**Deductive Proof**

**of the**

**F**.

**.**

__E__**.**

__D__*“*

**Fundamental Law of**”__Dialectical__Thought**from the**

*Axioms*of the

__N__**Axiomatic System of**

__Q___**.**

__Dialectical__Arithmetic[

__Note__: The algebraic notation used in this blog-entry may be lost or denatured in automatically-emailed summaries of it].

Dear Readers,

**. The F.E.D. “first dialectical arithmetic” is [also] a dialectical symbolic logic, whose algebra serves as a first [i.e., as a simplest, most abstract, least determinate in expressive capability] algebra for dialectical logic.**

__Introduction__This F.E.D. algebra for dialectical logic [a]mends the original, Boolean algebra for [syllogistic formal] logic, at its very root: it [a]mends the key flaw in Boole’s

*“*, which Boole expressed – in his 1847 work

**Fundamental Law of Thought**”*“*, and, again, in his 1854 work

**”**__The Mathematical Analysis of Logic__*“*, as explicated below.

**”**__The Laws of Thought__Boole used

**x**as an algebraic variable, or algebraic “unknown”, standing for a

**class or category – standing for**

*generic***class or category – and wrote his**

__any__*“*as a universal assertion of the reduction of “logical

**Fundamental Law of Thought**”**linearity” [e.g., second [or higher] degree or power] to “logical linearity” [first degree or first power]: for every category,**

__non__**x**–

**x^2**equals

**x^1**

-- or --

**x**squared equals

**x**

-- or simply --

**xx**=

**x**.

The Boolean versions of the numbers ‘

**0**’ and ‘

**1**’ conform to this

*“*:

**law**”**0**

**×**

**0**

**=**

**0**;

**1**

**×**

**1**

**=**

**1**.

Boole’s law thus captures only a purely external version of the

**“moment” – only the first “moment” of the dialectical «**

*conservation***» operation, which dialectical logic asserts to be the universal operation of nature-process, including of human[oid] mind-process.**

*aufheben*The other two “moments” or aspects of the dialectical «

**» operation are**

*aufheben***, meaning qualitative, determinate change or determinate negation, and**

*transformation***[in level/scale of existence].**

*elevation*The F.E.D.

*“*,

**Fundamental Law of**”__Dialectical__Thought*“*, or [just]

**Fundamental Law of**”__Dialectical__Logic*“*, [a]mends Boole’s law as follows.

**Fundamental Law of**”__Dialectical__ProcessF.E.D. uses

**as an algebraic variable, or “algebraic unknown”, standing for a generic ontological class or category – standing for**

__x__**primary “kind-of-being” class or category – as an «**

*any***» [self- or other-]operation, or operator.**

*aufheben*Note Tony Smith’s definition of a “category” in his 1990 work

*“*, in which he describes the systematic dialectical logic employed by Marx to construct his «

**”**__The Logic of Marx’s Capital__**»:**

*magnum opus*“. . . consider what a category is. It is a principle (a universal) for unifying a manifold of some sort or other (different individuals, or particulars). A category thus articulates a structure with two poles, a pole of unity and a pole of differences”. [p. 5].

That is, the meaning of a category, as standing for an “assemblage” or “set” or “population” of individuals, “contains” both the unity-moment, the aspect of what those individuals have in common, the

**that they all share, and also, equally, the differences-moment, the aspect of the**

*similarity*

*dis**-*similarities – the individual differences – that “differentiate”, “distinguish”, and “divide” each individual “in” a category from all others “in” its same class or category.

Given that context, F.E.D. writes their

*“*as a universal assertion that “logical nonlinearity” does

**Fundamental Law of**”__Dialectics__**reduce to “logical linearity”; that, on the contrary, “logical nonlinearity” makes a**

__not__**difference: for every primary ontological category,**

__qual__itative**--**

__x__**squared is**

__x__**to**

__qual__itatively__un__equal

__x__-- or --

__xx__**=**

__x__**+**

**_**

__delta__

__x__-- or --

__xx__**=**

__x__**+**

__q__**xx**

-- such that

**_**

__delta__**or**

__x__

__q__**xx**represents a category or class that is

**from that for which**

__qual__itatively different**stands, by way of an «**

__x__**»**

*aufheben***/**

*conservation***/**

*transformation***of the individuals for which**

*elevation***stands, and specifically, such that**

__x__**_**

__delta__**or**

__x__

__q__**xx**stands for a new ontological category, representing for a different category or class -- a different

**of individuals than those for which**

*kind***stands.**

__x__F.E.D.’s

*“*is a

**Fundamental Law of**”__Dialectical__Thought*“*to Boole’s

**”**__strong__contrary*“*.

**Fundamental Law of Thought**”A

*“*to Boole’s law would be –

__weak__contrary”“

**x**squared is

*to*

__quant__itatively__un__equal**x**”, e.g., “

**x**squared is

*greater than*

**x**”, which works for all “

**hole Numbers”**

__W__

__othe__r__than__**0**and

**1**.

That is, F.E.D.’s “law” requires ‘dialectical meta-numbers’ whose squaring or second power generates a value which is not even within the same “number-line” as its linear or first power; a nonlinear or second power value which resides in a

*different*perpendicular

*– in a*

__direction__**– from its first or linear power. F.E.D. has discovered such “meta-numbers”, and has axiomatized their workings [see Blog-entry**

*different*__dimension__**#**

**4**of this Blog].

In general,

__x__**--->**

__xx__**=**

__x__**+**

__q__**xx**expresses an abstracted, simplified, purely-qualitative description of what is in reality a

*“*

**-**__quant__o**”**__qual__itative**of**

__dialectical__process*“*.

**”**__quant__itative change leading to / becoming / turning into__qual__itative changeWhat is the general

*and*

**meaning***of the F.E.D.*

**import***? It is a general assertion of the ubiquity of "ontological dynamics" -- of "onto-dynamasis". It is the assertion that, whenever [the "cumulum" of] a given spectrum of ontological possibilities operates upon itself, an expanded ["cumulum", representing an expanded] spectrum of ontological possibilities results.*

**Fundamental Law of Dialectics**For example, if we “apply”

**to stand for a cosmological population of**

__x__**toms,**

__a__**, and**

__a__

__q__**aa**to stand for a cosmological population of

**olecules, also denoted by**

__m__**, then –**

__m__

__a__**--->**

__aa__**=**

__a__**+**

__q__**aa**

**=**

__a__

**+**

__m__-- describes a cosmological new-ontology-generating process in which the expanded self-reproduction of the population-quantity of

**toms, together with self-increase of the measure of their quantitative “density”, i.e., of their physical-spatial “concentration”, leads to a growing frequency and intensity of interactions between/among**

__a__**toms – also presented as an “int**

__a__

__ra__*-*action” with

**the population of**

__in__**toms, or as a “self-interaction” of the category “**

__a__**toms” – denoted by**

__a__**– which reaches a critical threshold, or “metafinite resonance singularity”, whereafter it gives rise, not only, possibly, to further increases in its own population-quantity, and quantitative density, but possibly also to a new kind of population of a new kind of “individuals” –**

__aa__**olecules.**

__m__This description implies that

__q__**aa**

**=**

**because**

__m__**olecules are the result of a self-«**

__m__**» operation of**

*aufheben***toms upon**

__a__**toms: each**

__a__**olecule is “meta-**

__m__**tom”, made up out of, and “internalizing”, a typically heterogeneous multiplicity of**

__a__**toms. Each**

__a__**olecule thus represents an «**

__m__**»**

*aufheben***/**

*conservation***/**

*transformation***of a sub-population of [“former”]**

*elevation***toms-only self-organizations.**

__a__Thus a "

__q__**aa**" means a

*"*

**meta**-**tom", or "**

__a__**tom of the second degree", made up out of a typically heterogeneous multiplicity of "**

__a__**toms of the first degree".**

__a__
Therefore, while the Boolean

In the language of the ancient Mediterranean original awakening of [Occidental] philosophy and science -- in the language of the original philosophy of mathematics, which both predates, and continues beyond, the time of Plato –

An «

A «

If we inwardly form a mental abstraction – an idealized object, or mental object – representing such a

Diophantus of Alexandria, the father of algebra, in his «

**x**operator --**x**grasped as a [potential] operation upon itself, or upon other Boolean operators/operations -- is only a*"*operation, omitting the other two aspects of the full «**preservation**"**»***aufheben***/***conservation***/***transformation***operation, on the contrary, the F.E.D.***elevation***operation --**__x__**gasped as a [potential] operation upon itself, or upon other dialectical-logic classes-or-categories-as-operators/-operations -- represents a full «**__x__**»***aufheben***/***conservation***/***transformation***operation, under the interpretation that***elevation*__q__**xx**denotes a "meta-«**»"***arithmos**relative to*the__q__**x**or**«**__x__**», that is, that***arithmos*__q__**xx**denotes an «**» each of whose «***arithmos***» is a***monads**"*«**meta**-**»***monad**-*, or a**ization**"*"*, of the «**meta**-**unit**-**ization**"**» of the***monads*__q__**x**or**«**__x__**».***arithmos***:**__Motivation__**Historical Context and Application**. It may assist comprehension and conceptualization of the full import and meaning of F.E.D.’s*“*a simple deductive demonstration of which is presented below -- if we locate that**Fundamental Law of**” --__Dialectical__Process*“*within the tide of human history – placing it in context within the development of the human phenome, of the “memes pool”, of the “psychohistorical material[ity]” of human history, and of the historical labor of the self-production of humanity.**Law**”In the language of the ancient Mediterranean original awakening of [Occidental] philosophy and science -- in the language of the original philosophy of mathematics, which both predates, and continues beyond, the time of Plato –

An «

**» is an assemblage of***arithmos*__qual__itative**– a “population” of individuals of a given***units***: a***kind**“*of individuals / individual**number**”**.***units*A «

**» is a***monad*__qual__itative**.***unit*If we inwardly form a mental abstraction – an idealized object, or mental object – representing such a

**, or**__qual__itative unit**«**__qual__itative**»,***monad***, then, by referencing assemblages of such**__in____gene__ral**, or**__gene__ric unit**«**__gene__ric**», idea-objects -- anciently named, by Plato, the «***monad***», the mental “assemblages”, or groups/populations, of ‘mentalized’, or***arithmoi*__monad__ikoi**, «**__gene__ric**»: abstract, idealized mental “numbers” of abstract, idealized, mental «***monads***» /***monads***-- then we can construct an «***units***», or «***arithmetiké***»***arithmoi**-*«**» -- a universal “«***techné***»***arithmos**-*[“art” of “number”], a universal “«**art**”**»***arithmos**-*[“craft” of “number”], or a universal “«**craft**”**»-***arithmos**tech**”*[*“*nique” or**tech***“*nology” of “number”]:**tech***“*.**theoretical arithmetic**”Diophantus of Alexandria, the father of algebra, in his «

**» 250 C.E. proto-algebraic manuscript entitled***circa**“*-- or the «**”**__The Arithmetica__**» in transliterated Greek – actually “abbreviatively” symbolized the generic «**__Arithmetiké__**», or abstract qualitative unit, long before the advent of the Hindu’s abstract purely-quantitative unit symbol,**__Mo__*nadikos***1**, by the abbreviation**Mo**[actually, the Greek capital letter “**M****u**” with the lower-case letter**o**micron,**o**, written atop it], so that he symbolized our modern**1****+****1****=****2**by something more like**a***’*Mo**&****a***’*Mo**=****b***’*Mo**[the ancient Mediterranean Greek/Hellenistic civilization used “primed” letters of the Greek alphabet as numerals: «****» or «***Gematria***»],***Isopsephy***just by**__not__**a***’***&****a***’***=****b**as would later, more abstract notations such as our own, reflecting our modern, far deeper phenomic immersion in the “meme-etic” paradigm of what Marx called*’**“*[**The Elementary Form of****[***Commodity**-*]]**Capital**-*Value**”*, that is, in the core capital-relation consequence of the [mostly**conscious]**__un__*“*[capital-]**law of***value**”*, first fully raised into widespread human consciousness as such by Marx.Our earlier example, in the introductory section above, concerned a cosmological population, or «

**», of**

*arithmos***tom**

__a__**: a**

*units**“*of

**number**”**toms – e.g., a galactic interstellar**

__a__*“*

**atomic***cloud”*, soon to transform itself into a proto-stellar

*“*.

**molecular**cloud”Unlike Boole’s “primary”

**[**

*algebra for***]**

*formal***-- as his “existential logic”, “existence logic”, or “logic of things” -- in which**

*logic***aa**

**=**

**a**, the F.E.D.

**does not get stuck in stasis –**

*algebra for*__dialectic__

__a__**--->**

__aa__**=**

__a__**+**

__q__**aa**

**=**

__a__**+**

__m__-- or –

“

**tomic cloud”**

__a__**--->**[goes to”, or, in time, “becomes”] “

**olecular cloud”.**

__m__That is, once sufficiently expandedly self-reproduced

__quant__*itatively*– once

*“number-*ous” or

*“numer*ous” enough, or “populous” enough, and “densified” enough -- the

**toms in the “**

__a__**tomic cloud” give rise to**

__a__**olecules as well, that is, to a**

__m__

__qual__itatively*-*reproduction; to the irruption of the new “cosmo-ontology”; to that new kind of being, that we call

**expanded****olecules.**

__m__The

**toms no longer conduce to mere**

__a__**, within a**

__quant__itative change*qual*itative

*“*, but to

**onto**-**”**__sta__sis**in the form of**

__qual__itative change*“*: to the irruption of a new, “

**onto**-**”**__dynam__asis**olecules” ontology, or kind of being, where there once was only the**

__m__**toms kind of being [**

__a__**+**its predecessor kinds].

The thus self-transformed, proto-stellar “

**olecular cloud” is an [“evolute”]**

__m__*“*of [at least] two ontologies; a “cumulum” of both the type

**cumulum**”**and the type**

__a__

__m__*“*of being” --

**kinds**“

**tomic cloud”**

__a__**=**

**. . .**

**+**

**;**

__a__“

**olecular cloud”**

__m__**=**

**. . .**

**+**

__a__**+**

**.**

__m__The Boolean

*“*,

**Fundamental Law**”**xx**

**=**

**x**, the law of eternal

__simple__**, is insufficient to construct a**

*reproduction***, a**

*more scientific*science**, a science that more fully**

*more empirically faithful*science*“*.

**follows nature**”To achieve the latter science –

**,**

*Marxian***, the positive fruition of the dialectical, immanent critique of [mostly]**

*dialectical science***consciously ideology-vitiated capitalist science – we need a[n a]mended**

__un__*“*, the

**Fundamental Law**”*“*of the

**Law**”**/ontologically self-**

__qual__itatively**self-**

__expanding__**of universe ontology –**

*reproduction*

__x__**--->**

__xx__**=**

__x__

**+**

__q__**xx**

-- or --

__q__**x**

**--->**

__q__**x**

**×**

__q__**x**

**=**

__q__**x**

**+**

__q__**xx**.

Juxtaposing the two “laws” – the Boolean and the “contra-Boolean” --

**xx**

**=**

**x**

-- versus –

__xx__

**=**

__x__

**+**

__q__**xx**

-- highlights how the F.E.D.

*“*still [externally] «**Fundamental Law**”**»-***aufheben*__contains__and [partially] retains the Boolean*“*, in the**Fundamental Law**”*first**term*of the RHS of the F.E.D. law’s equation --

__xx__

**=**

__x__

**. . .**

-- but how it also goes beyond that Boolean law, to [internally] «

**»-**

*aufheben***, [internally] «**

__conserve__**»-**

*aufheben***, and to**

__elevate__

__qual__*itatively*-transform,

*determinately**-change*, or «

**»-**

*aufheben***that “Boolean” term,**

__negate__**, via the**

__x__*second term*of the RHS of the F.E.D. law’s equation --

__xx__

**=**

**. . .**

__q__**xx**

-- which “double-contains”

**, as**

__x__**xx**, at the

*sub-*script level of its generic ideographical symbol,

__q__**xx**.

Thus, «

**», the F.E.D.***in toto**“*is a full-«**Fundamental Law of**__Dialectical__**Thought**”**»***aufheben**containment*/**,**__conservation____qual__*itative transformation*/*determinate*, and__negation__**[to a new scope/scale/level], of the “Boolean” term, of the**__elevation__**term – the “operand” or “argument” upon which**__x__**itself again, but this time as “function” or “operator – operates, or “functions” in –**__x__

__x__**(**

__x__)

**=**

__x__**+**

**_**

__delta__

__x__

**=**

__x__

**+**

__q__**xx**

--- and, indeed, the F.E.D. rule is the positive fruition of a

**,**__dialectical__**of the Boolean “law”, of “Boole’s Rule”, as expressed in the first triad of***immanent critique***which we explicated in an earlier blog-entry [wherein***the*__Dialectic__of the Algebras for Logic__W__**denotes the axioms-system of the Boolean arithmetic for logic, wherein**__E_____W____Q__**_**denotes the**hole numbers-based version – whose number-space is**__W__**W**__=__**{ 0, 1, 2, 3, . . .}**-- of the axioms-system of the F.E.D. “First Dialectical Arithmetic”, and wherein__W____q__**Q****E****denotes the axioms-system of the**_______dialectical__**of***synthesis*__W____Q__**_**and__W__**]:**__E___

__W__

__E___

**--->**

__W__

__E___**+**

__W__

__Q__**_**

**--->**

__W__

__E___**+**

__W__

__Q__**_**

**+**

__W__

__q__**Q**

**E**

**.**

_____
F.E.D. has identified four [qualitatively-]distinct «

**» of the «***species***»***genos***in their “[**__Dialectics__*Psycho*]*Historical*__Dialectic__*of the*__Dialectic__**[a**__Itself__**which itself also accords with the F.E.D.**__Dialectic__*“*], all four of which have been instantiated, together, so far, historically, only in the work of Karl Marx, not [quite] in the work of Hegel, or even in that of Engels, in my view.**Fundamental Law of**”__Dialectics__
These four «

**» are --***species*

__Systematic__

__Dialectics__

**&**

__Historical__*Dialectics***&**

__Meta____-__

__System__

__atic__

__Dialectics__

**&**

**.**

__Psychohistorical__*Dialectics*
For additional background regarding these «

**», see blog-entry***species***# 9**, and/or the following definitional JPGs in the F.E.D. Glossary [ http://www.dialectics.org/dialectics/Glossary.html ] --

__Systematic__**--**

__Dialectics__

__Historical__**--**

__Dialectics__

__Meta____-__

__System__

__atic__

**--**

__Dialectics__

__Psycho__

__historical__**--**

__Dialectics__
In the context of

**, and especially for an**__Historical__*Dialectics***which represents a**__x__**«***physical***», or empirically-given “population” of “individuals”, the formula***arithmos*__x__**^1****--->**__x__**^2****=**__x__**+**__q__**xx**is true primarily because__q__**xx**denotes an emergent phenomenon, an emergent property, an emergent quality, and an emergent ontology**, an addition of new ontology that is**__potential__**«***already present***»***qua***, [with]in**__potential__**. This is partially also the case for**__x____Psycho__**and for**__historical__*Dialectics*__Meta____-____System__**.**__atic__*Dialectics*
In the context of

**-- the**__Systematic__*Dialectics***of theoretical systems as generalized, “genericized” idealizations; of “idea-objects” or “mental objects” -- where**__Dialectics____x__**(**__x__)**=**__x__**+**__q__**xx**is a law of the “expanding reproduction” of ideas, and, especially, for an**which represents a**__x__*mental**-*«**only****», such as***arithmos***N**, or**W**, i.e., an abstractly-given “population” of “internal”, “mental objects”, most remote from physical/”external” experience, existing as such only for “internal”, “mental experience”, the fundamental formula of**,**__Dialectics____x__**^1****--->**__x__**^2****=**__x__**+**__q__**xx,**is true primarily because the category/concept/idea denoted by__q__**xx**is semantically, conceptually**[with]in -- is**__implicit__**as a***already present*__“____presupposition”__[with]in –**, but requires mental**__x__**[“self-***reflection**dialogue**”*, or “int__ra__*-*mind*dialogue**”*], or int__er__*-*personal**[“int***dialogue*__er__*-*mind*dialogue**”*], to become**; to be “lifted up’, or**__explicit__*“*, into “**elevated**”**”. This is partially also the case for**__explicitude____Meta____-____System__**and for**__atic__*Dialectics*__Psycho__**.**__historical__*Dialectics*
As with a holographic image, or hologram, in which each part of the whole image contains a “sided”, “tendentious”, “distorted”, “deformed”, “fuzzy” replica of that whole image, such that the image as a whole, as “made up out of its parts”, is, in this sense, “made up out of itself”, so it is also with a well-developed, matured theory, or conceptual system. That is, such a theory-system is

*“*, meaning that each of its category/concept parts “contains” – entails / involves / presupposes -- all of the other category/concepts of that system**”**__holonomic__**. This “**__implicitly__**” can be evoked, or “out-voked”, into “**__implicitude__**”, via introspective**__explicitude__**, or via inter-personal [“Socratic”]***reflection***, or via skillful oral, or written, “monologue-ic” presentation.***dialogue*
The pedagogical strategy of

**, as a**__Systematic__*Dialectics*__dialectical__**totality-theories***method of***, is to start with/from the simplest, most abstract of the categories/concepts in this “holonomic” network of mutually-involved, “inter-pre-supposing”, “intergenerate” categories/concepts, as «**__presentation__*arch***» / beginning, and to proceed from there, evoking a[n ever-lengthening] series of more complex, more “thought-concrete” [more “determinate”; more richly «***é***»-fied] categories/concepts, as successor series, from each predecessor series, -- from each less lengthy, less complex, less “thought-concrete”, series -- until the most lengthy series, ending with the most-complex, most-“thought-concrete” category/concept -- one which most richly in-itself represents the total human experience of the [sub-]totality of human experience so theorized -- is reached / presented / evoked.***speci***.**

__Demonstration__**A**

**Simple**

**Deductive Proof**

**of the**

**F**.

**.**

__E__**.**

__D__*“*

**Fundamental Law of**”__Dialectical__Thought**from the**

*Axioms*of the

__N__

__Q__**_**axioms-system of

**.**

__Dialectical__Arithmetic**:**

__Given__**n**is an element of the set

**N**

__=__**{ 1**,

**2**,

**3**,

**. . .}**of "

**N**atural" Numbers.

**:**

__To Prove__**"times"**

__x__**is**

__x__**to**

__qual__itatively__un__equal**,**

__x__for every

**in**

__x__**N**

__Q__

__=__

**{**

__q__**1**,

__q__**2**,

__q__**3**,

**. . .}**, the set of the

**F**.

**.**

__E__**. "First Dialectical Meta-Numbers".**

__D__**:**

__Proved__

__Assertion__**......................**

__#__**......**

__Assertion Content_____________________.__

__Assertion Justification__**1**......................

**n**is an element of

**N**......................................................Given

**2**......................

__q__**n**is an element of

**N**

**, for every**

__Q__**n**in

**N**..............

**Axiom**

**5**

**3**.....................

__q__**n**

*squared***=**

__q__**n**

**+**

__q__**2**

**n**, for every

**n**in

**N**.........

**Axiom 9**

**4**.....................

**n**

__does__

__not__

__equal__**2**

**n**, for every

**n**in

**N**................

*****

**5**......................

__q__**n**is

__qual__itatively*to*

__un__equal

__q__**2**

**n**,

............................for every

**n**in

**N**...............................................................

**Axiom 6**

**6**.......................

__q__**n**

**+**

__q__**2**

**n**is

__qual__itatively*to*

__un__equal

__q__**n**,

.............................for every

**n**in

**N**..............................................................

**Axiom**

**8**

**7**.......................

__q__**n**may be represented by

**,**

__x__.............................for

*any*

**n**in

**N**.................................................................Equivalent Substitution

**8**........................

__x__**is**

*squared*

__qual__itatively*to*

__un__equal

**..............Assertion**

__x__**# 3**

**&**

............................................................................................................................Assertion

**# 6**, above:

............................................................................................................................

**Q**.

**E**.

**D**.

*****The proposition "

**n**does not equal

**2**

**n**, for every

**n**in

**N**." is justifiable, for example, from a corollary of a theorem of the "

**N**atural Numbers" axiomatic system of arithmetic that derives from the "

**ultiplicative identity element" axiom for that system. This theorem asserts that for any element of the set of "**

__m__**N**atural Numbers", call it

**m**, that is

**the "**

__not__**ultiplicative identity element" of that set, and for**

__m__**element of the set of "**

__any__**N**atural Numbers", call it

**x**, it is true that

**x < mx**.

Here is the same theorem again, but this time expressed in something closer to the "full regalia"

__notation --__

**Encyclopedia Dialectica****Key Upshot**: The “contra-Boolean”

**F**.

**.**

__E__**.**

__D__*“*is not a direct premise, assumption, presumption, postulate, or

**Fundamental Law of**”__Dialectical__Thought**of the**

*axiom*

__N__**axiomatic system of**

__Q___*for*

**arithmetic**

**dialectical***: it is a*

**algebra***proven*

**of that system.**

*theorem*The proof above invokes the following Axioms of the

__N__**axiomatic system of arithmetic for dialectical algebra, posted previously to this Blog, in blog-entry**

__Q___**#**

**4**[with some new commentary included] --

**“**Dear Readers,

Thought it might be useful to reproduce here the core "axioms" -- the main rules -- of the "rules-system", or "axiom

__s__*-*system" ["axiomatic system"] of the

**F**.

**.**

__E__**. "First Dialectical Arithmetic".**

__D__**. . .**

**Axiom 5**. For every

**n**in

**N**,

__q__**n**is in

**N**

**.**

__Q__[this states the «

**» tie between the**

*aufheben***N**and the

**N**

**. The rest of the rules below state "Non-Standard" aspects of the**

__Q__**N**

**relative to the "Standard"**

__Q__**N**].

**Axiom 6**. For any

**j**and

**k**, both in

**N**:

If

**j**is

__quant__itatively__un__equal to**k**, then

__q__**j**is

__qual__itatively__un__equal*to*

__q__**k**.

[This axiom expands the

*"*chotomy principle" of the "Standard" arithmetics -- the principle that for any arithmetical objects

__tri__**a**and

**b**, just one of the following three relations obtains:

**a**

**<**

**b**, or

**a**

**=**

**b**, or

**a**

**>**

**b**

-- to a

*"*, that adds a fourth possibility, that of

**"**__tetra__chotomy principle**, to the basic possible relations between any pair of arithmetical objects].**

__qual__itative__in__equality**. . .**

**Axiom 8**. For all

**j**, and

**k**, both in

**N**:

If

**j**is

*to*

__quant__itatively__un__equal**k**, then

__q__**j**

**+**

__q__**k**is

**to**

__qual__itatively__un__equal

__q__**x**

for any

**x**in

**N**.

[This is the rule that makes dialectically interpreted

**N**

**expressions**

__Q__*"*

**ontologically**

__anti__*-*. It means that a "heterogeneous sum" of ontological categories, like

**reductionist**"

__C__

**+**

**in the previous blog-entry, does not reduce or collapse into any single ontological category at the same level of generalization as its summands. This rule too is crucial to the calculations summarized in that previous blog-entry.**

__M__Another way of understanding this axiom is to say that it denies "additive closure" for the arithmetic of the

**N**

**.**

__Q__Addition is not a "closed" operation for the

**N**

**.**

__Q__Indeed, every addition of a pair of [qualitatively-]

**distinct****N**

**'dialectical meta-numbers' leaps beyond the**

__Q__**N**

**space of 'dialectical meta-numbers'. Every such addition operation jumps you to a "place", to a value, that is outside of the set**

__Q__**N**

**.**

__Q__As the next axiom shows, by way of re-expressing '

**dialectical***, i.e., '«*

**multiplication**'**» multiplication', or 'ontological multiplication', in terms of '**

*aufheben*

**dialectical***, i.e., of*

**addition**'*'*/qualitative addition', the

**non**-**reductionist****N**

**'dialectical meta-numbers' form a**

__Q__*'*arithmetic.

**comprehensively**'__OPEN__That is, the

**N**

**'meta-numbers' arithmetic is**

__Q__*for either of the operations that these axioms define for it.*

__NOT__CLOSEDAny addition operation for distinct 'dialectical meta-numbers', and any multiplication operation among these 'dialectical numbers'

*, distinct or not, gives rise to*

**at all***'*.

**'**__diagonal__transcendenceAny such operation thus represents, generically, an

*'*.

**ontological**'__revolution__This means that any such operation gives rise to a value which corresponds to a 'meta-

*directed line-segment in the '''analytical-geometric representation''' of the*

__'__**diagonal****N**

**'''set''', or "space", i.e., to a directed line-segment, 'meta-vector', or 'dialector,' which**

__Q__*'"*, or resides beyond, that set of the

**transcends**"'*diagonal 'dialectors' which forms the '"space"' of the*

__-__**non****N**

**.**

__Q__For illustrations regarding this point, see --

http://www.dialectics.org/dialectics/Dialectic_Ideography_files/6_Dialectics-Part1c-Briefing_OCR.pdf

-- pages

**I**-

**147**;

**I**-

**152**.].

**Axiom 9**. For every

**j**and

**k**, both in

**N**:

[

__q__**k**]

**x**

**[**

__q__**j**]

**=**

**[**

__q__**j**]

**+**

**[**

__q__**(k+j)**].

[This rule is most crucial of all to the "purely-qualitative, algorithmic calculations" presented in the previous blog-entry. It defines what "multiplication" means among the

**N**

**"dialectical meta-numbers". It is called, by**

__Q__**F**.

**.**

__E__**.,**

__D__*"the double-conservation*«

**»**

*aufheben**.]*

__evolute__product rule"**. . .**

**”**

Regards,

Miguel

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