Monday, June 24, 2013

Part 08 of 29. THE DIALECTICA MANIFESTO. Nonlinearity, Autokinesis, and Dialectic.

Full Title:  Part 08 of 29 --

The Dialectica Manifesto



Dialectical Ideography 




the Mission of F.E.D.


Dear Readers,

I am, together with F.E.D. Secretary-General Hermes de Nemores, and F.E.D. Public Liaison Officer Aoristos Dyosphainthos, organizing to develop a new, expanded edition of the F.E.D. introductory documents, for publication in book form, under a new title --

The Dialectica Manifesto:  Dialectical Ideography and the Mission of Foundation Encyclopedia Dialectica [F.E.D.]

-- and under the authorship of the entire Foundation collective.
Below is the eighth installment of a 29-part presentation of this introductory material, which the F.E.D. General Council has authorized for serialization via this blog over the coming months, as we develop the material.

I plan to inter-mix these installments with other blog-entries, including the planned additional F.E.D. Vignettes, other F.E.D.  news, my own blog-essays, etc.

Links to the earlier versions of these introductory documents are given below.

Unlike the typical blog-entry, this series will attempt to deliver an introduction to the Foundation worldview as a totality, in a connected account, making explicit many of the interconnexions among the parts.




Nonlinearity, «Auto-Kinesis», and Dialectic.

If nonlinearity is the root cause of this mathematical difficulty and “intractability”, and if nonlinearity is the root cause of this present “closed-form unsolvability”, then what doesnonlinearity signify?

In its deepest meaning, nonlinearity signifies what Plato called «autokinesis» [see below].

¡Differential-equation nonlinearity is the mathematical name for mathematical, ideographical, ‘equational’, ‘purely-quantitative’ modeling of self-motion, of self-motion, of self-[induced] movement’; of self-induced change-of-state; of self-reflexiveness [cf. Russell], that is, of self-reflexive, self-referential action; of self-refluxiveness and of self-refluxive action, that is, of action which flows back to and changes its source, its agent, its subject -- and, thus, of self-developing process, or of developmentally self-propelling, self-developing eventity!

These equations -- equations that the presently absent, presently-unformulated ‘‘‘closed-form’’’ function-formulae solve -- are called nonlinear because, in them, the unknowns [which, for integro-differential equations, are function-unknowns, not single number-values as are the solutions of algebraic, diophantine equations], the so far unfathomed solving-functions, appear, with their function-values, or with the values of their derivative-functions, and/or with the values of their integrals, operating upon themselves, and/or operating upon one another.

Conjecture:  Such signifies the dynamical, i.e., the time-like [indeed, the chronogenic], interaction and self-interaction of the underlying actualities that these unknown solving-functions must mime.

Linear integro-differential equations, the kind that have for so long been so easily solved by Terran mathematicians, are characterized, in contrast, by function-unknowns which occur in “isolation”, singly, independently, without interaction, operating upon / multiplying neither self nor any other unknown / to-be-solved-for function(s).

A typical nonlinear “ordinary” or “total” differential equation is an ideographical state-ment’, that asserts -- that “states”, in effect -- that the instantaneous velocity of evolution of the generic, pure-quantitative value of the state of the system modeled by that equation for any, generic time-value, t — the state represented by x(t), thus denoting the generic state-function-value of the dynamical function-unknown to be solved for — is proportional to a higher power of that unknown, generic state-value itself, denoted x(t)n, n > 1, i.e., to a multiplicative self-application / self-operation / self-flexion or self-re-flexion; to a self-multiplication, of that state-value.

Such a purely-quantitativeself-multiplication signifies:  either (1) a [purely-quantitative’]self-magnification, or; (2) a [purely-quantitative’] self-diminution’, of one or more of the values of the state-variable(s) of x(t), for every [‘non-Boolean’] value of the state-variable-value components of that state-“vector” function, x(t)  =  x1(t) + . . . + xm(t)

That state-“vector” function is, in effect, a ‘“list”’, or, more precisely, a “non-amalgamative sum”, of m “state-variables”, x1(t) through xm(t), each one reflecting a model-predicted value of a different prime “vital sign” measurement of the system modeled by that state-“vector” function, x(t), as of a given value of t.  Let us herein call that system as a whole by the name X.

Such an equation describes a system’s “evolution” as an at least partially self-induced, or autokinesic’, self-driven, self-propelling motion in a state-space, or ‘space of states.

“State Space” an imagined space, or conceptually-constructed space, in which every point denotes a different possible state of that system, as a ‘“list”’ of that system’s vital signs.

For example, nonlinear differential equation models of predator-prey population-count dynamics, or bio-mass dynamics, within an ecological system, often contain a population-size, or bio-mass, self-limiting Verhulst term -- a ‘“self-interaction term”’. 

This term involves adding in, e.g., a self-multiplication of Ni(t), where Ni(t) denotes the population-count-as-state of the ith species as a function of time, t, such that a minus sign is applied to that self-product[ion], thus registering as a negative contribution to the momentary, “instantaneous” rate of growth of that species’ population-count — where that rate of growth is here defined as the metric for the rate of evolution, or velocity of evolution, of that species’ statewith respect to time, as the [“continuous”] time ‘‘‘count’’’ advances

The nonlinear correction term is referred to as a self-interaction . . . term [which term we also term self[-re]-flexive [‘‘‘bent back upon itself’’’] or self[-re]-fluxive[‘‘‘flowing [from self] back to self’’’] —  F.E.D.], of the form Ni(t)2 . . . where the terms of the form Ni(t)Nj(t), i ~= j, also quadratically nonlinear, are referred to as mutual interaction terms [which terms we also term hetero-flexive’ or ‘allo-flexive’, meaning ‘‘‘other-bent’’’, or ‘‘‘bent by other-than-self’’’F.E.D.].
[R. Dutt, P. K. Ghosh, “Nonlinear Correction to the Lotka-Volterra Oscillation in a Predator-Prey System”, in Mathematical Biosciences [27: 1975], pp. 9-16].

The equations of Einstein’s mathematical model of the ‘“universal gravitational field”’, that is, the equations of his General Theory of Relativity, are nonlinear  precisely because they must model the non-a-tom-istic, self-reflexive, auto-kinesic, self-changing self-interactivity of the cosmos-encompassing gravitic ‘‘‘field’’’ -- interaction is non-linear if the total force exerted by several bodies is not the sum of the forces each would exert if acting alone.”

Why is the gravitational-inertial interaction non-linear?”

“The reason is a fundamental one.”

“We saw at the end of the preceding chapter that all forms of energy have mass and so act as a source of gravitation and inertia.”

“This is true, not only of matter and of light, but also of gravitational potential energy.”

“We know that this form of energy has a real physical significance; it has to be included in a total energy balance . . ..”

This means that when two bodies act together as a source, in addition to their individual masses we must take their mutual gravitational potential energy as a source.”

“The total force is then not the sum of the individual forces.”

“... It follows that the exact interaction between [better, amongF.E.D.] an arbitrary number of bodies is going to have a complicated form.”

“Indeed, as we shall see, it has not been possible to formulate this interaction in an explicit way.”

In consequence, our previous calculation of the total inertial force due to all the matter in the universe is neither strictly correct nor easily correctable.”

“We can only hope that our linear approximation gives an answer that has the correct order of magnitude . . ..”

“In view of all these difficulties, how was Einstein able to write down a law general enough to specify all the properties of the non-linear gravitational-inertial interaction?”

“The answer is that he wrote down the local properties of the interaction, using the field point of view.”

“From this the global properties of the interaction between [better, amongF.E.D.]  distant bodies can be calculated in principle, although in practice no one has been able to do this exactly even for just two bodies, except in the limit when one of them has a mass negligible compared with the other . . ..”

“It is instructive to look at this self-interaction of the gravitational field from a slightly different point of view . . . inertial forces act on gravitational waves and, if the Principle of Equivalence is correct, so must gravitational forces. ...”

“This shows how essential is the self-interaction of gravitation. . ..”

“It is one manifestation of the fact that gravitation acts on everything . . ..”

“We then have a self-interacting gravitational field satisfying a non-linear field law.

[D. W. Sciama, The Physical Foundations of General Relativity; Doubleday [New York: 1969]; pp. 55-62].

Thus, for the past 300+ years human knowledge and industry have been partially paralyzed and vitiated by a perennial failure to “solve” general nonlinear integro-differential equations, that is, to attain the means by which the vast potential knowledge that especially the laws of nature equations among them encode can be explicitly extracted and practically applied.
Key instances of this incapacity include --

·         the Newton gravity-equations for more than two mutually-gravitating bodies, the Einstein universal gravitational field equations of General Relativity just addressed in the quotation above;

·         the Navier-Stokes equations for electro~neutral liquid / gaseous hydrodynamics, and;

·          the ‘‘‘electro-magneto-hydrodynamics’’’ of the Maxwell-Boltzmann-Vlasov equation for electro-dynamically non-neutral, ‘‘‘magneto-hydro-dynamical’’’ “plasmas”, e.g., for superheated, ionized gases — that is, for the very media in which nuclear fusion reactions, self-sustaining over ‘mega-macroscopic’ spatial and temporal scales, are observed to occur in extra-human nature, e.g., in the central core-regions of stars.

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