The ‘Meta-Peanic’, Qualitative, Categorial Successor Functions and the F.E.D. Dialectical Method.

 

 

 

 

 

 

 

 

 

 

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, q_a.

 

§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, ‘q_aa’, to your initial, arché» category-symbol, q_a; a second category-symbol with a double, repeat subscript: generically,

‘q_a Å q_aa’.

 

§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, q_b, 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, q_a.  Generically, your dialectical, ontological, categorial progression is now ‘q_a Å q_b’. 

§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 q_a Å q_b Å q_ba.  Then find, if you can, a single-word name for category description q_ba, 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: ‘q_a Å q_b Å q_g’.  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:

 

www.dialectics.info

 

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.

 

 

 

 

 

 

YOU are invited to post your comments on this blog-entry below!