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

A spacetime map?

The notion of a spacetime map emerged within my mind as I was reading page 42 of the Scientific American of June 2010 article, 'Is time an illusion?' p.41-47, by Craig Callender.

"Physicists, artists and graph makers of all kinds routinely depict time as another dimension of space, creating a unified spacetime-shown here as a three-dimensional block in which a ball bounces off a wall. Relativity theory holds that spacetime can be sliced up in various ways. But not all are equally sensible."

Spacetime that can be sliced up in various ways. Time only another additional dimension to the other three familiar of space, a map routinely drawn by reducing the height dimension to zero, effectively out of the three space dimensions only considering two of them in order to construct the map, a two-dimensional map. And as nothing prohibits what dimensions to choose in order to construct a map, a map is constructed by reducing any out of the three spatial dimensions to zero.

"The usual way takes slices of space at successive moments of time, creating a movie of the ball's motion. Each frame leads to the next, according to the familiar laws of physics."

Slices of space while considering all four dimensions of spacetime available, zeroing not only one but two out of the four spacetime dimensions. Zeroing one of space and the time dimension. The slices of space taken at successive moments of time but nevertheless at zero time. Frame-by-frame that altogether make up a movie, a mapping of an event unfolding, according to the laws of physics.

"An alternative considers slices not from past to future but from left to right. Each slice is part space, part time. To the left of the wall, the ball appears in two positions; on the right, it does not appear at all. If this slicing seems strange, it should: it makes the laws of physics very unwieldy."


..not the zeroing of time but one of the dimensions of space, a two-dimensional map out of time and one of space dimension

..the event looked from another perspective augmenting insights triggering imagination

Maps, slices of space and maps slices of spacetime. Maps constructed by choosing at will two out of the four now dimensions, as long as it produces our familiar two-dimensional arrangements.. at will


Objects and events. Spacetime maps for mapping events? Simultaneity being absolute, observer-independent? Events being observer-independent?

"No matter when or where an event occurs, classical physics assumes that you can objectively say whether it happens before, after or simultaneously with any other event in the universe. Time therefore provides a complete ordering of all the events in the world. Simultaneity is absolute-an observer-independent fact. Furthermore, time must be continuous so that we can define velocity and acceleration."

playing with words dragging along the concepts they are attached to, forcing in their way new conceptions to appear, to create new notions in my mind

Chaotic machines...

There are a few things I would like to ponder on and let the thoughts develop to whatever end. It is mentioned in page 48 of James Gleick “Chaos, making a new science” that

“A letter from a colleague informed him that many systems were not so well-behaved as he had imagined, and it described a counterexample, a system with chaos and stability, together. This system was robust. If you perturbed slightly, as any natural system is constantly perturbed by noise, the strangeness would not go away. Robust and strange - Smale studied the letter with disbelief that melted away slowly.”

And further

“Chaos and instability, concepts only beginning to acquire formal definitions, were not the same at all. A chaotic system could be stable if its particular brand of irregularity persisted in the face of small disturbances.”

Irregularities persisting in the face of small disturbances? How can this be used? Irregularities that do not go away? Appearing time and time again? Could that be all that chaos is? Just a curiosity? What about your claim about chaos being creative? The way that nature, and not only, seeks out solutions to problems presented? Any problems? The idea of stable chaos embodied in the persistent presence of irregularities, does not conform with the idea that immersing in chaos what comes out would be a novel way to deal with a problem. A new way of thinking about it.

Or is it that the difference is what a chaotic system is fed with. A chaotic machine, all-in-one, along with its persistent irregularities, an ensemble that acts upon its input to produce a certain output, and in-between the input is treated in various ways characteristic of the particular chaotic machine. The persistent irregularities the hallmark of its bearer, the particular chaotic machine.

 Chaotic machines being widespread and ubiquitous, operating under the principles of self-similarity and universality, for each activity natural or man-made, with their own particular range of parameters representing the input that feeds into them, which as it is mentioned in the same page

“The chaos Lorenz discovered, with all its unpredictability, was as stable as a marble in a bowl. You could add noise to this system, jiggle it, stir it up, interfere with its motion, and then when everything settled down, the transients dying away like echoes in a canyon, the system would return to the same peculiar pattern of irregularity as before. It was locally unpredictable, globally stable. Real dynamical systems played by a more complicated set of rules than anyone had imagined.”

Chaotic machines fractally connected, where their inputs being the outputs of other chaotic machines residing in activity levels below, hierarchically arranged in an intricate web that surpasses all reality. The range of parameters in chaotic machines in a particular level, being crucial, to the range of parameters in chaotic machines immediately below.

ascension, a term to use to define the origin of properties in emergent systems

baby ...
the tenets of emergence .. literally ..
level upon level ..
attain ..de-emergence .. simulated ..
ascension .. in properties ..

(ascension, a term to use to define the origin of properties in emergent systems ..dragging along the connotations, the relevant concepts, notions the word 'ascension' associates with.. ascending to another realm ..above and beyond .. unseen and ..untraceable .. and by accepting such a term .. bringing down ..to earth, that which can not be explained .. defy explanation)   

properties ..assumed .. with the tasks .. pre-ordained with what ..one knows they ought to do ..

..pre-knowledge hinders .. the properties ..that lower levels ..of constructed artificial intelligence .. ought to accomplish ..

based upon simple mechanisms ..
simple mechanisms ..exhausted ..for and up to the level .. invented ..

avoid what is .. in the back of the mind ..

System agents retain their stability throughout all the interactions pertained in a system

Understanding money from the viewpoint of systems. Based on the grounds of how systems are organised determining the flow which energy and momentum within the system represents carried by the relevant agents of the system.

Agents in a system are propelled within the system boundaries by a specific amount of momentum drawn out of the energy they possess. The higher the energy they possess the higher their momentum, the lower the energy the lower the momentum. The length of the path traversed depend on the amount of energy they have at their disposal and it is depleted in the course of their run.

Draw energy when energy levels are depleted? Could the agents be drained of their energy? Has the adiabatic principle anything to do with it? Prevents the total energy drainage of the agents in a system? Only controlled amounts of energy are lost by an agent, as the agent needs to retain its stability.

The embedded systems in an individual. The individual as a macroscopic system, the embedding system. One of the embedded systems in the macroscopic system individual is defined by its biological nature. Biological systems are nourished by food. Food represent the energy reservoir and define momentum. Momentum is expressed by mobility. The human individual a mobile entity.

Food which is required by the macroscopic system of the individual; .......

..sub-processes within sub-processes..

Chaotic logic p.29

"His idea is that lower levels of the hierarchy correspond to older, evolutionarily prior forms of perception and control."

.. so apt .. lower levels .. correspond to older evolutionary forms .. living fossils ..idea .. how ..perception and control .. started out .. and evolved .. and what it evolved out .. is still preserved .. though the word ..'preserved' doesn't do it credit .. as it is upon it .. the rest of the construction .is based upon .. is founded .. upon .. the attractors .. which present a solid base .. on top of which .. the rest of the construction .. erects upon ..

"Let us sum up. The multilevel control methodology, in itself, has nothing to do with patterns. It is a very simple and general way of structuring perception and action: subprocesses within subprocesses within subprocesses, each subprocess giving orders to and receiving feedback from its subsidiaries. In this general sense, the idea that the mind contains a multilevel control hierarchy is extremely non-controversial."

..subprocesses within subprocesses ..

..the finite field notion, the "wraparound" feature .. finite fields the systems existing

Finite fields the role of parameters, of constants, prediction.

In Ben Goertzel's 'Chaotic Logic', page 35, Chapter 3.4.1 Discrete Logarithms, mentions about finite fields. Its intuitive content generates ideas in my mind.

"3.4.1. Discrete Logarithms (*)
For a simple mathematical example, let us look to the theory of finite fields. 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 (the size must be a prime 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, although the results are often counterintuitive -- for instance, 12/8 = 3, and 2/3 = 5 (to see why, just multiply both sides by the denominator)."

..the finite field notion, the "wraparound" feature .. finite fields the systems existing .. for all systems possible and plausible .. physical but not only .. mental, social, psychological, emotional .. all to be seen as finite fields .. scale being not the limit .. and prediction worked out in the same way it is described in this chapter of Ben Goerzel's, chaotic logic.

The size of the fields given


"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. But in a finite field, exponentiation must be defined in terms of the "wraparound" 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? 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, 4.
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. The only way to do all 1000 steps in parallel, 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.
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."

in the beginning there are ..processes

"His team modelled a minimal black hole, defined only by having an inside and an outside, using quantum theory. To their surprise, they found that this object reproduces a lot of the features of real black holes that are thought to rely on gravity, including Hawking radiation, which could occur via a process called quantum tunnelling."

To their surprise,.. something ..unexpected they stumbled upon .. just by chance .. Thomas Kuhn ..in mind ..

the black hole model .. simulated upon the processes of quantum mechanics .. the processes ..fundamental .. quantum processes .. 'anneal' in any 'field' .. for the lack of a more elaborate concept in my mind .. in this case enable the creation of black holes in physical space and virtual space ..alike



processes ..and process space .. in the beginning there are ..processes ..and as the processes undergo ..create ..the space where they develop .. be that space ..physical space ..or any other 'space' that allows the processes to undergo ..

.. it reveals its very nature ..give us a glimpse of the fundamental .. in any way this is perceived

.. virtual .. what effectively is the space where their black hole model ..materialises .. produces ..as I gather .. the image on a computer screen ..

.. but what is significant are the algorithms fed into the machine that represent the processes .. the quantum processes .. the ..maths .. that generate the model black hole .. that in itself behaves as the real one ..

.. reproduces a lot of the features of real black holes .. and as the techniques .. and the algorithms are perfected the black hole models will rival the real black holes..

to study at leisure ..without the danger to be sucked into its vortex and vanish .. self and co-workers ..to the earth itself ..


"This chimes with suggestions that gravity is not a fundamental component of the universe but an emergent property of quantum mechanics, just as waves are an emergent property of water molecules."

.. an emergent property .. thinking ..takes strides ahead

Structure

"The term 'structure' may be defined as a pattern,.. "

.. why may be? What 'may be' denotes, what brings to mind, by saying 'may be', is that the distinction between 'might' and 'might not', is not as yet resolved. Confuses the mind, undermines the downright assertion required, to build further on solid foundations instead of a shaky foundation, ready to tumble down if the 'might not' element gains priority. The whole 'structure' collapses, leaving the individual in disarray.

Beyond the point at bay that 'structure' is a symbol, to be used at will to suit the mind in its attempt to organise its world and not the other way around, where the mind is required hopelessly and helplessly to fit itself under an umbrella that its self created, in the first place. Pre-eminence on the symbols used, and not on the symbol-creators.

Structure is a pattern. The pattern, the category into which 'structure' belongs to, therefore structure is a pattern.

proto-chaos ..and that primordial chaos is still around and consciousnesses continually collapse to further render the cosmos into what is going to be

proto-chaos .. quantum superpositions.. it fits with the ancient Greek belief about chaos .. lack of form that an observer ..a consciousness collapsed into the cosmos  ... and that primordial chaos is still around and consciousnesses  continually collapse to further render the cosmos into what is going to be
we are the descendants of gods ..gods ourselves the bicameral consciousness the voices in the head  fallen lost it
quantum superposition collapse what is right behind  gods awesome powers
religion as it is practiced slavery utter and complete, distance humans from their roots renders them stupid, trivialize their existence
time and space
time and space advanced and retarded waves in Feynman's thoughts of dealing with the collapse of superpositions shaped the cosmos acquiring the form it has now via the collapses effected by the consciousnesses in the future the waves traveling backwards to the beginnings to render the cosmos to suit their existence .. to be favourable for their existence

the notion of bootstrapped in mind .. future events bootstrapped to past .. past bootstrapped to future .. coincide
what can one say about the ideas we come up with is not but .. dictated by the future ..past bootstrapped to future .. how the collapse is effected.. what we envisage ourselves about our future is but the future states transmitted by advanced or retarded waves to the past
quantum mechanical interpretation of fate and destiny

that might explain what thought itself is ..quantum mechanical way process-wise .. advanced and retarded waves handshake

conditions abrupt termination  of handshake apathy resignation