Leveraging ‘Peanicity’.
-- Part 9: Seldon’s Worldview Series.
Dear Reader,
It
is my pleasure,
and my honor, as an elected member
of the Foundation Encyclopedia
Dialectica [F.E.D.] General Council, and as a voting member of F.E.D., to share, with you, from time to time, as they are approved for public release, by the F.E.D. General Council, key excerpts from the internal writings, and from the internal sayings, of our co-founder,
Karl Seldon.
The ninth release in
this new such
series is posted below
[Some E.D.
standard edits have been applied, in the version presented below, by the editors
of the F.E.D. Special Council for the Encyclopedia,
to the direct transcript of our co-founder’s
discourse].
In this 9th installment, Seldon reveals the essence of his new, easier algorithm
for the NQ dialectical method as that of ‘leveraging Peanicity’,
i.e., of utilizing the character of the ‘consecutivity’ of the ‘Peanic Ordinal “Natural” Numbers’,
whether as “purely”-quantitative ordinal numbers, or as “purely”-qualitative
ordinals.
Seldon –
“The new and easier to learn algorithm that we have recently propounded for the Encyclopedia Dialectica [E.D.] dialectical method works by way of what we call “leveraging Peanicity’, as presented in the final 'concluding commentary' of our recent book entitled Dialectic: Users’
Manual, and throughout our latest book, entitled Dialectics Made Easy.”
“ ‘Peanicity’
is our name for the quality of ‘ordinality’ and ‘consecutivity’ captured by the first four of the five Peano-Dedekind axioms, which were intended
to uniquely capture the “Standard Natural Numbers” only.”
“These
first four of the “Peano Postulates”, unlike the fifth “Peano Postulate”, are
all formulated in “first order logic”. One
significance of this fact is that it means that, per the joint implications of
the Gödel Completeness Theorem and First Incompleteness Theorem, these “first
order” postulates must have “non-standard” models.”
“The E.D. ‘first axiomatic
arithmetic for dialectic’, which we denote by the double-underscored symbol(s) N_Q_, as one of those Gödel-theorems-implied
“non-standard models of the “natural” numbers”, shares those
first four, “first order” Peano Postulates – which we denote, collectively, by the symbol N_, i.e., with a double underscore – equally
with the “standard “natural” numbers”, whose full, including “second-order logic”
axioms system we denote by the symbol N, i.e., with only a single underscore.”
“It is the ultra-simple
Peano “successor function”, in its N_Q_ “non-standard”
form, which can be leveraged to define our new and easier recipe for the
generation and the solution of dialectical ontological-categorial progressions.”
“Denoting
the “standard” Peano successor function by s,
such that for n, a variable which denotes
any “standard “natural” number” in the set of all “standard
natural numbers”, denoted by N, we define its function/operation as –
s(n) = n + 1
-- which is
thus an ordinal ‘vestigial «aufheben» function/operation’,
as it “conserves” n by determinately
“negating” n, i.e., by producing a
not-n
which includes n but which is also “elevated” from/above/beyond n
by one standard unit, indeed by the «arché»
unit of the “standard “natural” numbers”, 1, thus moving from ordinal number n
to the consecutive next ordinal “natural” number.”
“We may then define the N_Q_ version of the Peano successor function, which we denote via the symbol s, by the “purely”-qualitative equation(s) --
s[qn] = qs(n) = qn+1
-- for every qn in the ‘meta-number space’ of the N_Q_ axioms-system, the ‘‘‘space’’’ or “set” which we denote by the symbol NQ, and for every ‘n’ in N.”
“If a, also representable as qa, denotes a user’s chosen ontological ‘«arché»-category’ for a given Domain, D, for either a systematic-dialectical or a historical-dialectical analysis of that Domain’s ontology, then, in a Domain-specific or ‘‘‘applied’’’ version of the generic NQ arithmetic/algebra for dialectic, we associate [‘[----]’] that ‘«arché»-category’, starting-category symbol, qa, of that Domain-specific dialectic-algebra, with the generic NQ ‘«arché»-category’ symbol --
q1: qa [----] q1.”
“Thereby, adding 1 to the subscript of the generic ‘«arché»-category’ symbol, by Peanic succession, such that --
s[q1] = qs(1) = q1+1 = q2
-- is associated with appending the Domain-specific ‘«arché»-category’ subscript, ‘a’, to the Domain-specific ‘«arché»-category’ symbol, qa, yielding --
sD[qa] = qsD(a) = qaa [----] q2.”
“Likewise, if bi-vocal qaa is solved [‘|-º’] by the user as having the same categorial meaning as the univocal category-symbol qb --
qaa |-= qb [----] q2
-- then the next step of advance in the dialectical categorial progression model of the Domain is --
sD[qb] = qsD(b) = qba [----] q3.”
“And so on,
in a one new category at a time, category-by-category advance, until the ontological
content of the Domain has been categorially completely represented, comprehended, and
explained, and, perhaps, a ‘pre-construction’ hypothesis regarding the future
ontological, categorial content of the Domain
has been framed.”
“In such dialectical categorial progression analyses of a given Domain, the 2nd category --
qaa |-º qb [----] q2
-- is typically called by us the ‘first contra-category’ of that categorial progression, and the third category --
qba |-º qg [----] q3
-- is called its ‘first uni-category’.”
For more information regarding these Seldonian insights, please see --
For partially pictographical, ‘poster-ized’ visualizations of many of these Seldonian insights -- specimens of ‘dialectical art’ -- see:
https://www.etsy.com/shop/DialecticsMATH
¡ENJOY!
Regards,
Miguel Detonacciones,
Voting Member, Foundation Encyclopedia Dialectica [F.E.D.];
Elected Member, F.E.D. General Council;
Participant, F.E.D. Special Council for Public Liaison;
Officer, F.E.D. Office of Public Liaison.
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