__Full Title__: Part

**08**of

**29**--

#
__The Dialectica Manifesto__

__The Dialectica Manifesto__

##
__Dialectical Ideography__

__Dialectical Ideography__

## and

##
the Mission
of F.__E__.__D__.

__E__

__D__

**,**

*Nonlinearity*

<<

**>>,***Autokinesis*

*and***.**

__Dialectic__

Dear Readers,

I am, together with

**F**.**.**__E__**. Secretary-General Hermes de Nemores, and**__D__**F**.**.**__E__**. Public Liaison Officer Aoristos Dyosphainthos, organizing to develop a new, expanded edition of the**__D__**F**.**.**__E__**. introductory documents, for publication in book form, under a new title --**__D__

__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

**installment of a***eighth***29**-part presentation of this introductory material, which the**F**.**.**__E__**. General Council has authorized for serialization via this blog over the coming months, as we develop the material.**__D__
I plan to inter-mix these installments with other
blog-entries, including the planned additional

**F**.**.**__E__**. Vignettes, other**__D__**F**.**.**__E__**. news, my own blog-essays, etc.**__D__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

**, in a**

*totality***, making explicit many of**

*connected account***.**

*the interconnexions among the parts*Enjoy!!!

Regards,

Miguel

##
*Nonlinearity*,* *«*Auto**-**Kinesis*», *and *__Dialectic__.

If __Dialectic__

*“*

__nonlinearity__*”*is the

**of this mathematical difficulty and “intractability”, and if**

*root cause**“*

__nonlinearity__*”*is the

**of this present “closed-form**

*root cause***”, then what does**

__un__solvability*“*

__nonlinearity__*”*signify

*?*In its

**meaning,**

__deepest__**signifies what Plato called «**

__nonlinearity__**» [see below].**

*autokinesis***Differential-equation**

*¡**“*

__nonlinearity__*”*is the mathematical name for mathematical, ideographical, ‘equational’, ‘purely-quantitative’ modeling of

*‘*

__self__-*motion’*, of

*‘*

__self__-*motion’*, of

*‘*

**[**

__self__-*induced*]

**’; of**

__movement__*‘*

__self__-*induced*change-

*of-state*

*’*; of

*“*

**self**-**refl**

__e__xiveness*”*[cf. Russell], that is, of

*‘*

__self__*-refl*,

**xive’**__e__*“*

__self__*-referential”*

**; of**

__action__*‘*self-refl

**xiveness**

__u__*’*and of

*‘*self-refl

**xive**

__u__*’*

**, that is, of**

__action__**which**

__action__*flows back to and changes its*

**, its**

*source***, its**

*agent***-- and, thus, of**

*subject**‘*

__self__*-*

__developing__*, or of*

**process**’

*developmentally**,*

**self**-**propelling***‘*

__self__*-*

__developing__

__ev__ent

*ity**’*

**!**These equations -- equations that the presently absent, presently-unformulated ‘‘‘closed-form’’’ function-formulae solve -- are called

*“*

__non__linear*”*because, in them, the

**[which, for integro-differential equations, are**

__un__knowns

__function__*-*, not single number-values as are the solutions of algebraic, diophantine equations], the

__un__knowns*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,

**, and/or**

*operating*__upon____themselves__**.**

*operating upon one another***: Such signifies the**

__Conjecture__**, i.e., the**

*dynamical*

__time__*-*[indeed, the

**like***‘*

__chronogenic__*’*],

*interaction*

*and*

__self__*-*of the underlying actualities that these unknown solving-functions must mime.

**interaction****integro-differential equations, the kind that have for so long been so easily solved by Terran mathematicians, are characterized, in contrast, by**

*Linear*

__function__*-*which occur in “

__un__knowns**isolation**”,

**,**

*singly***, without interaction,**

*independently*

*operating upon / multiplying*__neither__self__nor__any other unknown

*/*

*to**-*(

**be**-**solved**-**for function****).**

*s*A typical

**“ordinary” or “total” differential equation is an ideographical**

__non__linear*‘*

**-ment’, that asserts -- that “**

__state__**”, in effect -- that the instantaneous**

__state__s**of the generic, pure-quantitative**

*velocity of evolution***of the**

*value***of the**

__state__*modeled by that equation for*

__system__**, generic time-value,**

*any***t**— the

**represented by**

__state__

__x__**(t)**, thus denoting the

*generic*

**-**

__state__**of**

*function-value**the*

*dynamical*__function__*-*— is proportional to a higher power of that unknown,

**to be solved for**__un__known*generic*

**-value itself, denoted**

__state__

__x__**(t)**,

^{n}**n**

**>**

**1**, i.e., to a

*multiplicative*

__self__-application /__self__*-*

*operation /**‘*or

**-**__self__**fl**’__e__xion*‘*

*self**-*; to a

**-**__re__**fl****’**__e__xion

*self**-*

**, of that**

*multiplication***-value.**

__state__Such a

*‘*

*purely-*signifies: either

**itative’**__quant__**self**-**multiplication****(1)**a [

*‘*

*purely-*]

**itative’**__quant__*‘*, or;

**self**-**magnification**’**(2)**a [

*‘*

*purely-*]

**itative’**__quant__*‘*

*self**-*’, of one or more of the values of the

**diminution****-variable(s) of**

__state__

__x__**(t)**, for every [‘non-Boolean’] value of the

**-variable-value components of that**

__state__**-“vector” function,**

__state__

__x__**(t)**

**=**

**x**

_{1}(t)**+**

**. . .**

**+**

**x**.

_{m}(t)That

**-“vector” function is, in effect, a ‘“list”’, or, more precisely, a “non-amalgamative sum”, of**

__state__**m**“state-variables”,

**x**through

_{1}(t)**x**, each one reflecting a model-predicted value of a different prime “vital sign” measurement of the

_{m}(t)*modeled by that*

__system__**-“vector” function,**

__state__

__x__**(t)**, as of a given value of

**t**. Let us herein call that

*as a whole by the name*

__system__**.**

__X__Such an equation describes a

*’s “evolution” as an at least partially*

__system__*‘*

*self-induced’*, or

*‘*

**’,**

__autokinesic__

*self**-driven*,

*self**-propelling*

**in a**

*motion***-space, or**

__state__*‘space of*

__state__s*’*.

“State Space” an

**, or**

__imagined____space__

__conceptually__*-*, in which every

__constructed__space**denotes a different**

__point__*possible*

**of that**

__state__*, as a ‘“list”’ of that*

__system__*’s vital signs.*

__system__For example,

**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,**

__non__linear

__self__*-*limiting

*Verhulst term --*a

*‘“*

__self__*-*term”’.

**interaction**This term involves adding in, e.g., a

*self**-*of

**multiplication****N**, where

_{i}(t)**N**denotes the

_{i}(t)

*population**-*

**count**-**as**-**of the**

__state__**i**th species as a function of

**ime,**

__t__**t**,

*such that a*__minus____sign__is applied to that*‘*[

**self**-__product__**]**

*ion**’*,

*thus registering as a*, or

**contribution to the momentary, “instantaneous”**__negative__**of that species’**__rate__of growth**population**-**count**— where that**of growth is here defined as the metric for the**__rate____rate__of evolution**,**

__velocity__of evolution*of that species’*

**—**

__state__*with respect to time, as the*[

*“continuous”*]

*time ‘‘‘count’’’ advances*:

**“**The

__non__linear**is referred to as a**

*correction term**“*

__self__*-*. . .

**interaction**

*term**”*[which term we also term

*‘*[

__self__*-*]

**re**

*-fl*__e__xive*’*[

*‘‘‘bent back upon itself’’’*] or

*‘*[

__self__*-*]

**re***-*[

**fl**’__u__xive*‘‘‘flowing*[

*]*

__from__self*] —*

__back____to__self’’’**F**.

**.**

__E__**.], of the form**

__D__**N**. . . where the terms of the form

_{i}(t)^{2}**N**,

_{i}(t)N_{j}(t)**i**

**~=**

**j**,

**, are referred to as**

*also*__quadratically____non__linear*“*

__mutual__interaction terms*”*[which terms we also term

*‘*’, meaning

**-**__hetero__**flexive**’ or ‘**-**__allo__**flexive***‘‘‘*bent’’’, or ‘‘‘bent by

__other__-*—*

__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

**precisely because they must model the**

__non__linear*‘*

__non__*-*,

**a**-**tom**-**istic**’*‘*

*self**-*,

**reflexive**’*‘*

*auto**-*,

**kinesic**’

*self**-*

**changing***‘*

*self**-*of the cosmos-encompassing gravitic ‘‘‘field’’’ --

**interactivity**’**“**...an

**is**

*interaction*

__non__*-*if the

**linear****exerted by**

*total force***is**

*several bodies*

__not__

__the__**of the**

*sum*

*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

**; it has to be included in a total energy balance . . ..”***real physical significance**“*,

**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

**force is then***total***of the**__not____the__sum**forces.”***individual*
“... It
follows that the

**between [***exact interaction**better*,**—***among***F**.**.**__E__**.] an**__D__**is going to have a***arbitrary number of bodies***.”***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

**that our***only hope***gives an answer that has the***linear approximation***. . ..”***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

**properties of the interaction, using the***local***point of view.”***field*
“From
this the

**properties of the***global***between [**__interaction__*better*,**—***among***F**.**.**__E__**.] distant bodies**__D__**,***can be calculated in principle***, except in the limit when one of them has a mass negligible compared with the other . . ..”***although in practice no one has been able to do this exactly*__even__for__just____two__bodies
“It is
instructive to look at this

__self__*-*__interaction__**from a slightly different point of view . . . inertial forces act on***of the gravitational field***and, if the Principle of Equivalence is correct, so must gravitational forces. ...”***gravitational waves*
“This
shows

**is***how essential**the*__self__*-*. . ..”__interaction__of gravitation
“It is
one manifestation of the fact that

*gravitation acts on**“**everything**”*. . ..”
“We
then have

*a*__self__*-*satisfying__interacting__gravitational field*a*__non__*-*.**linear field law****”**
[D. W. Sciama,

**; Doubleday [New York:**__The Physical Foundations of General Relativity__**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

**integro-differential equations, that is, to attain the means by which the vast**

__non__linear*potential knowledge*that especially the

*“*

*laws**”*of nature equations among them encode can be explicitly extracted and practically applied.

Key instances of this

**include --**

__in__capacity
·
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**-neutral,***non**‘‘‘magneto*-hydro-dynamical’’’ “plasmas”, e.g., for superheated,**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.**__ionized__
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