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Creativity and education

This is about creativity and education, and the use of education (or knowledge) in regard to practical creativity, and such practical creativity is often about technology, but surprisingly it is also about "the spirit" (religion).
The knowledge upon which the Holy-Puritan-Empire of the US has collapsed, it has become irrelevant,
[see The End of Science, J Horgan 1995, or read further in this paper, which provides a stronger condemnation of the authority of modern (2012) math and science, because modern science and math are in opposition to the development of knowledge, and they help to suppress the relation that creativity has to knowledge, where this relation is suppressed by a social structure in which only those intellects who win the academic competitions are considered to be qualified to be creative and that creativity is always expressed within a business whose activities are controlled by similar monopolistic businesses (however, one can interpret the life of S Jobs as proving this idea wrong, on the other hand, S Jobs was happy to exploit (and direct) a small set of technicians and engineers (whose interests were very narrowly defined by school business and society) in order to suit the interests of S Jobs {the interesting thing about Jobs was that he developed a wide range of technologies, big business does not typically do this}, but, in fact, creativity is best served "not by technology" (though technology can be put to good use for creative and communication purposes) but rather by the development of a wider range of knowledge brought about through equal free-inquiry (see below) [allow for many "S Jobs types" {but without the commanding high salaries used "by leadership" to force the technical help to develop the technologies which are to be placed onto the market}, but rather within a much wider range of creative possibilities. Placed into a truly free market in a society where everyone is an equal creator {if a person cannot accept such an egalitarian vision for the people of a society, then such a person is, in fact, supportive of totalitarian societies, but such intolerant and arrogant people, may themselves be victims of the propaganda system, in which the main message is that "people are not equal," and only the owners of society can develop (through investing) something which has high social and intellectual value, by exploiting how people are not equal]].
The formula for "value" in a totalitarian society is that an activity is narrowed and personal value is defined by one's discipline in regard to the high value defined on the narrowly defined activity, where value is considered to be easily identified in the narrow context of allowed (highly-valued) activity (although this is really the realm of indefinable randomness). This is the formula for art and it is the formula for math and science, where the narrow complicated subjects of the math-science language are best dealt with in a personal context of autism. This is quite similar to perpetuating one's famousness by the act of being famous, the life of both the winners (of the narrow contests in which high-value of a person is proven) and the people who compose the propaganda system (which creates the fanfare of famousness) is allowed (encouraged) since the process is either creating profits for the owners or the process is maintaining the social structure and the idea that people are not equal. People learned to love the bankers since they supported the high-brow artists, but the love the people show towards the gifted people who possess high-social-value is the illusion which propaganda (which focuses on the narrow realms of high-value) can sustain within the public.
The owners of society have very little value for life. As a civilization, America chose violence over equality shortly after the US nation arose (though the Declaration of Independence states that American law ids to be based on equality.), and now an inability (or lack of will) to care for deteriorating nuclear bomb factories, ie nuclear reactors, (where such care provides no profit) that one can surmise that the Fukushima-Chernobyl model of failed nuclear reactors will soon be the model (or signature) of a failed civilization, and they will liter the country-side, and be destroying life.
The social collapse is here, but the oligarchy is aided by a propaganda system as well as a (hidden) military group of psychopaths (included within, and shielded by, the justice system, but also in the social ranks of the so called business men, eg the extremely violent Tea-Party types) who are organized to terrorize the US public, such social organization (can) will push the society to the "absolute right" and to greater violence and oppression.
However, the social failure of the US is so obvious that this could be countered by a push toward the re-establishment of US law "based on equality" and developing a subsequent relation that equality... , this new law "based on equality"... , has to knowledge and creativity.
That is, creativity is related to equal free-inquiry, where the inquirer [and their desire to be creative based on new knowledge] is in the authoritative position (not the academics, whose authority reflects the selfish violent and authoritative interests of the owners of the [failed] US society, eg academic physics is all about nuclear bomb engineering).
That is, any knowledge and "the creativity to which it is associated" which provides either profits or support and maintenance for the oligarchs is a failed knowledge, any ideas which do not oppose the dogmatic absolute authority of the oligarchy are in support of a failed belief system, and it is a (failed) belief system (or set of ideas) which is opposed to equal creative freedom, and the knowledge of the oligarchy is all about militarism and violent inequality and irrelevant absolute ideas.

Note: The value of the things which "comprise the 'things of value'" in the oligarchical hierarchy should be measured in ounces of baloney.

Creativity is not enhanced all that much by technology, though technology could be used in creative ways (perhaps it could help organize an equal society), but instead it is used to enhance the power of the (apparently very frightened) oligarchs who define value (so very narrowly) within the US totalitarian society. Practical creativity is best enhanced by having command of the language of math and science, and command of this language is not possessed by the narrowly defined authorities of math and science whom possess so much high-social-value and authoritative social power. The subject (the knowledge of the math-science language) of this paper has a far deeper relation to creativity, than do those who possess the authoritative math-science knowledge of this society.

In a nutshell here is the structure of math-science.

Classical science (physics) is unnecessarily constrained by materialism, while quantum physics is troubled (and rendered useless) with a fundamental assumption of randomness. General relativity is non-linear and thus not practically useful.

Math is divided into the 7 simple categories of:
1. Arithmetic 2. Set containment 3. Algebra 4. Analysis 5. Topology 6. Geometry 7. Complex number techniques.

1. Arithmetic
(adding and multiplying operations)
2. Set containment
(dimension, continuity, functions, geometry [local vector structures, often associated to shapes], and indefinable randomness, ie function space techniques)

Measurable physical properties of a physical system are most often represented as the values of functions defined on a domain space (the variables of the function), which is a system-containing set of coordinates which are usually structured as a metric-space, where the function is found as a solution to a "system defining" differential equation where the differential equation has attributes associated to both the domain space and to the system's properties.
There are basically physical systems of the type:
Geometric (the linear solvable types, and the non-linear (mostly) unsolvable (chaotic) types),
Geometric-spectral (electromagnetic systems, and mechanical wave systems),
Spectral-random (indefinable randomness, see the end of the paper).
The "spectral-random systems" and the "non-linear geometric systems" have for the most part not been solvable, they are not accurate (precise and true to an acceptable level of precision) in regard to a wide range of general systems of these types nor are these (mostly unfound) descriptions practically useful. They are placed into feedback systems with some small ranges of success, especially the non-linear systems where critical points of the differential equations can be related to the vaguely measurable structures of limit cycles, which are often circles or system boundaries, the dynamics of non-linear systems approach in a chaotic manner the system's limit cycles, etc.

3. Algebra
(order of operations, inverses [needed for solving algebraic equations], and polynomials [where algebraic geometry focuses on polynomial solution sets and is a generalization {an unnecessary generalization} of the simpler discrete hyperbolic shapes] {matrices, linearity, linear independence, orthogonality, are also defined} algebra is used in conjunction with set containment properties which are needed to identify the structure of (function) invert-ability, eg local invert-ability for general non-linear structures)
4. Analysis
("function space" and "sets of operators" techniques are used in analysis [which has a strong relation to algebra] to solve differential equations, where transformations of function spaces to their "dual spaces" is used to transform differential-equations to algebraic-equations [hopefully in the context of complex-numbers where solutions to algebraic equations are easier to find] and often in the context of second-order differential equations, because of the relation that the differential equation has to the metric-function defined on the system containing domain space, wherein this context, the measurable properties of the system are related to the spectral values and the oscillations of the system (or its wave-structure) [which are associated to the set of functions], where if the oscillating system's wave properties have direct physical properties and the metric-function is simpler... , ie either a metric-invariant Euclidean space metric or a metric-invariant space-time space metric, then the difficult techniques of analysis, {that is, that of finding bounding properties on the system's spectral set, eg lower and/or upper bounds on the values of the spectra}, are easier to identify (when there is a simple physical structure)... , then (in the case when the wave has directly measurable physical attributes) these function space techniques have a better chance of finding a solution-function to differential-equations, but in the context of indefinable randomness (where there are no [direct] physical attributes to the wave-function whose properties are indirectly identified in a probability-spectral-event context), and/or the metric-function is complicated, eg the non-linear case, then these solution techniques are mostly irrelevant, ie they do not achieve finding valid solution functions to differential-equations (or to sets of operators), and the information provided has very limited value)
{analysis should be thought of as techniques used to solve differential-equations in the context of both algebra and the "derivative and integral" operators which act on function spaces.}
5. Topology
(continuous deformations of space, and the relation that these deformations have to holes in space. This is used in the context of differential-forms and their associated (dual) metric-space bounding simplex structures [where the association is made in the context of integrating differential forms on bounded regions of the system's (or shapes) containing coordinate space.)
Basically the new context within which to define the containing domain spaces for measurable (physical) systems is to make this topological context the simplest it can be, ie the cubical simplex geometries related to circle spaces, and to introduce new sets of (very simple) mathematical processes within this new (simple) system (or mathematical) containment context, but it is many-dimensional, but nonetheless, materialism is a subset within the new context.
In the new context "holes in space" are defined by discrete Euclidean shapes, and discrete hyperbolic shapes.
Continuity needs to also be adjusted a bit, but in the new context the spectral set... ,
which defines all existing metric-spaces (and their metric-space shapes, ie all material too) in the description's over-all high-dimension space
... , is a finite set.
That is, the containment set of the description does not get "too big beyond comprehension" and "too big" beyond mathematical (or logical) consistency (as it is difficult to determine what patterns exist and what these patterns mean in the current (2012) "very large set structure" context).
The proof of this is that: 1. particle-physics cannot be related to the stable spectral structures of general nuclei and 2. no one knows what string-theory means, other than that it is a "string" of very complicated math patterns pieced-together with the irrelevant focus on elementary particle properties and the useless non-linear general relativity being combined together.

6. Geometry
(coordinate functions define general shapes and the relation that the local [linear?] vector structure of these coordinate functions have to a metric-function and its geodesics and curvature define the shape of the coordinate functions with (or within) a non-linear structure, ie the local vector properties of the coordinate functions do not commute, thus coordinates are defined locally and then pieced together, but if the relations are non-linear then the quantitative structures which are being pieced together are chaotic and the local calculations are not reliable so the process of piecing together is not valid, ie shape cannot be determined in a reliable manner in this process)
Only the linear, metric-invariant (with non-positive constant curvature, see below), and diagonal (see below) shapes, defined by differential-forms, are stable and quantitatively consistent, and thus measurable in a reliable manner, and useful in regard to their relation to practical creativity (note: only geometric patterns can be used in a practical context). (these are essentially the [real] circle-spaces)

7. Complex number techniques applied to complex functions defined on complex domain spaces,
(which means that the algebra is diagonal (but within the complex numbers) and the spaces (or shapes) seem to be either "circle spaces" or "disc spaces" or "spherical spaces" (with emphasis on the Riemann sphere, ie a model of the complex number-plane, but "sphere spaces" are difficult to work with, they are non-linear {at least in their associated real-case}) and if a complex function is differentiable then it is also analytic, which seems to mean it has close connections to polynomials and to the structures of numbers in real number systems)

Complex number structure might be best associated to both metric-space states and discrete processes within a metric-space state structure... ,
... ., where metric-spaces (different dimensions and different metric-function signatures) are given physical-math properties, eg Euclidean space is associated to spatial displacement and inertia etc, and these physical properties have opposite metric-function states, eg Euclidean space has either fixed stars or rotating stars etc hyperbolic space (the equivalent of space-time) has positive and negative time metric-space states associated to itself, [dynamics of objects takes place in a discrete time structure, of very short time intervals, associated to the spin-rotations of metric-space states, while this dynamic process is mediated by discrete Euclidean shapes, which possess the property of action-at-a-distance {Einstein was wrong about action-at-a-distance and about the absolute frames of Newton, ie the frame of the fixed stars} (so the discrete Euclidean shapes are defined in each time interval, but they change size with each time interval, due to spatial displacements of the interacting objects), the property of action-at-a-distance has been confirmed in what are called non-local quantum experiments (also referred to in the propaganda system as teleportation)]
... , while real structure is what we tend to experience (though it is not clear why) and the real structure is related to an almost continuity (the time intervals are very small), but the real continuous structure seems to (definitely) be related to the continuity properties of conserved material and energy etc.

All (seven) of these math structures need to be re-evaluated in the context of the (real) metric-invariant spaces, wherein the metric-functions only have constant coefficients, and the spaces have non-positive constant curvature, where these are the stable shapes of the (discrete subgroups of) Euclidean and hyperbolic spaces (these are [also] circle spaces) and they are (to be considered) the bounding structures of metric-spaces to be used to model set containment, and within these closed metric-spaces (of constant curvature) are contained other closed bounded (and unbounded) metric-spaces with constant curvature, where these (lower dimension) contained closed-metric-spaces are models of material, but in this model these metric-spaces (or this material) can be any dimension.
The reason to re-consider quantitative description within such a context is that these spaces identify the attributes of the stable and consistent quantitative structures and the stable and quantitatively consistent geometric structures which are linear, metric-invariant, and diagonal, ie they are quantitatively consistent and stable, eg the discrete hyperbolic shapes are very stable and possess stable spectral properties (eg circles) which are defined around holes in the discrete hyperbolic shapes. These (hidden) stable, bounding geometric structures, associated to our metric-space set-containment structure, explain the stable spectral-orbital properties observed in our experience. This is something which the current (overly authoritative) math constructions cannot do.

Critical analysis of math (and physics) and how the professional authorities are used within society.

Here is a critical analysis of math and science in 2012, but it also outlines a process of how to use language at the level of: assumption, and context, and interpretation so as to develop new ways of framing both knowledge and creativity can be accomplished.

One needs to understand the precise descriptive languages in order to be able to create at a practical level, and the descriptions need to be based in geometry, and they need to be mathematically stable.

Mathematicians are not sufficiently critical of their own subject, they follow the traditional math constructions of the math professionals, but [apparently unknown to the mathematicians] these traditions are defined by powerful financial interests to which the mathematicians are (or have made themselves) pawns.

The mathematicians do not question or consider the fundamental ideas of math to a sufficiently deep level of: basic assumption, context, and interpretation, where ultimately math should consider:
1. Shape (the context of useful measurability), and
2. Quantity (the construction of measurable sets used for descriptions), and
3. Stability (the assurance that measuring is related to useful information and useful control of the
identifiable patterns).

Mathematicians are mostly motivated by their desire to carry-on and extend traditional math constructions, the constructions identified by the propaganda system as those "math constructions developed by the true intellectual masters." Furthermore, the mathematicians seem to believe that these constructions are leading to an "absolute math truth" (whatever this might mean, something about attaining great generality in regard to the traditional patterns of math) upon which the professional math community is induced to work, based on their wage-slave status.
However, many mathematicians are Aspergers autistic types, compulsive obsessive types, who still retain the use of language, and they are attracted to impossible complications requiring obsessive memorization of traditional, fixed, and very complicated math constructions, which are assumed to define a road to an "absolute math truth" of the aristocratic intellectual class [of so called superior (autistic) intellects].

Autism (Aspergers) is not "high intellect," rather it is "obsessive narrow intellect."

That is, the owners of society use the property of autism to identify a false sense of intellectual superiority, within a small intellectual-class within society, an elite "intellectual class" which is defined within a propaganda system, and this is done within the context of professional mathematicians and professional scientists.

The professional (wage-slave) authorities are unwilling to critically analyze the math constructs about which they have academically competed, so as to become professional authorities, and instead they wish to carry-on and extend the traditional authoritative math constructions, in a way which stabilizes their social positions. Thus, their capacity to develop new math ideas... , needed to solve many very fundamental questions which are going without answers (such as why is the nucleus stable?)... , does not exist. This is because the professional mathematicians exist within a social context which opposes the expression of such new ideas (they are pawns to the selfish interests of the owners of society).
The vague idea which seems to be held by professional mathematicians is that the authoritative math constructions will eventually lead to useful accurate description (or computations), and to an absolute mathematical truth. The wait is over 80 years, and the wait is not over for these complicated truths to be related to practical technical development, other than using (this) knowledge to further develop militarism.
This is the interpretation of Godel's incompleteness theorem in which new axioms are continually being added to the traditional axioms so as to extend the amount of the professional math literature concerning the particular types of favored, authoritative traditional math constructions which the social system (or equivalently, the propaganda system) allows and funds, and which serve investment interests.
But the correct interpretation of Godel's incompleteness theorem is that the language of math needs to be continually criticized and changed at the level of axioms, contexts, interpretations, containment structures, and these (new) elementary ideas need to be expressed at the most elementary level of what mathematics is about... ,
ie adding and multiplying applied within a geometric context where these math structures are assured to be stable,
... , so that the descriptions of measurable (physical) patterns can be both accurate (to an allowable level of precision) and practically useful.

There are great limitations as to the set of patterns which a precise descriptive language is capable of describing. This is the point of both Godel's incompleteness theorem and the example of Copernicus.
Godel showed that in a precise descriptive language in which there is a containing set, axioms are identified, definitions are given and interpretations made and then in such a precise context (concerning what words mean and how they are to be interpreted) there always exist precise patterns which are consistent with the context of the language, but they cannot be proven by logically relating these new patterns to the axioms.
On the other hand the example of Copernicus is the more common example, where two different types of precise descriptive languages are based on incompatible sets of axioms, ie either the sun has an orbit about the earth, or the earth has an orbit about the sun, so that these are two incompatible assumptions (though the containing sets have many similarities) where one set of assumptions is more accurate and has greater relation to practical creative development that the other set of assumptions.
However, the way in which Copernicus challenged authoritative assumptions has been thwarted by a re-labeling propaganda-procedure in which "the authority of science" is not the (old) "arbitrary authority of religion" rather the (new, narrowly defined dogmatic) authority of science is based on an "absolute scientific truth" which is verified by experiments, but science today (2012) is authoritative dogma, just as religion was an authoritative dogma in the age of Copernicus, ie it is authoritative dogma (about which people have a deep religious faith due to the propaganda system) with an elaborate set of math constructions which have been built around data so that the data can be interpreted (from narrow sets of cherry-picked data) to support these elaborate math constructions, science is narrowly defined arbitrary authority and it has this structure because it has been organized to fit the selfish interests of the owners of society, who want to control both knowledge and creativity within society.

Now consider the state of science and math in 2012.

There is a fixed traditional set of very authoritative math constructions into which the observed patterns of our experience are being forced, ie one is only allowed to describe "in a quantitative manner" the observed patterns of existence (only) within the "traditional authoritative set of math constructions," and, sure enough, the observed patterns [which must be described within an assumed containment set, associated to the axioms, definitions and interpretations of quantity and shape defined by the authorities] do not fit into these fixed authoritative math constructions, [yet the information (data) is manipulated in the propaganda system to make it appear that the observed patterns do fit these authoritative math constructions, but this is not true (see below)].
The focus in both math and science has been on "identifying a physical system's set of differential equations," and there are two approaches:
(classical physics) Local properties are used to identify a differential equation which, if solvable, provides precise, global information about the system, (Note: The set of linear solvable differential equations are quite useful, while the non-linear differential equations are not ""all that useful" since they are chaotic (though they can be used in feedback systems), but the there is the question, "Why is the solar system stable when its many-body differential equations are non-linear, and thus describe a chaotic solar system?")
(quantum physics) The system is represented by a function space, where the functions have no directly measurable physical properties, and sets of operators are supposed to identify the system's (measurable, real) physical properties which become identified as eigenvalues by the probability-spectral-component system's spectral-functions (where it is assumed that the set of operators can diagonalizable the function space so as to find the system's spectral-functions) where the spectral-functions identify random event "singularly measured" physical properties, which are the system's eigenvalues.
The eigenvalues for general quantum systems cannot be identified within this formula. Thus, a "good name" for such a very formal math process is "indefinable randomness."
Indefinable randomness has been the main focus of the professional math community for over 50 years, and only those descriptions related to waves which actually have physical properties which can be measured directly have any validity. Quantum physics is a failure, and the calculation of financial risks using the techniques of indefinable randomness are also failures, etc.
Nonetheless authoritative traditional math construction-ism "marches on," and further data, such as in the example of particle-physics, is forced into this set of math patterns (eg dark-energy), but they are math patterns which do not describe the observed patterns in regard to the large set of general quantum systems such as: general nuclei, general atoms, general molecules, etc.

One does not need to continue to try to force "one set of observed patterns" into "another set of traditional authoritatively constructed patterns," rather what is needed is a new type of publishing, a new type of equal and free-inquiry... ,
(not controlled by dominating monopolistic interests, ie kick-out the owners of society and the academics [who have been conditioned to support the owners of society] from determining what is published)
... , in which the main aspect of a person's idea is about the:
set of containment,
definitions, and
"interpretations of both one's new precise language and the observed patterns of our experience," and
within this clear structure of a new precise language one develops a precise descriptive structure.
Words need to be well-defined, but one does not want their definitions to be traditional definitions, though traditional definitions can be used, but the definitions need to be made explicit.
Faraday crafted the language of electromagnetism, not so much from technical jargon but rather from ordinary words which were given a structure of: containment, assumption, special use was identified, and interpretations were given. Faraday used his intuitive descriptive structures in a creative manner, eg he invented the electric motor around 1835. That is, formal math is not needed for a precise descriptive language to be related to creative acts.

[Then Maxwell put a technical polish on the descriptive language which Faraday developed, and the academics mostly give Maxwell, credit for the descriptive structure of electromagnetism, which he does not deserve. In fact, Maxwell checked if he was correct by seeking-out Faraday, furthermore, Faraday was using his precise descriptive language in a practical context.]
{The owners of society, as well as the academics who work for the owners of society, use the very creative people like Faraday... , and their example of both developing knowledge and using that knowledge for creative purposes... , in a very degrading and demeaning manner, and this is all about social domination and controlling knowledge and creativity within society.
The Faraday family should be one of the richest families in the world today (not the Rockefellers), since the electrical and computing industries were derived from his ideas.
But civilization is all about living within a den of murderous liars and thieves, ie the model of civilization since the creation of the Holy-Roman-Empire by Constantine (which the American Revolution tried to oppose by basing American law on equality [but the empire of oligarchs quickly won-out]).}

That is, one wants "the submission of written ideas to a filtering system" which filters ideas into the sets of assumptions which underlie these ideas, so that the new ideas can be easily separated from the old, where the old ideas have shown themselves to be unworkable "traditional mental constructions," which, because of their long duration, have become extremely complicated, and essentially incomprehensible. [technical polish makes a descriptive language less comprehensible, though there is advantage to a succinctly organized structure to a precise language]
The incomprehensibility of modern (2012) traditional science and math is demonstrated by the prevalence of these very complicated patterns and their being essentially unrelated to any practical development.
Furthermore, the complicated theories of quantum physics, particle physics, and non-linearity and all their derived structures such as string-theory, (are incomprehensible) because they do not describe anything which is close to being a thorough and complete (or comprehensive) description of the observed patterns of general quantum systems, and they are unrelated to practical development.
These traditional authoritative theories tend to linger on irrelevant issues, such as: big bangs, and black holes, and elementary particles, so as to try to not be detected as being pointless and useless descriptive structures.
The propaganda system cherry-picks data about which it is claimed that these descriptions are being verified (by the cherry-picked data).
However, though particle-collision data from particle accelerators does fit into unitary patterns, this does not mean that these unitary patterns have any relation (by means of the math structures constructed for quantum systems and/or particle-physics) in regard to describing the observed properties of general quantum systems (which go without valid descriptions). This is true for all general quantum systems: from nuclei, to atoms, to molecules, to crystals to the solar system, and beyond.

Do the mathematicians need to continue the endless considerations concerning:
"How 1/r can be related to spectral values within inconsistently defined math patterns?" When these speculations do not provide a valid description of the observed patterns.
"How non-linearity can be related to stable useful geometry or to 'what seem to be indescribable' observed spectral values of physical systems?"
Must mathematicians obsess about using established authoritative math constructions in their attempts to describe the observed properties of many very fundamental systems, but the mathematicians and physicists are continually failing at these attempts? Apparently, for mathematicians, it is enough to follow either authoritative math constructions, or to follow the math projects which are being funded.
Unfortunately, one gets a deep sense that underneath all the tradition authority and great complicated sets of detailed considerations that the mathematicians do not comprehend the point of many of their traditional authoritative constructions, ie they are used as "memorized rules," and not (or very seldom) considered in relation to elementary math concepts.

In probability-spectral (or quantum) a global quantum (spectral) system's properties are identified as having a relation to the (quantum) system's global wave-function's wave-phase (note that this means that quantum systems cannot be invariant to local phase-changes, since derivatives of such new wave-phases [caused by wave-changes] would change the global properties of the quantum system), whereas local properties of a probability-spectral system is related to the eigenvalues of individual spectral functions where this local property is associated to random spectral-component events in space, which when such local spectral-particle events are observed this causes the spectral system to collapse, and these math models of local system properties is not the result of a local vector structure associated to the properties of the spectral-functions, that is, the local physical properties in a probability-spectral system are the "global" spectral-functions (this turns the very useful math structures of classical physics [where local relations are used to determine global solution functions] but in the probability-spectral system global functions define the system's local properties).
Furthermore, it should be noted that the system's wave-function is an infinite series of these spectral functions, but the system's general wave-function collapses when a particular spectral-function's spectral-particle value is observed.
On-the-other-hand the Dirac delta function is also an infinite series of spectral functions, and the Dirac delta function identifies function values at a particular point in space. For example, the Dirac delta function can (also) be used to model the r=0 point of a 1/r potential energy term in radial equations of spherically symmetric physical systems, where the 1/r term is the term upon which the spectra of a probability-spectral system are, essentially, derived.

Note: It should be clear that atoms and molecules and nuclei as well as solar systems develop in a physical process which is highly controlled, and thus it is information [or precise description] which is controllable, since these systems can be identified by their precise, stable spectral-orbital properties which (in these mentioned systems) exist at all size scales; from nuclei to the solar systems; but the stable discrete spectral-orbital properties of none of these systems is being adequately described, yet these are the most fundamental of all the measurable systems which are associated to the idea of materialism.

It should be noted that quantity is about addition and a well defined uniform stable unit of measurement, where the counting process (or the addition operation) identifies a number scale, and addition depends on the type of things being added, while multiplication can change (by multiplication with a constant) a number scale in a consistent manner, ie in a linear manner, ie y=mx where m is a constant scalar is the simplest linear functions, furthermore, the arithmetic operation of multiplication is able to change number type).
It is a valid assumption that, shape (or observation) is intricately tied to the idea of quantity and measuring, since it depends on the shape of a well defined (stable) uniform unit of measuring (upon which counting is based).

Mathematicians currently base (ie the vast majority of the math patterns which professional mathematicians describe) mathematics on overly general contexts, which are not stable, nor consistent, their set structures are "too big," their containment context "too general," (eg much of topology is based on the general notion of holes in space rather than the more concrete model of the holes on discrete hyperbolic shapes), and the application of quantity to non-linear shape is not consistent with stable and fundamental quantitative structures of mathematics.

The math models which apply in a useful creative context (associated to measuring and controlling by means of calculated [described] information) to the widest range of useful generality is the correct key to be used to analyze in regard to determining the relevance of their precise (careful) descriptions.
That is, logical consistency, in regard to word agreement, is not a valid determination of mathematical truth. And it is certainly no guarantee that the authoritative math constructions followed by the professional mathematicians are on the way to absolute final truths about mathematics.
This is because the "correct interpretation of Godel's incompleteness theorem" implies that measurable description, ie useful measurable descriptions through which creative invention can be based, is the natural domain of description for both science and math, where truth is based on practical creative development about which a measurable description (a calculable set of information based on measuring shapes placed within the context of seeking to create new systems) is related.

The vast majority of math models of the professional mathematicians are based on:
1. Indefinable randomness (eg probabilities associated to spectra which cannot be fit to the observed patterns of quantum systems to a sufficient level of precision), and
2. Non-linearity (often about either general shapes or shapes believed to be related to spherical symmetry, but stable systems must fit into stable shapes and non-linear shapes are predominantly unstable).

This basic set of prevalent math patterns (indefinable randomness and non-linearity) used by professional mathematicians seems to exist because it is believed that these models cover the widest range in regard to the description of the observed details of both mathematical and material as well as random event experiences, which people perceive or conceive, but neither of these "models for measuring" is either logically consistent or quantitatively consistent.
Neither of these math constructs are quantitatively consistent and they are defined in contexts where measuring is unreliable, and subsequently, it has been shown to be "not useful" in regard to practical development. Yet, the vast range of math patterns which are associated to these false (or unstable, and thus unreliable) quantitative and geometric constructs keep the math professionals attached to these math constructs.

That is, there is good reason to believe that authoritative dogmas (or traditional math constructions) are leading to practically un-useful descriptions (or un-useful calculation scenarios) of a fantasy world, not of a world where new inventions can be created, based on measurable descriptions which are reliable and controllable, at some level of the descriptive structure (since the physical systems which one is [the professionals are] trying to describe are so stable and precise (and discrete) and thus they must fit into some math structure which is controllable).

Consider some fundamental questions concerning math:
1. What is a derivative?
Is it about local linear measuring of stable shapes?
Is it about probing a wave's phase to determine a stable system's global properties (wherein it is difficult to put a geometry onto (or within) the system to be used to determine the system's global properties)? Yet within quantum description only system's which possess a relation to a well defined geometry come close to describing (in a valid way) the stable quantum system's properties.
2. What is a differential equation? How is it related to the idea of measuring physical properties?
Should differential equations mainly be about relating the local measurable properties geometry so as to find a solution which provides controllable global information about the system's properties? Then would it not be best if the geometric patterns were related to the vary stable "discrete hyperbolic shapes" as well as the "discrete Euclidean shapes?" (yes, it would, see below)
3. Are quantitative sets best defined on finite, or simple countable, contexts? (Yes)
That is, are sets which are "too big" allow one to construct math patterns whose measurable (or counting) patterns are unreliable, ie both inconsistent and unstable?
4. Is the very general idea of holes in space [associated to very general geometries, ie associated to non-linear geometries as well as algebraic geometries (which are most often non-linear)], needed in math or is it better to reference holes in space to the very stable discrete hyperbolic shapes? (yes, it is better to be specific and simple)
5. Should math patterns first and foremost be about stable quantitative structures as well as consistent logical structures? (Yes)
5. How should elementary fundamental math questions be related to fundamental assumptions about contexts of meaning, set containment, logical and quantitative consistency, and the interpretations of observed patterns, about which one is trying to describe certain properties of these observed patterns? (Yes)
6. Should math description mostly be about wide ranging generality, or about useful reliable measurable patterns to be used in practical creative processes related to measuring? (it should be about identifying reliable measurable patterns)
7. Holes in space or discrete hyperbolic shapes? (discrete hyperbolic shapes!)
8. Does meaningful, precise descriptive, language require that it have a relation to practical creativity? (Yes)
9. What math patterns are important?
Only those math patterns which are related to practical creativity? (Primarily related to practical creativity)
10. Is a differential equation primarily about placing a quantitative (measurable) structure, in regard to both functions and their local linear measurable properties, onto a set of geometric patterns? (A geometric description is a practically useable description)
Thus, it is (might be) natural to believe that this structure should be based on the most common (prevalent) of geometric structures, ie the non-linear geometric structures, in regard to general metric-functions defining the most general geometries of physical descriptions? But this does not take into account the need for stability in regard to having descriptions which provide reliable measure structures for the description (for the calculations).
That is, the quantitative structures placed onto geometries are practically useful when the measuring structures are reliable, and this takes place with a math structure for a differential equation which is: linear, metric-invariant, contexts in which a function's domain properties are geometrically separable, so that the differential equations of the physical system can be placed into a diagonal matrix structure and solved, where the focus of the differential equation is about describing material positions in space, ie material geometries.
11. Does "over generalization" lead to description of patterns which exist only in illusionary worlds? (Yes)
12. Is the idea of materialism an unnecessary constraint on descriptive structures in regard to the patterns of existence which best relate to practical creativity? (It is)
13. Should oscillatory functions be required to be related to containing domain metric-spaces which have the proper number of holes in the space for the set of spectral values to exist? For example, should harmonic functions be related to spaces which contain discrete hyperbolic shapes? (harmonic functions should be related to [spaces which contain] discrete hyperbolic shapes)
14. Is a differential equation about sets of operators which acting on function spaces, so as to define global system properties, so that spectral functions, ie the spectral-functions determined by an "operator determined diagonalized function space," of a (quantum) global wave-function identify a (quantum) system's local properties (of an observed random spectral-particle event)?
(Such a set-up might have value if the wave-functions were associated to physical properties which could be measured directly, that is, the physical property is not measured by searching for random events)
Is this too complicated of a model (for probability waves) to be useable, or is its uselessness directly related to the fact that it is a probability descriptions of a system which is composed of relatively few components, so that no information can be pinned-down for the components, where the only information available is about probabilities associated to the components, but the measured values are not averages instead they deal with properties of specific components which compose the system?
15. How can one take the model of... . an indefinably random, non-linear, locally-determined correction to an indefinably random, linear, global-local wave-function (composed of spectral functions which cannot be identified)... .seriously? (It cannot be taken seriously) That is, "How can one take either particle-physics or quantum physics seriously?" (It cannot)
16. Should one be concerned about "how to navigate through all the details, which are observed, and which are assumed to have a relation to measurable patterns? (do they really?) (Apparently not, it is better to relate a math description to a stable math structure than to try to account for all the details)
17. Is the fundamental issue in math about, "What allows for the stability of measurable properties?" (Yes)
18. What is the purpose, ie what is the natural context, of the algebraic axioms, which are assumed to be the fundamental properties of quantities? Are the algebraic axioms focused on solving algebraic equations? (Yes, mostly) Why is the property of commutativity for (multiplication) operators so important in regard to solving differential equations? (Because it is a simple model which has a greater possibility for being solved, and controlled) (The issue is about multiplication, in regard to the local linear representation of the derivative (of the function). For each measurable set which composes the "system containing" domain space, one wants each of these measurable sets to be independent so that the local linear value of the function's derivative is to be independently related by a constant factor for (to) each one of these independent sets (in the domain space), ie each of these independent sets is related to the local linear approximation of the function's values by only one factor. If the function's values also have independent components then each of the function's directions needs to coincide with one of the independent coordinate directions. [If the function's value is a scalar then that scalar needs to be factorable so that each factor is associated to (or a function of) only one of the coordinate variables.] Thus, for the local linear properties of the derivative to depend on only one constant for each independent directions in the domain space then the coordinate directions need to be orthogonal and aligned with corresponding function directions [or function factors for scalar functions]. Furthermore, this simple independent multiplication property needs to hold for each point in the domain space.) Thus, in this case of orthogonal and linearly independent local representation the local linear approximation of a function given by a derivative can be represented as a diagonal matrix with constant coefficients at each point in the coordinate domain space. Such a context is stable, consistent with the metric-function, solvable, quantitatively consistent, and controllable. If one does not have these math properties within the descriptive measurable context then one's descriptions and measurements will not be reliable. This is essentially the underlying set of ideas which determines Thurston's (or Thurston-Perelman's) geometrization theorem, but places these stabilizing ideas at a more fundamental level upon which to build a stable, useable, measurable descriptive math structure.
19. Why is there a deep uncompromising belief in the math construction-ism which is built around the ideas of materialism and indefinable randomness, and which is related to an uncritical analysis of probability-spectral-component models of both quantum physics and particle-physics? (Because it is funded by the owners of society, and this knowledge (or this oppression of knowledge) serve the domineering interests of the owners of society [physics has come to be used by industry as military engineering, eg particle-physics is bomb engineering etc])
20. Does the observed set of stable patterns, which are also consistently measurable, eg the identification of the elements by observing their stable fixed spectral sets, imply that there is a bounding (bordering) stable context (in regard to measurable shapes and patterns) within which is contained the many stable, as well as the many unstable, patterns which are observed and are also related to measuring and quantitative structures?
21. Are these bordering stable structures, the discrete isometry and unitary subgroups (excluding non-linear shapes, ie and excluding the spherical shapes), so that these structures exist in a many-dimensional structure of both containment and "almost continuous"-discrete processes, which exist between these bounding (bordering) stable shapes? (These "almost continuous"-discrete processes have the same type of math structure as do the classical physics descriptions, but it is now within a new context for its dimensional structure.)
22. Does the existence of a bounding shape within a set which has a bounding (bordering) stable context, eg the stable nature of a new discrete shape emerging from an interaction into its containing space, depend on the (new) bounding shape being in resonance with some aspect of the spectra of the over-all containing (and bounding) set (or sets)?
This is how stable and quantitatively consistent (math and physical) structure automatically appears by being in resonance with the spectral properties of its containing space, or by being within the fundamental stable material systems upon which the stable, regular aspects of our world experience depend. It should be noted that this structure extends past the limits of materialism.
This also answers the question as to, "what allows for math stability?"

For example, in the new context, the wave-equation, partial differential equation structure, of "electromagnetic material interactions" are defined geometrically (by means of discrete Euclidean shapes) in Euclidean 4-space, so as to contain (or provide an approximate model in Euclidean space) a 3-dimensional discrete hyperbolic shape, but where there is also a set of "hidden" 2-dimensional discrete hyperbolic shapes in Euclidean 3-space, and the set of Euclidean differential-forms which are defined in Euclidean 4-space is given an imaginary-number structure so as to identify the electromagnetic properties which are observed in Euclidean 3-space (or in hyperbolic 3-space). [That is, the electromagnetic field 2-differential-form, F, in space-time is being calculated and adjusted in Euclidean 4-space.]
The discrete hyperbolic shapes which are contained in hyperbolic 4-space, ie 3-dimensional discrete hyperbolic shapes, and which are not of the size of the solar system (ie they are smaller than the solar system) interact only very slightly, this is due to the values of physical constants which are defined between adjacent dimensional levels of the many-dimensional containing space, eg when molecules of a gas are close to one another there are van der Waals forces (which exist between stable neutral components) etc.

The observed neutrality of stable quantum systems such as atoms comes about through a 3-dimensional discrete hyperbolic shape which is contained in hyperbolic 4-space, so that in 3-space the neutrality manifests on the intersection set of: (3-dimensional discrete hyperbolic shape) intersected with (hyperbolic 3-space).

That is, math constructions are simpler and more mathematically consistent when placed into a many-dimensional containing space of discrete isometry and unitary shapes.

23. Why do non-linear differential equations have limit cycles associated to their critical points? (The differential equation is defined in a measurable context but the observed patterns may not be linear [the underlying question is, why is stable linear structure so prevalent?] yet they are defined in a measurable context but the only aspect of the measuring which is stable is the differential equation (itself) thus its properties still reference measurable properties, ie its critical points can have stable measurable shapes associated to themselves, eg points, lines, circles etc which limit the measured properties of the system being identified by the differential equation if the differential equation become irrelevant after a while then the limit cycles (or the critical points) also lose their stable properties within the particular system originally identified by a particular differential equation.)
24. Has the propaganda system created a religious zeal... , based on the false idea of intellectual superiority (an icon of value surrounding an illusion), ie people are not equal... ., for the absurd math constructions of quantum and particle physics, as well as for the quantitatively inconsistent, non-linear theory of general relativity, as well as the derived constructionist math patterns such as string theory, which are based on these three absurd math constructions (Quantum Physics, Particle-Physics, General Relativity) etc? (The propaganda system has created such an irrational zeal for absurdities, because of how monopolistic business interests want to control and use knowledge within the society, eg the monopolistic business interests do not want new knowledge to create any unwanted competition for themselves, and they want their technicians to be narrowly focused)

In classical descriptions, the function (to be solved for) represents (the local geometrically) measurable properties of a physical system which fit into a (confining) geometric structure, so as to provide a global function... ,
(a solution function to an object's differential equation, the object which is both a part of the system and is interacting with the surrounding geometry)
... , of an interacting material component's position within a (containing) coordinate frame.
But this global solution function (and the ability to control this function by means of boundary and initial conditions) can only be found in the context of a linear, metric-invariant, and (geometrically) separable differential equation, ie the differential equation is (locally) represented by diagonal matrices.
But these math properties of a linear, metric-invariant, and (geometrically) separable differential equation also identify the properties of stable, consistent, measurable, and controllable math structures.
However, in quantum description the function represents a system which has a (direct) relation to the system's measurable properties only through the operators which act on the wave-phase of a quantum system's wave-function, where the function space, within which the wave-function... [which is formed by an infinite sum of spectral functions so as to form the general wave-function]... is defined (or contained), (the function space) really represents a quantum system's observed randomness, while the spectral-function-components of the system's wave-function represent a specific type of random local spectral-event of the quantum system. But this local spectral-component property is only observed at the expense of the total collapse of the quantum system's wave-function into one spectral function whose eigenvalue represents a random locally measured (spectral) property of the system (for that observation), ie the spectral function represents the local properties of the system. That is, spectral-functions represent the system's local properties, while a set of operators are used to identify the global properties of the quantum system (and this is essentially done by the operators acting on the wave-function's properties of wave-phase, eg the global properties of the (quantum) spectral-probability system are energy and angular momentum properties, but it is a (quantum) system in which only a local measurements determine the spectral values of the quantum system, and these local properties are associated to spectral-functions whose identification (or observation) destroys the global description of the system.
This is the use of differential equations and functions turned-on-its-head.
In fact, the set...
[or the complete (convergence to a wide class of functions) set of commuting (simultaneously diagonalizable) Hermitian (real measured [local] properties, and an assumption of the operators (in the continuous context of a Lie group) being unitary, ie the energy operator is unitary invariant]
... , of operators represent the system's global properties, where these global properties reside in the wave-phase of the (probability) wave-function for the quantum system, while the system's local properties are represented by the (observation of) spectral functions, but there is no actual vector structure associated to these spectral functions, as is the case for differential-forms, which have local vector properties associated to themselves when they are used in the descriptions of classical physics.

However, this dependence of a quantum system's global properties on the wave-phase implies that quantum physics is not locally-phase-independent, ie the global properties of the quantum system would change, dependent on the position in space in which the wave-equation is defined
(the argument is sometimes given that the wave-equation is unitary invariant, so that the real diagonal elements of a local unitary operator, ie its Lie algebra Hermitian operator, are unaltered by local phase changes in the continuous Lie group unitary structure, but this argument cannot be sustained if the global properties of the system are dependent on the derivatives of the wave's phase (since then the global properties of the quantum system would depend on where "in space," the derivative is taken, though formally this can be cancelled by introducing a non-linear connection term to the derivative [but the claim being made is that the unitary theory is locally phase independent, but the system's global properties depend on the wave-function's local phase properties] but if one allows for cancellation, due to "need" for a connection term, then two arbitrary structures (wave-phase and the connection) must be related to one another, where the reason for this correlation would be to try to fit data, but neither the wave-functions nor the particle-collision structures can be made consistent [in regard to fitting data] with the fundamental stable quantum systems which these math structures are trying to describe).
That is if one focuses on the details of the math this seems to be an absurd set of claims about wave-function invariance but the details can be so constructed, but if one looks at the over-all capabilities of this descriptive structure, which are essentially non-existent, then one sees an even more striking attempt to construct within a set of patterns which are useless (ie the description which is being constructed is not related to any form of practical creative development).
[Note: Because the patterns of particle-collisions found in particle-accelerators (locally) fit into a unitary pattern, which is applied to particle-states of particle families, it is claimed that this (overly complicated) descriptive structure has been experimentally verified, but the only conclusion one can "come to" in regard to the actual particle-collision patterns found in particle-accelerators...
(since they have not been shown to have any relation to the properties of relatively stable quantum systems, such as non-radioactive general nuclei)
... is that these component-events are unitary, and they hint at a hidden higher-dimensional containing space in regard to whatever structures these component-events are (might be) related.]

Thus, only "global phase-independence is the only possibility."

However, this, in turn, implies a structure to space which is commutative (or the math structures used in the descriptive (containment) sets for the material systems contained in the space need to be commutative), ie the stable math structures needed for valid physical descriptions must be diagonal.
That is, the data upon which particle-physics is based, ie the patterns of particle-collisions in particle-accelerators, can be interpreted to mean that functions which describe physical systems must have a unitary structure which is a part of the physical description, that existence is many-dimensional, ie dimensions defined beyond the dimension defined by the idea of materialism, and that the descriptive structure must be diagonal, ie phase invariance must be global, which in turn, means that the math used to describe a quantum existence and its components must be stable and consistent, ie linear metric-invariant, and geometrically separable, ie the descriptive structure (the containment space) must be related to the very stable discrete hyperbolic shapes.

Furthermore, the idea that the properties (or laws) of quantum physics can be derived from invariances (or symmetries), as "the property of 'spin' is derived from the space-time invariance of the wave-equation," cannot be sustained. That is, the claim that the physical description of particle-physics is based on local phase-invariance... ,
ie gauge invariance, and thus the need for a connection type derivative (to maintain gauge invariance) where in turn, the connection defines new sets of particle-states for the wave-equation,
... , is not valid.

In classical physics there are general rules, about defining a physical system's differential equation, which apply to a wide range of physical systems, and which relate motion measurements to distant geometric properties which nonetheless (these geometric properties) are defined in a local linear manner.

In quantum physics there are vague rules about the properties of a set of operators which are supposed to diagonalize a quantum system's function space, where the function space represents the random quantum system, ie diagonalize in order to find the spectral functions for a quantum system's function space. But these rules do not work in general.
This inability to identify a general quantum system's spectral set is the basis for calling such random structures "indefinably random."

An interesting thing about classical physics is that only a few systems are:
geometric separable (see below), and
metric invariant,
and thus they are solvable, thus a global "picture" of the system in time is available (the information about the properties of such a system is complete). Furthermore, these systems are controllable, and they are the basis for our civilization's technical development.

The solutions functions (to the system's differential equations) for such solvable structures are reasonably accurate (precise) and controllable,
and these classical systems are diagonal, ie the matrices used to describe the geometry of the system, translated into the language of a system's differential equation, are diagonal (in this simple context).
That is, quantum physics is modeling the solvable, diagonal subset of classical (physical) systems in regard to its general set of rules about the measurable properties of a quantum system. But this math structure cannot be used to fit the observed data in regard to these math models of quantum systems.
However, classical also has trouble in regard to identifying a math structure for the stable solar system.

It is the limiting context of materialism which causes classical mathematics to stop being able to formulate and solve for observed macroscopic properties, eg the stable solar system.
While, in regard to probability-spectral-component physical models, ie quantum physics, it is the non-existence of the mathematical property of stability and quantitative consistency as well as a lack of logical consistency (which is (assumed to be) a part of a measurable descriptive structure), as well as the dismissal of the classical interpretation of differential equations... ,
{and substituting a local (representation) model of a measured value as being the eigenvalue associated to the individual spectral-functions [or basis elements of a vector space] (which is to represent the locally measurable properties of a physical system), but these spectral-functions do not actually possess a local vector structure} [Note: This construction seems to be more like a wish than a coherent math structure.]
... , which has caused the quantum description to never have worked for general (but stable) quantum systems.

Both quantum and classical math structures need to be revised.

All technical development (of our civilization) depends primarily on the controlled structures which come from the limited diagonal structure of certain classical systems, but the ideas apply in a general manner within mathematics, ie when a classical system's differential equation has these properties then it is solvable and controllable, but the probability-spectral-component model of quantum physics even when placed in this simple context cannot be related to the stable, discrete spectral properties observed in quantum systems.
That is, in quantum physics the set of linear operators (placed in a Euclidean context) needed to diagonalize a general quantum system, in general, can never be found (for general quantum systems), yet these fundamental quantum systems (such as nuclei, atoms, molecules, crystals etc) do have observed stable, definitive, discrete spectral-orbital structures.

Mathematically, in order to "be able" to compare two measuring sets, ie the function values and the domain variables (of the system-containing coordinates (upon which the function's values are defined)), one needs the math properties of:
1. Linear local measuring relations, ie the differential equations needs to be linear,
2. Metric invariance, ie general metric functions are, in general, non-linear (and thus not quantitatively consistent),
3. Local coordinates for the containing coordinate set which are (or need to be) parallel and orthogonal at each point in the coordinate space, a property which can be called "geometrically separable" the local coordinate directions are always parallel and mutually orthogonal to one another at each point in the containing coordinate space (ie "geometrically separable" is specifically defined, and [most likely] unrelated to standard math terminology, though it is consistent with the described property of being a "separable differential equation" ).
This allows the matrices, which define the local differential equation structure, to be diagonal, ie the measuring direction of the coordinates are always independent of one another at each point in the coordinate space.
The descriptions... , ie the solutions to these types of relatively simple differential equations... , of these types of systems are very stable.
This "everywhere locally diagonal" math structure allows the solution function to be represented as a product (multiplication) of different functions, as many (function) factors as there are coordinates, where each function-factor (or product-function) is dependent on only one coordinate variable.
This allows a linear differential equation to be separated, and solved, in each separate (and independent) variable, and the function-factors are then multiplied together to identify the solution function. Furthermore, such systems can be controlled, by means of controlling the boundary and/or initial conditions of each separate function-factor of the differential equation, which is defined in a global coordinate structure which conforms to the system's geometry.

The three above mentioned properties (linear, metric-invariant, and geometrically separable) imply a (metric-invariant) metric-space with non-positive constant curvature, where the metric-functions have constant coefficients.
These properties can be interpreted to imply that both discrete Euclidean shapes, and discrete hyperbolic shapes, are naturally defined on such metric-spaces, and are (in fact) the geometric basis for the bounding math structures needed for physical description.
This simple descriptive structure is about cubical (or rectangular) shapes (which are related to discrete Euclidean and hyperbolic shapes), where the oppose-side faces of a cube (or rectangular block) are (in turn) related to circles, so as to form circle spaces whose dimension is the dimension of the cube (or rectangle).
The circle and the line are the only shapes which allow "geometrically separable" structures to be formed directly in the context of discrete isometry subgroups, ie the lattice groups of the three spaces of:
Euclidean, and
hyperbolic spaces (where hyperbolic space is equivalent to space-time),
but spherical space is non-linear, and thus not quantitatively stable, and thus it has limited value in regard to the descriptions of stable physical systems.
That is, this is all about the structure of the classical Lie groups and the relation of their discrete subgroups to circle spaces, and subsequently their relation to linear, metric-invariant, and geometrically separable shapes, which can be used to define a dimensional hierarchy of dimension and "dimension's subsequent relation to containment," within which the descriptions of fundamental stable physical properties can be bounded (and contained).

In the structures of indefinable randomness... ,
ie randomness where the probability properties (upon which the function space description is based) cannot be correctly identified with the observed (but random) spectra of quantum systems,
... , the local-global relations of the math structures have been reversed. ###...

That is, when the derivative has a "function space" as its domain space then the derivative operators become related to a system's global properties, eg the system's energy and angular momentum (associated to the wave-phase of the quantum system's wave-function [for an assumed spherically symmetric system]), while the spectral-functions now define local properties, ie random spectral-particle-events in space... , ie where space is the domain space of all the functions which compose the function space... , where now the point is to find the "local spectral properties" of the system. But spectral-functions are not local linear structures so it is not clear as to why one would believe that a spectral function can represent a system's locally measurable properties.
[Note: Spectral-functions are not local linear structures, but they may possess local linear properties (when they are differentiated), but when they are representing local physical properties of quantum systems they are not being used in the context of their being differentiated.]
However, now the system is described by probabilities for a relatively few components, eg atoms nuclei, molecules etc... ,
[eg quantum computers (defined on crystals) want to identify local quantum properties, ie properties of a few local components of the crystal and then to interact with these local properties which can only be identified along with the collapse of the crystal's wave-function, but then the computer can only function if it can use these local properties within the crystal's global wave-function (which must collapse when the local properties are observed)]
... , and the probabilities of the wave-function must collapse when the local random properties are observed and thus it is a description whose information is structured to oppose being able to control and use the system's properties.
Yet, these same systems form (when the various components of a particular type of a system interact, so as to form these [general, but] particular quantum systems) so each particular quantum system has the same stable, definitive, discrete set of spectral properties, ie they apparently form (or their components interact) in a very controllable context.

... ### whereas in classical descriptions the derivative (or its associated differential equation) is a local (linear) measure (of an object) which places quantitative properties onto (global) geometric structures, so as to find a global solution function for the system, to be used to control the system.

In the function space context there is a great amount of effort in regard to identifying specific spectral properties in regard to a very general term, eg of K/r, where K is a constant and r is the radius of a sphere.
The singularity for K/r, of r=0, is modeled as a Dirac delta function, in regard to the position of the charge which determines the K/r potential term, where the position, identified by r=0, is in the space which contains the quantum system.
The Dirac delta function is defined to be the infinite sum of the spectral functions of the quantum system.
Does this construct make any sense? (No)
Thus, one now has a set of functions supposedly defined by a wave-equation, wherein the functions are defined in regard to the limit processes of the derivatives, within the context of differential equations (or sets of differential operators which act on the quantum system), while a new set of density-functions, eg the Dirac delta function, are defined by a new limit process defined on the infinite sum of spectral functions, the Dirac delta function (or density-functional) is used to try to identify the spectral functions. Thus there are two levels of convergences which need to be defined, to try to fit the math structures with the observed data but this data-fitting is only related to a few descriptions and is not valid to a wide range of general quantum systems, and thus it is a description which is not verified by experiment, but only "verified" in relation to particular cherry-picked data.
Thus, it appears that limit processes are being used to arbitrarily adjust, essentially, undefined function properties, lost in a circular logical circuit of limit processes, so as to fit calculations to data.

Furthermore, that ddf=kf, where d is the exterior derivative, f is the wave-function, and k is the set of eigenvalues associated to the quantum system is considered valid without the need for "holes in space," ie ddf=0 if there are no holes in space. This shows how arbitrary this data fitting process actually is, ie wave-phases dominate over the geometric properties of a quantitative description, ie quantities associated to harmonic functions can be separate and independent of geometric structures (ie independent of holes in space, that is, holes in space are no longer required in regard to defining a wave-equation).

Consider atomic quantum systems, because there is supposed to be spherical symmetry within a separable differential wave-equation (for such a system), so that the radial equation has the, 1/r, term, there is assumed to be no reason to create an "artificial" bound on the system's geometry, ie r can have any very large value. However, it has been observed that quantum systems seem to be able to become neutrally-charged in many, if not most, of their inter-material interactions with other quantum systems, but this charge-neutrality seems to imply a geometrically bounded system.
Local molecular and atomic interactions (if the molecules or atoms are not ionized, ie the components of the gas are neutral) between molecules in gases are dominated by Van der Waals forces, not electromagnetic forces.
That is, the point-particle model of either charge or other elementary particles which possess spherical symmetry in an unbounded (and non-linear, ie spherical) geometric context seems to be quite a flawed model, since it is difficult to account for charge neutrality for atomic and molecular systems, but such charge-neutrality is common in many physical systems. Furthermore, it is also difficult to account for wide-ranging charge distribution in systems, such as clouds, if the only mechanism which determines the structure of charged systems (ie the structure of charge distribution within physical systems) is local point particles.

New mathematical context for physical descriptions

In fact, the whole issue about spectral properties and their relation to local measuring structures (such as differential equations) can be resolved by re-framing physics, not as a set of laws used to define a physical system's differential equations, but rather by re-defining (or re-framing) the context within which material and space are defined.
That is, physics is fundamentally about the stable geometry of the containing set structure, and it is within these stable bounding structures that measurable physical processes, ie differential equations, are to be defined.
This new metric-space structure has dimensions which are defined beyond the idea of materialism, so that materialism is but a subset of the new descriptive structure (where materialism is based on three spatial dimensions).
The proof that such a stable context is needed to understand the physical processes of physical description is that:
(1) classical solvable systems are very useful in regard to practical creativity, whereas its non-linear equations provide very limited (useful) quantitative information, essentially relegated to the limit cycles associated to a (non-linear) differential equation's critical points, and
(2) that quantum systems are observed to have such very stable properties associated to themselves and these properties are measured in a metric-invariant context.
Furthermore, there is evidence that action-at-a-distance is a verifiable physical property (though apparently remote from physical manipulation [unless perhaps it is given its proper descriptive context]), and thus the existence of absolute frames in relation to the fixed stars, as identified by Newton, also seem to be a part of the true aspects of physical description.
Whereas special relativistic frames seem to hold for hyperbolic space, but hyperbolic space is primarily characterized by its relation to energy, eg the conservation of energy, so that the sectional curvature of hyperbolic-space's discrete hyperbolic shapes is related to both potential energy and kinetic energy. However, its relation to potential energy is best modeled in the discrete Euclidean geometry, in turn, associated to the material interactions of discrete hyperbolic shapes, where discrete hyperbolic shapes model stable charged material systems, where the discrete Euclidean shapes model the spatial separation which exists between the discrete hyperbolic shapes which are interacting.
However, general relativistic frames, which are characterized by non-linearity, have no basis in the observed properties of existence (eg the solar system is stable), though in 3-space the material interactions are spherically symmetric in the new descriptive context, but the prevalence of orbital planes in regard to gravitational systems seems to imply that the gravitational interaction is best described in two-spatial-dimensions, so that the basic model of "mass" in the new geometric context would be a 1-dimensional loop (or a circle) [or perhaps several loops tangent to one another in a plane, eg a figure-eight (on its side) {in resonance with an underlying stable 2-dimensional discrete hyperbolic shape}].

Thus, the new idea is that a [stable geometric] containing structure (for math descriptions, which accurately [to an acceptable level of precision] and usefully describe material properties) must have a relation to stable geometric shapes, ie discrete hyperbolic and discrete Euclidean shapes, and it must be many-dimensional.

That is, both material and metric-spaces are based-on the very stable discrete hyperbolic shapes where n-dimensional (or lower) metric-space are contained in (n+1)-dimensional metric-spaces, so that these metric-spaces have discrete hyperbolic shapes and this containment structure continues up to 11-dimensional hyperbolic space, ie 12-dimensional space-time. These spaces, viewed as shapes are open-closed. Furthermore, material would be the bounded shapes which are 1-dimension less than their containing metric-space (which also possesses a shape, but it is higher dimensional than the material shapes, which is actually a closed metric-space). That is, a closed, bounded hyperbolic metric-space (which has a discrete hyperbolic shape) of dimension-n identifies "charged-material" in a hyperbolic metric-space of dimension-(n+1), while its mass is associated to a closed bounded (interaction) Euclidean metric-space of dimension-n (or of dimension-(n+1)) which is in resonance with the given (closed, bounded) hyperbolic metric-space.
In this new geometric context material interactions are determined by differential forms defined around discrete Euclidean shapes which are defined as averages, ie center of mass coordinates, between pairs of holes in the material space-form's discrete hyperbolic shapes, and this is done in a higher dimension so that the material containing metric-space so that a dominating property which determines the interactions in the material containing metric-space (in relation to the differential-form's geometric relations with the fiber group of the interaction metric-space, where the differential-forms are defined around discrete Euclidean shapes) while the averages over other holes (identified by the discrete Euclidean shapes of material interactions) contribute to the envelope of stability (related to the material interaction in the material containing metric-space), where this is possible in the higher-dimensional context of the description of the material interaction processes (that is, the interactions which contribute to the envelope of stability are in a direction [in the higher-dimensional interaction-space] which is tangent to the material containing metric-space).
This is all geometric, thus, if it is essentially correct, then it is useable and controllable information [since (if it is a stable system then) it is: linear, metric-invariant, and geometrically separable].
This interaction structure has a structure similar to a connection... , that is if the fiber group geometry and the system's geometry are not aligned (or not consistent with one another, so that geometric separability is not maintained)... , and thus it allows for the very prevalent non-linear structures observed for material interactions, but these non-linear interactions fit between a stable containing framework for physical description of discrete hyperbolic shapes which are related through material interaction by discrete Euclidean shapes.
The material systems have stable discrete hyperbolic shapes upon which stable spectra can be defined while also the metric-spaces have stable discrete hyperbolic shapes and it is the set of spectra... defined by "all the different stable discrete hyperbolic shapes which, in turn, define all of the metric-spaces" which are contained in the hyperbolic 11-dimensional over-all containing metric-space... of which the material spectral structures must be in resonance (so as to come into existence in a stable form during material interactions).

Ignoring the failed ideas about how to describe quantum physics as well as particle physics... ,
ie sets of operators and sets of random particle-collisions [which adjust the (unfound) solution to the wave-equation],
... , the useful laws of physics ie the laws of classical physics, are about "how discrete Euclidean shapes guide the structure of differential-forms" and the relation these differential forms have to discrete spatial displacements determined by a geometric relation (of the differential-forms) to the Euclidean fiber group (ie a geometric structure realized through the relation that the geometry of the differential-forms have to the geometry of the Euclidean fiber groups), within the new context of physical existence.

Consider physical law:

1. Rules which define a system's differential equation in a context of materialism (geometric properties of material defined in a fixed dimension) and frames of reference (accounting for the geometric distortions caused by moving frames). In this context a quantitative structure is applied to geometry, defined on a metric-invariant material containing space, both to the geometry of an object's motion and to the geometry of the material surrounding the object. The local structure upon which a physical system's differential equation is defined is either in regard to local measures of motion of an object in a domain of time (where the solution function is about identifying this global function of position in space [ie geometry]), or in regard to a local vector structure associated to a material distribution's relation to differential-forms and the holes in space implied by the orbits of the global material geometry (defined on either space or on space-time). These local structures are the properties upon which a system's differential equation is defined.
If motion is inertia defined on geodesics in curved space then the geodesics defined in regard to a geometry's general metric-function identify an object's global motions, but now the general metric-function is continually being deformed (except for the 1-body problem which possesses spherical symmetry, and no deformation of shape) and represents a non-linear math structure and this is a quantitatively inconsistent math structure, ie it cannot be determined (as a global function) by calculations.

2. Physical description is to be based on the idea of both randomness and well defined spectral sets, and it is assumed that this randomness can be defined on a function space of harmonic functions, in complex coordinates, where the spectral functions of the system's function space, solve a wave-equation, but this function space has a dual function space which by integral transforms change the differential wave-equation into an algebraic equation, which in the context of complex numbers can be solved algebraically. However, a quantum system's function space and its dual space identify a probability based uncertainty principle (in regard to statistical measures of probability distribution's width, ie standard deviation) which in the context of the pairs of physical dual function spaces exclude locally continuous geometry to be defined within such a quantum system. Yet it is only systems which possess a well defined geometry which can be solved so that the spectral structure identified by wave-equations correlates well with the quantum system's observed well defined, discrete, and very stable spectral properties. However, the wave-equation is about how sets of differential operators (associated to the system's physical properties by the relation that these differential operators have to the wave-function's phase), such as the energy operator (or wave-equation), relate to the quantum system's function space so as to find the quantum system's spectral functions. It is the spectral functions which identify a quantum system's local random spectral-component values of the quantum system. However, there is no (local) vector structure to this math construct to enable these spectral functions to have any relation to a local spectral-component value, rather it is a random event which magically identifies the quantum system's local spectral properties. The system wave-function structure collapses when a spectral-component value is identified, as a random event in (the domain) space [whereas local vectors associated to geometry affect an object's motion, in the classical context]. However, the set of spectral functions... , associated to general, but fundamental, quantum systems (ie nuclei, atoms, molecules, and crystals)... , cannot be found, and the wave-collapse means that this description (calculation) provides no control over the quantum system. Yet these general quantum systems must form in a highly controlled context, since they form in physical processes so as to always possess the same set of stable, definitive, discrete spectral properties. Why?

3. Then local random system-component collisions, which identify a non-linear operator, are used to adjust the global properties of the individual spectral functions, (by a perturbation series associated to the many types of system component-collisions which might identify the local structure associated to the adjustment process, in a descriptive context which prohibits the well defined geometry of component-collisions of point-particles). This means that this descriptive structure is both logically and quantitatively inconsistent and describes unstable meaningless patterns. Nonetheless, then it is claimed that this geometric collision structure, of the supposed system components, deforms the spatial structure around the local components of the system (but whose localness makes no logical sense within this probability based math structure), so that the adjustment needs to be re-adjusted, and this is done within an adjustment process which is associated to spectral-functions which cannot be found (for general quantum systems).
This is not physical law, rather this is math construction-ism based on authoritative arrogance, ie personal authority defines truth, and taken to a point of absurdity.

These probability based descriptive structures, in regard to the probabilities of colliding-components, only have relevance to statistical systems which possess a large number of components, where then the averages of the "probabilities of collisions" can be related to "rates of reaction," eg nuclear reactions.

This is not intellectual value, rather it is absurdity, about which these arrogant authorities should be ashamed. However, these authorities are actually autistic-types who are being used and manipulated by the owners of society to identify an idea (an icon) of intellectual-superiority, ie people are not equal, and "fundamental knowledge is not attainable by the public," so that this mis-representation of an icon of intellectual-value, is to be used within the propaganda system, so as to control knowledge and creativity, within a society owned and controlled by a few, where this control is based in the domineering monopolistic economically based social system [of control (over society)] which is possessed by the owners of society.

Indefinable randomness is about: unstable events, un-define-able (or incalculable) events, the elementary event space is not fully identified, new events are added to the elementary event space, where each or any of these properties for elementary events makes these event spaces uncountable, ie one cannot rely on the counting process, thus the probabilities cannot be determined for such elementary events spaces upon which probability must depend for its definition.
That is, even though it might be claimed that certain types of events are distinguishable, and thus countable (so as to be related to probability), however, if these events actually fit into a space which is "indefinably random" then one cannot rely on the probabilities which are determined from any counting process applied to such a probability space.

The definition of indefinable randomness causes such, "apparent quantitative structures" applied to random events, to be inconsistent with the quantitative structures into which they are being placed. Thus, the precise patterns which they are claiming to describe, are not reliable patterns, they are not stable, and their measured properties have no validity.