The Logic of Randomness

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Eodnhoj7
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The Logic of Randomness

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Certain studies have shown chaos to move towards order and develop pattern if certain conditions are applied to it. Four conditions for the Kuramoto model, Christiaan Huygens' pendulums, fireflies, neurons, the London Millennium Bridge resonance, and large arrays of Josephson junctions. (Strogatz) "Barnsley fern created using the chaos game. Natural forms (ferns, clouds, mountains, etc.) may be recreated through an Iterated function system." (Weisstein)

The same must be assumed with logic for all perceived randomness and contradiction reflects, symmetry and logic. Contradictions, paradoxes, and perceived randomness within certain Logistic must be assumed as moving towards stability in one respect while reflecting symmetry in another. This reflection of symmetry fundamentally points to all randomness as a gradation of form and not form itself, and in this respect gradated logic must be observed as logic nonetheless.

Albert Einstein stated that "as far as the laws of mathematics [logic] refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."[16] (Einstein) It is in this respect, using this quote as an example, that logic and reality must be viewed as relative to each other, and in this respect are defined not as "things" in and of themselves, but rather through their relations. It is in this respect that logic and reality are closely related, however are "relatives" none the less further implying a separation. This relation can further be defined by Marcus du Sautoy claiming mathematics [logic] "the Queen of Science ... the main driving force behind scientific discovery". (Sautoy, A Brief History of Mathematics: 10) with this relation between reality and logic manifesting as the space of exploration. It is in this respect that Relationalism is the observation of the relations of logistics through people by deficiencies of logistics as logistic space. It is the explorations of the deficiencies between logic and mathematics that a further understanding of their relations may be observed.


In the early era of modern logic relation of ideas was considered the cornerstone of semantics. Antoine Arnauld claimed that “after conceiving things by our ideas, we compare these ideas, and, finding that some belong together and some do not, we unite or separate them. This is called affirming or denying, and in general judging'. (Arnauld) It is in this respect that true and false statements manifest through relations as logistics and the logistics of which they are composed, and their continual divergence relates as a "root function" for all logistics. Generally, in mathematics, all continuous root functions lead to 1. A similar observation can be made for logistics.


The 1898 study of "Hadamard's billiards" (Hadamard) observing all particle trajectories exponentially diverging through a positive Lyapunov exponent, gives a physical degree of evidence through imaging as to the nature of logistic particles (atomic facts) being observed in a similar manner. The positive Lyapunov exponent implies a probabilistic degree of interpretation as to the nature of particle flux, as divergence, and a similar observation can be argued, at the abstract level, in regard to logistics particles.


"Take a number at random between one and a million. How likely is it to be prime? This is just another way of asking how many primes there are between one and a million. Further: how many prime divisors will it have, on average? How many divisors will it have altogether, and with what likelihood? What is the probability that it will have many more or many fewer divisors or prime divisors than the average?" (Px) It is in this respect that the observation of logistic particles can be viewed as probability densities.


"It is sometimes said that probabilistic combinatorics uses the fact that whatever happens with probability greater than 0} must happen sometimes; one may say with equal justice that many applications of probabilistic number theory hinge on the fact that whatever is unusual must be rare. If certain algebraic objects (say, rational or integer solutions to certain equations) can be shown to be in the tail of certain sensibly defined distributions, it follows that there must be few of them; this is a very concrete non-probabilistic statement following from a probabilistic one." (Px)

Viewing logical truths as particles in turn cycles back to logical atomism as a form of relationalism where definition is acquired through change as divergence of one logical particle from another. This divergence required for the manifestation of definition is in itself acausal in the respect it is an absence of structure. In this manner definition is in a state of continual flux through relationalism as the means of definition. In these respects the act of definition maintains an inherent degree of randomness.

Flux requires a degree of stability to be observed, and in this respect Relationalism must relate to Holistism, otherwise the flux of relationships can never truly be observed. Relationalism, due to its dependence of flux through the manifestation and relations of partials, is equivalent to an observation of the deficiency of a whole through relations. It is in this respect that Relational logic has similarity to "fuzzy logic contain immeasurable degrees of truth observed by a number between 1 and 0 (Hájek) as a gradation of non-existence and existence. This fuzzy logic implies a necessary duality of "Probabilities" and "Potentials" as fractals. Relationalism, in turn, can paradoxically be observed as a dimension of fractals manifesting fractals through divergence.


The interrelations of particles exist if and only if there is divergence of particles; therefore, relations exist if and only if there is divergence and can never be strictly observed as they are never really unifying. Relationalism is fundamentally an observation of logistic negative space forming logistic positive space.


It is in this respect that a "fact" or "atom" is fundamentally a reduction of a whole and not a thing in and of itself; therefore, logistic particulate, "facts", or "atoms" is an observation of "divergence" synonymous to a dimension of flux, which in itself is a negative dimension that cannot negate itself as there is "nothing" to negate. It is in this further respect that Relationalism, as an individual dimension" exists if it relates to "Holism". A further definition can be inferred as a "particulate", "fact", or "atom" by their vary nature are a "part" of a whole, and in this respect exist if and only if there is a relation. Logical Atomism can be implied as synonymous to Relationalism. As a further note it may be implied that the necessary linear only construct Klein implies, under infinitism as continual linear progression, as necessary for logistics is a psychological relation to physical laws of the universe, as unless the "logistics" ancestry is continually renewed it suffers from "entropy". However, this is a half-truth as all deficiency, such as entropy or randomness, moves toward stability so a necessary "circularity" is involved that manifests a stability.

It must be noted that Wittgenstein later rejected Logical Atomism, (Wittgenstein) along with Russel. However, it must be noted that this view is actually correct when viewed strictly as a "negative dimension" that moves towards structure as the relation of separate dimensions; thereby manifesting a greater whole as an extension of itself as each "particulate", "fact", or "atom" is a "temporal whole". In this respect, Logical Atomism must manifest as a micro state of divergence requiring a greater "whole" similar to Russel's Logical Holism. In this respect, Wittgenstein and Russel are in many respects paradoxically wrong about being wrong as what they observed was a dimension of Logistics, not a be all end all. Atomism was perceived to have failed because it was required to be "a whole law", not an "atomic law" and therefore could neither reflect a greater "Holism" nor relate to it. In many respects, Russel and Wittgenstein where right.

The independence of logistic quantums and particulate in turn argues that each exists as a degree of "truth", and in this respect, are "wholes". In this respect, and only in this respect, they exist, however it is because of their relations. An observation of Wittgenstein claims:
4.211 It is a sign of a proposition's being elementary that there can be no elementary proposition contradicting it. (Wittgenstein, Tractatus Logico-Philosophicus)

As each logistic particulate and quantum exists as a relation, relations do not contradict but rather defined. The logistic particle and logistic quantum only contradicts if it is relative to itself, as it fundamentally must keep diverging.
5.134 From an elementary proposition no other can be inferred. (Wittgenstein, Tractatus Logico-Philosophicus)

Relations are not causal, no inference can be made, only correlation. It is in this respect, all logistic particles and quantums cannot infer other logistic particles or quantums, they may only correlate to them as relations are acausal. Assuming the logic reflects a degree of all scientific theories, as it does in math, it may be observed through the theory of 6 degrees of separation (xy) that logics particles may all be related to each other within a certain number of degrees. If this is the case, it applies a stronger case for logistics having a numerical dimension. As current research has not observed this is yet it must be observed as theoretical.

Russell's observations that atoms are the limit of analysis” and “the final residue in analysis”. However, he simultaneously argued that “analysis could go on forever”, “ad infinitum” (Russel's Logical Atomism).

An equivalent pictorial metaphor would be: a black hole needing matter in order to remain as a relative state of being to matter.

Relationalism, can be implied as a continual contradiction within logistics as no real structure is ever really maintained. However, a more accurate picture may be one of "transcendence" as all randomness can be equally observed as moving towards stability. "Nothingness", as a gradation of "Existence", moving toward "Existence" in many respects argues a degree of logistic transcendentalism as the movement towards logistic unity. In this respect, all logistic particles as "probabilistic", manifest a clear cut "partiality" as a duality of existence/non-existence and further relate to Reflectivism in this manner.

Leibniz argues that language is subject to "countless ambiguities" (Bochenski), and because of these ambiguities it can never be a fully accurate form of calculation. The problem occurs as Leibniz is using logic and not mathematics to prove his point. A more accurate interpretation, of what Leibniz probably never intended to say is: These fluxuations within language most probably relate to a fuzzy logic as a perception/dimension of an ever-changing socio-linguistics. This further goes against Leibniz argumentation as fuzzy logic is an extension of fuzzy mathematics, which in turns points to deficiencies within mathematical calculation. The ambiguities within logic and language may be further defined/reduced through a many valued logic { (Chiara) (Dalla Chiara and Giuntini) (Baianu) (Georgescu and Vraciu) (G. Georgescu) } that is sustained by a continual observation of logistic quantum and particle relations. Language is calculable, however the degree of it's calculation is ad infinitum, with infinity being a stabilizing dimension as itself. The lack of calculation, in many respects, would cause a deficiency in symmetry as it may be infinite, in a separate respect equivocating logic to have geometric dimension automatically involves Pi as part of these calculations thereby including infinity as an inescapable dimension.

It can be implied that this "logistic vacuum state as a negative" seeks further relations in order to maintain stability as a particle and eliminate its "vacuum state". This would be logistically equivalent to an implied contradiction needing relations to non-contradictory logistics in order to be observed. In this respect, all contradictory logistics need relations to non-contradictory logistics. Relationalism as a dual negative of Holism can be argued as a logistic vacuum state of Holism.
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