## Math science and practical usefulness of a description

m concoyle
*08 Mar 2011 21:41 GMT*

The mathematics of both physical description and its subsequent relation to “practical” creativity in regard to general existence.

Math is about the measurable (or quantitative) descriptions of quantities and shapes, (or algebra, analysis [spectra or averages], and geometry).

Conjecture: Description based on shapes is more readily related to practical creative development.

Thus, it is the geometric context which should be the (dominant) focus of math and science, in particular the linear geometrically separable context (or discrete isometric subgroup context)…,

…where geometrically separable means that at each point in space the local coordinate directions are mutually perpendicular to one another.

Quantitative structures

Quantitative descriptions begin with a clear identification of a particular type of thing, then identifying a uniform measuring unit for those things, and a process of counting, wherein the idea of quantity or comparative size can be identified (or measured) for the things which one is counting (based on the values which the counting process identifies), in regard to particular values (of the count for the particular set of things).

The uniform measuring unit can be changed by dividing the unit into a set of equal but smaller parts so as to identify a new smaller “uniform unit” with which one can then count and subsequently measure the same type of things but now measured in a smaller “uniform unit.”

This is equivalent to multiplying the original uniform unit by a constant, where the constant can be either (uniformly) bigger than the original uniform unit or (uniformly) smaller than the original unit. When the old quantitative set…., the quantitative set which measures in the context of the original “uniform unit,”…. is represented as a variable, and then multiplying such a (domain) variable by a constant so as to identify both a linear map (to a new or different quantitative set) and a valid “change of units,” (or a “change of scale”) for the quantitative set within which one is measuring (or comparing the relative sizes of things of a particular type).

General linear maps

In a coordinate space, multiplying each local coordinate direction…, where each local coordinate direction is perpendicular to all the other coordinate directions…, by a fixed constant would also define a linear map between local coordinate spaces (or between tangent spaces).

If two sets of quantities, ie the set of function values and the domain set, are related to one another linearly then they are quantitatively consistent with one another and can relate measurable properties (between the two quantitative sets) in a consistent way.

Metric-invariant transformations (or metric-invariant maps)

The local linear maps which leave the metric function invariant, where the coefficients of the original metric function are all constants, are called metric invariant transformations, and which are also called isometry transformations. That is, when the length measured between two points measured by a given metric-function and then the local coordinates (as well as the local metric function) are transformed by a metric-invariant transformation (then) [so that when] the length between the same two points is the same after the local coordinate transformations then the local transformation is a metric-invariant transformation.

When the length in a coordinate space is preserved under transformations then the geometric measures between the two transformed sets remain consistent, thus if measuring is to remain a consistent idea within a description then the allowable transformations of a measurable physical description will be metric-invariant transformations (or isometry transformations). Thus one wants in a measurable consistent description of a physical system the local coordinate transformations (or local operators) to be both linear and metric-invariant.

It should be noted that the properties of the material systems in the physical world which are most commonly observed are all the stable, discrete, definitive, measurable spectral-orbital physical structures, or properties, (of material systems) which exist at all size scales.

Because there are many physical systems which are stable and measurable, this then means that the containment space (within which physical systems are both contained and subsequently described) are spaces of constant curvature, ie metric-invariant spaces whose metric functions have constant coefficients. Subsequently the natural shapes to describe within such spaces are (either the separable shapes or) the separable shapes associated to the discrete isometry, or unitary, subgroups of the coordinate space’s isometry, or unitary, fiber groups (of local coordinate transformations). Such discrete separable shapes (called space-forms) are naturally related to stable definitive discrete spectral-orbital properties.

Note: Hyperbolic space-forms are particularly stable in their geometric properties. Furthermore, general manifolds (of particular dimensions) continuously deform into these separable discrete geometries.

Fundamental algebraic (or quantitative) structures (as opposed to geometric structures)

When one can measure large and small things (of the same type) within the context of rational numbers then one can also succinctly list, by the means of using variables, the universal properties (or laws) of the rational numbers in regard to the arithmetic operations of adding and multiplying, such as the laws for:

“the order of operations,”

the existence of identity elements, and

the definitions of inverse operations,

for both addition and multiplication arithmetic operations, as well as

the distributive law for mixing the adding and multiplying operations.

This list of laws for the basic math operations (expressed with variables) has been the basis for algebra.

The existence of solutions to algebraic equations, eg polynomial equations, “within what (new) set of numbers do solution sets exist?” That is, expanding the size of the quantitative sets so that the solution values are within the set, a set which is governed by specific laws of operation, are the important questions in algebra.

The properties of numbers, such as the properties of prime numbers in regard to the operation of multiplication (or the “number-type properties” of solutions to algebraic equations [eg Fermat’s last theorem]), has come to be number theory.

Functions are defined between quantitative sets by means of formulas which identify new values based on the domain variables, as well as functions defined by new operations, eg exp(x) etc.

Convergences of sequences so that subsequently analysis was built out of calculus operators whose definitions depend on the existence of limits.

Sequences can be defined on sets of numbers, such as the real numbers, or by relating domain sequences to sequences determined by evaluating functions’ values in regard to these domain sequences, thus forming limits. Limits (or values of convergence) of the sequence or of a subsequence, ie number-values to which the bounded sequence (or subsequence) converges are often referred to as limits, can be found for bounded sequences, if the set of number-values is expanded to ensure the existence of a bounded sequence’s least upper bound within the set, again if one knows an element is in a particular set then one also knows things about the rules for operations for such an element.

Is the claim that least upper bounds always exist for converging sequences, and this makes the sets of quantities “too large” so as to no longer remain logically consistent, as the “set of everything” is too large a set upon which a consistent logic can be based. One might be concerned about the properties of convergence when the real number line has the same number of points (or elements) as the 2-plane has points [which compose its set (on the 2-plane)]. That is, very large sets blurr the distinction of being able to identify a quantity as being of a particular “type” of quantity, eg is it a real quantity or a quantity of the 2-plane?

Within the context of finding such limits of sequences, the derivative operator and its inverse integral operator of calculus can be defined, so that, in turn, a context for solving differential equations is identified.

So what is a differential equation?

Is it an example of a local linear measurement being related to a (separable) geometric property, so as to be solved in the linear, geometrically separable cases.

In classical (physics) description geometric properties of geometric measures and the identifications of position and motion of the material (which compose physical systems) are the fundamental properties of description, and these geometric properties relate to practical creativity,

or

Alternatively, are calculus operators part of the set of operators which act on function spaces where operator sets are to be related to a function space’s spectral sets of functions? (Are differential equations mainly about a system’s spectral sets, and subsequently related to optics and to the calculus of variations?)

Is measuring about geometric measures of length and area, etc or is measuring (of a physical system) about identifying the physical system’s spectral set (and subsequently related to the average [geometric] properties of the system)?

Fundamental problems with physical description

It should be noted that the properties of the physical world which have, as yet, no valid physical description: are all the stable, discrete, definitive, measurable spectral-orbital physical structures (of material systems) which exist at all size scales.

Thus, when one asks what aspect of mathematical description best identifies a math description of a physical system to widely applicable and very useful creativity in the practical sense of the idea of creativity then one has to say that “the geometric descriptions of physical systems” are best or easiest related to great practical creative usefulness. In particular it is the linear, separable geometric descriptions related to the differential equations of classical physics which are best related to widely applicable, great practical creative usefulness, where this very useful property of geometric description has been demonstrated as to its wide applicability and its great usefulness in regard to practical creativity in classical physics.

That is, the geometric description is much more useful (than is the spectral description) in regard to widely applicable, practical creativity.

It should be noted that the properties of the material systems in the physical world which are most commonly observed are all the stable, discrete, definitive, measurable spectral-orbital physical structures, or properties, (of material systems) which exist at all size scales.

Because there are many physical systems which are stable and measurable this then means that the containment space (within which physical systems are both contained and subsequently described) are spaces of constant curvature, ie metric-invariant spaces whose metric functions have constant coefficients. Subsequently the natural shapes to describe within such spaces are either the separable shapes or the separable shapes associated to the discrete isometry, unitary subgroups of the coordinate spaces isometry, unitary fiber group of local coordinate transformations.

However,

In math, the descriptive process goes straight to the most general and usually the most difficult descriptive context so that this descriptive context is filtered through the main descriptive structures known to math, eg algebra, analysis, and general (non-linear) geometry.

In regard to physical description, math focuses on non-linear geometries and general spectral structures which math tries to relate (though unsuccessfully) either to general (non-linear) geometry or to an (incalculable) spectral structure which cannot be (have not [as yet] been) related to the general structures of function space operators.

The focus, in math, is always on properties of operators and converging sequences within sets of containing structures.

Thus the focus (within math) on very simple linear, separable, discrete geometries is very minimal in regard to the (above mentioned) other more general math structures of set containment, algebra, and limits on quantitative structures often only slightly related to simple geometric patterns.

The non-linear geometries identify…,

with a few exceptions (most likely [still] in a separable geometric context),

….. a fundamentally random descriptive structure which has no practical uses other than in the context of feedback (to a system which can adjust itself based on knowledge of the system’s perceivable context) which is related to the properties of the system’s non-linear differential equation (as opposed to being related to the solution function of the non-linear differential equation, which essentially cannot be found), and the general spectral context of the system, where (most often) neither the sets of operators nor the geometric structure can be identified.

Yet, in a definitive spectral context spectral systems can be described well.

Perhaps the definitive spectral context that is observed in the world, but is as yet un-described, can only be associated to definitive, separable geometric context of a many dimensional space. Such a context would imply a definitive spectral context. [Supplying a wide variety, but within a fixed spectral set, of spectral properties of dimensions 5, 4, 3, 2, 1-spectra.]

Summary

The mathematics of both physical description and its subsequent relation to “practical” creativity in regard to general existence.

Math is about the measurable (or quantitative) descriptions of quantities and shapes, (or algebra, analysis [spectra or averages], and geometry).

The fundamental math property which most directly relates to useful practical creativity is the property of geometry.

One needs to realize that the geometric properties to consider are the linear, separable geometries and the stable, definitive, discrete spectral-orbital properties that are observed in the world are to be related to the spectra (and orbits) of linear, separable, bounded, space-form geometries…, which are used to model both material systems and (of material-containing) metric-spaces…., associated to a (the) many-dimensional containing spaces of constant curvature. The spectra of both the material systems and the metric-spaces are inter-related by means of resonances which exist between the dimensional levels, and between the different subspaces of the same dimension, which are all a part of the high dimensional containing space, where the higher dimensional containing spaces are composed of linear, discrete, separable geometric shapes.

Both general non-linear geometries and general incalculable spectral structures (or spectral systems) [associated only to sets of function space operators] are not describable in a meaningful way, ie not related to practical creative usefulness. Nonetheless it is these general structures which the math community spends most of its efforts trying to describe, in a… as yet to be found…. actual description of any observed systems and subsequently the math descriptions of these (too) general systems have no practical meaning.

General (non-linear) manifolds are related by continuous deformations to linear, separable, discrete geometries which exist within coordinate spaces of constant curvature, where these linear, separable, discrete geometries are characterized by the number of holes in the separable geometries’ shapes.

These holes in the (linear) separable geometric shapes are naturally related (on hyperbolic space) to stable spectral (or orbital) properties of linear, discrete, separable geometric structures.

Whereas the non-linear geometries and the general spectral structures which remain unidentifiable by (valid) math processes (or math methods) are unstable and either decay or deform to stable structures, but such a deformation process is without any definition of a structured deformation process in traditional authoritative math descriptions. That is, the general processes of deformation (or change) used by mathematicians is defined over general spectral and geometric structures (but they are structures which in theory may exist, but in calculation they cannot be identified).

That is, the generalities about which the math community considers when trying to describe general geometric and spectral structures, ie algebraic properties and manipulating the convergence values of quantities and functions, as well as a viewpoint of generality which ignores other viewpoints of generality, other viewpoints which are associated to generalizing the idea of:

(1) materialism, and

(2) how higher dimensional structures can relate to material and to the fundamental aspects of both geometry and spectra.

(3) That is, the general geometric and general spectral systems to which observed spectral and geometric structures naturally reduce (either decay or continuously deform), namely, to separable discrete definitive geometric shapes

(4) and these separable, discrete, definitive geometric shapes are associated to discrete isometry, unitary etc subgroups,

(5) defined on spaces of constant curvature, defined over various dimensional levels,

(6) where, at each dimensional level, one can consider a variety of ways to divide the spatial and temporal subspaces of such metric-invariant coordinate spaces,

(7) where the metric-functions have constant coefficients, and each subspace subdivision is associated to new material types and associated new physical properties.

A deformation process (in the new descriptive structure) so as to either result in decay or to deform into stable structures can take place by means of a non-local, discrete partition of the interaction-dynamic process into linear separable (Euclidean) geometries which are periodically deformed in a process involving (1) spin-rotation of metric-space states (2) small time-interval periods (associated to the spin-rotation of metric-space states) and (3) spin-group local transformations of either material-space-form positions or spatial space-form shapes, (4) where the geometry of the dynamic structure is related to the 2-forms defined on the interaction space-form, (5) where the dimension of the 2-forms is equal to the dimension of the spin-group.

As the dynamical process continues at some point the interaction space-form (during a spin-rotation of state time period) will begin to resonate with the spectra of the over-all containing space. If the energy values are “correct” then the deforming interaction space-form can enter into a stable space-form state, or if the energy properties are not correct then there will be the usual dynamics, eg it will be the system of colliding material space-forms which stay separate, where the collision can either let the interacting space-forms remain stable, or the collision can cause the interacting material space-forms (one or the other, or both) to break apart (depending on the energy of the collision). This is consistent with classical physics and it actually describes a quantum interaction process which determine stable quantum systems (so as to not be full of huge logical gaps in the descriptive structure, eg no global wave-functions yet non-local processes, divergences of the solution to the radial equation of the H-atom etc). Note: The point-like properties of observed random particle-spectral quantum interactions are related to the distinguished point of the material space-forms.

Note: This model of geometric interactions can be related, at the atomic level, “to random behavior” from which the idea (or property) of quantum randomness emerges, and both the interaction structure and the physical constants (ie relative sizes of interacting space-forms of a given dimensional level) cause the interaction structure to (appear to) make the adjacent dimensional levels to be geometrically independent of one another.

The vast majority of what the math community does is either “pointless” or it is mathematical description based on illusions (or beliefs) about what the general structures of mathematics is capable of describing. This delusional aspect of math description is based on extreme adherence to a belief in algebraic structures and a belief that convergences defined on quantitative sets and on sets of functions can be used to describe any general geometric or spectral structures which can ever exist, and thus it can be used to uphold…., by cunning convergence structures which can be used to fit data (thus the warning about the sets upon which convergence is defined are too big and thus logically compromised)…., assumptions which are illusions, such as the idea of materialism so that the idea of materialism controls the dimensional structure of physical description.

The general geometric and spectral structures either decay or deform, they are not true fundamental shapes of a measurable existence, and materialism is wrong, yet math adheres to these illusions, and finds Platonic truths within a particular structure of mathematical language which should not be maintained, since it is a descriptive language which has virtually no relation to either observed properties (other than transient relations) or to practical creativity (it is widely applicable only to transient illusions).

A main idea, which seems to unnecessarily limit the descriptive context, is the idea of materialism, an idea which unnecessarily restricts the dimensional and subsequent geometric context of physical description.

Whereas, an idea which is equivalent to materialism…, so that the stable, discrete, definitive spectral-orbital properties of material systems can be described…., in a new context of a higher dimensional model of existence, requires that…:

(1) material systems be stable compact (bounded) space-forms, and that

(2) light be stable, infinite-extent space-forms,

…defined on spaces of constant curvature (what we observe to be) at a particular dimensional level…,

(3) so that, different numbers of metric-space states (at that particular dimensional level) are considered (where, for example, matter and anti-matter identify one pair of opposite metric-space states)

…. in the descriptive properties of material systems (at the given dimensional level).

For example, one can consider different numbers of metric-space states (of the material property associated to the particular metric-space) one-state, two-states, four-states, and eight-states would be contained in: R(n) (real), C(n) (complex), H(n) (quaternion), and O(n) (oction) coordinates of n-dimensions, respectively, where these metric-space states are rotated amongst themselves by the respective fiber groups of these coordinate spaces. (Note: for R(n), C(n), H(n), and O(n) coordinates the dimension of the temporal subspace is: 0, 1, 2, 3, respectively, ie 2^0=1, 2^1=2, 2^2=4, 2^3=8-dimensions associated to R=1, C=2, H=4, O=8, in real-dimensions, respectively)

The general dimension, n, can be decomposed into spatial subspaces, of dimension-s, and temporal subspaces, of dimension-t, so that s+t=n.

For any given value n the value s-t=(the metric-function’s signature).

For any given n the value of t determines the type of material which is contained in the metric-space and the physical properties associated to that material type, and thus the dimension, t, is related to the number of two opposite metric-space states associated to that metric-space (see above).

For example, when t=0 then the material is mass, and the two metric-space states are the translational and rotational states of the fixed stars, and these metric-space states correspond to matter and anti-matter.

Each different value for t is also associated with a particular allowed set of discrete isometry or unitary (or quaternion related, or oction related) subgroup structures, which are separable geometric shapes (sometimes called space-forms). That is, these discrete subgroups identify space-forms of separable geometric shapes associated to material systems.

However, such discrete separable geometric shapes can also be associated to the geometric structure of metric-spaces themselves.

The hyperbolic space-forms have stable definitive discrete properties of spectral flows defined on the space-form.

Hyperbolic space-forms are stable, but discretely separated from one another, while the Euclidean space-forms can adjust to continuous changes of the natural spectral properties of the Euclidean space-form shapes.

The hyperbolic shapes can be represented as sets of Euclidean space-forms attached together. That is, dynamics is naturally described in the Euclidean context of continuous deformation of space-form shapes, but the Euclidean context is also consistent with the stable, but discretely separated hyperbolic shapes.

Thus interactions can be approximated by combining stable Euclidean approximations of hyperbolic shapes to changeable Euclidean space-form approximations of both spatial separations and spatial changes (see above).

D Coxeter has identified many important properties of hyperbolic discrete separable geometries of various dimensional levels.

Hyperbolic space-forms have stable spectral-orbital properties associated to themselves. Other such math properties are that: (1) there are no hyperbolic space-forms of dimension-eleven, and (2) the last bounded hyperbolic space-form (in the hierarchy of dimension) is a 5-dimensional hyperbolic space-form, ie all 6-dimensional (and higher dimensional) hyperbolic space-forms are of infinite extent (but of finite volume).

It should be noted that the properties of the physical world which have no valid physical description are all the stable, discrete, definitive, measurable spectral-orbital physical structures which exist at all size scales.

Because there are many physical systems which are stable and measurable this then means that the containment space is related to coordinate spaces of constant curvature, ie metric-invariant coordinate spaces whose metric-functions have constant coefficients.

Space-forms as elements of a “number field,” eg the real numbers are a “number field”

When math people consider space-forms these space-forms are often seen in the context “rational representations of” complex polynomial functions with complex variables, so that these rational functions form a field of numbers, just as the real numbers form a “field of numbers.” This high level algebraic abstraction is the focus of rational functions composed of ratios of complex polynomial functions in math, whereas these rational holomorphic functions are also space-forms. Thus, the useful simple geometry of space-forms tends to be ignored while the abstract field structure of these complicated functions is (becomes) the focus of math.

But why do the space-forms form into a “field of numbers”? The mathematician says to themselves, it must be an algebraic structure which will be important to understanding the formation of “number fields.”

An alternative way in which to view certain of these rational complex functions which also represent hyperbolic space-forms, is to consider that the set of stable spectral flows on the hyperbolic space-forms can take the same place (in the process wherein quantities were defined) as the uniform unit which is now defined within a set of stable spectral values (which identifies the type of number) in the (above mentioned) process when numerical value was developed (or constructed).

Thus, because they are defined as rational functions means that the different arithmetic operations will also be defined, thus stability and a uniform unit of a particular type of thing associated to number allows for the properties of numbers to be defined.

That is, instead of increasing the complexity and abstractness of a math pattern perhaps looking at the simpler relation might be used to identify the basic stability needed for quantities to enter into being (into consideration).

e-mail:: martinconcoyle@hotmail.com