Friday, September 29, 2017

Notational Basics #1, for the Q_ Arithmetic/Algebra for Dialectics, for its ‘Universal Algorithmic-Heuristic Method’.








Notational Basics #1, for the 

NQ Arithmetic/Algebra for Dialectics, for its Universal Algorithmic-Heuristic Method.







Dear Readers,



The diagrams with “call-out” notational definitions and explanations, pasted-in below were also recently posted to the www.dialectics.org web site, to its Application Page, under the heading of Karl Seldon’s forthcoming essay, entitled Universal Algorithmic Heuristic Method.

These diagrams have been prepared for use in that forthcoming essay.



ENJOY!



Regards,


Miguel Detonacciones,

Member, Foundation Encyclopedia Dialectica [F.E.D.],
Participant, F.E.D. Special Council for Public Liaison,
Officer, F.E.D. Office of Public Liaison.

























Wednesday, September 27, 2017

Proudhon’s ‘‘‘Serialism’’’ -- ¿Precursor/Anticipation of the Seldonian Dialectic?





Proudhons ‘‘‘Series-ism’’’ -- ¿Precursor/Anticipation of the Seldonian Dialectic?








Dear Reader,



FYI:  This is a follow-up to our blog-entry, here, of 27 August 2017. 

A passage from a new, 2017 book on Marx’s «Das Kapital» “caught the eye” of the Foundation Encyclopedia Dialectica [F.E.D.] research community, with a passage on its page 36.

That book is:  Marx’s Inferno:  The Political Theory of Capital, by William Clare Roberts, Princeton:  Princeton University Press, 2017.

This passage intimates the existence of a Fourierist, and a Proudhonist, ‘serial laws’ anticipation of the Seldonian Dialectic, one that may have also influenced Marx in his ‘‘‘systematic-dialectical’’’ construction of his «Das Kapital».

More specifically, this passage intimates that the CHARLES-Fourier-ist, and Proudhonist, concepts of ‘serial laws’ may have anticipated the multi-categorial, poly-qualinomial series qualitative superpositionings’ [“superpositions”, or “sums”, of heterogeneous, qualitatively differing terms -- not of purely quantitative terms, as per JOSEPH Fourier’s famous series], or non-amalgamative sums -- i.e., the multi-ontic cumula -- of ontological-category-representing symbols, as generated by a ‘‘‘Seldon Function’’’ -- i.e., by a self-reflexive function of an «arché» ontological category, or by an even more exotic such mathematical function.

Footnote 63 to William Clare Roberts’ page 36 refers to a passage in another book:  Revolutionary Justice:  The Social and Political Theory of P.-J. Proudhon, by Robert L. Hoffman, University of Illinois Press, 1972, pp. 106-109.

I have reproduced, with commentary [given, below, within brackets, and in magenta-colored text], excerpts from a superset of this passage -- from pp. 105-117, below.  I have extracted only the passages that contain this author’s descriptions of the Proudhon’s beliefs, and hopes, about his ‘‘‘serial method’’’, not that author’s criticisms of Proudhon’s attempts at such a method.


The findings of the F.E.D. research on the claims of these passages -- one way or another -- are likely to influence the further development of the content of the www.dialectics.info ‘methodology page’ --




Dialogically Yours,


Miguel Detonacciones,

Member, Foundation Encyclopedia Dialectica [F.E.D.];
Participant, F.E.D. Special Council for Public Liaison;
Officer, F.E.D. Office of Public Liaison.







[Excerpts from pp. 105-117.  Paragraph breaks of the original have often been modified, by M.D.]  Proudhon’s two major books in these years [were] the Système des contradictions économiques and De la Création de l’ordre dans l’humanité...some have thought his methodological efforts here important... .”

“... Like most of those in the intellectual tradition stemming from the French Enlightenment, Proudhon is very much impressed with natural science’s powers of discovery.”

“He is especially drawn to the taxonomic accomplishments of biology, which do not rely on controlled experiment and mathematical analysis, as do developments in the physical [sic -- M.D.] sciences.  He consciously tries to do the same kind of thing for social science.”

[Here comes to mind, from a Seldonian point of view, the question as to whether or not Proudhon considered “social science” taxonomy to be a fixed and eternal one, or a dynamical one, owing to an undergirding ontological dynamism in [human] Nature.]

“The elements of this taxonomy are to be discovered through the observation of social structure in historical development:  “the experience of the past is the science of the future.”  These elements cannot be comprehended independently, but only when put together in a synthesis.  To make the synthesis Proudhon tries ... to adapt the “serial method” of [Charles, not Joseph -- M.D.] Fourier, using it in combination with his own version of dialectic philosophy.”

“While both methods claim an empirical basis in the study of historical process, in fact he concentrates more on contemporary society, with his interpretation of the past based more on that of the present than the other way around.”

[Note thus, in our terms, the ‘synchronic dialectic’ leaning or bias of Proudhon’s attempted method, vis-a-vis ‘diachronic dialectic.]

“... his conception of humanity’s growth through time is vital to his theory of society.  The conception is one of man making himself through the course of history in a succession of stages.”

[At this level of generalization, we -- and Marx -- are in complete resonance with Proudhon’s, epochal, ‘historical labor of the self-production of humanity’ conception.]

“The stress on mankind’s active role in the process is essential; Proudhon rejects both fatalism and any idea that history is made by exceptional men or by God.”  [So far, so good!]

“His efforts at developing a systematic method turn upon determination of the mechanism by which the historical stages succeed one another.  This would simultaneously define the essential character of social dynamics in any particular stage.” [This at least rings of Seldonian ‘onto-dynamical immanentism’.].

“In Création de l’ordre he takes notions of an ordered series of connected developments from Fourier, together with an idea of “antinomy” (contradiction) which comes from his reading of Kant, but differs from the German’s conception.  By these he describes a process of historical movement through the interaction of contradictory opposites.  This method he further refines in the Système des contradictions économiques, with a pseudo-Hegelian dialectic employed to complete the theory.”

[Again, from a Seldonian point-of-view, one would want to ask -- ¿where do these opposites supposedly come from?  ¿Are they conceived in eternal, Manichean mutual externality, as separately self-subsistent, or is each new opposite seen as emergent from within that which it later opposes? -- and finally -- ¿Does this “interaction’ of opposites lead to the possibility of the ‘‘‘evolute’’’, external conservation of both opponents, as well as of the irruption of their internalization/“synthesis” in a «tertium quid»?]

“Marx’s dialectical and historical materialism [actually, Marx did not use these terms -- M.D.] is similar in approach though not in execution, but he knows better what a systematic social science should be.  Although it has its own faults, the German’s theory is far more compelling -- as a system.”

“... He [Proudhon] says that men must go through the experience of successive historical stages in order to realize how to better order their lives.  They must know through actual experience all the varieties of injustice and contradictory absurdities possible in social relations if they are to understand fully what really is required for justice.  Any particular social structure entails its own peculiar forms of injustice and absurd contradiction, and it does not develop substantially different ones without becoming qualitatively different:  a new set of conditions defines a new stage.  Thus historical change is a succession of significant stages to be passed if progress is to be realized.”

[This account of Proudhon’s views, if correct, highlights an ‘ideal-ist’ tendency in Proudhon’s thought, or at least an uncritically ideological tendency, though it might also be seen to contain seeds of a Marxian, and of what we call a ‘‘‘psychohistorical’’’, perspective, of how developing incompatibilities between predominant social relations of production and growing social forces of production, due to the often ‘uncognized’ growth of the latter, are fought out in ideological terms, which misconstrue and mystify the real roots of the conflict, until, toward the end of the epoch of the predominance of the capital social relation of production, ideology predictedly begins to give way to [Marxian] social science, in and after the transition to the successor system, i.e., to the prevailing of the successor social relation of production, successor to the capital system/relation.  However, it is not only “absurd contradictions”, but psychophysically operative, albeit often ‘uncognized’, objective/subjective human-social “contradictions”, or ‘intra-dualities’, such as that of capital as “self-expanding value” versus capital as ‘self-contracting value’, the latter due to growth-of-the-productive-forces induced ‘technodepreciation’ of fixed capital plant and equipment, that drives the self-transcendence of a given epoch into its successor epoch, one of a new prevailing ‘socio-ontology’ -- of a new prevailing “social relation of production”.  But a human-social-scientific paradigm of human-historical change as a primarily immanently-driven progression of “substantially different” -- of “qualitatively different”, ‘socio-ontologically different’ -- historically-specific, ‘historical species’ epochs, driving themselves, internally, to be “passed” / surpassed / transcended, is a congenial one, for us, and with Marx.  However, the way in which the ‘“old set of conditions”’ transforms itself, primarily immanently, into the “new set of conditions”, needs to be correctly specified if this paradigm is to work, for us, as [dialectical] “social science”.]

“This reflects Proudhon’s conviction that society is a dynamic, constantly changing entity.” [This ‘‘‘rhymes’’’ with Marx’s statement, in his Preface to the first German edition of Capital, Volume I, that “...the present society is no solid crystal, but an organism capable of change, and is constantly changing.”].”

“The change as he conceives it is not merely that of gradual evolution, or growth like an organism’s, where each new form is very like the one preceding and stems directly from it.”

[Note that, as per Karl Seldon’s view, the “new form” may ‘“stem directly from the one preceding’’’, and still be qualitatively, ontologically different from, and advanced/progressed with respect to, and even supplementarily opposite to, “the one preceding”.  This is the case in a sudden ‘‘‘revolution’’’ within, and punctuating, a preceding, gradual, ‘‘‘evolution’’’, i.e., in an irruption of (a) new ontology -- of an ‘ontological singularity’ -- from out of the continuity of the quantitative growth of the old, preceding ontology, when that growth crosses a definite quantitative threshold of qualitative, ontological self-transformation.]

“Rather, a society changes by becoming quite another thing than it was -- a dialectic of historical stages.”

[It is not necessary, in forming a theory of a ‘“dialectic of [pre]historical stages”’ of human society, in order to affirm the actuality of the ‘self-other-ization’ kind of change of natural formations-in-general [including of human-social formations specifically], to disaffirm the actuality of long periods of the ‘non-other-ization’ kind of change, e.g., of human-social formations.  Indeed, the one kind of change flows into, and causes, the other kind of change, and vice versa.  The quantitative growth of the social forces of production, progressing at first still within the identity of a given human-social formation, and of its predominant social relations of production ‘socio-ontology’, gives rise, eventually, to an irruptive self-transcendence of that identity of that human-social formation, and to the emergence of new and unprecedented prevailing social relations of production ‘socio-ontology’, and, thereby, to a new mode of human-social reproduction, and, thus, to a new, higher form[ation] of human society, one which «aufheben»-include/internalizes modified elements of all of its predecessors.]

“This is one reason he [Proudhon] rejects not only conservatism and conventional progressivism, but also the utopias of contemporary socialists.  Their utopias are conceived as final, ideal forms, to remain fixed in basic structure if not in detail.  To Proudhon this is no less static than the absolutism of monarchs and Church, and scarcely less likely to prevent continuing human self-actualization.

[¡Kudos to Proudhon on this realization, contra fetishizing eternalisms, Platonian idealisms, and Parmenidean staticisms and immutable-isms, etc., of all kinds!].


“His views of man making himself throughout history is significant because it adds a dimension to his individualism.  The ideal of freedom to strive and grow can be conceived in terms of merely private desires, or in typical terms of liberal humanism, that mankind develops through the achievements of free individuals.”

“Proudhon’s vision builds upon the latter, seeing the growth still more as a collective, social process, the joint product of the efforts of all individuals rather than [of] a select number.”

[Proudhon’s vision, as described above, one also highly resonant with the views of Marx and the best moments of the views of Hegel, and even of Vico as well, is, in our eyes, a remarkable achievement, a remarkable advance upon ‘precedingly predominant’ social ideologies, and indicates that the dialectic of collectivism versus individualism was at work in his thinking in a much less suppressed/one-sided way than was the case for so many others before, during, and since his lifetime.  Human society was no mere Lucreatian/linear “gas”, or “sum”, of “human atoms”, as so many prisoners of bourgeois ideology have pretended.  Just recall the reprehensible views, in this regard, of Maggie Thatcher, for one!]

“[Proudhon was reaffirming the old notion] that “man makes himself,” that out of the common efforts of mankind, out of the struggles and collaborations among men [and women!!! -- M.D.], largely without design, intent, or self-consciousness, come the institutions, beliefs, and all that goes to make up civilization, purely human creations which humanity adds to what nature and/or God has made.”*

[The foregoing quote is resonant with the Hegelian concept of “the cunning of Reason”, with the Seldonian concept of ‘onto-dynamasis’ -- of human ‘Nature-al’ [self-re]productive activity adding new ontology to pre-human / exo-human Nature -- and, if the “what God has made” phrase truly were to hold for Proudhon, to Tolkein’s concept of [a] human “sub-creation”.].

“He speaks of society as a “living being” whose existence is manifested by “the concert and intimate solidarity of its members.”

“Society has its own identity distinct from that of individuals, because it is what they can be only collectively.  Yet its being is not beyond that of individuals, not something with, not something with its own demands and capacity to make them what they are, as it is in the social metaphors of conservative organicism.”

[Here, per this description, Proudhon reveals his sensitivity to the graded, scaled -- even ‘qualo-fractal’ -- features of the identities of real objects, in particular, of human «monads», and of the ‘societary’ «arithmos» that they constitute, and that co-constitutes them, capturing the ‘identity consolidation’ of those «monads» in the unit[y] of their «arithmos», but also the «aufheben» conservation of those «monads» as identities in their own right, achieving an ‘‘‘organismic’’’ view of, e.g., human society, a ‘‘‘totality’’’ view thereof, but one without totalitarianism; without the one-sided, absolutist, and coercive consolidation of human individual identities into the “community”, canalized in the everything-owning singular “Leader”, that characterizes “organicist” fascist [and even Lenino-Stalinoid, pseudo-socialist] state-capitalist ideologies.  Proudhon’s grasp of human society as a “living being”, produced conjointly by its human «monads», shows his discernment of what we would call ‘the human phenome’, in its intimate, intricate, interaction -- its ‘‘‘complex unity’’’ -- with ‘“the human genome”’.]

Social science is the reasoned and systematic understanding, not of what has been in society, nor of what will be, but of what IS in its whole life, that is, its totality of successive manifestations, for it is only thus that there can be reason and system.  Social science must embrace the order of humanity, not only in such or such period of its duration, nor in some of its elements, but in all its principles and in the integrality of its urgent needs. . . . Such must be the science of all living and progressive reality; such incontestably is social science.”**

[The direct quote from Proudhon, above, Proudhon’s bias toward what we call ‘‘‘synchronic, systematic dialectic’’’, as opposed to ‘‘‘diachronic, historical dialectic’’’, is again exhibited.  Proudhon seeks “social science” not merely as a presentation of the actual present, including that present’s -- that present system’s/period’s/manifestation’s -- «aufheben» internalization/conservation of “manifestations”, “periods”, “principles”, and “elements” of the past, even to the point of a ‘present-ization’ of the extincted, unconserved moments of that past -- a kind of “supra-historical” ‘meta-temporal summarization’, presently, experientially existing/“manifesting” only as a human-mental abstraction.  Of course, Marx also embraced ‘‘‘systematic dialectics’’’, especially in his construction of his multi-volume treatise «Das Kapital», but also integrated historical content -- e.g., historical examples -- therein, and described, in some of his methodological comments in his Grundrisse draft, the relationship between historical accounts and systematic accounts:  the -- dialectical -- relationship between ‘‘‘historical dialectics’’’ and ‘‘‘systematic dialectics’’’ in ‘the dialectic of the dialectic itself’.]

“Society is not only a vital being, it is “the essential environment for the unfolding and realization of purely human, individual goals.” ... Knowing this environment means understanding the intricate web of social relationships and the forces of the collectivities and individuals constituting society.  Proudhon is very conscious of how diverse are its elements and how varied the relations between [better:  among -- M.D.] them.  A scientific approach typically (though not necessarily) would attempt to reduce connections between [again, better among -- M.D.] elements to rather mechanical terms. . .”

“Proudhon cannot admit such a mechanical treatment.”

“For him, social relationships have all the vitality of society and the men [and the women!!! -- M.D.] who make it.”

“But the essential qualities of living things are not usually precise or regular.”

“”They defy attempts at mechanical analysis which seek to identify and exactly describe the regular.  By reducing men to their universal terms, science [sic -- M.D.] tends to eliminate the idiosyncratic qualities that gives things their vitality.”

[Such science is bourgeois science, ideologically compromised, in its scientificity -- in its empirical, experiential faithfulness and accuracy -- by the largely unconscious projection of atomistic, radically separatist, ‘pointal’ bourgeois social self-identities onto all reality, i.e., by the ‘‘‘psychohistorical’’’ limitations of human self-consciousness within the historical specificity of the capitalist horizon.  Such science is that of linear dynamical systems theory, suppressing more realistic, nonlinear-dynamical integro-differential equationsdescriptions by linearized approximation -- approximation which excises the very hearts of the phenomena to be modeled.]

“With his notion of “serial order” he [Proudhon -- M.D.] seeks to integrate distinct elements into a coherent synthesis, discovering a “unity in the multiplicity” of individuals and relations which is the order of society. ... Given his view of the separate elements, it could hardly be otherwise.”

[Here, Proudhon’s “prose” method of discerning and describing distinct elements, and how they have already integrated themselves into coherent syntheses, concretizing the potential moment of the unity of their multiplicities, converges with the Seldonian algebraic, algorithmic-heuristic method.  If we represent, by qA, the first-evoked or first-manifested “distinct element”, or ‘“distinct quality”’, then the proceedings of the  generic dyadic Seldon function sequence of “series” can be represented as follows --  

qA --->

qA + qAA   |-=   qA + qB --->

qA + qB + qBA + qBB   |-=   qA + qB + qC + qD --->

qA + qB + qC + qD + qDA + qDB + qDBA + qDD   |-=  . . .  .

-- such that category-terms in the above generic algebraic series-sum, such as qDA, qDB, and qDBA represent “integrations”, i.e., “combinations”, or “coherent syntheses”, of, e.g., qDwith qA, qDwith qB, and qDwith qA and qB, respectively.  Of course, these terms represent the generic potential for “coherent synthesis”.  The actualized achievement of the representation and description, in detail, of such syntheses, depends upon the knowledge and skill of the user of this algorithmic-heuristic, ‘‘‘serial’’’ method, in selecting the best definition for the beginning term/category of the ‘‘‘series’’’, qA, and in ‘‘‘solving’’’ ['|-='] for the most pregnant definitions of the meanings of all of the subsequent terms/categories in the ‘‘‘series’’’, each generated, ‘meta-genealogically’, and in the last analysis, by/from that starting term/category, qA.]

“...his [Proudhon’s -- M.D.] pursuit of a method makes clear how definitely he believes that the bases of social order are to be discovered in what man makes of himself [of what men and women together make of themselves -- M.D.], rather than conceived in abstraction and imposed through authority.  Re-creation of that order into better forms can be knowing and intelligent, but it must be spontaneous and self-directed by the whole being of society.”

[Proudhon’s view, as described above, regarding the ‘‘‘spontaneous self-organization and aperiodic self-re-organization’’’ of human societies as “wholes within the whole” of Nature, is both an insight into the phenomena that are characteristic of the more realistic, nonlinear dynamical systems of contemporary mathematical modeling, and into the inherently and essentially democratic nature, and destiny, of the “nonlinear dynamical system” that is humanity itself.]


TO BE CONTINUED.



*“[Aaron Noland, “History and Humanity:  The Proudhonian Vision,” in The Uses of History, ed. Hayden V. White (Detroit, 1968), 69ff.].

**“[Oeuv., I, vol. 1, 123.  He also uses the phrase “homme collectif” and “personne collectif”, and he emphasizes the reality of “la personnalité de l’homme collectif” throughout his work].











Sunday, September 24, 2017

‘‘‘Rosetta Stone’’’ -- For ‘Diachronic Dialectics’ Specifically, in the ‘Marxian Format’: The Seldonian ‘Generic Dialectics / Specific Dialectics / Applied Dialectics’ Interconnection.







‘‘‘Rosetta Stone’’’ -- For Diachronic Dialectics Specifically, in the ‘Marxian Format’:

The Seldonian Generic Dialectics / Specific Dialectics / Applied Dialectics Interconnection.







Dear Readers,



The dialectogram pasted-in below was also recently posted to the www.dialectics.org web site,
to its Application Page, under the heading of Karl Seldon’s forthcoming essay entitled Universal Algorithmic Heuristic Method.

This dialectogram has been prepared for use in that forthcoming essay.



ENJOY!



Regards,


Miguel Detonacciones,

Member, Foundation Encyclopedia Dialectica [F.E.D.],
Participant, F.E.D. Special Council for Public Liaison,
Officer, F.E.D. Office of Public Liaison.






















Friday, September 22, 2017

‘‘‘Rosetta Stone’’’ -- For ‘Synchronic Dialectics’ Specifically, in the ‘Marxian Format’: The Seldonian ‘Generic Dialectics / Specific Dialectics / Applied Dialectics’ Interconnection.







‘‘‘Rosetta Stone’’’ -- For Synchronic Dialectics Specifically, in the ‘Marxian Format’:

The Seldonian Generic Dialectics / Specific Dialectics / Applied Dialectics Interconnection.







Dear Readers,



The dialectogram pasted-in below was also recently posted to the www.dialectics.org web site,
to its Application Page, under the heading of Karl Seldon’s forthcoming essay entitled Universal Algorithmic Heuristic Method.

This dialectogram has been prepared for use in that forthcoming essay.



ENJOY!



Regards,


Miguel Detonacciones,

Member, Foundation Encyclopedia Dialectica [F.E.D.],
Participant, F.E.D. Special Council for Public Liaison,
Officer, F.E.D. Office of Public Liaison.






















Wednesday, September 20, 2017

Part 03: Seldon’s Insights Series -- ‘Dialectical Meta-Equations’ and ‘Dialectical Meta-Models’.


Part 03:  Seldon’s Insights Series --

Dialectical Meta-Equations and Dialectical Meta-Models.







Dear Readers,



It is my pleasure, and my honor, as an Officer of the Foundation Encyclopedia Dialectica [F.E.D.] Office of Public Liaison, to share with you, from time to time, as they are approved for public release by the F.E.D. General Council, key excerpts from the internal writings, and from the internal sayings, of our co-founder, Karl Seldon.
The third such release in this new series is entered below [Some E.D. standard edits have been applied, in the version presented below, to the direct transcript of our co-founder’s discourse].

For more about these key concepts, see --

http://www.dialectics.info/dialectics/Glossary_files/Glossary,Equations_vs._Meta-Equations,19JUN2014.jpg

http://www.dialectics.info/dialectics/Glossary_files/Glossary,Models_vs._Meta-Models,19JUN2014.jpg

ENJOY!



Regards,


Miguel Detonacciones,

Member, Foundation Encyclopedia Dialectica [F.E.D.],
Participant, F.E.D. Special Council for Public Liaison,
Officer, F.E.D. Office of Public Liaison.







...The ‘‘‘categorial progressions’’’ -- the qualitative, non-amalgamative’ ‘‘‘sums’’’ of increasingly thought-concrete, increasingly determinate category symbols -- that we express via equalities using the NQ arithmetic and its algebra:  these we refer to as ‘[dialectical] META-equations.”

We do so because each such unity-to-diversity equating expression is ‘made up out of’, and thus ‘‘‘«aufheben»-contains’’’, a heterogeneous multiplicity of qualitatively different, ontologically different mere equations.”

“Indeed, they ‘‘‘«aufheben»-contain’’’ one such mere equation for each distinct, consecutive, relevant value of that expressions independent variable -- e.g., s [step number] for a synchronic, presentational dialectic; t [epochal time count] for a diachronic, historical dialectic.”

“Moreover, when each such equation formulates a distinct -- in these cases, a qualitatively distinct, ontologically distinct -- dialectical-mathematical model, then their single meta-equation also formulates a META-model, ‘made up out of a heterogeneous multiplicity of internally-related mere models; ‘‘‘«aufheben»-containing’’’ a heterogeneous multiplicity of such related models.”

“Now, of course, meta-equations do not arise from the NQ arithmetic/algebra alone.”

“For example, the general planular linear equation algebraic equalityy  =  mx + b’, given that its two parameters, m and b, denote ‘‘‘known variables’’’ [relative to the unknowns /variablesx and y] both range over R, the “Real numbers, is a purely”-quantitative meta-equation, ‘made up out of’ a multiplicity of linear mere models.”

“Likewise, the ‘«arché»-ic nonlinear differential equation, ‘dx(t)/dt  =  ax(t)^2’, about which we have spoken before, and will again, is also a purely”-quantitative meta-equation, given that its single control-parameter, a, ‘‘‘modifying’’’ its ‘function-unknown’, x(t), is taken as denoting a ‘‘‘variable parameter’’’, or ‘‘‘parameter variable’’’, e.g., again ranging over R, the “Real numbers, and co-determining the location of its “movable pole” singularity.  This meta-equationis again ‘‘‘contains’’’ a different nonlinear differential equation for each distinct value of a.”

“A characteristic which distinguishes NQ arithmetic/algebra meta-equations from the other kinds of meta-equations noted above is this:  because of the ‘‘‘evolute’’’, «aufheben» nature of the NQ multiplication operation, the categorial progression of each successor equation in an NQ meta-equationcontains all of the category symbol term(s) of all of its predecessor equations, as well as, and non-amalgamatively added-to, the new category symbol term(s) that this successor equation introduces for the first time in this succession.