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Can a number system be both the new kid on the block and older than written history?

The real number system as it exists today has been with us for a few centuries. In foundation it is monovalent, monophasic, and sequential.

The probable number system dates to prehistory but was lost in the mists of time until recently rediscovered and resurrected. In contrast to the real number system it is foundationally bivalent, biphasic, and cyclic.

The probable number system has considerably more structure than the real number system and is therefore more robust. In this sense, it is similar to the complex number system.

In contrast to the complex number system, the probable number system in its foundation presupposes that numbers can assume wavelike forms capable of constructive and destructive interference operationally through the compositing of higher to lower dimension.

By means of compositing of dimension probable numbers are able to distribute throughout the entire mandalic unit vector cube (which is structurally a superposition of the 6-dimensional unit vector hypercube on the 3-dimensional unit vector cube) a function analogous in important ways to that performed in the complex number system by the centralized imaginary unit i.

Another important way in which the probable number system differs from both the real number system and the complex number system is the absence of nothingness and the zero representing it. In its place we find the concepts of balance and equilibrium. Nullification still exists in form of annihilation and its opposite in the form of creation. But the Cartesian coordinate system of ordered pairs and ordered triads is transformed by this approach to handling number and dimension from a ring into a field of hyperdimensional numbers over real numbers in three dimensions.

(to be continued)

Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 315-

@mandalicgeometry

Sun, 29 May 2016 12:16:04

Toward a new geometry

Definitions and guiding principles

Dimension is a primitive concept which for purposes here can be defined as any linearly independent parameter or variable.
Mathematics is most fundamentally about measurement.
Rules of mathematics are conceptual tools developed by the human mind and are subject to change if and when such should prove necessary or utilitarian.
Points, lines and planes are products of the human mind. Neither they nor the realities they represent have continuous existence in space or time but can recur intermittently at intervals which may be either regular or irregular. As a matter of convenience for purposes of description, however, any or all of these may be represented as existing in some continuous abstract sense.
The whole is greater than the part and has emergent properties not found in any single part or any number of subdivisions of the whole.
The consequences of number structure and interaction cannot be explained entirely on a local basis but emerge from holo-interactive dynamic processs throughout an entire system or any portion thereof in conceptual focus.
It is always possible to express a reflection as a rotation and a rotation as a reflection. The two are isomorphic via properly chosen operations.
Symmetry is of great importance but not always obvious.
Two foundational guiding principles of reality are continuity and change. These then provide the basis and primary focus of mandalic geometry.
Of the two types of change, cyclic and sequential, cyclic change is the more fundamental.
Neither coordinates nor coordinate systems are a feature of nature. They are man-made devices which are pragmatic and utilitarian means by which to grasp a reality which itself has no need of them. Geometry nevertheless requires these crutches to exist and execute its functions.
Geometry frequently also requires conventions of expression to promote widespread understanding of content but should strive to be as convention -free as possible.
Ambiguity is a permissible feature, in fact a necessary feature, of mandalic geometry. This is related to its probabilistic nature and multiple-valued logic which reflect what we, from our limited vantage point, misunderstand to be paradoxes of nature.

Axioms

Numbers can be characterized by dimension of context.
A number may be embedded in multiple dimensions concurrently, existing in variant forms specific to the dimension(s) of context.
The same number may function differently in different dimensional contexts. The dimension of context of a “point” described by a number or subsidary number delimits its expression. In other words, the expression of a number in spatiotemporal terms is determined by its dimensional context. In referring to a number or subsidiary number in any of their variants, therefore, the dimension of context must always be specified.
It may not always be possible to identify the full dimensional context of a number. It is sufficient to determine and elucidate those contexts essential to elaboration of the specific operation(s) under present consideration.
Numbers are not necessarily elemental. They may consist of parts or subsidiary numbers which refer to various different dimensional contexts.
A point is not dimensionless extension in space. It is an emergent feature of the system-as-a-whole which appears intermittently at the common intersection of three or more dimensions. Points, lines and planes are evanescent occurrences in terms of geometry and spacetime. They, and the things they represent, are fleeting events which come and go. Repetition is possible (in a conceptual sense certainly; possibly in a material/energetic sense as well) but not inevitable.

Rejected definitions

The definition of a point found in Euclid’s Elements: A point is that which has no part. [This definition may or may not have been in Euclid’s original Elements.]
The definition of a line found in Euclid’s Elements: A line is breadthless length.
The definition of a surface found in Euclid’s Elements: A surface is that which has length and breadth only.

Rejected axioms

The real numbers are order-isomorphic to the linear continuum of geometry. In other words, the axiom states that there is a one to one correspondence between real numbers and points on a line. This has been described as the Cantor–Dedekind axiom. The Cartesian coordinate system implicitly assumes this axiom which then becomes the cornerstone of analytic geometry.

Rejected notions

Things which coincide with one another equal one another. [Euclid’s Common notion 4]

Also important to note:

Euclid’s first postulate states that any two points can be joined by a straight line segment. It does not say that there is only one such line; it merely says that a straight line can be drawn between any two points.

Image credit: James Gyre-Naked Geometry

Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 314-

#mandalic geometry #geometry #axioms #definitions #guiding principles #algebra #mathematics #toward a new geometry #new geometry #euclid #descartes #philosophy of mathematics #physics #quantum logic #interdisciplinary #transdisciplinary

@mandalicgeometry

Sat, 21 May 2016 10:59:06

Beyond the Enlightenment Rationalists:
From imaginary to probable numbers - VI

(continued from here)

“O Oysters, come and walk with us!” The Walrus did beseech. “A pleasant walk, a pleasant talk, Along the briny beach: We cannot do with more than four, To give a hand to each.”

* * *

“The time has come,” the Walrus said, “To talk of many things: Of shoes–and ships–and sealing-wax– Of cabbages–and kings– And why the sea is boiling hot– And whether pigs have wings.”

-Lewis Carroll, The Walrus and the Carpenter

In this segment, probable numbers will be shown to grow out of a natural context inherently rather than through geometric second thought as transpired in the history of Western thought with imaginary numbers and complex plane. To continue with development of probable numbers it will be necessary to leave behind, for the time being, all preoccupation with imaginary numbers and complex plane. It will also be necessary to depart from our comfort zone of Cartesian spatial coordinate axioms and orientation.

Probable coordinates do not negate validity of Cartesian coordinates but they do relegate them to the status of a special case. In the probable coordinate system the three-dimensional coordinate system of Descartes maps only one eighth of the totality. This means then, that the Cartesian two-dimensional coordinate plane furnishes just one quarter of the total number of corresponding probable coordinate mappings projected to a two-dimensional space.^[1] It suggests also that Cartesian localization in 2-space or 3-space is just a small part of the whole story regarding actual spatial and temporal locality and their accompanying physical capacities, say for instance of momentum or mass, but actually encompassing a host of other competencies as well.

Although this might seem strange it is a good thing. Why is it a good thing? First, because nature, as a self-sustaining reality, cannot favor any one coordinate scheme but must encompass all possible - if it is to realize any. Second, because both the Schrödinger equationandFeynman path integral approaches to quantum mechanics say it is so.^[2] Third, because Hilbert space demands it. This may leave us disoriented and bewildered, but nature revels in this plan of probable planes. Who are we to argue?

So how do we accomplish this feat? Well, basically by reflections in all dimensions and directions. We extend the Cartesian vectors every way possible. That would give us a 3 x 3 grid or lattice of coordinate systems (the original Cartesian system and eight new grid elements surrounding it), but there are only four different types, so we require only four of the nine to demonstrate. It is best not to show all nine in any case because to do so would place our Cartesian system at direct center of this geometric probable universe and that would be misleading. Why? Because when we tile the two-dimensional universe to infinity in all directions, there is no central coordinate system. Any one of the four could be considered at the center, so none actually is. Overall orientation is nondiscriminative.^[3]

LOOKING GLASS CARTESIAN COORDINATE QUARTET

The image seen immediately above shows four Looking House Cartesian coordinate systems, correlated within a mandalic plane. This mandalic plane is one of six faces of a mandalic cube, each of which is constructed to a different plan but composed of similar building blocks, the four bigrams in various positions and orientations. A 2-dimensional geometric universe can be tiled with this image, recursively repeating it in all directions throughout the two dimensions.^[4] It should not be very difficult for the reader to determine which of the four mandalic moieties references our particular conventional Cartesian geometric universe.^[5]

It remains only to be added here and now that potential dimensions, probable planes, and probable numbers arise immediately and directly from the remarks above. In some ways it’s a little like valence in chemical reactions. We’ll likely take a look at that combinatory dynamic in context of mandalic geometry at some time down the road. Next though we want to see how the addition of composite dimension impacts and modifies the basic geometry of the probable plane discussed here.^[6]

(to be continued)

Top image: The four quadrants of the Cartesian plane. These are numbered in the counterclockwise direction by convention. Architectonically, two number lines are placed together, one going left-right and the other going up-down to provide context for the two-dimensional plane. This image has been modified from one found here.

Notes

[1] To clarify further: There are eight possible Cartesian-like orientation variants in mandalic space arranged around a single point at which they are all tangent to one another. If we consider just the planar aspects of mandalic space, there are four possible Cartesian-like orientation variants which are organized about a central shared point in a manner similar to how quadrants are symmetrically arranged about the Cartesian origin point (0,0) in ordinary 2D space. But here the center point determining symmetries is always one of the points showing greatest rather than least differentiation. That is to say it is formed by Cartesian vertices, ordered pairs having all 1s, no zeros. That may have confused more than clarified, but it seemed important to say. We will be expanding on these thoughts in posts to come. Don’t despair. For just now the important takeaway is that the mandalic coordinate system combines two very important elements that optimize it for quantum application: it manages to be both probabilistic and convention-free (in terms of spatial orientation, which surely must relate to quantum states and numbers in some as yet undetermined manner.) At the same time, imaginary numbers and complex plane are neither.

[2] Even if physics doesn’t yet (circa 2016) realize this to be true.

[3] It is an easy enough matter to extrapolate this mentally to encompass the Cartesian three-dimensional coordinate system but somewhat difficult to demonstrate in two dimensions. So we’ll persevere with a two-dimensional exposition for the time being. It only needs to be clarified here that the three-dimensional realization involves a 3 x 3 x 3 grid but requires just eight cubes to demonstrate because there are only eight different coordinate system types.

[4] I am speaking here in terms of ordinary dimensions but it should be understood that the reality is that the mandalic plane is a composite 4D/2D geometric structure, and the mandalic cube is a composite 6D/3D structure. The image seen here does not fully clarify that because it does not yet take into account composite dimension nor place the bigrams in holistic context within tetragrams and hexagrams. All that is still to come. Greater context will make clear how composite dimension works and why it makes eminent good sense for a self-organizing universe to invoke it. Hint: it has to do with quantum interference phenomena and is what makes all process possible.

ADDENDUM (12 APRIL, 2016)
The mandalic plane I am referring to here corresponds to the Cartesian 2-dimensional plane and is based on four extraordinary dimensions that are composited to the ordinary two dimensions, hence hybrid 4D/2D. It should be understood though that any number of extra dimensions could potentially be composited to two or three ordinary dimensions. The probable plane described in this post is not such a mandalic plane as no compositing of dimensions has yet been performed. What is illustrated here is an ordinary 2-dimensional plane that has undergone reflections in x- and y-dimensions of first and second order to form a noncomposited probable plane. The distinction is an important one.

[5] This is perhaps a good place to mention that the six planar faces of the mandalic cube fit together seamlessly in 3-space, all mediated by the common shared central point, in Cartesian terms the origin at ordered triad (0.0.0) where eight hexagrams coexist in mandalic space. Moreover the six planes fit together mutually by means of a nuclear particle-and-force equivalent of the mortise and tenon joint but in six dimensions rather than two or three, and both positive and negative directions for each.

[6] It should also be avowed that tessellation of a geometric universe with a nondiscriminative, convention-free coordinate system need not exclude use of Cartesian coordinates entirely in all contextual usages. Where useful they can still be applied in combination with mandalic coordinates since the two can be made commensurate, irrespective of specific Cartesian coordinate orientation locally operative. Whatever the Cartesian orientation might be it can always be overlaid with our conventional version of the same. More concretely, hexagram Lines can be annotated with an ordinal numerical subscript specifying Cartesian location in terms of our local convention should it prove necessary or desirable to do so for whatever reason.

On the other hand, before prematurely throwing out the baby with the bath water, we might do well to ask ourselves whether these strange juxtapositions of coordinates might not in fact encode the long sought-after hidden variables that could transform quantum mechanics into a complete theory. In mandalic coordinates of the reflexive nature described, these so-called hidden variables could be hiding in plain sight. Were that to prove the case, David Bohm andLouis de Broglie would be immediately and hugely vindicated in advancing their pilot-wave theory of quantum mechanics. We could finally consign the Copenhagen Interpretation to the scrapheap where it belongs, along with both imaginary numbers and the complex plane.

ADDENDUM (24 APRIL, 2016)
Since writing this I’ve learned that de Broglie disavowed Bohm’s pilot wave theory upon learning of it in 1952. Bohm had derived his interpretation of QM from de Broglie’s original interpretation but de Broglie himself subsequently converted to Niels Bohr’s prevailing Copenhagen interpretation.

Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 311-

@mandalicgeometry

Sun, 10 Apr 2016 20:09:10

Beyond the Enlightenment Rationalists:
From imaginary to probable numbers - V

(continued from here)

The four Cartesian quadrants provide the two-dimensional analogue of the number line and its graphic representation in Cartesian coordinate space. This is the true native habitat of the square and, by implication, of square root. Because Enlightenment mathematicians found fit to define square root in a different context inadvertently -that of the number line- we will find it necessary to devise a different name for what ought rightly to have been called square root, but wasn’t. I propose that we retain the existent definition of tradition and refer to the new relationship between opposite numbers in the square, that is to say, opposite vertices through two dimensions or antipodal numbers, as contra-square root.^[1]

Modified from image found here.

Given this fresh context - one of greater dimension than the number line - it soon becomes clear with little effort that a unit number^[2]ofany dimension multiplied by itself gives as result the identity element of that express dimension. For the native two-dimensional context of the square the identity element is OLD YANG, the bigram composed of two stacked yang (+) Lines, which corresponds to yang (+1), the identity element in the one-dimensional context of the number line. In a three-dimensional context, the identity element is the trigram HEAVEN which is composed of three stacked yang (+) Lines. The crucial idea here is that the identity element differs for each dimensional context, and whatever that context might be, it produces no change when in the operation of multiplication it acts as operator on any operand within the stated dimension.^[3]

As a corollary it can be stated that any number in any dimension n composed of any combination of yang Lines (+1) and yin Lines (-1) if multiplied by itself (i.e., squared) produces the identity element for that dimension. In concrete terms this means, for example, that any bigram multiplied by itself equals the bigram OLD YANG; any of eight trigrams multiplied by itself equals the trigram HEAVEN; and any of the sixty-four hexagrams multiplied by itself equals the hexagram HEAVEN; etc. (valid for any and all dimensions without exception). Consequently, the number of roots the identity element has in any dimension n is equal to the number 2ⁿ, these all being real roots in that particular dimension.

Similar contextual analysis would show that the inversion element of any dimension n has 2ⁿ roots of the kind we have agreed to refer to as contra-square roots in deference to the Mathematics Establishment.^[4]

That leads us to the possibly startling conclusion that in every dimension n there is an inversion element that has the same number of roots as the identity elementandall of them are real roots. For two dimensions the two pairs that satisfy the requirement are bigram pairs

For one dimension there is only a single pair that satisfies. That is (surprise, surprise) yin(-1)/yang (+1). What it comes down to is
this:

If we are going to continue to insist on referring to square root
in terms of the one-dimensional number line, then

+1 has two real roots of the traditional variety, +1 and -1
-1 has two real roots of the newly defined contra variety,
+1/-1 and -1/+1

So where do imaginary numbers and quaternions fit in all this? The short answer is they don’t. Imaginary numbers entered the annals of human thought through error. There was a pivotal moment^[5] in the history of mathematics and science, an opportunity to see that there are in every dimension two different kinds of roots - - - what has been called square root and what we are calling contra-square roots. Enlightenment mathematicians and philosophers essentially allowed the opportunity to slip through their fingers unnoticed.^[6]

Descartes at least saw through the veil. He called the whole matter of imaginary numbers ‘preposterous’. It seems his venerable opinion was overruled though. Isaac Newton had his say in the matter too. He claimed that roots of imaginary numbers “had to occur in pairs.” And yet another great mathematician, philosopher opined. Gottfried Wilhelm Leibniz, in 1702 characterized √−1 as “that amphibian between being and non-being which we call the imaginary root of negative unity.” Had he but preserved such augury conspicuously in mind he might have elaborated the concept of probable numbers in the 18th century. If only he had truly understood the I Ching, instead of dismissing it as a primitive articulation of his own binary number system.

(continuedhere)

Image: The four quadrants of the Cartesian plane. By convention the quadrants are numbered in a counterclockwise direction. It is as though two number lines were placed together, one going left-right, and the other going up-down to provide context for the two-dimensional plane. Sourced from Math Is Fun.

Notes

[1] My preference might be for square root to be redefined from the bottom up, but I don’t see that happening in our lifetimes. Then too this way could be better.

[2] By the term unit number, I intend any number of a given dimension that consists entirely of variant elements of the number one (1) in either its positive or negative manifestation. Stated differently, these are vectors having various different directions within the dimension, but all of scalar value -1 (yin) or +1 (yang). All emblems of I Ching symbolic logic satisfy this requirement. These include the Line, bigram, trigram, tetragram, and hexagram. In any dimension n there exist 2ⁿ such emblems. In sum, for our purposes here, a unit number is any of the set of numbers, within any dimension n, which when self-multiplied (squared) produces the multiplicative identity of that dimension which is itself, of course, a member of the set.

ADDENDUM (01 MAY 2016): I’ve since learned that mathematics has a much simpler way of describing this. It calls all these unit vectors. Simple, yes?

[3] I think it fair to presume that this might well have physical correlates in terms of quantum mechanical states or numbers. Here’s a thought: why would it be necessary that all subatomic particles exist in the same dimension at all times given that they have a playing field of multiple dimensions, - some of them near certainly beyond the three with which we are familiar? And why would it not be possible for two different particles to be stable and unchanging in their different dimensions, yet become reactive and interact with one another when both enter the same dimension or same amplitude of dimension?

[4] Since in any contra-pair (antipodal opposites) of any dimension, either member of the pair must be regarded once as operator and once as operand. So for the two-dimensional square, for example, there are two antipodal pairs (diagonals) and either vertex of each can be either operator or operand. So in this case, 2 x 2 = 4. For trigrams there are four antipodal pairs, and 2 x 4 = 8. For hexagrams there are thirty-two antipodal pairs and 2 x 32 = 64. In general, for any dimension n there are 2 x 2ⁿ/2 = 2ⁿ antipodal pairs or contra-roots.

[5] Actually lasting several centuries, from about the 16th to the 19th century. Long enough, assuredly, for the error to have been discovered and corrected. Instead, the 20th century dawned with error still in place, and physicists eager to explain the newly discovered bewildering quantum phenomena compounded the error by latching onto √−1 and quaternions to assuage their confusion and discomfiture. This probably took place in the early days of quantum mechanics when the Bohr model of the atom still featured electrons as traveling in circular orbits around the nucleus or soon thereafter, visions of minuscule solar systems still fresh in the mind. At that time rotations detailed by imaginary numbers and quaternions may have still made some sense. Such are the vagaries of history.

[6] I think an important point to consider is that imaginary and complex numbers were, -to mathematicians and physicists alike,- new toys of a sort that enabled them to accomplish certain things they could not otherwise. They were basically tools of empowerment which allowed manipulation of numbers and points on a graph more easily or conveniently. They provided
their controllers a longed for power over symbols, if not over the real world itself. In the modern world ever more of what we humans do and want to do involves manipulation of symbols. Herein, I think, lies the rationale for our continued fascination with and dependence on these tools of the trade. They don’t need to actually apply to the world of nature, the noumenal world, so long as they satisfy human desire for domination over the world of symbols it has created for itself and in which it increasingly dwells, to a considerable degree apart from the natural world’s sometimes seemingly too harsh laws.

Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 310-

@mandalicgeometry

Sun, 03 Apr 2016 14:24:07

Beyond the Enlightenment Rationalists:
From imaginary to probable numbers - II

(continued from here)

When a geometric interpretation of imaginary numbers was at last proposed, long after they were invented, it was as though accomplished by central committee. The upshot was easily enough understood but also simplistic. In broad brushstroke here is what seems to have gone down.

The 3 dimensions of Descartes’ coordinate system-a number already deficient from the perspective of mandalic geometry-were reduced to just one. Of the real number axes then only the x-axis remained. This meant from the get-go that any geometric figure that ensued could encompass only linearity in terms of real numbers and dimensions. It was applicable only to a line segment, so the complex plane that resulted could describe just one real dimension and one imaginary dimension. It consecrated the number line in a single dimension, to exclusion of its proper habitation in two others besides. Strike one for imaginary numbers.^[1]

With that as background let’s look now at the rotations described by this geometric interpretation of imaginary numbers in the context of the complex plane.

i in the complex or cartesian plane. Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axis By Loadmaster (David R. Tribble) (Own work) [CC BY-SA 3.0orGFDL],via Wikimedia Commons

The number 1 is the multiplicative identity element for real numbers and the number -1 is the reflection inversion element for real numbers. Put another way, the number one times any number equals that number; the number -1 times any number is a negative of that number or the inverse number through a reference point, usually taken as zero. Multiplying by 1 then leaves 1, -1, i and -i all unchanged. Multiplying by -1 changes -1 to 1, 1 to -1, i to -i, and -i to i. In terms of rotations in the complex plane, these changes all involve a rotation through 180 degrees. Multiplication of the number 1 by i changes it to i; i by i changes it to -1; -1 by i to -i; and -i by i to 1. These changes all involve rotations through 90 degrees. And finally, multiplication of 1 by -i changes it to -i; -i by -i changes it to -1; -1 by -i to i; and i by -i to 1: all changes involving rotations through -90 degrees.

The figure below shows another way to interpret these rotations that amounts to the same tbing: i¹ = i; i² =-1; i³ = -i; i⁴ = 1. Click to enlarge.

Four numbers on the real line multiplied by integer powers of the imaginary unit, which corresponds to rotations by multiples of the right angle. By Keφr [CC0],via Wikimedia Commons

I think a committee of some sort must have come up with this resplendent plan. For certain it was an Academy of Mathematics and Sciences that endorsed and enthroned it. All bow to central authority.

I had planned to include a comparison of imaginary numbers and probable numbers in this post as well but because that is a long discussion itself, it will have to wait till the next post. I might add it should prove well worth the wait.

(continuedhere)

Image: A drawing of the first four dimensions. On the left is zero dimensions (a point) and on the right is four dimensions (A tesseract). There is an axis and labels on the right and which level of dimensions it is on the bottom. The arrows alongside the shapes indicate the direction of extrusion. By NerdBoy1392 (Own work) [CC BY-SA 3.0orGFDL],via Wikimedia Commons

Notes

[1] Mathematician William Rowan Hamilton subsequently addressed this deficiency in 1843 with his quaternions, a number system that extends the complex numbers to three-dimensional space. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or, in other words, as the quotient of two vectors. This complicated matters even more by introducing a non-commutative multiplication operation to the system, though to be fair the quaternion coordinate system has found some useful applications mainly for calculations involving 3-dimensional rotations, as in 3-dimensional computer graphics,computer vision, and crystallographic texture analysis. Still it becomes problematic when theoretical physics attempts use of quaternions in calculations pertaining to atomic and subatomic spaces where rotations do not actually take place. The conclusion to be drawn here is that quaternions can be usefully, if somewhat clumsily, applied to 3-dimensional macro-spaces but are inapproriate for accurate description of higher dimensional spaces. What is here unfortunate and misleading is that quaternions apparently do describe outcomes of events in the quantum realm to some partial degree, if not the mechanisms of the events themselves. Physicists would not long tolerate them were that not so.

[ADDENDUM (24 APRIL, 2016)
Since writing this I’ve learned that quaternions are not currently used in quantum physics nor were they ever, to any great degree, in the past.]

In other words, sometimes the right answer can be reached by a wrong method. In the case under discussion here, we should note that it is possible for a rotation to mimic inversion (reflection through a point). A 90° rotation in two dimensions can mimic a single inversion in a single plane through an edge of a square, and a 180° rotation in two dimensions can mimic a single inversion through a diagonal of a square or two successive inversions through two perpendicular edges of a square. A 180° rotation in three dimensions can mimic three inversions through three mutually perpendicular edges of a square; a combination of one inversion through a diagonal of a square and another through an edge perpendicular to the plane of the first inversion; or a single inversion through a diagonal of the cube. Subatomic paricles exist as discrete or quantized entities and would follow such methods of transformation rather than rotations through a continuous space. Of course, transformations involving a diagonal would require more transformative energy than one involving a single edge.

Such patterns of relationship and transformation could no doubt be described in terms of quantum states and quantum numbers without too much difficulty by a knowledgeable theoretical physicist. Surely doing so could be no more difficult than using quaternions, which may give a correct answer while also misleading and limiting knowledge of the the true workings of the quantum realm by using an incorrect mechanism, one non-commutative to boot. Nature doesn’t approve of hat tricks like that.

Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 307-

@mandalicgeometry

Tue, 22 Mar 2016 17:40:42

A Short Philosophical Aside

The scrupulous 3-dimension world we humans inhabit is in fact biological, not physical, in origin. Its limitations are determined by our specific sensory, motor and mental apparatus and abilities. It only hints at the real world, and while doing so it combines some highly erroneous observations as well. Molluscs and insects and arachnids all have a very different perspective of their environment. We would find discomfort in the world view of an octopus, as we do in the quantum world view.^[1]^[2]

Dimension is a term laymen toss about haphazardly. Mathematicians and physicists have a more precise interpretation concerning dimension. For them, any independent parameter constitutes a separate dimension. But when it comes down to the nitty-gritty, what if anything can truly be separate and independent? Those are both relative terms. Nothing that exists is really fully isolate and independent. That is one of the substratal premises from which mandalic geometry evolves: relationships invariably exist. And relationships can always change. Mandalic geometry therefore is a geometry of process - a spacetime geometry, not one of space alone.

For those who created the primal I Ching relationship was considered a fundamental aspect of reality. When they thought of dimension - - - and they did, in their own way - - - relationships were always involved. Flash-forward a few thousand years - quantum mechanics accomplishes much the same with its view of interacting particles in continual motion, ever-changing, and incessantly forging transient effective links with numerous other particles of similar and different type under the influence of various fields of force.

Kant thought that human concepts and categories determine our view of the world and its laws. He held that inborn features of our minds structure our experiences. Since, in his view, mind shapes and structures experience, at some level of representation all human experience shares certain essential operational features. Among these according to Kant are our concepts relating to space and time, integral to all human experience. The same might be said about our concepts of cause and effect.

Kant further asserts that we never have direct experience of things, referred to in his writings as the noumenal world. All we experience is the phenomenal world that is relayed to us by our senses. Kant views noumena as the thing-in-itself or true reality and phenomena as our experience or perception of that thing, filtered through our senses and reasoning. According to Kant science can be applied only to things that can be observed and studied. The entire world of noumena is beyond the scope and reach of science. As an heir to Enlightenment philosophy Kant respects the value of reason but believes the noumenal world to be beyond its scope and reach. So are we fated then never to experience the noumena directly? Not by a long shot. Kant claims the noumena to be accessible but only by intellectual intuition without the aid of reason.^[3]

In the world of phenomena nothing is self-existent. Everything exists by virtue of dependence on something else. Point to something, anything at all, that refutes that view and I’ll tell you you’re out of your mind - and in the noumenal world. What, pray tell, are you doing there and how did you get there anyway? If you can clearly communicate the how I may give it a try myself.^[4]

Image:

One of a set of illustrations by Emma V. MooretitledNoumena - Collages © Emma V Moore 2013 courtesy of the artist. More of her exceptional art can be found at http://www.emmavmoore.co.uk. Follow also on Bēhance Please do not remove credits.

Notes

[1] The world view granted us by our inherited biologic capacities has been millions of years in the making. Indeed. But that makes it still not a whit truer than had we groped it only yesterday. Evolution seems to have sacrificed a full immersive sense of reality to grant a greater degree of interoperability essential to dealing with vicissitudes of a material world and confer durability within that domain. The quest after true apprehension we feel impelled to pursue is a siren not without danger.

“The search for reality is the most dangerous of all undertakings, for it destroys the world in which you live.”
-Nisargadatta Maharaj

[2] Regarding the origin and transformations of the word “scrupulous”:

Scrupulous and its close relative “scruple” (“an ethical consideration”) come from the Latin noun scrupulus, the diminutive of “scrupus.” “Scrupus” refers to a sharp stone, so scrupulus means “small sharp stone.” “Scrupus” retained its literal meaning but eventually also came to be used with the metaphorical meaning “a source of anxiety or uneasiness,” the way a sharp pebble in one’s shoe would be a source of pain. When the adjective “scrupulous” entered the language in the 15th century, it meant “principled.” Now it also commonly means "painstaking" or “careful.” [Source]
Sad to say, this fascinating word that so successfully wended its way through several related incarnations in a number of different Indo-European languages prior to its appearance in English, c.15th century, appears to be passing out of usage among English speakers in modern times. We will likely be left with the occasional utterance of “scruples” but “scrupulous” itself seems destined for oblivion.
Curiously, my election of the word here was not rationally motivated. As I was framing the thought expressed in the paragraph in my mind, the word just appeared out of nowhere and seemed to insist, “I belong here though you may not yet understand why. You really need a word with my complex heritage of multiple meanings here.” And so I went with it, not fully knowing why. Funny thing about it, my rational mind is quite unable now to come up with any other single word that suits as well.

[3] Kant’s epistemology recognizes three different sources of knowlege: sensory experience, reason, and intuition. He views intuition as independent of the other two and the only one of the three with direct access to the world of noumena. This may present as suspect at first, but then how do we explain things like what Einstein did a century ago? Einstein himself has hinted in his writings at the essential role of intuition and imagination in his thinking.

Slide 25 of 48

Clickhere for more slides on Kant’s philosophy by William Parkhurst from Introduction to Philosophy Lecture 13, source of the above slide reproduction.

[4] Our human penchant for categorization inevitably leads to dismemberment of holistic reality into an endless number of manifest objects, many of which we no longer recognize as essentially related.

“People normally cut reality into compartments, and so are unable to see the interdependence of all phenomena. To see one in all and all in one is to break through the great barrier which narrows one’s perception of reality.”
-Thích Nhất Hạnh

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Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
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-Page 305-

@mandalicgeometry

Sun, 13 Mar 2016 18:07:58

Neo-Boolean - II: Logic Gates
Thinking Inside the Lines

(continued from here)

We have already looked briefly at three of the more important Boolean operators or logic gates: AND, OR,andXOR.NOT just toggles any two Boolean truth values (true/false; on/off; yes/no). Here we introduce two new logic gates which do not occur in Boolean algebra. Both play an important role in mandalic geometry though.

We’ll refer to the first of these new operators or logic gates as INV standing for inversionorinvert. This is similar to Boole’s NOT except that it produces toggling betweeen yang/+ and yin/- instead of 1 and 0. Because it is based on binary arithmetic, Boole’s NOT has been thought of as referring to inversion also (as in ONorOFF). Although both ANDandINV act as toggling logic gates they have very different results in the greater scheme of things, since nature has created a prepotent disparity between a -/+ toggle and a 0/1 toggle in basic parameters of geometry, spacetime, and being itself. This makes Boole’s AND just a statement of logical opposition, notinversion.

Recognition of this important difference is built into mandalic geometry structurally and functionally, as it is also into Cartesian coordinate dynamics and the logic of the I Ching, but lacking in Boole’s symbolic logic. This is necessarily so, as there is no true negative domain in Boolean algebra. The OFF state of electronics and computers, though it may sometimes be thought of in terms of a negative state, is in fact not. It relates to the Western zero (0), not the minus one of the number line. Where Boolean algebra speaks of NOT 1 it refers specifically to zero and only to zero. When mandalic geometry asserts INV 1 it refers specifically to -1 and only to -1 . The inversion of yang then is yin and the inversion of yinisyang.^[1] In the I Ching, Taoist thought, and mandalic geometry the two are not opposites but complements and, as such, interdependent.

The second added logic gate that will be introduced now is the REV operator standing for reversionorrevert. This operator produces no change in what it acts upon. It is the multiplicative identity element (also called the neutral elementorunit element), as INV is the inverse element. In ordinary algebra the inverse element is -1, while the identity element is 1. In mandalic geometry and the I Ching the counterparts are yinandyang, respectively. If Boolean algebra lacks a dedicated identity operator, it nonetheless has its Laws of Identity which accomplish much the same in a different way:

A = A
NOT A = NOT A

Again, Boolean algebra has no true correlate to the INV operator. There can be no sign inversion formulation as it lacks negatives entirely. Although Boolean algebra may have served analog and digital electronics and digital computers quite well for decades now, it is incapable of doing the same for any quantum logic applications in the future, if only because it lacks a negative domain.^[2] It offers up bits readily but qubits only with extreme difficulty and those it does are like tears shed by crocodiles while feeding.

(to be continued)

Image: Boolean Search Operators. [Source]

Notes

[1] Leibniz’s binary number system, on which Boole based his logic, escapes this criticism, as Leibniz uses 0 and 1 simply as notational symbols in a modular arithmetic and not as contrasting functional elements in an algebraic context of either the Boolean or ordinary kind.

In the field of computers and electronics, Boolean refers to a data type that has two possible values representing true and false. It is generally used in context to a deductive logical system known as Boolean Algebra. Binary in mathematics and computers, refers to a base 2 numerical notation. It consists of two values 0 and 1. The digits are combined using a place value structure to generate equivalent numerical values. Thus, both are based on the same underlying concept but used in context to different systems. [Source]

[2] Moreover, I expect physics will soon enough discover that what it now calls antimatter is in some sense and to some degree a necessary constituent of ordinary matter. I can already hear the loudly objecting voices declaring matter and antimatter in contact necessarily annihilate one another, but that need not invalidate the thesis just proposed. My supposition revolves around the meaning of “contact” at Planck scale and the light speed velocity at which subatomic particles are born, interact and decay only to be revived again in an eternal dance of creation and re-creation. Material particles exist in some kind of structural and functional homeostasis, not all that unlike the anabolic and catabolic mechanisms that by means of negative feedback maintain all entities of the biological persuasion in the steady state we understand as life. Physics has yet to get a full grip on this aspect of reality, though moving ever closer with introduction of quarks and gluons to its menagerie of performing particles.

Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
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-Page 304-

@mandalicgeometry

Tue, 01 Mar 2016 12:30:36

A Recap of Some Important Ideas Regarding Mandalic Geometry

Mandalic geometry (MG) is a new kind of mathematical methodology based on a worldview having roots that predate written history.
It is a discrete geometry which currently consists of just a coordinate system but can be extended as Descartes did his to encompass an entire analytic geometry.
Mandalic geometry introduces and is based on a new number system, the probable number system (or probabilistic number system.)
Just as the complex number system combines real numbers and imaginary numbers and is more robust than either, the probable number system combines real numbers and probable numbers and is more robust than either.
The probable number system is also more robust than the complex number system. Complex numbers combine real numbers with imaginary numbers to form the single complex plane. Composite numbers combine real numbers with probable numbers to form six interdependent composite planes.
Axiomatic to the system is the contention that numbers can exist in different dimensions and therefore can be described as being of some particular dimension. Numbers are always viewed and treated within context of a stated dimension.
Probable numbers are an extension of the real numbers to higher dimensions and are independent of imaginary and complex numbers.
Mandalic geometry does not admit the existence of square root of -1 in the real world other than in mathematics invented by the human mind. In place of square root of negative numbers, MG introduces the new concept of contra-square root. In brief this involves substitution of a combination form of interactive two-dimensional analogues of +1 and -1 for -1 as currently used in imaginary number contexts. This is more fully explained elsewhere in the blog.
Put another way, in place of imaginary numbers MG posits the existence of probable numbers. These can be considered the result of what is essentially wavelike interactions of higher dimensional numbers to form the real numbers we know in the 3-dimensional world.
Higher dimensional numbers can interact with one another through wavelike constructive and destructive interference to generate ordinary
3-dimensional numbers. Numbers are not viewed as constants to be acted upon as Descartes so views them but rather as being themselves active and changeable. They participate in process. This feature alone enables composite numbers to mediate between mathematics and physics better than either real or complex numbers can.
The interactions of higher dimensional numbers in the process of dimensional compositing to yield 3-dimensional numbers is a function of time and therefore probabilistic from our limited ordinary point of view. From this perspective, certain probablity distributions are the result of dimensional compositing and the consequent mandalic form. MG considers the probabilistic nature of quantum mechanics likely to be based on such.
The probabilistic nature in three dimensions of what are here called probable numbers is what gives rise to the mandalic form which can in a sense be considered the 3-dimensional evolution of 6-dimensional numbers from protean representations through progressive differentiation of form to the stage of maximal differentiation and back again to the undifferentiated state of greatest probability.
The mandalic form has a geometric progression of its line structures in the three Euclidean/Cartesian dimensions such that series of numbers of the form 1:2:1, 2:4:2, and 4:8:4 occur throughout all of those dimensions when a hybrid 6D/3D coordinate system results from performing 2:1 compositing from six to three dimensions.
Mandalic geometry views points and lines in three dimensions as convenient fictions that exist only as evanescent probabilistic concurrences of analogous entities in higher dimensions.
The probabilistic nature of MG makes it ideal for investigations and descriptions of quantum mechanics.
The exclusion of imaginary and complex numbers and substitution of probable and composite numbers which are easily reducible to ordinary algebraic/arithmetic forms and can be worked with using the same methods as those mathematical disciplines makes MG more utilitarian and appropriate to application to quantum mechanics than are complex numbers. All operations performed are based on simple inversion (reflection through a point) and on real numbers, maintaining all the usual rules and properties of ordinary arithmetic, including commutativity (which quaternions fail to preserve.)
MG is currently based on discrete numbers and is concerned mainly with the positive and negative integers. Fractions and irrational numbers are not excluded from the system but do not currently play a significant role. Future incarnations of MG will extend it outward beyond the unit vector cube to tile the geometric universe and inward to encompass fractional entities and fractals.
It is a hybrid geometry resulting from superposition of 6-dimensional numbers and 3-dimensional numbers and is fully commensurate with
3-dimensional Cartesian geometry.
It describes a linear mapping of two dimensions to one dimension which forms a field of probable numbers over the field of real numbers, analogous to the field of complex numbers but constructed on a different principle and extending to the real numbers in all three Cartesian dimensions rather than just one. The two independent higher dimensions so mapped become dependent variables in the mandalic “line” that results from the compositing of the two. This is expressed, in a sense, as two sine waves 180 degrees out of phase that mutually intersect a common Cartesian axis (x,y or z) at Cartesian +1 and -1 and are maximally separated at Cartesian 0.
This phase difference produces wave interference of both constructive and destructive varieties. So-called “points” or “particles” they represent come into existence only discretely and intermittently at Cartesian -1, +1, and 0, the locations of intersection or confluence (-1 and +1) and maximum separation, the maxima/minima of the two entangled sine waves that occur at Cartesian 0.
As the unit vector cube corresponds to and describes only half of each of the two sine waves, two unit vector cubes are required for a full cycle. Mandalic geometry as currently formulated with a single unit cube then needs to be extended to at least two of these. Extension in both directions of all three Cartesian axes is easily accomplished by repeatedly inverting the current single unit vector cube.
This means that mandalic coordinates alternate positive and negative on both sides of Cartesian 0. The extensions can be continued to infinity in both directions, but not, properly speaking, positive and negative infinity since the manner of extension has created what is essentially a convention-free coordinate system which consists of repeated units of consecutively inverted unit vector cubes in which positive and negative alternate ad infinitum and every Cartesian even-numbered coordinate becomes a “zero equivalent” , or better, a neo-zero in this extended mandalic coordinate system.
The resulting geometry is a dynamic one with “points”, “lines”, and “planes” coming into and passing out of existence intermittently in a time-sharing of corresponding Cartesian entities. It “persists” in time and space by means of continuous creation, destruction and re-creation and is “held together” by “force fields” produced and maintained by means of tensegrity which is based ultimately on dimension and number, and by a process that.might best be described as a “weaving of reality” with warp and woof.
The 2:1 compositing of dimension involved creates a new number system the members of which are like the real integers in all ways except that they map differently to a Cartesian geometric space. Whereas Decartes assumes that one number maps to one point, MG does not make this assumption which is just an unproved axiom that Descartes makes implicit use of.
The method of dimensional compositing automatically results in a mandalic formation having a geometric progression through three Euclidean/Cartesian dimensions from periphery to center (origin).
Currently MG is limited to a description of unit vectors in a composite hybrid 6D/3D geometry but can be extended to include all scalar values and any even number of dimensions.
The notation system used is borrowed from Taoism and foreign to most Western mathematicians. It is, however, basically equivalent to Cartesian coordinate signs (yin=minus; yang=plus); ordered pairs (=bigrams); and ordered triads (=trigrams); and extends these concepts to include ordered quads (=tetragrams) and ordered sextuplets (=hexagrams).
This notation system is used rather than the usual Cartesian notation because it is much easier for the mind to manipulate dimensional numbers using it. It takes only a little practice to become accustomed to using it. Without its use, understanding of mandalic geometry becomes extremely difficult, if not impossible.
As MG views a point as a concurrence of various different dimensions, it interprets Cartesian ordered pairs and triads, and their extensions to higher dimensions, as tensors and treats them as such. This makes it possible to apply operations of addition and multiplication to these mathematical entities in a manner analogous to the way William Rowan Hamilton applied these operations to complex numbers by way of what he called “algebraic couples”.
The probabilistic mandalic form that is the hallmark of MG conveys and necessitates a new interpretation of zero(0). In MG “zero” is not the empty null that it is in Cartesian geometry and Western mathematics generally, but rather a fount of being, so to speak, and a logic gate spanning dimensions. Wherever a zero occurs in Cartesian coordinates two Cartesian-equivalent forms are found in mandalic coordinates. So in the mandalic cube based on unit vectors the twelve edge centers, having a single Cartesian zero, have two Cartesian-equivalent forms (hexagrams); the six face centers, having two Cartesian zeros, have four Cartesian-equivalent forms; and the single cube center, the Cartesian origin point with three zeros, has eight Cartesian-equivalent forms.
Thisalternative zero and the mandalic structure it inhabits force the creation of four different amplitudes of dimension in the 6-dimensional unit vector cube. These are not independent but all mutually dependent and holo-interactive within the composite 6D/3D coordinate system. All of this occurs in a context reminiscent of the one inhabited by nuclear particles. The mapping proposed by MG may in fact model the elementary force fields, electromagnetism and quantum chromodynamics. It suggests a possible mechanism for formation of the state of matter known as a quark-gluon plasma. Hidden within it may even be the secret of quantum gravity.

Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 312-

#mandalic geometry #interdisciplinary #transdisciplinary #philosophy

@blindmen6

Tue, 31 May 2016 17:59:43

Beyond the Enlightenment Rationalists:
From imaginary to probable numbers - VI

(continued from here)

“O Oysters, come and walk with us!” The Walrus did beseech. “A pleasant walk, a pleasant talk, Along the briny beach: We cannot do with more than four, To give a hand to each.”

* * *

“The time has come,” the Walrus said, “To talk of many things: Of shoes–and ships–and sealing-wax– Of cabbages–and kings– And why the sea is boiling hot– And whether pigs have wings.”

-Lewis Carroll, The Walrus and the Carpenter

In this segment, probable numbers will be shown to grow out of a natural context inherently rather than through geometric second thought as transpired in the history of Western thought with imaginary numbers and complex plane. To continue with development of probable numbers it will be necessary to leave behind, for the time being, all preoccupation with imaginary numbers and complex plane. It will also be necessary to depart from our comfort zone of Cartesian spatial coordinate axioms and orientation.

Probable coordinates do not negate validity of Cartesian coordinates but they do relegate them to the status of a special case. In the probable coordinate system the three-dimensional coordinate system of Descartes maps only one eighth of the totality. This means then, that the Cartesian two-dimensional coordinate plane furnishes just one quarter of the total number of corresponding probable coordinate mappings projected to a two-dimensional space.^[1] It suggests also that Cartesian localization in 2-space or 3-space is just a small part of the whole story regarding actual spatial and temporal locality and their accompanying physical capacities, say for instance of momentum or mass, but actually encompassing a host of other competencies as well.

Although this might seem strange it is a good thing. Why is it a good thing? First, because nature, as a self-sustaining reality, cannot favor any one coordinate scheme but must encompass all possible - if it is to realize any. Second, because both the Schrödinger equationandFeynman path integral approaches to quantum mechanics say it is so.^[2] Third, because Hilbert space demands it. This may leave us disoriented and bewildered, but nature revels in this plan of probable planes. Who are we to argue?

So how do we accomplish this feat? Well, basically by reflections in all dimensions and directions. We extend the Cartesian vectors every way possible. That would give us a 3 x 3 grid or lattice of coordinate systems (the original Cartesian system and eight new grid elements surrounding it), but there are only four different types, so we require only four of the nine to demonstrate. It is best not to show all nine in any case because to do so would place our Cartesian system at direct center of this geometric probable universe and that would be misleading. Why? Because when we tile the two-dimensional universe to infinity in all directions, there is no central coordinate system. Any one of the four could be considered at the center, so none actually is. Overall orientation is nondiscriminative.^[3]

LOOKING GLASS CARTESIAN COORDINATE QUARTET

The image seen immediately above shows four Looking House Cartesian coordinate systems, correlated within a mandalic plane. This mandalic plane is one of six faces of a mandalic cube, each of which is constructed to a different plan but composed of similar building blocks, the four bigrams in various positions and orientations. A 2-dimensional geometric universe can be tiled with this image, recursively repeating it in all directions throughout the two dimensions.^[4] It should not be very difficult for the reader to determine which of the four mandalic moieties references our particular conventional Cartesian geometric universe.^[5]