F.E.D. Vignette #16: The [Systematic] Dialectic of 'Modern Computerware' -- A Simple Example of Dialectic Modeling in Action.

Dear Readers,

An updated version of the recent blog-entry, here, on 'The Dialectic of Modern Computerware' --

 http://feddialectics-miguel.blogspot.com/2013/08/part-iii-b-interlude-dialectic-of.html

-- has been posted to the F.E.D.-affiliated website www.dialectics.org, in a typographical context not subject to the coercive edits, regarding size, font, and other typographical features, associated with the Google software here.

You can access this typographically-reformed content via the following links --

http://www.dialectics.org/dialectics/Welcome.html

http://www.dialectics.org/dialectics/Vignettes.html

http://www.dialectics.org/dialectics/Welcome_files/Miguel_Detonacciones,v.2.0,F.E.D._Vignette_16,The_Dialectic_of_Modern_Computerware,A_Simple_Example,28AUG2013.pdf



Regards,

Miguel

F.E.D. Vignette # 15: The [Systematic] Dialectic of TV Series -- An Ultra-Simple Example of Dialectic Modeling In Action.

Dear Readers,

An updated version of the recent blog-entry, here, on 'The Dialectic of TV Series' --

http://feddialectics-miguel.blogspot.com/2013/08/part-iii-interlude-dialectic-of-tv.html

-- has been posted to the F.E.D.-affiliated website www.dialectics.info, in a typographical context not subject to the coercive edits, regarding size, font, and other typographical features, associated with the Google software here.

You can access this typographically-reformed content via the following links --

http://www.dialectics.info/dialectics/Welcome.html

http://www.dialectics.info/dialectics/Vignettes.html

http://www.dialectics.info/dialectics/Welcome_files/Miguel_Detonacciones,v.2.0,F.E.D._Vignette_15,The_Dialectic_of_TV_Series,Ultra-Simple_Example,28AUG2013.pdf



Regards,

Miguel

An Address to Remember!

Dear Readers, I find myself unable to resist sharing, with you, the finale of the latest address by Karl Seldon to the Assembly of the Foundation. Here it is: "To research dialectics; to advance humanity's comprehension of -- indeed, to advance humanity's self-comprehension -- and to work for the dissemination of, dialectical reason, is to advance the very essence of humanity; is to advance what is most unique, and most essential, about the human, amid all of the myriads of kinds of being to which our cosmos has, so far, to our knowledge, given birth:  human cognition. For dialectics is both the ancient and the modern name for the highest stage of human creative cognitive power, one all the more worthy to be advanced because "the dialectical operations phase of adult human cognitive development" is the phase in which human reason, and human passion, unite."  [Emphases added, per Seldon's enunciations, by M.D.]. In its own special way, this statement "says it all"! Regards, Miguel

The Historical Dialectic of the Capital-Relation -- Thorstein Veblen's Version of Marx's "Law of the Tendency of the Rate of Profit to Fall". A Commentary.

Full Title:

The Historical Dialectic of the Capital-Relation --

Thorstein Veblen’s Version of Marx’s “Law of the Tendency of the Rate of Profit to Fall”.

A Commentary on a Potent Passage from Veblen’s circa 1904 Theory of Business Enterprise.     

Introduction.  In F.E.D. Dialectics, the driver of the immanent aspects of the historical dialectic of a given dialectical ‘‘‘eventity’’’ is held to be what F.E.D. names as the ‘self-duality’, or ‘intra-duality’, of that ‘‘‘eventity’’’ itself.

F.E.D. conceives such ‘intra-dualities’, generically, as a kind of ‘‘‘self-opposition’’’, or ‘‘‘internal antithesis’’’, that inheres in the innermost ‘internity’ of such an ‘‘‘eventity’’’, as an ineluctable aspect of its very essence -- of its thus inescapably dual essence -- and that energizes the immanent aspect of its development -- its ‘‘‘self-development’’’, via its «causa immanens», or «causa sui» --  in counterpoint with the forces impinging upon it from its ‘externity’, from its external environment, from other ‘‘‘eventities’’’ -- its «causa transiens».  Together, in combination, «causa immanens» and «causa transiens» co-determine the total development, the full life-history, of the individual ‘‘‘eventity’’’.

The classic examples of such ‘intra-duality’ for F.E.D., in the context of pre-human physics -- of the pre-human and extra-human «physis» -- are those ‘hot, shining orbs’, the stars, the ‘stellar eventities’.  Their immanent development is driven by the ‘intra-dueling’ of their self-gravitational self-implosion and their thermonuclear fusion self-explosion, twin, self-generated -- internally-generated -- ‘self-forces’ that oppose one another at every point of the stellar body ‘internity’, as one core fusion fuel is exhausted, followed by resumed self-implosion, compressing and igniting the core fusion “ash” into a new -- the next -- core fusion fuel, until, typically, a stellar core of consisting of iron arises, terminating the very existence of the star as such in a combined core implosion and ‘exo-core’ explosion, enriching the interstellar medium with the dying star’s evolved, “metallic”, higher atomic species from which, eventually, ‘contra-stars’ -- “planets”, cold[er], ‘shine-less’ orbs -- are formed, in subsequent generations of stellar/planetary system-formation.

But the «genos» of ‘intra-duality’ manifests «species» and instances also within the domain of ‘human physics’ as well -- within the human-social epoch of cosmological ‘self-meta-evolution’.

A prime, and salient example of such a systemic ‘intra-duality’ for that epoch, is the primary ‘intra-duality’ of “the capital-relation” as the predominating “social relation of production” of modern society, i.e., of “capital-value”, and its “law of motion” -- the ‘‘‘law’’’ of its motion of “accumulation” [and of ‘‘‘dis-accumulation’’’] -- as uncovered by Karl Marx.  For “the capital-relation” [Marx], the primary ‘intra-duality’ is the internal opposition, within the movement of the “total social capital” [Marx], between capital as “self-expanding value”, and capital as ‘self-contracting value’, which is such that both of these opposing processes are ineluctably self-caused, self-imposed, by capital, upon capital, as a result of the very nature of the capital ‘‘‘eventity’’’ itself.

Capital as “self-expanding value” [Marx] arises via the productive reinvestment of produced surplus-value -- e.g., in the form of profits of enterprise -- in the enterprise that produced that surplus-value, or in other productive enterprises -- thus expanding the capital-value of total social capital-assets.

Capital as “self-contracting value” [Seldon] arises via the competitive-survival-incentives that the capitals-system imposes on individual capitals in general -- on those individual “personifications” of the capital-relation, the capitalist -- to boost the [relative] surplus-value / profits that their enterprises produce, by inventing and/or purchasing and installing -- investing in -- new, better capital plant and equipment, capital plant and equipment which produces more units of output per unit time than do older vintages, or that costs less to purchase while delivering the same output rate, or that costs less to operate while delivering the same output rate, or which realizes some combination of such “better” features.

The consequence of the installation and operation of such “better” capital plant and equipment, by one enterprise, upon the other enterprises that compete with it, and which have not [yet] so-installed, is to lower their profit margins as a result of the underselling of their prices of output by the prices of the installing competitor-enterprise, or to force them to de-install and write-off their old capital plant and equipment, and to install the new, to remain price-competitive, causing a drop in their net profits, at least for the accounting-period(s) in which that write-off, and the initial expenses of purchasing and installing the new capital plant and equipment, occur, or some combination of the two, after which their rate of return on the new equipment itself should recover.

Over time, in the ongoing continuity of innovation, installation, and operation of progressively improving industrial capital plant and equipment -- the continuity of “the growth of the social forces of production” [Marx], i.e., of the rising rate of human societal self-reproduction, i.e., of the rising rate of ‘self-productivity’ of humanity/of ‘human socio-mass’ -- this process that we have named ‘the self-depreciation of capital’, and ‘technodepreciation’, drives a continual devaluation of accumulated capital-value, a ‘dis-accumulation of capital’, that is “netted out” against the equally ongoing self-expansion of capital value that is driven by the reinvestment of industrial, etc., profits, into industrial, etc., production.

During the initial phase of capitalist development, which we term its ‘‘‘ascendance phase’’’, when the capital plant and equipment composition of total capital is still relatively low, reflecting lower productivity, so that the proportion of capital that is exposed to devaluation is also relatively low, the expansion of profits, and of capital, tends to outstrips their ‘technodepreciation’-driven contraction.

The accumulating, growing proportion of capital plant and equipment in total capital, and the rising rate of occurrence of ‘technodepreciating innovation’, as a result of the capitalism-immanent incentives already cited, and reflecting a rising rate of rising social productivity, eventually generates a turning point, after which the technodepreciation-driven contraction of the total social capital value outstrips capital’s co-occurring expansions.  This second main phase of capitalist development is that which we name the “descendance phase” of capitalist development, which, to our data, appears to have occurred circa 1887, at least for North American industrial capital [ for more about this determination of timing, see http://capitalismsfundamentalflaw-wayforward.blogspot.com/2012/12/part-5.html ].

It is, psychohistorically, very telling, indeed, to see how these processes, and their consequences, are observed by Thorstein Veblen, writing circa 1904, some 17 years after the silent turning point that we have ascertained, but at a time by which the profound consequences of that turn had become more widely -- and more “loudly” -- evident.

Extract and Commentary.  Below we extract a particularly pregnant passage from Veblen’s 1904 Theory of Business Enterprise, adding our own commentary regarding his perceptions, and regarding the -- psychohistorically predictable -- ideological, policy, and institutional responses of the ruling plutocracy in light of Veblen’s insights, and those of Marx, and those of others, into the inherent self-destructive destiny of capitalism -- responses which we have characterized elsewhere under the headings of ‘Anti-Marxian Marxianism’, ‘Capitalist Anti-Capitalism’, and ‘Human Anti-Humanism’.

Veblen predicted “chronic depression” as the new norm for the global capitalist economy, as a result of what we would term the turn from the ‘‘‘ascendance phase’’’ to the ‘‘‘descendance phase’’’ of the global capitals-system:  

“Chronic depression, however, does not seem to belong, as a consistent feature in the course of things, in this nineteenth-century period, prior to the eighties or the middle of the seventies.” 

“The usual course, it is commonly held, was rather:  inflation, crisis, transient depression, gradual advance to inflation, and so on over again.”

[M.D.]:  We interpret the passage above as noting a change in the dynamics of capitalism, reflecting what we would describe as the turn from the ‘‘‘ascendance phase’’’ into the ‘‘‘descendance phase’’’ of the global capitalist system.

Veblen continues --

“On the view of these phenomena here spoken for, an attempt at explaining this circuit may be made as follows: 

A crisis, under this early nineteenth century situation, was an abrupt collapse of capitalized values, in which the capitalization was not only brought to the level of the earning-capacity which the investments would have shown in quiet times, but appreciably below that level.”

[M.D.]:  In the passage above, Veblen defines what we would term “ascendance phase crisis” as a critical and precipitous ‘self-contraction of capital-value’, overshooting even the restoration of a normal ratio of net earnings to fixed-capital value -- a normal rate of “return on [capital plant and equipment] investment” -- that would register the accumulated ‘technodepreciation’ of capital plant and equipment, accumulated over the course of the “inflationary” period.  The very non-registration, up until the crisis, of this ‘technodepreciation’, was, itself, a major hidden cause of that crisis -- of that eventual precipitous contraction/deflation of plant-and-equipment capital-value, and of other, related, capital-value, which is the crisis. 

The crisis-induced drop in the value of the “return on investment” ratio’s capital-plant-and-equipment-value denominator, relative to that ratio’s net-earnings [“return”] numerator, has a post-crisis salutary effect.  It raises the magnitude of that “profit-rate” ratio as a whole.  That ratio had tended to fall as a whole, during the “non-crisis” phase.  It tended to fall because productivity-increase-induced, competitive drops in prices, hence in the net earnings numerator of that ratio, owing to the productivity-increase-induced drop in unit costs of production, failed to be compensated by a commensurate drop in the valuation of the ratio’s capital plant and equipment value denominator, such as would have registered the ‘technodepreciation’ of that capital plant and equipment value denominator, owing to the accumulated ‘non-crisis period’ -- or ‘pre-crisis-period’ -- productivity improvements.

Veblen then advances his argument as follows --

“The efficiency and the reach of the machine industry in the production of productive goods was not then so great as to lower the cost of their production rapidly enough to overtake the shrinkage in capitalization and so prevent the latter from rising again in response to the stimulus of a relatively high earning capacity.”

[M.D.]:  Above, Veblen locates the key locus of ‘technodepreciation’ in what Marx termed “Department I”, the department of “the production of means of production”, of ‘the machine production of production machinery’, for what Marx termed “Department II”, the department of the production of the means of population-sustaining consumption. 

Veblen holds that, in what we term the ‘ascendance phase’ of the capitals-system, the velocity of improvement in the efficiency [and in the design] of the means of production, thus technodepreciating older vintages of those means of production, already installed, was insufficient to match and, indeed, to exceed the crisis-induced over-shrinkage of the “capitalization” -- of the capital-valuation -- of the already-installed, ‘technodepreciated’ means of production.  Thus, there remained a margin of that overshoot, leading to a higher net-earnings numerator value relative to that ‘over-shrunk’ capital plant and equipment denominator value, after the crisis-shock waned, sufficient to stimulate a rise in the valuation of that denominator, partially restoring the pre-crisis capital-valuation of that denominator, given the higher-than-normal earnings relative to that ‘over-shrunk’ denominator.  The latter rise tended to restore a normal/expected value of that net-earnings-divided-by-capitalization ratio as a whole, correcting the initial post-crisis ‘high-side super-normal value’ of that ratio, owing to the crisis-induced undervaluation of that ratio’s denominator.

Veblen then summarizes these dynamics of ‘ascendance phase’ capitalist crisis as follows --

“The shock-effect of the liquidation passed off before the cheapening of the means of production had time to catch up with the shrinkage of capitalization due to the crisis, so that after the shock-effect had passed there still remained an appreciable undercapitalization as a sequel of the period of liquidation.”

“Therefore there did not result a persistent unfavorable discrepancy between capitalization and earning capacity, with a consequent chronic depression,”

“On the other hand, the earning-capacity of the investments was high relatively to their reduced capitalization after the crisis.”

“Actual earning-capacity exceeded nominal earning-capacity of industrial plants by so appreciable a margin as to encourage a bold competitive advance and a sanguine financiering on the part of the various business men, so soon as the shock of liquidation had passed and business had again fallen into settled channels.”

[M.D.]:  Veblen then sets forth his view of the causes -- and of the consequences -- of the then-recent, “permanent” change in those dynamics of ‘‘‘ascendance phase’’’ crisis, that characterize what we term the pre-1913, early ‘‘‘descendance phase’’’ ‘neo-dynamics’ of capitalist crisis --

“Since the [M.D.:  eighteen-]seventies, as an approximate date and as applying particularly to America and in a less degree to Great Britain, the course of affairs in business has apparently taken a permanent change as regards crises and depressions.”[M.D.:   The ‘transition from the ascendance phase to the descendance phase of the global capitals-system’.  ].

“During this recent period, and with increasing persistency, chronic depression has been the rule rather than the exception in business.”

“Seasons of easy times, “ordinary prosperity”, during this period are pretty uniformly traceable to specific causes extraneous to the process of industrial business proper.”

“In one case, the early [M.D.:  eighteen-]nineties, it seems to have been a peculiar crop situation, and in the most notable case of a speculative inflation, the one now (1904) apparently drawing to a close, it was the Spanish-American War, coupled with the expenditures for stores, munitions, and services incident to placing the country on a war footing, that lifted the depression and brought prosperity to the business community.”

“If the outside stimulus from which the present prosperity takes its impulse be continued at an adequate pitch, the season of prosperity may be prolonged; otherwise there seems little reason to expect any other outcome than a more or less abrupt and searching liquidation.”

“ ...It was said above that since the [M.D.:  eighteen-]seventies the ordinary course of affairs in business, when undisturbed by transient circumstances extraneous to the industrial system proper, has been chronic depression.  The fact of such prevalent depression will probably not be denied by any student of the situation during this period, so far as regards America and, in a degree, England ...”

“The explanation of this persistent business depression, in those countries where it has prevailed, is, on the view here spoken for, quite simple.”

“By an uncertain date toward the close of the [M.D.:  eighteen-]seventies the advancing efficiency and articulation of the processes of the machine industry reached such a pitch that the [M.D.:  fall in the] cost of production of productive goods [M.D.:  i.e., of capital plant and capital equipment, functioning as “means of production ”] has since then persistently outstripped such [M.D.:  downward] readjustment of capitalization as has from time to time been made [M.D.:  e.g., due to crises].”

[M.D.]:  Thus, it is the increase, past a certain critical point, in the “pitch” -- the acceleration -- of the rate of improvement of the overall productivity of the means of production, for Veblen, that explains this change,  which we term the turn, from ‘‘‘ascendance phase’’’, self-overcoming aperiodic economic crises, to ‘‘‘descendance phase’’’, self-perpetuating, chronic depression-crises -- change in the crisis-dynamics of capitalism.

This change, per Veblen, manifests also as a “persistent decline” in [the rate of] “profits” [in the value of profit returns divided by the value of the “industrial apparatus” used in producing those profits] --

“The persistent decline in profits, due to the relative overproduction of industrial apparatus, has not permitted a consistent speculative expansion, of the kind which abounds in the earlier half of the nineteenth century, to get under way.”

“When a speculative movement has been set up by extraneous stimuli, during this late period, the inherent and relatively rapid decline in earning-capacity on the part of older investments has brought speculative inflation to book before it has reached such dimensions as would bring on a violent crisis.

“And when a crisis of some appreciable severity has come and has lowered the capitalization, the persistent efficiency and facile balance of processes in the modern machine industry has overtaken the decline in capitalization without allowing time for recovery and subsequent boom.”

“The cheapening of capital goods has overtaken the lowered capitalization of investments before the shock effect of the liquidation has warn off.”

“Hence depression is normal to the industrial situation under the consummate regime of the machine, so long as competition is unchecked and no deus ex machina interposes.”.

[Thorstein Veblen, The Theory of Business Enterprise, Charles Scribner’s Sons [New York: 1904], pp. 248-255, emphases added].

The questions which should leap to mind, in the aftermath of reading this circa 1904 description of the ‘‘‘law’’’ of a ‘technodepreciation-induced’ fall in the rate of profit on industrial capital, since a ‘‘‘turning point’’’ after the 1870s in the U.S., and, in lesser degree, also in the U.K., leading to a condition of “chronic depression”, of “persistent business depression”, include the following --

1.  ¿What was the capitalist class moved to enact -- in terms of new economic policy, new “popular” ideologies, and new political-economic institutions -- in response to the prospect of ‘‘‘permanent depression’’’, and the specter of a popular search for alternatives to capitalism -- for alternatives to the rule of that ruling class -- leading to the overthrow of the power and the “perks” of that ruling class?

2.  ¿Did the circa 1913 imposition in the U.S., by that ruling class, of the Federal Reserve System, of the Federal Income Tax, and of World War I, serve to mitigate the trend to “chronic depression” that Veblen observed, and essayed to explain, and to further mutate the dynamics of the global capitals-system itself, leading to the principle historical phenomena that humanity has experienced since World War I?

Perhaps key clues to the answers to these questions were captured in the paper that we have cited here before [ Geert Reuten, "The Incompatibility of Prolonged Technical Change and Competition: Concurrence and the Socialization of Entrepreneurial Losses through Inflation" http://www1.fee.uva.nl/pp/bin/642fulltext.pdf ], in which the author, Geert Reuten, summarizes as follows [emphases added]:

“To the extent that technical change accelerates, price competition precludes the full amortization of capital investments.”

“In contrast with the common opinion that both technical change and competition are key characteristics of the capitalist system, they are incompatible, at least when technical change accelerates.”

“Such acceleration then gives rise to forms of concurrence — abstinence from price competition, price leaderships, cartels.”

“The particular form depends on the structure of production of enterprises (i.e. the make-up of the stratification of capital).”

“Concurrence is a major determinant of the inflationary form of the accumulation of capital.”

“Because it is in their interest, banks tend to accommodate the concurrent price settings of enterprises and so to accommodate a socialisation of private losses that would be due to the devaluation of capital in the case of price competition.”

“Price inflation also puts enterprises in a relatively advantageous position vis-á-vis labour.”

Amplifying upon Reuten’s final point, above, we note that “permanent inflation” is, precisely, a permanent tendency to reduce real wages and salaries, if nominal wages and salaries remain constant, or even if wages and salaries increase, but at a rate of increase (s)lower than the rate of general consumer price inflation.

And exponential “permanent inflation” [excepting the 1930’s “Great Depression” ‘Great deflation’ aftermath to the 1920’s “roaring” inflation] is exactly what the U.S. has had, ever since the “Fed” was imposed, by the ruling class, in 1913, as shown by the data unified in the following remarkable, and little-known, graph, also cited here previously --

[source:  http://oregonstate.edu/cla/polisci/faculty-research/sahr/pl1665.htm -- see  http://oregonstate.edu/cla/polisci/sites/default/files/faculty-research/sahr/inflation-conversion/pdf/price-levels_1774-2012.pdf for an update to 2012, which shows a slight down-tick in ~2008 for “The Great Recession” ].

Certainly, the continual de facto reduction of wages, by Fed-managed chronic, exponential inflation, helps to shore up capitalist profits, and their rates, and to mask, to delay, or even to avert technodepreciation losses.

For a ruling class hell-bent on averting the massive technodepreciation of its older-vintage, legacy capital plant and equipment in its U. S. and U. K. core, due to the price competition of newly-industrializing, lower-wage nations in its periphery, installing the latest, most advanced vintages of capital plant and equipment from the start -- i.e., for a ‘Capitalist Anti-Capitalist’ ruling class, hell-bent on suppressing capitalist industrialization in the periphery of its core, hence hell-bent on creating a “Third-World” of military dictatorships and rising poverty -- it certainly helps to have a “Federal Income Tax”.

That tax allows that ruling class to, in effect, create a new category of “surplus-value”, draining away part of the wages of the working class as personal income losses, paid to the national state, and using the vast proceeds of those income taxes to force the core working class to pay for “foreign aid” to the military junta’s that the ruling class sets up in those “peripheral” nations, to suppress industrial development there, and to massacre democratic nationalists there who oppose that suppression.

Veblen pointed out how episodic wars can temporarily mitigate the tendency to chronic depression that he describes:  “it was the Spanish-American War, coupled with the expenditures for stores, munitions, and services incident to placing the country on a war footing, that lifted the depression and brought prosperity to the business community.”

World War I, launched by the core ruling class about the same time that it imposed the Fed, and the Federal Income Tax, upon the U. S. working class, certainly represented a big boost for manufacturers of munitions, and for the financiers who financed them, enabling them to shamelessly sell military “goods” to all sides of the conflict -- military goods that would not “last” on shelves, but that, instead, would be used -- and used up -- in short order, requiring rapid replacement, hence new repeat sales in rapid succession.

Making war, and preparation for war, into a permanent institution, a “military-industrial complex” [Eisenhower], massively supported by the working-class income, and by the profits of subordinated capitalists, taxed away from them by the Federal Income Tax, would give a lasting boost to “business”, at the cost of diverting vast former forces of production into forces of destruction of forces of production, whenever those resulting military “goods” were used, or to mere waste of productive forces when those military “goods” simply sit idly in arsenal.

¿But what new kinds of capitalist “crises” -- of global “Great Depressions” and “Great Recessions” -- and the Totalitarian Dictatorships, the Genocides, and the Global Wars to which they lead -- arise out of the dynamics of a capitalism mutated by National Income Taxes, National “Military-Industrial Complexes”, and National, fiat-currency-foisting “Central Banks”, a la the U.S. “Fed”?

To the answering of those -- and related -- questions, we plan to devote many subsequent blog-entries.

Part III. C.: Interlude. Dialectic of Arithmetical Operations. The F.E.D. Psychohistorical-Dialectical 'Meta-Equation' of Human-Social Formation(s) 'Meta-Evolution' Series.

Part III. C.:   Interlude.  The F.E.D. Psychohistorical-Dialectical 'Meta-Equation' of Human-Social Formation(s) 'Meta-Evolution' Series.

Modeling the Systematic Dialectic of ‘‘‘The Basic Operations of Arithmetic’’’ using _CQ_ -- 

Systematically Presented via a 5-Symbol Expression.

Dear Reader,

Below is the third and final of the three simpler examples of dialectical models, presented as an interlude, before concluding the series on the F.E.D. Psychohistorical-Dialectical 'Meta-Equation' of Human-Social Formation(s) 'Meta-Evolution', with the problematics of the nation-state social formation, and with its meta-model-predicted successor-formations.

Enjoy!


Regards,

Miguel

Introduction.  This model is more “Complex” [pun intended] than the models of “TV-Series”, and of ‘Modern Computerware’, presented earlier in this sub-series, because it requires some “domain-expertise” -- or, at least, some “domain familiarity” -- with respect to the domain of the so-called “Complex Numbers”, the set standardly denoted by the symbol C.  

The ‘axioms-system’ of the arithmetic of the C numbers, which we denote by C, is the 6th system of arithmetic in the following standard order of standard arithmetics, with our light-spectrum ordinal color-coding added --
  

N, W, Z, Q, R, C, 

-- for the “Natural”, “Whole”, “Integer”, “Rational”, “Real”, and “Complex” arithmetics, respectively.  About the ‘Goedelian Dialectic’ of these systems, see:  http://www.dialectics.org/dialectics/Vignettes.html, Vignette #4.

We will, in this blog-entry, use the Complex-Numbers-subsuming version of the F.E.D. ‘first dialectical algebra’ to construct, and to “solve”, a “heuristic”, ‘intuitional’ model of a systematic presentation of the domain of “the basic operations of arithmetic” -- encompassing both its ‘‘‘verse’’’ [e.g., addition, multiplication, exponentiation] and its “inverse” [e.g., subtraction, division, root-extraction] operations, jointly, via qs with C subscripts, which we also reference as _Cqs.

The models that we usually narrate here are constructed by interpreting the generic _NQ_ version of the F.E.D. ‘first dialectical algebra’ [ see E.D. Brief # 5 and its Preface ], or, at most-advanced, by interpreting the generic _WQ_ version of that algebra [ see E.D. Brief #6 and its Preface ], with the subscripts of the _Nq or _Wq ‘meta-numerals’ drawn from the number-space N = { 1, 2, 3, ... }, or from the number-space 

W = { 0, 1, 2, 3, ... }, respectively.

This time, the subscripts of the _Cqs will be drawn from the standard number-space 

C = { R + Ri }, 

wherein R denotes the space of the standard so-called “Real” numbers, and where the i unit stands for so-called "imaginary" unity, the positive square root of -1.  

FYI:  The generic Complex number is often expressed as a + bi, with a an element of R, and also with b an element of R, or as z  =  x + yi, with x an element of R, and also with y an element of R.

That is, we will be constructing our example model using the generic _CQ_ version of the F.E.D. ‘first dialectical algebra’. 

We use the _CQ_ language this time, as it allows us to present both ‘‘‘verse’’’ & ‘‘‘reverse’’’ operations in a single model.

Herein we mean, by the word, ‘‘‘systematic’’’ in the phrase ‘‘‘systematic presentation’’’, a presentation of the major kinds of “entities” that exist in this 'Human-Phenomic', 'Meme-etic' domain, the domain of the basic operations of arithmetic -- by means of categories that classify those entities by their “kinds”, i.e., as ‘‘‘[ideo-]ontology’’’, or as “kinds of [idea-]things, or "meme-things"” -- and in strict order of rising complexity, starting from the simplest category, and moving, step-by-step, from lesser to greater, i.e., more inclusive, complexity, until we reach the most complex/inclusive extant category of this domain, or for the purposes of this example.

The model that we build will describe these categories in that strict, systematic order of rising operational complexity / inclusivity.

This will be, once again, like the previous two “interlude” models, a “snapshot” model, a “synchronic” model that takes the contemporary slice of time -- or at any rate, a recent-past slice of time -- and algorithmically generates descriptions of categories for entities that presently exist, or that might possibly presently exist, for the model’s domain, in their systematic order of inclusivity, as described above.

Our model here will not be a “chronology” model, or “diachronic” model, like the previous, major model, narrated in this series, in which the units of earlier categories are described as actually, e.g., physically, constructing, through their activity as “causal agents”, i.e., as “subjects”, the units of later categories, categories whose units did not exist until that construction took place.  

That is, it will not be a model of a ‘self-advancing’ historical progression of ontology, with each historical epoch containing both old ontology, inherited from past historical epochs, plus new ontology, ontology that had never appeared before -- in past historical epochs -- until the later epoch in question, plus ‘hybrid categories’, combining / synthesizing the old with the new.

We will apply a documented, standard procedure to “solve” this ‘‘‘algebratric’’’ model -- to determine what actual category each of these generated category-descriptions refers to, and to determine which, if any, of these category-descriptions describe “empty categories”, i.e., represent ‘combinatorially’ possible entities that actually do not exist "within" this domain -- at least not presently.

To get started, we must determine the starting-point -- the point-of-departure -- for our systematic model. 

This starting category will be the seed of our whole progression of generated category-descriptions, influencing every category that follows, as the “controlling source”, and as the “ever-present origin”, of all that follows from it.

The rule for getting started is to ask oneself “¿What is the least complex kind of thing, the simplest kind of thing, the least inclusive kind of thing, which inheres in this domain?” -- in our case, in the domain of ‘basic arithmetic operations’ -- and to then find the answer to that question, based upon one’s prior knowledge of, or familiarity with, this domain.

The answer to this starting question that we will pursue in this example is the following:  The ‘‘‘verse’’’ operations of “Additions”, and its “inverse operations”, or ‘‘‘reverse operations’’’, or “Subtractions”, are the simplest ancestors, the ultimate units, of basic arithmetical operations, ingredient in every one of the more complex operations of that domain. 

A letter that the spelled names of these two kinds of operations have in common is “t”.

Therefore, we shall name/symbolize our starter category as _Ct, or as _Cq_t, denoting the “Complex” combination of the “Additions” sub-category of elementary Real arithmetic basic operations, with the sub-category of “Subtractions”, and identifying that combination of specific sub-categories with the generic first category symbol of our generic category-arithmetic model, namely, with the symbol --

_Cq_{[1 + 1i]}, 

in an “identification”, an “interpretation”, or an “assignment” [ ‘[---)’ ] that we indicate by writing:  

_Ct     =    _Cq_t    =   _Cq_{[A + Si]}    [---)  _Cq_{[1 + i]}.

Our model then, will take the form of an “interpeted”, specific equation, assigned to the generic equation, like this --

_C)-|-(_s   =   _Ct^{2^s}   =   ( _Cq_{[A + Si]} )^{2^s}    [---)   

_C|-|-|_h   =   _Ch^{2^h}   =   [ _Cq_{[1 + 1i]} ]^{2^h} 

-- with the variable s indicating the step in our systematic method of presentation that the ‘accumulation of categories’, denoted by _C)-|-(_s, represents.  

We will not, here, further recount the [Marxian] method of systematic discovery that was used to arrive at the starting category of this systematic presentation.  

For more regarding that method of discovery, see Marx, Grundrisse, Penguin Books [London:  1972], pp. 100-101. 

Stage 0.  Our initial step -- step s = 0 -- contains only our starting category, 

_Ct     =    _Cq_t    =   q_{[A + Si]} --

_C)-|-(₀   =   _Ct^{2^0}   =   _Ct¹   =   _Ct    =   q_{[A + Si]}    [---)  _Cq_{[1 + 1i]}   

-- because 2 “raised” to the power 0 -- 2⁰ -- is just 1, 

and because _Ct “raised” to the power 1 is just _Ct.

Stage 1.  It is when we get to the next step after step s = 0, namely, to step s = 1, that our equation-model gives us back something initially “unknown” -- and, therefore, something ‘‘‘algebraical’’’, not merely something ‘‘‘arithmetical’’’:  something to “solve-for” --

)-|-(₁   =   _Ct^{2^1}   =   _Ct²   =   _Ct  x  _Ct    =   _Cq_{[A + Si]} x _Cq_{[A + Si]}   =  

_Cq_{[A + Si]}    +  _Cq_{[AA + SSi]}   =   _Cq_t  +  _Cq_tt

-- because 2 “raised” to the power 1 -- 2¹ -- is just 2, and because our rule for multiplying a generic category, call it  _Cq_{[X + Yi]}   =   _Cq_Z =  _CZ, “by”, or “into”, itself, is, for subscripts X and Y denoting sub-category symbols, and for subscript Z denoting a category-symbol, simply --  

_Cq_{[X + Yi]} x _Cq_{[X + Yi]}    =   _Cq_{[X + Yi]}  +  _Cq_{[XX + YYi]}   =   _CZ + _Cq_ZZ

-- and for x and y denoting “Real” numbers --

_Cq_{[x + yi]}  x  _Cq_{[x + yi]}     =   _Cq_{[x + yi]}   +  _Cq_{[(x + x) + (y + y)i]}     =    

_Cq_{[1x + 1yi]}   +  _Cq_{[2x + 2yi]}.

Note again:  Herein, _Cq denotes the generic category ‘qualifier’ with “Complex” subscripts. 

The subscripts that come after it are specific category descriptors.

¿But how do we discover what the resulting, initially “unknown”, or ''algebraical'', ‘category-description’, here _Cq_tt, means?

Well, the generic rule to “solve-for” the categorial meaning of such symbols is that, if we know what is meant by category _Cq_Z  =  _CZ, then the symbol _Cq_ZZ describes a category each of whose units is a ‘_CZ OF _CZs’, that is, a category for a different kind of units, called ‘meta-_CZs’, each such unit being made up out of a multiplicity of those units of which the category of the _CZs is made up.

To be specific with this rule, in our example-model, _Cq_ZZ specifies a “Complex” of two sub-categories. 

Each of the units of the first sub-category, the sub-category of the ‘‘‘verse’’’ operations, must be an ‘Addition OF Additions’ that is, must be a ‘meta-Addition’, such that each ‘meta-Addition’ is made up out of a multiplicity of “mere” Additions.

Each of the units of the second sub-category, the sub-category of the ‘‘‘inverse’’’ operations, must be a ‘Subtraction OF Subtractions’, that is, must be a ‘meta-Subtraction’, such that each such ‘meta-Subtraction’ is made up out of a multiplicity of “mere” Subtractions.

Our experiences of / "in" the domain of 'the basic operations of arithmetic' suggest that such operations do “presently” exist in the domain of “Real” arithmetic.

“Multiplication” is a basic arithmetical operation that is “made up out of multiple [repeated] additions”, viz. -- 

4 x 5  =  5 + 5 + 5 + 5  =  4 + 4 + 4 + 4 + 4  =  5 x 4  =  20

-- a sum of four fives, or a sum of five fours:  either order will do [a characteristic called “commutativity of addition”]! 

In a partial reverse likeness, “division” is a basic arithmetical operation that is “made up out of multiple [repeated] subtractions”, viz., 5 "goes ["evenly", i.e., with 0 remainder] into 20" 4 times;  4 "goes ["evenly"] into 20" 5 times -- 

20 ÷ 5  =  4;  20 - 5 - 5 - 5 - 5  =  0  =  20 - 4 - 4 - 4 - 4 - 4; 20 ÷ 4  =  5 

-- to see how many fours there are in twenty [not the same as how many twenties there are in four]; how many “times” four “goes ["evenly"] in to” twenty, or to see how many fives there are in twenty, [not the same as how many twenties there are in five]; how many “times” five “goes ["evenly"] in to” twenty:  but, in this case, either order will not do!

A letter that the spelled names of these two kinds of operations have in common is “n”.

Therefore, we shall name/symbolize our second category as _Cn, or as _Cq_n, denoting the “Complex” combination of the “muLtiplications” sub-category of elementary Real arithmetic basic operations, with the sub-category of “diVisions”, and identifying that combination of specific sub-categories with the generic second category symbol of our generic category-arithmetic model, namely, with the generic category-symbol _Cq_{[2 + 2i]}.

We may “assert” our solution as follows: 

_Cq_tt  =  _Cq_n  =  _Cn =  _Cq_{[L + Vi]}   =  _Cq_{[AA + SSi]}  [---) _Cq_{[2 + 2i]}.

Again, what is dialectical about the relationship between _Ct and  _Ct², or  _Ct x _Ct, or _Ct of _Ct, or _Ct(_Ct), the relationship of what we call ‘meta-unit-ization’, or  ‘meta-«monad»-ization’, between _Ct and its already presently existing, ‘supplementary other’, _Cn, is that this relationship is a synchronic double-«aufheben» relationship. 

That is, each single “unit” of the “muLtiplications” sub-category of category _Cn, i.e., each typical individual “multiplication” operation, is a negation, and also a preservation, by way of also being an elevation to the / forming the “higher” / more inclusive “muLtiplications” sub-category / level / scale, of a whole [sub-]group of units of the “Additions” sub-category / level / scale of the _Ct category.

Likewise, each single “unit” of the “diVisions” sub-category of category _Cn, i.e., each typical individual “division” operation, is a negation, and also a preservation, by way of also being an elevation to the / forming the “higher” / more inclusive “diVisions” sub-category / level / scale, of a whole [sub-]group of units of the “Subtractions” sub-category / level / scale of the _Ct category.

So, our full solution to the step s = 1 equation of our model is --

_C)-|-(₁  =  _Ct   + _Cn   =  Additions & Subtractions  + MuLtiplications & DiVisions  

[---)  _Cq_{[1 + 1i]}   +  _Cq_{[2 + 2i]}.

If this model is working right, Additions & Subtractions is the simplest category of the domain of ‘basic arithmetical operations’; MuLtiplications & DiVisions is the next more complex category of that domain.

Stage 2.  ¿What additional ‘category-specifications’ do we generate in our next step, step s = 2, that need “solving-for”? 

Let’s find out:

_C)-|-(₂   =  _Ct^{2^2}  =  _Ct⁴  =  ( _Ct² )²  =  ( _Ct   + _Cn )²  =  

( _Ct   + _Cn ) x ( _Ct   + _Cn )   =  

_Ct   + _Cn   +  _Cq_nt  +  _Cq_nn.

This result arises by way of two key rules of categorial algebra, plus the general rule for multiplication when one category-symbol is multiplied by a different category-symbol [we used a special case of this general rule, for the case where the same category-symbol is multiplied by itself, in step s = 1, above] --

1.  general case:  _Cq_Y x _Cq_X   =   _Cq_X  +  _Cq_YX   =   _CX  +  _Cq_YX;

    special case:    _Cq_X x _Cq_X   =   _Cq_X  +  _Cq_XX   =   _CX  +  _Cq_XX.

2.  _Cq_x + _Cq_x   =   _Cq_x; the same category-symbol, added to itself, does not make “two” of that category-symbol; one “copy” of each category is sufficient; two or more copies of any category would be redundant, for the purposes of this dialectical-categorial algebra.

3.  There is no _Cq_w such that _Cq_x + _Cq_y    =   _Cq_w; different category-symbols, added together [as opposed to being ‘‘‘multiplied’’’], do not reduce to a single category-symbol, just like in the proverbial case of ‘apples + oranges’, or a + o.

Well, we already know how to “solve-for” _Cq_nn. 

It describes a category “containing” two sub-categories, the first sub-category being one of ‘muLtiplications OF muLtiplications’, and the second sub-category being one of ‘diVisions OF diVisions’.

The first sub-category is one each of whose units / operations is a ‘muLtiplication OF muLtiplications’, i.e., each of which is a ‘meta-muLtiplication’, such that each such ‘meta-muLtiplication’ operation is made up out of a multiplicity of  muLtiplication operations. 

Our experiences of / "in" the ‘basic arithmetical operations’ domain suggest that such arithmetical operations do indeed presently exist.

That sub-category-description describes the sub-category of ‘multi-muLtiplication’ operations -- i.e., of “exPonentiations”:   “exPonentiation” is a basic arithmetical operation which is “made up out of multiple [repeated] muLtiplication operations, viz. -- 

2³  =  2 xx 3  =  2 x 2 x 2  =   8    ≠    9   =    3 x 3  =  3 xx 3  =  3².

I.e., “two cubed”, or “two raised to the exponent three”, is “made of” a product involving three twos, that yields eight, whereas “three squared”, or “three raised to the exponent two”, yields nine:  in general, the order of “base” and “exponent” cannot be reversed without changing the result as well.  

Generally, each order will return a different result. 

The second sub-category should be, per our standard method, one each of whose units / operations is a ‘diVision OF diVisions’, i.e., each of which is a ‘meta-diVision’, such that each such ‘meta-diVision’ operation is made up out of a multiplicity of  diVision operations. 

That is, the second sub-category should be one of ‘multi-diVision’ operations, “made up out of multiple [repeated] diVision operations.  

We interpret this to be the “inverse” operation of ‘de-exPonentiation’, or of “nth Root extraction”.  

The “log” operation, which returns exponents, not bases or roots, is also a candidate for this “inverse” operation, but is not as fully this inverse operation as is the Root extraction operation. 

Given our experience of / "in" the domain of ‘basic arithmetical operations’, this sub-category description may, at this point, give us pause.

Many of us may be unfamiliar with the algorithms by which the “nth” root(s) of a given number are “extracted”.  

In what sense, if any, can an exponentiation be reversed, the “root” “extracted” from its “power”, by repeated division?

But let us consider the method of extracting square roots that is perhaps the oldest such method still known.  

It is called “The Babylonian Method”, and also “Heron’s Method”, because the storied Heron of Ancient Alexandria is the most ancient source known to have written an explicit account of this method.  

This method is, by the way, a special case of the more general “Newton’s Method”, but predates the discovery of “Newton’s Method” by many centuries.  

The method involves guessing a “starting estimate” for the square root sought, followed by repeated stages of, well, diVision -- division of the square by the current best estimate of its square root -- followed by, well, diVision again -- this time division of the sum of the previous consecutive pair of estimates by two, thus averaging them -- to obtain the next better estimate of the square root, all leading to an improving estimate for the square root with each iteration of the ‘double diVision’ just described. 

‘Formulaically’, the next better estimate of the square’s square root, x_n+1, is derived from the previous best estimate, x_n, by dividing the square, S, by the previous best estimate, x_n, summing x_n and S÷x_n, then dividing that sum by 2:   

x_n+1    =    ( x_n  +  S÷x_n ) ÷ 2.

Let us apply this method to “extracting” the “square root” from the “square”, 9, with “starting estimate” of x₁ = 2:

n....Current Best Estimate ( x_n )...DiVide Square by That Estimate ( S÷x_n )..DiVide Their Sum by 2 for new est.

1....2............................................9÷2 = 4.5....................................................(2+4.5)÷2 = 3.25

2....3.25.....................................9÷3.25 ≈ 2.769........................................(3.25+2.769)÷2 ≈ 3.01

3....3.01.....................................9÷3.01 ≈ 2.99...........................................(3.01+2.99)÷2 ≈ 3.000

4....3.000..................................9÷3.000 = 3.000......................................(3.000+3.000)÷2 = 3.000

After n = 3, with "rounding" as shown above, the method reaches a “fixed point” / “equilibrium” at x_3+... =  3, which is the positive square root of 9. 

Thus we see in what sense, in this method at least, square root extraction is made up out of repeated diVisions.

We may thus “assert” our solution as follows: 

_Cq_nn  =  _Cq_e  =  _Ce  =  _Cq_{[P + Ri]}   =  _Cq_{[LL + VVi]}   [---)  _Cq_{[4 + 4i]}.

Our step s = 2 equation-model, as we have solved it so far, thus now looks like this --

_C)-|-(₂   =  _Ct^{2^2}  =  _Ct⁴   =   _Ct   + _Cn   +  _Cq_nt  + _Ce   

[---)   _Cq_{[1 + 1i]}   +  _Cq_{[2 + 2i]}   +  _Cq_{[3 + 3i]}  +  _Cq_{[4 + 4i]}

-- since we have not yet determined which actual category of the ‘basic arithmetic operations’ domain is described by the algorithmically-generated symbol _Cq_nt -- if any, i.e., if _Cq_nt is not an “empty category”, “inoperative” for this domain.

When, as a component of ( _Ct   + _Cn ) x ( _Ct   + _Cn ), the “higher-complexity” category, _Cn, operates upon / “multiplies” the “lower-complexity” category, _Ct --

_Cn  x  _Ct      =    _Ct  +  _Cq_nt      =       _Cq_{[A + Si]}  + _Cq_{[LA + VSi]}

-- generically speaking, the categorial relationship to be called to the user’s attention by this operation, in this ‘categorial arithmetic’, is, again, a synchronic «aufheben» relationship, this time, that between _Ct and _Cq_nt. 

It calls the user to search that user’s knowledge and memory of the domain in question -- in this specific case, the domain of ‘basic arithmetical operations’ -- for a category which represents an “uplift” of category _Ct entities to the level of the entities native to category _Cn, thereby “canceling” the _Ct-type entities concerned, at their own native level, but, by the same token, “preserving” those category _Ct entities at the _Cn level, combining _Cn and _Ct qualities, in the relationship of “elevation” of those category _Ct entities within the level typical of category _Cn entities.  

Thus, the additional category thereby presented, _Cq_nt, signifies a category whose units are the operational interactions of the _Ct operations with the _Cn operations, as codified in the axioms, and/or theorems, and/or corollaries, and/or lemmas, and/or ‘‘‘rules’’’ of the “Real Numbers” system of arithmetic.

The first sub-category of the category _Cq_nt  = _Cq_{[LA + VSi]} answers to a sub-category description which connotes the way in which, or the ‘‘‘rules’’’ by which, the operation of muLtiplication “subsumes” the operation of Addition, denoted herein by ‘L | A’.

To our lights, this sub-category-description connotes the elementary arithmetical phenomenon often named '''Distribution''', or the “Distributive law”, e.g., of “Real” arithmetic, an axiom of that system of arithmetic, which codifies the interaction of the addition operation with the multiplication operation -- the rule that the multiplication operation “distributes over” the addition operation.  This “law” involves two components, often called “left distributivity” and “right distributivity”, respectively:

·       [“left distributivity”]:    For all elements a, b, c of R, 

c x (a + b)   =   (c x a) + (c x b).

·       [“right distributivity”]:  For all elements a, b, c of R, 

(a + b) x c  =   (a x c) + (b x c).

The second sub-category of the category _Cq_nt  = _Cq_{[LA + VSi]} answers to a sub-category description which connotes the ‘‘‘rules’’’ by which the operation of diVision “subsumes” the operation of Subtraction, denoted ‘V | S’.

To our lights, this sub-category-description connotes a “non-distributive rule” of “Real” arithmetic for ‘diVision / Subtraction’, although this rule is, typically, not an explicit one in presentations and in axiomatizations of “Real” arithmetic.  It is learned informally, as a joint consequence of other rules, i.e., as [partly] already subsumed under, or included in, the “distributive law”, or is encountered as a theorem, corollary, or lemma.  First of all, note that diVision does not fully “distribute” over [ ‘|’ ] Addition:

·       [‘‘‘left non-distributivity’’’, ‘V | A’]:  For some a, b, c of R, (a + b) ≠ 0, 


c ÷ (a + b) ≠ (c ÷ a) + (c ÷ b).

·       [‘‘‘right distributivity’’’, ‘V | A’]:       For all a, b, c of R, c ≠ 0, 


(a + b) ÷ c   =   (a ÷ c) + (b ÷ c).

The (a + b) ≠ 0 and c ≠ 0 proviso’s are necessary, in these assertions about ‘V | A’, because diVisions by zero invoke a value that resides beyond the ‘‘‘number-space’’’of the set R.

But the second sub-category of _Cq_nt  = _Cq_{[LA + VSi]} pertains directly to the interaction of the diVision operation with the Subtraction operation, not with the Addition operation [although, given that the set R includes “signed numbers” with “negative” signs, i.e., “additive inverses”, as well as the subtraction operation-sign, additions can also express subtractions, i.e., if b  =  -d, then a + b   =   a - d], e.g.:

·       [‘‘‘left non-distributivity’’’, ‘V | S’]:  For some a, b, c of R,

     (a - b) ≠ 0, c ÷ (a - b)   ≠   (c ÷ a) - (c ÷ b), 

e.g., 3 ÷ (1 - 2 )  =  -3    ≠   1.5  =  ( 3÷1) - (3÷2).
  

·       [‘‘‘right distributivity’’’, ‘V | S’]:       For all a, b, c of R, c ≠ 0, 

     (a - b) ÷ c   =   (a ÷ c) - (b ÷ c).

If we re-express subtractions as additions, and divisions as multiplications -- which the “Real” number arithmetic enables us to do, since it includes ratios and “multiplicative inverses”, as well as “additive inverses”, we see that the “Real” arithmetic’s '''rules''' for ‘V | S’ are partly implicit in the ‘L | A’ '''rules''', e.g., if we set b  =  -d and c  =  1÷e  =  1/e, e ≠ 0:

·       [an aspect of ‘L | A’]:  For a, -d, 1/e of R, (a + d) ≠ 0, 

1/e x 1/(a + d)   =   1/((e x a) + (e x b)),

e.g., 1/(1/3) x 1/(1 + -2 )    =   -3   =   1/( 1/3 x 1)  + ( 1/3 x -2) ). 
 

·       [‘‘‘right distributivity’’’, ‘L | A’]:  For all a, -d, 1/e of R, e ≠ 0, 

(a + d) x 1/e  =  (a x 1/e) + (d x 1/e),

e.g., (1 + -2) x (1/(1/3))   =   -3   =   ( 1 x (1/(1/3)) )   +   ( -2 x (1/(1/3)) ).

The subscript ‘_VSi’ component of the subscript ‘_{[LA + VSi]}’ of category-symbol --

 _Cq_{[LA + VSi]}  = _Cq_nt 

-- can thus be interpreted as calling attention systematically and explicitly, if somewhat redundantly, to the specific ‘V | S’ rules, which differ from the generic ‘L | A’ rules, in that the ‘V | S’ rules require the making explicit of special restrictions [e.g, 0 denominators not allowed], etc., as we have seen above.


A better interpretation of / solution for the meaning of the  _Cq_[VSi]  sub-category of the  _Cq_{[LA + VSi]}  category would be the [sub-]category of/for '''DiVided Differences''', which can form a portal to the differential calculus, involving the Leibnizian 'infinitesimal difference' operator, d -- and, which did, in part, [psycho]historically, actually serve as such, for Isaac Newton, in his pathway to that discovery:


(Y - y) / (X - x)  =  

((y + delta(y)) - y) / ( (x + delta(x)) - x )  =  

( (f(x + delta(x)) - f(x) ) / ( (delta(x) )  =

delta( f(x) ) / delta(x), such that --


d( f(x) ) / dx  =  

limit as delta(x) --> 0(delta( f(x) ) / delta(x).

We may therefore write our full solution for step s = 2 as --

_C)-|-(₂      =    _Ct^{2^2}     =     _Ct⁴      =      

_Ct   + _Cn   +  _Cq_nt  + _Ce      

additions & subtractions   +  

multiplications & divisions   +  

n & t interactions   +    

exponentiations & de-exponentiations.

Our categorial progression so far can be summarized textually as below.

The ‘qualo-fractal’ content-structure of this psychohistorical dialectic to step 2 can be summarized as follows --

exponentiations & de-exponentiations “contain” multiplications & divisions,

which, in turn, “contain” additions & subtractions.

The “five symbolic-elements expression” for this model is thus _Ct^{2^2} [five if we count the underscore under the t as a separate “symbolic-element”].

The systematic dialectic of the basic operations of arithmetic ‘‘‘presently’’’ and “standardly” ends here, at step s = 2, with the category named exponentiations & de-exponentiations as its ‘meta-meristemal’ category, or ‘‘‘vanguard’’’ category.

We like the compactness of the _CQ_ representation of this systematic dialectic, whose “final step” can be modeled via a single equation --

_C)-|-(₂      =      _Ct   + _Cn   +  _Cq_nt  + _Ce.

However, there is also the alternative of expressing each step of this dialectic by two separate equations, using the _NQ_ dialectical algebra instead, e.g. --

_verse)-|-(₂      =      _CA   + _CL   +  _Cq_LA  + _CP, for the ‘‘‘verse’’’ operations;

_inverse)-|-(₂     =      _CS   + _CV   +  _Cq_VS  + _CR, for the ‘‘‘inverse’’’, or ‘‘‘reverse’’’, 

operations.



To isolate the first triads of categories from these paired dialectical categorial-combinatoric progressions, we can use twin step s = 1 Triadic Seldon Function '[meta-]model [meta-]equations' as follows, using the modified notation also employed in the two images below --

_verse)-|-(₁   =    _O+)-|-(₁   =   _O+A^{3^1}   =    O+A³   =



_O+A   + _O+M   +  _O+q_MA     =     _O+A   + _O+M   +  _O+D, 


for ‘‘‘verse’’’ operations;

_inverse)-|-(₁     =    _O-)-|-(₁   =   _O-S^{3^1}   =    _O-S³    =  



_O-S   + _O-D   +  _O-q_DS     =     _O-S   + _O-D   +  _O-V,  


for ‘‘‘inverse’’’/‘‘‘reverse’’’, operations

-- which can then be depicted as follows --

Stage 3.  To iterate our _CQ_ ‘meta-equation’, 

_C)-|-(_s   =   _Ct^{2^s}   =   ( _Cq_{[A + Si]} )^{2^s}, 

for step s = 3, is to iterate the systematic presentation of the domain of basic arithmetical operations beyond the “basic”, beyond the “present”, beyond the conventional conclusion of that presentation, and beyond the “systematic reconstruction” of this domain at present, to a somewhat “preconstructive” -- somewhat “predictive” -- extrapolation of its possible future.  However, as we shall see, we have already encountered units of the “vanguard” term of step s = 3, in this very text. 

Let’s see what are the additional category-descriptions that this step s = 3 ‘self-iteration’ generates:

_C)-|-(₃   =  _Ct^{2^3}  =  _Ct⁸  =  ( _Ct⁴ )²  =  

(_Ct   + _Cn   +  _Cq_nt  + _Ce )²   = 

(_Ct   + _Cn   +  _Cq_nt  + _Ce ) x (_Ct   + _Cn   +  _Cq_nt  + _Ce )   =  

_Ct   + _Cn  + _Cq_nt  +  _Ce   +  _Cq_et  +  _Cq_en  +  _Cq_ent  +  _Cq_ee     

[---)   

_Cq_{[1 + 1i]}   +  _Cq_{[2 + 2i]}   +  _Cq_{[3 + 3i]}  +  _Cq_{[4 + 4i]}  +  

_Cq_{[5 + 5i]}   +  _Cq_{[6 + 6i]}   +  _Cq_{[7 + 7i]}  +  _Cq_{[8 + 8i]}.   

We know -- from past experience, narrated above -- how to “solve-for”category 

_Cq_ee   =  _Cq_{[PP + RRi]}. 

It describes a category “containing” two sub-categories, the first sub-category being one of ‘Powers OF Powers’, and the second sub-category being one of ‘Root-extractions OF Root-extractions’.

The first sub-category is one each of whose units / operations is an ‘exPonentiation OF exPonentiations’, i.e., each of which is a ‘meta-exPonentiation’, such that each such ‘meta-exPonentiation’ operation is made up out of a multiplicity of  exPonentiation operations. 

But that is precisely the new operation that we have encountered in this text, at the heart of the Seldon Functions in general, and at the heart of our _CQ_ ‘meta-equation’ -- 

_C)-|-(_s   =   _Ct^{2^s}

-- specifically.

A unit increment in the ‘meta-exponent’ of the ‘starting-category’ symbol of that ‘meta-equation’, corresponding to a unit increment in its step-value, s, is equivalent to a two-fold exponentiation of that ‘starting-category’ symbol. e.g. -- 

_Ct² = _Ct^{2^1};   

(_Ct² )²  =  (_Ct^{2^1} )²   =   (_Ct^{2^1} )^{2^1} = _Ct^{2^(1+1)}  =  _Ct^{2^2}

-- because repeated exponents mutually multiply, and because 'meta-exponents' of exponents add together when those 'meta-exponents' have the same exponents as their bases, and are multiplied together.  

Let’s call this sub-category Hyper-exponentiation, or H for short.  

The second sub-category should be for operations which are ‘de-exponentiations OF de-exponentiations’, i.e., which are ‘meta-de-exponentiations’, such that each ‘meta-de-exponentiation’ operation is made up out of a multiplicity of  de-exponentiation operations.

That is, the second sub-category should be one of ‘multi-de-exponentiation’ operations, “made up out of multiple [repeated] de-exponentiation operations, each denoted by '√', the sign of the "square-root"-extracting operation.  

We can use the ‘self-example’, exemplified in this very text, to illustrate this process: 

√√√_Ct  + _Cn  +_Cq_nt + _Ce  + _Cq_et + _Cq_en + _Cq_ent + _Cq_ee      = 

√√√_Ct^{2^3}    = 

√√_Ct   + _Cn  + _Cq_nt  +  _Ce    =

√√_Ct^{2^2}    =

       _____________

√ _Ct   + _Cn            = 

√_Ct^{2^1}    =     

_Ct^{2^(3-3)}  =  

_Ct^{2^0}  =  

_Ct.   

Let’s call this sub-category De-Hyper-exponentiation, or D for short.
 

We may thus “assert” our solution as follows: 

_Cq_ee  =  _Cq_m  =  _Cm   =  _Cq_{[H + Di]}   [---)  _Cq_{[8 + 8i]}.

Our step s = 3 equation-model, as we have solved it so far, thus now looks like this --

_C)-|-(₃   =   _Ct^{2^3}  =  _Ct⁸   =  

_Ct   + _Cn  + _Cq_nt  + _Ce   + _Cq_et  + _Cq_en  + _Cq_ent  +  _Cm 

-- since we have not yet determined which actual categories of the ‘basic arithmetical operations’ domain are described by the algorithmically-generated ‘category-description’ symbols _Cq_et, _Cq_en, and _Cq_ent, if any.

But we already know how to characterize the possible categories that these three category-symbols “call for”, viz.:

·       _Cq_et   [---)  _Cq_{[5 + 5i]} “calls for” a '''hybrid''' category for the kind of ‘meta-operation’, or ‘operation of operations’, that combines the e and t operations.

·       _Cq_en  [---)  _Cq_{[6 + 6i]} “calls for” a '''hybrid''' category for the kind of ‘meta-operation’, or ‘operation of operations’, that combines the e and n operations.

·       _Cq_ent [---)  _Cq_{[7 + 7i]} “calls for” a '''hybrid''' category for the kind of ‘meta-operation’, or ‘operation of operations’, that combines the e and the _Cq_nt.

We may thus write our full solution for step s = 3 as --

_C)-|-(₃   =  _Ct^{2^3}  =  

_Ct⁸   =   

_Ct   + _Cn  + _Cq_nt  + _Ce   + _Cq_et  + _Cq_en  + _Cq_ent  +  _Cm   =   

additions & subtractions        +      

multiplications & divisions       + 
                                             
n with t interactions        +

                                            
exponentiations & de-exponentiations      +   
                                             
e with t interactions      +     

e with n interactions      +  

                                             
e with n & t interactions       + 

                                             
meta-exponentiations & de-meta-exponentiations.

Our categorial progression so far can be summarized textually as below.  



The ‘qualo-fractal’ content-structure of this psychohistorical dialectic through step 3 can be summarized as follows --

meta-exponentiations & de-meta-exponentiations “contain”

exponentiations & de-exponentiations, which “contain”

multiplications & divisions, which “contain”

additions & subtractions.

The “five symbolic-elements expression” for this model, up to this step, is thus _Ct^{2^3}.

The meaning mnemonically compressed into the 5 symbolic-element expression _Ct^{2^3} can be depicted as follows --