Monday, December 07, 2015

Seldon Speaks: ‘Contra-Boolean Computers’.

Full Title -- 

Seldon Speaks:  Contra-Boolean Computers’, ‘Singularity Semantification, & The Nonlinearity Barrier.

Dear Readers,

It is my pleasure to share with you, from time to time, seminal selections from the sayings, shared by him among we of F.E.D., by F.E.D.’s co-founder, Karl Seldon, such as the following [E.D. standard edits applied] -- in this case, said recently, during one of the induction seminars for new F.E.D. recruits:

As we have so often noted before, the greatest mathematical, scientific, engineering, cognitive, intellectual challenge facing Terran humanity today, across essentially the entire spectrum of the sciences, is what we of F.E.D. have come to call -- somewhat in keeping with the similar phrase coined by Ladis D. Kovach [in his article entitled “Life Can Be So Nonlinear”] -- ‘The Nonlinearity Barrier’.”

“This epochal obstruction to human progress, this ‘Barrier’, often comes to attention by way of the contemporary, and longstanding, inability to solve “law of Nature”-expressing nonlinear differential equations -- in many cases after centuries of efforts in attempts to do so, including by some of the greatest masters of mathematics in mathematics history -- an inability to solve due precisely to their nonlinearity.   This unconscious ideological, psychohistorical disability began with the beginning dynamical model of modern science, that of the Newton gravity equations for more than two gravitationally-interacting bodies, e.g., for more than two planets. ... as well as in more quotidian loci, such as the epidemic unconscious inability, in modern society, in our natural languages, e.g., in modern English, for many of us to sensibly formulate comments/descriptions that involve totalities within which we ourselves, the ‘commenter’s’/‘describers’, are also included. ... .”

“The NQ arithmetic for dialectics -- despite its [relative] utter simplicity, as the [meta-]systematically first explicitly dialectical arithmetic in our presentational progression of dialectical arithmetics -- already exhibits, in embryo, the solution to one of the most essential symptoms of ‘The Nonlinearity Barrier’:  infinite singularity in one or more of the metrics, the “vital signs”, of the dynamic system modeled by a nonlinear differential equation, irrupting at a real finite time parameter value, let’s call it t*, of that nonlinear differential equation model of that dynamic system.”

“The Foundation has recently published an account of our solution to this aspect of ‘The Nonlinearity Barrier’, in the realm of such nonlinear differential equation models --

-- as a further development of earlier work [see pages A-6 to A-14] --

-- when such equations are «aufheben»-elevated into the context of the seventh system for the mathematics of dialectics in our presentational progression, via the ‘metrical-qualifiers re-qualification’ of such equations.”

“Such ‘re-qualified’ classical dynamical differential equations can, at best, describe the state of affairs that arises at/after their singularity, as/by ‘·’, the axiomatically regulated value which we call ‘full zero’.”

This value, ‘full zero’, ‘·’, a value far different from, although deeply related to, the traditional ‘empty zero’ value, 0, means that the ontologically new/different state of affairs that arises, at/after/as, singularity, cannot be described within the terms and via the other ‘ideo-resources’ available within the de facto model specification of the nonlinear differential equation model, given the implicit, or explicit, ‘‘‘ontological commitments’’’, the implicit, or explicit, ‘‘‘ontology’’’, of that model specification -- its ontology-descriptive limits.”

“That is, the ‘full zero’ value expresses the inexpressibility of the post-singularity state of affairs by/in the language of the pre-singularity model.”

“For example, in a dynamical model of a three-body -- e.g., of a three-planet -- gravitationally-interacting dynamic system, for which the initial conditions are such that two of the planets are predicted to collide, and since collisions take the form of division-by-zero, “infinite”-force singularities in Newton-equations models, this model cannot describe the coalescence/disintegration process of the interplanetary collision, in part, because this model is committed to the existence of the two planets as integral parts of its ontology, and cannot describe processes that result in their ‘dis-existentiation’, or a state of affairs after their ‘dis-existentiation’.”

“Likewise, a “purely-quantitative” nonlinear differential equation model of a main-sequence, Hydrogen-burning star, might “go singular” at the point of the exhaustion -- or of the complete conversion into Helium -- of the star’s core Hydrogen “fuel”, so that the core Hydrogen concentration has abruptly dropped to 0, as a result, not of any action upon the star, from its exterior, by other [ev]entities, outside of it, but as a result of the self-gravitational/implosive versus thermonuclear self-explosive ‘intra-duality’, and of the self-interactive ‘‘‘self-activity’’’, of this star itself.”

“The NQ dialectical ideography, despite its rudimentary descriptive capabilities, can describe, connotatively, the new ontology that arises at/with/after/as such a singularity.”

“If the ontology, ‘pre-singularity’ -- just prior to the t = t* moment of that singularity/- division-by-0 -- is denoted by the ontological-quality-connoting ideographical qualifier, X, or, equivalently, by qX, then the incremental new ontology that irrupts with/as/after the [ontological] singularity is denoted by qXX.”

“This new symbol, generated by the symbolic representation of the self-interaction of that which is denoted by qX, or by X [a ‘‘‘self-action’’’ represented by the self-juxtaposition of its symbol, or by that symbol’s “squaring” or self-multiplication, via the operation denoted by ‘*’]  --

XX  =  X * X  =  qX * qX  =  qX + qXX  

-- ‘‘‘solves’’’ for that singularity, “purely”-qualitatively, and ontologically, to the extent that the connotations of qXX can be aptly identified with the ontology, and consequent phenomenology, known to manifest, or expected to manifest, after such a singularity.  More advanced systems of dialectic mathematics, and the more advanced mathematical models that they can express, are needed to ‘‘‘compute past singularity’’’ in quantitative detail.”

“Our typical interpretation of the connotations of a term/‘categorogram’ like qXX, is that it connotes a category for “individuals” which are ‘meta1-individuals’ in relation to the individuals constituting the meaning of the category denoted by qX.”

“Thus, in the context of our second example, if qX connotes the star in question, as a whole, but emphasizing the composition of that star’s core as, initially -- in its qX ‘meta-state’, or ‘ontology-state’ -- consisting principally of Hydrogen atoms/nuclei as its defining “individuals”, then qXX could be ‘‘‘solved-for’’’ as connoting the dialectical, or «aufheben», negation/conservation/elevation of that core content, into a core content that, ‘post-singularity’, consists principally of those ‘meta1-individuals’, relative to the Hydrogen atoms/nuclei as its former defining “individuals”, called Helium atoms/nuclei, in that each individual Helium nucleus is created by a process involving the fusion of two Hydrogen nuclei/protons --

XX  =  ~X  =  qX[qX]  =  ~[qX] =  qX2   =  qX + qXX

-- interpreted/ ‘‘‘solved’’’ as --

HH  =  ~H  =  qH[qH]  =  ~[qH]  =  qH2   =   qH + qHH   =   qH + qHe   =   H + He

-- so that the category He refers to the ontologically different composition of the -- now Helium -- core of the star, after the singularity.  The carried-forward category, H, now refers to the possibility or actuality of H type “individuals” remaining inside the star, e.g., in (a) concentric layer(s) surrounding the now-Helium core.”

“Then, the next singularity-bounded epoch, whose description is generated by means of the ‘‘‘self-interacting’’’, or squaring/‘dialectical self-negating’, of the description of the result of the first singularity, (H + He) --

~(H + He)   =   (H + He)(H + He)   =   (H + He) * (H + He)   =   (H + He)2   =  

(H + He + qHeH   + qHeHe)      [-->        q1 + q2 + q3 + q4      

-- describes, via, qHeH, the processes and loci of the continuing conversion of Hydrogen ‘‘‘individuals’’’ into Helium [meta1-]individuals, e.g., in concentric outside-core layers of the star, &, via qHeHe, describes the new, second singularity, of the irruption, e.g., transiently and intermediately, of the [locally] new ontology of Beryllium atoms/nuclei, of atomic number 4 [e.g., "PPII" variant of "p-p process"] -- i.e., with two pairs of protons in their nuclei, each resulting from processes involving the effective fusion of two Helium atoms/nuclei -- and thus as ‘meta1- meta1-individuals’, or ‘meta2-individuals’, in the core of the star.”

“Because the NQ dialectical ideography is aptly describable as a contra-Boolean algebra for [dialectical] logic, and since the [revised] Boolean algebra models the logic of today’s digital computers, one might well wonder what capabilities might especially and uniquely accrue to a computer architecture whose core logic was that modeled by the NQ contra-Boolean algebra.”

F.E.D. has not, to-date, made public much of our ‘‘‘wondering’’’ on this topic, in the context of this era of, e.g., the Watson supercomputer, the internet, “Big Data”, etc.” 

“For today, we will leave the solution of this topic to you, our newest members, as an “extra credit” assignment, to enable you to test your mettle on an issue which may be even of the very greatest importance.”

“Apparently, such a computer might be capable of ‘‘‘computing past singularity’’’, i.e., of ‘computing post singularity’’’ [including, in particular, of computing up to, and beyond, the ‘‘‘Singularity’’’ of our predicted irruption of the android robotics, second species of the ‘meta-humanity’ that we foresee], its calculations not ‘‘‘losing their way’’’ at the t* point of singularity; of division-by-zero ‘‘‘explosion’’’.”

“Also, albeit in a “purely”-qualitative, “purely”-ontological way, the NQ way of modeling such veritable progressions of nonlinearity-induced [‘‘‘self-reflexivity’’’-induced] singularities, also captures, if, again, only in a “purely” non-quantitative way, the other of the most essential symptoms of ‘The Nonlinearity Barrier’, namely, the ‘irrepetitivity’ of the behavior of, e.g., nonlinear dynamical systems’ total differential equation models; their phenomena of causal, deterministic, exact, but never-repeating order.

“This kind of order -- little recognized as such within the prevailing «mentalité» -- is akin to the never-repeating order of the decimal expression of an algebraic or transcendental “irrational” “Real” number, i.e., the exact determinism that determines each digit in the decimal sequence of digits representing that “irrational” number, as determined by the individual nature/definition of that “irrational” number, coupled with the non-repeating character of that sequence of digits, which many are wont to -- quite mistakenly -- call “chaos”, or “randomness”.

“Consider, for example, the ‘‘‘deterministic non-periodic flows’’’ -- the never-repeating state-space trajectories -- generated by the approximate, “numerical” solutions of the nonlinear, “total-differential” equations that model such nonlinear dynamical systems, and that constitute “attractors”, or ‘dynamical solution-geometries’, that are unique to non-linear differential equations, not found among linear differential equation models, e.g., “limit-cycles” and “strange attractors”. 

“This trajectorial ‘irrepetitivity’ is most obvious in the case of the so-called “strange” attractors, which are the most “normal”, the most ubiquitous ‘dynamical solution-geometries’ for nonlinear total-differential equations.  These “strange” attractors are ‘‘‘asymptotically aperiodic’’’.”

“However, the “asymptotically periodic attractors, e.g., the “limit-cycles”, also exhibit, though not so obviously, this phenomenon of trajectorial ‘irrepetitivity’;  their actual trajectories never, ever actually “cycle”, i.e., “until” the “limit”, of the |t| in W “moment”, a “moment” which can never, ever arrive.”

“For all such strictly ruled, determinate, determined, deterministic trajectories, however non-periodic” -- however ‘irrepetitive’ -- “chaos” is an ideologically, psychohistorically misleading misnomer.”

“Those who want you to believe that shit [ just ] happens are those who do not want you to notice that they are the real source of the shit that is happening to you, as they “pull the strings”, from “behind the scenes”.

“The dyadic or triadic ‘self-iterative’ ‘self-reflexion’, and consequent «autokinesis», of an initial, initiating population of “individuals”, as represented by the «arché», or starter, ontological category, in an NQ ‘meta-model meta-equation’ that generates an NQ-algebraic categorial progression, is also qualitatively, ontologically never-repeating, is qualitatively different for each and every value of its self-iteration parameter, e.g., for t in W.”

“For each value of this historical, epochal time-representing parameter, a unique ‘cumulum’ of ontological category-symbols is generated, as representing possible existences during that epoch -- qualitatively, ontologically different from those for all other epochs -- and is asserted as being representative of the ‘ontology-state’ of the modeled Domain during that epoch.”

“True, each such cumulum, from t = 1 onward, does «aufheben»-include the representation of the possibility of the continued existence of the content represented by the ontological categories irrupted in all preceding ‘self-iterations’, for preceding values of t, i.e., corresponding to earlier, smaller values of this ‘self-iteration parameter’, or ‘‘‘time-like’’’ “independent variable”.”

“This «aufheben»-inclusion represents the ‘‘‘self-similarity’’’ aspect of the content represented by such ontological-categorial progressions, an aspect whose representation is provided for via the second conservation moment of the double-conservation «aufheben» evolute product rule, axiom §9.”

“However, each such ‘self-iteration’ also includes [one or] more new, unprecedented possible ontological categories -- representing possible new qualities, new kinds of «arithmoi», or of populations, made up out of new kinds of [meta-]«monads», i.e., new kinds of individuals -- that make that cumulum, as a [sub-]totality, ontologically unprecedented, unique, and, as a whole, never-to-be-repeated -- irrepeatable.



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