Tuesday, April 07, 2015

E. D. Notation. ‘Full Zero’ -- via a New ‘[Meta-]Number’ Concept, in a New, ‘Quanto-Qualitative’, Fully-Ideographic/-Algorithmic Dialectical Arithmetic for “Dimensional Analysis”, with Non-Problematic Division by Zero [‘Semantification’ of Dynamical [Nonlinear] Differential Equation "Singularities"].









Dear Readers,



On this 19th anniversary of Seldon’s April 7th, 1996 breakthrough -- his sudden discovery of the NQ_ First Dialectical Arithmetic for contra-Boolean Algebra, after years of [re-]searching for, and of slow progress toward finding, a mathematics of dialectics -- the F.E.D. General Council has cleared, for public dissemination, its eight specifications sheets defining the meta-number value that we call Full Zero, ‘.’ -- as distinct from ordinary zero, 0, which we, in this context, call Empty Zero, 0 -- and elaborating upon the candidate postulate(s) to govern the use of this new 'ideo-ontology', this new dialectical-ideographical symbol, within the Seldonian seventh, or 'Mu', dialectical arithmetic.




I have, in my dissertation-contribution to the Foundation, for my induction-into-membership in the Foundation, entitled The Gödelian Dialectic of the Standard Arithmetics [which is accessible via the following links, on the Vignettes Page, as Vignette #4, Parts 0, I, and II, at --






-- described the internal inadequacies/‘self-incompletenesses’ of each successive axioms-system of the standard arithmetics -- how each standard arithmetic is marked by algebraic, “diophantine equations” which, grounding an ‘‘‘immanent critique’’’, or ‘‘‘self-critique’’’, and an ideo-intra-duality, or ideo-self-duality, of each such system by itself, are -- syntactically -- well-formed within that arithmetic, but for which, semantically, no ‘semantification’ of the unknown, x, of that algebraic equation, i.e., no solution(s) of that equation, are available/expressible within that standard arithmetic’s axioms-system, i.e., within the, partially tacit, ‘‘‘ideo-ontological commitments/presumptions/self-limitations’’’ of that system.


Thus, the equation x + 1  =  1 is unsolvable within the system of arithmetic of the so-called “Natural” numbers, N, wherein N = {1, 2, 3, ...}, and indeed this equation asserts a psychohistorical paradox for the concept of addition native to that system. 
Thus, in a sense, within the limitations of the N system, x = ., although this equation of x and/to . must be considered a 'meta-arithmetical', '''meta-mathematical''' assertion, because . is not an element of -- is not a "number" within -- N.  

this equation, x + 1  =  1, marks the presentational transition from the N system to the W system, the axioms-system of the so-called “Whole” numbers, W = {0, 1, 2, 3, ...}, wherein that equation is readily solvable:  x = 0.


However, the equation x + 1  =  0 is unsolvable within the W system, and indeed asserts a psychohistorical paradox for the concept of addition native to that system. 

Thus, within the limitations of W system, in a sense, x = . [again, as a 'meta-arithmetical' assertion, because . is not an element of -- is not a "number" within -- W] and this equation marks the presentational transition from the W system to the Z system, the axioms-system of the so-called integers --

Z = {..., -3, -2, -1, ±0, +1, +2, +3, ...}

-- wherein that equation is readily solvable:  x = -1.


However, the equation 2x  =  1 is unsolvable within the Z system, and indeed asserts a psychohistorical paradox for the concept of multiplication native to that system.   
Thus, within the limitations of the Z system, in a sense, x = . [again, as a 'meta-arithmetical' assertion, because . is not an element of -- is not a "number" within -- Z], and this equation marks the presentational transition from the Z system to the Q system, the axioms-system of the so-called rational numbers --

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

-- wherein that equation is readily solvable:  x = +1/2.


However, the equation x2  =  2 is unsolvable within the Q system, and indeed implies a psychohistorical paradox for the concept of exponentiation native to that system -- that x must be either both odd and even, or neither odd nor even. 
Thus, within the limitations of the Q system, in a sense, x = . [again, as a 'meta-arithmetical' assertion, because . is not an element of -- is not a "number" within -- Q], and this equation marks the presentational transition from the Q system to the R system, the axioms-system of the so-called Real numbers --

R = {.....-pi....-e....-\/2....±0.000.......+\/2....+e....+pi.....}

-- wherein that equation is readily solvable:  x = ±\/2.


However, the equation x2 + 1  =  0 is unsolvable within the R system, and indeed implies a psychohistorical paradox for the concept of inverse values native to that system -- for that equation implies that, for that x, its additive inverse value and its multiplicative inverse value must be equal:  
  
-x  =  +1/+x. 
Thus, within the limitations of the R system, in a sense, x = . [again, as a 'meta-arithmetical' assertion, because . is not an element of -- is not a "number" within -- R], and this equation marks the presentational transition from the R system to the C system, the axioms-system of the so-called Complex numbers --

C = {R + Ri}

-- wherein that equation is readily solvable:  x = ±i.  And so on . . ..



However, notice also that, in NONE of these systems -- [not in N], not in W, not in Z, not in Q, not in R, not in C, ... -- is division by ZERO workable;  is an equation of the form  x = c/0 solvable [in the N system, such an equation is not even ‘‘‘well-formed’’’, because the number 0 is not even part of the ideo-ontology -- of either the syntax, or the semantics -- of that system].


This internal, immanent inadequacy and ‘‘‘incompleteness’’’ of ALL systems in the progression of the systems of the standard arithmetics is evidently of a far deeper sort than the inadequacies and ‘‘‘incompleteness’’’ that drive that progression, and that were progressively solved in that progression, as outlined above.


The [implicitly-dialectical] first-order-logic, Peano axioms system of the “Natural” numbers, which Encyclopedia Dialectica denotes by N_, and sees as being standardly interpreted as a purely-quantitative arithmetic, is the first, «arché» category/system of arithmetic in the Seldonian progression of non-standard, dialectical arithmetics. 

The Seldonian First [explicitly-]Dialectical Arithmetic, which Encyclopedia Dialectica denotes by NQ_, and interprets as a purely-qualitative ordinal arithmetic, is the second category/system of arithmetic in the Seldonian progression of non-standard, dialectical arithmetics, the first contra-category/ contra-system, in that progression.


The seventh system of dialectical arithmetic in that Seldonian progression, which Encyclopedia Dialectica connotes by Rq_MQN  =  Rq_MU  =  Rm_  -- the second uni-category system of dialectical arithmetic -- arises naturally as the first non-syncopated, fully-ideographic, fully-algorithmic arithmetic for dimensional analysis”.

I that seventh system, questions leading to the Full Zero meta-number’ concept also arise naturally, and yield, at long last, an arithmetic in which division by zero appears to become non-problematic.

As a result of that ‘‘‘rectification’’’ and ‘‘‘regularization’’’ of division by zero, dynamical singularities, presently manifesting as infinity residuals, i.e., as infinite errors, in the predictions of [especially nonlinear] dynamical differential equations, including of those which represent this humanitys presently most advanced scientific-consensus expressions of the “laws” of Nature, can be ‘semantified’ by correct solution-values, under intuitively satisfying new axioms, which can be stated, briefly, as: 

“ ‘Empty Zero’ “times” a metrological unit qualifier yields ‘Full Zero’ ”.

-- and --

“ ‘Full Zero’ is operationally dominant, in multiplication and division, with respect to all other ‘[meta-]number’ values in this Rm_ system, that is, multiplication and division operations if they involve ‘Full Zero’, yield only ‘Full Zero’ ”.

-- or --
0mo = ..      


-- and --

[ for all mo in Rm_ ][ [.xmo = mox. = .] & [./mo = mo/. = .] ].


Unlike in the cases of the systems of arithmetic -- of the N, W, Z, Q, R, and C, ..., systems of arithmetic -- considered earlier, above, in this case, the present case, the case of the system of arithmetic,  Rm_, . IS, finally, an element of the set -- is, at last, a number within the '''number-space''' -- Rm.   Thus --

0mo = ..

-- is no longer a '''meta-mathematical''' assertion.



The classic published rendition of an earlier version of this theory is available via --




-- on pages A-7 through A-21 of the latter.



I have posted the eight sheets of the new Full Zero specification, below, for your convenience.



May you much enjoy this deeper glimpse into the world-historical fruition of these arithmetics of dialectic!




Regards,

Miguel











































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