calculus, as simulated by an arithmetic restricted to the endpoints of the unit interval, i.e., restricted to 0 and 1, namely, the WE_ "<<arche'>>-system".

The WU_ "uni-system", on the contrary, "descends" from a "full-multiplicity" "<<arche'>>-system", the W_ system of "Whole numbers" arithmetic, and, consequently, shows that "inheritance" in its "full-W-multiplicity quantifier functions", the un(t).

The two dialectical systems-progressions considered in this post, per F.E.D., coincide neither in their first, "<<arche'>>-system", nor in their third, "uni-system", but only in their second, "contra-systems":  wQ_ in both cases.



F.E.D. source materials for this post --

http://www.dialectics.org/dialectics/Briefs_files/_Brief1-29JUL2008_OCR.pdf
[page A-42].




Regards,

Miguel


Friday, October 28, 2011

Aphorisms aligned with the F.E.D. Dialectical "Theory of Everything": #3

Aphorisms aligned with the F.E.D. Dialectical "Theory of Everything":  #3






Dear Readers,


Aphorism #3, for the F.E.D. "Dyadic Seldon Function" --

"Onto-<<dynamis>> both symbolizes and simulates "onto-dynamasis".




1.  An "onto" in F.E.D. parlance means "ontological category", i.e., an <<arithmos>>, or population of <<monads>>, or of "ontological units", or of "ontological individuals", all belonging to the same "kind", e.g., to the same <<genos>>.

For more about this and related terms, see --





2.  The word <<dynamis>>, the etymological root of the modern English term "dynamics" -- in the ancient Greek language of Diophantus of Alexandria, circa 250 C.E., as it is used in the surviving portions of the manuscripts of the first known work of ["syncopated"] ideographical-symbolical algebra, which was entitled The <<Arithmetike'>> -- meant "power", in general, and "squaring" in particular. 

Diophantus wrote, in those ancient <<arithmetike'>> manuscripts, famous among historians of mathematics [and among F.E.D. psychohistorians], as follows --

"All numbers [<<arithmoi>> -- M.D.] are made up of some multitude of units [<<monads>> -- M.D.] . . . . Among them are -- squares, which are formed when any number [<<arithmos>> -- M.D.] is multiplied by itself; the number itself is called the side of the square; . . . Now each of these numbers [<<arithmoi>> -- M.D.], which have been given abbreviated names, is recognized as an element in arithmetical science; the square [of the unknown quantity] is called dynamis, and its sign is [Greek capital letter delta -- M.D.] with the index [superscript -- M.D.] Y [the Greek capital letter psi], that is [delta]^Y . . .". 

I.e., Diophantus's "syncopation", or abbreviation, for the ancient Greek word <<DYnamis>> was the capitalization of its first Greek letter, followed by the superscription of the capitalization of its second Greek letter.

[Victor J. Katz, A History of MathematicsAn Introduction, second edition, Addison-Wesley [NY: 1998], p. 174, typography adapted to that available for Google blogs by F.E.D.].



3.  The word "onto-dynamasis" is an F.E.D. coinage which means the production of new "kinds" of being/becoming, of new, "meta", "ontology", from -- and as a result of the "self-intra-action" of, or "squaring" of -- "old", already extant "kinds" of being/becoming, e.g., the generation of new <<gene>> as a result of the interaction, or "squaring", of the <<monads>> of previously-/presently-existent, earlier-first-extant <<gene>>, or "numbers" [<<arithmoi>>], of <<monads>>.

For more about this and related terms, see --





So, in sum[mary], this aphorism means that the burgeoning of new ontology in the universe as a whole -- or in subsumed [sub-]universes of discourse within that whole -- can be both symbolically represented by, and simultaneously symbolically simulated by, the self-juxtapositioning, or [dyadic] "squaring", of the "dialector" symbols which represent the predecessor "self-hybrid" ontology.

"Self-iterated" dyadicity of, self-juxtapositioning of, or "squaring" of ontological category symbols constitutes the very "semantico-syntactical" heart of the Dyadic Seldon Function in general, and of the F.E.D. Dialectical "Theory of Everything" Equation in particular.




Regards,

Miguel