Theory of criticality
From Wikipedia, the free encyclopedia
| This article is being considered for deletion in accordance with Wikipedia's deletion policy. Please share your thoughts on the matter at this article's entry on the Articles for deletion page. Feel free to edit the article, but the article must not be blanked, and this notice must not be removed, until the discussion is closed. For more information, particularly on merging or moving the article during the discussion, read the guide to deletion. Steps to list an article for deletion: 1. {{subst:afd}} 2. {{subst:afd2|pg=Theory of criticality|cat=|text=}} ~~~~ (categories) 3. {{subst:afd3|pg=Theory of criticality}} (add to top of list) 4. Please consider notifying the author(s) by placing {{subst:adw|Theory of criticality}} ~~~~ on their talk page(s). |
[edit] Preface:
Here is a framework of 10 propositions collectively called the Theory of Criticality (ToC).It is my hope that this framework can be use to simplify modeling of the real world. ToC provides a natural way of looking at the world by removing the human bias of time impressed upon us since young by our own mortality. The ToC framework serves to remove the preconditioned lens and allow us to view world events from a different perspective. Some of the implications that arise from ToC are controversial within today’s paradigm. For instance, ‘war’ is reduced to a mechanism that moves the system to some state. ‘War’, ‘Famine’, ‘sickness’, ‘death’, in the ToC framework are neither good nor bad. They are but natural mechanisms in a system. Another contentious point that arises from the ToC is that people are dispensable and not unique. People are considered either a resource or conduits for channeling energy to produce events. This can be shocking for some of us as we like to think of ourselves as unique. The ToC erases our individuality and uniqueness and tells us that given the right condition, many such as ourselves will exist.
[edit] Motivation of ToC
What similarities does predicting lightning strikes have with political upheaval, war, famine, disease? All these events arise from complex systems. Modeling the underlying system to these events will test the boundaries of mathematics. Even with today’s powerful computers, their computational limitations are easily reached when it comes to modeling simple world events. The disparity of today’s computation technologies versus the real world modeling needs is still large despite the exponential evolution of technology. Instead of waiting for technology to breach the gap, rethinking the way approach complex problems by examining the assumptions behind the mathematics could be an alternative of modeling the world in a much simpler term. My hope is that by using the ToC framework, predicting events that arise from complex systems are made possible and in very simple terms.
[edit] Scope of the ToC framework
The ToC framework needs to be tested against several types of systems. As of this moment, I do not possess enough data or resources to get the data to demonstrate that the ToC framework is applicable to all systems be they social, chemical, biological or otherwise. I have tried to set the ToC in such away that it is scalable from a sub-system to a very large system.
[edit] Definitions of Entities and Enerzons
In the 10 proposals that will ensue there a couple of terms that are used will be loosely defined here.
A System refers to a collection of holons. A holon is a system when viewed in isolation and a sub-system when viewed within a larger system. An Entity within the System is a holon. An Entity need not necessary be independent or self-contained.
The definition of Enerzon has two parts to it. First, Enerzon is a context dependent measure. Second, Enerzon is the fundamental energy in the system that enables an entity to process inputs into outputs.
The reason for the invention of the term ‘Enerzon’ rather than using the conventional word ‘energy’ is to recognize that by the law of energy conservation, there are many reincarnations of energy. In physics, enerzon is energy and is often measured in joules, watts, or Newton meter. In the socio-economic context, enerzon can be money, gold, oil or any tradable proxies between entities. In our body, enerzon can be calories. The second but subtle departure from the traditional definition energy is that enerzon is not just the ability to do work but the ability to process inputs into some output. Though the reader can interchange the words ‘enerzon’ and ‘energy’ to facilitate easier understanding, the meanings are different.
[edit] Proposition 1: partitioning property of Enerzons
Proposition 1: Enerzon can be partitioned wholly and the sum of its parts will add up the whole again subject to the 4 conditions below.
First condition: Enerzon will strictly follow the law of conservation as stipulated by Noether's theorem. Consequently, Enerzon can be summed and divided without any loss.
Second condition: by the first part of the definition of enerzon, the partitioning has to be done in the same context and all the parts of the enerzon must be measured in the same unit. For instance, if the total enerzon is money in euros currency; its compartmentalization must consistently be in euros currency as well.
Third condition lies in the duration where the enerzon partitioning occurs. An important consideration of enerzon partitioning is the duration of interest. The duration always starts from time index zero where it is the instance the desired event ‘x’ is achieved. Time index zero is designated by the symbol ‘t’. Duration, of course, requires an end-point. The end-point would be the point in time before t and is usually designated by ti where 'i' is any whole number greater than 0, hence duration in which the total cumulative enerzon is considered would be symbolized as di = [ti,t). Note that the bracket ‘[‘indicate that the endpoint is included in the duration while the start point t is not included in the duration. This makes perfect sense as only the duration prior to the instantiation of the desired event is or interest.
Fourth condition: the choice of the event changes the perspective of how to the enerzon is being used. Thus the choice of event X must be determined at the very beginning.
If the 4 conditions above are met, let X=x where X is a random variable and ‘x’ (small x) designates the desired event, then the total enerzon that has entered into a particular system within the duration di can be broken down as:
Ftotal(x,di) = Fentalphy(x,di) + Fentrophy(x,di) + Fe(x,di)[equation 1.1].
Ftotal(x,di) is the total cumulative enerzon put into the system in di.
Fe(x,di) is the cumulative enerzon used in a system within di for purposes other than the production of event ‘x’ and the maintenance of the system’s integrity.
Fentrophy (x,di) represents the enerzon used within the duration di to prevent the system from achieving the desired event ‘x’.
Borrowing from Gibbs free energy concept in Thermodynamics, let
Fentalphy(x,di) =Fs(x,di) + Fa(x,di)[equation 1.2].
Fs(x,di) is the minimum enerzon needed to maintain the system in the duration di in order to achieve the desired event x.
Fa(x,di) is the required enerzon used towards achieving the event x within the duration di.
If Fp(x,di) is the potential enerzon that has not been used up by the system, i.e. enerzon that cannot be binned into Fs(x,di), Fa(x,di),Fentrophy(x,di), Fe(x,di), then we have
Fdelta(x,di) = Ftotal(x,di) - Fp(x,di)
Fdelta(x,di) = Fentalphy(x,di) - Fentrophy(x,di) - Fe(x,di) - Fp(x,di)
Fdelta(x,di) = Fentalphy(x,di) - (Fentrophy(x,di)+ Fe(x,di)) - Fp(x,di)
[equation 1.3]
Let Fw(x,di) = Fentrophy(x,di)+Fe(x,di), then from equation 1.3,
Fdelta(x,di) = Fentalphy(x,di) – Fw(x,di) - Fp(x,di)[equation 1.4]
Fdelta(x,di) = Fs(x,di) + Fa(x,di) - Fw(x,di) - Fp(x,di)[equation 1.5]
From equation 1.4, we can see that Fdelta(x,di) is the cumulative enerzon needed to achieve the desired event ‘x’ within the duration di.
[edit] Proposition 2: Sufficiency Condition for Criticality
Proposition 2: if the we sum up the cumulative enerzon applied towards achieving the event, the cumulative enerzon needed by the system to maintain its integrity, and the cumulative enerzon needed by all entities to come together to interact, and if that sum exceeds the minimum required enerzon, then the probability of achieving the event converges to 1. In addition to the enerzon invested in the system, the system’s capabilities do play a critical role in realizing the desired event. Enerzons are wasted if the entities within the system do not possess the right quality to channel the enerzon to the produce the desired event. Borrowing heavily from thermodynamics, mathematically Proposition 2 would read as follows:
If P(Fdelta(x,di) > min(Fdelta(x) ) → 1 and Q(F(x,di)) → 1,
then P(X=x) → 1.
X is a random variable to represent the desired events; and
min(Fdelta(x) represents the minimum enerzon needed to produce the desired event ‘x’.
P(Fdelta(x,di) > min(Fdelta(x) ) is the probability that the enerzon being pumped into the system within the duration di is more than the minimum enerzon needed for achieving the desired event ‘x’;
Q(F(x,di)) represents the entities’ ability to channel the enerzon for the purpose of achieving the desired event ‘x’ during the duration di.
P(X = x) is the probability of attaining the desired event ‘x’;
[edit] Definition: ‘Q’ for quality
An entity endowed with a lot of enerzon would not be able to produce the desired event if it does not possess the right property or quality to channel the enerzon for the purpose of achieving the desired event. Let us consider an improbable example of anti-gravity. Further to our example, let us assume that if we are the world’s richest person and desire to possess a vehicle that utilizes anti-gravity as a means of propulsion. However this type of technology is beyond our current repertoire of knowledge. So even if enerzon in terms of money is futilely spent, that piece of technology will still be beyond today’s reach. Hence we have to build the Q function into proposition 1. The Q function essentially measures if the entities collectively possess right property to effectively channel the enerzon to realize the event. As an input the Q function takes on a collection of entities within the system and returns a percentage or a number from 0 to 1. A number close to 0 or 0% indicates that the entities have no ability to produce the desired event. A value of 100% means that the entities within the system possess the capability to manufacture the event.
If we equate the Q function as a proxy of knowledge, we could apply already existing tools in statistics for the Q function such as the r2 (r-square) or the coefficient of determination found in regression analysis. Associated to regression analysis, we could also use the degree of freedom (d.f.) as a means of measuring complexity. The higher the d.f., the more complex the event. It remains to be proven that the relationship between complexity of the event and the amount of enerzon required in achieving the event.
[edit] Definition: Event
An EVENT is an outcome in the system that resulted from the expenditure of enerzon by a set of entities within the system.
Note that the choice of an event or sets of events is pivotal in this framework. To make Proposition 3 useful by avoiding triviality, there’s a need to distinguish the major events from the minor ones. One of the ways to mark a significant event would be to require that the event consume at least 80% of the system’s enerzon. Why 80%? The approach of putting a threshold of 80% stemmed from three sources. The first is influenced from Vilfredo Pareto’s work on wealth distribution. The second is from Joseph Juran’s use of the 80:20 principle to prioritize problems. Third source of inspiration is from Genichi Taguchi’s application of the 80:20 principle in the field of engineering to distinguish the signal from the noise. The by product of choosing a significant event that consumes 80% of the system’s resources, I suspect that accounting for the enerzon would be easier due to ability to aggregate.
There will be situations where there will be no event that can eat up at least 80% of the system’s enerzons. What happens then? I can recommend two approaches. One is that the events can be grouped within some duration. In this paper, ‘event’ and ‘set of events’ are used interchangeably. A second approach is to reduce the scope of what is considered a system. The scope of what is considered a system can be scaled down enough to such a degree that the Pareto principle can be used.
[edit] Definition: State
A STATE of a system is the condition of a system when no event within the system is significantly changing it.
Further to the definition, a state of a system not only refers to a condition of a system; its reference is relative to the universe of all the possible conditions of that particular system. For illustration purposes, let’s consider a sovereign-economic setting where the system is our planet, entities are countries, and the different states of the system are measured by GNP. Hence the set of numbers from $0 above would constitute the entire universe of the possible state of the system.
[edit] Proposition 3: Uniqueness
Only the different states of a system can be unique.
The conventional use of time is as an indexing function for past events; e.g. dates in history. For history, time plays a vital role and is treated as unique. This uniqueness explains a lot today’s contemporary preoccupation with time travel. Contrary to convention, in the ToC, only the states of a system can be unique and not the time indices. This implies that the state of the system is used as a unique indexing function instead. By placing the uniqueness on the state of the system, time now plays a subsidiary or supporting role. To illustrate how the state of the system and time are treated in this framework, let’s say the event of interest is “water in the kettle boiling”. Additionally, let’s assume that the person boiling the water lives in an environment where all factors can renew or reset itself prior to the event of ‘water boiling’. ‘Renew’ can mean that the utilized resources can be recycled100% whilst ‘reset’ means that the infrastructure that is used to boil water is not subjected to wear and tear. The person boiling the water will find himself similar to the movie Ground Hog's Day where he can boil the water in the kettle again and again. Time in this scenario would be quite meaningless. The only way to ‘break-out’ of this fictitious loop is to change the state of the system. Note that though there can be multitude of methods use to boil water. The varied methods will still result in the same event. Later in the proposition, the methods will produce different efficiencies of the utilization enerzon. The mathematics underlying the statistical distribution should be very interesting.
[edit] Proposition 4: System mutation
Only an event or a set of events will change the state of a System.
Any event will change a state of a system. Let’s use Heisenberg’s principle to carry Proposition 4 to an extreme, even the act of using enerzon to observe the system will change the state of the system. Hence if no events transpired in a system, the state of the system will not change.
[edit] Proposition 5: Future state and past event dependencies
'
The possible future states in the set Si+1 in ti+1 are dependent of the event at ti.
The proposition simply says that current events directly influence the number of possible future states that a system can occupy. Another way of looking at this is the number of elements in Si+1 is directly dependent on current events. The adage of ‘investing for the future’ is a very good illustration for this proposition.
[edit] Proposition 6: contiguity of the possible states of the system
Let Si and Si+1 be the sets containing all the possible states of a system at time index ti, ti+1 respectively, ti < ti+1.
'
If there are no other time frames in between ti and ti+1 , then Si∩Si+1 ≠ {}.
This proposition is a cornerstone assumption in this philosophy. A simple way of interpreting Proposition 6 is that the possible future states of a system depend on its current status. Imagine a Cartesian map where the x-axis is the time index and the y-axis contains all the possible states of some system, the resulting chart would be a contiguous polygon that stretches through the time indices. 
[edit] Proposition 7: Past events give birth to future events
'
Past events influences future events
'
By Proposition 5 and 6, future events are made possible by past events. A past event refers to an event that has occurred to change the state of the system. Future events refer to the multitude of potential events that have yet to occur. This new statement that resulted from combining Proposition 5 and 6 will most likely be the cause of lots of contention. There will be some who will argue that future events happen randomly and have no association with any past events. This is where I believe a lot of work for this philosophy needs to be done to prove that the disassociation of past and future to be a fallacy.
[edit] Proposition 8: Entities Dispensability
All entities within a System are not unique and only the state of the System can be unique. This is a difficult concept for human beings to accept as the word entities also encompass human beings. We humans have always had the penchant of viewing ourselves as a unique individual. In the ToC framework, we human beings are as dispensable much like the napkin that we use or the chicken that we consume for dinner. There is no doubt that this proposal will definitely encounter lots of controversy and opposition. This is where I humbly beg that in order to wear a different lens; the reader needs to proceed with the utmost open mind. I have no doubt in my mind that there will be a lot of people that argue that the Picasso, Einstein, Edison, and so forth embody the essence of uniqueness. I argue that these individuals are only labels much like the brands we associate to our clothes. Given the right system state, there will be more of these individuals.
[edit] Proposition 9: Critical state
Let S represent the universe of all possible states of a system.
Let Si Í S where Si is the set containing all the possible states of a particular system at time index ti.
Let sijÎSi where sijrepresents some state of the system at time ti.
If the following conditions prevail:
'
Condition 1: Q(F(x,di)) is the maxima (i.e. 
),
Condition 2: Fw(x,di) is the minima (i.e. 
), and
Condition 3: sij is the state where Condition 1 and 2 are met;
'
Then P(X=x) is the maximum.
Condition 1 is the basic condition that the entities with the System has the sufficient quality to produce the desired event x. Condition 2 basically states that the entities is very efficient in using enerzons to produce the desired event x. Condition 3 is the most state where all these optimum conditions take place.
[edit] Proposition 10: Critical convergence
If the ToC Propositions 2,5,6 and 9 are true, then there will exist at least a state sij(x,d) ÎSi(x,d) such that
is the minima and êSi(x,d)ê is the size of the set .
Essentially Proposition 10 states that that if the entities within a system have the right ability to produce the desired event ‘x’ and the enerzon invested to producing the desired event ‘x’ exceeds the minimum requirement, then the system will converge on a state where the system has the highest probability of producing the desired event ‘x’.Diagrammatically, the x-axis will be time index and y-axis will be the enumeration of the possible states of the system. The polygon’s area will get denser and denser as it gets close to t.
[edit] ToC prologue
The implications of 10 propositions are controversial. We play around with two scenarios to just to understand some of the contention. Let us imagine that if Hitler was not around and the condition that started WWII persisted, there would be another to take his place. From the ToC framework Hitler is but another agent the bigger scheme of things set in motion by forces larger than him.
To illustrate the complexity behind Proposition 7 and Proposition 9, let us consider the popular science fiction movie titled ‘Back to the Future’ where the protagonist goes back to the past to try to change the future. Assuming for moment that it was possible to go back in time, and it was possible to change the past events, the time traveler would need to design a whole new set of events that will divert enerzon to change the future state of the system. The protagonist would need to divert the enerzon for the undesired event ‘u’ and desired event ‘x’ in the following manner:
i.Fw(u,di) > Fs(u,di) + Fa(u,di) ;
ii.Fs(x,di) + Fa(x,di) > Fw(x,di) + Fp(x,di)
If the protagonist cannot meet the condition i and ii, the event will still happen. Even if he protagonist succeeded in eliminating a few entities, the event may still happen. The protagonist may have just delayed the inevitable.
The difficulty for thinking in the ToC framework is due to our mortality. If we were immortal, life would take on a different meaning for us. We probably would not be obsessed with time. The ToC framework offers a different perspective of reality. With the ToC framework, our obsession with time and our self-centeredness are removed from our world view.
 |
This article is uncategorized. Please categorize this article to list it with similar articles. (May 2008) |
