Karl Seldon on 'The Fundamental Theorem of Dialectics' -- The Significance of the Theorem: Curing the Chief Defect of Ancient 'Arithmism' & 'Monadism'.

Dear Reader,



I have excerpted, below, for your enlightenment, a passage from a recent "lecture to the troops" by Karl Seldon, which addresses the significance of the Seldonian 'Fundamental Theorem of Dialectic'.


I have also appended, to the excerpt below, the simple proof of that 'Fundamental Theorem'.  The N Q_ axioms used in that proof are posted above, for your convenience.


Enjoy!


Regards,

Miguel

“... This ‘contra-Boolean’ theorem, that constitutes our ‘Fundamental “Law” of Dialectics’, in ideograms --

x + Delta[ x ]    =    x²    ~< & ~= & ~>    x¹

-- is deductively implied by the axioms that we have presented, as per the proof also presented -- all seemingly so simple -- provides nothing less than a rectification of the chief defect of Ancient Mediterranean ‘Arithmism’ and ‘Monadism’, i.e., of the principle that animated Ancient Mediterranean philosophy and science from the Pythagoreans, circa 360 B.C.E., through Plato and beyond, all the way forward at least to Diophantus’s ‘proto-ideographical’ algebra, circa 250 C.E.

That chief defect was the radical dualism of «arithmos» vis-a-vis «monad», i.e., of ‘‘‘assemblages of units’’’ versus ‘‘‘individual units’’’, viz.: “ EUCLID defines in the Elements, VII, 2, a number as “the multitude [K.S.:  «arithmos»] made up of units [K.S.:  «monad»]” having previously (Elements, VII, 1) said that a unit is “that by virtue of which each of existing things is called one.”  As a unit is not composed of units [K.S.:  but, on the contrary, a unit is, typically, “made up of” sub-units, e.g.,  a ‘meta-«monad»’ is “made up of” «monads», as we have seen], neither EUCLID nor ARISTOTLE regard a unit as a number, but rather as “the basis [K.S.:  «arché»] of counting, or as the origin [K.S.:  «arché»] of number.” ”

[H.-D. Ebbinghaus, et al., Numbers, Springer Verlag [NY: 1991], p. 12, bold/italic/color emphases added].

Under the spell of that radical diremption [‘<--|-->’] between «arithmos» and «monad», only a radically ‘statical’, ‘Parmenideanoid’, ‘early-Platonoid’ cosmos could be conceived, as an eternal ‘meta-«arithmos» cumulum’ of eternally fixed, radically distinct, genetically unrelated «arithmoi» of «monads».

True, an element of [“purely”-quantitative] dynamism could enter this world picture as “genealogy” -- as the begetting of new «monads» by old[er] «monads» of the «arithmoi aisthetoi», the sensuously-empirical “units” -- but all and only within a given kind, i.e., within a single, “eternal”, “eternally fixed” «genos»-«arithmos», or «species»-«arithmos»:  never as ‘meta-genealogy’.

That is, no «genos»-«arithmos» -- not even any «species»-«arithmos» -- could cross its boundaries of kind, its ontological boundaries, to give birth to «monads» of even a different, but already existing other «genos»-«arithmos», or «species»-«arithmos», LET ALONE give birth to a previously unprecedented, brand new «genos»-«arithmos», or «species»-«arithmos».  That is, “genealogy”, but no ‘meta-genealogy’, was admissible for such a world picture.

Thus, no cosmological ‘meta-dynamical meta-evolution’ was even conceivable for such a drastically ‘«arithmos»  <--|-->   «monad»’ «mentalité».

But the deductively-derived ‘contra-Boolean’ rule --

x[ x ]   =   x²   ~< & ~= & ~>   x   ==>  

~x   =   x + Delta[ x ]    =    x²   |   Delta[ x ]  ~< & ~= & ~>  x¹

-- interpreted in such a way that each of the «monads» of the successor «arithmos», denoted by Delta[ x ], is constituted out of a [sub-]«arithmos» of [some of] the [former] «monads» of the predecessor «arithmos», denoted by x, i.e., such that each [meta-]«monad» of the Delta[ x ] [meta-]«arithmos» is “made up of” a heterogeneous multiplicity of the «monads» of the x «arithmos», each Delta[ x ] unit thus a ‘meta-«monad» of those x-type units / «monads», tells a dramatically different story.

This rule can make possible the compact, ideographical description of Domains, of universes-of-discourse -- including of the universal [whole cosmos] universe-of-discourse, the universe-of-discourse of the total universe [as a whole], as a single, still-further-unfolding ‘meta-genealogy’.

Such a description thus formulates a ‘dialectical theory of everything’.

But even also for subordinate Domains, this rule makes possible the ultra-condensed, ideographical description of ‘sub-universes’, causally and ‘meta-genetically’ connecting predecessor «arithmoi»-kinds with their ‘‘‘offspring’’’ new-«arithmoi»-kinds -- their successor kinds -- consisting of both meta- and hybrid «arithmoi»-of-«monads», to which those predecessor «arithmoi»-of-«monads»-kinds give birth, describing a universe-of-discourse-‘universe-al’ ‘meta-genealogy’ of ongoing, recurring ontological innovations, i.e., of ‘onto-dynamases’, expressible / describable by / in / via a single ‘dialectical meta-equation’.  ...  ”