Part 6 of 29. THE DIALECTICA MANIFESTO. Examples of Dialectic, Example #0.: 'The Goedelian Dialectic'.

Full Title:  Part 06 of 29 --

The Dialectica Manifesto

 

Dialectical Ideography and 

 

the Mission of F.E.D.

Dear Readers,

I am, together with F.E.D. Secretary-General Hermes de Nemores, and F.E.D. Public Liaison Officer Aoristos Dyosphainthos, organizing to develop a new, expanded edition of the F.E.D. introductory documents, for publication in book form, under a new title --

The Dialectica Manifesto:  Dialectical Ideography and the Mission of Foundation Encyclopedia Dialectica [F.E.D.]

-- and under the authorship of the entire Foundation collective.

Below is the sixth installment of a 29-part presentation of this introductory material, which the F.E.D. General Council has authorized for serialization via this blog over the coming months, as we develop the material.

I plan to inter-mix these installments with other blog-entries, including the planned additional F.E.D. Vignettes, other F.E.D.  news, my own blog-essays, etc.

Links to the earlier versions of these introductory documents are given below.

http://www.dialectics.info/dialectics/Primer_files/1_F.E.D.%20Intro.%20Letter,v.2,New_OCR.pdf

http://www.adventures-in-dialectics.org/Adventures-In-Dialectics/DiaRith/Intro/Dialectical-Ideography_An-Introductory-Letter.htm

http://www.dialectics.info/dialectics/Primer_files/2_F.E.D.%20Intro.%20Letter,%20Overall%20Prefactories%20%26%20Abstract,%20Supplements%20A.%20%26%20B.,%20v.2_OCR.pdf

http://www.dialectics.info/dialectics/Primer_files/3_F.E.D.%20Intro.%20Letter,%20Supplement%20A-1_OCR.pdf

http://www.dialectics.info/dialectics/Primer_files/5_F.E.D.%20Intro.%20Letter,%20Supplement%20B.,%20Prefatories,%20v.2_OCR.pdf

http://www.dialectics.info/dialectics/Primer_files/4_F.E.D.%20Intro.%20Letter,%20Supplement%20B-1,%20v.2_OCR.pdf

Unlike the typical blog-entry, this series will attempt to deliver an introduction to the Foundation worldview as a totality, in a connected account, making explicit many of the interconnexions among the parts.

Enjoy!!!

Regards,

Miguel

Part 06 of 29 --

The Dialectica Manifesto

 

Dialectical Ideography and 

 

the Mission of F.E.D.

 

 

The Gödelian Dialectic

 

and

 

F.E.D.’s ‘Dialectical Meta-Axiomatics’ Method of Presentation

Examples of Dialectic, Example #0: 

The Gödelian Dialectic.

Kurt Gödel, the contributor of, arguably, the greatest leaps forward in the science of logic since classical antiquity, described, in effect, an ‘axiomatic dialectic’ of mathematics, albeit in “[‘early’- ]Platonic”, ‘a-psychological’ terms, and in ‘a-historical’ terms, hence also in ‘a-psychohistorical’ terms, as follows:

“It can be shown that any formal system whatsoever — whether it is based on the theory of types or not, if only it is free from contradiction — must necessarily be deficient in its methods of proof.” 

“Or to be more exact:  For any formal system you can construct a proposition — in fact a proposition of the arithmetic of integers — which is certainly true if the system is free from contradiction but cannot be proved in the given system [the foregoing summarizes Gödel’s “First Incompleteness Theorem” — F.E.D.].”

“Now if the system under consideration (call it S) is based on the theory of types, it turns out that exactly the next higher type not contained in S is necessary to prove this arithmetic proposition, i.e., this proposition becomes a provable theorem if you add to the system the next higher type and the axioms concerning it.”

[Kurt Gödel; "The Present Situation of the Foundations of Mathematics (*1933o)", in S. Feferman, et. al., editors., Kurt Gödel: Collected Works (Volume III: Unpublished Essays and Lectures), Oxford University Press (NY: 1995), page 46; bold, italics, underline, and color emphasis added by F.E.D.].

Again:

“If we imagine that the system Z  [a formal, logical, propositional-/predicate-calculus system inclusive of “Natural” Numbers’  Arithmetic, not the full system of the positive and negative Integers, and zero [which is both, [or neither] positive [n]or negative], standardly also denoted by Z — F.E.D.] is successively enlarged by the introduction of variables for classes of numbers, classes of classes of numbers, and so forth, together with the corresponding comprehension axioms, we obtain a sequence (continuable into the transfinite) of formal systems that satisfy the assumptions mentioned above, and it turns out that the consistency (ω-consistency) of any of these systems is provable in all subsequent systems.”

“Also, the undecidable propositions constructed for the proof of Theorem 1 [Gödel’s “First Incompleteness Theorem” — F.E.D.] become decidable by the adjunction of higher types and the corresponding axioms; however, in the higher systems we can construct other undecidable propositions by the same procedure.”

“...To be sure, all the propositions thus constructed are expressible in Z (hence are number-theoretic propositions); they are, however, not decidable in Z, but only in higher systems...”

[Kurt Gödel; On Completeness and Consistency (1931a), J. van Heijenoort, editor, Frege and Gödel: Two Fundamental Texts in Mathematical Logic, Harvard University Press (Cambridge: 1970), page 108; bold, italic, underline, and color emphasis and [square-brackets-enclosed commentary] added by F.E.D.].

The cumulative «aufheben»-progression of ‘‘‘conservative extensions’’’, i.e., the advancing ‘ideo-cumulum’ of axiomatic systems which Gödel describes above was viewed ahistorically by him.

Gödel, as a ‘Parmenidean’, and a professed “mathematical Platonist” [in the sense of the earlier rather than of the later Plato; see below], didn’t intend this ‘meta-system’ — this cumulative diachronic progression of [axioms-]systems — to serve as a temporal or psychohistorical model of the stages of human mathematical understanding, as reflective of the stages of the self-development of humanity’s collective cognitive powers as a whole; of the knowledges to which each such epoch of those powers renders access, and of the “historically-specific” ideologies [or pseudo-knowledges] to which human thinking is susceptible within each such epoch.

But we do wish to explore its efficacy as such. 

Note how, as Gödel narrates above, each successor system «aufheben»-contains its immediate predecessor system, and, indeed, all of its predecessor systems; how each higher logical type «aufheben»-contains all predecessor logical types. 

¿Can Gödel’s theory of this cumulative, ‘‘‘evolute’’’, «aufheben» progression of axioms-systems, which we term ‘The Gödelian Dialectic’, or ‘The Gödelian [Idea-Systems’ Ideo-]Metadynamic’, provide at least an idealized [i.e., a distorted] image of this humanity’s actual history, of the actual psychohistorical struggle, process, and progress of mathematical aspects of the self-development of a humanity’s collective cognitive capabilities, hence of its knowledges and ideologies?

We shall see.

Each of Gödel’s “undecidable” propositions of arithmetic that plague each ‘‘‘epoch’’’ of this formal axiomatic expansion are propositions each asserting the unsolvability of a different, specific “diophantine” [referencing Diophantus’ «Arithmetica»; see more on this below] unsolvable equation. 

I.e., each “Gödel formula”, or “Gödel sentence”, which formally asserts the self-incompleteness-or-self-inconsistency of its axioms-system, “deformalizes” to one asserting the unsolvability of a specific, algebraic “diophantine equation”:

“... The Gödel sentence φ... asserts its own undeducibility from the postulates... .”

“Deformalizing φ... we see that under the standard interpretation it expresses a fact of the form [for every n-ary list of number-components of the “diophantine” algebraic equation’s unknown, x, such that each number-component is a member of the set of ‘‘‘diophantine numbers’’’, or “Natural” Numbers, in use — F.E.D.]  

...ƒx ≠­ gx..., 

where ƒ and g are n-ary polynomials... .”

“An equation ƒx = gx, where ƒ and g are two such polynomials, is called diophantine [see below for further information regarding Diophantus of Alexandria  — F.E.D.] ... .”

 

“By a solution of the equation we mean an n-tuple α of natural numbers such that 

ƒα = gα... .”

“So φ... asserts the unsolvability of the...equation ƒx = gx, and the proof of [Gödel’s “First Incompleteness Theorem” — F.E.D.] produces... a particular diophantine equation that is really unsolvable, but whose unsolvability cannot be deduced from the postulates...”

[Moshé Machover, Set Theory, Logic, and their Limitations, Cambridge University Press (Cambridge: 1996), pp. 268-269; emphasis added].

 

Per the standard modern definition, a “diophantine equation” is an equation whose parameters [e.g., coefficients] and whose solutions are restricted to the “natural” numbers [“positive integers”]. 

Each “Gödel sentence”-encoded equation truly is unsolvable within the given axioms-system. 

However, the proposition that it is so, also cannot be deductively proven within that axioms-system — but it can be so proven within the next axioms-system, the given axioms-system’s immediate successor-system — the latter, expanded axioms-system being created by means of the «aufheben» [self-]internalization’  of the ‘‘‘vanguard’’’, ‘‘‘meristemal’’’, highest [in logical type] set idea-objects of the universe of discourse of that predecessor axioms-system. 

It can also be so proven within all subsequent successor-systems, created by yet-further such «aufheben» ‘[self-]internalizations’. 

If the “logical individuals” or ‘arithmetical idea-objects’ “existing”[constructed] per the “comprehension axioms” of a given axioms-system are limited to “natural” numbers, classes of “natural” numbers, ... , all the way up to classes of classes of... of “natural” numbers, e.g., to ‘class-objects’ up to a given “logical type”, then the next system will cumulatively expand those ‘‘‘existential’’’ limits by one step, to include also classes of classes of classes of classes of... of “natural” numbers, i.e., ‘class-objects’ of still higher “logical type”.

Each successive higher class-inclusion of previous ‘class-objects’ can model [including via adjunction of its corresponding “comprehension axioms”, defining the ‘computative behavior’ of these new entities] a new kind of arithmetical ‘idea-object’; indeed, a new, higher kind of number. 

Thereby, this qualitative expansion of each predecessor axioms-system, in the formation of its successor axioms-system, this adjunction of the additional, “comprehension axioms” to the previous, predecessor axioms, corresponds to a qualitative expansion of the ‘idea-ontology’ -- of the ‘arithmetical ontology’, i.e., of the ‘number-ontology’ -- of that axioms-system.


CONJECTURE:  Specifically, the diophantine algebraic equation that was unsolvable as such within the predecessor axioms-system may itself become solvable, albeit in a non-diophantine sense, within the next [as well as within all subsequent] successor axioms-systems in this cumulative sequence of axioms-systems, precisely by means of these next new kinds of numbers, which will not be ‘diophantine numbers’, i.e., not “natural” numbers; not in the number-set N.

We can see a kindred  ‘unsolvability-to-solvability dialectic’ -- i.e., an ‘unsolvability-turning into-its-opposite’ dialectic  -- at work in the following examples, presented in a systematic order, rather than in the [psycho]historical order in which they arose for, and were solved by, our ancestors. 

The systematic order is that of the “filling in” of the so-called “Real” Number-Line, prior to the «aufheben» dialectical negation/elevation/conservation of that Number-Line in the irruption of the new ‘ideo-ontology’ of the “imaginary” numbers, and of the “Complex Plane”.

The progression of ‘‘‘number-spaces’’’ depicted above can be encompassed by a single Seldonian dialectical ‘meta-equation’ -- which “contains” a qualitatively different dialectical equation for every value of the presentational step parameter, the “Whole number” s -- which describes the progression of the axioms-systems of the arithmetics founded upon these ‘ideo-ontologically distinct’ number spaces, and which is written in terms of the Seldonian first dialectical arithmetic, viz.:

We shall not explicate the ‘dialectical meta-equation meta-model’ above in terms of its detailed workings at this juncture, letting that wait until readiness for this explication has been further cultivated in the course of this presentation. 
 
For those who wish to leap ahead, a thorough narration of this ‘meta-model’ is available via the links below.

http://www.dialectics.info/dialectics/Vignettes_files/v.4.5,Part_0.,Prefatories,Miguel_Detonacciones,F.E.D._Vignette_4,The_Goedelian_Dialectic_of_the_Standard_Arithmetics,14FEB2013.pdf

http://www.dialectics.info/dialectics/Vignettes_files/v.4.4,Part_I.,Miguel_Detonacciones,F.E.D._Vignette_4,The_Goedelian_Dialectic_of_the_Standard_Arithmetics,last_updated_29NOV2012.pdf

http://www.dialectics.info/dialectics/Vignettes_files/v.4.4,Part_II.,Miguel_Detonacciones,F.E.D._Vignette_4,The_Goedelian_Dialectic_of_the_Standard_Arithmetics,last_updated_29NOV2012.pdf



I.  The Paradox of Gainless Addition.  The equation [2 + x  =  2 or x  =  2 - 2] asserts a paradox:  ¿How can the addition of a number, x, produce a result, a sum, that is not bigger than that ‘known’ number, here 2, to which that “unknown” number, x, is added?

 
Given the N «genos» of number, addition always means increase.  It never means no increase.
This equation is not solvable within the set and within the system of arithmetic of the cardinal, or, sometimes, “Natural”, numbers, N  =  { 1, 2, 3, . . . }.
 
However, this equation is solvable -- specifically by the ‘non-diophantine number’ 0 -- within the ‘ideo-ontologically’ expanded system of the “Whole numbers”, W  =  { 0, 1, 2, 3, . . . }.

The “number” 0 belongs to a new set of numbers -- to a new kind of number -- which we denote by a, which stands for the set of the ‘‘‘aught’’’ numbers.

[¡Adjunction of this zero concept may seem trivial to us, yet it entailed a great and protracted conceptual travail for our ancient Mediterranean ancestors, and, with respect to issues surrounding division by zero, and the related issues of the singularities of, especially, the nonlinear integrodifferential equations, remains fraught with unresolved problems, “even” among we moderns today!].

II.  The Paradox of 'Subtractive' Addition.  The equation [2 + x  =  1] asserts a paradox:  ¿How can the addition of a number, x, produce a result, a sum, that is less than that ‘known’ number, here 2, to which that “unknown” number, x, is added?

Within the W «genos» of number, addition always means a change that increases, or, at minimum, that results in no change at all, but it never means a decrease.

Our second equation thus finds no number among the so-called “Wholes” to solve it/satisfy it.

However, that equation does find a solution among the “integers” or ‘‘‘integral’’’ numbers, the expanded numbers-set Z  ≡  { . . . ,  -3,  -2,  -1, ±0, +1, +2, +3, . . . }, which is a qualitatively, that is, ‘ideo-ontologically expanded, new-kinds-of-numbers-expanded, meaning-of-“number”-expanded, or ‘meme-ing’-of-“number”-expanded -- semantically-expanded -- universe-of-discourse of “Number”, vis-à-vis  the preceding «genos», the W universe-of-discourse.

The equation is solved / “satisfied” by the ‘non-diophantine number’  

-1 is an E[lement] of m which is C[ontained] in Z.

The “number” -1 belongs to a new set of numbers -- to a new kind of number -- which we denote by m, which stands for the set of the ‘‘‘minus’’’ numbers.

III.  The Paradox of 'Decreasive' Multiplication.  Next, the equation [2x  =  1] also asserts a ‘‘‘paradox’’’:  ¿How can the multiplication of any number, namely that of the “multiplicand”, denoted here by the algebraic “variable” or “unknown”, x, by another, known, number, the “multiplier”, here 2, produce a “product” which is less than that “multiplier”, e.g., in this case, the “product” 1?
 
Multiplication, within the Z «genos» of number, always produces a “product” which is either increased in absolute value relative to the “multiplicand” “factor”, or leaves the multiplicand unchanged, or turns it into zero.
 
But Z multiplication can never turn a (+)2 into a (+)1.

Such an equation is not solvable within the system of arithmetic of the “integers”; of the Z.
This equation is solvable, however, via ‘ideo-ontological expansion’ to encompass the qualitatively different system of arithmetic of the “Quotient numbers”, the “ratio-numbers”, the “ratio-nal” numbers, or “fractions”, denoted by Q, i.e., by an expansion that encompasses yet a new kind of ‘non-diophantine number’ [per the modern sense of the term “diophantine”], the ‘split a-tom’ [i.e., the ‘cut uncuttable’], ‘monad-fragment’, or “fractional value”, +1/2, part of --

Q   =  {....-4/2...-3/2...-2/2...-1/2...±0/2...+1/2...+2/2...+3/2...+4/2....}.

The “number” +1/2 belongs to a new set of numbers -- to a new kind of number -- which we denote by f, which stands for the set of the ‘‘‘fractional’’’ numbers.

IV.  The Paradox of the Odd Number Unknown That Must Also Be an Even Number, or That Must Be NEITHER an Even Number NOR an Odd Number.  The [algebraically] nonlinear equation [ x²  =  2 ] asserts a ‘‘‘paradox’’’ too.   

This equation requires x to be a kind of number which is ‘both odd and even at the same time’ [see the classic «reductio ad absurdum» proof of the “ir-ratio-nality”, or ‘non-ratio-ness’ [in terms of ratios of integers], of the square-root of 2, \2/2.
 
This equation is not solvable ‘‘‘ratio[n]ally’’’.


This equation is solvable via ‘ideo-ontic’ expansion to the so-called “Real” numbers, this time by two distinct ‘non-diophantine numbers’, due to the nonlinear, “2nd degree” character of this “unsolvable” equation, rather than by just one ‘non-diophantine number’, as were the preceding, [algebraically] linear, degree 1 “unsolvable” equations / ‘‘‘paradoxes’’’.

 
The two solutions are the so-called “irrational” values -\2/2 and +\2/2, both elements of the set --

R   =   {.....-pi....-3....-e....-2....-\2/2....-1....±0....+1....+\2/2....+2....+e....+3....+pi.....}.


The “numbers”-\2/2 and +\2/2 belong to a new set of numbers -- to a new kind of number -- which we denote by d, which stands for the set of the ‘diagonal numbers’ [i.e., for the set of the so-called “irrational numbers”].

V.  The Paradox of Additive Inverse  =  Multiplicative Inverse 'Identicality'.  Finally -- but “finally” only for the limited purposes of this selected spectrum of examples of “unsolvable diophantine equations” -- the algebraically nonlinear equation --

x² + 1  =  0
 
-- asserts a ‘‘‘paradox’’’ as well.   


This equation implies that -x  =  +1/x, requiring a kind of number whose additive inverse, -x, equals its multiplicative inverse -- 1/x, or x^-1 -- whereas, among so-called “Real” numbers --


-2     ~=   +1/2, 
-3     ~=   +1/3,
-pi    ~=­   +1/pi

-- etc., etc.

¿On what basis do we hold that the number x described by the equation [x² + 1  =  0] must be such that its additive inverse is always equal to its multiplicative inverse? 

On the basis that this characteristic of x is implicit in that x-defining equation itself.

To see that this is so, let us apply a little algebra to this equation, [x² + 1  =  0], always performing exactly the same operation on both the Left Hand Side [LHS] and the Right Hand Side [RHS] of this equation, so that the relationship of equality is always maintained between those two sides, even as their form and content changes as a result of the operations performed on them.

First, let us subtract 1 from both sides --   x² + 1 - 1  =  0 - 1  --  yielding  --   x² + 0   =   -1.

Next, let us divide both sides of the resulting equation, [x²   =   -1], by x. 

This action requires the assumption that x  ~=  0, to preclude dividing by zero, but we already know this to be true, because the equation [0² + 1  =  0] is a false assertion.

That is, [0² + 1  ~=  0], or, more specifically, [0² + 1  =  1] is the true propositional negation of that false 'posit-tion' / assertion. 

So, we obtain thereby --

x² /   x   =   -1 / x  

-- which is equivalent to --

+x   =   -1/x

-- then, multiplying both sides of that equation by -1, we obtain that which we had asserted to be implicit in the equation [x² + 1  =  0], namely, the equation between the additive inverse and the multiplicative inverse of x --   

-x   =   +1/x.

 

The equation [x² + 1  =  0]  is not solvable, or “satisfiable”, within any of the foregoing «gene» of number, or of arithmetics, all the way up through the «gene» of the so-called “Real” numbers. 

The equation [x² + 1  =  0]  is ‘non-diophantinely solvable’, via expansion to the “Complex” numbers, which we denote by [given that r  =  +1.0..., and that \2/-1 denotes the square-root of negative Real unity]  --

C   =   { Rr + R·\2/-1 }   =   { R + Ri }

-- by, again, due to its 2nd degree algebraic nonlinearity, two ‘non-diophantine’ numbers, in the case of this equation, by two so-called “pure imaginary” numbers --
 
x  =  +\2/-1    =   0r  +  1\2/-1    =    0r  +  1i    =    +i

-- and --
 
x  =  -\2/-1    =   0r  -  1\2/-1    =    0r  -  1i    =   -i. 

The “numbers” -i and +i belong to a new set of numbers -- to a new kind of number -- which we denote by i, which stands for the set / totality of the so-called “imaginary numbers”.

Note how each successor «genos», or universe, of number ‘‘‘«aufheben»-contains’’’ all of its predecessor universes of number -- i.e., is a ‘‘‘conservative extention’’’ of all of its predecessor ‘‘‘universes-of-discourse’’’ about number and about arithmetic -- while also determinately changing/negating & elevating into an ‘ideo-ontologically’ richer level, all of its predecessors.


Such a [‘Qualo-Peanic’ or ‘Meta-Peanic’] «aufheben» ‘consecuum’ of number-«gene» -- of number ‘ideo-ontology’ -- evinces part of the essence of what we mean by a ‘dialectic’; by a ‘dialectical process’, i.e., by an ‘[ideo-]meta-dynamical, meta-[ideo-]system[at]ic, and also [ideo-]meta-evolutionary [self-]progression of [axioms-]systems’, [self-]launched from an original, «arché» [ideo-]system, in this case, from the axioms-system of “Natural Numbers” arithmetic. 



¿But how might this potentially-infinite, constructed progression of «gene» and of ‘species’ of number, required for equational solvability, map to “sets of sets of ... of sets”?


One way that sets of higher “logical type” can model ['(---)'] higher, later kinds of [non-diophantine] numbers is as ordered pairs of earlier kinds of numbers / earlier kinds of sets.

We have already noted, above, that ordered pairs can be modeled via certain kinds of sets.

  

“Integers”, for example, can be modeled as ordered pairs of “Whole numbers”, i.e., as sets of logical type 2 — if we take the “Whole numbers” to be our ‘base objects’ — with the Zs, the “integers”, being defined, via their “comprehension axioms”, as differences, i.e., as subtractions, viz., as --

{ {1}, {1, 0} }   =   <1, 0>   (---)  1 - 0   =  +1       ~=      -1   =   0 - 1  (---) <0, 1>   =   { {0}, {1, 0} }.

Rational numbers can, in turn, be modeled as ordered pairs of “integers”, defined, via their “comprehension axioms”, as quotients, i.e., as divisions, viz., as --

<+1, +2>  (---)   (+1) / (+2)    =   +(1/2)         ~=        +(2/1)    =    (+2) / (+1) (---)  <+2, +1>. 

Thus, they also translate to ordered pairs of ordered pairs of “Whole numbers”, or to ‘sets-of-sets of sets-of-sets’ of “Whole numbers”, that is, to ‘sets-of-sets of “Whole numbers” squared’, meaning that these sets-of-sets operate upon these very sets-of-sets themselves, per a 'multiplicand-ingestion' set-product rule, so --
+(1/2)  (---)  <+1, +2>   (---)   {{+1}, {+1, +2}} 
 
  

< <1, 0>, <2, 0> >   (---)    {{<1, 0>}, {<1, 0>, <2, 0>}}

-- which translates to --

{{ { {1}, {1, 0} } }, { { {1}, {1, 0} }, { {2}, {2, 0} } }}

-- which is a class-object of logical type 4, i.e., of logical type 2², or “two squared”, w.r.t. the Whole numbers” taken to be the ‘base-objects’.

Further on in this axioms-systems progression, ordered pairs of “Real numbers” may, in turn, model the space of the “Complex numbers”, C, viz. --

<+1.000, +2.000>   (---)  1r + 2i         ~=        2r + 1i  (---)  <+2.000, +1.000>

-- such that C can be modeled as the two-dimensional space of a special kind of direction-denoting, as well as magnitude-denoting, ‘‘‘directed line segment’’’, or ‘‘‘2-dimensional vector’’’.

... and so on, to the axioms-systems for the arithmetics of the Hamilton Quaternionic [H] hypercomplex numbers, the Cayley/Graves Octonion [O] hypercomplex numbers, the Clifford hypernumbers [K], the Grassmann hypernumbers [G], the Sedenions [S], etc., etc. ... .


Thus, e.g., the “rational” numbers are seen as “analogues” -- as ‘meta-fractal similants’ -- of the “integers”; the “integers” as ‘meta-fractal similants’ of the “whole numbers”.


Even though these successive numbers-systems are of different kind, differing in quality, their base ‘idea-objects’, or numbers — their universes-of-discourse — may be constructed and presented, systematically, in and as different steps in a progressive-cumulative ‘[self-]iteration’ of one and the same «aufheben» operation of ‘self-internalization’,  of ‘self-incorporation’, of ‘self-subsumption’, of ‘self-combination’ / ‘self-combinatorics’, of sets, i.e., of ordered pairs.

This number-systems progression, modeled by the self-iteration of the ‘ordered pairs of’ or ‘sets of’ operations, constructs a “logical” or ‘‘‘idea-object’’’ version of what we term an «aufheben», ‘meta-fractal [ideo-]cumulum’.
 
This cumulum is a ‘[meta-]fractal ideo-cumulum’ because it constructs and presents a structure which is self-similar at successive “scales”. 

This cumulum is a ‘meta[-fractal] ideo-cumulum’ because its “scales” or ‘‘‘levels’’’ are not purely quantitative, as they supposedly are for “mere” fractals, but are ‘quanto-qualitative’, or ‘quanto-ontological’.


That is, such a ‘cumulum’ [self-]constructs and/or [self]-presents as a diachronic progression of ‘[ideo-]meta-«arithmoi»’ -- of sets of ordered pairs -- and persists as a synchronic, nested “tree” of an «arché»-«arithmos» -- e.g., the set of “Whole numbers” as base «monads», or base “elements” -- pooled with a graduated scale of  ‘[ideo-]super-«arithmoi»’ -- of sets of higher, more-inclusive ordered pairs -- each one made up out of ordered pairs of ‘[ideo-]sub-«arithmoi»’-as-‘meta-«monads»’ -- e.g., the set of “integers” made up out of elements which are ordered-pair sets of “Whole numbers”, the set of “rational” numbers made up out of elements which are ordered-pair sets  of “integers”, ... , the set of “Complex” numbers made up out of elements which are ordered-pair sets of “Real” numbers, etc. -- and also such that each constituent ‘[ideo-]«arithmos»’ of number-«monads» -- e.g., N, W, Z, Q, R, C, H, ...  --  is, in turn, ‘‘‘populated’’’ via a different kind of ‘[ideo-]«monad»’; in this example, by different kinds of number [via different unit[y][ie]s, viz. --   

1, versus 


(0...01), versus 


(+0...01), versus


(+0...01/+0...01), versus

 (+0...01.0...), versus 


(+0...01.0...)r vs. (+0...01.0...)i

-- etc.].

This may be seen in that the later ‘similants’ involve adjunctions of new [idea-]ontology, of new, higher logical types of sets; new kinds of sets; new kinds of ordered pairs; new kinds of numbers, qualitatively, ‘ideo-ontologically’ different from all of their earlier ‘similants’, not just quantitatively different therefrom, because «aufheben»-‘‘‘containing’’’ all of their earlier ‘similants’ in a nested fashion, and thus, and thereby, also ‘meta-fractally’ ‘scale-escalated’, i.e.,  «aufheben»-‘‘‘elevated’’’, with respect to all of their earlier ‘similants’.


We also find, in the known history of nature to date, physical, ‘external-objective’ ‘meta-fractal’ structures; a cosmological ‘physio-cumulum’, of ‘physio-meta-«arithmoi»’, each one made up out of multitudes of ‘physio-sub-«arithmoi»’ -- made of different kinds of ‘physio-«monads»’:
  

·         . . .;

·         molecules as ‘meta-atoms’, each one made up out of a heterogeneous multiplicity of atoms;

·         atomic nuclei as ‘ meta-sub-atomic “particles” ’, each one made up out of a heterogeneous multiplicity of nuclear sub-atomic “particles”;

·         nuclear sub-atomic “particles” as ‘ meta- pre-nuclear-“particles” ’, each one made up out of a heterogeneous multiplicity of pre-nuclear-“particles”; 

·         . . ..

On the basis of the foregoing, we therefore hold that —

(1) the ‘human-minds-internal’, ‘inter-subjective’, ‘idea-object-ive’, mathematical-progress-driving, conceptual process of ‘The Gödelian Dialectic’, i.e., of ‘ideo-onto-dynamasis’ [as modeled, using the ‘‘‘algebra’’’ of dialectical ideography, via generalized self-multiplication, ‘quadraticity’, or ‘ideo-onto-dynamis’ — to appropriate Diophantus’s term for ‘‘‘squaring’’’, i.e., for the second, or “quadratic”, “degree”, or “power”, of a variable], and;

(2) the ‘human-minds-external’, “objective”, “natural” process driving the self-development of ‘physical Nature’, or «physis» — both of ‘‘‘pre-human Nature’’’, diachronically, and also of ‘‘‘extra-human Nature’’’, synchronically, as well as of ‘‘‘human Nature’’’ — i.e., ‘physio-onto-dynamasis’, share a similar,  «aufheben» / ‘meta-fractal’ logic — i.e., a generally single, singular dialectical logic — or pattern of ‘followership’, which we term ‘meta-monadization’.



‘Dialectical Models’ of such ‘ideo-onto-dynamasis’ are presented in Supplement A to this text [see:  http://www.dialectics.info/dialectics/Primer_files/3_F.E.D.%20Intro.%20Letter,%20Supplement%20A-1_OCR.pdf ], and also in F.E.D. Vignette #4, entitled “The Gödelian Dialectic of the Standard Arithmetics”.

[see: http://www.dialectics.info/dialectics/Vignettes_files/v.4.5,Part_0.,Prefatories,Miguel_Detonacciones,F.E.D._Vignette_4,The_Goedelian_Dialectic_of_the_Standard_Arithmetics,14FEB2013.pdf 

http://www.dialectics.org/dialectics/Vignettes_files
/v.4.4,Part_I.,Miguel_Detonacciones,F.E.D._Vignette_4,The_Goedelian_Dialectic_of_the_Standard_Arithmetics,last_updated_29NOV2012.pdf

http://www.dialectics.info/dialectics/Vignettes_files/v.4.4,Part_II.,Miguel_Detonacciones,F.E.D._Vignette_4,The_Goedelian_Dialectic_of_the_Standard_Arithmetics,last_updated_29NOV2012.pdf ].

‘Dialectical Models’ of such ‘physio-onto-dynamis’ are presented in Supplement B to this text.
 [see:  http://www.dialectics.info/dialectics/Primer_files/4_F.E.D.%20Intro.%20Letter,%20Supplement%20B-1,%20v.2_OCR.pdf ].



A fuller exploration — and a ‘‘‘dialectical model’’’ — of the‘Gödelian Ideo-Meta-Evolution’, as actually observable in the [psycho]history of arithmetics, is forthcoming in volume II. of Dialectical Ideography, entitled The Meta-Evolution of Arithmetics].

F.E.D.’s ‘Dialectical Meta-Axiomatics’ Method of Presentation:

A Bi-Systematic Method of Presentation.

Gathering together, and integrating/synthesizing what we have reviewed so far regarding the Platonian «arché» of dialectic, and regarding the subsequent historical development of dialectic, and regarding the dialectical, immanent critique of the ideology impairing modern mathematics, now calls for us to frame a new perspective on mathematical systems and their optimal mode of exposition, as well as, in later exposition, a new, more optimal mode of mathematical discovery. 
 
The new, rationally and pedagogically advantaged mode of exposition that we advocate has come to be called, in our internal deliberations, ‘Dialectical Meta-Axiomatics’.

‘Dialectical Meta-Axiomatics’ names a new, complex unity, or dialectical synthesis, of the traditions of axioms/postulates/primitives/definitions/lemmas/theorems mode of exposition, of which Euclid’s Elements provide the «arché», with that of the, later, ‘‘‘Systematic Dialectics’’’ and ‘Meta-Systematic Dialectics’ modes of exposition.  

 Plato’s proposal cited above, Hegel’s achievement, in his «Logik» and «Encyklopädie der philosophischen Wissenschaften», and Marx’s achievement, in his Capital, provide the chief models of the latter modes extant to-date. 

F.E.D. advocates ‘Dialectical Meta-Axiomatics’ as the preferred mode of exposition for Dialectical Science, including for the ‘psych[e]ohistorically’ expanded Science of Mathematics.



‘Dialectical Meta-Axiomatics’ «aufheben»-conserves, without apology, the full rigor of formal-logical/mathematical-logical deductive proof, of «verstand» / ‘«dianoetic»’ reason, within each Axioms-System of a Gödel-Incompleteness-driven, ‘Gödelian-ideo-dialectical’ progression of Axioms-Systems [that systematically-ordered systems-progression of Axioms-Systems constituting what we term the Gödelian, diachronic ‘Meta-System’ for those successive Axioms-Systems]. 
 
But it also applies dialectical reason in the trans-deductive realm of the necessarily non-deductive derivation of the possible, candidate Axioms, and to the rational justification of the choice of Axioms from among those possibilities.

Moreover, it applies dialectical logic also to the «aufheben» transitions between predecessor /-successor pairs of Axioms-Systems, from a predecessor Axioms-System to its ‘‘‘conservative extention’’’ in its successor Axioms-System. 


Such transitions partly «aufheben»-conserve the Axioms of the predecessor Axioms-system in the Axioms of the successor Axioms-System, while also adding, via «aufheben»-transformation / elevation, the new comprehension Axioms, and the new ‘ideo-ontology’ that they implement, rendering “decidable”, within the successor Axioms-System, the “undecidable” propositions of the predecessor Axioms-System, which was thus Gödel-Incomplete with respect to [at least] those propositions. 
 
E.g., such transitions form new kinds of numbers, able to solve the diophantine equations that are unsolvable within the number ‘ideo-ontology’ of the predecessor Axioms-System -- equations the unsolvability of which is asserted by the “deformalized” undecidable propositions of that predecessor Axioms-System. 

Such transitions form these new, higher kinds of numbers as new kinds/higher logical-types of sets, qualitatively, ‘ideo-ontologically different’ from the predecessor logical-types of sets, within the “power-set” «aufheben» ‘self-internalization’ / ‘subsets-subsumption’, of those sets of the predecessor Axioms-System’s ‘ideo-ontology’, which formed the farthest horizon of the number-concept extant within that predecessor Axioms-system and its implied  ‘ideo-ontology’. 


This «aufheben» process renders the relative truth of the formerly undecidable propositions provable via the new “comprehension Axioms” added to form the new-Axioms-component of the successor Axioms-System, and renders the formerly unsolvable diophantine equations solvable within the successor Axioms-System, using the new kinds/higher logical-types of sets, defining the new, higher kinds of numbers thus «aufheben»-created within the successor Axioms-System. 


‘Dialectical Meta-Axiomatics’ drops the pretence that each Axiom in the Axioms-set of an Axioms-System can be “self-evident”, and uniquely-determined, with no possible Axiom-alternatives. 
By this pretense, the function of dialectical reason, as the non-deductive derivation of multiple candidates for a given key Axiom, and as the justification of the choice of one Axiom from among those candidates, has for so long been dogmatically denied [¡ever since Plato, for the anti-dialectical traditions of academia, for which the Occidental Dark Ages have never yet ended!].

‘Dialectical Meta-Axiomatics’ admits that axiomatic ‘alternativity’ veritably abounds, and that Axiom-choice needs to be justified dialectically, i.e., ‘self-reflexively’ / ‘self-refluxively’, that is, in light of each candidate Axiom’s consequences within the context of the «arithmos» of Axioms it is candidate to join, and in light of the purposes for which the Axioms-System it is candidate to join is designed. 
 
Examples of such ‘alternativity’ -- of the “independence” or Gödel-undecidability of key Axioms with respect to the rest of the Axioms of a given Axioms-System -- include the choice of the Euclidean ‘fifth Axiom’, the Parallels Postulate, versus one of its possible contraries, for the Axioms-System of Euclidean Geometry versus for those of the several Non-Euclidean Geometries, and the choice of the Generalized Continuum Hypothesis, versus one of its possible contraries, and/or of the Axiom Of Choice, versus one of its possible contraries, for the Axioms-Systems of Totality Theories [“Set Theories”]. 

‘Dialectical Meta-Axiomatics’ also rejects any pretence that first-order-logic Axioms-Systems have but one possible, “categorical”, unique “model”, or “interpretation”, an old dogma that has been refuted both by the Löwenheim-Skolem Theorem, and by the first order co-applicability of the Gödel Completeness Theorem and the Gödel Incompleteness Theorem.

‘Dialectical Meta-Axiomatics’ is also based on a grasp of the ‘intra-duality’ / ‘intra-multiality’ of the ‘interpretabilities’ of a first order Axioms-System as marking a potential «arché» for a ‘meta-systematic dialectical’, ‘categorial-progression’, ‘Axioms-Systems-progression’ exposition, and for a dialectical-algebraic modeling, of alternative, “non-standard” models of that first order Axioms-System. 

‘‘‘Diachronically’’’, between each predecessor/successor pair of Axioms-Systems, the ‘Dialectical Meta-Axiomatics’ methodology practices an expository, pedagogical discipline, using an heuristic, intuition-involving, “intensional” derivation of the «aufheben» progression of Axioms-Systems:  of the ‘Axioms-Meta-System’. 
 
‘‘‘Synchronically’’’, within each, successive Axioms-System in that ‘Axioms-Meta-System’, ‘Dialectical Meta-Axiomatics’ justifies the theorems implied by that Axioms-System’s ‘Axioms-collective’, definitions, primitives, and rules of inference via rigorous deductive logic. 

Theorems are also justified, and explained conceptually and intuitively [«begrifflichkeit»], without apology.  


Indeed, the main text, in a work of ‘Dialectical Meta-Axiomatics’, is the intuitive/conceptual exposition, but with a parallel stream of formal-logical, algorithmic/mechanical deductive proof [which may often compel a human mind to assent to a proposition without comprehension] relegated to a subordinate narrative -- e.g., to End-Notes, or to [an] Appendi[x][ces] -- as a necessary verification check on the conceptual/intuitive narrative’s flow/progression of claims/assertions.  

The two parallel texts should thus each contain ‘“pointers”’ -- i.e., cross-references and bridging commentary; ‘‘‘transversals’’’ and asides -- linking from the deductive proofs to the intensional-heuristic / intuitive narrative, and from the intensional-heuristic / intuitive narrative to the deductive proofs. 

We can therefore ‘visualize’ the ‘content-structure’ of an exposition constructed in accord with the tenets of ‘Dialectical Meta-Axiomatics’ as follows --

The present work, overall, remains an ‘intuitively-ordered narrative’.  

We plan for volume 2 of our treatise, A Dialectical Theory of Everything, to contain instantiations of ‘Dialectical Meta-Axiomatics’ method of exposition.




‘Dialectical Meta-Axiomatics’ «aufheben»-conserves the full logical rigor of deductive proof-based «dianoesis», without apology.

But ‘dialectical Meta-Axiomatics’ also exceeds that «dianoesis» in rigor by virtue of its unified recognition of:
 

(i) the non-self-evidence of appropriate and optimal axioms generally; the exercise of choice and skillful design required in their development and selection, and the abounding ‘alternativity’ which that activity confronts;

(ii) the axioms-dependence or assumptions-relativity of all formal proofs, hence of all formal “truths”;

(iii) the logical ‘equi-coherence’ of non-standard models of “first order logic” axioms-[sub]systems with respect to the standard models with which those non-standard models are associated, and with reference to which they are defined as “non-standard”;

(iv) the formal independence, or Gödel-undecidability, of key axioms of “higher-than-first-order-logic” axioms-systems with respect to the rest of the axioms, hence the logical ‘equi-coherence’ of alternative axioms-systems, built on contraries of those key axioms, and especially;

(v) ‘The Gödelian Dialectic’; the ‘‘‘psychohistorical-dialectical’’’, «aufheben»/evolute-cumulative progression of de facto Axioms-system within the social and ‘socio-cognitive’ [‘‘‘psychohistorical’’’], ‘Phenomic’ progression of a human species.

That is, the controversial, dialogical, dialectical process of discovery, exploration, comparative evaluation, and rational selection of assumptions [of premises, postulates, axioms, definitions, primitives, and rules of inference] is not a final, once-for-all, ‘finishable’, synchronic human activity. 

Not all possible alternative and/or incremental axioms are known, or even knowable,  for humanity, at any given moment in human history -- i.e., at any given stage in the self-development of the complex unity of the ‘‘‘Human Phenome’’’/“Human Genome”.
This ‘meta-axiomatic’ dialectic process is, on the contrary, an ever-renewed, ongoing, and cumulative process, a diachronic activity of expansion of the axiomatic and ‘ideo-ontological’ foundations that are accessible to humanity -- i.e., that have newly become new parts of ‘‘‘The Human Phenome’’’.  

It produces an «aufheben»-progressive [‘Qualo-Peanic’ or ‘Meta-Peanic’ ] historic[al] sequence of systems of logic and mathematics.


That [‘Qualo-Peanic’ or ‘Meta-Peanic’ ] «aufheben»-progression reflects the immanent emergence, within human society, of psycho-cultural ‘readiness’ for each next new epoch of cognitive, axiomatic, and ‘ideo-ontological’ expansion, borne in the interconnexion between:
  
(1) “technical” or ‘technique-al’, “technological-ontological” self-expansion of the activities/practices of a generally acceleratedly-expanding human-societal self-reproduction; i.e., of “human species social-reproductive praxis” [“growth of the social forces of production”], and;

(2) maturation in the prevailing level of ‘exo-somatically’ acquired, ‘trans-genomically’ transmitted cognitive and affective development within the typical “social individual”, hence of the global human culture and “memes pool” [or ‘‘‘Phenome’’’].



We describe ‘Dialectical Meta-Axiomatics’ as a ‘A BI-Systematic Method of Presentation’, because both the ‘Meta-Systematic Dialectical’ method of presentation, and the Euclidean/Newtonian method of presentation -- recall the ordering of the theorems, etc., presented in Euclid’s Elements and in Newton’s Principia --  can exhibit a rigorous, determinate, systematic ordering of their contents.

Thus, both the intuitive expository stream, and the deductive-‘dianoic’ expository stream -- as depicted in the graphical visualization above -- are systematically-ordered narrative streams.