Thursday, July 11, 2013

The Historic Breakthrough to an Algebra of Dialectical Logic.

Dear Readers,

Here’s the story.

During 1665-1666 -- while Newton was discovering the integro-differential calculus [his “Method of Fluxions”] -- Leibniz, who was later to independently [re-]discover that calculus, was discovering something else:   the mathematics of formal logic; the ability of an arithmetic limited to the values 0 and 1 to model key characteristics of Aristotelian, syllogistic formal logic.

For Leibniz, this discovery constituted a starting point for his lifelong quest to find and to found a <<characteristica universalis>> -- a “universal character-language”, that is, a universal algebra, a universal ideography to facilitate both a scientific method of discovery and a scientific method of presentation of scientific discoveries. 

But Leibniz’s voluminous manuscripts on the algebra of formal logic, and on the larger <<characteristica universalis>> beyond it, were never published in Leibniz's lifetime.

Sometime before 1847, when he published The Mathematical Analysis of Logic, George Boole, with no knowledge of Leibniz's unpublished manuscripts on the algebra of formal logic, also [re-]discovered Leibniz’s discovery:   the capacity of a {0, 1} arithmetic to mimic much of formal logic.

Boole found, and then publicly founded, “Boolean Algebra”, the algebra of formal logic, that was to serve also the algebra of later electronic digital computer logic.

¿What do 0, which Boole took to model the “Nothing” category, and 1, which Boole took to model the extreme opposite category of the “Nothing” category, namely, the Everything category -- or “Universe” as Boole called it -- have in common?

Multiplicatively, they are “idempotent”:  their self-multiplication produces only themselves again as their only product --

0 x 0  =  0

1 x 1  =  1 

--  ‘Nothing times Noting equals Nothing’; ‘Everything times Everything equals Everything’:  identical in form, maximally different in meaning.  

Boole took an algebraic equation, universally true in an arithmetic limited to {0, 1}, as his “fundamental law of thought”, or “law of duality” --

X x X   =  X.

For Boole, this equation meant that when a category interacts with itself, nothing new results:  such self-interaction simply reproduces the category itself, without any conceptual or intellectual gain [or loss].

Such a “law” describes an eternally statical conceptual universe, an intellectually stagnant universe of thought.

On April 7th, 1996, Karl Seldon -- who later, in 1999, became a co-founder of Foundation Encyclopedia Dialectica [F.E.D.] -- discovered an “ ontologically dynamical ”, “contra-Boolean” algebra of dialectical logic, an algebra of qualitatively, ontologically expanding reproduction, after a nearly 40-year search, triggered by a childhood dream, in which he saw an “Equation of Universal Evolution”.

The “fundamental “law” of dialectical thought which Seldon formulated, using this new “contra-Boolean algebra” is a dialectical, <<aufheben>> negation of Boole’s “fundamental law of [formal-logical] thought”:  Seldon’s law conserves, and elevates, and transforms Boole’s “fundamental law” --

X x X   is qualitatively [ontologically] unequal to   X; unequal to X in kind [not in quantity]

-- and, more specifically --

X x X   =   X + DX

-- where DX denotes a new category, for a different kind of things than the kind of things category X denotes, but such that the DX new kind “grows out of” the X old kind.

The first term on the right-hand-side of the equation above, X, is the “Boolean” term:  the self-interaction of X reproduces X again. 

The second term on the right-hand-side of the equation, DX, is the “contra-Boolean” term:  the self-interaction of X also produces something new.

¿Can Seldon’s “contra-Boolean algebra” model aspects of experienced, empirical reality that Boole’s algebra cannot, reconstructing the historical past?

Consider the history of the “ meta-evolution ” [,EVOLUTION_vs._%27META-EVOLUTION%27,19DEC2012.jpg ] of the Marxian “social relations of production”. 

As the “social forces of production” -- as the “productivity of labor” -- grows, the social relations of production category of barterable Commodities, the interactions of the rising, expandedly-reproduced population and density of commodities and of barter-activities, gives rise to a “singularity” in human social relations, with irreversible changes in human activity and mentality:  the irruption into existence of monies and of the “Monies” category:

C   --->   C x C   =   C + DC    =   C + M.

Or, consider the history of the “meta-evolution” of pre-human ontological categories. 

When the rise in the populations and local ‘densifications’ / ‘concentrations’ of atoms, i.e., the expanded reproduction of atoms -- e.g., in the original proto-stellar ‘atomic clouds’ -- crosses a critical threshold, the first “molecules” are born, and “atomic clouds” become “molecular clouds”:

a   --->   a x a   =   a + Da    =   a + m.

This “fundamental law of dialectical thought also applies to the realm of mathematical ideation, including to the case at hand. 

If we take B to denote the category of Boolean Algebras, and C to denote the category of Seldonian Contra-Boolean algebras, we can present the following progression of categories of the algebras of logic, beginning with the dialectical antithesis of B and C --

B  --->  B x B    =   B^2   =   B + DB  =  B + C  

-- which, by going one step further, by multiplying again by B, yields --

B x (B + C)   =   B + C + qCB

-- all of which we can summarize as “the negation of the negation of B, by B, i.e.,
by ~~B  =  BBB  =   B x (B x B)   --

B^3   =   B x (B x B)   =   B x (B + C)   =   B + C + qCB

-- wherein qCB signifies a dialectical synthesis of C with B, i.e., here the q by itself signifies any kind of logical algebra category, and qCB signifies a kind of logical algebra that combines the C kind with the B kind.

In the F.E.D. rendering, the qCB algebra of logic is an actualization logic”, whereas B is a certainty logic”, and C is a possibility logic”.

The B algebra tells you what is certainly true about categories, or “classes”, by virtue of their form alone, ignoring their content. 

The C algebra tells you what ontological categories are possibly populated in a given epoch, versus which are not [yet] possible to be populated in that epoch.

The qCB algebra allows you to tell whether a category actually manifested in a given past epoch or not, because it combines B type coefficients with C type ontological categories:  bc(t) x c. 

If bc(t) = 1, then category c was actually manifest / populated / instanced in epoch t. 

If bc(t) = 0, then category c was not extant / existent in epoch t.

¿Can Seldon's "Contra-Boolean algebra" of dialectical logic predict -- or "pre-construct, symbolically" -- possible future historical developments?

A dramatic possible instance of such dialectical prediction is Seldon's hypothesis that the ontological category of "humanity", h, if it continues to grow and "densify", will give rise to a new cosmological ontological category, which he calls "meta-humanity" --

h ---> h x h = h + Dh = h + y

-- which, in terms of individual "meta-human" body-types, will develop three species, a "thesis" species category, a "contra-thesis" species category, and a "uni-thesis" species category --

g ---> g^3 = g x (g x g) = g x (g + r) = g + r + qrg

-- such that --

g = bodies resulting from human-genome self-re-engineering;

r = bodies resulting from android robotics [], and;

qrg = c = bodies built by the combination of r with g -- cyborg prosthetics / bionics [ ].



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