F.E.D. Dialectics

Wednesday, November 11, 2026

The Dialectic of the Kinds of ORDER.

The

Dialectic

of the

Kinds

ORDER.

GLOBAL STRATEGIC HYPOTHESES.

Dear Reader,

There are two kinds of temporal orderedness with which most of us are familiar –

(1) Changelessness, e.g., “eternal” stasis, of the kind which Platon and Parmenides thought to be the immutable reality behind the “delusory” sensuous appearances of things, and;

(2) Cyclical, circular, periodic, “ever”-repeating change.

In the state-space geometrical metaphor of dynamical systems theory, for mathematical models of “dynamics” expressed in the language of “ordinary differential” – i.e., of total-differential – equations, kind (1) is expressed by fixed points in the state-space of, e.g., a linear differential equation, the only kind that can have a single fixed point as its solution, its, motionless, state-space [non-]trajectory.

Dynamical systems can also have “neutrally-stable” state-space trajectories, which neither attract nor repel nearby state-points in their “basin”. This provides a state-space metaphor for kind (2), the purely-cyclical, ever-repeating kind of change.

Only nonlinear total-differential equations can exhibit limit-cycle attractors as their state-space ‘solution-geometry’. The “limit” in their name, “limit-cycles”, alludes to the fact that they never become truly “cyclical”, and purely “repetitive”, “until” t = +¥, a “time” that never arrives.

For all finite – actual – time, the state-space trajectories of limit-cycle attractor solution-trajectories approach the “cycle” inside them ever-closer, “asymptotically”, but never actually touch that limit-cycle closed curves in their state-spaces.

This means that, nonlinear, limit-cycle solutions represent a first step toward the third kind of order, which many people have never considered as a kind of orderliness: non-repetitive; even never-repeating order. Many people see such order as dis-order.

More deeply nonlinear equations have a kind of attractor, called, wouldn’t you know it, a “strange” attractor, that is not asymptotically periodic, as is a limit-cycle attractor. “Strange attractors” are asymptotically aperiodic; non-periodic. They exhibit what is called “non-periodic deterministic flow”.

Such nonlinear dynamical system-models are completely causal – completely deterministic. Their entire future is determined by their initial, t = 0 state-point, if we abstract from all externally-sourced so-called “perturbations”.

But they are popularly called “chaotic” attractors, with “chaotic” trajectories, because, to calculate their future states in advance, one would have to know the coordinates of the initial state-point of each with infinite precision.

However, in our view, these never-repeating trajectories provide a mathematical metaphor for a third category of order, which is a dialectical synthesis of the category of (1) static [intrinsically-repetitive] order, and/with the category of (2) cyclical/strictly-repetitive order, by drawing ‘negatively’ upon both order-kind category (1) and order-kind category (2): that is, by being a not statical and a not cyclical kind of order – neither statical nor cyclical – i.e., a non-repeating kind of order.

Such a species of order, and such a name for that kind of order, may sound, at first, overwhelming exotic.

However, in actuality, we are surrounded and permeated, both physically and culturally – ‘human-phenomically’ – by cases of this kind of order.

The “Natural” Numbers –

e.g., 1, 1+1 = 2, 2+1 = 3, 3 +1 = 4, etc. –

may seem like the nearly simplest ‘ideo-ontological’ product of the human mind.

Yet the distribution of the “prime” natural numbers among the “composite natural numbers” exhibits a never-repeating, yet not random, pattern.

If the Riemann Hypothesis proves to be true, then the Riemann Zeta Function apparatus may have already captured their non-repeating pattern, the order of the primes. That function-apparatus is, have no doubt, a very complex mathematical expression to write down via our present mathematical languages.

But, already, via solution of Hilbert’s Tenth Problem, in March 1970, algebraic equations have been found that, with 26 or 24 or 21 or even as few as 10 algebraic variables, capture the positive prime numbers as their positive integer solutions. These finite, algebraic equations capture the order of the primes in, perhaps, a simpler, algebraic way, compared to the complexity of the Riemann function.

Moreover, consider the decimal representation of “Natural Numbers”, of the space named N, of “Rational Numbers”, of the space named Q, and of “Irrational Numbers”, of the space named R.

“Natural Numbers” can be mapped to “ever-repeating” 0s to the right of “their” decimal points, because “Natural Numbers” are non-fractional, e.g., 1 maps to –

1.0000000000000000000000… .

Their decimal representations are analogous to dynamical system state-space fixed point solution-geometries.

“Rational Numbers” exhibit “ever-repeating” sequences of digits to the right of their decimal points. For example, the old standard rational number approximation for the, “transcendental”, “Real”, “irrational” number pi is –

22/7 = 3.142857 142857 142857 142857 142857… .

They are analogous to dynamical system limit-cycle attractor state-space solution-geometries/trajectories, “at” t = +¥, and to “neutrally-stable”, purely-cyclical, purely-repetitive, closed-curves state-space trajectory solution-geometries.

But “irrational” numbers, like that “algebraic irrational”, the positive square root of 2, or that “golden” algebraic irrational, phi, or those “transcendental” irrational numbers, like pi and e, are analogous to “strange attractor”, so-called “chaotic”, NEVER-repeating state-space trajectory solution-geometries, in their decimal expressions/expansions, because their right-of-decimal-point digits-sequences never repeat.

Yet, change even a single digit, say the billionth digit, in the decimal expansions of the positive square root of 2, of pi, of phi, or of e, and you no longer have a representation of the positive square root of 2, of pi, of phi, or of e any more. Every digit of the decimal expansion of any “irrational” number is strictly “deterministic”, strictly determined. No other sequence of digits will do to be those numbers!

For more information regarding these Seldonian insights, and to read and/or download, free of charge, PDFs and/or JPGs of Foundation books, other texts, and images, please see:

www.dialectics.info

and

https://independent.academia.edu/KarlSeldon

For partially pictographical, ‘poster-ized’ visualizations of many of these Seldonian insights -- specimens of ‘dialectical art’ – as well as dialectically-illustrated books published by the F.E.D. Press, see –

https://www.etsy.com/shop/DialecticsMATH

¡ENJOY!

Regards,

Miguel Detonacciones,

Voting Member, Foundation Encyclopedia Dialectica [F.E.D.];

Elected Member, F.E.D. General Council;

Participant, F.E.D. Special Council for Public Liaison;

Officer, F.E.D. Office of Public Liaison.

YOU are invited to post your comments on this blog-entry below!

Sunday, November 08, 2026

“Hypercomplex” Numbers-Systems’ Dimensionalities and The Seldon Numbers-System’s Dimensionality.

“Hypercomplex”

Numbers-

Systems’

Dimensionalities

and

The

Seldon

Numbers-

System’s

Dimensionality.

GLOBAL STRATEGIC HYPOTHESES.

Dear Reader,

www.dialectics.info

and

https://independent.academia.edu/KarlSeldon

https://www.etsy.com/shop/DialecticsMATH

¡ENJOY!

Regards,

Miguel Detonacciones,

Voting Member, Foundation Encyclopedia Dialectica [F.E.D.];

Elected Member, F.E.D. General Council;

Participant, F.E.D. Special Council for Public Liaison;

Officer, F.E.D. Office of Public Liaison.

YOU are invited to post your comments on this blog-entry below!

Sunday, November 01, 2026

A KEY, NEGLECTED PASSAGE by Marx on the, critical, Profit-Rate Formula.

KEY,

NEGLECTED

PASSAGE

by Marx

on the,

critical,

Profit-Rate

Formula.

Dear Reader,

In volume three of Marx’s treatise Capital: A Critique of Political Economy, in its part three, “The Law of the Tendency of the Rate of Profit to Fall”, in its Chapter XIII, “THE LAW AS SUCH”, near the end of that chapter [p. 229 in the 1967 New World edition], resides a key paragraph whose opening sentence marks a major ‘de-abstraction’ – a major advance in the realism and ‘thought-concreteness’ – of Marx’s model of the rate of profit return on industrial capital invested. To our knowledge, this crucial passage has been completely neglected in the Marxian literature on Marx’s “The Law of the Tendency of the Rate of Profit to Fall” to-date.

That opening sentence – which opens up, for us, a new, trans-algebraic understanding of the mechanism of the “Law of the Tendency of the Rate of Profit to Fall” – is as follows:

“The rate of profit must be calculated by measuring the mass of produced and realized surplus-value not only in relation to the consumed portion of capital, reappearing in the commodities [M.D.: (c + v)], but also to this part plus that portion of unconsumed but applied capital which continues to operate in production [M.D.: i.e., “fixed capital” machinery/plant and equipment, that has not yet entered into the “circulations-process of capitals”, i.e., that has not yet been depreciated by “wear and tear”: f].”

[Italics emphasis per source; underscored bold italics emphasis added by M.D.].

Thus, Marx’s profit-rate formula, in this latest step towards “the surface of society”, in its deepening of the ‘thought-concreteness’ and realism; in its further ‘de-abstraction’/‘de-simplification’, in the flow of Marx’s subtle systematic-dialectical method of presentation, becomes –

s’_t / (f_t + c_t + v_t)

– wherein the terms above, for annual production “cycle”/whorl number t, are defined [‘=’] as follows –

s’_t = period t net surplus-value, net of interest payments, rent payments, taxes, tax payments, etc. per Marx’s law of value, this is the economic value-added/created by the living labor component of “productive capital”, represented in value-terns by v_t, for “variable capital”, i.e., the wages portion of productive capital investment;

f_t = fixed capital value unconsumed but applied during production period t;

c_t = constant capital value invested during annual production period t;

= (c_d_,_t + c_r_,_t + c_a_,_t);

v_t = variable capital value [wages] invested/applied during year t;

c_d_,_t = depreciation by “wear-and-tear” consumption of fixed capital

plant and equipment applied during annual production year period t;

c_r_,_t = raw materials capital-value invested/applied during period t;

c_a_,_t = auxiliary materials capital-value invested/applied during period t.

Note: Putting this profit-rate formula on the annual production period “cycle”/whorl [“whorl” because capitalist production is a non-repeating iteration; not a circle or cycle, in detail, but a ‘qualo-quantitative’, helical spiral, with both quantitative changes/-expansions and qualitative/ontic changes/expansions encompassed within each annual production period] basis averts problems of combining value-“stocks” with value-“flows”.

This further ‘de-abstracted’ version of the Marxian profit-rate formula, the formula – s’ / (f + c + v) – is a big step closer to the capitalist profit-rate formula, the formula that reflects capitalist consciousness of the profit rate, which is the “Return On Capital Investment”, or “ROI”, ratio, R/I. This ratio is, approximately, $R/$I » $s’/$f, or, using our generalized currency metrological qualifier, X, is Xs’/Xf. Per the, “fetishistic”, delusion of the capitalist mentality, this ratio excludes, and denies that, capital investment in wages, v, is a co-cause of the production of the net profit Returns on capital Investment, Xs’, treating both wages, v, and wear-and-tear on fixed capital plant and equipment, c_d, as mere expenses.

This ‘de-abstracted’ component of the Marxian profit-rate formula,

s’ / f,

will tend to converge to the dominant component of

s’ / (f + c + v)

if both f rises non-linearly, and f >> (c +v) as capitalist productive-force progresses.

All value components invested/applied in each year’s “productive capital” must count equally as portions of the capital invested to produce the year’s commodity-capital-value output.

Marx’s profit-rate ratio is a kind of “input-output ratio” or ‘capital-value amplification factor”.

It is, as s/(c + v), a gross Output over [‘/’] Input ratio: O/I.

As s’/(c + v), it is a net output over input ratio:

(O – I)/I.

It is based upon the causal relation between the input and the output. That is, the capital-value invested, now (f + c + v), measures, in value-terms, the cause of the net-profit effect, s’, measuring the effect relative to its cause – measuring the profits-producing power of that input/cause; its ‘profits-productivity’.

This value-cause includes the value of the fixed-capital machinery/-plant and equipment “applied” [Marx] in, to and during the time-period, t, of the value-effect’s production, i.e., including all of the not-yet-consumed, not-yet-“wear-and-tear”-depreciated portion of the value of that, applied, fixed-capital machinery/plant and equipment, because it too, f_t, is part of the cause of the s’_t effect.

Implicitly, the ‘de-abstractive’, ‘de-over-simplifying’ progression from –

s’_t / (c_t + v_t)

s’_t / (f_t + c_t + v_t)

– opens the way to consider the effect of what Marx called the ongoing “moral depreciation” of fixed capital – technological obsolescence depreciation of fixed capital, or ‘techno-depreciation’ for short – which is caused, not by the, “wear-and-tear”, partial consumption of that fixed capital in each production period, but by the growth of the social forces of production, enforced when that growth is embodied in productivity-improved plant and equipment/-machinery, and applied, in their production, by other, competing individual capitals.

To remain competitive/viable, the individual capitals still employing the ‘obsoleted’ fixed capital must partially or wholly scrap that obsoleted fixed capital – the loans by which that ‘obsoleted’ fixed capital was originally purchased usually not yet being fully paid-off at the time of their enforced obsolescence/-scrapping – and take out new loans to purchase the more technologically-advanced, fixed capital to replace the scrapped fixed capital, so as to render their individual industrial capitals once again competent/competitive.

This means that the capital-value loss on the ‘obsoleted’ fixed capital must be subtracted from both the numerator and the denominator of the, now further enhanced, profit-rate ratio, and the old annual sum of debt service [interest + principal] payments on the old and the new fixed-capital loans subtracted from the numerator of this, enhanced, net-profit-rate ratio.

That more-realistic, even-more ‘thought-concrete’ profit-rate ratio thus becomes –

(s’_t + Ds_t- DL_f_,t - f_o_,t - L_o_,t)/((f_t - f_o_,t + Df_t) + c_t + v_t)

– wherein the algebraic terms above are defined [‘=’] as –

Ds_t = incremental relative surplus-value via new/replacement fixed capital;

f_o_,_t = remaining, undepreciated value of period t obsoleted/retired fixed capital;

L_o_,_t = period t Loan debt-service still due on obsoleted fixed capital;

DL_f_,_t = period t new Loan debt-service due on new/replacement fixed capital;

Df_t = capital-value of new/replacement fixed capital applied during period t.

As you can see from the revised profit-rate/ROI formula defined above, the debt-service expense for replacement fixed capital, to replace the, e.g., scrapped – ‘techno-depreciated’ – fixed capital, an expense which is subtracted from period gross profits, is a key cause of the profit-rate/ROI fall. This cause of productive-force profit-rate fall increases its dominance as the accumulation of capital, and of social productive force, grows, as any self-financing of fixed capital purchases, including such replacement fixed capital purchases, becomes increasingly ‘im-practice-able’, as the cost of fixed capital units rises, e.g., as the ratio f/(s’ + c + v ) rises.

Moreover, the formula above still understates the impact of bank-loan replacement financing, and consequent debt-service expenses subtraction from gross profits. This is because the techno-depreciation of fixed capital assets is not a one-time event. Such events recur from annual production period to next annual production period, albeit not in every single annual production period, as industrial technological progress continues and even accelerates. Thus typical industrial-capital “going concerns” accumulate a growing sequence of layers of such replacement debt-service annual net profit reductions – growing in size with productive-force growth and with the growing scale of world-market industry – and that continue for many years, until the, e.g., 10-year, or 30-year or 50-year, etc. bank loans are paid-up. The typical fixed-capital-including profit-rate formula is thus –

(s’_t + Ds_t - DL_f_,_t - f_o_,_t - S_k(DL_o_,_t,_k))/ ((f_t - f_o_,_t + Df_t) + c_t + v_t).

Note that the profit-rate formulas defined above can “aggregate”, i.e., can “scale”, preserving “similarity”, ‘quanto-qualo-fractally’, not “infinitely”, but up from “individual capital”, to “branch of industry”, to “department of industry”, to nation-state “total social capital”, and, finally, to the, “world market”, “total social capital”.

Marx himself did not live to fully write-out these next steps, leaving the explanation for the profit-rate fall as if a merely “algebraic mechanism”, by the relative fall of v – relative to c, albeit both increasing in absolute terms – resulting in a relative fall of s, without locating precisely where the resulting relative dearth of s registers and manifests empirically. It manifests – the growth of the social forces of production manifests, the growth of ‘use-value productivity’ manifests – in losses and write-offs, subtracted from gross profits, to the capital-value of productivity-growth-‘obsoleted’ fixed capital, which thus loses some-to-all of its original purchase-cost value, and its use-value to capitalists, for their profit-making, by becoming uncompetitive, due to growth of productivity/-productive-force, elsewhere in the industry to which it belongs, as well as in general, throughout the capitalist, world-market economy.