Sunday, April 19, 2015

The First Ever Arithmetic in which Division-by-Zero is Unproblematic? 'Singularity Semantification'.









Dear Readers,



I wanted to call your attention to the following eight recently released F.E.D. image files, reproduced below, which define the first arithmetic, to our knowledge, for which, and in which, division-by-zero, "zero division", is unproblematic, in a practical, concrete way -- in a way that renders, e.g., the fundamental equations of present-day physics more meaningful, rather than less meaningful, or, even worse, apparently "meaningless" -- or even 'infinitely erroneous' -- specifically with regard to the "singularities" of those equations, and/or of their solutions.

This arithmetic is the seventh arithmetic in the Seldonian series of dialectical arithmetics, and is named the "Mu" arithmetic.

This arithmetic also provides the first example, to our knowledge, of a non-"syncopated", fully-ideographical arithmetic for dimensional analysis.


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




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



Regards,

Miguel











































Friday, April 17, 2015

Karl Seldon's Sagacious Sayings Series -- #3. The Mission of Our Foundation.









Dear Readers,




From time to time, I like to share with you some of the gems of insight that leap from out of the '''multilogues''', among Karl Seldon and other members of the Foundation, and from the transcribed versions, published internally, including of those '''multilogues''' in which I did not happen to participate, once those transcripts are cleared for public sharing by the Foundation's General Council.


Here's an excerpt from the transcript of a recent orientation presentation by Karl Seldon for new recruits  --



[Karl Seldon]:

"The mission of Foundation Encyclopedia Dialectica -- in the midst of theproto-New-Dark-Ages, descendance-phase, capitalist global civilization/system of present-day planet Terra -- is to propagate, anonymously, and/or pseudonymously, a meta-system of new memes which will, potentially, help Terran humanity to avert that New Dark Age entirely, to avert that ‘‘‘mutual ruin of the contending classes’’’."

   

. . .

 

"If it proves not to be possible to avert that impending New Dark Age completely, then our mission is to diffuse memes which will, if possible, help to shorten the duration of that Dark Age -- the duration to humanity’s recovery."





Regards,

Miguel







Wednesday, April 15, 2015

Karl Seldon's Sagacious Sayings Series -- #2. The 'present-ce' of the Past and the Future in the Present.








Dear Readers,




From time to time, I like to share with you some of the gems of insight that leap from out of the '''multilogues''', among Karl Seldon and other members of the Foundation, and from the transcribed versions, published internally, including of those '''multilogues''' in which I did not happen to participate, once those transcripts are cleared for public sharing by the Foundation's General Council.


Here's a recent such '''jewel''' --



[Karl Seldon]:

"If you describe the 'present-ce' of something present deeply enough, then you will describe not just its present condition, but something of its past condition, and also something of its predictable future condition as well.

I think that this is part of what is implicit in Marx's remarkable methodological mentioning in his Grundrisse, regarding the interconnexion of his version of [Meta]Systematic Dialectics, with his version of [Psycho]Historical Dialectics, as follows --



...our [F.E.D.:  systematic-dialectical] method indicates the points where historical investigation must enter in, or where bourgeois economy as a merely historical form of the production process points beyond itself to earlier historical modes of production.”

“In order to develop the laws of bourgeois economy, therefore, it is not necessary to write the real history of the relations of production.”

“But the correct observation and deduction of these laws, as having themselves become in history, always leads to primary equations -- like the empirical numbers e.g. in natural science -- which point towards a past lying behind this system.”

“These indications [Andeutung], together with a correct grasp of the present, then also offer the key to the understanding of the past -- a work in its own right which, it is hoped, we shall be able to undertake as well.
[F.E.D.:  alas, not!].”

“This correct view likewise leads at the same time to points at which the suspension [i.e., at which the «aufheben» -- F.E.D.] of the present form of production relations gives signs of its becoming -- foreshadowings of the future.”

“Just as, on one side the pre-bourgeois phases appear as merely historical, i.e. suspended [i.e., as ‘«aufheben»-ed’ or as ‘«aufheben»-ated’ -- F.E.D.pre-suppositions, so do the contemporary conditions of production likewise appear as engaged in suspending themselves [i.e., in ‘«aufheben»-ating themselves -- F.E.D.] and hence in positing the historic presuppositions for a new state of society.

[Karl Marx, GrundrisseFoundations of the Critique of Political Economy (Rough Draft), translated by Martin Nicolaus, Penguin Books [Middlesex:  1973], pp. 460-461, emphases added by F.E.D.].

-- although, clearly enough, this statement by Marx, both explicitly and implicitly, also goes far beyond my immediate point here. ..."



For Your Continued Cognitive Enjoyment and Expansion!




Regards,

Miguel







Tuesday, April 14, 2015

E. D. Notation: Categorial ADDITION Operation Signs Designs Defined.








Dear Readers,



A new 'text+ module', specifiying the Encyclopedia Dialectica designs & definitions for its Categorial ADDITION Signs, has been cleared for posting to the www.dialectics.org  Glossary Page by the F.E.D. General Council.


I have also, posted this 'text+ <<monad>>' below, for your convenience.



Regards,

Miguel






















Saturday, April 11, 2015

E. D. Notation: Categorial Progression '''Arrow''' Signs Defined.








Dear Readers,



A new 'text+ module', specifiying the Encyclopedia Dialectica definitions for its Categorial Progression '''Arrow Signs''', has been cleared for posting to the www.dialectics.org  Glossary Page by the F.E.D. General Council.


I have also, posted this 'text+ <<monad>>' below, for your convenience.



Regards,

Miguel






















Friday, April 10, 2015

E. D. Notation: Propositional Status Signs Defined.








Dear Readers,



A new 'text+ module', specifiying the Encyclopedia Dialectica signs for '''Propositional Statuses''', has been cleared for posting to the www.dialectics.org  Glossary Page by the F.E.D. General Council.


I have also, posted this 'text+ <<monad>>' below, for your convenience.



Regards,

Miguel






















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, «archĂ©» 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