"What happens when the brakes are applied? Let us look first from the frame of reference of the outside world."
I do not know why .. but I feel there is a connection there .. the .. reference of the outside world .. earth moving .. an accelerated frame of reference .. the bus itself .. by looking at it .. from outside .. having the outside as reference .. think what will happen ..when earth decelerates
though that is not the intriguing point .. as much as .. think this whole subject of inertial .. fictitious forces .. from the perspective of processes .. systems instantiating by the ..action? ..of processes .. inertial forces .. in that framework
forces .. that develop in systems .. what makes entities in systems .. to advance forth .. to move from state to state .. the forces that develop within the system .. which makes them .. unique? .. for that system alone .. fictitious forces
fictitious forces .. whose their existence .. is bounded within the system .. they were born in .. they are real .. for its occupants .. but .. un-real for .. outside the system
where are you getting it .. that even the forces .. that are at large .. in the universe .. the ones called fundamental .. they are no different .. than the fictitious forces .. the universe .. a system .. as the many systems within .. and what governs all systems .. rules universe too .. the universe .. the reference frame .. and outside the universe .. the reference of the outside world .. all the forces .. within the universe .. fundamental to us .. but still fictitious .. from the frame of reference of the outside world
forces .. apparent .. created .. within the system universe .. and their existence .. makes the universe possible .. powers up .. drives through .. all the processes .. that form galaxies .. stars .. worlds .. despite their mega-scale .. nevertheless ... fictitious .. the mega-scale .. and their growth .. explained as the growth of the objects within .. demand .. greater and greater forces .. sustained .. suspended .. by the very presence .. their existence creates
from the other point .. as they are self-creating .. they do not require .. any reservoir from which they draw .. their power .. they are supplied with ..
.. gives reason .. to the universe born out of nothing ..
Monday, 26 April 2010
Nothing .. the origin of everything. Thoughts
nothing .. the origin of everything .. any line of processes .. emanate from nothing .. and that is where .. the line of processes .. instantiate .. materialise .. when there is .. a standstill? .. when the line of processes .. has lost its momentum? .. it is not powered up? .. it is not driven up .. forward?
the idea .. that came to me .. after I watched .. this advert .. of orange mobile .. nothing .. the user have nothing to say .. to the world .. sort to speak .. to announce to the world .. what is thinking .. at a particular instance .. in the social network .. is part of .. which taking this modern activity aside .. is a state that the mind enters .. when everything in one's life .. appears empty .. a void pressing deep in one's existence .. where one can not see .. anything at all .. when what really happens .. is .. reverting? .. to its origin .. to nothing
nothing being .. the origin of the very universe .. of universes .. the quantum fluctuation .. powered by the uncertainty principle .. the seed .. the quantum seed .. that gave birth to the universe .. the big bang .. that created order .. out of nothing
bootstrapped? .. suspended ..
universe spread .. making it possible .. for countless processes .. to spring up .. themselves following the same blueprint .. originating from nothing .. the human individual .. representing such a line of processes .. which in the same way .. like its largest counterpart .. started from nothing .. itself suspended .. bootstrapped .. by the very own processes .. that gave birth to it .. suspending it .. to its origin to nothing ..
thinking .. what powers .. the bootstrapped .. suspended processes ..
or .. probably .. or potentially .. we are in that state .. the state of nothing .. and we are kept .. suspended? .. via the processes .. which we are into .. letting go of the processes .. we are left .. in the void .. at the ..mercy .. of our origin .. nothing
processes .. branching out .. from a central .. trunk .. bifurcating .. to an array .. of other processes .. each dealing with .. to a different aspect .. aspects .. dimensions .. opening up .. new areas .. for activity to emerge ..
engaging in .. the processes .. engagement failing .. or not materialising .. leaves us .. at the midsts .. of the void .. that our existence sprang out of .. and as this attribute .. is universal .. applies to all processes .. instantiating in the universe
surrounded by the void .. the .. nothing .. an idea .. that's brought forth .. in that kid's film .. where the fantasyland .. was destroyed .. by the void ..as people stopped believing in fairy tales .. the processes that gave birth .. created fantasia .. were not engaged in .. and as a result .. fantasia was .. receding into oblivion ..
disengage .. from the processes.. and you are lost .. you are left with nothing ..
god .. the solace .. a refuge .. running to it, to shelter in .. the angst .. felt ..
the idea .. that came to me .. after I watched .. this advert .. of orange mobile .. nothing .. the user have nothing to say .. to the world .. sort to speak .. to announce to the world .. what is thinking .. at a particular instance .. in the social network .. is part of .. which taking this modern activity aside .. is a state that the mind enters .. when everything in one's life .. appears empty .. a void pressing deep in one's existence .. where one can not see .. anything at all .. when what really happens .. is .. reverting? .. to its origin .. to nothing
nothing being .. the origin of the very universe .. of universes .. the quantum fluctuation .. powered by the uncertainty principle .. the seed .. the quantum seed .. that gave birth to the universe .. the big bang .. that created order .. out of nothing
bootstrapped? .. suspended ..
universe spread .. making it possible .. for countless processes .. to spring up .. themselves following the same blueprint .. originating from nothing .. the human individual .. representing such a line of processes .. which in the same way .. like its largest counterpart .. started from nothing .. itself suspended .. bootstrapped .. by the very own processes .. that gave birth to it .. suspending it .. to its origin to nothing ..
thinking .. what powers .. the bootstrapped .. suspended processes ..
or .. probably .. or potentially .. we are in that state .. the state of nothing .. and we are kept .. suspended? .. via the processes .. which we are into .. letting go of the processes .. we are left .. in the void .. at the ..mercy .. of our origin .. nothing
processes .. branching out .. from a central .. trunk .. bifurcating .. to an array .. of other processes .. each dealing with .. to a different aspect .. aspects .. dimensions .. opening up .. new areas .. for activity to emerge ..
engaging in .. the processes .. engagement failing .. or not materialising .. leaves us .. at the midsts .. of the void .. that our existence sprang out of .. and as this attribute .. is universal .. applies to all processes .. instantiating in the universe
surrounded by the void .. the .. nothing .. an idea .. that's brought forth .. in that kid's film .. where the fantasyland .. was destroyed .. by the void ..as people stopped believing in fairy tales .. the processes that gave birth .. created fantasia .. were not engaged in .. and as a result .. fantasia was .. receding into oblivion ..
disengage .. from the processes.. and you are lost .. you are left with nothing ..
god .. the solace .. a refuge .. running to it, to shelter in .. the angst .. felt ..
Tuesday, 6 April 2010
Quantum entanglement computations from the perspective of variable connection points
Chaos, emergence, complexity concepts risen and spread in the minds of people inextricably intertwined, each one built out of the ideas that build the other, notions coming out from the, still fresh in my mind, thoughts expressed by John Holland, in his book 'emergence, from chaos to order', what is mentioned in Chapter 9 titled Variation, page 167,
"With these provisions a mechanism is described by a string over the alphabet {1,0,#}."
an alphabet referring to the elements that comprise an entity and determines its state, a universal alphabet, so simple in its inception, (1) if it exists, (0) if it does not exist and hash (#) if it "doesn't care"
By taking the current stage of the concepts developed for each of these entities, as being their current states, and as for every and any state possible, what it depends upon, therefore owes its existence, being a necessary condition for the state's own existence, and by giving rise to the state's existence, its own existence instantiated, effected therefore verified and proven. Its value (1) for existing.
For what the afore-said state does not depend upon, it doesn't exist. Its presence is not required or even its absence is a must, vital for the state's existence. Other states might require its existence, therefore though its existence is not instantiated in that case and can not be verified or proven, it can be verified and proven by other existent states. Its value (0) for non-existing.
And the "doesn't care" element of the alphabet for all other elements that can be taken as the scenery, since their being there is irrelevant to the state's existence only filling up space in the state's background. Its value (#) for "doesn't care".
And all in the context of the hierarchical structure of nature, being built by levels upon levels of entities all the states that makes such super-structure to exist.
Quantum entanglement, a special case of connections, connections in the sense .. John Holland puts it in Chapter 9, Variation. What is mentioned in the article “Sudden Death” Threatens Quantum Computing, ..
".. physicists need only figure out how rapidly sudden death can occur, then design systems that finish quantum calculations within that window."
a window .. a time-step
Entanglement, underlying nature, one of the mechanisms in its deep deep recesses. Fundamental mechanisms, its ultimate expression, a mechanism or mechanisms ruled by the very same cgp's and especially the penultimate case of cgp's, the cgp-v's .. the variable constrained generating procedures.
Constraints that generate procedures, procedures acting upon variable connection points, an ever changing landscape step-by-step, but for each time-step they are as good as acting on fixed connection points presented as a discreet time-step and in that time-step, the window referred to above, to perform the quantum calculations.
As for storing any outputs, for further use, for posterity, the notation derived and presented by John Holland in Chapter 9 Variation could provide a leeway in accomplishing the entire computation.
Treat the whole entanglement phenomenon as an emergent phenomenon, and use the angles opening up by such an approach to derive the quantum calculations, performed in a series of time-steps as the emergent phenomenon evolves.
Entanglement at the threshold of time, a time that lingers between the conundrum of existence, entanglement as such, its weird properties, a result of how we experience distance.
Distance which becomes a concept on the back of time, and that is the (horse)-back equivalent, as the distance connected it is only comprehended as a derivative of time. The time and by the time it takes, namely the speed, for a message to be transmitted. For the connection to be effected. A connection which, for all concerns it appears as violating physics, a physics developed from our only standpoint, what we have, what is allowed for us, and we can not do anything, but use it and interpret the world through it, though it looks as if it is not the only one that nature has.
What if time is not fundamental, but derived and what appears weird, run against our common sense, our physics, it is not weird at all, but a commonality that we are unable to fathom, as yet.
"With these provisions a mechanism is described by a string over the alphabet {1,0,#}."
an alphabet referring to the elements that comprise an entity and determines its state, a universal alphabet, so simple in its inception, (1) if it exists, (0) if it does not exist and hash (#) if it "doesn't care"
By taking the current stage of the concepts developed for each of these entities, as being their current states, and as for every and any state possible, what it depends upon, therefore owes its existence, being a necessary condition for the state's own existence, and by giving rise to the state's existence, its own existence instantiated, effected therefore verified and proven. Its value (1) for existing.
For what the afore-said state does not depend upon, it doesn't exist. Its presence is not required or even its absence is a must, vital for the state's existence. Other states might require its existence, therefore though its existence is not instantiated in that case and can not be verified or proven, it can be verified and proven by other existent states. Its value (0) for non-existing.
And the "doesn't care" element of the alphabet for all other elements that can be taken as the scenery, since their being there is irrelevant to the state's existence only filling up space in the state's background. Its value (#) for "doesn't care".
And all in the context of the hierarchical structure of nature, being built by levels upon levels of entities all the states that makes such super-structure to exist.
Quantum entanglement, a special case of connections, connections in the sense .. John Holland puts it in Chapter 9, Variation. What is mentioned in the article “Sudden Death” Threatens Quantum Computing, ..
".. physicists need only figure out how rapidly sudden death can occur, then design systems that finish quantum calculations within that window."
a window .. a time-step
Entanglement, underlying nature, one of the mechanisms in its deep deep recesses. Fundamental mechanisms, its ultimate expression, a mechanism or mechanisms ruled by the very same cgp's and especially the penultimate case of cgp's, the cgp-v's .. the variable constrained generating procedures.
Constraints that generate procedures, procedures acting upon variable connection points, an ever changing landscape step-by-step, but for each time-step they are as good as acting on fixed connection points presented as a discreet time-step and in that time-step, the window referred to above, to perform the quantum calculations.
As for storing any outputs, for further use, for posterity, the notation derived and presented by John Holland in Chapter 9 Variation could provide a leeway in accomplishing the entire computation.
Treat the whole entanglement phenomenon as an emergent phenomenon, and use the angles opening up by such an approach to derive the quantum calculations, performed in a series of time-steps as the emergent phenomenon evolves.
Entanglement at the threshold of time, a time that lingers between the conundrum of existence, entanglement as such, its weird properties, a result of how we experience distance.
Distance which becomes a concept on the back of time, and that is the (horse)-back equivalent, as the distance connected it is only comprehended as a derivative of time. The time and by the time it takes, namely the speed, for a message to be transmitted. For the connection to be effected. A connection which, for all concerns it appears as violating physics, a physics developed from our only standpoint, what we have, what is allowed for us, and we can not do anything, but use it and interpret the world through it, though it looks as if it is not the only one that nature has.
What if time is not fundamental, but derived and what appears weird, run against our common sense, our physics, it is not weird at all, but a commonality that we are unable to fathom, as yet.
Thursday, 11 March 2010
Emergence, consciousness .. and physical substrates.
How can these, seemingly different notions, can be reconciled? What notions? For once is what Steven Johnson refers to in his O'Reilly network interview, on the subject of emergence..
"Emergence is what happens when the whole is smarter than the sum of its parts. It's what happens when you have a system of relatively simple-minded component parts -- often there are thousands or millions of them -- and they interact in relatively simple ways. And yet somehow out of all this interaction some higher level structure or intelligence appears, usually without any master planner calling the shots. These kinds of systems tend to evolve from the ground up."
Emergence appearing(?)out of systems of simple component parts which they interact in relatively simple ways. This interaction brings about a new structure, (or may be not a new structure, but the old structure revamped), that as a whole, possess qualities not exhibited by any of its component parts.
And the other notion?
Each neuron cell represents a state amidst a multitude of superposed states, the whole brain in a superposition status? The brain in quantum superposition?
From Jonathan CW Edwards, in 'Is Consciousness Only a Property of Individual Cells?' paper
" .... consciousness is a basic correlate of function, but a function that only certain fundamental physical substrates can subserve. To know the true function is to know the substrate."
A fundamental physical substrate that serves as the ground that consciousness arises from? What substrate this might be?
"Emergence is what happens when the whole is smarter than the sum of its parts. It's what happens when you have a system of relatively simple-minded component parts -- often there are thousands or millions of them -- and they interact in relatively simple ways. And yet somehow out of all this interaction some higher level structure or intelligence appears, usually without any master planner calling the shots. These kinds of systems tend to evolve from the ground up."
Emergence appearing(?)out of systems of simple component parts which they interact in relatively simple ways. This interaction brings about a new structure, (or may be not a new structure, but the old structure revamped), that as a whole, possess qualities not exhibited by any of its component parts.
And the other notion?
Each neuron cell represents a state amidst a multitude of superposed states, the whole brain in a superposition status? The brain in quantum superposition?
From Jonathan CW Edwards, in 'Is Consciousness Only a Property of Individual Cells?' paper
" .... consciousness is a basic correlate of function, but a function that only certain fundamental physical substrates can subserve. To know the true function is to know the substrate."
A fundamental physical substrate that serves as the ground that consciousness arises from? What substrate this might be?
Saturday, 2 January 2010
Seeking out the depths of reality, its utter complexity within the grasp of the human brain
John H. Holland .. 'emergence from chaos to order', Chapter 5, 'Neural Nets', page 81-114, while reading through insights instantiate, glimpses flicker, pointing towards answers for questions in fundamental processes of the workings of the human brain.
in p. 86
"To construct a useful model of the CNS using this information, we begin with the idea that the activity of the CNS is described by which neurons are firing during a given time-step. Because of the uniformity of the pulses emitted, we can simply say that each neuron is either "on" (emitting a pulse) or "off" (quiescent) during a time-step."
.. the uniformity of the pulses emitted .. the basis upon which the brain works .. all the cognitive tasks undertaken .. the mind that comes out of .. the ideas, thoughts, knowledge .. all start as pulses .. uniform pulses .. indistinguishable from one another .. whose only effect is to turn a neuron 'on' or 'off'
in p. 90
"Because of its distributed nature and the simplicity of its elements, a neural net is a satisfactory vehicle for studying emergence: sophisticated behaviors emerge from the aggregation of simple actions."
simple actions, being .. the uniform pulses emitted .. and they give rise to sophisticated behaviors .. emergence
in p. 86
"We can further simplify the model by assuming that all pulses arriving within a time-step have equal effect in determining whether or not the neuron fires (in technical jargon, we ignore the phase of the pulses)."
simplifying matters for the sake of making more usable the model employed .. uniformity of pulses .. is what is only necessary .. ignoring all other aspects of the pulses-waves as amplitude, wavelength, frequency, .. phase .. coherent or incoherent .. is that allowable? .. it makes no difference .. the characteristics of one pulse from another .. for the emerging outcome? .. for the thoughts, the ideas, the mind instantiating? ..
.. or the unique features of each pulse do matter .. but at a deeper level. Each pulse, no matter the unique features it carries, is capable of eliciting .. the responses that emerge into sophisticated behaviors .. making up minds ..
.. and the unique features carried are significant at another, deeper level .. a level analogous, to what crossed my mind .. while I was contemplating .. on human resolving time compared with the resolving time of a fast electronic device.
The point - entering my mind right now - is, that the features of waves, which according to the model John H. Holland uses, namely the phase of the pulses-waves as irrelevant for the behaviors they instantiate, are what is appropriate with the reality the human brain is able to extract, engineered into the specific mechanics of the human brain, and not, for that matter, of a fast electronic device.
Equipped with the resolving time of a fast a electronic device the mind, is expected, to be different. Another reality or another aspect of reality, appearing before the eyes of the human individual. Whether, it would end up being another totally different reality or another aspect of reality, it remains to be seen.
What someone can surmise, is that the pulses-waves will be processed in the same way, posited by John H. Holland's model, namely pulses-waves being uniform, represented as simple actions which out of their aggregation, the sophisticated behaviors emerge, something that increases the complexity of reality alongside the apparent capability - or potential - of the human brain, to fathom into this deeper reality in its full extent.
in p. 86
"To construct a useful model of the CNS using this information, we begin with the idea that the activity of the CNS is described by which neurons are firing during a given time-step. Because of the uniformity of the pulses emitted, we can simply say that each neuron is either "on" (emitting a pulse) or "off" (quiescent) during a time-step."
.. the uniformity of the pulses emitted .. the basis upon which the brain works .. all the cognitive tasks undertaken .. the mind that comes out of .. the ideas, thoughts, knowledge .. all start as pulses .. uniform pulses .. indistinguishable from one another .. whose only effect is to turn a neuron 'on' or 'off'
in p. 90
"Because of its distributed nature and the simplicity of its elements, a neural net is a satisfactory vehicle for studying emergence: sophisticated behaviors emerge from the aggregation of simple actions."
simple actions, being .. the uniform pulses emitted .. and they give rise to sophisticated behaviors .. emergence
in p. 86
"We can further simplify the model by assuming that all pulses arriving within a time-step have equal effect in determining whether or not the neuron fires (in technical jargon, we ignore the phase of the pulses)."
simplifying matters for the sake of making more usable the model employed .. uniformity of pulses .. is what is only necessary .. ignoring all other aspects of the pulses-waves as amplitude, wavelength, frequency, .. phase .. coherent or incoherent .. is that allowable? .. it makes no difference .. the characteristics of one pulse from another .. for the emerging outcome? .. for the thoughts, the ideas, the mind instantiating? ..
.. or the unique features of each pulse do matter .. but at a deeper level. Each pulse, no matter the unique features it carries, is capable of eliciting .. the responses that emerge into sophisticated behaviors .. making up minds ..
.. and the unique features carried are significant at another, deeper level .. a level analogous, to what crossed my mind .. while I was contemplating .. on human resolving time compared with the resolving time of a fast electronic device.
The point - entering my mind right now - is, that the features of waves, which according to the model John H. Holland uses, namely the phase of the pulses-waves as irrelevant for the behaviors they instantiate, are what is appropriate with the reality the human brain is able to extract, engineered into the specific mechanics of the human brain, and not, for that matter, of a fast electronic device.
Equipped with the resolving time of a fast a electronic device the mind, is expected, to be different. Another reality or another aspect of reality, appearing before the eyes of the human individual. Whether, it would end up being another totally different reality or another aspect of reality, it remains to be seen.
What someone can surmise, is that the pulses-waves will be processed in the same way, posited by John H. Holland's model, namely pulses-waves being uniform, represented as simple actions which out of their aggregation, the sophisticated behaviors emerge, something that increases the complexity of reality alongside the apparent capability - or potential - of the human brain, to fathom into this deeper reality in its full extent.
Labels:
emergence,
human brain,
reality
Tuesday, 29 December 2009
Information, state of motion, frame of reference. Concepts, applied, context. The stuff of nature.
'Special Relativity' by A.P. French, Chapter 3 'Einstein and the Lorenz-Einstein transformations', p. 63.
" .. -that nature would apparently yield up no information about our state of motion with respect to a supposed fundamental frame of reference."
.. in that sentence connected, information, state of motion, frame of reference.
Information, the stuff that concepts are build out with, concepts in-turn materialise, instantiate reality, responsible for the state of motion, how nature unfolds within the bounds of frames of reference. A sentence that ties up, in a specific, dare say unique way, these three entities: information, state of motion, frame of reference.
Specific, unique way, out of the myriad (ignorance exacerbates imagination) ways available for information to be tied up, only a few combinations are allowed, survive to become the stuff for concepts, concepts that define reality, depict the way nature materialises. Information a fundamental entity that each minute detail of its existence, of its state of motion in all frames of reference brings forth. Information as conveyed in all manners of media, for all possible states, but what matters more, for information to assume its fundamental status, is to become the stuff to build the concepts that reality adheres to, is woven out of.
A notion that comes out from the processes involved in generating concepts by conversations,as it is mentioned in SCHOOLS AND EDUCATION :: LEARNING AND TEACHING, VII Conference of Foundation 2020, Thursday, 15 March 2007, Hotel Regent Esplanade, Zagreb, Education in Mind – Mind in Education: An Invitation to A Conversation. Graham Barnes, Chapter 'Education: A Story', p.16
"We constitute our concept of education (or any other concept)
(1) through conversations,
(2) in which concepts or procedures are shared,
(3) through sharing these concepts we apply them.
(4) our application of a concept creates a context, and applying a concept as a procedure produces a product. The context or product is relevant to the concept for the participant(s) who find meaning in the context.
(5) conversations are between participants but the participants are not necessarily, and are certainly not limited to, individual persons as the participants are also concepts. The participants in the conversation as concepts are also constituted by the conversation."
Concepts, as they are applied create a context, the frame of reference, produce a product. The context or product being relevant to the concept by virtue of the meaning it confers, when applied. And what comes out of, when participants in conversations are thinking about concepts, is that concepts are participants also. Concepts are thinking too, what ties human consciousness with nature. The meaning sought out by individuals when thinking the concepts inextricably tied up with the way nature is.
Concepts, in all their manifestations, propositions, theories, axioms, laws, in all the various ways they are used for, to indicate or prove or lead to, employed in thinking, the process that emulates nature's workings, guides our consciousness. It achieves its full potential when individuals make use of chaotic processes entrenched deep into our brains. Where nothing is overlooked, no bit of information possessed or coming across, is irrelevant or insignificant, and coupled with its exponential non-linear mode of being applied, lead to radical overhauls in conceptualising effortlessly.
Able to guide the mind to the right information amidst the penumbrae, seemingly disorganised bits of information, chaos apparent, and out of the penumbrae into brand new insights.
" .. -that nature would apparently yield up no information about our state of motion with respect to a supposed fundamental frame of reference."
.. in that sentence connected, information, state of motion, frame of reference.
Information, the stuff that concepts are build out with, concepts in-turn materialise, instantiate reality, responsible for the state of motion, how nature unfolds within the bounds of frames of reference. A sentence that ties up, in a specific, dare say unique way, these three entities: information, state of motion, frame of reference.
Specific, unique way, out of the myriad (ignorance exacerbates imagination) ways available for information to be tied up, only a few combinations are allowed, survive to become the stuff for concepts, concepts that define reality, depict the way nature materialises. Information a fundamental entity that each minute detail of its existence, of its state of motion in all frames of reference brings forth. Information as conveyed in all manners of media, for all possible states, but what matters more, for information to assume its fundamental status, is to become the stuff to build the concepts that reality adheres to, is woven out of.
A notion that comes out from the processes involved in generating concepts by conversations,as it is mentioned in SCHOOLS AND EDUCATION :: LEARNING AND TEACHING, VII Conference of Foundation 2020, Thursday, 15 March 2007, Hotel Regent Esplanade, Zagreb, Education in Mind – Mind in Education: An Invitation to A Conversation. Graham Barnes, Chapter 'Education: A Story', p.16
"We constitute our concept of education (or any other concept)
(1) through conversations,
(2) in which concepts or procedures are shared,
(3) through sharing these concepts we apply them.
(4) our application of a concept creates a context, and applying a concept as a procedure produces a product. The context or product is relevant to the concept for the participant(s) who find meaning in the context.
(5) conversations are between participants but the participants are not necessarily, and are certainly not limited to, individual persons as the participants are also concepts. The participants in the conversation as concepts are also constituted by the conversation."
Concepts, as they are applied create a context, the frame of reference, produce a product. The context or product being relevant to the concept by virtue of the meaning it confers, when applied. And what comes out of, when participants in conversations are thinking about concepts, is that concepts are participants also. Concepts are thinking too, what ties human consciousness with nature. The meaning sought out by individuals when thinking the concepts inextricably tied up with the way nature is.
Concepts, in all their manifestations, propositions, theories, axioms, laws, in all the various ways they are used for, to indicate or prove or lead to, employed in thinking, the process that emulates nature's workings, guides our consciousness. It achieves its full potential when individuals make use of chaotic processes entrenched deep into our brains. Where nothing is overlooked, no bit of information possessed or coming across, is irrelevant or insignificant, and coupled with its exponential non-linear mode of being applied, lead to radical overhauls in conceptualising effortlessly.
Able to guide the mind to the right information amidst the penumbrae, seemingly disorganised bits of information, chaos apparent, and out of the penumbrae into brand new insights.
Thursday, 17 December 2009
Crisis induced sudden death of chaotic attractors ... Contemplating.
I read the abstract of the paper "Multiple attractors and crisis route to chaos in a model food-chain" by Ranjit Kumar Upadhyay, of the Department of Applied Mathematics, Indian School of Mines ,Dhanbad 826 004, Jharkhand, India published in the journal Chaos, Solitons & Fractals, intrigued my senses, contemplated upon its contents, and the thoughts sparked in my mind, I laid them down as they occurred, as I read through the abstract, line-by-line.
"An attempt has been made to identify the mechanism, which is responsible for the existence of chaos in narrow parameter range ..
narrow parameter range .. the existence of chaos in narrow parameter range .. narrow signifying the small change necessary for chaos to exist .. existence of chaos.
The notion of existence of chaos, is new to me. Is it a way to describe chaos, occurring? It seems as it is extending the properties of chaos, or may be it is better to refer to as extending the concept of chaos.
Whatever, in that particular mention, significant is the reference to a mechanism. The mechanism that is responsible for bringing into existence after, well narrowly manipulating a given connected or attached parameter range. Or, maybe just, parameter. The existence of chaos, a result of manipulating or adjusting the parameter within a narrow range. Values given of the parameter fall into a narrow stretch of the parameter's total possible range of values.
in a realistic ecological model food-chain. Analytical and numerical studies of a three species food-chain model similar to a situation likely to be seen in terrestrial ecosystems has been carried out.
ecological model food-chain .. a situation likely to be seen in terrestrial ecosystems
The study of the model food chain suggests that the existence of chaos in narrow parameter ranges is caused by the crisis-induced sudden death of chaotic attractors.
'chaos existence caused' .. 'by crisis-induced sudden death of chaotic attractors' .. profound notion. First, is the statement, the best way I can put it, sudden death of chaotic attractors. Which, come to think of it, it accurately puts in the right perspective. Attractors, which, weight added, they are chaotic, their origin is chaos too, suddenly die. Suddenly die. Cease to exist. For chaos being dynamic, it postulates, if that term is correct, the necessity of what brought them into existence, in the first place. For the attractors to die, meaning, they lack of what they were fed with, what sustained them. What was necessary for their existence. Once, the condition that was keeping them alive, cease to be, they ceased to be, too.
That what was brought, was induced by a crisis. And their sudden death caused chaos. The existence of chaos, in narrow parameter range. Minute changes in the parameters involved brought chaos into existence. One would assume that this fresh bout of chaos would lead to new chaotic attractors. Chaos, let's say, methodology.
Varying one of the critical parameters in its range while keeping all the others constant, one can monitor the changes in the dynamical behaviour of the system, thereby fixing the regimes in which the system exhibits chaotic dynamics.
'critical parameters' .. implying sensitive dependence situations. In order to identify the result, each of the crucial parameters have, in the overall state of the system, the referred to, as dynamical behaviour, you vary one at a time. Monitoring the changes, varying one of the critical parameters, brings forth. 'Fixing the regimes', what combinations of critical parameters, combined values within their ranges, that exhibit chaotic dynamics, in the sense that certain regimes are more chaotic than others. As, they would be regimes, where the system exhibits no, behaviour at all, by taking behaviour to mean, the tendency for changes, among the members of a population. If, there are no changes, there is no behaviour to speak of, and with the sudden death of the already existing chaotic attractors, there is nothing to keep the system alive. Extinction follows.
Which points towards the realisation that its submersion into chaos, is vital, in order to keep the system alive, to avoid extinction.
The computed bifurcation diagrams and basin boundary calculations indicate that crisis is the underlying factor which generates chaotic dynamics in this model food-chain.
'computed bifurcation diagrams' .. 'basin boundary calculations' .. bifurcations induced by crisis, test the boundaries of the attracting basins. Basins, within which, the chaotic attractors reside, and once these boundaries are surpassed, chaotic attractors can sustain themselves no-more. Their purpose ends, the system requires to spring up, rejuvenate itself, if not, the members of the population, in crisis, will perish along with the death of their attractors that was keeping them alive. Chaos is necessary for their survival.
We investigate sudden qualitative changes in chaotic dynamical behaviour, which occur at a parameter value a1=1.7804 at which the chaotic attractor destroyed by boundary crisis with an unstable periodic orbit created by the saddle-node bifurcation.
'saddle-node bifurcation', 'created an unstable periodic orbit' .. unstable, there is no consistency, its trajectory changes between runs, cause a crisis manifesting at the boundaries, the chaotic attractors disintegrates. It is destroyed. As it occurs when a parameter acquires a specific value. An instability embedded in the system as the parameter fluctuates beyond its allowed range. The range that was necessary to keep the system stable, by keeping the periodic orbits stable.
Multiple attractors with riddled basins and fractal boundaries are also observed.
riddled basins, no clear-cut basins with fractal boundaries, many attractors, chaotic states, unstable
If ecological systems of interacting species do indeed exhibit multiple attractors etc., the long term dynamics of such systems may undergo vast qualitative changes following epidemics or environmental catastrophes due to the system being pushed into the basin of a new attractor by the perturbation.
Perturbations push the systems into the basin of a new attractor. Perturbations brought forth by epidemics or environmental catastrophes.
Coupled with stochasticity, such complex behaviours may render such systems practically unpredictable."
"An attempt has been made to identify the mechanism, which is responsible for the existence of chaos in narrow parameter range ..
narrow parameter range .. the existence of chaos in narrow parameter range .. narrow signifying the small change necessary for chaos to exist .. existence of chaos.
The notion of existence of chaos, is new to me. Is it a way to describe chaos, occurring? It seems as it is extending the properties of chaos, or may be it is better to refer to as extending the concept of chaos.
Whatever, in that particular mention, significant is the reference to a mechanism. The mechanism that is responsible for bringing into existence after, well narrowly manipulating a given connected or attached parameter range. Or, maybe just, parameter. The existence of chaos, a result of manipulating or adjusting the parameter within a narrow range. Values given of the parameter fall into a narrow stretch of the parameter's total possible range of values.
in a realistic ecological model food-chain. Analytical and numerical studies of a three species food-chain model similar to a situation likely to be seen in terrestrial ecosystems has been carried out.
ecological model food-chain .. a situation likely to be seen in terrestrial ecosystems
The study of the model food chain suggests that the existence of chaos in narrow parameter ranges is caused by the crisis-induced sudden death of chaotic attractors.
'chaos existence caused' .. 'by crisis-induced sudden death of chaotic attractors' .. profound notion. First, is the statement, the best way I can put it, sudden death of chaotic attractors. Which, come to think of it, it accurately puts in the right perspective. Attractors, which, weight added, they are chaotic, their origin is chaos too, suddenly die. Suddenly die. Cease to exist. For chaos being dynamic, it postulates, if that term is correct, the necessity of what brought them into existence, in the first place. For the attractors to die, meaning, they lack of what they were fed with, what sustained them. What was necessary for their existence. Once, the condition that was keeping them alive, cease to be, they ceased to be, too.
That what was brought, was induced by a crisis. And their sudden death caused chaos. The existence of chaos, in narrow parameter range. Minute changes in the parameters involved brought chaos into existence. One would assume that this fresh bout of chaos would lead to new chaotic attractors. Chaos, let's say, methodology.
Varying one of the critical parameters in its range while keeping all the others constant, one can monitor the changes in the dynamical behaviour of the system, thereby fixing the regimes in which the system exhibits chaotic dynamics.
'critical parameters' .. implying sensitive dependence situations. In order to identify the result, each of the crucial parameters have, in the overall state of the system, the referred to, as dynamical behaviour, you vary one at a time. Monitoring the changes, varying one of the critical parameters, brings forth. 'Fixing the regimes', what combinations of critical parameters, combined values within their ranges, that exhibit chaotic dynamics, in the sense that certain regimes are more chaotic than others. As, they would be regimes, where the system exhibits no, behaviour at all, by taking behaviour to mean, the tendency for changes, among the members of a population. If, there are no changes, there is no behaviour to speak of, and with the sudden death of the already existing chaotic attractors, there is nothing to keep the system alive. Extinction follows.
Which points towards the realisation that its submersion into chaos, is vital, in order to keep the system alive, to avoid extinction.
The computed bifurcation diagrams and basin boundary calculations indicate that crisis is the underlying factor which generates chaotic dynamics in this model food-chain.
'computed bifurcation diagrams' .. 'basin boundary calculations' .. bifurcations induced by crisis, test the boundaries of the attracting basins. Basins, within which, the chaotic attractors reside, and once these boundaries are surpassed, chaotic attractors can sustain themselves no-more. Their purpose ends, the system requires to spring up, rejuvenate itself, if not, the members of the population, in crisis, will perish along with the death of their attractors that was keeping them alive. Chaos is necessary for their survival.
We investigate sudden qualitative changes in chaotic dynamical behaviour, which occur at a parameter value a1=1.7804 at which the chaotic attractor destroyed by boundary crisis with an unstable periodic orbit created by the saddle-node bifurcation.
'saddle-node bifurcation', 'created an unstable periodic orbit' .. unstable, there is no consistency, its trajectory changes between runs, cause a crisis manifesting at the boundaries, the chaotic attractors disintegrates. It is destroyed. As it occurs when a parameter acquires a specific value. An instability embedded in the system as the parameter fluctuates beyond its allowed range. The range that was necessary to keep the system stable, by keeping the periodic orbits stable.
Multiple attractors with riddled basins and fractal boundaries are also observed.
riddled basins, no clear-cut basins with fractal boundaries, many attractors, chaotic states, unstable
If ecological systems of interacting species do indeed exhibit multiple attractors etc., the long term dynamics of such systems may undergo vast qualitative changes following epidemics or environmental catastrophes due to the system being pushed into the basin of a new attractor by the perturbation.
Perturbations push the systems into the basin of a new attractor. Perturbations brought forth by epidemics or environmental catastrophes.
Coupled with stochasticity, such complex behaviours may render such systems practically unpredictable."
Friday, 6 November 2009
Understanding what mass is, in view of expanding space.
Weird notions come into my mind and I like to ponder on them and see to where they take me.
Universe expanding with the speed of light, as is mentioned here too, space expanding.
Outer space? The space between galaxies and stars or all space? Even to the space we occupy ourselves. Blowing up, expanding like balloons in all three dimensions.
It is space itself that expands, is found in NASA's Imagine the universe website,
".. but actually it is space itself that is expanding faster than the speed of light, driving objects further apart at an increasing rate."
The example of raisins in a cake as it is baked and raised up, the raisins on the surface grow distant from one another, but the raisins themselves are built on space, the concept of space is not limited only to the space observed by an individual, but includes as well the space the individual itself occupies, the raisins themselves occupy, even the space their atoms occupy.
The atom built by protons and neutrons enclosed in the nucleus and far far away, in relative terms, the electrons orbiting, an enormous amount of space included and include the point-like entities of electrons, protons and neutrons, attributed with mass equals energy, that space that lies within objects expands too, and expands with the speed of light, or, as it is claimed faster than the speed of light, in order to compensate for the ultimate speed, the speed of light, the limit all objects in the universe can be accelerated to.
We ourselves, as objects of this universe expand in space, or the space that our atoms occupy expand with or faster than the speed of light.
And all done equally, we are none the wiser. The only thing that changes is the distribution of energy in space. Which in a sense is diluted. How this notion would accommodate all what we think about what mass is? Inertial mass, rest mass of a body for zero speed; at any other speed the inertial mass is greater.
According to the equations of special relativity the mass of a body increases as it is gaining speed, it is about three times as much of the inertial mass for speeds at 0.95 of the speed of light and the graphs point to infinite mass as it gets nearer and nearer to the speed of light, asymptotic curve that never touches the y axis as the speed increases to ridiculous decimal increments to the speed of light.
What if space, all around the universe environs expands at the same time? Would an infinite mass postulate still hold? Would the beginnings of the universe from a singularity, an infinitesimal point of infinite energy be more plausible? Energy and space fundamental entities that combining the two together to produce mass? Mass an emergent entity, an entity that becomes apparent only if it is looked at from above a threshold, a macro-threshold, our familiar macro-world loosing completely that attribute when it is looked at from a point below that threshold. Mass concept becomes meaningless from the standpoint of the micro-world, the quantum realm, being only extremely concentrated foci of enormous amounts of energy kept apart by even greater distances of empty space?
What difference would that make if we take the point that this space expands at the speed of light in the same way that the universe itself expands at the speed of light.
Universe expanding with the speed of light, as is mentioned here too, space expanding.
Outer space? The space between galaxies and stars or all space? Even to the space we occupy ourselves. Blowing up, expanding like balloons in all three dimensions.
It is space itself that expands, is found in NASA's Imagine the universe website,
".. but actually it is space itself that is expanding faster than the speed of light, driving objects further apart at an increasing rate."
The example of raisins in a cake as it is baked and raised up, the raisins on the surface grow distant from one another, but the raisins themselves are built on space, the concept of space is not limited only to the space observed by an individual, but includes as well the space the individual itself occupies, the raisins themselves occupy, even the space their atoms occupy.
The atom built by protons and neutrons enclosed in the nucleus and far far away, in relative terms, the electrons orbiting, an enormous amount of space included and include the point-like entities of electrons, protons and neutrons, attributed with mass equals energy, that space that lies within objects expands too, and expands with the speed of light, or, as it is claimed faster than the speed of light, in order to compensate for the ultimate speed, the speed of light, the limit all objects in the universe can be accelerated to.
We ourselves, as objects of this universe expand in space, or the space that our atoms occupy expand with or faster than the speed of light.
And all done equally, we are none the wiser. The only thing that changes is the distribution of energy in space. Which in a sense is diluted. How this notion would accommodate all what we think about what mass is? Inertial mass, rest mass of a body for zero speed; at any other speed the inertial mass is greater.
According to the equations of special relativity the mass of a body increases as it is gaining speed, it is about three times as much of the inertial mass for speeds at 0.95 of the speed of light and the graphs point to infinite mass as it gets nearer and nearer to the speed of light, asymptotic curve that never touches the y axis as the speed increases to ridiculous decimal increments to the speed of light.
What if space, all around the universe environs expands at the same time? Would an infinite mass postulate still hold? Would the beginnings of the universe from a singularity, an infinitesimal point of infinite energy be more plausible? Energy and space fundamental entities that combining the two together to produce mass? Mass an emergent entity, an entity that becomes apparent only if it is looked at from above a threshold, a macro-threshold, our familiar macro-world loosing completely that attribute when it is looked at from a point below that threshold. Mass concept becomes meaningless from the standpoint of the micro-world, the quantum realm, being only extremely concentrated foci of enormous amounts of energy kept apart by even greater distances of empty space?
What difference would that make if we take the point that this space expands at the speed of light in the same way that the universe itself expands at the speed of light.
Labels:
expanding space,
mass
Tuesday, 25 August 2009
Systems adiabatic? Understanding subsystem-system interactions.
It is mentioned in page 166, of the book Experimental techniques in low-temperature physics by Guy Kendall White, Philip J. Meeson
"Most properties of materials are categorised in terms of the phonon system, the electron system and the spin system (i.e. the nuclear or electronic magnetic moments)."
Three subsystems that make up for most of the properties of materials, the physical and chemical properties, matter organised into the macroscopic system of the physical world, determine how the physical world behaves.
"These different subsystems, although intimately mixed together in any one material, can be separated in the imagination and analysed as independent systems."
Determine physical behaviour as the embedded sub-systems are intimately mixed, which brings into mind attractors and synchronization, attractors achieved by the ever so slight tweaking of relevant parameters, the electron-phonon interactions synchronized, by even the transient fluctuation enhancement referred to as peculiar to macroscopic systems, triggered by the exposure to additional parametres brought in, by the other embedded sub-systems.
These three subsystems separated and analysed independently? Their connections severed, each one imagined on itself? An approach which can be followed on all subsystems, lower-dimensional systems embedded in higher dimensional systems, the macroscopic systems? Understanding the relationships between embedded and embedding systems, all systems in all realms, portable via the self-similarity principle of chaos.
And all the while, each sub-system being dynamic, ever so 'touchy' to any changes introduced by either sub-system, via the electron-phonon interaction.
"Consider for example the separation of the phonon system from the electron system in a metal. This proceeds from the knowledge that the phonon system is 'heavy' arising from the motion of ions in the lattice whilst electrons are light. The mass difference (and differing statistics) influences the available energy levels in each system and ultimately limits the flow of momentum and energy between the two systems."
The phonon system 'heavy', the electron system 'light' a matter of size. The phonon system arises from the motion of ions, the electron system from electrons. This mass difference limits the flow of momentum and energy between the two systems, practically isolated from one another, each system with its own agent, phonon electron, interactions between their agents determine their behaviour.
Accordingly, it can be assumed that any introduction of extra agents, like nanoparticles upsets this balance, the flow of momentum and energy, between the two systems, as the mass of the nanoparticles being similar in size, could introduce novel interactions, leading to the emergence of properties, alien to the subsystems. The observation of novel chemical and physical properties in nanoparticles, when introduced into the nanoscales, indicative of this process.
"This 'adiabatic' or Born-Oppenheimer approximation regards the two subsystems as separate but interacting via an electron-phonon interaction."
Embedded sub-systems separate but interacting via an electron-phonon interaction? Interacting as sub-systems? Preserving their integrity, their independence of a sort. An adiabatic boundary, a barrier that prevents the direct flow of energy and momentum from an embedded system to another or even the embedding macroscopic system apart from the electron-phonon interaction, each sub-system the 'whole' by itself and as a whole interacts.
Embedded sub-systems adiabatic. Embedding systems, by virtue of being embedded themselves, sub-systems to higher dimensional systems, adiabatic too. All systems adiabatic. Biological, emotional, psychological, social systems adiabatic? Understanding systems by using the adiabatic system property?
"Most properties of materials are categorised in terms of the phonon system, the electron system and the spin system (i.e. the nuclear or electronic magnetic moments)."
Three subsystems that make up for most of the properties of materials, the physical and chemical properties, matter organised into the macroscopic system of the physical world, determine how the physical world behaves.
"These different subsystems, although intimately mixed together in any one material, can be separated in the imagination and analysed as independent systems."
Determine physical behaviour as the embedded sub-systems are intimately mixed, which brings into mind attractors and synchronization, attractors achieved by the ever so slight tweaking of relevant parameters, the electron-phonon interactions synchronized, by even the transient fluctuation enhancement referred to as peculiar to macroscopic systems, triggered by the exposure to additional parametres brought in, by the other embedded sub-systems.
These three subsystems separated and analysed independently? Their connections severed, each one imagined on itself? An approach which can be followed on all subsystems, lower-dimensional systems embedded in higher dimensional systems, the macroscopic systems? Understanding the relationships between embedded and embedding systems, all systems in all realms, portable via the self-similarity principle of chaos.
And all the while, each sub-system being dynamic, ever so 'touchy' to any changes introduced by either sub-system, via the electron-phonon interaction.
"Consider for example the separation of the phonon system from the electron system in a metal. This proceeds from the knowledge that the phonon system is 'heavy' arising from the motion of ions in the lattice whilst electrons are light. The mass difference (and differing statistics) influences the available energy levels in each system and ultimately limits the flow of momentum and energy between the two systems."
The phonon system 'heavy', the electron system 'light' a matter of size. The phonon system arises from the motion of ions, the electron system from electrons. This mass difference limits the flow of momentum and energy between the two systems, practically isolated from one another, each system with its own agent, phonon electron, interactions between their agents determine their behaviour.
Accordingly, it can be assumed that any introduction of extra agents, like nanoparticles upsets this balance, the flow of momentum and energy, between the two systems, as the mass of the nanoparticles being similar in size, could introduce novel interactions, leading to the emergence of properties, alien to the subsystems. The observation of novel chemical and physical properties in nanoparticles, when introduced into the nanoscales, indicative of this process.
"This 'adiabatic' or Born-Oppenheimer approximation regards the two subsystems as separate but interacting via an electron-phonon interaction."
Embedded sub-systems separate but interacting via an electron-phonon interaction? Interacting as sub-systems? Preserving their integrity, their independence of a sort. An adiabatic boundary, a barrier that prevents the direct flow of energy and momentum from an embedded system to another or even the embedding macroscopic system apart from the electron-phonon interaction, each sub-system the 'whole' by itself and as a whole interacts.
Embedded sub-systems adiabatic. Embedding systems, by virtue of being embedded themselves, sub-systems to higher dimensional systems, adiabatic too. All systems adiabatic. Biological, emotional, psychological, social systems adiabatic? Understanding systems by using the adiabatic system property?
Thursday, 20 August 2009
Low-dimensional systems embedded in macroscopic systems. Enhanced fluctuations?
I feel I have stumbled once again onto something which will augment my knowledge, my perspective. It will drag me away from my current endeavours to bring me back enhanced and ready for new challenges.
Writing unhindered by any considerations deemed irrelevant for my pursuits. Enough introduction, back to the point.
Šil'nikov phenomena, are referred to, in this paper titled 'Poincaré versus Boltzmann in Šil'nikov phenomena'. It was the climax of my google searching, following terms like, subordinate Šil'nikov bifurcations, Sil'nikov-saddle-node interaction, Šil'nikov phenomena, Sil'nikov chaos.
Analyse the terms and deepen my knowledge by examining the effects, the mentioned concepts had on the knowledge already held in my reverberated neural nets and in the process ascend whatever fragments are there, and emerge in new knowledge, new ways of looking at things.
In the abstract of the paper, 'Poincaré versus Boltzmann in Šil'nikov phenomena', it mentions
"By suitable adjustment of the control parameters in a CO2 laser with feedback we show experimental evidence of Sil'nikov chaos, ..."
... control parameters, as in parameters and variables. Parameters, of which their values are considered if not unchanged, changing within a very narrow limiting and limited range, as in any fractal-making program, what you tweak ever so slightly, to get all kinds, versions of the initial fractal. Suitably adjusting the parameters, deemed as controlling in the CO2 laser. Feedback providing the iteration part, the recursiveness in function, for chaos coming.
Experimental evidence of Sil'nikov chaos? What kind of chaos is this?
It further mentions,
".. characterized by intensity pulses almost equal in shape, but irregularly separated in time."
Intensity pulses irregularly separated in time, their experimental setup signature of chaos. Irregularity signifying chaos.
"The times of return to a Poincare section are statistically spread,.."
Poincare sections traversing its flow in time, incisions right through its fractal space, revealed statistically spread times of return, the order hidden amidst chaos. Chaos generating order.
Chaos as described in mathematics, measuring fluctuations in the intensity of forces, stresses, pressures plotted into graphs or even plain mathematical functions, self-referential, iterated loosing themselves into chaos. Translated into all kinds of forces exhibited in all systems possible. Physical, mental, emotional, psychological, social, spiritual systems analysed in the very same way as mathematical chaos.
"... however their iteration map is one-dimensional and in close agreement with that arising from Sil'nikov theory."
The map of the iterated points is one-dimensional? All falling onto a single line? Or the map, being a function and the one-dimensional referred to, its mathematical property. And all this, .. 'in close agreement with that arising from Sil'nikov theory'.
"Thus, the iteration map of the time intervals becomes the most appropriate indicator of this chaos."
An appropriate indicator of this chaos? Sil'nikov chaos. The one-dimensional iteration map of the time intervals?
"The residual width of the experimentally measured maps is due to a transient fluctuation enhancement peculiar to macroscopic systems, which is absent in low-dimensional chaotic dynamics."
Macroscopic systems distinct from low-dimensional chaotic dynamics? Underlying systems different than systems above? Chaotic dynamics of underlying systems, systems embedded within systems above, low-dimensional as embedded systems are bound to have less dimensions, than the system they are embedded in, embedding system which includes more than one low-dimensional systems.
The transient fluctuation enhancement peculiar to macroscopic systems? Due to the influence of additional parametres a low-dimensional system is exposed to, by the other low-dimensional system(s) that together comprise the macroscopic system they are embedded in?
Each low-dimensional system compensating for the introduced parameters, tweaked. Embedded low-dimensional systems integrated and together now determine the properties, attributes, the behaviour of the macroscopic system.
My very own thoughts developing in a similar process. All systems possible, physical psychological, social developing likewise.
Writing unhindered by any considerations deemed irrelevant for my pursuits. Enough introduction, back to the point.
Šil'nikov phenomena, are referred to, in this paper titled 'Poincaré versus Boltzmann in Šil'nikov phenomena'. It was the climax of my google searching, following terms like, subordinate Šil'nikov bifurcations, Sil'nikov-saddle-node interaction, Šil'nikov phenomena, Sil'nikov chaos.
Analyse the terms and deepen my knowledge by examining the effects, the mentioned concepts had on the knowledge already held in my reverberated neural nets and in the process ascend whatever fragments are there, and emerge in new knowledge, new ways of looking at things.
In the abstract of the paper, 'Poincaré versus Boltzmann in Šil'nikov phenomena', it mentions
"By suitable adjustment of the control parameters in a CO2 laser with feedback we show experimental evidence of Sil'nikov chaos, ..."
... control parameters, as in parameters and variables. Parameters, of which their values are considered if not unchanged, changing within a very narrow limiting and limited range, as in any fractal-making program, what you tweak ever so slightly, to get all kinds, versions of the initial fractal. Suitably adjusting the parameters, deemed as controlling in the CO2 laser. Feedback providing the iteration part, the recursiveness in function, for chaos coming.
Experimental evidence of Sil'nikov chaos? What kind of chaos is this?
It further mentions,
".. characterized by intensity pulses almost equal in shape, but irregularly separated in time."
Intensity pulses irregularly separated in time, their experimental setup signature of chaos. Irregularity signifying chaos.
"The times of return to a Poincare section are statistically spread,.."
Poincare sections traversing its flow in time, incisions right through its fractal space, revealed statistically spread times of return, the order hidden amidst chaos. Chaos generating order.
Chaos as described in mathematics, measuring fluctuations in the intensity of forces, stresses, pressures plotted into graphs or even plain mathematical functions, self-referential, iterated loosing themselves into chaos. Translated into all kinds of forces exhibited in all systems possible. Physical, mental, emotional, psychological, social, spiritual systems analysed in the very same way as mathematical chaos.
"... however their iteration map is one-dimensional and in close agreement with that arising from Sil'nikov theory."
The map of the iterated points is one-dimensional? All falling onto a single line? Or the map, being a function and the one-dimensional referred to, its mathematical property. And all this, .. 'in close agreement with that arising from Sil'nikov theory'.
"Thus, the iteration map of the time intervals becomes the most appropriate indicator of this chaos."
An appropriate indicator of this chaos? Sil'nikov chaos. The one-dimensional iteration map of the time intervals?
"The residual width of the experimentally measured maps is due to a transient fluctuation enhancement peculiar to macroscopic systems, which is absent in low-dimensional chaotic dynamics."
Macroscopic systems distinct from low-dimensional chaotic dynamics? Underlying systems different than systems above? Chaotic dynamics of underlying systems, systems embedded within systems above, low-dimensional as embedded systems are bound to have less dimensions, than the system they are embedded in, embedding system which includes more than one low-dimensional systems.
The transient fluctuation enhancement peculiar to macroscopic systems? Due to the influence of additional parametres a low-dimensional system is exposed to, by the other low-dimensional system(s) that together comprise the macroscopic system they are embedded in?
Each low-dimensional system compensating for the introduced parameters, tweaked. Embedded low-dimensional systems integrated and together now determine the properties, attributes, the behaviour of the macroscopic system.
My very own thoughts developing in a similar process. All systems possible, physical psychological, social developing likewise.
Sunday, 26 April 2009
The tremendous speeds earth is traveling in space. Is gravity a classical force? Is the search for gravitational waves a futile endeavour?
In Philosophic Wonders
"The Sun and its family of planets are orbiting the galaxy at about 135 miles per second."
The solar system moves at 135 miles per second, relativity's essence. Where the hell do we draw the certainty, be so sure, letting ourselves feel peace and tranquility, when we relax bathing in the sunshine lying in oblivion, on a beach lounger. In the knowledge, that at the same time, planet earth along with us, are propelled into the vacuum of space, in shearing shattering speeds. If we subject, the hapless fragile human being, in these tremendous unreal speeds, we will be lucky to find traces of the bones to verify its past existence.
And yet, here we are, alive and well, despite the speed we travel in space, to have the arrogance to enjoy, a peaceful morning. Peaceful. Anything else apart.
Centripetal force, the essence of gravity, what keep us pinned on the ground. That do not let us, being swept away from the face of planet earth, and not only us, but what keeps the dust, the rocks on the face of the moon and all the rest of the planets and other so-called heavenly bodies.
Gravity, that sucks us inwards, towards the centre of planet earth, to spare us the oblivion of our precious being, by the shattering speeds, our very own nurturing earth, is itself subjected to.
Apart from the solar system traveling at 135 miles per second, we have as well that
"The Earth rotates on its axis at about 1,100 miles an hour, a motion that creates day and night."
and
"The Earth orbits the Sun at about 67,000 miles an hour, a motion that takes one year."
and
"The Sun circles the Milky Way at a speed of about 486,000 miles per hour."
and
"And every object in the Universe is moving apart from the other objects as the Universe expands at a constantly accelerating rate."
Devastating forces that would have left nothing standing, long enough, that not only life nurtured, in the warm and welcome abode of planets, but even the existence of these wonderful incubators, if it wasn't for the protection offered by the gravitational forces, that not only effectively counteract against the shearing power of the speeds developed, by heavenly bodies in their endless journey in the vast space but even overcomes it, for us to enjoy the tranquility of nature.
Heavenly bodies in their endless journey around the centre of... The centre of .... their very own system or any other system they are part of. Earth orbiting around itself, moon orbiting around itself and moon orbiting around earth, the earth-moon system. Earth orbiting around the sun, the planets orbiting around the sun, the solar system around... Circularity.
Driven by gravity itself and by gravity is counteracted. The shearing speeds that the motion, its very own existence propagates and as it is counteracted by, at the same time. A dependency built-in, in the system, in each system by itself, the speed and the forces developed cannot be greater, than what the heavenly body can withstand. The speed of rotation around itself, as well as around the larger body in the system, would not be allowed to reach levels that would jeopardize its very own integrity. The seemingly hazardous journey, a gentle stroll in comparison.
How can we comprehend the nature of gravity, when we think of its appearance, as it vanishes down to the centre of a heavenly body. Attraction, of every piece of matter upon each other. Additive, as more and more pieces of matter accumulate, the stronger it becomes. Heavenly bodies borne out of dust, stardust as it has been found, accumulating into foci driven by chance (chance?) ... for more and more dust particles to gather up into the heavenly bodies which they become. What they are now. Matter may be pushed into coalescing, ... by the very vacuum surrounding it ... via pressures they are subjected to. Is it just a vacuum pressure effect or an actual attraction of matter objects. The actual rotation of a heavenly body around itself, a vacuum pressure effect. The same would hold valid, regarding the rotation of a heavenly body around its bigger counterpart in the system, rendering the presumed action at a distance character meaningless, what elevates gravity from a purely classical force into a fundamental force. Is the search for the graviton justified, or even the existence of gravitational waves feasible, the search for gravitational waves a futile endeavour.
The whole case, can be thought of in terms of systems. Systems within systems, a guiding principle universal for all systems. Systems that are dynamic ...
Oh, well fractal. Is there anything apparent in the galaxy system that would justify the term fractal? May be not merely in its spatial dimensions, but taking into account its evolution through time too. Energy dissipation. Energy being apparently high, at the very centre of a galaxy, stellar nurseries, black holes. Dissipating to lower and lower energies towards its perimeter. Fractal being characterized by its non-integral nature, non-integral dimensions. Is there any indications of non-integral dimensions in the galaxial system organisation?
What about warped spacetime, wormholes and the like. Does that not point towards a fractal evolution for the galaxy system. Gravity a mere manifestation of energy dissipation governed even by the Feigenbaum series concept.
"The Sun and its family of planets are orbiting the galaxy at about 135 miles per second."
The solar system moves at 135 miles per second, relativity's essence. Where the hell do we draw the certainty, be so sure, letting ourselves feel peace and tranquility, when we relax bathing in the sunshine lying in oblivion, on a beach lounger. In the knowledge, that at the same time, planet earth along with us, are propelled into the vacuum of space, in shearing shattering speeds. If we subject, the hapless fragile human being, in these tremendous unreal speeds, we will be lucky to find traces of the bones to verify its past existence.
And yet, here we are, alive and well, despite the speed we travel in space, to have the arrogance to enjoy, a peaceful morning. Peaceful. Anything else apart.
Centripetal force, the essence of gravity, what keep us pinned on the ground. That do not let us, being swept away from the face of planet earth, and not only us, but what keeps the dust, the rocks on the face of the moon and all the rest of the planets and other so-called heavenly bodies.
Gravity, that sucks us inwards, towards the centre of planet earth, to spare us the oblivion of our precious being, by the shattering speeds, our very own nurturing earth, is itself subjected to.
Apart from the solar system traveling at 135 miles per second, we have as well that
"The Earth rotates on its axis at about 1,100 miles an hour, a motion that creates day and night."
and
"The Earth orbits the Sun at about 67,000 miles an hour, a motion that takes one year."
and
"The Sun circles the Milky Way at a speed of about 486,000 miles per hour."
and
"And every object in the Universe is moving apart from the other objects as the Universe expands at a constantly accelerating rate."
Devastating forces that would have left nothing standing, long enough, that not only life nurtured, in the warm and welcome abode of planets, but even the existence of these wonderful incubators, if it wasn't for the protection offered by the gravitational forces, that not only effectively counteract against the shearing power of the speeds developed, by heavenly bodies in their endless journey in the vast space but even overcomes it, for us to enjoy the tranquility of nature.
Heavenly bodies in their endless journey around the centre of... The centre of .... their very own system or any other system they are part of. Earth orbiting around itself, moon orbiting around itself and moon orbiting around earth, the earth-moon system. Earth orbiting around the sun, the planets orbiting around the sun, the solar system around... Circularity.
Driven by gravity itself and by gravity is counteracted. The shearing speeds that the motion, its very own existence propagates and as it is counteracted by, at the same time. A dependency built-in, in the system, in each system by itself, the speed and the forces developed cannot be greater, than what the heavenly body can withstand. The speed of rotation around itself, as well as around the larger body in the system, would not be allowed to reach levels that would jeopardize its very own integrity. The seemingly hazardous journey, a gentle stroll in comparison.
How can we comprehend the nature of gravity, when we think of its appearance, as it vanishes down to the centre of a heavenly body. Attraction, of every piece of matter upon each other. Additive, as more and more pieces of matter accumulate, the stronger it becomes. Heavenly bodies borne out of dust, stardust as it has been found, accumulating into foci driven by chance (chance?) ... for more and more dust particles to gather up into the heavenly bodies which they become. What they are now. Matter may be pushed into coalescing, ... by the very vacuum surrounding it ... via pressures they are subjected to. Is it just a vacuum pressure effect or an actual attraction of matter objects. The actual rotation of a heavenly body around itself, a vacuum pressure effect. The same would hold valid, regarding the rotation of a heavenly body around its bigger counterpart in the system, rendering the presumed action at a distance character meaningless, what elevates gravity from a purely classical force into a fundamental force. Is the search for the graviton justified, or even the existence of gravitational waves feasible, the search for gravitational waves a futile endeavour.
The whole case, can be thought of in terms of systems. Systems within systems, a guiding principle universal for all systems. Systems that are dynamic ...
Oh, well fractal. Is there anything apparent in the galaxy system that would justify the term fractal? May be not merely in its spatial dimensions, but taking into account its evolution through time too. Energy dissipation. Energy being apparently high, at the very centre of a galaxy, stellar nurseries, black holes. Dissipating to lower and lower energies towards its perimeter. Fractal being characterized by its non-integral nature, non-integral dimensions. Is there any indications of non-integral dimensions in the galaxial system organisation?
What about warped spacetime, wormholes and the like. Does that not point towards a fractal evolution for the galaxy system. Gravity a mere manifestation of energy dissipation governed even by the Feigenbaum series concept.
Friday, 24 April 2009
.. a ... significance of Lyapunov exponents? ... to absorb chaos?
... to absorb chaos?
.. a ... significance of Lyapunov exponents? In their predictive(?) power to show whether a certain trajectory converges or diverges? Convergence or divergence determining the way that chaos is viewed by individuals? Diverging pathways, or for that matter constantly diverging pathways, result in what as seen as chaos and converging pathways create attractors, which can be seen as order?
... determining a measure of Lyapunov exponents out of the processes involved which will show whether the processes undertaken will bring about order or chaos? In almost everything which evades a thorough analysis by any other prescribed procedures but exhibiting chaotic tendencies?
Rigorous mathematics, scientific proof, all part of cognition which is taking place outside, from where it should be taking place. The human mind. As meaning is derived by mind itself, it can not materialise in spheres other than what the mind provides for. And attempting to frame meaning, with processes that the language provides it is a futile endeavor, because at the end the whole process must be accepted by the mind itself.
.. a ... significance of Lyapunov exponents? In their predictive(?) power to show whether a certain trajectory converges or diverges? Convergence or divergence determining the way that chaos is viewed by individuals? Diverging pathways, or for that matter constantly diverging pathways, result in what as seen as chaos and converging pathways create attractors, which can be seen as order?
... determining a measure of Lyapunov exponents out of the processes involved which will show whether the processes undertaken will bring about order or chaos? In almost everything which evades a thorough analysis by any other prescribed procedures but exhibiting chaotic tendencies?
Rigorous mathematics, scientific proof, all part of cognition which is taking place outside, from where it should be taking place. The human mind. As meaning is derived by mind itself, it can not materialise in spheres other than what the mind provides for. And attempting to frame meaning, with processes that the language provides it is a futile endeavor, because at the end the whole process must be accepted by the mind itself.
Space, a result of forces? Interaction of forces? It is bound to be.
New thoughts light up, in my mind... Exciting.
Space, a result of forces? Interaction of forces? It is bound to be.
As all these I have been working out, put into these posts .... are getting pieced up together, become coherent.
(Regression). I have to analyse this. The way it happens. What makes a thought developed independently, loose as is, with no ties (obvious?) with any other thoughts (residing in some other realm, context). But out of no-where, without warning, nevertheless click (systems exhibiting chaos' self-similarity?), or hook, or whatever, one with the other. They become connected, tag and drag along the entirety of the concept, notion, thought, idea; the bonus being, all the rest of the properties and attributes, these mental structures, are associated with. A bonus since, as the thinking develops, all these properties and attributes, branching-like extensions on all sides, are engaged in the thinking process. Being tags themselves, either concepts, or notions, or thoughts, or ideas, potentially concise and coherent mental structures, on their own right, they have all what is needed, to form by themselves, novel associations. The word 'concepts', as it comes into mind, with their properties (as they develop), provide the associations, and by virtue of their associations, cohere from ..(humble, incoherent, haphazard) beginnings ... into more inclusive concept circa thoughts.
Space, (as I have surmised in that post) was non-existent before the big bang, or in the manner we experience it right now. (Vast, solid objects within, immense, three dimensional, containing or expressed with or give substance to the word-concepts of macrocosm and microcosms). Space, distance developed fractally the world we live in, the universe, just a thin fractally developed sheet. Space, a reference frame fractally developed, 'sitting' on another deeply fundamental space
Space, a result of forces? Interaction of forces? It is bound to be.
As all these I have been working out, put into these posts .... are getting pieced up together, become coherent.
(Regression). I have to analyse this. The way it happens. What makes a thought developed independently, loose as is, with no ties (obvious?) with any other thoughts (residing in some other realm, context). But out of no-where, without warning, nevertheless click (systems exhibiting chaos' self-similarity?), or hook, or whatever, one with the other. They become connected, tag and drag along the entirety of the concept, notion, thought, idea; the bonus being, all the rest of the properties and attributes, these mental structures, are associated with. A bonus since, as the thinking develops, all these properties and attributes, branching-like extensions on all sides, are engaged in the thinking process. Being tags themselves, either concepts, or notions, or thoughts, or ideas, potentially concise and coherent mental structures, on their own right, they have all what is needed, to form by themselves, novel associations. The word 'concepts', as it comes into mind, with their properties (as they develop), provide the associations, and by virtue of their associations, cohere from ..(humble, incoherent, haphazard) beginnings ... into more inclusive concept circa thoughts.
Space, (as I have surmised in that post) was non-existent before the big bang, or in the manner we experience it right now. (Vast, solid objects within, immense, three dimensional, containing or expressed with or give substance to the word-concepts of macrocosm and microcosms). Space, distance developed fractally the world we live in, the universe, just a thin fractally developed sheet. Space, a reference frame fractally developed, 'sitting' on another deeply fundamental space
Sunday, 29 March 2009
Higgs boson existence?
About the Higgs boson in ars technica article 'Narrowing in on the Higgs boson'
"Since the LHC is anticipated to produce its first collisions later this year, it may not be long before we obtain clear evidence of the existence of the Higgs boson. I tend to think that this would be one of the least exciting answers possible, though. The real scientific treat would be if the LHC and other colliders can't find any Higgs particles. This would mean that the standard model, one of the shining examples of the power of particle physics and a theory on which a lot of physics rests, might be wrong; we would have to go back to the drawing board and invent something new."
Attributed as 'real scientific treat' if the colliders do not find any proof of the Higgs boson existence? Why would my mind stop at that particular sentence?
It is not its existence that is doubted, ... but its existence in a ... free form in the universe? ... that the universe is inhabited by a vast number of Higgs bosons, hiding obscuring their presence? ... it is what part of the universe they inhabit, it's what matters.
While this is unlikely, seeing as how the experimentalists have found 60 of the 61 particles that appear within the standard model, it sure would make the next few years interesting, both for scientists in the field and interested third parties like science journalists."
"Since the LHC is anticipated to produce its first collisions later this year, it may not be long before we obtain clear evidence of the existence of the Higgs boson. I tend to think that this would be one of the least exciting answers possible, though. The real scientific treat would be if the LHC and other colliders can't find any Higgs particles. This would mean that the standard model, one of the shining examples of the power of particle physics and a theory on which a lot of physics rests, might be wrong; we would have to go back to the drawing board and invent something new."
Attributed as 'real scientific treat' if the colliders do not find any proof of the Higgs boson existence? Why would my mind stop at that particular sentence?
It is not its existence that is doubted, ... but its existence in a ... free form in the universe? ... that the universe is inhabited by a vast number of Higgs bosons, hiding obscuring their presence? ... it is what part of the universe they inhabit, it's what matters.
While this is unlikely, seeing as how the experimentalists have found 60 of the 61 particles that appear within the standard model, it sure would make the next few years interesting, both for scientists in the field and interested third parties like science journalists."
Labels:
Higgs boson
Monday, 23 March 2009
Archimedes actual infinities, as revealed in his lost manuscript.
Archimedes actual infinities, as revealed in his lost manuscript.
I read in the article 'A Prayer for Archimedes', about the long-lost text by the ancient Greek mathematician which shows that he had begun to discover the principles of calculus.
In the claim put forward by Reviel Netz, an historian of mathematics at Stanford University who transcribed the newly found text, says that the recent discoveries show that Archimedes indeed used the notion of actual infinity.
Infinities, as defined by Aristotle, mentioned here
"The Greek philosopher Aristotle built defenses against infinity's vexing qualities by distinguishing between the "potential infinite" and the "actual infinite." An infinitely long line would be actually infinite, whereas a line that could always be extended would be potentially infinite. Aristotle argued that the actual infinite didn't exist."
An infinitely long line would be actually infinite, hence actual infinity, whereas a line that could always be extended would be potentially infinite, hence potential infinity. Aristotle argued that the actual infinity didn't exist.
I read further
"Archimedes found a relationship between the full area of that slice, which was a section through the plane-sided volume, and the smaller area within it, which was a section through the curved shape. Then he argued that he could use that relationship to calculate the entire volume of the curved shape, because both the curved figure and the straight one contained the same number of slices."
and
"That number just happened to be infinity—actual infinity."
and
"The interesting breakthrough is that he is completely willing to operate with actual infinity," Netz says, but he adds that "the argument is definitely not completely valid. He just had a strong intuition that it should work." In this case, it did work, but it remained for Newton and Leibniz to figure out how to make the argument mathematically rigorous."
Archimedes being willing to operate with actual infinity, an infinitely long line, instead of a line that could always be extended. A potentially infinite line, what Aristotle argued that exists, whereas the actual infinity did not exist.
So, there is not infinity as such, whatever name it can be given, actual or potential, since an infinite line can always be extended. There are no boundaries.
But the infinity of decimal numbers, lying between two integer numbers, it has boundaries. The two integer numbers that lies within. Does having boundaries determine the kind of infinity it is? The infinity of decimal numbers between two integers, can not be extended beyond the integer boundaries.
It cannot be potentially infinite, it can only be actual infinity, what the calculus uses. What has driven Archimedes "strong intuition that it should work", and why "In this case, it did work". And it works when dealing with infinities that the boundaries are known or postulated beforehand.
What "it remained for Newton and Leibniz to figure out, how to make the argument mathematically rigorous.
Confusing the issues amidst vague statements, if "the argument is definitely not completely valid." What is valid, what is completely valid, what is not completely valid, and what is definitely not completely valid. How is validity defined, what is required to make an argument valid.
I read further in the same article
"Newton and Leibniz also worked with actual infinity. Leibniz went so far as to say in a letter, "I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author."
Since Newton and Leibniz also worked with actual infinity and produced calculus, would that not validate Archimedes intuition and his willingness to operate with actual infinity?
And what is meant by Leibniz' statement that nature makes frequent use of it, everywhere. Does that not imply the fractality inherent in all nature's objects, chaos driven processes surpassing, traversing fractal dimensions, from the microscales to the macroscales weaving the perfections of its author, chaos. Actual infinities trapped within the delimiting boundaries of any nature's object, all objects.
The statement 'modern calculus no longer makes use of the actual infinite; it sticks with Aristotle's distinction', a matter of taste?
I read in the article 'A Prayer for Archimedes', about the long-lost text by the ancient Greek mathematician which shows that he had begun to discover the principles of calculus.
In the claim put forward by Reviel Netz, an historian of mathematics at Stanford University who transcribed the newly found text, says that the recent discoveries show that Archimedes indeed used the notion of actual infinity.
Infinities, as defined by Aristotle, mentioned here
"The Greek philosopher Aristotle built defenses against infinity's vexing qualities by distinguishing between the "potential infinite" and the "actual infinite." An infinitely long line would be actually infinite, whereas a line that could always be extended would be potentially infinite. Aristotle argued that the actual infinite didn't exist."
An infinitely long line would be actually infinite, hence actual infinity, whereas a line that could always be extended would be potentially infinite, hence potential infinity. Aristotle argued that the actual infinity didn't exist.
I read further
"Archimedes found a relationship between the full area of that slice, which was a section through the plane-sided volume, and the smaller area within it, which was a section through the curved shape. Then he argued that he could use that relationship to calculate the entire volume of the curved shape, because both the curved figure and the straight one contained the same number of slices."
and
"That number just happened to be infinity—actual infinity."
and
"The interesting breakthrough is that he is completely willing to operate with actual infinity," Netz says, but he adds that "the argument is definitely not completely valid. He just had a strong intuition that it should work." In this case, it did work, but it remained for Newton and Leibniz to figure out how to make the argument mathematically rigorous."
Archimedes being willing to operate with actual infinity, an infinitely long line, instead of a line that could always be extended. A potentially infinite line, what Aristotle argued that exists, whereas the actual infinity did not exist.
So, there is not infinity as such, whatever name it can be given, actual or potential, since an infinite line can always be extended. There are no boundaries.
But the infinity of decimal numbers, lying between two integer numbers, it has boundaries. The two integer numbers that lies within. Does having boundaries determine the kind of infinity it is? The infinity of decimal numbers between two integers, can not be extended beyond the integer boundaries.
It cannot be potentially infinite, it can only be actual infinity, what the calculus uses. What has driven Archimedes "strong intuition that it should work", and why "In this case, it did work". And it works when dealing with infinities that the boundaries are known or postulated beforehand.
What "it remained for Newton and Leibniz to figure out, how to make the argument mathematically rigorous.
Confusing the issues amidst vague statements, if "the argument is definitely not completely valid." What is valid, what is completely valid, what is not completely valid, and what is definitely not completely valid. How is validity defined, what is required to make an argument valid.
I read further in the same article
"Newton and Leibniz also worked with actual infinity. Leibniz went so far as to say in a letter, "I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author."
Since Newton and Leibniz also worked with actual infinity and produced calculus, would that not validate Archimedes intuition and his willingness to operate with actual infinity?
And what is meant by Leibniz' statement that nature makes frequent use of it, everywhere. Does that not imply the fractality inherent in all nature's objects, chaos driven processes surpassing, traversing fractal dimensions, from the microscales to the macroscales weaving the perfections of its author, chaos. Actual infinities trapped within the delimiting boundaries of any nature's object, all objects.
The statement 'modern calculus no longer makes use of the actual infinite; it sticks with Aristotle's distinction', a matter of taste?
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