Sunday, October 14, 2012

Q: What Drives the Dialectic of Arithmetics? A: "Unsolvable" Equations Becoming Solved.

Full Title --
QWhat Drives the Dialectic of Arithmetics?
A:  "Unsolvable" Equations Becoming Solved.

Dear Readers,

Quoted below is a passage from the Background section of F.E.D. Vignette #4 --,Part_I.,Miguel_Detonacciones,F.E.D._Vignette_4,The_Goedelian_Dialectic_of_the_Standard_Arithmetics,last_updated_29NOV2012.pdf

-- on the driver of the 'Meta-Systematic Dialectical' Presentation of the Standard Arithmetics.

"The ‘‘‘Driver’’’ for the Progression of the Standard Arithmetics”:   The Solution of Unsolvable Equations.

Despite those limitations of its scope, the ‘meta-model’ presented herein does capture The Gödelian Dialectic of this progression of the axiomatic systems of the Standard Arithmetics.

That is, this ‘meta-model’ captures a pedagogically-selected, pedagogically-optimized progression of “Gödel formulae” analogues for this progression of arithmetics.

Each such Gödel formula “deformalizes” into a proposition asserting that a certain “diophantine” algebraic equation cannot be solved within the [“incomplete”] ‘number-ontology’, or ‘number-kind ideo-ontology’, of the given arithmetical axiomatic system, inside which it arises as a ‘well-formed equation’ of that system.

Also, as a ‘Gödel formula proposition’, asserting the unsolvability of such an equation in such an axioms-system, such a proposition would be true of the axioms-system within which it arises, but would be neither deductively provable nor deductively dis-provable -- i.e., would be undecidable” -- from the axioms of that axioms-system, i.e., within the logical-language resources, syntactical and semantic, of that axioms-system.

Derivation of such a well-formed unsolvable equation, and unprovable theorem, within such an axioms-system precipitates an immanent formal-logical «aporia»,  impasse, quandary, or predicament, for that axioms-system:   a logical inadequacy -- the a-pore’, or not-pore’; the absence of a “passage”, of an “opening”, beyond the present “impasse”, which appears, at first, to be a hopeless dead end.

However, such an equation would be solvable within the consecutive next arithmetical axioms-system, the “successor system” of that “predecessor” arithmetical axioms-system, by means of the new ‘ideo-ontology’ -- by means of the new kind of ‘number ontology’ -- which first fully arrives in that successor system.

That same equation would also be solvable in all subsequent arithmetical axioms-systems in this progression, using that new ‘ideo-ontology’, which is conserved, as well as qualitatively/‘ideo-ontologically’ further extended, in all subsequent arithmetical axioms-systems in this ‘meta-system’, or systems-progression.

Such a ‘Gödel formula proposition’, asserting the unsolvability of that equation in that predecessor system, will become provably true, via the expanded logical-language resources of that successor arithmetical axioms-system, as well as in all of its “successor systems”, in which those new logical-language resources are also both conserved & determinately-negated/changed/elevated/further-extended in a qualitative/‘ideo-ontological’ sense."

A rendition together with examples of such "solved unsolvables" can be found here --

-- and in the following sub-section of the Background section of Vignette #4:

"The Pedagogical Strategy Guiding System Order Choices for Our Presentational Meta-Model.

F.E.D. could have chosen to present the 6 kinds of “unsolvable diophantine equations” listed below in another possible order than the order in which they are listed below, which is also that in which our ‘meta-model’ actually presents them --

1. [n + x1  =  n], for n in N, posing the paradox, for N#’s notion of number ‘Natural-ness’, of 'non-increasive addition. Solution[-set]:   x1 = 0, or x1 is contained in a is contained in W.
This equation jumps us from N#  to W#;

2. [w + x2  =   0; x2, w ~= 0], for w > 0 in W, posing the paradox, for W’s notion of number ‘Whole-ness’, of decreasive addition.
Solution-set: x2 is contained in m is contained in Z. This equation jumps us from W# to Z#;

3. [ |x3 x z| < |z|; x3, z ~= ±0 ], for z in Z, posing the paradox, for Z#’s notion of number ‘‘‘integ[e]r-ity’’’, ofdecreasive multiplication.  Solution-set: x3 is contained in (±0,+1)* is contained in f is contained in Q. This [in]equation in Z# jumps us into Q#;

*[Note: ‘(±0,+1)’ means all fractions strictly between ±0 & +1, i.e., excluding ±0 & +1].

4. [ x4^2 - p  ±0 ], ±0 < p in Q a Q prime number, posing the paradox, for Q#’s notion of number ratio-nality’, of incommensurability.   Solution-set:  x4 is contained in {±square-root of p} is contained in d is contained in R. This equation jumps us from Q# to R#;

5. [ x5^2 + 1.0...    =     ±0.0...  ], or -x5   =   (+1.0...)/x5   =   x5^-1  ], posing the paradox, for R#’s notion of number ‘Real-ity’, of the i unit’s additive inverse/multiplicative inverse equality or identity. Solution-set: x5 is contained in {±square-root of-1} such that ±i is contained in i is contained in C. This equation jumps us from R# to  C#;

6. [ +x6y6   =  -y6x6;   x6,y6   ~=  0 ], posing the paradox, for C’s notion of number ‘Complex-ity’, of multiplicative anti-commutativity, or of sign-reversal as a result of multiplicative factor-reversal. ... . This equation jumps us from C# to H#.

Equation 2. is inexpressible in N#, because it involves 0, not in N. It would take us next to Z#.

If we started with equation 3., which is not well-formed in N#, because it uses the useless- or meaningless-in-N# “absolute value” operator, ‘|...|’, it would take us next to Q#.

Equation 4. is inexpressible in N#, because it involves 0, not in N.    It would takes us next us to R#.

Equation 5. is also inexpressible in N#, because it involves 0, not in N.   It would takes us next us to C#.

If we started with equation 6., which is not well-formed in N#, because it uses the useless- or meaningless-in-N#“signs”, ‘+’ and ‘-’, it would take us next to H#!

F.E.D. came to the conclusion that N#, W#, Z#, Q#, R#, C#, & H#... were the right «species» for the «genos» of [counting] number, the best progressive  partitioning of the generic “Standard” number concept, the best division [«diairesis»] of that «genos» of number into «species» of number, the best ‘‘‘speciation’’’ of number-kinds for ready assimilation by those to whom we planned to present the “Standard Arithmetics”, given the contemporary view  of the standard arithmetics, and of standard mathematics in general, and given the total psychohistorical / phenomic / ideological cognitive context of ‘recent-modern’ humanity.

F.E.D. came to the further conclusion that the sequence given above was the right sequence of presentation of these number-«species», representing the right simplicity-to-complexity, abstractness-to-thought-concreteness gradient, with the right “consecutive step-sizes”, in terms of the ‘‘‘sizes’’’ of the qualitative increments in ‘ideo-ontology’, for optimal ease of assimilation.

The inspiration for the order of presentation that F.E.D. has selected for this Standard Arithmetics’ systems-progression is partly pictorial.

It is the perceived coherence of the order of filling-in of ‘Standard-Numbers-Space’, as expressed by the ‘spaces-progression’ of the diagrams/depictions of the ‘number-spaces’, or the ‘‘‘analytical geometries’’’, of the “Standard” systems of arithmetic, as shown via the graphics below ... .

Note: We are herein rendering explicit throughout, in both our ideographic renderings of the individual numbers/numerals of the various numbers-systems, and in our depictions thereof... certain key features [e.g., the “leading zeros” place-holders in 0...01 within W, for example], i.e., various syntactic attributes -- signifying semantic attributes of the concepts of those numbers -- which are usually, in common uses of these numbers, left implicit, abbreviated-out, or ignored, but which are of crucial conceptual significance in tracking the cumulative changes in ‘number ideo-ontology’ from each stage and predecessor system of arithmetic to its successor stage and system, throughout the entire progression of systems of “Standard Arithmetic”.

We also, in the depictions below, use ‘(---)’ as ‘assignment sign’, or ‘interpretation sign’, from one “pure” ‘ideo-system’ to another.

We note, in passing, one further dimension of the gradient -- the gradual gradations, escalating from the «arché»’s maximal relative simplicity, to the maximal relative complexity of the final ‘counter-supplement’ culminant that we consider, that runs through the entire inclusive ‘qualitative interval’, [N#, h#], in the progress from N# to h#.

This further so-‘gradated’ dimension is that of the increasing relative thought-complexity of the arithmetical operations ‘aporialized’, or ‘paradoxicalized’, in the succession of “unsolvable” diophantine algebraic equations that catapult our core section discourse from arithmetical system to next higher arithmetical system.

Equations 1. & 2. involve the paradoxes of the arithmetical addition operation.

Equation 3. involves the paradoxes of the arithmetical multiplication operation, definable, in the E.D. dialectic of such operations, as the meta-additionoperation.

Equations 4. and 5. involve the paradoxes of arithmetical exponentiation, definable, in that same dialectic of arithmetical operations, as the meta-multiplication operation.

Together with equation 6., equations 4. and 5. involve quadratic nonlinearity -- 2nd degree algebraic equation nonlinearity [albeit not integrodifferential equation nonlinearity]."



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