Friday, December 07, 2012

Recent Dialogue on the "Purely-Qualitative Mathematics" that is also a "Mathematics of Dialectics"

Below is a paraphrased excerpt from a recent dialogue, in which I participated, regarding the NQ_ "arithmetic of pure, unquantifiable ontological qualifiers" as an "arithmetic for dialectics".

Q:  How can you have "qualitative mathematics"?    Math is all about manipulating quantities.

M.D.:  Thank you for your excellent question!

What mathematics appears to be, up to a certain period in human history, may not be identical to what it, in essence, really is.

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.D., that the historically general definition of mathematics should be --

'Mathematics = Ideometry via [phono-picto-]Ideography'

-- with 'Ideometry' naming a very general concept of "the measurement of ideas"; any method of "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 <<mentalite'>>, 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. --

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 unequal to, but not quantitatively unequal to the set {c, d}; therefore

E. the set {a, b} is qualitatively unequal to the set {c, d}.

2. Mathematical Logic ["first order predicate calculus"]: Fa ==> Ga, meaning "If the individual a exhibits the quality F, then it must also exhibit the quality G". Nothing quantitative about this "calculus".

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. +i is not = +1, and,

C. +i is not < +1; therefore

D. +i is unequal to, but not quantitatively unequal to, +1; therefore

E. +i is qualitatively unequal to +1.

4. Dimensional Analysis. 3 centimeters [3 cms.] are greater than [>] 2 cms., but, regarding the relationship between 1 linear centimeter [1cm. = 1cm.^1] and 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 unequal to, but not quantitatively unequal to, 1cm.^1; therefore

E. 1cm.^2 is qualitatively unequal to 1cm.^1.

The above provide just a few of the manifold evidences of the fact that modern mathematics is not just about your "manipulating quantities", but is about, at the very least, the 'qualo-quantitative manipulation of 'qualo-quantities'.

The ancient Mediterranean makers and discoverors of mathematics had, at the very least, deeply insightful inklings of this 'qualo-quantitative essence of mathematics.

The most insightful of these insights, I would argue, was that of Plato, who placed his <<Arithmoi Eidetikoi>> -- his 'idea [<<Eide>>]-numbers [<<Arithmoi>>]' -- at the very heart of his Dialectics.

But, at least until Jacob Klein's Greek Mathematical Thought and the Origin of Algebra, most Plato scholars, blinded by the unconscious 'capital mind' of the modern human <<mentalite'>>, seem to have had not a clue as to what Plato meant by <<Arithmoi Eidetikoi>>:

"arithmos: number; arithmêtikê; the science of number.

Zero was unknown as a
number and one also was not counted as a number, the first numberbeing the duas – two.

From the Pythagoreans, ton arithmon nomizontes arkhên einai – who consider number to be the first principle (Ar. Met. 986a15) – number played a great part in metaphysics, especially in Plato’s unwritten doctrines, involving obscure distinctions of e.g. sumblêtoi and asumblêtoiaddible and non-addible numbers."

J. O. Urmson, The Greek Philosophical Vocabulary, Gerald Duckworth & Co., Ltd. [London: 1990], pp. 31-32, [emphasis added]

In the ancient concept, "Number" -- <<Arithmos>> -- means "an assemblage of qualitative units".

The <<Eide>>-units -- the <<Eide-monads>>, the 'idea-units' -- are <<asumbletoi>>, 'unsumable', because they are purely-qualitative units, not purely-quantitative units, "numbers".

The <<Arithmoi Eidetikoi>> are 'purely-qualitative numbers', not our modern "purely-quantitative numbers", such as the "Natural" Numbers -- which are "Natural" only to our modern, "Elementary Form of Value" formed <<mentalite'>> and ideology, but not to the <<mentalite'>> and ideology of ancient humanity.

Natural language, the ultimate foundation of all of mathematics, is built upon 'qualo-quantitative' phrases such as "ten silas of olive oil".

It took the genius of Karl Seldon to see this, and to see through the ideology of the "purely-quantitative", the ideology of "The Elementary Form of [Commodity[-Capital-]]Value" formed <<mentailite'>>, and to actualize the latent, fuller, explicitly dialectical stage of mathematics.

Seldon calls the "ten" component of the phrase above "the metrical quantifier". He calls the "silas" component the "metrical unit qualifier". He calls the "olive oil" the "ontological unit qualifier".

And it was Seldon who first discerned that there is not only an ideographical arithmetic of "pure quantifiers", such as that of the standard "Natural" Numbers, N, but also that there is an ideographical arithmetic of metrical qualifiers -- of a fully-algorithmic "dimensional analysis" -- and an ideographical arithmetic of "pure, unquantifiable ontological qualifiers", that reveals itself to be a "purely-qualitative arithmetic of dialectics", a "dialectical ideography", an "ideography of/for dialectics", as well as a dialectical synthesis of all three of these, initially separate-appearing, arithmetics, an arithmetic of quantified metrical qualifiers qualo-quantifying their ontological qualifiers, and more.

It took Seldon to see, in the four, "first order logic" axioms of "Natural" arithmetic, the first four "Dedekind-Peano Postulates" [generalized] --

(1) The first entity is a constituent of a generic succession of entities.

(2) The immediate consecutive successor entity of any constituent entity of this consecutive succession ofentities is itself also a constituent entity of this consecutive succession of entities.

(3) No two distinct constituents of this consecutive succession of entities have the same successor entity.

(4) The first entity in this 'consecuum' has successor entities in this 'conseccum', but no predecessor entities in this 'consecuum',i.e., it is the «arché» entity of this therefore archeonic consecuum of entities.

-- an internal tension between ordinality as the purely-quantitative ordinality of the 'consecuum' --

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

-- and ordinality as the purely-qualitative ordinality of the consecuum' --

the quality of first-ness ---> the quality of second-ness ---> the quality of third-ness ---> the quality of fourth-ness ---> the quality of fifth-ness ---> ...

Thus, in Seldon's dialectical presentation of the dialectic of the dialectical arithmetics, the standard model of the "first-order Natural Numbers" arithmetic, N_, the model of an arithmetic of numbers which are "pure, unqualified quantifiers", externalizes, under dialectical, immanent critique [self-critique] , its previously only internal, implicit, suppressed other, an arithmetic of "[meta-]numbers" which are "pure, unquantifiable ontological qualifiers" --

N_ ---> N_(N_)   =   N_^2   =   N_ + Delta_N_   =   N_ + NQ_

-- wherein NQ_ turns out to be, on further interpretation, a "non-standard model" of the "Natural" 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

NQ   =   { q1, q2, q3, ... }

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

q1 represents the general quality if first-ness in a progression;

q2 represents the general quality if second-ness in a progression;

q3 represents the general quality if third-ness in a progression, etc., etc.

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 qualifiers", the qN qualifiers of the 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 qualifiers", the qN qualifiers of the NQ_ "First Dialectical Arithmetic" already show their potential to model systematic dialectics.

When the next layer of interpretation, the second, "sub-ordin[aliz]ation" layer of interpretation, is super[im]posed upon of the first layer of interpretation, the "qualitative ordinality" layer of interpretation, yielding --

q1 represents the quality of the first thesis in a systematic-dialectical presentation of a progression of ontological categories, or of the first <<physis>> in an historical-dialectical model of the chronological progression of the emergence of physical kinds-of-things;

q2 represents the quality of the first contra-thesis in a systematic-dialectical presentation of a progression of ontological categories, or of the first meta-<<physis>> in an historical-dialectical model of the chronological progression of the emergence of physical kinds-of-things;

q3 represents the quality of the first full uni-thesis in a systematic-dialectical presentation of a progression of ontological categories, or of the first uni-<<physis>> in an historical-dialectical model of the chronological progression of the emergence of physical kinds-of-things, etc., etc.

-- then the capability of the -- "purely qualitative [meta-]numbers of the -- qN qualifiers of the NQ_ "First Dialectical Arithmetic" to model both systematic dialectics and historical dialectics is even more explicitly and concretely conveyed.

Regards,

Miguel