Finite fields.. to a theory of levels. The "wrap-around" feature, and the infinity contained. The implications. Premises afforded to societies. Parallel processing, people parallel processors.

Chaotic Logic: Language, Thought and Reality From the Perspective of Complex Systems Science

Ben Goertzel. 

Chapter Three - The Structure of Thought 

3.4 PREDICTION

3.4.1. Discrete Logarithms (*) 

"For a simple mathematical example, let us look to the theory of finite fields[E1] [E2] . A finite field is a way of doing arithmetic on a bounded set of integers. For instance, suppose one takes the field of size 13[E3]  (the size must be a prime [E4] or a prime raised to some power). Then, in this field the largest number is 12. One has, for example, 12 + 1 = 0, 10 + 5 = 2, 3 x 5 = 2, and 8 x 3 = 12. One can do division in a finite field as well[E5] , although the results are often counter-intuitive -- for instance, 12/8 = 3, and 2/3 = 5 (to see why, just multiply both sides by the denominator)."

 [E1]

 the theory of finite fields .. a level reached .. literally a finite field .. the state of a system .. a system with far-wide implications .. society .. the level .. the levels .. the states reached .. the boundaries ..allowed .. extended to .. connections .. communications ..allowing .. language barriers .. geographical boundaries .. national boundaries .. fragmented .. the push that technologies afford .. boundaries surpassed .. the inclusivity parameter ..  

 [E2]
the multiverse idea .. the universe .. a finite field ..of immense proportions ..yet a finite field in itself .. as all the finite fields that the systems comprising the universe .. are themselves .. too .. to the tinniest of systems within it.

 [E4]
the implication of the prime numbers .. a factor that seems ..uncanny .. yet inane .. cannot really one see .. appraise its significance in the whole case .. but taken in account the size of a field in every system in the universe .. the universe included .. that must involve huge numbers .. in terms of societies that must equate to the populations each society is made out of .. though in terms of individuals in societies .. this should be determined upon the participating individuals .. the ones that make use of what technology makes available for people ..  
 

 [E5] 

the arithmetic .. of the numbers within the finite field .. the extra twist .. when computing is taken place .. the fractal ..worlds ..produced within the finite field bounds .. numbers continue to be infinite .. but their extend do not exceed the boundaries of the field .. the implications of taken up the same values that the same process takes in other stages of their cycle .. and the size of their cycle as it is delimited by the size of the field .. intriguing .. remain the question whether this bears any significance .. in the derivation of outcomes .. or by being fractal .. this indicates that they are separate .. develop in fractional dimensions so they do not meet ..

In finite field theory there is something called the "discrete logarithm" of a number, written dlogb(n). The discrete logarithm is defined just like the ordinary logarithm, as the inverse of exponentiation[E6] . But in a finite field, exponentiation must be defined in terms of the "wrap-around" arithmetic illustrated in the previous paragraph. For instance, in the field of size 7, 34 = 4. Thus one has dlog3(4) = 4. But how could one compute the log base 3 of 4, without knowing what it was[E7] ? The powers of 3 can wrap around the value 7 again and again -- they could wrap around many times before hitting on the correct value[E8] ,4.

[E6] 

logarithm .. exponentiation .. more complicated ..computations .. included .. afford deep analysis .. how physical properties can be calculated measured .. examining their complicated associations .. associations where their ..expression ..surpasses .. simple addition .. complex ..parameters .. variables .. their connection ..association being of a greater degree .. multiplied instead of being additive ..

networks .. whole networks .. creating patterns .. adding networks .. their addition requires ..higher arithmetic operations .. where ..multiplication ..division .. exponentiation .. logarithms .. enter .. are required .. the continued spectrum of numbers .. require ..  is the addition of networks .. afford us the following of their ..developments .. matrices .. how can a mind manage to follow .. the adding of networks .. how can observation fathom te addition of networks .. it can sense them .. intuition but not pinpoint their alternation .. from input to outcome .. there are too many in-between processes that is impossible to ..observe ..  

  

[E7] 

without knowing what it was .. the implications of that ..phrase .. sentence .. the assumption .. brought forth .. imprinted in the human mind ..explaining our inability ..inherent inability .. mathematically expressed .. not knowing what we want .. intuition .. at bay ..

  

[E8] 

wrap around many times .. before hitting the correct value ..

The problem of finding the discrete logarithm of a number is theoretically easy, in the sense that there are only finitely many possibilities. In our simple example, all one has to do is take 3 to higher and higher powers, until all possibilities are covered. But in practice, if the size of the field is not 7 but some much larger number, this finite number of possibilities can become prohibitively large.

So, what if one defines the dynamical system nk = dlogb(nk-1)? Suppose one is given n1, then how can one predict n1000? So far as we know today, there is better way than to proceed in order: first get n2, then n3, then n4, and so on up to n999 and n1000. Working on n3 before one knows n2 is essentially useless, because a slight change in the answer for n2 can totally change the answer for n3[E9] . The only way to do all 1000 steps in parallel[E10] , it seems, would be to first compute a table of all possible powers that one might possibly need to know in the course of calculation. But this would require an immense number of processors; at least the square of the size of the field.

[E9] 

the wrap around feature complicates computations .. how can this be interpreted .. the numbers .. their sequential nature ..is disrupted by the nature of the wraparound field .. unfit .. unable to take into consideration the wraparound feature of the finite field .. they do not stop where the size of the field stops .. continuous in nature .. maths prohibitive .. could not have been otherwise if the computations would have some sense .. make sense .. ought to .. should be of that nature .. beyond ..human intervention .. wish .. will .. that is how the universe requires them to be .. the added twist in its ..ongoings .. words .. symbols .. thought processes .. underlying .. are fulfilled by the same premises ..

out of field of vision .. visual field .. and yet there .. the essence of .. finite fields .. levels .. the wraparound feature .. fundamental ..considerations ..

  

[E10] 

1000 steps in parallel .. what people are doing .. parallel processing .. all that are presented as problems .. all that is required to be solved .. society a parallel processing computer ..

This example is, incidentally, of more than academic interest. Many cryptosystems in current use are reliant on discrete logarithms. If one could devise a quick method for computing them, one could crack all manner of codes; and the coding theorists would have to come up with something better. 

 

3.4.2. Chaos and Prediction

More physicalistic dynamical systems appear to have the same behaviour[E11] . The classic example is the "logistic" iteration xk = cxk-1(1-xk-1), where c[E12] =4 or c assumes certain values between 3.8 and 4[E13] , and the xk are discrete approximations of real numbers. This equation models the dynamics of certain biological populations, and it also approximates the equations of fluid dynamics under certain conditions.

It seems very, very likely that there is no way to compute xn from x1 on an ordinary serial computer[E14] , except to proceed one step at a time. Even if one adds a dozen or a thousand or a million processors, the same conclusion seems to hold. Only if one adds a number of processors roughly proportional to 2n can one obtain a significant advantage from parallelism. 

[E11] 

behaviour .. out of all the properties of the unit .. all .. should be taken into account .. the enormity of the combinations .. and how much little have been afforded .. by the systems .. the filters being put .. the laws .. the perspectives ..open .. inevitable .. to seek out .. their instantiations .. to fulfil all .. what can be called aspirations .. out of their very own properties that struggle to confirm their existence .. societies in turmoil .. joy ..happiness .. the fulfillment ..sought after .. the integration of these properties in the whole they comprise .. the desire to incorporate .. these attributes .. into the whole .. to blend in an environment that is deemed suitable .. the properties of the environment .. the level ..that .. c .. represents .. the finite field .. the nature itself ..  

 [E12] 

c .. represents the level reached .. and the ..fluctuations afforded ..out of that level .. the level .. being the finite field .. the wraparound feature .. the fractal development .. that affords self-similarity .. what chaotic systems are prone to exhibit .. 

 [E13] 

c ..assumes certain values between ..3.8 to 4 .. the small range out of the level at bay .. that produces much larger outputs from the given ..tiny inputs .. a chaotic systems feature .. the dependent variables .. one must determine the associations of the variables involved .. the abstract .. mathematical relations ..out of the ..physical .. instantiations .. of the abstract mathematical properties .. out of the quantity .. the measured abstract quantity .. the quality .. how the features are coming into existence .. dependent variables ..derived variables .. being able of quantification .. multiplicity .. relationships .. relate to one another .. the unit bearing properties are added .. emergent .. emerging out of the unit .. what the unit can do ..

  

[E14] 

serial computer .. its comprehension ..out of .. serial and parallel circuits .. electric bulbs connected .. what governs their function.. serial connection.. from the same source .. when the source is off ..all bulbs are off.. parallel connection.. is not the source ..it is the connection with the source .. the route it takes .. the source is common in both cases .. the route differs .. in serial ..the source is connected through the same route .. each unit is making up the route .. the output diminishes .. sequential .. parallel ..different route .. for each unit .. output is un-diminished by the other ..

 

In general, all systems of equations called chaotic possess similar properties[E15] . These include equations modelling the weather, the flow of blood through the body, the motions of planets in solar systems, and the flow of electricity in the brain. The mathematics of these systems [E16] is still in a phase of rapid development. But the intuitive picture [E17] is clear. To figure out what the weather will be ninety days from now[E18] , one must run an incredibly accurate day-by-day simulation [E19] -- even with highly parallel processing[E20] , there is no viable alternate strategy. 

 [E15] 

chaotic ..systems of equations .. similar properties .. 

 [E16]

 mathematics of these systems .. chaotic systems .. in a phase ..of rapid development .. 

 [E17] 

the intuitive picture .. intuition ..at bay ..tuition .. inner .. being tutored by inner self ..being taught by inner self ..

 [E18] 

now .. the now .. which it should take aboard .. what was on .. in the appropriate unit of time ..before now .. a cycle .. a complete cycle .. 

 [E19] 

simulation .. accurate ..day-by-day .. cycle-by-cycle .. 

 [E20] 

highly parallel processing .. simply .. being processed ..on its own .. independent of other processes .. but in the ..computing .. the completion of the algorithm at bay .. .. simultaneously .. without meaning that the processes are truly independent .. that they are not ..being affected by the conditions .prevailing ..

---

Augmentation of our arsenal of senses required.

a matter that becomes..  prescient .. extra senses demand their inclusion ..

extra sensory perception .. studied with the scientific method .. to understand emergence ..fathom it ..

another sense employed by the human entity to shape the systems that is part of .. what has been around with such terms as sixth sense .. or ..extra sensory perception ..

build our worlds .. experience our worlds .. define our worlds and our place within their walls .. via our senses .. our five senses provides with an incomplete model .. hinders the definitions employed .. the exclusion of further senses involuntary or deliberate .. whose ample manifestations  are pushed away .. ignored .. not taken into consideration .. are referred to ..beyond science .. therefore beyond the scope of the scientific method .. stagnant fields allowing growth only horizontally ..

science penetrating vertically lost its sharp edge ..

mainstream science stays away .. but in the fringes of science .. there is a renaissance .. undeterred by arteriosclerotic considerations .. under processes running parallel

soon to overwhelm their artificial boundaries .. surpass them and spread out in ways unfathomable as yet ..

another paradigm shift at bay .. equivalent to Skinner’s misconception .. decades ago .. where what was driving the human individual was ignored .. totally .. only what was obvious, what one could sense, was taken into consideration ..

they didn’t go too far .. now is the time to go a step too further .. love .. one of the emotions .. emergent, as such .. was and is ..untouchable .. despite its ubiquitous existence .. its spiritual connotations prohibited its study .. the realm of religion .. subjective .. objective .. dichotomy  .. the persistence of the objective criterion a drawback for science .. obstructing its application .. its application considered a hurdle .. an obstruction ..

religion being driven out of its pedestal .. re-interpreted .. its role re-assigned .. leaving the field empty .. welcome to new interpretations ..  

the attempt to define religiosity in a vague formation in a brain area .. detached from other manifestations .. has been abandoned .. its connections indeterminate .. left unexamined .. a result of the motive behind that led to its discovery .. its exploitation for financial gains ..

it appears with the signature of force ..as it compels individuals to act in ways that cannot be explained by the existing prevailing principles .. nevertheless presents itself in all contexts possible and potential .. the bonds that hold people together in all settings possible at every level that are instantiated ..vertically and horizontally .. expressed in similar ways .. metaphor explained .. despite being kept apart .. in the fragmented manner that appear .. making its communication very difficult .. unrecognised .. misinterpreted .. today’s technological advances overcome the barriers between individuals .. the approach of one another is enabled .. it requires the conceptual tools to set it into a new level .. its growth is expected to reach exponential levels of achievement .. once understanding its existence sinks in into the human psyche .. acquires firm footing .. roots and is used as a ploy to explain under its newly-found perspective ..the world anew ..

The ..Starting Point ..Nothing ..from........

New Scientist Live: Come to a New Scientist event about Nothing.

such an approach ..alleviates .. the intrinsic ..inescapable ..human ..lack of perspective .. of guiding ..

to build from ..Nothing .. whatever .. it has to be build ..

at any level of organization .. structural or process ..

any ..whatsoever.. structure and process ..our ..being ..is part of..

or ..even .. and particularly .. the very essence of being .. in its ..deepest self ..

right down to the universe .. to existence ..proper ..

ought not to take for granted ..what is given .. endowed ..inherited .. thrown at ..

whatever way ..It ..is ..

it ..can be thought off .. one ..can think of it ..

instead of .. following ..obediently or otherwise .. what is given ..

.. in any ..whatsoever way ..given ..is surmised .. is ..taken as ..

a journey to the roots of being ..as it has been .. since the dawn of humanity .. our very ..purpose of existing at all

what,how,why ..we achieved, what we have achieved ..  up to now ..

..what we strive for ..all along ..

Nothing ..ought to be the beginning .. where ..any ..starting ..begins from ..

Boltzmann brains?..???..

"The paradox arose a decade or so ago, when physicists realized their models led to a future filled with Boltzmann brains: fully formed conscious entities that pop out of the vacuum. It sounds bizarre, but there's nothing to stop matter sometimes randomly arranging itself in just the right way for this to occur. The problem arises when you add in the universe's accelerating expansion."







Death by Higgs rids cosmos of space brain threat - physics-math - 04 September 2013 - New Scientist

..a decade ago .. future filled with .. Boltzmann brains.. what a name ..

.. fully formed conscious entities that pop out of the vacuum .. pop out .. out of vacuum .. out of nothing .. uncertainty principle implied .. evolution .. to its .. uttermost evolving stage .. an ultimate level .. penultimate

.." By contrast, the Higgs boson can do it using well-accepted physics."

..well-accepted physics .. mind-numbing .. blowing-up ..minds ..


Spooks in space - physics-math - 17 August 2007 - New Scientist


.. to contemplate about ..Boltzmann brains ..than ..fucking ..finance, turn a profit, money .. plaguing our very ..souls .. out of existence ..

and think about ...

a mathematical ..sort of .. algorithmic connection .. the span of entity existence ..shared by all entities .. at every level .. no matter how big or small .. all ..having the same duration .. alongside the ..human years .. the cat years .. the ..flea years .. then ..the stars ..years .. the galaxy years .. the .. superclusters ..years .. the universe years .. what .. for each level .. existence of entities in an upper level seems incomprehensible .. innumerable life cycles .. countless generations .. equivalent to mere seconds .. and even less .. for an upper level entity's lifespan ..

what equates their life cycles .. of the relevant entities .. from the lowermost levels of existence to the uppermost levels .. differing little .. time in lockstep fashion with space .. space expanse .. reflected into a .. time expanse .. being seeing with equivalent implements that space is observed .. microscopes ..telescopes ..observing the time dimensions .. out of which ..time-machines can be ..conceived and ..devised .. designed ..

a universe pops out .. every ..80 human years ..before it withers and dies .. for another universe to take its place .. replacing the old with the new .. in some sort of mathematical ..arithmetic or geometric progression .. calculated in magnitudes level-by-level ..

all sharing ..what ..the same ..destiny ..

the deterministic ..foundations of being ..

"The chemical universe comes in quantitative lumps. Moreover, chemical reactions just redistribute the existing lumps, with every single lump neatly accounted for."

 ...with every single lump neatly accounted for.. the deterministic ..viewpoint ..perspective .. quality .. the intrinsic property ..

"It is far more plausible that there are 101 guests, and each has a place setting of 1 knife, 2 forks, and 4 glasses. Slightly strange numbers, but clearly a terrific party."

each one content .. and out of this contention ..the terrific-ness .. properties emergent .. each one is accounted for .. what is lost in the hierarchy of levels .. the deterministic quality ..at the bottom of the level .. still persists .. but is clearly not evident .. as each level acquires its own set of properties that obscure the properties of its fundamental .. ingredients ..

but .. they are still there .. and because of them .. the properties of the higher levels .. emerge ..

a leap in mankind..

a leap in mankind .. in thought .. thought, as it can only be described as, at that particular time and space .. that requires .. demands of us .. to rid of ..past structures .. tempted to add ..all .. all past structures .. petty .. the reasons to retain ..to hold on to .. to any of them .. a severe drawback in entering the next stage of evolution of our species ..

something that it will come no matter .. whether wanted or not .. it will come quicker .. if prepared ..instead of stalling .. prolonging unnecessarily the process ..

the connections between .. and out of them the reason why should let ourselves embrace  wholeheartedly ..what is coming ..

emerges out of disparate fields of human endeavour ..that evolve unknowingly to one another .. yet .. individually progress only to meet in a particular cusp of time in the future .. our efforts should aim to accomplish such feat ..nearer in the future ..

the next stage in the evolutionary ladder .. entered gradually by different ..communities .. indicated by ..
Mathematicians think like machines for perfect proofs

"You wouldn't necessarily know it, but all of maths is currently founded on set theory, the study of collections of objects,.."

collections of objects ..

"Think of the elements of a set as grains of sand," says Andrej Bauer of the University of Ljubljana, Slovenia, who worked on the project. "You can build a sandcastle, but you have to work hard to make it."

..elements of a set as grains of sand .. the collections of objects .. to build sandcastles .. worlds ..

"The elements of a type are more like different arrangements of a collection of building blocks, their structure limited by the basic shape of the blocks. "The type is the stuff we can make following the rules of construction."

elements of a type .. different arrangements of collections of building blocks .. building blocks out of objects .. following the rules of construction .. the laws upheld .. the rules of constructing social groups .. social group .. the collection of objects .. objects ..the human individuals

"Assuming the underlying code – and the automated proof assistants that verify everything as the mathematicians goes along – is bug-free, it's impossible to write a proof that isn't correct."

impossible to write a proof that isn't correct .. a tool in the hands ..of everyone .. mathematician or not ..relying upon competent knowledge of the program itself .. only requires the parameters specific to any problem at hand ..

all bearing in mind

"Mathematics is the language scientists rely on to describe everything in the physical world, from the inner space of elementary particles to the outer space of distant galaxies. Most scientists believe deeply in the mathematical structure of nature; they see their task as simply discovering what this structure is. They view mathematics as an unambiguous business which leaves no scope for subjective contributions, no individuality of choice; it develops inexorably by a logical process in which old symbols are traded for new. But a closer look reveals that it is not quite so simple to pin down exactly what 'mathematics' is."

Shaking the foundations of mathematics: There are many interpretations of mathematics. The one we use has proved itself. But what would happen to our view of the Universe if we experimented with another?

nature, out of processes .. processes that differ in duration .. processes that take place .. complete in billionths of a second .. theorized Planck time its potential down limit.. instantaneously for human scales ..and processes that take eons to reach closure ..its completion .. the universe itself its potential upper limit .. our surroundings ..ongoing processes .. what underlies nature .. processes described by mathematics .. that leads the discovery .. of what already exists .. knowledge already existing .. simply arriving to knowledge .. an arduous journey for the individual .. since its birth ..

processes .. dressed as .. via the properties attributed to them .. in each system .. mathematics the naked reality beneath of what nature appears to us

Brazil uprising points to rise of leaderless networks

human individuals the grains .. to build sandcastles .. worlds .. free ..individuals from all other concerns that plague humanity since its inception milennia ago ..

the past structures ..and all that is adhered by them .. that present a severe drawback to humanity's evolution .. are dealt with.. in the only way possible ..not externally ..  but by the very same powers emanating from within .. following steadfastly .. the very rules that nature itself ..obeys ..

"That may not be surprising. Dan Braha at the New England Complex Systems Institute (NECSI) in Cambridge, Massachusetts, says civil unrest spreads like other spatial phenomena such as epidemics and forest fires: different factors control susceptibility, resistance, and rates of outbreak and transmission."

epidemics .. forest fires .. civil unrests .. the vocabulary used .. drawn out of the perspective of the very structures that undergo ..reconstruction .. as there are no other words to describe .. due to lack of them ..

emergence .. self-organisation .. out of the dynamics inherent .. processes that follow their natural completion ..driven by, out of all parts included .. that can only be described mathematically .. the whole out of the properties of all parts involved .. the emergence of such properties that reflect all that participate .. which it would make its own ..new .. more than the sum of its parts ..whole

and out of its mathematical description .. would emerge a new language to be used ..to describe such processes .. being rid of ..of all the remnants out of the past structures that linger menacingly

an ongoing process that demand its completion ..and it will have it .. irrespective of any obstacles confronted ..

Terry Young: Using Big Data to Spot Big Trends

identifying the properties .. out of the masses of big data .. the burden of complexity .. cleansed and cleared ..

the analytical power that computers afford ..provide .. out of the huge chunks of data digitally gathered and stored .. free for all to access ..  the properties emerging ..that would endow the systems to be .. the worlds ..sandcastles for people to build .. only but ..directed towards such purpose that would serve humanity at its best ..

all ..deeply ..fundamentally based .. human heart and soul .. on the grounds that each grain ..matters ..

The journey that people are on ..

..culminating into a world sparked from profound ideas .. among others ..what that article refers to ... one of the aspects open to inquiry .. by the human kind as a whole ..

Are there parallel universes?

..constructing out an uttermost glove-fit reality to the human individual..

.. the human individual evolves and rightly so the consensus reality should follow suit .. 

.. a constant struggle .. the world in turmoil .. a clash between different approaches .. to what the human individual is .. is represented as .. as it is the signature that determines the state the world ..humans included .. is ..

..to convince others .. as if, it is the case .. but it is not

".. are not commonplace only because we have not programmed our minds with the beliefs that would make them so."

The Universe as a Hologram by Michael Talbot

.. it is already there .. and it will happen ..

.. it only requires .. to make it happen .. a seed .. in each and every stage in the upcoming human mind development ..

".. Perhaps we agree on what is "there" or "not there" because what we call consensus reality is formulated and ratified at the level of the human unconscious at which all minds are infinitely interconnected."

reality .. current ..consensus reality .. constructed ..fabricated .. under parochial guidelines prevalent in earlier stages of the human mind evolution .. therefore unfit to humans .. did not evolve on a par to the evolution of the human mind ..

"What we perceive as reality is only a canvas waiting for us to draw upon it any picture we want."

.. on the road to constructing out a consensus reality .. fitting to the abilities of the human individual .. who is inseparable to the universe that contains its human entity ..

esoteric thoughts .. it is bound to ..

the concept of phase space extended

Understanding phase space ... using phase space to understand processes

p.134 James Gleick's "Chaos making a new science"

"In phase space the complete state of knowledge about a dynamical system at a single instant in time collapses to a point. The point is the dynamical system-at that instant. At the next instant, though, the system will have changed, ever so slightly, and so the point moves. The history of the system time can be charted by the moving point, tracing its orbit through phase space with the passage of time."

.. phase space provides the complete state of knowledge about a dynamical system ... the complete set of states of a dynamical system .. any state that a dynamical system can attain is included in the phase space for that particular system ... a phase space that can be defined by the variables and parameters of the dynamical system .. something that allows the mathematical treatment .. manipulation .. of the phase space of a system .. and can be used to explain dynamical systems

any dynamical system .. whatever its origin and field upon which it develops .. unfolds, it possesses its very own phase space and it will unravel its course within that particular phase space .. it will not fall outside the boundaries .. or limits of its phase space .. that phase space as determined by the variables and parameters of the dynamical system

what falls outside the phase space does not influence the progress of the dynamical system .. whatever falls outside the limits of the phase space is ignored .. it is not taken into consideration

where are you driving at .. what do you want to say, be specific .. phase space can be thought as being isolated ..from one another .. whatever happens within the phase space of a dynamical system .. has not any relevance any connection with the going-ons in another phase space of another dynamical system

phase spaces might intersect .. happen to intersect .. by virtue of unfolding at the same space or even time or a combination of space and time put together as spacetime .. should the two dimensions coincide for an intersection to occur? .. maybe should extend further to include of any variables and parameters that dynamical systems are defined by .. hence whenever there is coincidence .. then the phase spaces mix

when mixing occur the phase spaces .. are shared? .. the extend that each phase space possess becomes the phase space of their dynamical systems .. each dynamical system of the mixed phase spaces surpass its limits .. something that attributes nonlinearity in dynamical systems?

nonlinear functions hard to solve .. solvable only for special cases .. the only approach to solve nonlinear equations .. that is because nonlinear happens when the phase spaces of dynamical systems mix .. the particular phase spaces that mix should be taken into account in order to account for the nonlinearity in the development

the isolation of phase space used to explain why humans as dynamical systems have limited perceptions as the extend of their perceptions are determined by the phase spaces their dynamical systems unfold .. our perceptions are limited by the phase spaces that the variables and parameters set have set .. variables and parameters as are influenced by knowledge ..limited knowledge limited scope variables and parameters .. determine the extend of the phase spaces . and these phase spaces do not mix .. strangers but close together