__Full Title__: Part

**07**of

**29**--

#
__The Dialectica Manifesto__

__The Dialectica Manifesto__

##
__Dialectical Ideography__ and

__Dialectical Ideography__

##
the Mission
of F.__E__.__D__.

__E__

__D__

*The Nonlinearity Barrier*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***seventh***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

Part

**07**of**29**--#
__The Dialectica Manifesto__

__The Dialectica Manifesto__

##
__Dialectical ____Ideography__ and

__Dialectical__

__Ideography__

##
the Mission
of F.__E__.__D__.:

__E__

__D__

*The Nonlinearity Barrier*Of course, all of the above

*“*algebraic / diophantine equations may, today, appear to us to be “trivial” to solve, and their solutions may appear all too familiar to us, given that those solutions were all pioneered long ago, by our remote ancestors.

**solvable”**__un__**But are there still**

*¿**“*solvable equations” in our own day

__un__

*?*

**Are there still new kinds of numbers, beyond the**

*¿***G**[the

**G**rassmann hypernumbers] yet to be discovered, that will provide the

*‘*

*ideo**-*wherewithal’ -- the

**ontological****-- necessary to**

*new kinds of numbers***such equations**

__solve__

*?***Is there yet a new arithmetic, right now on the verge of being discovered / constructed**

*¿*

*?*If Gödel is right, that this

*‘*

__dialectic__*’*of

**completeness /**

__in__**decidability /**

__un__**solvability is “inexhaustible”; [**

__un__**] “continuable into the transfinite”, then there**

*potentially***still be such.**

*must***If so, how far has this**

*¿**‘*

__Gödelian____dialectic__*’*progressed, to date, in Terran human history

*?*

**As mapped into**

*¿**the history of the collective human psyche*per its

*‘*

*collective**, anthropological /- ‘psyche-ological’, ‘‘‘*, how far along into it are we as of today

**psychohistorical**’’’ conceptual**readiness**-**gradient**’

*?*

**Does our present stage of this**

*¿**‘*

__Gödelian____dialectic__*’*have any scientific relevance

*?*And, if there are, today, still, some equational «

**», would their solution — garnered by moving into the next higher stage of this**

*insolubilia**‘*

__Gödelian____dialectic__*’*— have any

**, e.g.,**

*practical value***; any**

*engineering value***; any contribution to make to the growth of the society-**

__urgent__technological application

__productive__**of humanity, i.e., any contribution to make to the**

__forces__**,**

*viability**‘*

__qual__o*-*

**’**__quant__itative

__self__*-*and

**productivity****of the global human species?**

__prosperity____:__

**Our Conjecture**

*Yes to all!*
Indeed, the very equations which
formulate this humanity’s most advanced collectively-recognized formulations of
the so-called

*“**laws**”*of nature -- of the**legislated but habitual patterns of natural action -- are generally of the type that is named**__un__**[**__non__linear**]***partial***.***differential equations*
They also remain, for the most part —
especially when they are

**— chronically**__non__linear**solved by “standard” mathematics, typically a century or more after their first formulation.**__un__
They are also often — and without proof
— simply declared, by “standard” mathematicians, to be, not just ‘so far
unsolved’, but [

**] “unsolvable” in “exact” or “analytical” or “closed” “form”.***forever*
This conclusive-sounding phrase is
actually anything but.

It merely means that their solutions
apparently cannot be expressed in terms of the “elementary”, or fundamental, “algebraic”
and ‘trans-algebraic’, or “transcendental” functions or operations

**-- as “elementary” -- even if their solutions***currently recognized as such***be expressed in ‘‘‘open form’’’, involving [**__can__**]***potentially**“***sums”, i.e., [***infinite***]***potentially**“***series” or [***infinite***]***potentially**“***polynomials” — ever improvable approximators — made up out of finite and “closed-form” terms.***nonlinear to the infinite degree*
The “unsolvability”, or so-called “non-integrability”,
of these

__non__linear**may also mean that the “integration”, or solution, of these equations encounters***differential equations*__zero__-__division__*“*__singularities__*”*, which apparently lead to “function-values of**magnitude”, so that their solution “diverges” or attains***infinite**“**infinite**”*or “undefined” / “indeterminate” values corresponding to**values of the time parameter; that the “***finite***” of their**__limit__*“***series” sums, forming their integrals, appears to be without [finite] quantitative limit; appears to be***infinite***“**__quant__itatively**less” or**__limit__*‘***un**-**-ed’.**__limit__This

*‘‘‘*

__Nonlinearity____Barrier__*’’’*of modern, “standard” mathematical science massively blocks this humanity’s capability for further scientific and technological / engineering advance around its entire perimeter with the

__un__*-*known; with its present

*‘*knowledge’, viz. --

**-**__un__**“**That is the way I explained non-linearity to my son.”

“But,
why was this so important that it had to be explained at all?”

“The
complete answer to this question cannot be given at present, but some people feel that the answer, if known, would shake the
very foundations of mathematics and science . . .”

“. . . practically
all of classical mathematical physics has evolved from the hypothesis of linearity.”

“If it
should be necessary to reject this hypothesis because of the refinements of
modern experience, then our linear
equations are at best a first and inadequate approximation.”

“It was
Einstein himself who suggested that the basic equations of
physics must be non-linear,
and that mathematical physics will have to be done over
again.”

“Should
this be the case, the outcome may well be a mathematics totally
different from any now known.”

“The
mathematical techniques that might be used to formulate a
unified and general non-linear
theory have not been recognized . . .”

“. . . we are now at the threshold of

*the***.***nonlinear barrier***”**
[Ladis Kovach; “Life Can Be So Nonlinear”,
in

**[**__American Scientist__**48**:**2**, June**1960**], pp.**220**-**222**,*emphases***by***added***F**.**.**__E__**.].**__D__No less than

**of modern, ‘‘‘mathematico-science’’’ — a problem that was also a central focus and motivation of ancient science — today takes the form of a system of**

*the founding problem*

__non__linear

*integro**-*which have, to this day, in both their Newtonian and Einsteinian, General Relativistic versions, remained essentially unsolved [the ingenious

**differential equations****1991**, slow convergence, “open-form”,

__singularity__*-*

*“*/

**infinitely**”-**delaying****i.e., planetary-collisions-**

*evading*

*infinitely**-*/

**delaying****— series solution by Qiu-dong Wang notwithstanding], because of their**

*evading***.**

__non__linearityThis

**is the fundamental problem of astronomy, the problem of the mutual-determination,**

*founding problem***including the**

**-objects-mediated-**

__other__

__self__*-*determination, of the motions of celestial objects, when any more than two such objects are admitted into the mathematical model of the celestial cosmos:

**“**The

**n**-body problem is the name usually given to the problem of the motion of a system of many particles attracting each other according to Newton’s law of gravitation.”

“This is the classical problem of mathematical natural
science, the
significance of which goes far beyond the limits of its astronomical
applications.”

“The

**n**-body problem has been the main topic of celestial mechanics from the time of its inception as a science.”
“The
fundamental dynamical problem for a system of

**n**gravitating bodies is the investigation and pre-determination of the changes in position and velocity that the [bodies] undergo as the time varies.”
“However,

*this is a complex**non**-*.**linear****problem****whose solution has not been possible under the present**-**day status of mathematical analysis****”**
[G. F. Khilmi,

__Qualitative Methods in the Many-Body Problem__*,*Gordon & Breach [**1961**], page**v**.,*emphases***by***added***F**.**.**__E__**.].**__D__Indeed, the models of nature that modern mathematical science has favored are profoundly flawed and misleading in crucial aspects of their ‘descriptics’ of nature, due to

**inadequacy of the mathematics that Terran humanity has evolved so far:**

*this specific*
“It is
an often-stated truism that

*nature is inherently*__non__*-***.”***linear**“*particularly

**Biological systems**

*are full of***. . .**

__non__*-*

**. . .”**

*linearities*
“The
reason that we go to the trouble of building

**models when we are really interested in***linear*__non__*-**linear***is that we then acquire the power to evaluate the dynamic performance of the system***systems***. . ..”***analytically*
“In
fact, we can

**solve for the response of a***analytically**linear***to any conceivable input function, however complicated.***system***”**
[Bernard C. Patten,

**[**__System Analysis and Simulation in Ecology__**volume I**], Academic Press [NY:**1971**], p.**288**,*emphases***by***added***F**.**.**__E__**.].**__D__However, in the

__non__*-*

**domain:**

*linear***“**In general, the

*analytical study of*__non__*-*has been developed only to a very limited extent, owing to the

**linear differential equations****of the subject.**

*inherent mathematical difficulties*
There
does not exist, in this field, a suitable technique for attacking

**problems as they arise in practice.***general*__non__-linear**”**
[John Formby,

__An Introduction to the Mathematical Formulation of Self__*, Van Nostrand [NY:*__-__**Organizing Systems****1965**], p.**115**,*emphases***by***added***F**.**.**__E__**.].**__D__

*General*__non__*-*

*linear***cannot presently be solved in “closed form”, because the [‘‘‘elementary’’’] functions that would solve them have so far, for the most part, “resisted” discovery and formulation within the extant tradition of Terran human mathematics:**

*integrodifferential equations***“**...

*the***in**

*assumption of linearity***underlies most applications of**

*operational processes***to the problems of the natural world.”**

*analysis*
“. . .

**, with scant regard for the desires of the mathematician, often seems to delight in formulating her mysteries in terms of***Nature*__non__*-**linear systems**of equations*. . .”
“ . . .

*the theory of functions***. . .***has been developed largely around classes of functions in which**the linearity property**is an***. . .***essential factor*
. . .

*most*__non__*-**linear**equations*__define__*whose properties have*__new____functions__**been***not***nor for which tables exist...***explored***”**
[Harold T. Davis,

**, Dover (NY:**__Introduction to Nonlinear Differential and Integral Equations__**1962**); pp.**1**,**7**,**467**,__emphases____added__*by***F**.**.**__E__**.].**__D__In the light shed by the foregoing statements, the oft-decried

*‘‘‘*

*mechanistic**’’’*bias of modern mathematics, and of modern science in general, is seen in altered perspective.

This new perspective is strengthened by the observation that the more ‘

*’ and*

**organitic***“*

**organismic***”*qualities of

**, which classical**

*Nature**“*

*mechanism**” / ‘*excludes — phenomenologies such as those of

**linearism**’

__non__*-*

**and [meta-]evolutionary [meta-]dynamics; of**

*equilibrium**,*

**holistic**

**synergistic***,*“whole-more-than-sum-of-parts” self-organization; of the qualities of self-determination and self-development, and of sudden and qualitative

__self__*-*

**change**— find a native and potent expression in the

__non__*-*

**domain.**

*linear*It thus emerges that science has been

*‘‘‘*

*mechanistic**’’’*only to the extent that it has failed to be

**enough — failed to be**

__scientific__**enough, or true-enough-to-observation/-experience.**

__empirical__Mathematics has been

*‘‘‘*

*mechanistic**’’’*and

*‘*

*linearistic**’*only to the extent that it has failed to be

**enough.**

__mathematical__Modern science and applied mathematics have fallen short of a more adequate description of experiential / empirical truth through suppression or neglect of the immanent truth already enshrined within themselves.

*Not even scientific mechanics itself is truly**‘‘‘*

*mechanistic**’’’*:

**“**. . .

**Mechanics as a whole is**-

__non__**; the special parts of mechanics which are**

__linear__**may seem**

*linear***nearer to**

**common sense**, but all this indicates is that

**good sense in mechanics is uncommon**.”

“We
should not be resentful if materials show character instead of docile obedience.”

“. . .

*Although**mechanics is essentially*__non__*-***, it is little exaggeration to say that**__linear__*for 150 years only***.”***linear mechanics and its mathematics were studied*
“It
became

**practice, after deriving the equations for a phenomenon, to replace them at once by a***standard**linear**so**-*“approximation”.”**called**
“It
would be wrong to regard

**as being in the original tradition of mechanics...***this mangling***”**
[C. Truesdell, “Recent Advances in Rational
Mechanics” in

**[**__Science__**127**:**3301**,**04**April**1958**], p.**735**,__emphases____added__*by***F**.**.**__E__**.].**__D__
Closed-form-function solutions for our

*-equation-expressed so-called “laws” of nature would provide ready-calculation of*__non__linear__global__**, for the total domain of initial conditions.***solutions*
A “computer
simulation solution” or “numerical solution” —
the only kind of “solution”, if any, presently available for most of these

*“laws” of nature — merely “simulates”*__non__linear**of the implications of the***some***solved equation, and is limited to a single solution-trajectory or solution-history, from a single initial condition, a single “point”, or**__un__*“**starting state**”*, leaving all other*starting points***solved-for.**__un__
Such simulation-“solutions” also suffer
severe limitations of computer calculation time
[computation-speed] and storage capacity [memory space],
as well as all of the limitations of the computational and “qualitative”
[in-]accuracy of “numerical” algorithms, particularly with regard to the
detection of

*“**essential**”***.***singularities***Could it be that what is really brewing here -- in this protracted, chronic, centuries-spanning failure of modern science to solve its primary “laws-of-nature” equations -- is another ‘Gödelian Crisis’; a crisis of the Gödelian-incompleteness and diophantine-equation-unsolvability of these “laws-of-nature” equations within the**

*¿**de facto*most advanced axiomatic system of arithmetic /- algebra

**+**that is so far extant and that is presently in use for all attempts to solve these equations

*?***Could it be that what is required to make these equations solvable is, precisely, a new, unprecedented ‘ideo-ontology’ -- new comprehension axioms; new, higher logical types of sets of ordered pairs, or of ordered**

*¿***n**-tuples;

*new kinds of numbers?*
Many prevalent presumptions militate
against

*“**yes**”*answers to any of the questions above.
Unlike what is the case with

**,**__algebraic__equations**require more than**__differential__equations**to solve them.***individual numbers***require**

__Differential__Equations**--**

*functions***of the time-variable,**

*functions***t**, in the case of

**-- e.g., whole**

__dynamical__differential equations*‘‘‘*of individual numerical values -- to solve them.

**continua**’’’
Moreover,

**belong to***differential equations**“*, not to**analysis**”*“*.**algebra**”
It thus does not seem, at first glance, that

**[***any***]***system of***(***nonlinear differential equation***) could be represented by any algebraic, diophantine equation, the assertion of whose unsolvability would constitute the deformalization of the incompleteness-or-inconsistency-asserting Gödel Formula immanent to any “Natural”-arithmetic-or-more-encompassing axioms-system.***s*
But the fruition of

**,**__dialectical__**typically requires far more than first glances.***immanent critique*
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