Thought it might be useful to reproduce here the core "axioms" -- the main rules -- of the "rules-system" of the "First Dialectical Arithmetic".

In this post, I am basing my account on the following

**F**.

__.__

**E**__. writings --__

**D**http://www.dialectics.org/dialectics/Correspondence.html

http://www.dialectics.org/dialectics/Correspondence_files/Letter17-06JUN2009.pdf

http://www.dialectics.org/dialectics/Dialectic_Ideography.html

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

[pages

**I**-

**144**through

**I**-

**150**especially].

http://www.dialectics.org/dialectics/Primer.html

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

[pages

**A**-

**1**through

**A**-

**35**].

These eleven dialectical-arithmetical rules, or axioms, are rendered below in a language closer to everyday English -- less in the language of symbolic logic than in the sources cited above, or in my immediately-previous post to this thread, and with some commentary from yours truly.

Throughout, I am using visible-light-spectrum-order color-coding --

**red**,

**orange**,

**yellow**,

**green**,

**blue**,

**indigo**,

**violet**-- to call attention to relative,

*.*

**dialectical**,**qualitative ordinality**The "Peano Postulates" are the Standard axioms for the Standard

__atural Numbers, the "set" or "space"__

**N****N = {1, 2, 3, . . . }**.

An arithmetic that follows these postulates, yet is qualitatively different from the Standard

__atural Numbers, is called a "Non-Standard Model" of the Peano Postulates.__

**N**Such "Non-Standard Models" have been relatively little explored in academic mathematics.

Such a "Non-Standard Model" is the

**F**.

__.__

**E**__. "First Dialectical Arithmetic", whose "meta-number" space they denote by__

**D****N**.

__Q__= {__q__/1,__q__/2,__q__/3, . . . }A key concept for the "Peano Postulates" is that of the

__uccessor function, which, for the__

**s****N**, can be expressed, for all

**n**in

**N**, as

**s(n) = n + 1**.

The successor function for

**N**incorporates the function

__Q__**s**in its own version of the successor function, denoted by

__:__

**s****.**

__s__[__q__/n] =__q__/(s(n)) =__q__/(n+1)**1**.

**is an element of**

__q__/1**N**. [this is the

__Q__**N**Non-Standard version of Peano Postulate #

__Q__**1**]

**2**. For any

**n**in

**N**, if

**is in**

__q__/n**N**, then

__Q__

__s__**(**

__q__**/n)**

**=**is in

__q__/(n+1)**N**as well. [i.e., the successor of any element of

__Q__**N**is also an element of

__Q__**N**].

__Q__[this is the

**N**version of Peano Postulate #

__Q__**2**].

**3**. For any

**j**and

**k**, both in

**N**: If

**is not equal to**

__q__/j**, then the successor of**

__q__/k**is not equal to the successor of**

__q__/j**. [i.e., no two, distinct elements of**

__q__/k**N**have the same successor].

__Q__[this is the

**N**version of Peano Postulate #

__Q__**3**].

**4**. For all

**x**in

**N**: There is no

**in**

__q__/x**N**such that the successor of

__Q__**is**

__q__/x**. [i.e.,**

__q__/1**has successors in**

__q__/1**N**, but has no predecessor(s) in

__Q__**N**;

__Q__**is the**

__q__/1*"*of the

**arche'**"**N**"meta-numbers"]. [this is the

__Q__**N**version of Peano Postulate #

__Q__**4**].

**5**. For every

**n**in

**N**,

**is in**

__q__/n**N**. [this states the

__Q__*"*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**].

**6**. For any

**j**and

**k**, both in

**N**: If

**j**is

__quantitatively____equal to__**un****k**, then

**is**

__q__/j

__qualitatively__

__equal to__**un****.**

__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:

**, or**

a < b

a < b

**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].*

__qualitative__inequality**7**. For all

**n**in

**N**:

**.**

__q__/n +__q__/n =__q__/n[the kind of "addition of likes" specified by this rule is called "idempotent addition". It is true not only for the

**N**

__Q__*, but also for the modern Boolean algebra of*

__dialectical__logic

*formal**logic*, in which not only does

**0 + 0 = 0**, but also

**1 + 1 = 1**. In the context of dialectically interpreted

**N**expressions, it means that repeated "summed" occurrences of the same ontological category symbol is redundant, i.e., per our previous post,

__Q__**,**

__C__+__C__=__C____"__

**not****2**".

__C__*This is the rule that makes the*

**N**

__Q__*arithmetic "*. The "count" of the presence(s) of any ontological category symbol can never exceed "

**-**__non__**quantitative**", or "**purely qualitative**"**1**". This rule is crucial to the calculations summarized in that previous post].

**8**. For all

**j**, and

**k**, both in

**N**:

If

**j**is

*to*

__quantitatively__unequal**k**, then

**is**

__q__/j +__q__/k*to*

__qualitatively__unequal

__q__/xfor 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**"**in the previous post, 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 post].**

__C__+__M__**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 post. It defines what "[ontological] multiplication" means among the

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

__Q__**F**.

__.__

**E**__.,__

**D***"the double-*.]

**aufheben**__evolute__product rule"**10**. For all

**j**, and

**k**, both in

**N**:

__q__/j +__q__/k**=**

__q__/k +__q__/j[This rule -- the

*"law" of addition -- is shared, by the*

**commutative****N**dialectical arithmetic, with most of the more familiar "Standard" arithmetics. The indifference of the

__Q__*of an*

**value****N**"qualitative sum" to the order of its terms does not mean, however, that there is not a preferred order for sums of

__Q__**N**"meta-numbers" / "dialectors". That preferred order -- the "progressive order" -- is their "ordinal order":

__Q__**]**

__q__/1 +__q__/2 +__q__/3 +__q__/4 +__q__/5 +__q__/6 +__q__/7 +__q__/8 + . . .**11**. For all

**i**,

**j**, and

**k**, all in

**N**:

**[**

__q__/i +__q__**/j ] +**

__q__/k**=**

__q__/i + [

__q__/k +__q__/j ][This rule -- the

*"law" of addition -- is shared, by the*

**associative****N**dialectical arithmetic, with most of the more familiar "Standard" arithmetics.]

__Q__**.**

__Extended Commentary__This product rule is called

*"*

*double**, I think, because the*

**aufheben**"**j**denominator/subscript of the "multiplicand" or "operand",

**, is**

__q__/j*"*-

**aufheben***twice, in*

**conserved**"__both__the first, "Boolean", term of the product, and, once again, in the second, "ontology-increment" term of the product, where it is also, simultaneously

*"*-

**aufheben***and*

**negated/determinately-changed**"*"*by the addition, to that

**aufheben**-**elevated**",**j**denominator/subscript, of the

**k**denominator/subscript from the "multiplier" or "operator",

**.**

__q__/kThis product rule is called

*"*because it encodes, in generic syntax, an empirical observation of great generality: when a new, higher level of organization irrupts within a given universe[-of-discourse],

__evolute__"__all still-extant units [if any, if not completely "extincted"] of the previously-irrupted levels of organization are converted into units of the new level [as they would be in__

**not***"*

__convolute__*tion"]. Some of the previous levels' units remain unconverted to the new, higher level units.*

**product**-E.g., not all Commodity units disappear / are converted into Money units, once the concentrated ferment of Commodity-barter reaches the quantitative threshold where qualitative/ontological change irrupts; where the new, previously-unprecedented units of Money-Commodit(y)(ies) irrupt, forming the next

*<<*[assemblage-of-units] of social-relations-of-production human-social ontology. Units at the Commodity level of organization remain present, "even" after the "Money-ontology revolution" irrupts in any locus of human history.

**arithmos**>>E.g., not all atoms disappear / are converted into molecules, once the concentrated ferment of atoms-formation reaches the quantitative threshold where qualitative/ontological change irrupts; where the new, previously-unprecedented units called "molecules" irrupt, forming the next

*<<*[

**arithmos**>>-of-<<**monads**>>

**assemblage**-of-*of cosmological, dialectic-of-nature ontology]. Units at the atom level remain present, "even" after the "molecules-ontology revolution" irrupts in any locus of cosmological natural history; of the "meta-evolution" of the cosmos.*

**units**In the Boolean algebra for

*,*

__formal__logic**x**times

**x = x**, e.g.,

**1**times

**1 = 1**, and

**0**times

**0 = 0**.

In fact, Boole called this rule

*"the fundamental law of*[

__dialectical]__

**un***thought"*.

The "Boolean term" is still there in this rule

**9**., for

**F**.

__.__

**E**__.__

**D****.**

__dialectical__logicThe "multiplicand",

**, returns in the**

__q__/j*"*product.

**aufheben**"That is a part of the

*"*operation: a "conservation" moment.

**aufheben**"However, this rule for

*not only includes but also goes beyond that for*

__dialectical__logic*in adding a second term -- a term that*

__formal__logic*simultaneously*

*"*, determinately changes/

**conserves***, and*

**negates***the "multiplicand", or "operand",*

**elevates**"**.**

__q__/jThat second term, qualitatively distinct from

**q****/j**, per Rule

**6**., and capable of representing the

*"*

**aufheben**

*"*revolutionary irruption of new ontology, of a new ontological category, is

**, which still "contains"/**

__q__/(k+j)*"*

**conserves**"**j**in its denominator, but, also, adds

**k**to it, thereby also

*/ determinately*

**changing***"*and

**negating**"*"*that denominator

**elevating**"*vis-a-vis*that of

**.**

__q__/jIf a

**-- say**

__q__/j**-- is "interpreted" for, or "assigned" to, a specific ontological category, for example, to**

__q__/1__, denoting the__

**C**__ommoditie__

**C**__s__<<

*>>*

**arithmos***of <<*[

**monads**>>**=**assemblage of

__ommodity unit__

**C**__s__], then --

**[**

__q__/1] x [__q__/1] = [__q__/1] + [__q__/(1+1)] =__q__/1 +__q__/2-- stands for the "interpreted" dialectical-ontological process --

**.**

__C__x__C__=__/C] x [____[__q__q__/C] = [__q__/C] + [__q__/CC] = [__q__/C] + [__q__/M] =__C__+__M__The "interpreted" expression can be read as --

"the self-reflexion / self-confrontation / self-operation of a sufficiently numerous and spatially-concentrated "population" of

__ommodities continues to expandedly-reproduce itself, but also <<__

**C***>>-irrupts a new ontological category/<<*

**aufheben***>>/"population": that of "*

**arithmos**__onies"; an assemblage of a new kind of units; whose units are__

**M****oney units".**

__M__Each unit of

__oney is, in the minds of the human agents of the__

**M****M**oney-social-relation-of-production, a

__-__**meta**__ommodity unit.__

**C**Each unit of

__oney, as__

**M***, is made up out of a heterogeneous multiplicity of the*

**meme**__ommodities that__

**C**__oney will trade-for, a mental "__

**M**__oney-price-list" for the various purchasable__

**M**__ommodities,__

**C***the lists in "value-form" sections*

**a la****C**. and

**D**. in Chapter

**I**. of Volume

**I**. of Marx's

__.__

**Capital**Another example: Assigning

**to the ontological category "**

__q__/4__toms", denoted__

**a**__, in an__

**a****N**model of

__Q__*, then we have --*

**the dialectic of nature****[**

__q__/4] x [__q__/4] = [__q__/4] + [__q__/(4+4)] = [__q__/4] + [__q__/8]-- standing for the "interpreted"

*--*

**dialectical**-**ontological process****.**

__a__x__a__= [__q__/a] x [__q__/a] = [__q__/a] + [__q__/aa] = [__q__/a] + [__q__/m] =__a__+__m__The "interpreted" expression can be read as --

-- "the self-reflexion / self-confrontation / self-operation of a sufficiently numerous and sufficiently spatially-concentrated, or "self-densified", "population" of

__toms continues to expandedly-reproduce itself, but also__

**a***-irrupts a new ontological category/*

**aufheben***"*/"population": that of "

**arithmos**"__olecules"; the assemblage of a__

**m***of unit, the irruption of a new kind of assemblage, whose units are*

**new**__kind__**olecules".**

__m__Each typical

__olecule unit is a__

**m***tom.*

__-__**meta****a**Each typical

__olecule unit is made up out of a heterogeneous multiplicity of its predecessor,__

**m**__tom, units.__

**a**Regards,

Miguel

**F**.

__.__

**E**__. definitions for special terms applied in the narrative above --__

**D**<<

*>>*

**arithmos**http://www.point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Arithmos/Arithmos.htm

<<

*>>*

**aufheben**http://www.point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Aufheben/Aufheben.htm

**convolute**

http://www.point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Convolute/Convolute.htm

**evolute**

http://www.point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Evolute/Evolute.htm

[

**the**]

**dialectic of nature**

no definition is as yet available in the Clarifications Archive, but see pp.

**B**-

**20**through

**B**-

**22**in --

http://www.dialectics.org/dialectics/Primer_files/4_F.E.D.%20Intro.%20Letter,%20Supplement%20B-1,%20v.2_OCR.pdf

**dialectical meta**-

**numbers**

no definition is as yet available in the Clarifications Archive

**dialectical logics**

http://point-of-departure.org/Point-Of-Departure/ClarificationsArchive/DialecticalLogics/DialecticalLogics.htm

**formal**

**logics**

http://point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Logics/Logics.htm

<<

*>>*

**meme**no definition is as yet available in the Clarifications Archive

**ontological reductionism**, and [ontological]

**anti**-

**reductionism**

no definition is as yet available in the Clarifications Archive

**ontological category**

http://www.point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Onto/Onto.htm

**ontology**

http://point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Ontology/Ontology.htm

[neo-]

**ontology**-

**increment**

**term**

no definition is as yet available in the Clarifications Archive

[dialectical, qualitative]

**ordinality**

no definition is as yet available in the Clarifications Archive

**purely**-

**qualitative**

no definition is as yet available in the Clarifications Archive

**units**[<<

*>>]*

**monads**http://www.point-of-departure.org/Point-Of-Departure/ClarificationsArchive/Monad/Monad.htm

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