Introducing ‘The Hegelian Triadic-Tetradic Product Rule’ for Seldonian ‘Dialectical Algebras’.

Introducing ‘The Hegelian Triadic-Tetradic Product Rule’ for Seldonian ‘Dialectical Algebras’.

Dear Reader,

Hegel wrote, in Part I [The Science of «Logik»] of his Encyclopaedia of the Philosophical Sciences, as follows, regarding the ‘‘‘numerology’’’ of the rational [«vernunft»] division of contents, or of ‘‘‘subject-matters’’’, in the “philosophical sciences” -- “Any division is to be considered genuine when it is determined by the Concept.  So genuine division is, first of all, tripartite; and then, because particularity presents itself as doubled, the division moves on to fourfoldness as well.” [G.W.F. Hegel, The Encyclopedia Logic, translated by T.F. Geraets, W.A. Suchting, and H. S. Harris, Indianapolis:  Hackett Publishing Company, Inc., 1991, p. 298, §230, Addition [«Zusätze»]].

While we of Foundation Encyclopedia Dialectica [F.E.D.] in no way wish to claim any Hegelian orthodoxy for any of our work, let alone to import any of Hegel’s ‘‘‘massive’’’ mystifications into that work, we call attention, here, to the passage above, because of certain resonances between Hegel’s description, in that passage, of his version of dialectical «diairesis» [ontological-categorial division, per Platon], and some recent results of our ongoing research into possible alternative axioms, for Seldonian ‘dialectical algebras’, that go beyond our classical ‘‘‘Seldon Functions’’’, dyadic and triadic alike, in their efficacy for ‘dialectical meta-modeling’.

Those recent results center upon what we are provisionally calling ‘The Hegelian Triadic-Tetradic Product Rule’, which is showing signs of more apt ‘dialectical meta-modeling’ with regard to a certain super-Domain of dialectical instances, both at the ontological-categorial level, e.g., at the «species-arithmos» level, and also, with unprecedented efficacy, in our work, at the «monadic» level, that of ‘‘‘individual historical objects’’’.

I have included, here, below, a ‘pictogramical/ideogramical/phonogramical’ illustration, in the form of a new kind of E.D. ‘dialectogram’, applied to a particularly relevant case in point:  that of the currently consensus-hypothesized ‘cosmo-historical’ genesis of the Earth-Moon system in which we have our existence at this very moment.

As an axiom, that can be substituted for axiom §9 of the core axioms of the _NQ axioms-system for the Seldonian ‘First Arithmetic for Dialectic’, the formal essence of this ‘Hegelian Triadic-Tetradic Product Rule’ is the following --

‘A.  If the current ‘ontology-state’ of the dialectic is expressed via a single dialectical ‘meta-number’ value, call it

q_n,

then the next (E[_]) ‘ontology-state’ of the dialectic is expressed via the ‘non-amalgamative sum’ of the next two, consecutive, qualitatively distinct dialectical ‘meta-number’ values,

E[q_n]  =  sq_n + ssq_n  = q_sn + q_ssn  =  q_n+1 + q_n+2,

wherein s denotes the successor function for these dialectical ‘meta-numbers’, and wherein s denotes the Peano successor function for the Peano “Natural” Numbers;

B.  If the current ‘ontology-state’ of the dialectic is expressed via a ‘non-amalgamative sum’ of two, consecutive, qualitatively distinct dialectical ‘meta-number’ values,

q_n+1 + q_n+2,

then the next (E[_]) ‘ontology-state’ of the dialectic is expressed via a single dialectical ‘meta-number’ value, namely, that value which is the next consecutive value after the value of that one value of that summed pair of values which bears the larger Peano “Natural” Number subscript: 

E[q_n+1 + q_n+2]   =  s(max-subscript(q_n+1, q_n+2))  =  q_n+3.’

An illustration of the generic dialectical progression that arises per this candidate axiom is also pictured below.

The candidate axiom above represents what we call the ‘convolute product’ version of the ‘Triadic-Tetradic Product Rule’ for dialectical algebras.  Per it, all previous ontology uniformly vanishes when each next, incremental new ontology makes its irruption.

We are also investigating ‘evolute product’ versions of this candidate axiom, wherein previous ontology is conserved, at least ‘possibilistically’, external to and coeval with each ‘newly-irruptant’ increment of ontology.

So far in our research, the ‘convolute product’ version, given above, appears to work better for the ‘dialectical meta-modeling’ of individual «monads», units, or ‘‘‘holons’’’.  The ‘evolute product’ versions appear to work better for certain «arithmos»-level, categorial dialectics.

FYI:  Much of the work of Karl Seldon, and of his collaborators, including work by “yours truly”, is available for free-of-charge download via --

www.dialectics.info

Regards,

Miguel Detonacciones,

Member, Foundation Encyclopedia Dialectica [F.E.D.],

Officer, F.E.D. Office of Public Liaison