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Commentary :: Education
Why do stable, definitive, and discrete spectral-orbital, many-component, material systems form at all size scales?
19 Nov 2010
Why are new math categories, such as the new: (many-dimension)-(metric-invariant)-(separable geometries)-category, not considered so that more than the statement…,
“The stable, discrete, definitive, many-component physical systems are simply too complicated…, based on the laws of physics and the math categories which are used as the basis for precise descriptive languages…, for professional math and physics to be able to describe their properties in a useful manner.”
…can be made about “How these physical systems are to be described?”
That is, there are major problems with the capacity of physical description to describe the observed properties of the world, yet there are no other, new, original, creative ideas about how the descriptive language of physics should be organized…what they now consider to be “true” does not work…, it does not lead to valid descriptions…, and it does not lead to greater technical development.
One needs a society which has a wide range of creative possibilities, a wide range of descriptive patterns upon which can be built useful new creations (or a truth can be built [or determined] which leads to new creations).

If one product (such as oil) or one application of a (limited) descriptive language should dominate in markets (or channels of communication) over the many possibilities that can exist, then knowledge ceases to be a virtue rather it stifles development and creativity, and the exploitation and depletion of resources associated to a narrow way of using a specific set of technologies, upon which the entire society depends, becomes a basis for upholding a society, and a basis for the domination of the society by those who control (key) resources and knowledge. (Yet this is what society now does (2010)).

In a new society, the few people of “high virtue” are defined by their selfless desire to learn and decide what is true so as to selflessly create gifts for society (or for everyone else), yet these few people can be everyone (Are these few people, not everyone?).

Only in social structures which hold onto one set of values and only one set of (absolute) truths, do people learn to feel inadequate.

All Platonic truths (ie truths based on assumptions and on word agreement, and subsequent deduction), including religious truths, have great limitations as to what one can describe with these Platonic truths (with these fixed structures of descriptive language). If the limits…, as to “what can be described” with a fixed, but precise, language…, are not respected, then this can lead to great catastrophes, because development of knowledge stops, and thus creative capacities in relation to this knowledge stays limited.
Development of useful knowledge associated to new realms of creativity results from equality and free inquiry, in an educational context in which the limitations of knowledge are presented to the learner to a greater degree than the educational institution presents authoritative Platonic truths (or precise descriptive languages, ie descriptions based in mathematics) which are associated with measurable verifications, ie this is (one definition of) science.
But remember that Ptolemy’s ideas were measurably verified. That is, measurability is more about the context within which a description can couple to other aspects of the world such as measuring a property or coupling to and using a (geometric, or energetic) property, so that the information which is given in a description can be used in relation to a newly created system, by coupling properties of one system to the properties of another system, so that the new system can be used in a practical manner.
Platonic truths (in mathematics) are clear precise patterns which can form a basis for a useful description. But not all Platonic truths are useful, and Platonic truths which are not changed, ie fixed dogmas, lead to unbalanced use of language (similar to unbalanced markets, such as when all aspects of an economy become dependent on oil) dominated by too limited (or to few) ways in which talk about (or to do) things, and this means unbalanced ways of using already existing Platonic truths (or ways to organize language, or material resources) which means that there are unbalanced ways in which people are allowed to create (no new descriptions of material and space, where descriptions of material in space can be either a geometric based descriptive language or a probability based descriptive language…
[where probability is about a description of random events within the context of the system, where measurable operators act on a function space of spectral functions, but it gives a very limited context in which to relate a system’s spectra to a geometric or physical context for the system. Probability based descriptions only provide expected values over a wide range of possible random particle-spectral events].
That is, the institutions of society create within “too narrow a range of possibilities, or too narrow a range of descriptive contexts” and this means (or results in) an unbalance in the development of (new) knowledge.
This unbalanced state, is the state in which our society exists today (2010).

In regard to descriptive knowledge there is a dilemma provided by the example of Copernicus and Ptolemy, were both measurably verifiable (mathematical) models (and at first Ptolemy‘s model was much more precise than the model of Copernicus), but one led to greater development while the other led to illusions.
That is, authoritative Platonic truths, such as religion as well as today’s physical models of existence… (it is difficult to explain observed data within a descriptive structure (or measurable descriptive language), so today we should not believe ourselves to be immune from the same type of errors of applying Platonic truths in regard to describing observed patterns, as Ptolemy made errors?)…., most often lead to precise descriptions of illusions rather than precise descriptions of useful knowledge.
This is the main reason one should have education and knowledge based on equality and free inquiry, where each person’s inquiry must be allowed to have its own authority.
It is only from our own honesty and our own errors, which we recognize by means of our being honest, that we learn and develop knowledge.
But fixed dogmatic use of knowledge in the same narrow structures of business use causes new knowledge to cease to develop.
The main issues which the expressions in this book put forth (The main issues about which the expressions in this book deal) is concerning, “How Platonic truths can be used to help describe observed patterns of existence.”

The evidence of usefulness of a descriptive structure, and subsequent creative developments which can be derived from the information of the description, suggests that the currently accepted descriptive language of physics is failing, ie the knowledge of our society is stuck within the fixed context of knowledge which only allows for the essential set of ideas about, or inventions of:
TV (1910),
programmable computers (1936),
particle collision probability model of explosive transitions between two stable systems (before A Noble),
bacteria, virus (L Pasture) DNA-protein (1954-2000) model of disease,
classical physics used to do surgery (20th century),
transistors (1948), and
lasers (1958).
Transistors were developed by using a thermal annealing process of metals applied to silicon crystals and placing metal conductors into the annealed crystal (thus adjusting experimentally the observed energy properties of crystals), while the laser resulted from an understanding about how light fills quantum states in crystals (a pattern originally described by Einstein).

Note: Occasionally DNA based ideas for controlling an abnormality do work, but more often there are many more factors involved in an abnormality (or disease) than just DNA.

While all aspects of technology related to classical physics continue to develop, the models of DNA and quantum physics have only very limited relation to technical development, except where classical systems have enough controllable sensitivity to use properties of quantum systems. But quantum description has virtually nothing to do with these technical developments.

Indeed quantum physics cannot describe:
general nuclear orbits,
general atomic spectra,
Spectra of a general molecule,
general crystalline structures (getting smaller micro-chips is being obstructed by lack of a valid (or lack of useful) quantum description of the quantum properties of a general crystal).
Furthermore, “Clean energy extraction from a nuclear fusion reaction,” will be believed when it is actually accomplished, ie in 1950 the administrators of science projects (the atomic energy commission) said fusion was 5-years away. It is claimed that today energy can be extracted from fusion but is the amount of energy extracted a useable amount and is the equipment resilient enough to sustain a useable amount of such energy generation?

The evidence of (a lack of) usefulness, and subsequent (lack of) creative development within society, suggests that the Platonic truths being used today (2010) to describe observed patterns of existence, in quantum physics or in general relativity, etc are following descriptive patterns associated to illusions, rather than following a set descriptive structures (or following a Platonic math truth) which are useful.
Whereas classical physics is following a descriptive pattern which is useful.
This book provides an evaluation of both failed descriptive patterns (ie failed Platonic truths) as well as useful patterns which also exist in the realm of Platonic truths, but which are not being considered by both institutional structures (by the professional help) and institutional interests (by the business interests).
It provides a way to realize, or construct (or relate), the known useful aspects of physical description, ie classical physics, within a new set of math patterns. That is to construct a new descriptive language (or a new set of Platonic truths) so as to hold onto the (old) useful aspects of the classical descriptive patterns and to identify (new) Platonic truths into which the old (useful) descriptive structures can fit.
Furthermore, the stable discrete spectral structures of mostly small material systems (but actually of systems of all size scales, eg the stable solar system also has no valid description) are put into a new descriptive framework which accounts for (or can account for) all the observed properties, eg apparent randomness, as well as to the stable, definitive discrete, or stable, definitive discrete properties for physical systems of all size scales. One of the two (many) main failings of the current descriptive structures of quantum physics is an inability to describe these stable discrete existing structures, the other being that currently accepted descriptions of highly ordered (small) quantum systems are not useful and are not leading to new creative developments.

The formal representation of a model of existence which is organized to be best related to formal calculations (or solution techniques) used for finding the implied measurable properties of the model of existence, eg spectral properties of material systems and interaction inertial properties of material systems and the relation these properties have to the various dimensional levels of… and to the many components of various dimensions which compose… the full (over-all) containing set of existence.
That is, it is important to find the coordinates (or the sets of coordinates) used in the descriptive structures which are important to both the described patterns (What measured properties are important to a description or to a descriptive context?) and they facilitate deriving information from the formal model.
A space-form identifies a natural separable coordinate system. A space-form can be a model of either a material space-form (which is one-dimensional less than the dimension of the metric-space within which the material space-form is contained) or a metric-space. In the next higher dimension, than the dimension of a metric-space modeled as a space-form, the metric-space becomes a material space-form. In each dimensional level there exist infinite extent space-forms, where the geometric shape of the infinite space-form extends out to infinity. This can be used to identify a particular subspace of a particular dimension in the higher dimensional over-all containing space. It is also consistent with a particle as well as a wave model for these infinite extent space-forms interacting (by contact) with other material space-form systems.
According to Coxeter hyperbolic space-forms of dimension-11 do not exist, and the hyperbolic space-forms of dimension-6 and above are all infinite extent. The hyperbolic space-form whose dimension is 5 or less, and it is not an infinite extent space-form, has a very stable shape and a well defined, and stable, spectral set.
An odd-dimensional hyperbolic space-form with an odd-number of holes in its shape, and its spectral flows are all occupied with “charge,” will have an unbalanced charge distribution which will cause it to start oscillating so as to generate its own energy, because the oscillation will push together spectral flows which are of the opposite state from one another.
This will also define a forcing oscillation upon which resonances, with its spectra, manifest with other spectra of lower dimension…, in the over-all containing space…, so that resonances between two spectral sets can be defined.
That is, the over-all containing set has a spectral set associated to itself, which depends on both the spectral properties of the high dimensional space-forms as well as depending on the spectral properties of the low dimensional space-forms. That is, there is a spectral inter-relationship between distinct low dimensional subspaces, and the spectral properties of all the different dimensional levels defines the spectral structure of the over-all containing space. When the over-all containing space is an oscillating (energy generating) space-form, then the spectral energy of the oscillating space-form can be the basis for a forcing spectral set with which other spectral sets can resonate.
Between adjacent pairs of dimensional levels there exist conformal factors which affect the relative sizes of the interacting material space-forms which exist in the different adjacent dimensional levels, where (what is wanted is that) the size of the space-form of the lower dimensional metric-space is such that, the size of the material space-forms which interact in the adjacent higher dimensional level for a model of a metric-space are very large [for any given (or for a particular) dimensional level], thus the (smaller) interaction space-forms (see next paragraph) of the lower dimensional level of the two adjacent dimensional (metric-space) levels do not interact (as material space-forms in the higher dimensional adjacent level), rather the smaller interaction space-forms are confined to the boundary of the space-form shape of the metric-space of the lower dimensional level, of the two adjacent dimensional levels. These conformal factors are the same as physical constants.

Between adjacent dimensional levels there exist interaction space-forms defined between interacting material space-forms so that the interacting material space-forms identify some of the faces which compose the interaction space-forms (or its “cubical” simplex representation) and the space between the interacting material space-forms compose the other faces of the interaction space-form. Equations of relations between differential-forms, which represent different faces on the interaction space-form, result in the definition of force-fields which exist between the interacting material space-forms. The inertial response to the force fields are determined by the geometric relationship that exists between the Lie algebra of local coordinate transformations (in the fiber group) and the 2-forms defined on the interaction space-form (where the Lie algebra and the space of 2-forms have the same dimension) so that this geometric relation determines the direction of the local coordinate transformations which act on the spatial face of the interaction space-form, thus determining an inertial response of the material interaction.
On 2-interaction space-forms, which are defined in 3-Euclidean-space, the dynamical interaction is spherically symmetric, but each dimensional level will have its own geometric properties in regard to inertial interaction directions related to the force field 2-forms of the interaction space-form. These are relatively abstract constructions which are related to both higher dimensional existence and how to understand material and its interactions in these higher dimensions.
These space-form models of:
its interactions, and
the containing metric-space,
are also related to a simple model of life (an energy generating oscillating space-form).
Yet, the math substructures which are being used to create such an abstract description are very simple separable geometries.
Are these descriptions about reality or about illusions? The usefulness of the description is the fundamental property which determines if a description is about a real state of existence, or “if it is not useful” then the description is about an illusion.
The interaction structure is consistent with classical physics, and the interactions between small space-form systems results in Brownian motion. Such Brownian motion was shown by E Nelson to be equivalent to the random wave-function properties attributed to being fundamental and thus the basis for the quantum description of randomness.

Note: A property of autism is being manipulated to create the illusion that there is a real property called intelligence, where intelligence must be about having the energy to perceive the world as “it really is,” the state defined by the Buddha as a state of enlightenment.

[the (a) particular authoritative science institution research (or interests) and limited development of industry] the people selected [based on fixed dogmatic authority over models of science]

Obsessive (and/or autistic) people, ie who are also very competitive and aggressive, are selected as both representing the pinnacles of “high value,” and as “leaders,” ….
(these winners of a (the) competition concerning authoritative dogma which is consistent with the “science dogma” that is associated to the current narrow use of science in (an) industry, ie dogmatic science follows the selfish interests of business, eg physics is associated to the random particle-collision model of bomb engineering. That is, the obsessive, authoritarian, domineering people who are selected (by business interests) to be the leaders of science are competitive followers of dogma, similar to the authoritarian, dominating, self-righteous, self-centered, fundamentalist religious figures who are faithful in their obsessions to an abstract idea which is based on a fixed set of assumptions, eg the fundamentalist religious figure talks about sin and “what God has told him,” while the fundamentalist theoretical physicist talks about the big bang and particle physics, both ideas are irrelevant to the problem in science as to “Why stable definitive and discrete spectral-orbital, many-component material systems form at all size scales?”)
… that the social system encourages and rewards obsessive, aggressive people with institutional positions and prizes for their work in science and math. This recognition of high value is given in return for their obsessive-ness concerning particular narrow topics which are “densely abstract,” but the abstractions which are now put forth by science as authoritative truths, are really related to illusions which are without any practical relation to creativity. Excessive authority which is related to narrow obsessions keeps new ideas from emerging so as to mire the technical state of society so that lucrative material resources (such as oil, and military products) can be exploited.
This type of obsessive-ness is encouraged and fostered in all the professional categories of society, ie all institutions related to creativity and all institutions related to protecting the narrow selfish interests of the owners of society, ie the Big-Business interests.
The social institutions which uphold such obsessive narrowness are social institutions which make policy as to how to apply violence and coercion to “the many people” of society…. who are not the few owners of society and/or their minions (or their henchmen)…. so that this violence and oppression against the many is administered and sustained by a combination of factors, most notably corruption and an expression and interpretation of the property rights…, which relate to the (big) business interests of society…, within the institutions of “justice,” where any type of property-rights or copy-rights or rights due to contracts associated to “the people without any power” in society are not upheld by the corrupt “justice” institutions, while in turn, the same “justice” institutions uphold and brutally enforce the property rights for the very powerful and their minions.
These narrow obsessions which are cultivated by the business interest within society are attributed to desirable qualities which are associated to the subjective notion of “high value” in society, ie in both the media and the education system, such as intelligence…,
(What is intelligence? “The ability to discern truth,” But, Who can judge this? Answer: Only one who is already omniscient.)
…., determination, “belief in oneself,” business success, and having a desirable media image, but this identification of individual traits of obsessive people with the attribute of “high value” associated to these obsessive traits is a result of a simple process of branding…
(or manipulating people using icons of “high value,” eg Princeton Institute of Advanced Study, Theoretical Physics, etc)
……, and has no meaning as it stands.
Yet they are symbols (or icons) which direct how people are motivated within society, ie how people are to be obsessive, what they are to “obsess over” (or the subjects about which people are led to become obsessive).
Because people are trapped by their descriptive structures of existence, they do not perceive “the world as it really is,” and thus people are susceptible to being controlled by, and motivated by, an understanding of the world which depends entirely on descriptive language, where this descriptive language has come to be organized around these “icons of high value,” it is also a language which is most often consistent with the idea of materialism.
Thus, those who control how language is used in society…
(the business interests, and subsequently, the media and the education system, where the education system [in turn], follows the high valued icons of the media and business interests)
…..can control the ideas about which people will (tend to) believe simply by manipulating “icons of high value” in the language which they use.

However, all of these descriptions have grave limitations, which are not explored because free inquiry and equality are not allowed in a system in which only the owners of society are allowed to be associated to creative actions in society. The creative actions of the business interests upholds their own social positions.

The limitations of language are most noticeable in science and math, but branding particular ideas with iconic symbols of “high social value” takes the population out of the context of free inquiry (and its subsequent relation to the social property of “equality between people who seek knowledge” (which in turn, is related to the greater capacity to create)), and thus the limitations of math and science are either “not seen” or they are ignored.

In science and math a descriptive language’s “value in truth” can be associated to a description’s wide applicability and great usefulness, but descriptive knowledge has limitations due to the properties of language itself, as the example of both Classical physics as well as the example of Ptolemy and Copernicus, both demonstrate these limitations related to the assumptions upon which the language is built, as well as the conclusion of Godel’s incompleteness theorem, ie that a given precise language has limitations as to the range of patterns which it can describe (and this is due to the limitations caused by the set of assumptions upon which a language is built).
However, these limitations of what a fixed precise language can describe requires that science and math be developed in a context of equality, where the fundamental assumptions upon which a descriptive language is based are always open to change. That is, an intuitive development of new ideas not oppressed by an excessive requirement of deductive rigor, ie demonstration of Platonic truth, since as often as not obsession with rigor results in Platonic truths which describe illusions, and not a widely applicable and very useful descriptive language.
That is, identify a set of assumptions placed within a clear context and intuitively develop such a new descriptive language’s range of applicability and usefulness.
One will have a more versatile grasp of knowledge if one is familiar with the relationship between assumptions and the descriptive context to which the are related and the possible usefulness of such a new context than if one spends excessive amounts of effort languishing in a narrow context constrained by the limitations of a set of assumptions upon which a descriptive language is based.

But this is not allowed by those owners of society who dominate society.
Thus, science is no longer about creativity in general, but rather it has come to be associated to the creativity of big institutions, which serve the selfish interest of the owners of society.
Thus, physics becomes a description of existence based on “the probability of particle collisions” the main idea behind the building atomic and thermonuclear bombs, and then this narrow vision of physics (as manifested by particle physics)… (despite its great limitations as to applying these ideas to creative efforts)…. is an idea which is associated with “high iconic value,” and thus they are ideas which are difficult to change.

The rules of the game of domination within science:
Ideas about science can only be changed by the obsessive authoritarian “leaders” of science, but this is like asking the Pope who is following the ideas of Ptolemy to suddenly change so as to follow the ideas of Copernicus.
Any one with a new idea must be willing to accept the authority of the current paradigm, and subsequently prove one’s new ideas based on the assumptions of the system of language which is in opposition to the new ideas, ie the basic notion of “peer review,” but the new ideas depend on their being a part of a new descriptive language based on a new set of assumptions, as the example of Ptolemy and Copernicus demonstrates.

The authoritative ideas of science and math cannot be changes simply by pointing out the great limitations of…, quantum and particle physics, as well as general relativity…., as descriptions of existence, because these ideas have become “icons of high value,” supported by data from high energy particle-colliders, in a society where people are not allowed to reflect and question, rather they are only allowed to obsess on the narrow interests which the owners of society (as well as the oligarchs of intellect, who in turn, are the “high paid” technical helpers of the owners of society) are directing the people of society to consider.

Short history of western science:
Copernicus questions the authoritative of the orthodoxy of his age, and provides an alternative geometric context for the properties of existence in relation to the earth, ie a new set of assumptions for a new descriptive language.
Galileo identifies the properties of motion and forces,
Kepler (essentially) identifies the inverse square rule for gravity, and provides the answers which solutions to differential equations which the solution to the two-body orbital problem can later verify,
Newton (in his obsessive manner) tracks the inverse relation between the derivative and the integral and defines force as being a local change of momentum…, ie a derivative (which is also a local linear measuring relation between a function and its domain space)…, so as to be related to an inverse square geometric (vector) force field, by means of an equation (a differential equation), so that subsequently, the solutions to Newton’s differential equations are consistent with Kepler’s quantitative properties.
[This leaves the question as to how to understand the derivative and the differential equation and the context of its definition.
Is the derivative an expression of a fundamental model of (local, linear) measurement (between two quantitative sets) so that the derivative depends on a local region of coordinates, which in turn, have a set of properties associated to themselves, so that all similar regions in the coordinates also have these properties associated to themselves?
Is the derivative an operator on a domain of functions so that the functions in the function space have certain properties associated to themselves, so that in turn, there are certain operations which can be preformed on the function space (analogous to the operations on the vector properties of local coordinate structures)?
What are the advantages and limitations of these two different contexts within which to interpret and use the derivative in a descriptive language?
The function space viewpoint is placed in a “vastly larger set” than is the local measuring idea. Thus, the idea of convergences and consistency with quantitative structures, and thus the coherence of the descriptive structure, is much more questionable on the function-space-operator view of a derivative, or its differential equations.
The derivative as an “abstract model of a measurable quantity” has never been fully developed, while the derivative as a local, linear measuring relation between a measurable property, the (solution) function, and its domain space relates measuring to the domain space, while an operator (which is a measurable property) acting on an infinite dimension function space, in the case of a discrete system, eg a quantum system (because the spectra of a discrete [or quantum] system seems to be finite), has never been a consistent idea (discreteness seems to be better modeled by means of discrete geometries, eg discrete isometry subgroups (or space-form geometric shapes)), but when the same idea of a function space is applied to a continuous context, as in classical physics, the function space’s properties related to wave-properties, such as frequencies, has been a more valid model for solving the differential equations associated to continuous waves and continuous values for frequencies, eg spectral variety (many frequencies) and the infinite dimensions of a function space seem to form into a more consistent inter-related quantitative structure.]
Faraday defines the force fields of the electromagnetic field in space-time, he is helped by Maxwell, and subsequently the electromagnetic field is identified with the properties of differential-forms (thermal physics is also formulated as relations between differential-forms at about this time)
Einstein points out the importance of the metric-function in the context of metric-invariance…, but his theory of a general metric-function cannot be (has not been) applied, ie general relativity does not work,

[Einstein is opposed to having science based on probability, yet it is he who does the most to develop quantum physics, he identified properties of the photon light particle, he developed the statistics of Bose particles (beginning the idea that quantum physics is to be based in probability, it is also the idea upon which the laser is based), he used the property of Brownian motion to argue for the atomic hypothesis (but Brownian motion can also be used to explain apparent quantum randomness), and he allowed the ideas of deBroglie about “matter-waves” to be published…. (professional scientists involved in “peer review” were opposed to publishing the ideas of deBroglie, this is because professional scientists are only interested in their reputations and careers (their relations with those who will pay their wages) they are not concerned about ideas, but Einstein was interested in ideas)…. thus beginning the formal structure of quantum physics and its relation to wave-functions associated to probabilities, his ideas about general relativity led to the non-linear connection forms developed mathematically and subsequently used in particle physics, he allowed Kaluza’s ideas to be published and this led to string theory (which seems to be proving that quantum and particle physics are descriptions of illusions, with only a slight relation to reality)]
It seems that most of Einstein’s ideas were wrong (general relativity, quantum physics, particle physics, and string theory [despite all the claims of a few highly precise “verifying“ measurements for these ideas, remember Ptolemy‘s ideas were also very precisely verified]), but Einstein was central to allowing ideas to be expressed, and thus he was central to the development of ideas in science. Unfortunately, they have mostly been incorrect ideas, thus they have led science in the wrong directions.

Quantum physics shows the Hermitian-invariance, and the subsequent unitary-invariant nature of existence, but quantum physic’s basis in the idea about the fundamental randomness of small quantum systems and their components, is wrong (as Einstein suspected).

(but the fundamental property of Hermitian invariance is correctly interpreted in the context of the spin-rotation of metric-space states [an idea which requires a new descriptive context]),

Particle physics, in its relation to the data of high energy particle collision experiments, shows the “ghost remnants” of higher dimensional spaces,

(but this property of the existence of higher dimensions needs to be placed in the context of space-form geometries, which in turn are associated to all dimensional-levels of existence, so that these space-form (or discrete) geometries can be both small and large [where the relative sizes of space-forms partly depends on the particular dimensional level within which physical properties are being described], so that the space-form geometries identify both material and metric-spaces depending on the context of the dimensional level of the metric-space within which the description is to take place).

The basic math structure of quantum and particle physics is that of placing a non-linear (finite dimensional unitary) connection-operator into a linear model of an infinite dimensional function space representation of a probability based descriptive language, where probability is “defined on” an undefined elementary event space (the spectra of general quantum systems cannot be calculated) composed of an infinite set (the quantum numbers which identify the spectral set are infinite in number) of unstable events (elementary particles are [usually, or mostly] unstable), where the spectral-set is (theoretically) defined by a complete set of commuting, linear Hermitian operators, thus it is a unitary-invariant context for the quantum system’s defining (Hilbert, or L^2) function space (which demands that materialism be preserved, even though the data of particle physics suggests the possibility of (real) higher dimensional structure to existence). The basic math processes are defined in a context in which the “connection terms”….
(where the connection operators change the hidden particle-states in a particle-collision process, which in turn, define Feynman diagrams, which in turn, define perturbation terms in a series (an infinite sum) which [when renormalized] adjusts (or perturbs) the original wave-function)
…..result in a series of perturbation terms which change (or are added to) the original quantum system’s wave-function.
This basic structure is absurd but it gets worse.
The perturbing terms (or series) are determined at a collision point, which is also a point in a (new) “discrete model of a smooth, global wave-function” (this relation between smooth and discrete models of the same object [the wave-function] are not valid, ie not consistent with one another at the level of relating simple quantitative sets to one another, one set is a smooth wave-function which satisfies a set of linear operators, but these wave-function properties must be defined on a region of the domain space (not at a single point), while the other set is a very large set of independent discrete systems (of harmonic oscillators) which satisfy a different set of linear operators than does the given smooth wave-function), where the collision has an elaborate model of elementary-particles having their collision-dynamics modeled in a context of irreducible representations of the Poincare group, so that the collision point has a hidden set of particle-states associated to itself, where the connection operators act on the particle states, which together with the collision dynamics form the set of Feynman diagrams, which in turn, identify terms in the perturbation series which adjusts the value of the wave-function at the particle-collision point in the domain space of the wave-function. That is, the connection operator transforms between irreducible representations of the Poincare group, but the connection operator is either U(1) or SU(2) or SU(3), while the Poincare group is [SU(2) x iSU(2) x (translations)]. Does such an operation have a valid description?
(If so, does such a descriptive relation… (where the quantitative sets upon which the description depends are changed, from finite dimensional particle-states to many infinite dimensional representation structures) …still define a relation between particle-collisions and changes of particle-states, or is it simply an abstract relation between different groups which is unrelated to finding terms to a perturbation (due to particle-collision interactions) of a linear wave-equation?)
It is a finite dimensional set, either U(1) or SU(2) or SU(3), being identified with infinite dimensional sets (the infinite dimensional irreducible representations of the Poincare group).
But in order for this model “of perturbing an original wave-function by a particle-collision, and changes of particle-state, model of quantum interactions” to not define a diverging series, then the perturbed value of the wave-function at a particle-collision point (due to the series of corrections associated to many types of particle-collisions) must be renormalized, in relation to the values of the masses of the colliding elementary particles, where an undefined value is related to a finite value. In the linear operator representation of the quantum system’s wave-function mass is a constant, but suddenly there is a function in the variable of mass, which identifies a decreasing function.
There are at least three set-structures (supposedly functions, or math processes) which relate finiteness and/or discreteness with infinite sets. These seem to be relations which, in fact, identify abstract illusions, and they do not determine a structure which identifies a coherent quantitative description of a quantum system’s adjusted wave-function.
In fact, the problem is really identified at the level of identifying the spectral properties of general quantum systems. The complete set of linear, Hermitian, commuting operators (each of which is to identify a measurable quantity in a context of smoothness) cannot be found, so particle physics adjustments are irrelevant.
Discreteness needs to be introduced at a more fundamental level in relation to both its actual properties and its descriptive structure.
There is no way one should believe that a set of finite non-linear operators applied to a linear, but undefined, “infinite set” should have a valid relationship with the original (infinite) set, so as to change the original set.
That is, random changes in a finite dimensional particle-state space, which identifies an action of a non-linear operator…., where these processes (or actions) are being determined by (or associated to) energy levels (or ranges of energy) in relation to the value of a wave-function at a point in space…, which is adjusting the linear spectral properties of a global wave-function.
This is an incoherent descriptive structure, but in developing the inter-relationship there are too many places where it is not believable that infinite sets, whose elements are difficult to define with a finite amount of information, can be related to finite sets in a manner which is consistent with the elementary properties of quantitative sets, as well as in a way which is consistent with the elementary properties of the elementary event spaces (for a probability based descriptive language).
That is, convergences, which are defined on very large sets, can be such that it is not clear to which descriptive structure the points “to which the convergences are defined,” are (in fact) a part. For example, does a perturbed wave-function, perturbed by a non-linear structure, have any relation to the original wave-function which fits into a linear structure? Since the perturbations, related to elementary-particle collisions, are determined by a range of energy values of the wave-function (a linear, global property) at a point, and the same range of energy values can be attributed to many different wave-functions. That is, why does such a limited set of non-linear changes in particle-states apply so universally to vast sets of global wave-functions which fit into linear math structures?

Particle physics is simply a “red herring.” There is no reason to take it seriously, yet if one wants to be an authority in physics one must take it seriously. But none of the general abstractions about a quantum system’s wave-equation, and quantum interactions based on particle-collisions within either atoms or nuclei have been used to describe such general quantum systems with coherent spectral-orbital properties.

When the real numbers are considered to have the same “set size” as does the “set size” of the points on a plane, then one wonders, “To which of the two real lines, which identify the points in a plane, does an arbitrary point in the real numbers (might be considered to) belong?”
Then there is the property that a general element of the real numbers requires an infinite amount of information to identify (G Chaitin). Thus the identity of an element of a real line, may equivalently not be a member of the real line but rather be a member of the plane different from the real line, but this can only be determined from an infinite amount of information, ie it is not determinable.
That is, Platonic truths about large sets are properties which can lead into math structures which identify (what appear to be) illusions, rather than defining useful patterns of description.
It is clear that sets which have a large number of elements also lead to logical inconsistencies in regard to descriptive cohesion. That is, the set which “contains everything” is too big and it leads to logical inconsistencies. Similarly very large sets have elements which cannot be identified, except possibly with an infinite amount of information. Thus, it is not clear “what type of quantity (it is)” which an unidentifiable element identify in a very large set, actually is. That is, which type of set to which an unidentifiable element actually belongs is not determinable.
Thus, it seems to be true that using pure sets of quantities where the descriptive language is continually moving between quantitative sets of different quantitative types can have some serious logical uncertainties and/or inconsistencies.
It seems that there needs to be some smallest measurable value which can be identified, in regard to a quantitatively based descriptive language, so that the quantitative set can only be a finite set which can be associated to the smallest measurable value, but that other math structures which are a part of a system’s (or structure’s) descriptive context can be allowed (such as smoothness) but concepts of convergence can only be to points in the finite quantitative set, and convergence is defined by finite sequence of approximations. That is, one can assume the properties of continuity or smoothness are valid, but the approximations in relation to convergence can be approximated only to the precision of an actual smallest measurement outcome. That is, “in topology” the metric-function can be thought of as being (essentially) continuous (or smooth) but convergence is to be confined within a finite set. This is almost requiring that a quantitative set be consistent with the finite property of a probability description’s elementary event space.
When set theory is used to describe a math structure then the type of sets used and that the uniform structure of the sets be of the same type needs to be required.
Thus, what is fundamental to a description within a quantitative context are considerations about “What is measured?” “What is its quantitative type?” and “The math process which identifies a valid measured value, for a given type of measurable quantity, needs to be identified.”
That is, “Is measurement about local linear approximations of a measurable property (or function)?” or “Is measurement about a set of operators which act on function spaces so that these operators represent measurable values*, and are these operators: linear, smooth operators? And “If not?” “Is the measuring process consistently defined between the two quantitative (or measurably descriptive) sets which the operators relate to one another?”
*How do these operators represent measurable values?
Must it be in relation to (the operators being used in) finding a set of spectral functions for the function space (in relation to the set of operators)? If the answer to this question is, “Yes,” then the (given) set of operators must be commuting (or there will not be a set of spectral functions defined by the entire set of operators, ie the operators will not be able to diagonalize the function space) and then the measured values would be defined in relation to the eigenvalues of the eigenvalue equations which result when the given set of commuting operators act on the spectral functions (or eigenfunctions), ie functions in the function space which diagonalize the set of commuting operators. For these eigenvalues to be real numbers then the operators must be Hermitian (or self adjoint). If the domain space (for the functions in the function space) is metric-invariant then (if the values of the functions and the values of the coordinates are to be consistently related to one another in regard to the properties of quantities then) the operators must be linear and (if they are differential operators [or multiplication operators] then they must be) given in relation to separable coordinates within the domain space.
Ultimately, if a description is simply an abstract relationship between many quantitative sets of many different types…, eg function spaces, vectors spaces, domain spaces, geometries, etc…, then it is very difficult to determine the meaning and consistency of such a description, especially if unrelated descriptive structures are forced into a relation with one another, eg non-linear connection operators being related to a wave-function which satisfies a linear (eigenvalue) equations. On the other hand, if the structure which is guiding a descriptive language is geometric then the meaning and consistency of the inter-related math structures are easier to determine.
When considering the set structure used for measuring the physical world (or for measuring existence) then the beginning point is the idea of materialism. With the existence of material, it is assumed (in materialism) that there is a space which has a dimensional structure, and thus an inner product should be consistent with the dimensional structure, and a set of geometric measures associated to itself (the space) which are consistent with a length measurement or a metric-function, which is a symmetric operator and thus there is always a local coordinate system in which the metric-function is diagonal, thus for the (coordinate) geometry (geometric measures) to be consistent with the metric-function it (the geometry) must be separable. If the (local) measures of two sets of values are the be consistent with one another then they must be related to each other in a linear relation. Changes in a metric-function must be linear and this leads to metric-invariance.
**Math processes which are dependent on a metric-function are (1) spatial displacements (related to both position and motion), and (2) the existence of a maximal velocity on a metric-space. These are math processes which can occur on any “particular dimensional” metric-space. When the math processes are related to changes in the dimension of a metric-function then these math processes are related to rotations of metric-space states which can be observed from a higher dimension, where rotation operators do not have a symmetric structure (whereas the metric-function is a symmetric operator), eg (3) when changing from a metric-space at one dimensional level to another metric-space at another (adjacent) dimensional level, the only property observed in the higher dimensional level is the spin-rotation between metric-space states on the lower dimensional metric-space (or lower dimensional space-form, which is a (closed) metric-space contained in the (a) higher dimensional metric-space (or contained in an adjacent higher dimensional metric-space)), eg the spin-½ property of rotating the metric-space states of a 2-dimensional material space-form which is contained in a higher dimensional (3-dimensional) metric-space, so that the spin-properties are observed from the higher dimensional metric-space.

When the descriptive structure of a system is partitioned into sets, so as to use all of these sets to describe the system as a whole, then the sets all need to be composed of the same type of math structures so that they can be knit together to a whole structure which is consistent with both the system itself and the system’s description.
However, quantitative sets which are “too big” can lead to non-resolvable problems, just as the set of all things has no consistent relation to a particular thing.
It seems that only geometries lead a descriptive math language to a condition where it is clear what is being “discussed” within the descriptive language, and in this context abstract properties can be allowed yet the smallest measured value can also be a property of the description.

In many ways it is better to identify sets of assumptions and their respective contexts of applicability and then to develop the natural ideas (or patterns) of these assumptions in an intuitive manner, so as to see if the descriptive structure can lead to a useful context of description, rather than being obsessed with rigor, so as to lose sight of a descriptive patterns relation to usefulness.

How would one use information about set containment in regard to a domain set and an image (or co-domain) set of a function, (ie the topic of topology)?
The context of set-containment is related to:
covering sets,
boundary points,
the structure of set containment on the domain and co-domain sets in regard to a function
(usually in relation to the sets defining the property of being close to some given point (in the domain space)), and
the subsequent restrictions (or limitations) of the image set, which is contained in the co-domain space.
The above mentioned properties associated to one set may be determined if properties of the other set are known, and properties of containment in relation to (open sets) in the image space and the domain space are given, eg such set containment properties are identified if the function is continuous.
In the same context of continuity, restrictions on the structure of the image set and other set properties of the image set can also be determined.
These set containment properties, which relate the image and domain sets (by means of a function which is continuous), are typically set properties defined in relation to relative distance (or closeness) of the points in some (small) region to some given point (in the domain space), where these small distances are identified by a metric-function. Thus, abstract topology is either related to “less structured” sets than metric-spaces, or it can give further information about sets which have a bigger cardinality (or are spaces defined with a general metric-function, ie outside of the idea of metric-invariance) and with different structure than (metric-invariant) metric-spaces.
However, “Is any such larger set, or set with different structures than metric-invariance, mostly related to un-useable illusions, because the values of the general metric-function are not locally-linearly consistent with the values (or quantitative properties) of the domain space?”

On the other hand, comparing the quantitative properties of the domain and co-domain sets (or spaces) (or comparing the quantitative properties of the domain and image of the co-domain space) [can*] requires both a derivative and its (almost) inverse “integral operator.” These operators are defined on regions (or [open] sets) in the domain space, by means of a definition of a limit and the existence of a limit, or a sequence, defined on these regions of closeness. These two operators must be defined on regions because the quantitative properties of the two sets (domain and image of the co-domain sets) can only be compared if a linear function can be defined on the regions defined by the two sets, so that the linear function becomes an ever better approximation to the values of the given function when the linear function is defined on ever smaller regions. Subsequently geometric measures on the two sets (domain and co-domain sets) can be compared and inter-related, eg in relation to a derivative operator and its (almost) inverse integral operator.
For physical description there is the “set relationship” that “the system is contained in the domain space” and its measurable properties are defined in an image space. For classical physics, the “image space (of the function)” can be the property of spatial-position, and thus such an image is also contained in the domain set, or the property could be that of energy and thus energy, both motion and position, can be related to the domain space (in classical physics).
*However, operators defined on function spaces might have new “set relations.” If the operators are (linear) differential operators then the above analysis about regions and geometric measures also applies to these differential operators, but sets of commuting operators can represent a measured value for a system, such as the value of energy, but energy is related to properties of “motion and position and geometry,” which in turn, can be related to space-form properties, thus these relations can also be referenced back to geometric measures.
That is, if operators are to be defined on function spaces so that the properties defined by the operators are not contained in the domain space (but where the system is contained in the domain space) then the sets of commuting operators must reference a measurable property which cannot be contained or referenced (or modeled) so as to be contained in the domain space. But the theory of measurement in quantum physics, so far, defines measurable properties in relation to classical measurements, ie measurements which are properties of the domain space.
However, in the new descriptive category (ie the many-dimension, metric-invariant, separable geometry category), there is an entirely new set of: geometries, spectral structures, materials, and containment sets to consider, an entirely new set of ideas about measurable properties in a geometric context with various levels of dimension to consider, along with various new ideas about set containment. For example, “set containment” can depend on spectral-dimensional-geometric-(material-type) types of distinctions. Many relations concerning set containment based on descriptive structures, eg differential equations or sets of commuting operators acting on function spaces, and functions get turned into Platonic truths which describe illusions because of the psychopathic interpretation of materialism which plagues physical description and subsequently plagues the structures of Platonic truths, the basis for precise descriptive languages, which math considers.

There are new ideas based on space-form models of many-dimensional, metric-invariant metric-spaces which finally corrects classical physics so as to account for stable, definitive, discrete physical systems at all size scales: from nuclear, to atomic, to molecular, to crystalline, to classical macroscopic, to life-forms, to stable planetary orbits, to the structure of stars, to the shapes of galaxies, the motions of galaxies, and beyond.
The math relation between life and mind and existence can now be better explored, where life-forms can now be given a relatively simple hierarchical model for their being.

Newton was wrong, it does matter how “the causes of force fields” are modeled. That is, measurable, and even usefulness, as a verification standard is not enough.
What is the answer to the question: Why do linear, separable differential equations defined on metric-invariant metric-spaces relate to (very) useful descriptions of physical properties? Answer: Because metric-invariance, and many dimensions, and separable (space-form) shapes, for both material and its containing spaces, are the fundamental sets of (in regard to) measurable properties of a precise description, which are also fundamental to the (actual) structure of existence.
This leads to considering the geometry of separable shapes defined on many dimensions, so that differential equations for physical systems always relate to this type of space-form geometric structure for all dimensional levels.

What appear to be particle-spectral events in space are the distinguished points of space-form geometries, while apparent randomness of micro-material structures is generated by the Brownian motions related to many microscopic interactions of material space-forms in space (this idea was developed by E Nelson, Princeton 1957).
The differential forms of classical physics extend to interaction space-forms defined in higher dimensional metric-spaces, and resonances…, during an interaction…, between an interacting system and the entire many-dimensional many space-form structured containing space…, cause such a resonating and interacting system, which is within the correct energy range, to stop interacting and instead settle into being a stable material-spectral space-form system in relation to the dimension of the interaction space-form.

Conformal factors (of particular values) between dimensional levels ensure that the adjacent dimensional levels are geometrically, interaction-ally independent.
Note: There are many independent but equivalent high dimensional over-all containing spaces.
Finally, in the new descriptive structure, materialism is discarded in favor of a multi-dimensional space-form structured abstract description which is based on the descriptions of geometry, but still in a many atomic-component context.

(That is, physical description is not based on describing elementary event spaces which never had set structures which were consistent with the elementary properties of quantitative sets, ie elementary events in quantum physics were always defined as, an infinite number of unstable events which are undefined).

This work is in the public domain
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