As observed in
http://www.internationalskeptics.com/forums/showpost.php?p=12029582&postcount=2784 the cardinality of the places that are needed in order to represent each unique sequence (whether it has finite or infinite members) is equal to
x, which is used as a placeholder of the power value of any expression of the form
[base]x, whether
x is finite or infinite cardinal number, where
[base] is a placeholder for any cardinal number >
1 .
Moreover, even if strong limit cardinals are defined as inaccessible to the members of the set of cardinals below them (for example: the members of {
0,
1 ,
2,
3,
4,
5, ...} are inaccessible to
ℵ0), the members of the set of these strong limit cardinals are inaccessible to a strong limit cardinal above them (for example: the members of {
ℵ0 ,
ℵ1,
ℵ2,
ℵ3,
ℵ4, ...} are inaccessible to
ℵℵ0).
...
etc.
So, a collection of members with a common property can be taken as a whole, yet it is incomplete since it is inaccessible to a given limit above it, and this inaccessibility is essential to each one of the members with the common property in the given collection.
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So the notion of inaccessibility as an essential property of collections, actually goes beyond the multiple (the domain of collections), and we do not need the transfinite universe in order to deal with this essential property, as follows:
As I understand it, useful science is actually done by achieving results that can be tested and used beyond some particular observation.
It means that the observation is not-one-of-many observation, which naturally can be used as the substance of any one-of-many observation (not-one-of-many observation AND one-of-many observation will be explained later by using the model of Möbius strip).
I find Mathematics as one of the most powerful tools that enable to develop the natural linkage among not-one-of-many observation AND one-of-many observation.
In my opinion, in order to actually be such a tool, mathematicians can't partially use their own brain during mathematical activity.
Unfortunately, for the past 250 years almost every mathematical work has to be translated into verbal-symbolic form in order to be accepted as a valid mathematical work.
As I see it, in order to correct it, both visual_spatial AND verbal_symbolic brain skills have to be used during mathematical activity, where mathematical results are accepted as the association among both visual_spatial AND verbal_symbolic brain skills.
For example, let's use both visual_spatial AND verbal_symbolic brain skills in order to define the natural linkage among not-one-of-many observation AND one-of-many observation.
The minimal amount of one-of-many things is two things.
Question: What enables the knowledge that there are two things, in the first place?
By using almost only verbal_symbolic brain skills, this question is not asked at the basis of fundamental mathematical concept like Set, where {5,pi} is some example (without loss of generality) of two things (the outer "{" and "}" is not mathematically defined).
So, by Modern Mathematics (which is currntly mostly verbal_symbolic_only framework) the ability to gather at least two things is arbitrarily done.
A simple visual_spatial AND verbal_symbolic tool, which naturally enables to answer to this question (by also define the outer "{" and "}" as an essential thing of fundamental mathematical concept like Set), is Möbius strip:
By locally observe it, it has two edges (represented here by brown color), where each edge is one-of-many thing.
This observation is actually possible because also global observation is done, which provides the knowledge that there is one edge along the Möbius strip (which is not-one-of-many thing) where this knowledge is represented by the the outer "{" and "}" (which is not any one-of-many thing between them).
The visual_spatial knowledge of not-one-of-many thing (represented here by one edge along the Möbius strip), which is also represented by the outer "{" and "}", actually enables to gather one-of-many things like 5,pi into a set (for example: {5,pi}).
So, fundamental mathematical concept like Set is not rigorously defined, unless both visual_spatial AND verbal_symbolic brain skills are used.
The model of Möbius strip naturally demonstrates the linkage among being aware of one thing at once (the visual_spatial brain skill) as the substance of being aware of multiple things step-by-step (the verbal_symbolic brain skill).
Moreover, this awareness (may be known also as Unity consciousness) is exactly the core of useful scientific work, which actually achieving results that can be tested and used beyond some particular observation.
Furthermore, the observed, the observer and the tool of observation are actually one thing, which is not the sum of its expressions, and as a fundamental result we get non-entropic realm, which is essentially open to new expressions (infinitely many things can't be defined by a fixed limit, fixed cardinality, etc., as unfortunately done by the current main stream of Modern Mathematics).
Generally, by this notion one's awareness is fully activated only by simultaneously using his\her visual_spatial AND verbal_symbolic brain skills.
As I understand it ,without it, no actual scientific progression is possible.
I do not think that mathematical work that is done by using only verbal_symbolic brain skills (where visual_spatial brain skills are used only as tools for demonstration, and not as fundamental entities of mathematical work) enables to actually gather the observed, the observer and the tool of observation into Unity consciousness.
Moreover, in my opinion, the challenge is to develop natural Unity consciousness among us as human beings
directly within our biological systems, instead of seeking after our external agents in terms of Artificial Intelligence.
Uncertainty by this notion, is exactly the field of creativity for any brain that enables to define the linkage among Simplicity and Complexity in daily life (where the simple is not the trivial and the complex is not the complicated, such that the wrong pair is trivial_complicated and the right pair is simple_complex).
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According to latest scientific researches (for example: [1],[2],[3],[4],[5],6]) visual_spatial AND verbal_symbolic brain skills are activated during mathematical work.
I wish to share with you some model (some analogy based on visual_spatial AND verbal_symbolic brain skills) that demonstrates the linkage between Logic and Physics.
The attached schema (
https://c1.staticflickr.com/4/3929/15309487328_d8023ed9f0_b.jpg) illustrates the possible associations between Bose-Einstein statistics (even function) and Fermi-Dirac statistics (odd function) and the mathematical concepts of contradiction (expressed as sine-like wave) and tautology (expressed as cosine-like wave).
By this model contradiction is the separator among universes, where each universe has its own tautology (its own natural constants and, so called, the laws of nature).
The simplest form in this model is illustrated by non-composed endless straight line, which is essentially not the sum of its vibrations simply because being non-composed endless straight line is independent of its sine-like/cosine-like vibrations, yet they are depend on it.
This asymmetry of dependency defines a non-entropic realm since no amount of vibrations is the non-composed line in itself.
Being aware of the non-composed line during its vibrations, is actually Unity consciousness that is, in my opinion, the ultimate goal of life phenomena, that has no limits to its developed expressions, exactly because no amount of vibrations is the non-composed line in itself.
By not being limited by contradiction and tautology, life phenomena is beyond AND among the expressed universes, such that they become harmonious organs of an organism, which is ever developed exactly because no organ harms the developments of the other organs, since the development is naturally derived from cross-universes' source, which is their cause.
From this notion, Mathematics is a fulfilled tool only if it supports the development of Unity consciousness among the expressed universes, such that they become ever developed organs of one harmonious self aware realm.
As I see it, Contemporary Mathematics that defines collections of infinitely many members by using fixed sizes like |N|, |R| , or defines the sum of infinitely many numbers, or multiplication of infinitely many (0,1] members by a single finite result, blocks the ever developed nature of multiplicity and the direct awareness of Unity as the cause of multiplicity, which can't express the full potential of Unity, since, unlike Unity, multiplicity is limited by contradiction and tautology.
I wish to clarify something.
Given a non-composed endless straight line and a point not on that line, there are endless larger non-composed circles that are smaller than that line, and there are endless smaller non-composed circles that are larger than that point.
Yet
pi is a proportion among the endless larger and the endless smaller non-composed circles.
So a fixed value can be related to infinitely many things as long as it is not used to define their amount, their sum or any other fixed value that contradicts their property of being endless larger or endless smaller things.
For more details, please search for Nicholas of Cusa.
Here is an example of a mathematical work that is done by using visual_spatial AND verbal_symbolic brain skills:
Please observe the diagram in
http://farm5.staticflickr.com/4015/4430320710_686e9e991b.jpg.
X>0 AND X>2(a+b+c+d+...) are inseparable in the following diagram (where X is the constant positive value of the infinitely many orange figures in that diagram, whether they are straight (the first orange figure) or bent (the rest infinitely many orange figures, which their endpoints are projected upon the straight orange figure, and define 2(a+b+c+d+...) as an endless collection of added positive values that are < X, exactly because no infinitely many orange figures with constant value X>0 can have value 0)).
By using mathematics that is done only by verbal_symbolic brain skills, X=2(a+b+c+d+...), which means that X>0 AND X>2(a+b+c+d+...) are separable in the considered diagram, which (in my opinion) it is the result of separating visual_spatial and verbal_symbolic brain skills during mathematical work.
As I see it, Mathematics is actually fulfilled in case that (by using visual_spatial AND verbal_symbolic brain skills) it also goes beyond contradiction and tautology, and enables one to directly be aware of Unity as their cause (or in terms of Physics, it enables one to directly be aware of Unity as the cause of Bose-Einstein statistics (even function) and Fermi-Dirac statistics (odd function)).
Direct awareness of Unity is not achieved at the realm of thoughts process, which is closed under multiplicity.
As I get it, one enables to transcend multiplicity by being aware of finer vibrations' levels of thoughts' process without being restricted to their semantics (they are taken only in terms of different levels of vibrations)) which enables quantum leap into the non-composed source of multiplicity and quantum leap back to multiplicity.
More such bidirectional quantum leaps are exercised in one's brain, more one's brain enables the awareness of Unity as the source of multiplicity during daily life.
In my opinion, Mathematical science has to deeply be involved in such exercises in order to really be fulfilled, where reasoning at the level of thoughts (which is restricted by contradiction and tautology, like any multiple phenomena) as currently done by Contemporary Mathematics, can't fulfill the actual abilities of this science among human beings.
