The
‘Meta-Peanic’,
Qualitative,
Categorial
Successor
Functions
and the
F.E.D.
Dialectical
Method.
Dear Reader,
The Encyclopedia
Dialectica ‘Qualo-Peanic
Dialectical Method’ is founded on a qualitative, ontological-categorial version
of the Peano ordinal successor function of the “Peano-Dedekind Postulates”, the
four, “first-order-logic” axioms for the Standard “Natural” Numbers, N.
The simple essence of this
‘Qualo-Peanic Dialectical Method’, is to append the subscript of the ordinally first category-symbol in your
category-symbols progression – the «arché»-category’s symbol –
next to and on the right-hand-side of the subscript(s) of a duplicate of the
category-symbol that you have most recently formed, in order to form the
consecutive next category-symbol in your progression, be it a ‘self-hybrid’
or ‘merely-hybrid’ next category-symbol. This
procedure is equivalent to applying an ‘ordinal-qualitative’ version
of the, ultra-simple, ordinal-quantitative, Peano successor function –
s(n) = n+1, for any “Natural” number n.
This qualitative
successor function inheres in the Seldonian NON-Standard Model of the first-order-logic
axioms of the “Natural” Numbers,
the NQ Model.
As we have often noted before, the Standard Peano successor function, s, is already an ultra-simplified, vestigial case of the dialectical «aufheben» function itself. That is, when s operates on n –
s(n) = n + 1
– where n is a variable standing for any Standard “Natural” Number, the effect of s on n is to determinately negate n, making it into a not-n in the determinate form of –
n + 1
– which also, visibly, conserves n, but also elevates n, by one ordinal unit, ‘1’, up into the consecutively-next, next-higher – the successor – Standard, Peano “Natural” Numbers ordinal scale.
The rules of the Encyclopedia
Dialectica,
algorithmic-heuristic, ‘Qualo-Peanic Dialectical Method” are as follows:
§1. First write down your category-symbol for what you take to be the starting-category, or ‘«arché»-category’, for the subject-matter field that you want to, ‘condensedly’, model; to learn; or to present – call it, generically, qa.
§2. Next, place, to the right
of that first category-symbol, right after placing one blank space, a non-amalgamative
“plus sign”, ‘Å’, at script-level, and then, after placing
another blank space to the right of that “plus sign”, place a second category-symbol,
but with the subscript of your very first category-symbol,
‘a’, re-appended, again, next to and to the
right of the subscript of your very first, or «arché»,
category-symbol, also ‘a’.
The ‘Å’ sign, for “physical addition” – e.g.,
for “non-amalgamative” addition – goes, at script level, between those, now
two, similarly-subscripted category-symbols, thus ‘‘‘adding’’’ a second category-symbol,
‘qaa’, to
your initial, arché»
category-symbol, qa; a second category-symbol with a double,
repeat subscript: generically,
‘qa Å qaa’.
§3. Next, replace that
repeat-subscript of that second category-symbol, if you can, with a single, different subscript,
generically, ordinally, ‘b’, that abbreviates
a name for your meaning for the new units-kind
represented by that new, second category-symbol, and replacing its two, repeat subscripts, ‘aa’, with that subscript, ‘b’. This second, new units-kind
category-symbol, qb, is so as
to ‘‘‘contain’’’, referentially, multiple new units, each of which is a new, unifying combination of two or more former units of your starting-category, qa. Generically, your dialectical,
ontological, categorial progression is now ‘qa Å qb’.
§4. From then on, to construct each next category-symbol in your category-symbols progression for your chosen subject-matter Domain, just copy your last/most-recent, right-most category-symbol, and place that copy to the right of itself, after placing a non-amalgamative addition operation-sign, ‘Å’, there first, between them, at script level. Then, simply append the subscript for your starting category – generically, here, ‘a’ – to the right of the subscript(s) of that copy of your last category-symbol. Next, try to find a single character that ‘character-izes’, or abbreviates the name of, and thus can mnemonically replace, the resulting multiple subscripts, by a lesser number of subscripts; best by a single subscript. For example, the next step to our generic example, above, is to form the sequence qa Å qb Å qba. Then find, if you can, a single-word name for category description qba, and abbreviate that name by its initial letter, if non-redundant – perhaps “g”, the ordinally third letter of the Greek alphabet – as subscript, thus obtaining: ‘qa Å qb Å qg’. Continue iterating this step, together with applying rule §5., below, whenever rule §5. is applicable, until a category-symbol results whose potential meaning you do not recognize, and cannot define or name.
§5. If this adding of the
subscript of the starting category-symbol, generically ‘a’ , creates a repeat subscript, then, for consistency,
replace that repeat subscript with the single subscript by which
you replaced that repeat subscript previously, earlier in your category-symbols
progression so far. Or, if this repeat
subscript is new – has not arisen before in your category-symbols
progression so far – then replace it with, if you can, a single-character
subscript, abbreviating the name of the category that you recognize as the next
higher ‘self-hybrid’ units-kind that fits for what this
new category-symbol describes for you.
The JPG images posted at the top of this blog-entry
may help to clarify this ‘algorithmic-heuristic’ method further.
For more
information regarding these
Seldonian insights, and to read and/or download, free
of charge, PDFs and/or JPGs of Foundation books, other texts, and images, please see:
and
https://independent.academia.edu/KarlSeldon
For partially pictographical, ‘poster-ized’ visualizations of many of these Seldonian insights -- specimens of ‘dialectical art’ – as well as dialectically-illustrated books
published by
the F.E.D. Press, 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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your comments on this blog-entry below!
