Below is a paraphrased excerpt from a recent dialogue, in which I participated, regarding the

__"arithmetic of pure,__

**N****Q_****ifiable ontological**

*unquant*__ifiers" as an "arithmetic for__

**qual***.*

__"__**dialectics**__: How can you have "qualitative mathematics"__

**Q***Math is all about manipulating quantities.*

**?**__: Thank you for your excellent question__

**M**.**D**.

**!**What mathematics appears to be, up to a certain period in human history, may not be identical to what it,

*, really is.*

**in essence**Later, future historical appearances, as well as early but deep insights, may reveal that mathematics is more than it has mainly appeared to be in the past, up to a certain historical period.

I agree, with

**F**.

__.__

**E**__., that the historically__

**D****ral definition of mathematics should be --**

*gene*'

**Mathematics**[phono-picto-]

__=__*Ideometry via*

*Ideography**'*

-- with

*'*naming a very general concept of "the measurement of ideas"; any method of

**Ideometry**'*"accounting for*ideas".

And, defining Mathematics as an

*"accounting for*human ideas' places mathematics squarely in the possession of

*'*.

**psychohistorical materialism**'The historical development of mathematics has lately demonstrated -- although not with much noticing, given the purely-quantitative <<

*>>, the 'money-mind' or 'capital-mind', that prevails; the unconscious domination of modern human cognition by the unconscious paradigm of what Marx named "The Elementary Form of Value" -- that mathematics is, immanently and necessarily, far more than your "manipulating quantities", viz. --*

**mentalite'****1**.

__Set Theory__: if

**a**,

**b**,

**c**, and

**d**denote four

*distinct*set elements, then:

**A**. the set

**{a, b}**is

**not****>**the set

**{c, d}**, and

**B**. the set

**{a, b}**is

**not****=**the set

**{c, d}**, and

**C**. the set

**{a, b}**is

**not****<**the set

**{c, d}**; therefore

**D**. the set

**{a, b}**is

*, but*

__un__equal to

**not**__itatively__

**quant**__equal to the set__

**un****{c, d}**; therefore

**E**. the set

**{a, b}**is

__itatively__

**qual**__equal to the set__

**un****{c, d}**.

**2**.

__Mathematical Logic__["first order predicate calculus"]:

**Fa ==> Ga**, meaning "If the individual

**a**exhibits the

__ity__

**qual****F**, then it must also exhibit the

__ity__

**qual****G**". Nothing

__itative about this "calculus".__

**quant****3**.

__The Fundamental Theorem of Algebra__. Without the recognition of the value

**i**-- the positive square-root of the negative unit[y],

**-1**-- the "Fundamental Theorem of Algebra" -- roughly "Every algebraic equation of degree

**n**has

**n**solutions" -- is not true. But --

**A**.

**+**

**i**is

**not****>**

**+1**, and

**B**.

**is**

**+**i

**not****=**

**+1**, and,

**C**.

**+**

**i**is

**not****<**

**+1**; therefore

**D**.

**is**

**+**i*, but*

__un__equal to

**not**__itatively__

**quant**__equal to,__

**un****+1**; therefore

**E**.

**+**

**i**is

__itatively__

**qual**__equal to__

**un****+1**.

**4**.

__Dimensional Analysis__.

**3**

__enti__

**c**__eter__

**m**__[__

**s****3 cms**.] are greater than [

**>**]

**2**

**cms**., but, regarding the relationship between

**1**linear centimeter [

**] and**

**1cm. =**1cm.^1**1**square[d] centimeter [

**1cm.^2**] --

**A**.

**1cm.^2**is

**not****>**

**1cm.^1**, in the linear centimeters sense and

**B**.

**1cm.^2**is

**not****=**

**1cm.^1**, and,

**C**.

**1cm.^2**is

**not****<**

**1cm.^1**; therefore

**D**.

**1cm.^2**is

*, but*

__un__equal to

**not**__itatively__

**quant**__equal to,__

**un****1cm.^1**; therefore

**E**.

**1cm.^2**is

__itatively__

**qual**__equal to__

**un****1cm.^1**.

The above provide just a few of the manifold evidences of the fact that modern mathematics is

__just about your "manipulating quantities", but is about, at the very least, the__

**not***'*itative manipulation of

**-**__qual__o**quant***'*ities'.

**-**__qual__o**quant**The ancient Mediterranean makers and discoverors of mathematics had, at the very least, deeply insightful inklings of this

*'*itative essence of mathematics.

**-**__qual__o**quant**The most insightful of these insights, I would argue, was that of Plato, who placed his <<

*>> -- his*

**Arithmoi Eidetikoi***'*[<<

**idea***>>]*

**Eide***-*

*[<<*

**numbers***>>]' -- at the very heart of his*

**Arithmoi**__.__

**Dialectics**But, at least until Jacob Klein's

__, most Plato scholars, blinded by the unconscious 'capital mind' of the modern human <<__

**Greek Mathematical Thought and the Origin of Algebra***>>, seem to have had not a clue as to what Plato meant by <<*

**mentalite'***>>:*

**Arithmoi Eidetikoi**"

__:__

*arithmos***;**

*number*

*arithm*

*ê*

*tik***; the science of**

*ê***.**

*number*

Zero was unknown as a

Zero was unknown as a

**and**

*number**one also was not counted as a*

**, the**

*number**first*

*number**being the*

*duas*

*– two*.

From

*the Pythagoreans*,

*ton*

*arithm**on nomizontes arkh*

*ê*

*n einai*– who

*consider*

*number**to be*

*the first principle*(

**Ar**.

**.**

*Met***986a15**) –

**played a great part in metaphysics, especially in**

*number**Plato’s*

*unwritten doctrines*, involving

*obscure*distinctions of e.g.

*sumbl*

*ê***and**

*toi*

__a__

*sumbl*

*ê***–**

*toi***and**

*addible*

__non__*-*

**."**

*addible numbers*J. O. Urmson,

**, Gerald Duckworth & Co., Ltd. [London:**

__The Greek Philosophical Vocabulary__**1990**], pp.

**31**-

**32**, [

__emphasis__**]**

*added*In the ancient concept,

*"*-- <<

**Number**"*>> -- means "an assemblage of*

**Arithmos**__itative units".__

**qual**The <<

*>>-*

**Eide**__-- the <<__

**units***>>, the*

**Eide**-**monads***'*-- are <<

**idea**-__'__**units***>>, '*

__a__sumbletoi

__un__*able', because they are purely-*

**sum***, not purely-*

__qual__itative__units__*, "numbers".*

__quant__itative__units__The <<

*>> are 'purely-*

**Arithmoi Eidetikoi**__itative__

**qual***, not our modern "purely-*

**numbers**'__itative numbers", such as the "__

**quant**__atural" Numbers -- which are "Natural" only to our modern, "Elementary Form of Value" formed <<__

**N***>> and*

**mentalite'***, but*

**ideology**__to the <<__

**not***>> and*

**mentalite'***of ancient humanity.*

**ideology**Natural language, the ultimate foundation of all of mathematics, is built upon

*'*phrases such as "

**qual****o**-**itative'**__quant__**ten**

**silas**of

**olive oil**".

It took the genius of Karl Seldon to see this, and to see through the ideology of the

*"purely-*, the ideology of "The Elementary Form of [Commodity[-Capital-]]Value" formed <<

__itative"__**quant***>>, and to actualize the latent, fuller, explicitly*

**mentailite'**__stage of mathematics.__

**dialectical**Seldon calls the "

**ten**" component of the phrase above "the metrical

__ifier". He calls the "__

**quant****silas**" component the "

**metrica**

**l**

**unit**__ifier". He calls the "__

**qual****olive oil**" the

*"*

**ontological**__unit____ifier".__

**qual**And it was Seldon who first discerned that there is not only an ideographical arithmetic of "pure

__ifiers", such as that of the standard "__

**quant****atural" Numbers,**

__N__**N**, but also that there is an ideographical arithmetic of

*-- of a fully-algorithmic "dimensional analysis" -- and an ideographical arithmetic of "pure,*

**metrical**__qual__ifiers__ifiable ontological__

**un****quant**__ifiers", that reveals itself to be a "purely-__

**qual**__itative__

**qual***", a*

**arithmetic of**__dialectics__*"*, an

**"**__dialectical__ideography*"*, as well as a

**ideography**of/for**"**__dialectics__*of all three of these, initially separate-appearing, arithmetics, an arithmetic of*

__dialectical__synthesis*, and more.*

**-**__quant__ified metrical__qual__ifiers__qual__o__quant__ifying their ontological__qual__ifiersIt took Seldon to see, in the four, "first order logic" axioms of "

**atural" arithmetic, the first four "Dedekind-Peano Postulates" [generalized] --**

__N__**(**

**1**

**)**The

**is a constituent of a generic succession of**

*first entity***.**

*entities***(**

**2**

**)**The

**immediate consecutive successor***of any constituent*

*entity**of this consecutive succession of*

*entity**is itself also a constituent*

*entities**of this consecutive succession of*

*entity**.*

*entities***(**

**3**

**)**No two distinct constituents of this consecutive succession of

*have the same successor*

*entities**.*

*entity***(**

**4**

**)**The

*in this 'consecuum' has successor*

*first entity**in this 'conseccum', but no predecessor*

*entities**in this 'consecuum',i.e., it is the «*

*entities*

*arch***»**

*é**of this therefore*

*entity**‘*

__arche__

*onic consecuum**’*of

*.*

*entities*-- an internal tension between ordinality as the purely-

__itative__

**quant***of the 'consecuum' --*

**ordinality****1st ---> 2nd ---> 3rd ---> 4th ---> 5th ...**

-- and ordinality as the purely-

__itative__

**qual***of the consecuum' --*

**ordinality**the

__ity of__

**qual***ness*

**first**-**--->**the

__ity of__

**qual***ness*

**second**-**--->**the

__ity of__

**qual***ness*

__-__**third****--->**the

__ity of__

**qual***ness*

**fourth**-**--->**the

__ity of__

**qual***ness*

**fifth**-**--->**...

Thus, in Seldon's dialectical presentation of the dialectic of the dialectical arithmetics, the standard model of the "first-order

__atural Numbers" arithmetic,__

**N**

**N****, the model of an arithmetic of numbers which are "pure,**

_______ified__

**un****qual**__ifiers", externalizes, under dialectical, immanent critique [self-critique] , its previously only internal, implicit, suppressed other, an arithmetic of "[meta-]numbers" which are "pure,__

**quant**__ifiable ontological__

**un****quant**__ifiers" --__

**qual**

__N___ --->__N___(__N___) =__N___^2 =__N___ +__Delta_____N___=__N___+__N____Q___-- wherein

__N____Q__**turns out to be, on further interpretation, a "non-standard model" of the "**

_______atural" Numbers -- of their "first-order Dedekind-Peano Postulates", and the Seldonian "First Arithmetic of Dialectics", such that the number-space of that dialectical arithmetic is__

**N****N**

__Q__

**=****{**,

__q__1**,**

__q__2**,**

__q__3**... }**

-- where in, per its first layer of interpretation --

**represents the general**

__q__1__ity if__

**qual***ness in a progression;*

__-__**first****represents the general**

__q__2__ity if__

**qual***ness in a progression;*

__-__**second****represents the general**

__q__3__ity if__

**qual***ness in a progression, etc., etc.*

__-__**third**Historically, [any speed-of-light-contiguous region of] Nature exhibits a rigid chronological ordinality, a necessary "order of birth", or "order of appearance", of the kinds of things that it "contains" -- of the ontological units that it exhibits, and of the ontological categories to which we humans assign them.

E.g., sub-nuclear "particles" must sustainedly appear before "sub-atomic particles" can sustainedly appear, "sun-atomic particles" must sustainedly appear before "atoms" can sustainedly appear, "atoms" must sustainedly appear before "molecules" can sustainedly appear, "molecules" must sustainedly appear before "prokaryotic living cells" can sustainedly appear, and so on.

Likewise, with the arising of "critical densities" reflecting the growth of the social forces of production, the social relations of production known as --

"Goods" must sustainedly appear, and in a certain critical density, before bartered "Commodities" can sustainedly appear, "Commodities" must sustainedly appear, and in a certain critical density, before "Monies" can sustainedly appear, "Monies" must sustainedly appear, and in a certain critical density, before "Capitals" can sustainedly appear, and "Capitals" must sustainedly appear, in a certain critical density before the "Generalized Equities" of Socialist Society can sustainedly appear.

Thus, even in their interpretation as merely "ordinal

**ifiers", the**

*qual*

__q__N**ualifiers of the**

__q__

__NQ__**"First Dialectical Arithmetic" already show their potential to model**

_____*.*

**historical**__dialectics__Even a "synchronic" systematics, or taxonomization, of the ontological categories of a system like capitalism, shows a "natural", systematic ordering of those categories, in a gradient, or 'consecuum', from the simplest, most abstract such category -- e.g., for capitalism, Marx's '"Elementary Form of Commodity-[Capital-]Value"' -- all the way to the most complex, most thought-concrete such socio-ontological category -- e.g., that of the '''Expanded [Self-]Reproduction of the Total Social Capital Value, per the "Law of Capital-[Value]", including of the Revenues of the Three Major Classes of Capitalist Society [Wages, Rent, and Profit], together with that of its "underlying" human society, under the Equalization of the General Rate of Profit and the Resulting Transformation of Values, and under the Law of the Tendency of the Rate of Profit to Fall In Response to the Growth in the Social Forces of Production Driven by the Profit-Increase Motive That Drives the Pursuit of the Increase of Relative Surplus-Value ...".

That is, "The Elementary Form of Value" should be presented, in a systematically-ordered presentation of the theory of Capitalism, before presenting "The Expanded Form of Value", which should be presented before presenting "The General Form of Value", which should be presented before presenting '"The Money Form of Value''', ... which should be presented before presenting the '''

**The Capital Form of Value**''', ...

Thus, even in their interpretation as merely "ordinal

**ifiers", the**

*qual*

__q__N

**q****ifiers of the**

*ual*

__NQ__**"First Dialectical Arithmetic" already show their potential to model**

_____*.*

**systematic**__dialectics__When the next layer of interpretation, the second,

*"*ordin[aliz]ation" layer of interpretation, is super[im]posed upon of the first layer of interpretation, the

__sub__-*"*itative ordinality" layer of interpretation, yielding --

**qual****represents the**

__q__1__ity of the__

**qual**

**first***in a systematic-dialectical presentation of a progression of ontological categories, or of the*

**thesis***<<*

**first***>> in an historical-dialectical model of the chronological progression of the emergence of*

__phys__is__ical kinds-of-things;__

**phys****represents the**

__q__2__ity of the__

**qual**

**first***in a systematic-dialectical presentation of a progression of ontological categories, or of the*

**-**__contra__**thesis**

**first***<<*

__-__**meta***>> in an historical-dialectical model of the chronological progression of the emergence of*

__phys__is__ical kinds-of-things;__

**phys****represents the**

__q__3__ity of the__

**qual**

**first**full*in a systematic-dialectical presentation of a progression of ontological categories, or of the*

**-**__uni__**thesis**

**first***<<*

__uni__*-**>> in an historical-dialectical model of the chronological progression of the emergence of*

__phys__is__ical kinds-of-things, etc., etc.__

**phys**-- then the capability of the -- "purely

__itative [__

**qual***]*

**meta**-*of the --*

**numbers**

__q__N

**q****ifiers of the**

*ual*

__NQ__**"First Dialectical Arithmetic" to model both**

_____*and*

**systematic**__dialectics__*is even more explicitly and concretely conveyed.*

**historical**__dialectics__Regards,

Miguel

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