#symbolic logic

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.

© 2016 Martin Hauser

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 311-

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.

© 2016 Martin Hauser

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 310-

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

Imaginary numbers arose in the history of mathematics as a result of misunderstanding the dimensional character of numbers.  There was a failure to acknowledge that numbers exist in a context of dimension. This has earlier been addressed at length.^[1]  Simply put, numbers exist always in a particular dimensional context.  Square numbers pertain to a context of two dimensions and therefore to a plane,  not a line.  Square roots then ought justly reference a two-dimensional geometrical context rather than the linear one mathematics has maintained ever since mathematicians of the Age of Enlightenment decreed it so.  Square roots contrary to the way mathematics would have it can neither exist in nor be found in any single line segment,  because they do not originate in the number line but in the two-dimensional square.

Algebra, not geometry, provided the breeding ground for imaginary numbers.  They were given a geometric interpretation as an afterthought only, long after the fact of their invention. Rationalist algebraists, feeling compelled to give meaning to equations of the form b² = -4 came up with the fantastic notion of imaginary numbers. Only indirectly did these grow out of nature, by way of minds of men obsessed with reason.^[2]

Descartes knew of the recently introduced square roots of negative numbers. He thought them preposterous and was first to refer to the new numbers by the mocking name imaginary, a label which stuck and which continues to inform posterity of the exact manner in which he viewed the oddities.  It is one of the ironies of history that when at last a geometrical interpretation of square root of negative numbers was offered it involved swallowing up Descartes’ own y-axis. Poetic justice? Or ultimate folly?

Had the essential dimensional nature of numbers been recognized there would have been no need to inquire what the square root of -1 was. It would have been clear that there was no square root of -1 nor any need for such as +1 also has no square root.  As linear numbers,  neither -1 nor +1 can legitimately be said to have a square root.  Both, though, have two-dimensional analogues and these do have square roots, not recognized as such unfortunately by the mathematics hegemony.^[3]

In the next post we will look at a comparison between imaginary numbers,  which were formulated in accordance with this misconstrual about how numbers relate to dimensions,  and probable numbers which grow organically out of a consideration of how numbers and dimensions actually relate to one another in nature.^[4]  The first of these approaches can be thought of as rational planning by a central authority; the second, as the holistic manner in which nature attends to everything, all at once, and without rational forethought.

(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] See the series of about nine posts that begins here.

[2] The Rationalists missed here a golden opportunity to relate number and dimension by defining square root much too narrowly. They seem to have been so mesmerized by their algebraic equations that they failed to pursue the search into deeper significance pertaining to essential linkages between dimension and number that intuition and imagination might have bestowed.

[3] As Shakespeare correctly pointed out, a rose by any name would smell as sweet. Plus one times plus one certainly equals plus one but that has nothing to do with actual square root really, just with algebraic linear multiplication.  Note has often been made in these pages of the difference between mathematical truth and scientific truth. Whereas mathematics demands only adherence to its axioms and consistency,  science requires empirical proof.  Mathematics defined square root in a certain manner centuries ago, and has since been devoutly consistent in its adherence to that definition.  In so doing it has preserved a cherished doctrine of mathematical truth, as though in formaldehyde.  It has also for many centuries contrived to be consistently scientifically incorrect.  The problem lies in the fact it has converted physicists and near everyone else to its own insular worldview.

[4] For an early discussion of the probable plane, potential dimensions, and probable numbers see here.

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.

© 2016 Martin Hauser

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 304-

Bootstrapping Neo-Boolean - I

(continued from here)

So yes. This is very much a work in progress. And we have strayed now as it happens  into  unfamiliar territory.  Terra incognita.  Therewill be dragons.^[1]  Dragons  are  errors.  Errors  are  dangerous,  and we must slay them.  But  all  in  good  time.  First,  we should scout out the terrain. That would be prudent.

Descartes in constructing his system of coordinates built upon the bedrockofelementary algebra and the number line. We’vepreviously called attention to the important  but mostly overlooked issue of the 1:1 congruence between number and geometric/spatial position he incorporated implicitly in the logic of his coordinates and questioned the validity of such correspondence, at least with respect to subatomic scales.

Working two centuries later but very much under the influence of Descartes’ thought,  George Boole introduced his own unique brand of algebra.  A second major influence on the development of his symbolic logic was the binary number system of Leibniz, himself influenced to a large degree by Descartes. We need to carefully follow and connect the dots here. Great advances in human cognition rarely,  if ever,  occur in isolation and seclusion. There is a fine line to tread though. If progress requires the shoulders of giants to stand on,  it is still difficult at times not to be overly influenced by those who came before.

Boole’s new logic, constructed in the wake of what by his time were firmly entrenched systematizations of thought by two of the most highly regarded philosopher mathematicians, was devised in such a manner as to conform to both of these conventions of system design.  Significant to our purposes here are the facts that first, Boolean logic echoes Cartesian convention of attributing to each and every location in geometric space a single unique number,  and second, it adheres to Leibniz’s convention of using a modulo-2 number system based on binary elements 1 and 0.^[2]

The symbolic logic systems of mandalic geometry and the I Ching do not abide by either of these conventions.  Instead they are based on what is best described as  composite dimensions with four unique truth values (or vector directions) each, ranging from -1 through two distinctive zeros (0_a; 0_b) to +1, and assignment of numbers to spatial locations through all dimensions by means of probability distributions in place of a simple and simplistic 1:1 distribution.  To accommodate these alternative conceptual concepts, we will need to expand and modify traditional Boolean logic as we have already done as regards Cartesian coordinate theory.

For starters here we should doubtless add, the mandalic form is the probability distribution through all dimensions, and the probability distributions are the mandalas.  And movement through either or both can only be accomplished by  discretized stepwise maneuvers  between different amplitudes of dimension separated by obscure quantum leaps of endless being and becoming and being and unbecoming, toward and away from  the centers and subcenters of holistic systems,  the parts of which are always aiming towards some kind of equilibrium never quite within reach. Which then makes error also a necessary aspect of reality and not simply the fearful monster we imagined.  It is error that makes achievement possible.^[3][4]

(continuedhere)

Image:Here Be Dragons Map. Detail of he Carta marina (Latin “map of the sea” or “sea map”), drawn by  Olaus Magnus  in 1527-39.  This is the earliest map of the Nordic countries that gives details and place names, by Olaus Magnus [Public domain], via Wikimedia Commons. The map was in production for 12 years.  The first copies were printed in 1539 in Venice. [Wikipedia]

Notes

[1] Mapmakers during the Age of Exploration sometimes placed the phrase “here be dragons” at the edges of their known world,  presumably to warn of the dangers lying in wait for sailing vessels  and  travelers by land who strayed too far from well-traveled routes.  Here is a list of all known historical maps on which these words appear.

[2] Or in Boole’s case, we might say,  attributing to each proposition in concept space a single truth value:  TRUE or FALSE (var YES or NO;  or, in electronics applications,  ON or OFF.)  What we have here, I believe, is in many instances a false dilemma  or the old Aristotelian dichotomy of  either/or.  Quantum physics demands and deserves better.  OK, true enough,  Boole gets around to extending possibilities  by means of multi-term propositions,  which his system can readily handle.  The question here, though,  is whether  nature  can or does  handle such similarly.  I think not.  I think it approaches the question  at a more fundamental level of reasoning and reality: at the most basic level of spacetime itself.

[3] This echoes the view of cybernetics,  a transdisciplinary approach for exploring regulatory systems, their structures, constraints, and possibilities.

Cybernetics is relevant to the study of systems, such as mechanical, physical, biological, cognitive, and social systems. Cybernetics is applicable when a system being analyzed incorporates a closed signaling loop; that is, where action by the system generates some change in its environment and that change is reflected in that system in some manner (feedback) that triggers a system change, originally referred to as a “circular causal” relationship. [Wikipedia]

[4] This entire blog and its predecessor are in some sense the chronology of a journey from the familiar shoreline into largely uncharted waters.  Hesitant at first, increasingly more daring as time has gone on and I’ve come to see  errors  to be stepping stones along the way. And there have beenmanyerrors along the way. Some I am not yet cognizant of.  But of those I am aware,  I have left most intact in spite of since being superseded by ideas superior, more correct or better formulated.  I’ve done this  because I think it  important  to  map the course  of a conceptual journey,  how the ideas evolved from A to B to C to D.  It also allows readers to participate,  to a degree,  in the thrill of an exciting adventure of mind, should they so choose. Happy travels.

© 2016 Martin Hauser

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 303-

Mandalic geometry, Cartesian coordinates and Boolean algebra: Relationships - I

(continued from here)

In attempting to understand the logic of the I Ching it is important to know the differences between ordinary algebra  and  Boolean algebra and how Boolean algebra is related to the binary number system.^[1]

In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted
1 and 0 respectively. Instead of elementary algebra where the values of the variables are  numbers,  and the  main operations  are  addition and multiplication,  the main
operations of Boolean algebra are the conjunctionand, denoted ∧, the disjunctionor, denoted ∨, and the negationnot, denoted ¬. It is thus a formalism for describing logical relations in the same way that ordinary algebra describes
numeric relations. [Wikipedia]

Whereas in elementary algebra expressions denote mainly numbers, in Boolean algebra they denote the truth values false and true. These values are represented with the bits (or binary digits), namely 0 and 1.  They do not behave like the integers  0 and 1,  for which
1 + 1 = 2, but may be identified with the elements of the two-element field GF(2), that is, integer arithmetic modulo 2,  for which 1 + 1 = 0.  Addition and multiplication then play the  Boolean roles  of  XOR  (exclusive-or)  and  AND  (conjunction)  respectively, with disjunction  x∨y  (inclusive-or)  definable as  x + y + xy. [Wikipedia]^[2]

Mandalic logic already occurs fully in the structure and manner of divinatory practice of the I Ching,  if some of it only implicitly.  Although mandalic geometry does not originate from either Boolean algebra or the Cartesian coordinate system but from the primal I Ching which predates them by millennia, it does combine and augment aspects of both of these conceptual systems. It extends Boole’s system of symbolic logic to include an additional logic value represented by the number -1.  This necessitates modification of some of Boole’s postulates and rules,  and increases their total number through introduction of some new ones.  The hexagrams or native six-dimensional mandalic coordinates of the I Ching are related to Cartesian triads composed of the numbers -1, 0, and 1,  making these two geometric systems  commensurate  by means of composite dimension,  a 6D/3D hybridization or mandalic coordination of structure and function (or space and time).^[3]

The introduction of composite dimension produces four distinct dimensional amplitudes  and  is solely responsible for the mandalic form. For anyone reading this who might be down on sacred geometry,  itself a subject which I respect and admire, let it be known that I am talking here about genuine mathematics and symbolic logic,  and my suspicion is that there is some genuine physics involved as well.

Kalachakra Mandala

The mandalic number system, then, is a quasi-modular number system, different from Leibniz’s binary number system which is fully modular.  Boole’s rule  1 AND 1 = 1  still holds true in mandalic logic.  However we must add to this the new logic rule that  -1 AND -1 = -1.  Individually the two rules are modular,  based on a clock arithmetic using a modulo-3 number system rather than Leibniz’s modulo-2 or binary number system, but with yet another added twist.

Together the two rules prescribe a compound system, one which is not singly modular but doubly modular.  The two components, yinandyang, are complementary and are inversely related to one another in this unified system.  This  logic organization  appears based on the figure 8 or sine wave and its negative,  allowing for periodicity, for recursive periods of interminably repeating duration,  and,  perhaps most importantly,  for wave interference,  of  constructive  and  destructive  varieties. These two geometric figures also engender an unexpected decussation of dimension not recognized by Western mathematics.  This is so because 1 AND -1 = 0 and  -1 AND 1 = 0.  The surprise here  is that  there are two distinct zeros: 0_a and 0_b.^[4] In two- or three-dimensional Cartesian terms there exists no difference between these two zeros.  However,  in terms of 6-dimensional aspects of mandalic geometry  and  the hexagrams of the I Ching, the two are clearly distinct structurally and functionally.^[5]

This arithmetic system is the basis of the logic encoded in the hexagrams of the I Ching. Each hexagram uniquely references a single 6- dimensional discretized point, of which there are 64 total. These 64 6- dimensional points of the mandalic cube are distributed among the 27 discretized points  of the ordinary 3-dimensional cube  through the compositing of dimensions  in such manner  that a mandala is formed which positions  1,  2,  4  or  8 hexagrams at each 3-dimensional point according to the   dimensional amplitude  of the particular point.  This necessarily creates a concurrent probability distribution of hexagrams through each of the three Cartesian dimensions.

TheI Chinguses a dual or composite three-valued logic system.  In place of truth values,  the variables used are yin,  yang  and the two in conjunction.  These fundamentally represent vector directions.  Yin is represented by -1, yang by 1, and their conjunction, using Cartesian or Western number terminology, by zero (0). This symbol does not occur natively in the I Ching though where the representation used is simply a combination of yin and yang symbols, most often in form of a bigram containing both  and  regarded as representing a composite dimension, namely 0_[1]  or  0_[2].^[6]

The two bigrams that satisfy the requirement are

young yang

for 0_[1]

and

young yin

for 0_[2].

Although mandalic logic is in Cartesian terms a 3-valued system, in native terms it is 4-valued.  It is not a simple modulo-3  or  modulo-4 number system, but two interrelated modulo-3 systems combined.  The best way to think about this geometric arrangement is possibly to view it as a single composite dimension having four distinct vector directions: a negative direction represented by mandalic composite yin (Cartesian -1); positive direction represented by mandalic composite yang (Cartesian 1); and two decussating relatively undifferentiated directions in some sort of equilibrium, represented by mandalic 0_[1] (composite yin/yang) and 0_[2] (composite yang/yin).  both of which  devolve  to  Cartesian 0  (balanced vector direction of the origin or center).^[7]

So we’ve seen that the number system used in the I Ching is not binary as Leibniz believed but instead doubly trinary with the two halves, in simplest terms,  inversely related and intertwined.  Still, it was an easy mistake to make because the notation used is binary.  We’ve seen too that all trigrams and hexagrams in the system can be rendered commensurate with the Cartesian coordinate system:  trigrams by simple transliteration, hexagrams by dimensional compositing. What, then, of George Boole and his eponymous logic?  How do they fit in the logic scheme of the I Ching? I’m glad you asked. Stay tuned to find out.

(continuedhere)

Images: Upper: TRANSFORMATION OF THE SYMBOL OF YIN (LINE split in two) AND YANG (STRAIGHT-LINE). BLEND: 4 bigrams, THEN 8 trigrams. (MORAN, E. ET AL. 2002: 77). Found here. Lower: Modified from an animation showing how the taijitu (yin-yang diagram) may be drawn using circles, then erasing half of each of the smaller circles. O'Dea at WikiCommons [CC BY-SA 3.0orGFDL],via Wikimedia Commons

Notes

[1] Boole’s algebra predated the modern developmentsinabstract algebra and  mathematical logic  but is seen as connected to the origins of both fields. Similarly to elementary algebra, the pure equational part of the theory can be formulated without regard to explicit values for the variables.

[2] If you are new to Boolean algebra these definitions may be confusing because in some ways they seem to fly in the face of ordinary algebra.  I’ll admit, I find them somewhat daunting.  Let me see if I can clarify the three examples given in this quote. Those of you more familiar with the language of Boolean algebra might kindly correct me in the event I err.  I’m growing more comfortable with being wrong at times.  And this is after all a work in progress.

• Boolean XOR (exclusive-or) allows that a statement of the form (x XOR y) is TRUE
  if either x or y is TRUE but FALSE if both are TRUE or if both are FALSE.  Since Boolean algebra uses binary numbers and represents  TRUE by 1,  FALSE by 0,  then
            for  x = TRUE,   y = TRUE    x + y = 1 + 1 = 0 ,    so FALSE
            for  x = FALSE,  y = FALSE   x + y = 0 + 0 = 0 ,  so FALSE
            for  x = TRUE,    y = FALSE   x + y = 1 + 0 = 1 ,   so TRUE
            for  x = FALSE,   y = TRUE    x + y = 0 + 1 = 1 ,   so TRUE

• Boolean AND (conjunction) allows that a statement of the form (x AND y) is TRUE
  only if both x is TRUE and y is TRUE. If either x or y is FALSE or both are FALSE
  then x AND y is FALSE. Here algebraic multiplication of binary 1s and 0s plays the
  role of Boolean AND. (Incidentally, binary multiplication works exactly the same
  way as algebraic multiplication. There’s a gift!)
            for  x = TRUE,    y = TRUE      xy  =  1(1) = 1,    so TRUE
            for  x = FALSE,   y = FALSE     xy = 0(0) = 0,   so FALSE
            for  x = TRUE,    y = FALSE      xy = 1(0) = 0 ,  so FALSE
            for  x = FALSE,    y = TRUE      xy = 0(1) = 0 ,  so FALSE

• Boolean OR (inclusive-or) is the truth-functional operator of (inclusive) disjunction,
  also known as alternation. The OR of a set of operands is true if and only if one or
  more of its operands is true. The logical connective that represents this operator is
  generally written as ∨ or +. As stated in the Wikipedia article logical disjunction x∨y
  (inclusive-or) is definable as x + y + xy [(x OR y) OR (x AND y)] as shown below.
  [Note: x AND y is often written xy in Boolean algebra. So watch out whichalgebra
  is being referred to, ordinary or Boolean. Are we confused yet?]
            for  x = TRUE,    y = TRUE      x + y = 1 , xy = 1 ,    so TRUE
            for  x = FALSE,   y = FALSE     x + y = 0 , xy = 0 ,   so FALSE
            for  x = TRUE,     y = FALSE     x + y = 1 , xy = 0 ,   so TRUE
            for  x = FALSE,    y = TRUE      x + y = 1 , xy = 0 ,   so TRUE
  

[3] Fundamentally, though,  the  coordinates of mandalic geometry  refer to vector directions alone, rather than to both vectors and scalars (or direction and magnitude) as do Cartesian coordinates. Yin specifies actually the entire domain of negative numbers rather than just the scalar value -1. Yang similarly refers to the entire domain of positive numbers rather than the scalar value 1 alone. Their conjunction  through the compositing of dimensions,  though represented by the symbol zero (0)  in the format commensurate with Cartesian coordinates,  refers actually to a  state or condition  not found in Western thought  outside of certain forms of mysticism  and other outsider philosophies like alchemy;  equilibration of forces in physics; equilibrium reactions in chemistry; and the kindred concept of homeostasis mechanisms of living organisms found in biology.

[4] This is to Westerners counterintuitive. Our customary logic and arithmetic allows for but a single zero. That two different zeros might exist concurrently or consecutively is - to our minds - irrational and we wrestle mightily with the idea. To complicate matters still more,  neither of these zeros is  conveniently  like our familiar Western zero.  So which should win out here?  Rationality or reality?  In fact,  the decision is not ours.  In the end nature decides.  Nature always decides. It stuffs the ballot box  and  casts the deciding vote much to our chagrin,  leaving us powerless to contradict what we may interpret as a whim. Our votes count for bupkis.

[5] This calls to mind also the Möbius strip which involves a twist that looks very much like a decussation to me.  The decussation or  twist in space  we are talking about here though has a sort of wormhole at its center that connects two contiguous dimensional amplitudes. I can’t say more about this just now. I need to think on it still. It seems a promising subject for reflection. (1,2,3)

[6] It needs to be pointed out here that in mandalic geometry, and similarly in the primal I Ching as well,  a bigram can be formed from any two related Lines of  hexagrams,  trigrams,  and tetragrams. The two Lines need not be (and often are not) adjacent to one another. I would think such versatility might well prove useful for modeling and mapping quantum states and interactions.

[7] Note that yin and yang in composite dimension can each take the absolute values 0, 1, and 2  but when yin has absolute value 2, yang has absolute value 0; when yang has absolute value 2,  yin has absolute value 0.  This inverse relation in fact is what makes the arrangement here a superimposed, actually interwoven, dual modulo-3 number system. It also makes the center points of mandalic lines,squares,  and cubes  more protean and less differentiated  than their vertices and elicits the different amplitudes of dimension.

The composite dimension value at the origin points(centers) of all of these geometric figures is  always  zero  in  Cartesian  terms  since the values of the differing Lines  in  the  two entangled 6-dimensional hexagrams  located here add to zero. But neither of these 6-dimensional entities is in its ground state at the center.  Both  have absolute value 1  at Cartesian 0.  Let me say that again: composite dimension values at the center or origin are zero in Cartesian terms but the values of both individual constituents are non-zero.Yin is in its ground state when yang is at its maximum and vice versa. At the center, since the two are equal and opposite they interfere destructively. This results in a composite zero ground state.

So from the perspective of  Cartesian coordinate dynamics, which is after all the customary perspective in our subjective lives,  we encounter only emptiness. But it is this very emptiness that opens to a new dimension. In the hybrid 6D/3D mandalic cube  only line centers and the cube center  have direct access through change of one dimension to face centers and only the face centers have a similar direct access through a single dimension to the cube center and edge centers. All coexist in an ongoing harmony of tensegrity. There is method to all this madness then.

© 2016 Martin Hauser

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 302-

Mandalic Line Segments,
Entanglement and Quantum Gravity
Part I

(continued from here)

We are going to consider once again now geometric line segments of mandalic geometry  and  their relation to Cartesian line segments and the Western number line. Yes,  this is sort of a detour from what I stated we would look at next. But this is not unrelated and lies at the very heart of mandalic geometry, and I’m not yet ready to address what I projected in the last remark of my previous post.

I keep returning to this subject because of its extreme importance. Beyond its significance to understanding the logic encoded in mandalic geometry and the I Ching, I believe it may also hold the key to quantum entanglement and quantum gravity.  Despite the fact that mandalic line segments are really fundamentally mental constructs,  a fiction of sorts, it is still important to understand how they are composed and how their components interact.  Though they may themselves be fictions,  the line segments and the points that compose them do in fact map a number of physical entities,  realities that may be related to quantum numbers and quantum particles and states.

When Descartes invented his coordinate system, with its points and line segments,  he based his system on the number line extended to two or three dimensions. In modeling it on the number line the space he described was imagined to bear a  necessary  one to one correspondence to the real numbers.^[1]  However this  1:1 mapping  of geometric space to the real numbers was a premise implicitly assumed by Descartes.  It was in fact axiomatic,^[2]  but apparently not stated as such.^[3]  As a result, the presumed relation has become a blind spot^[4] in Western thought,  never proved nor disproved, at least not at subatomic scales.^[5]

Neither mandalic geometry nor the primal I Ching make such an assumption. In place of Descartes’ 1:1 correspondence of geometric space and the numbers on the number line, we find a mandalic arrangement in which there are different categories of spatial location which can host one or more discrete numbers in a probabilistic manner.  This creates various dimensional amplitudes and a multidimensional waveform of component entities.^[6]

To my mind these characteristics of the mandalic coordinate system in combination with others described elsewhere make it more relevant to investigation and interpretation of many quantum phenomena which are as yet poorly understood than Cartesian coordinate dynamics may be and without need for recourse to imaginary numbers and complex plane.

(continuedhere)

Image: 6 steps of the Sierpinski carpet, animated. By KarocksOrkav (Own work) [CC BY-SA 3.0],via Wikimedia Commons

Notes

[1] Real numbers are numbers that can be found on the number line. This includes both the rational and irrational numbers.

[2] That is to say, taken for granted as self-evident.

[3] See Note [4] here.

[4] We have lived with this unproved premise so long that we no longer even question it,  or imagine that there might be an alternative which conforms better to reality at certain scales, notably subatomic scales.  The I Ching also seems to suggest  that a complete true description of complex relationships that involve a large number of dimensions,  including complex societal relationships,  requires more than a simple 1:1 correspondence between the notational symbols involved and the realities they represent.

[5] And from what I can see, no one seems to have much interest in proving or disproving this assumption.

[6] When speaking about hexagrams the number of dimensions involved is six as each Line of the hexagram encodes a value for a single distinct dimension in a 6-dimensional space.  In a hybrid 6D/3D compositing of dimensions though, two such Lines in relation reference a single Cartesian dimension in 2- or 3-space.  A concept not to be missed here is that  interactions of quantum particles  may well involve such  integration of dimension,  of dimensions  we are not even aware of beyond the unsettling fact  they upset the neat applecart of customary conceptual categories.

© 2016 Martin Hauser

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 301-

Beyond Taoism - Part 5
A Vector-based Probabilistic
Number System
Part II

(continued from here)

Taoism and the primordial I Chingare in agreement that temporal changes have two different aspects: sequent and cyclic.  Western thought in general follows suit. The I Ching differs from the other two in asserting that  the direction of change - for both sequent and cyclic change - is fully reversible,  with the proviso  that sufficiently small units of measurement are involved.^[1]  The probability that reversal can be achieved  diminishes proportionately to the magnitude of change that has taken place.^[2]

Taoist appropriation of bigrams and trigrams of the I Ching to model such phenomena as change of seasons and phases of the moon  is plausible if not quite legitimate. The natural phenomena so modeled are macroscopic and vary continuouslyandinexorably throughout an ever-repeating cyclic spectrum. And there’s the rub.

As they occur and function in the I Ching bigrams and trigrams are dicontinuous discrete elements,  formed by other similarly discontinuous discretized entities,  and they follow evolutionary courses which are most often nonrepetitive. So the Taoist usage is misleading at best, annihilative at worst. Unfortunately, as the I Ching itself evolved through centuries of commentaries and reinterpretations,  it became  ever more contaminated and tainted by these Taoist corruptions of meaning, at the same time that it was being inundated by  Confucian sociological and ethical reworkings.  What we have today is an amalgam, the various parts of which do not sit well with one another.^[3]

Though it may in part be hyperbole to prove a point,  the stark difference between the two approaches,  that of Taoism and that of the I Ching, is epitomized by comparison of the Taoist diagram of the cycle of seasons with diagrams at the top and bottom of the page,  which are based on  the  number,  logic,  and coordinate systems of   The Book of Changes.^[4] The increased complexity of the latter diagrams should not prove a stumbling block, as they can be readily understood in time with focus and attention to detail.  The  important take-away  for now is that in the I Ching bigrams  exist within a larger dimensional context  than the Taoist diagram avows,  and this context makes all their interactions more variable,  conditional,  and complex. As well, the same can be said of trigrams and hexagrams.

One of the more important aspects of these differences has to do with the notion of equipotentiality.  As bigrams and trigrams function within  higher dimensional contexts  in the  I Ching,  this introduces a possibility of multiple alternative paths of movement and directions of change.  Put another way,  primordial I Ching logic encompasses many more  degrees of freedom  than does the logic of Taoism.^[5]  There is no one direction or path  invariably decreed or favored.  An all-important element of conditionality prevails.  And that might be the origin of what quantum mechanics has interpreted as indeterminism or chance.

Next up, a closer look at equipotentiality and its further implications.

Section F_H(n)^[6]

(continuedhere)

Notes

[1] There are exceptions. Taoist alchemy describes existence of certain changes that admit reversibility under special circumstances.  Other than the Second Law of Thermodynamics (which is macroscopic in origin,  not result of any internally irreversible microscopic properties of the bodies), the laws of physics neglect all distinction between forward-moving timeandbackward-moving time. Chemistry recognizes existence of certain states of equilibrium in which the rates of change in both directions are equal. Other exceptions likely occur as well.

[2] Since change is quantized in the I Ching, which is to say, it is divided into small discretized units,  which Line changes model,  the magnitude of change is determined by the number of Line changes that have occurred  between Point A and Point B in spacetime.  Reversal is far easier to achieve if only a single Line change has occurred than if three or four Lines have changed for example.

[3] Ironically, Taoism itself has pointed out the perils of popularity. Had the I Ching been less popular, less appealing to members of all strata of society, it would have traveled through time more intact.  Unless,  of course,  it ended up buried or burned. What is fortunate here is that much of the primordial logic of the I Ching can be reconstructed by focusing our attention on the diagrammatic figures and ignoring most of the attached commentary.

[4] These diagrams do not occur explicitly in the I Ching. The logic they are based on, though, is fully present implicitly in the diagramatic structural forms of hexagrams, trigrams, and bigrams and the manner of their usage in  I Ching divinatory practices.

[5] Or, for that matter, than does the logic of Cartesian coordinate space if we take into account the degrees of freedom of six dimensional hexagrams mapped by composite dimensional methodology to model mandalic space. (See Note [4] here for important related remarks.)

[6] This is the closest frontal section to the viewer through the 3-dimensional cube using Taoist notation.  See here for further explanation.  Keep in mind this graph barely hints at the complexity of relationships found in the 6-dimensional hypercube which has in total 4096 distinct changing and unchanging hexagrams in contrast to the 16 changing and unchanging trigrams we see here. Though this model may be simple by comparison,  it will nevertheless serve us well as a key to deciphering the number system on which I Ching logic is based as well as the structure and context of the geometric line that can be derived by application of reductionist thought to the associated mandalic coordinate system of the I Ching hexagrams. We will refer back to this figure for that purpose in the near future.

© 2016 Martin Hauser

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 300-

Beyond Taoism - Part 4
A Vector-based Probabilistic
Number System
Introduction

(continued from here)

Leibniz erred in concluding the hexagrams of the I Ching were based on a number system related to his own  binary number system.  He had a brilliant mind but was just as fallible as the rest of us.  He interpreted the I Ching in terms of his own thought forms,  and he saw the hexagrams as a foreshadowing of his own binary arithmetic.^[1]

So in considering the hexagram Receptive,  Leibniz understood the number 0; in the hexagram Return, the number 1; in the hexagram Army, the number 2; in the hexagram Approach, the number 3;  in the hexagram Modesty,  the number 4;  in the hexagram  Darkening of the Light, the number 5;  and so on, up to the hexagram Creative, in which he saw the number 63.^[2]  His error is perhaps excusable in light of the fact that the Taoists, though much closer to the origin of the I Ching in time, themselves misinterpreted the number system it was based on.^[3]

From our Western perspectiveI Ching hexagrams are composed of trigrams, tetragrams, bigrams, and ultimately yinandyang Lines. From the native perspective of the I Ching this order of arrangement is putting the cart before the horse.  Dimensions  and their interactions  are,  in the view of I Ching philosophy and mandalic geometry,  antecedent logically and materially to any cognitive parts we may abstract from them. Taoism in certain contexts has abstracted the parts and caused them to appear as if primary. It has the right to do so if creating its own philosophy,  but not as interpretation of the logic of the I Ching. It is a fallacy if so intended.^[4]

The Taoists borrowed from the I Ching two-dimensional numbers, treated them as one-dimensional and based their quasi-modular number system on  the dimension-deficient result.  This is the way they arrived at their seasonal cycle consisting of bigrams:   old yin (Winter),  young yang (Spring), old yang (Summer), young yin (Autumn), old yin (Winter),  and so forth. This represents a very much impoverished and impaired version of the original configuration in the primal strata of the I Ching.^[5]

The number system of the I Ching is not a linear one-dimensional number system like  the positional decimal number system  of the West; nor is it like the positional binary number system invented by Leibniz. It is not even like the quasi-modular number system of Taoism.  The key to the number system of the hexagrams is located not in the 64 unchanging explicit hexagrams,  but rather in the changing implicit hexagrams found only in the divination practice associated with the I Ching. These number 4032.^[6]  The manner in which these operate,  however,  is actually  fairly simple and is uniform throughout the system.  So once understood,  they can be safely relegated to the implicit background, coming into play only during procedures involving divination or in attempts to understand the system fully, logically and materially.  When dealing with more ordinary circumstances just the 64 more stable hexagrams need be attended to in a direct and explicit manner.

The Taoist sequence of bigrams is in fact a corruption of the far richer asequential multidimensional arrangement of bigrams that occurs in I Ching hexagrams and divination. There we see that change can occur from any one of the four stable bigrams to any other.  If this is so then no single sequence can do justice to the total number possible. The ordering of bigrams presented by Taoism is just one of many that make up the real worlds of nature and humankind.  Taoism imparts special significance to this sequence; the primal I Ching does not. It views all possible pathways of change as equally likely.^[7]

Next time around we will look further into the implications of this equipotentiality and see how it plays out in regard to the number system of the I Ching.

Section F_H(n)^[8]

(continuedhere)

Notes

[1] By equating yang with 1 and yin with 0 it is possibletosequence the 64 I Ching hexagrams according to binary numbers 0 through 63.  The mere fact that this is possible does not, however, mean that this was intended at the time the hexagrams were originally formulated. Unfortunately, this arrangement of hexagrams seems to have been the only one of which Leibniz had knowledge. This sequence was, in fact, the creation of the Chinese philosopher Shao Yong (1011–1077). It did not exist in human mentation prior to the 11th century CE.

This arrangement was set down by the Song dynasty philosopher Shao Yong (1011–1077 CE), six
centuries before Wilhelm Leibniz described binary notation. Leibniz published ‘De progressione
dyadica’ in 1679. In 1701 the Jesuit Joachim Bouvet wrote to him enclosing a copy of Shao Yong’s 'Xiantian cixu’ (Before Heaven sequence). [Source]

Note also that the author of Calling crane in the shade, the source quoted above, calls attention to confusion that exists about whether the “true binary sequence of hexagrams” should begin with the lowest line as the least significant bit (LSB) or the highest line. He points out that the Fuxi sequence as transmitted by Shao Yong in both circular and square diagrams takes the highest line as the LSB, although in fact it would make more sense in consideration of how the hexagram form is interpreted to take the lowest line as the LSB. My thinking is that either Shao Yong misinterpreted the usage of hexagram form or, more likely, the conventional interpretation of the Shao Yong diagrams is incorrect. Here I have chosen to use the lowest line of the hexagram as the LSB,  and I think it possible  Leibniz may have done the same.

If one considers the circular Shao Yong diagram,  the easier of the two to follow,  one can reconstruct the binary sequence,  with the lowest line as LSB,  by beginning with the hexagram EARTH at the center lower right half of the circle, reading all hexagrams from outside line (bottom) to inside line (top),  progressing counterclockwise to  MOUNTAIN over WIND at top center, then jumping to hexagram  MOUNTAIN over EARTH  bottom center of left half of the circle,  and progressing clockwise to hexagram  HEAVEN  at top center.  Of the two,  this is the interpretation that makes the more sense to me and the one I have followed here, despite the fact that it is not the received traditional interpretation of the Shao Yong sequence. Historical transmissions have not infrequently erred. Admittedly it is difficult to decipher all Lines of some of the hexagrams  in the copy Leibniz received due to passage of time and its effects on paper and ink.  Time is not kind to ink and paper, nor for that matter to flesh and products of intellect.

In the final analysis, which of the two described interpretations is the better is moot because neither conforms to the logic of the I Ching which is not binary to begin with. Moreover,  there is a third interpretation of the Shao Yong sequence that is superior to either described here.  It is not binary-based.  And why should it be? After all the Fuxi trigram sequence  which Shao Yong took as model for his hexagram sequence  is itself not binary-based. Perhaps we’ll consider that interpretation somewhere down the road. For now, the main take-away is that Leibniz, in his biased interpretation of the I Ching hexagrams made one huge mistake.  Ironically,  had he not some 22 years prior already invented  binary arithmetic, this error likely would have led him to invent it.  It was “in the cards” as they say. At least in certain probable worlds.

[2]ReceptiveandCreative are alternative names for the hexagrams EarthandHeaven, respectively. The sequence detailed can be continued ad infinitum using yin-yang notation, though of course this takes us beyond the realm of hexagrams into what would be, for mandalic geometry and logistics of the I Ching, domains of dimensions numbering more than six.  Keep in mind here though that Leibniz was not thinking in terms of dimension but an  alternative method  of expressing the prevalent base 10 positional number system notation of the West.  He held in his grasp the key to unlocking an even greater treasure but apparently never once saw that was so.  This seems strange considering his broadly diversified interests and pursuits in the fields of  mathematics,  physics,  symbolic logic,  information science,  combinatorics,  and in the nature of space.  Moreover,  his concern with these was not just as separate subjects of investigation.  He envisaged uniting all of them in a  universal language  capable of expressing mathematical,  scientific, and metaphysical concepts.

[3] Earlier in this blog I have too often confused Taoism with pre-Taoism. The earliest strata of the I Ching belong to an age that preceded Taoism by centuries, if not millennia.  Though Taoism was largely based on the philosophy and logic of the I Ching,  it didn’t always interpret source materials correctly,  or possibly at times it intentionally used source materials in new ways largely foreign to the originals. The number system of the I Ching is a case in point.

In the interest of full disclosure, I am not an expert in the history or philosophy of Taoism.  Taoist philosophies are diverse and extensive. No one has a complete set or grasp of all the thoughts, practices and techniques of Taoism. The two core Taoist texts, the  Tao Te ChingandChuang-tzu,   provide the philosophical basis of Taoism which derives from the eight trigrams (bagua) of Fu Xi, c. 2700 BCE, the various combinations of which created the 64 hexagrams documented in the I Ching.  The Daozang,  also referred to as  the Taoist canon,  consists of around 1,400 texts that were collected c. 400, long after the two classic texts mentioned. What I describe as Taoist thought then is abstracted in some manner from a huge compilation, parts of which may well differ from what is presented here. Similar effects of time and history can be discerned in Buddhism, Christianity, Islam and secular schools of thought like Platonism,Aristotelianism,Humanism, etc.

[4] Recent advances in the sciences have begun to raise new ideas regarding the structure of reality. Many of these have parallels in Eastern thought.  There has been a shift away from the reductionist view in which things are explained by breaking them down then looking at their component parts, towards a more holistic view. Quantum physics notably has changed the way reality is viewed. There are no certainties at a quantum level, and the experimenter is necessarily part of the experiment. In this new view of nature everything is linked and man is himself one of the linkages.

[5] It is not so much that this is incorrect as that it isextremelylimiting with respect to the capacities of the I Ching hexagrams.  A special case has here been turned into a generalization that purports to cover all bases. This may serve well enough within the confines of Taoism but it comes nowhere near elaborating the number system native to the I Ching. We would be generous in describing it as a watered down version of a far more complex whole.  Through the centuries both Confucianism and Taoism  restructured the I Ching to make it conducive to their own purposes.  They edited it and revised it repeatedly,  generating commentary after commentary,  which were admixed with the original,  so that the I Ching as we have it today,  the I Ching of tradition,  is a hodgepodge of many convictions and many opinions. This makes the quest for the original features of the I Ching somewhat akin to an archaeological dig.  I find it not all that surprising  that the oracular methodology of consulting the I Ching  holds possibly greater promise in this endeavor than the written text.  The  early oral traditions  were preserved better,  I think,  by the uneducated masses who used the I Ching as their tool for divination than by philosophers and scholars who,  in their writings,  played too often a game of one-upmanship with the original.

[6] A Line can be either yin or yang, changing or unchanging. Then there are four possible Line types and six Lines to a hexagram.  This gives a total of 4096 changing and unchanging hexagrams (4⁶ = 4096). Since there are 64 unchanging hexagrams (2⁶ = 64) there must be 4032 changing hexagrams (4096-64 = 4032).

[7] This calls to mind the path integral formulation of quantum mechanics which was developed in its complete form by Richard Feynman in 1948. See, for example, this description of the path integral formulation in context of the double-slit experiment, the quintessential experiment of quantum mechanics.

[8] This is the closest frontal section to the viewer through the 3-dimensional cube using Taoist notation.  See here for further explanation.  Keep in mind this graph barely hints at the complexity of relationships found in the 6-dimensional hypercube which has in total 4096 distinct changing and unchanging hexagrams in contrast to the 16 changing and unchanging trigrams we see here. Though this model may be simple by comparison,  it will nevertheless serve us well as a key to deciphering the number system on which I Ching logic is based as well as the structure and context of the geometric line that can be derived by application of reductionist thought to the associated mandalic coordinate system of the I Ching hexagrams. We will refer back to this figure for that purpose in the near future.

© 2016 Martin Hauser

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 299-

Beyond Taoism - Part 1
A Lost Logic of Chinese Antiquity

The 64 Hexagrams of the I Ching
(for interactive version click here)

(continued from here)

In speaking of “Taoist thought” I have often throughout this work used the term as a convenient shorthand for “primeval Chinese thought.” Strictly speaking, this usage is historically incorrect. Laozi is traditionally regarded as the founder of Taoism and is associated with “primordial” or “original” Taoism. Whether he actually existed is disputed.  According to tradition the classic text attributed to him, the Tao Te Ching, was written around the 6th century BCE. The oldest extant text, however, dates to the late 4th century BCE. The earliest strata of the I Ching predate both these historical periods by many centuries, if not millennia.  Taoism derived its cosmological notions from the philosophy of yin and yang,  and from that of the  Five Phases  or  Five Elements. Both these schools of thought were overwhelmingly  influenced and shaped  by exposure to the oldest known text of ancient Chinese classics, the I Ching.^[1]

The actual symbolic logic of Taoism,  although derived from the I Ching is extremely simplistic compared with that of the original upon which it is based. Whereas the philosophy of yinandyang as presented in the Tao Te Ching comprises little more than a two-dimensional cycle of two-valued elements,  in the I Ching these two represent vectors in a six-dimensional combinatorial manifold of 64 hexagrams (1,2). Clearly, it is a difference like that  between night and day.  It is,  in fact,  a literal comparing of 2² with 2⁶, the latter holding many more possibilities. The actual difference^[2]  in the  logic and geometry  emerging from the two is greater even than it appears at first. It eventuates not from just a simple geometric progression but from a mandalic intertwining and association of logical elements that give rise to different amplitudes of dimension as well as to a greater number of dimensions.  This mandalic interweaving leads also to a richer catalogue of relationship types.^[3]

Long viewed as mainly an ancient text of Chinese divination,theI Chingencompasses many more categories of thought - - - among them symbolic logic, geometry, and combinatorics.  As a treatise which deals with combinatorics alone, it soars without equal, the first known compendium of combinatorial elements and still one of the finest. The logic and geometry  that are embedded in the  hexagram system  of the I Ching are best understood in terms of dimensions and vectors akin to those in Cartesian systematics, and of logic gates analogous to the truth tables of Boolean algebra. And still the cognoscente will want to explore beyond the pertinency of these disciplines as also beyond Taoism to find the full meaning and intent of the I Ching.^[4]

Having existed for millenia,  and itself a treatise regarding change^[5] in its many aspects, it would be inconceivable that the I Ching as we have it today is as it was in its beginnings. Popular at all societal levels through its entire existence,  reinterpretations and reworkings  have been myriad. Confucianism in particular interlaced its own brand of philosophical and “ethical-sociopolitical teachings”  during and after  the fifth century BCE. Other schools of thought added their unique perspectives to what became essentially  a massive melting pot of schematization,  one not always self-consistent by any means.

When one attempts to uncover the original face of the I Ching the difficulties encountered soon appear insurmountable. If involved in such a venture,  it is imperative to bear in mind the bedrock strata of the work were in some ways more ingenuous, and in some more intricate, than the traditional version that has come down to us.  The earliest layers arose in context of a preliterate oral tradition with all the many unique aspects of being that entails. In some ways the golden age of the I Ching ended with coming of the written word and literacy. The multidimensional logic that was readily accommodated by an oral tradition foundered and eventually was all but lost in the unrelenting techno-sociological onslaught of script with its associated inevitable linearity. Anyone who hopes to excavate the buried multidimensional logic of the primordial I Ching can expect to do a good deal of laborious digging.

(continuedhere)

Image:Source. Originally from Richard Wilhelm’s and Cary F. Baynes translation “I Ching: Or, Book of Changes” [3rd. ed., Bollingen Series XIX, (Princeton NJ: Princeton University Press, 1967, 1st ed. 1950)]

Notes

[1] Two diagrams known as bagua (or pa kua) that figure prominently in the I Ching and its Commentaries predate their appearances in the I Ching. The Lo Shu Square is sometimes associated with the  Later Heaven arrangement  of the bagua or trigrams, and the  Yellow River Map  is sometimes associated with the Earlier Heaven arrangement of trigrams. Both are linked to astronomical events of the sixth millennium BCE. Although part of Chinese mythology, they played an important role in development of Chinese philosophy.  The Lo Shu Square is intimately connected with the legacy of the most ancient Chinese mathematical and divinatory traditions.  The Lo Shu is the  unique normal magic square (1,2) of order three (every normal magic square of order three is derived from Lo Shu by rotation or reflection). [Wikipedia]

[2] Taking into account both changing lines and unchanging lines of hexagrams there are four possible variants for each line:  unchanging yin,  unchanging yang, changing yin,  and changing yang.  This results in a total of  4⁶  or 4096  possible different line combinations for each six-line figure.  This allows for an enormous number of logical / geometric configurations,  all of which map to various points of the mandalic cube or, in terms of  Cartesian coordinates,  to discretized points of the three-dimensional cube bounded by  the eight Cartesian triads which have coordinates of  +1  and/or  -1  in all possible combinations (corresponding to the eight trigrams.)

To this point changing lines have not been discussed to avoid overcomplicating already complicated matters too soon.  Changing lines play an indispensable role in all changes of yin lines to yang lines and vice versa,  and therefore, in changes of one hexagram to another.  They are also essential elements in formation of the geometric line segment generated by the I Ching hexagrams which I have earlier referred to as the  "Taoist line“  and which we have yet to broach fully. Mandalic line segments uniformly comprise sixteen interrelated elements,  hexagrams with changing and/or unchanging lines.  Though various mandalic line segments have different compositions in terms of six-dimensional hexagrams,  these hexagrams can always be reduced in logical and geometrical terms to  sixteen bigram forms containing changing and/or unchanging lines. These bigram sets are all identical. No other variants are possible, since 4² equals 16. In this sense there is a single species of mandalic line segment but one which takes on different characteristics in different dimensional contexts.  Every hexagram has a commentary appended to each of its six lines,  which annotation is intended to be reflected upon only if the line is a changing one at time of consulting the oracle. Justly put, this system is brilliant beyond belief.

[3] Understand here that ‘relationship types’ may variously refer to human relationships in a society, to particle relationships in context of the atom, or to any other species of relationship one might imagine.

[4] For an exhaustive listing of linkstoI Ching related materials on the Web see here.

[5] Indeed, an alternative name of the I Ching in English is Book of Changes. The ensconced multidimensional logic encoded in the original work purports to be a microcosm describing all possible pathways of change, and their incessant changing relationships in the greater macrocosm of the universe.

© 2015 Martin Hauser

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 295-

Beyond Boole - Part 1
Symbolic Logic for the 21st Century

Boolean Algebra:
Fundamental Operations

(continued from here)

Looking back on how we arrived at this stage of reconstruction of Western thought,  I see the difficulty arose in attempting to explain the “missing zero” of Taoism. Blame our troubles on Leibniz. It was he who introduced binary numbers to the West,  and made the fateful choice of using zero(0) instead of -1 to counter with +1.  Leibniz knew full well of the I Ching, but did not understand it well. He missed the point, seeing in it only a resemblance to his own newly devised system of numbers.

By Leibniz’s time negative numbers were firmly entrenched in the European mind.  Why did  Leibniz  ignore them completely?  In doing so he blazed a new trail that led eventually to the digital revolution of recent times. It also led to a dead end in the history of Western thought, one the West has not yet come fully face to face with. It will, though. Give it a few more years.^[1]

George Boole, the inventor of what we know today as Boolean logic or Boolean algebra, was one of the thinkers who followed in the footsteps of Leibniz, building on the trail he blazed.^[2]  When he came to devise his truth tables,  he also chose zero(0) as the counterpart to one(1).  This led to certain resounding successes.  And ultimately,  to certain failures  that introduced yet another layer to the  blind spot  of Western symbolic logic. Here we are, almost two centuries later,^[3] saddled with and hampered by the unfortunate fallout of that eventful decision still.^[4]

Most arguments in elementary algebra denote numbers. However, in Boolean algebra, they denote  truth values  falseandtrue.  Convention has decreed these values are represented with the  bits (or binary digits), namely 0 and 1.  They do not behave like the integers 0 and 1 though, for which 1 + 1 = 2,  but are identified with the elements of the  two-element field GF(2), that is, integer arithmetic modulo 2, for which 1 + 1 = 0. (1,2) This causes a substantial problem when we attempt correlation of Taoist logic and Boolean logic. As we will soon discover, Taoist logic is a hybrid logic that is based on both vector inversion and arithmetic modulo 2.  As such,  it ought prove relatable to both Cartesian coordinates and Boolean algebra, though it may necessitate “forcing a larger foot in a smaller glass slipper.”

Taoism chose ages ago to use ‘yin’ and 'yang’ as its logical symbols. Although this appears, at first, to be a binary system, like those of Leibniz and Boole, on closer inspection it proves not to be.  It is one of far greater logical complexity, alternatively binary or ternary with intermediate third element understood. This implied third element is able to bestow balance and equilibrium throughout all of the Taoist logical system.  This is where the 'missing zero’ of Taoism went.  Only it is a very different zero than the 'zero’ of Western thought.  It is a zero of infinite potential rather than one of absolute emptiness.  It is a  zero  of  continual beginnings and endings, not of finality. It is one of the things that make the I Ching totally unique in the history of human cognition.  All these hidden zeros are wormholes between dimensions and between different amplitudes of dimension.

So where does this all lead to, then? We’ve seen that the Taoist 'yin’ can readily be made commensurate with 'minus 1’ of Western arithmetic, the number line,  and  Cartesian coordinates.^[5]  But if it is to remain true to Taoist logic,  it cannot be made commensurate with the Western 'zero’. We’ve found the Taoist number system and geometry to be Cartesian-like but not Cartesian. Now we discover them to be Boolean-like, not Boolean. Sorry, Leibniz,  they are not so much as remotely like your binary system. You were far too quick to disesteem the unique qualities of the I Ching.^[6]

This all has far-reaching consequences for Western thought in general. Especially though, for symbolic logic, mathematics, and physics. More specifically for our purposes here it means that when we create our Taoist notation transliteration of Cartesian coordinates, we will need also to translate Boolean logic into terms compatible with Taoist thought, that is to say, from a two-value system based on '1s’ and '0s’ into a three-value system based on '1s’, ’-1s’, and the ever-elusive invisible balancing-act '0s’ of Taoism.^[7] We turn to that undertaking next.

(continuedhere)

Image: Fundamental operations of Boolean algebra.  Symbolic Logic, Boolean Algebra and the Design of Digital Systems. By the Technical Staff of Computer Control Company, Inc.  Other logical operations exist and are found useful by non-engineer logicians.  However, these can always be derived from the three shown. These three are most readily implementable by electronic means. The digital engineer, therefore,  is usually concerned only with these fundamental operations of conjunction, disjunction, and negation.

Notes

[1] It is at times like this that I am thankful I am not a member of Academia. Were I so, I could not afford, from a practical standpoint, to make claims such as this. Tenure notwithstanding.

[2] A knowledge of the binary number system is an important adjunct to an understanding of the fundamentals of Symbolic Logic.

[3] If we look back far enough in time, it was the introduction of “zero” as a number and a philosophical concept that led us down this tangled garden path, though the history of human thought is nothing if not interesting.

[4] Far out speculative thought here:  Were binary numbers and Boolean logic based on +1s and -1s instead of +1s and 0s,  might it not be possible to construct today a software-based quantum computer requiring no fancy juxtapositions and superpositions of subatomic particles?  Think on it for a while before dismissing the thought as irrational folly.

[5] More correctly expressed, it can be made commensurate with the domain of negative numbers, since it is a vector symbol, properly speaking, concerned only with direction, not magnitude.

[6] Unfortunately there is still little understanding of the true nature of the symbolic logic encoded in the I Ching, as exemplified by this quote:

The I Ching dates from the 9th century BC in China. The binary notation in the
I Ching is used to interpret its quaternary divination technique.

It is based on taoistic duality of yin and yang.Eight trigrams (Bagua) and a set of 64 hexagrams (“sixty-four” gua), analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China.

The contemporary scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, although he did not intend his arrangement to be used mathematically. Viewing the least significant bit on top of single hexagrams in Shao Yong’s square and reading along rows either from bottom right to top left with solid lines as 0 and broken lines as 1 or from top left to bottom right with solid lines
as 1 and broken lines as 0 hexagrams can be interpreted as sequence from 0 to 63.

[Wikipedia]

It was this Shao Yong sequence of hexagrams (Before Heaven sequence) that Leibniz viewed six centuries after the Chinese scholar created it, so maybe he can be forgiven his error after all.

The more significant point here might be that an important  Neo-Confucian philosopher, cosmologist, poet, and historian of the 11th century either was no longer able to access the original logic and meaning of the I Ching or, at the very least, was hellbent on reinterpreting it in a manner contradictory to its original intent.  The latter is a distinct possibility,  as Neo-Confucianism was an attempt to create a more rationalist secular form of Confucianism by rejecting superstitious and mystical elements of  Taoism and Buddhism that had influenced Confucianism since the Han Dynasty (206 BC–220 AD).

[7] Taoist logic and mandalic geometry share some of the characteristics of both Cartesian coordinates and Boolean logic,  but not all of either.  Descartes’ system is indeed a ternary one when viewed in terms of vector direction rather than scalar magnitude. That fits with the requirements of Taoist logic.  It is, on the other hand, dimension-poor,  as Taoist logic and geometry require a full six independent dimensions for execution.  Boolean logic lacks the necessary third logical element -1, which causes inversion through a central point of mediation. But we shall see, it does bestow the ability to enter and exit a greater number of dimensional levels by means of its logical gates. Used together in an appropriate manner, these two can provide a key to understanding Taoist logic and geometry. Speculating even further, Taoist thought might provide a key to interpretation of quantum mechanics, the same quantum mechanics devised in the early twentieth century that no one can yet explain. Well,  I mean, actually,  Taoist thought in the formulation given it by mandalic geometry.  Why feign modesty, when this work will likely linger in near-total obscurity for the next hundred years gathering dust or whatever it is that pixels gather in darkness undisturbed.

© 2015 Martin Hauser

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 294-

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

(continued from here)

My objection to the imaginary dimension is not that we cannot see it.  Our senses cannot identify probable dimensions either, at least not in the visually compelling manner they can the three Cartesian dimensions. The question here is not whether imaginary numbers are mathematically true. How could they not be? The cards were stacked in their favor. They were defined in such a manner, – consistently and based on axioms long accepted valid, – that they are necessarily mathematically true. There’s a word for that sort of thing. –The word is  tautological.– No,  the decisive question is whether imaginary numbers apply to the real world; whether they are scientifically true, and whether physicists can truly rely on them to give empirically verifiable results with maps that accurately reproduce mechanisms actually used in nature.^[1]

The geometric interpretation of imaginary numbers was established as a belief system using the Cartesian line extending from  -1,0,0  through the origin  0,0,0 to 1,0,0  as the sole real axis left standing in the complex plane. In 1843,  William Rowan Hamilton introduced two additional axes in a quaternion coordinate system.  The new jandk axes,  similar to the i axis, encode coordinates of imaginary dimensions.  So the complex plane has one real axis, one imaginary; the quaternion system, three imaginary axes, one real, to accomplish which though involved loss of commutative multiplication. The mandalic coordinate system has three real axes upon which are superimposed six probable axes. It is both fully commensurate with the Cartesian system of real numbers  and  fully commutative for all operations throughout all dimensions as well.^[2]

All of these coordinate systems have a central origin point which all other points use as a locus of reference to allow clarity and consistency in determination of location.  The  mandalic coordinate system  is unique in that this point of origin is not a  null point of emptiness as in all the other locative systems,  but  a point of effulgence.  In that location  where occur Descartes’ triple zero triad (0.0.0) and the complex plane’s real zero plus imaginary zero (ax=0,bi=0), we find eight related hexagrams, all having neutral charge density,  each of these consisting of  inverse trigrams  with corresponding Lines of opposite charge, canceling one another out. These eight hexagrams are the only hexagrams out of sixty-four total possessing both of these characteristics.^[3]

So let’s begin now to plot the points of the mandalic coordinate system with  the view  of comparing its  dimensions and points  with  those of the complex plane.^[4]  The eight  centrally located hexagrams  all refer to  and are commensurate with the Cartesian triad (0,0,0). In a sense they can be considered eight  alternative possible states  which can  exist in this locale at different times. These are hybrid forms of the four complementary pair of hexagrams found at antipodal vertices of the mandalic cube.  The eight vertex hexagrams are those with upper and lower trigrams identical. This can occur nowhere else in the mandalic cube because there are only eight trigrams.^[5]

From the origin multiple probability waves of dimension radiate out toward the  central points of the faces of the cube,  where these divergent force fields rendezvous and interact with reciprocal forces returning from the eight vertices at the periphery. converging toward the origin.  Each of these points at the six face centers  are  common intersections  of another eight particulate states or force fields analogous to the origin point except that four originate within this basic mandalic module and four without in an adjacent tangential module. Each of the six face centers then is host to four internal resident hexagrams which  share the point in some manner, time-sharing or other. The end result is the same regardless, probabilistic expression of  characteristic form and function.  There is a possibility that this distribution of points and vectors  could be or give rise to a geometric interpretation of the Schrödinger equation,  the fundamental equation of physics for describing quantum mechanical behavior. Okay, that’s clearly a wild claim, but in the event you were dozing off you should now be fully awake and paying attention.

The vectors connecting centers of opposite faces of an ordinary cube through the cube center or origin of the Cartesian coordinate system are at 180° to each other forming the three axes of the system corresponding to the number of dimensions.  The mandalic cube has 24 such axes, eight of which accompany each Cartesian axis thereby shaping a hybrid 6D/3D coordinate system. Each face center then hosts internally four hexagrams formed by  hybridization of trigrams  in  opposite vertices  of diagonals of that cube face,  taking one trigram  (upper or lower)  from one vertex and the other trigram (lower or upper) from the other vertex. This means that a face of the mandalic cube has eight diagonals, all intersecting at the face center, whereas a face of the ordinary cube has only two.^[6]

The circle in the center of this figure is intended to indicate that the two pairs of antipodal hexagrams at this central point of the cube face rotate through 90° four times consecutively to complete a 360° revolution. But I am describing the situation here in terms of revolution only to show an analogy to imaginary numbers.  The actual mechanisms involved can be better characterized as inversions (reflections through a point),  and the bottom line here is that for each diagonal of a square, the corresponding mandalic square has  a possibility of 4 diagonals;  for each diagonal of a cube,  the corresponding mandalic cube has a possibility of 8 diagonals. For computer science, such a multiplicity of possibilities offers a greater number of logic gates in the same computing space and the prospect of achieving quantum computing sooner than would be otherwise likely.^[7]

Similarly, the twelve edge centers of the ordinary cube host a single Cartesian point,  but the superposed mandalic cube hosts two hexagrams at the same point. These two hexagrams are always inverse hybrids of the two vertex hexagrams of the particular edge.  For example,  the edge with vertices  WIND over WIND  and  HEAVEN over HEAVEN  has as the two hybrid hexagrams  at the  center point  of the edge  WIND over HEAVEN  and HEAVEN over WIND. Since the two vertices of concern here connect with one another  via  the horizontal x-dimension,  the two hybrids  differ from the parents and one another only in Lines 1 and 4 which correspond to this dimension.  The other four Lines encode the y- amd z-dimensions, therefore remain unchanged during all transformations undergone in the case illustrated here.^[8]

This post began as a description of the structure of the mandalic coordinate system and how it differs from those of the complex plane and quaternions.  In the composition,  it became also  a passable introduction to the method of  composite dimension.  Additional references to the way composite dimension works  can be found scattered throughout this blog and Hexagramium Organum.  Basically the resulting construction can be thought of as a  tensegrity structure,  the integrity of which is maintained by opposing forces in equilibrium throughout, which operate continually and never fail,  a feat only nature is capable of.  We are though permitted to map the process  if we can manage to get past our obsession with  and addiction to the imaginary and complex numbers and quaternions.^[9]

In our next session we’ll flesh out probable dimension a bit more with some illustrative examples. And possibly try putting some lipstick on that PIG (Presumably Imaginary Garbage) to see if it helps any.

(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] For more on this theme,  regarding quaternions,  see Footnote [1]  here. My own view is that imaginary numbers, complex plane and quaternions are artificial devices, invented by rational man, and not found in nature.  Though having limited practical use in  representation of rotations  in  ordinary space they have no legitimate application to quantum spaces,  nor do they have any substantive or requisite relation to square root, beyond their fortuitous origin in the Rationalists’ dissection and codification of square root historically, but that part of the saga was thoroughly misguided.   We wuz bamboozled.  Why persist in this folly? Look carefully without preconception and you’ll see this emperor’s finery is wanting. It is not imperative to use imaginary numbers to represent rotation in a plane. There are other, better ways to achieve the same. One would be to use sin and cos functions of trigonometry which periodically repeat every 360°.  (Read more about trigonometric functions here.)  Another approach would be to use polar coordinates.

[SOURCE]

A quaternion, on the other hand,  is a four-element vector composed of a single real element and three complex elements. It can be used to encode any rotation in a  3D coordinate system.  There are other ways to accomplish the same, but the quaternion approach offers some advantages over these.  For our purposes here what needs to be understood is that mandalic coordinates encode a hybrid 6D/3D discretized space. Quaternions are applicable only to continuous three-dimensional space.  Ultimately,  the two reside in different worlds and can’t be validly compared. The important point here is that each has its own appropriate domain of judicious application. Quaternions can be usefully and appropriately applied to rotations in ordinary three-dimensional space, but not to locations or changes of location in quantum space.  For description of such discrete spaces, mandalic coordinates are more appropriate, and their mechanism of action isn’t rotation but inversion (reflection through a point.) Only we’re not speaking here about inversion in Euclidean space, which is continuous, but in discrete space, a kind of quasi-Boolean space,  a higher-dimensional digital space  (grid or lattice space). In the case of an electron this would involve an instantaneous jump from one electron orbital to another.

[2] I think another laudatory feature of mandalic coordinates is the fact that they are based on a thought system that originated in human prehistory, the logic of the primal I Ching. The earliest strata of this monumental work are actually a compendium of combinatorics and a treatise on transformations,  unrivaled until modern times, one of the greatest intellectual achievements of humankind of any Age.  Yet its true significance is overlooked by most scholars, sinologists among them.  One of the very few intellectuals in the West who knew its true worth and spoke openly to the fact, likely at no small risk to his professional standing, was Carl Jung, the great 20th century psychologist and philosopher.

It is of relevance to note here that all the coordinate systems mentioned are, significantly,  belief systems of a sort.  The mandalic coordinate system  goes beyond the others though,  in that it is based on a still more extensive thought system, as the primal I Ching encompasses an entire cultural worldview.  The question of which,  if any,  of these coordinate systems actually applies to the natural order is one for science, particularly physics and chemistry, to resolve.

Meanwhile, it should be noted that neither the complex plane nor quaternions refer to any dimensions beyond the ordinary three, at least not in the manner of their current common usage.  They are simply alternative ways of viewing and manipulating the two- and three-dimensions described by Euclid and Descartes. In this sense they are little different from  polar coordinatesortrigonometry  in what they are attempting to depict.  Yes, quaternions apply to three dimensions, while polar coordinates and trigonometry deal with only two.  But then there is the method of  Euler angles  which describes orientation of a rigid body in three dimensions and can substitute for quaternions in practical applications.

A mandalic coordinate system, on the other hand, uniquely introduces entirely new features in its composite potential dimensions and probable numbers which I think have not been encountered heretofore. These innovations do in fact bring with them  true extra dimensions beyond the customary three  and also the novel concept of dimensional amplitudes.  Of additional importance is the fact that the mandalic method relates not to rotation of rigid bodies,  but to interchangeability and holomalleability of parts  by means of inversions through all the dimensions encompassed, a feature likely to make it useful for explorations and descriptions of particle interactions of quantum mechanics.  Because the six extra dimensions of mandalic geometry may, in some manner, relate to the six extra dimensions of the 6-dimensional Calabi–Yau manifold, mandalic geometry might equally be of value in string theoryandsuperstring theory.

Itis possible to use mandalic coordinates to describe rotations of rigid bodies in three dimensions,  certainly,  as inversions can mimic rotations, but this is not their most appropriate usage. It is overkill of a sort. They are capable of so much more and this particular use is a degenerate one in the larger scheme of things.

[3] This can be likened to a quark/gluon soup.  It is a unique and very special state of affairs that occurs here. Physicists take note. Don’t let any small-minded pure mathematicians  dissuade you from the truth.  They will likely write all this off as “sacred geometry.” Which it is, of course, but also much more.  Hexagram superpositions  and  stepwise dimensional transitions  of the mandalic coordinate system could hold critical clues  to  quantum entanglement and quantum gravity. My apologies to those mathematicians able to see beyond the tip of their noses. I was not at all referring to you here.

[4] Hopefully also with dimensions and points of the quaternion coordinate system once I understand the concepts involved better than I do currently. It should meanwhile be underscored that full comprehension of quaternions is not required to be able to identify some of their more glaring inadequacies.

[5] In speaking of  "existing at the same locale at different times"  I need to remind the reader and myself as well that we are talking here about  particles or other subatomic entities that are moving at or near the speed of light,- - -so very fast indeed. If we possessed an instrument that allowed us direct observation of these events,  our biologic visual equipment  would not permit us to distinguish the various changes taking place. Remember that thirty frames a second of film produces  the illusion of motion.  Now consider what  thirty thousand frames  a second  of  repetitive action  would do.  I think it would produce  the illusion of continuity or standing still with no changes apparent to our antediluvian senses.

[6] Each antipodal pair has four different possible ways of traversing the face center.  Similarly,  the mandalic cube has  thirty-two diagonals  because there are eight alternative paths by which an antipodal pair might traverse the cube center. This just begins to hint at the tremendous number of  transformational paths  the mandalic cube is able to represent, and it also explains why I refer to dimensions involved as  potentialorprobable dimensions  and planes so formed as probable planes.  All of this is related to quantum field theory (QFT), but that is a topic of considerable complexity which we will reserve for another day.

[7] One advantageous way of looking at this is to see that the probabilistic nature of the mandalic coordinate system in a sense exchanges bits for qubits and super-qubits through creation of different levels of logic gates that I have referred to elsewhere as different amplitudes of dimension.

[8] Recall that the Lines of a hexagram are numbered 1 to 6, bottom to top. Lines 1 and 4 correspond to, and together encode, the Cartesian x-dimension. When both are yang (+),  application of the method of  composite dimension results in the Cartesian value  +1;  when both yin (-),  the Cartesian value  -1. When either Line 1 or Line 4 is yang (+) but not both (Boole’s exclusive OR) the result is one of two possible  zero formations  by destructive interference. Both of these correspond to (and either encodes) the single Cartesian zero (0). Similarly hexagram Lines 2 and 5 correspond to and encode the Cartesian y-dimension; Lines 3 ane 6, the Cartesian z-dimension. This outline includes all 9 dimensions of the hybrid  6D/3D coordinate system:  3 real dimensions and the 6 corresponding probable dimensions. No imaginary dimensions are used; no complex plane; no quaternions. And no rotations. This coordinate system is based entirely on inversion (reflection through a point)  and on constructive or destructive interference. Those are the two principal mechanisms of composite dimension.

[9] The process as mapped here is an ideal one.  In the real world errors do occur from time to time. Such errors are an essential and necessary aspect of evolutionary process. Without error, no change. And by implication, likely no continuity for long either, due to external damaging and incapacitating factors that a natural world devoid of error never learned to overcome.  Errors are the stepping stones of evolution, of both biological and physical varieties.

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.

© 2016 Martin Hauser

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 304-

Mandalic geometry, Cartesian coordinates and Boolean algebra: Relationships - I

(continued from here)

In attempting to understand the logic of the I Ching it is important to know the differences between ordinary algebra  and  Boolean algebra and how Boolean algebra is related to the binary number system.^[1]

In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted
1 and 0 respectively. Instead of elementary algebra where the values of the variables are  numbers,  and the  main operations  are  addition and multiplication,  the main
operations of Boolean algebra are the conjunctionand, denoted ∧, the disjunctionor, denoted ∨, and the negationnot, denoted ¬. It is thus a formalism for describing logical relations in the same way that ordinary algebra describes numeric relations. [Wikipedia]

Whereas in elementary algebra expressions denote mainly numbers, in Boolean algebra they denote the truth values false and true. These values are represented with the bits (or binary digits), namely 0 and 1.  They do not behave like the integers  0 and 1,  for which
1 + 1 = 2, but may be identified with the elements of the two-element field GF(2), that is, integer arithmetic modulo 2,  for which 1 + 1 = 0.  Addition and multiplication then play the  Boolean roles  of  XOR  (exclusive-or)  and  AND  (conjunction)  respectively, with disjunction  x∨y  (inclusive-or)  definable as  x + y + xy. [Wikipedia]^[2]

Mandalic logic already occurs fully in the structure and manner of divinatory practice of the I Ching,  if some of it only implicitly.  Although mandalic geometry does not originate from either Boolean algebra or the Cartesian coordinate system but from the primal I Ching which predates them by millennia, it does combine and augment aspects of both of these conceptual systems. It extends Boole’s system of symbolic logic to include an additional logic value represented by the number -1.  This necessitates modification of some of Boole’s postulates and rules,  and increases their total number through introduction of some new ones.  The hexagrams or native six-dimensional mandalic coordinates of the I Ching are related to Cartesian triads composed of the numbers -1, 0, and 1,  making these two geometric systems  commensurate  by means of composite dimension,  a 6D/3D hybridization or mandalic coordination of structure and function (or space and time).^[3]

The introduction of composite dimension produces four distinct dimensional amplitudes  and  is solely responsible for the mandalic form. For anyone reading this who might be down on sacred geometry,  itself a subject which I respect and admire, let it be known that I am talking here about genuine mathematics and symbolic logic,  and my suspicion is that there is some genuine physics involved as well.

Kalachakra Mandala

The mandalic number system, then, is a quasi-modular number system, different from Leibniz’s binary number system which is fully modular.  Boole’s rule  1 AND 1 = 1  still holds true in mandalic logic.  However we must add to this the new logic rule that  -1 AND -1 = -1.  Individually the two rules are modular,  based on a clock arithmetic using a modulo-3 number system rather than Leibniz’s modulo-2 or binary number system, but with yet another added twist.

Together the two rules prescribe a compound system, one which is not singly modular but doubly modular.  The two components, yinandyang, are complementary and are inversely related to one another in this unified system.  This  logic organization  appears based on the figure 8 or sine wave and its negative,  allowing for periodicity, for recursive periods of interminably repeating duration,  and,  perhaps most importantly,  for wave interference,  of  constructive  and  destructive  varieties. These two geometric figures also engender an unexpected decussation of dimension not recognized by Western mathematics.  This is so because 1 AND -1 = 0 and  -1 AND 1 = 0.  The surprise here  is that  there are two distinct zeros: 0_a and 0_b.^[4] In two- or three-dimensional Cartesian terms there exists no difference between these two zeros.  However,  in terms of 6-dimensional aspects of mandalic geometry  and  the hexagrams of the I Ching, the two are clearly distinct structurally and functionally.^[5]

This arithmetic system is the basis of the logic encoded in the hexagrams of the I Ching. Each hexagram uniquely references a single 6- dimensional discretized point, of which there are 64 total. These 64 6- dimensional points of the mandalic cube are distributed among the 27 discretized points  of the ordinary 3-dimensional cube  through the compositing of dimensions  in such manner  that a mandala is formed which positions  1,  2,  4  or  8 hexagrams at each 3-dimensional point according to the   dimensional amplitude  of the particular point.  This necessarily creates a concurrent probability distribution of hexagrams through each of the three Cartesian dimensions.

TheI Chinguses a dual or composite three-valued logic system.  In place of truth values,  the variables used are yin,  yang  and the two in conjunction.  These fundamentally represent vector directions.  Yin is represented by -1, yang by 1, and their conjunction, using Cartesian or Western number terminology, by zero (0). This symbol does not occur natively in the I Ching though where the representation used is simply a combination of yin and yang symbols, most often in form of a bigram containing both  and  regarded as representing a composite dimension, namely 0_[1]  or  0_[2].^[6]

The two bigrams that satisfy the requirement are

young yang

for 0_[1]

and

young yin

for 0_[2].

Although mandalic logic is in Cartesian terms a 3-valued system, in native terms it is 4-valued.  It is not a simple modulo-3  or  modulo-4 number system, but two interrelated modulo-3 systems combined.  The best way to think about this geometric arrangement is possibly to view it as a single composite dimension having four distinct vector directions: a negative direction represented by mandalic composite yin (Cartesian -1); positive direction represented by mandalic composite yang (Cartesian 1); and two decussating relatively undifferentiated directions in some sort of equilibrium, represented by mandalic 0_[1] (composite yin/yang) and 0_[2] (composite yang/yin).  both of which  devolve  to  Cartesian 0  (balanced vector direction of the origin or center).^[7]

So we’ve seen that the number system used in the I Ching is not binary as Leibniz believed but instead doubly trinary with the two halves, in simplest terms,  inversely related and intertwined.  Still, it was an easy mistake to make because the notation used is binary.  We’ve seen too that all trigrams and hexagrams in the system can be rendered commensurate with the Cartesian coordinate system:  trigrams by simple transliteration, hexagrams by dimensional compositing. What, then, of George Boole and his eponymous logic?  How do they fit in the logic scheme of the I Ching? I’m glad you asked. Stay tuned to find out.

(continuedhere)

Images: Upper: TRANSFORMATION OF THE SYMBOL OF YIN (LINE split in two) AND YANG (STRAIGHT-LINE). BLEND: 4 bigrams, THEN 8 trigrams. (MORAN, E. ET AL. 2002: 77). Found here. Lower: Modified from an animation showing how the taijitu (yin-yang diagram) may be drawn using circles, then erasing half of each of the smaller circles. O'Dea at WikiCommons [CC BY-SA 3.0orGFDL],via Wikimedia Commons

Notes

[1] Boole’s algebra predated the modern developmentsinabstract algebra and  mathematical logic  but is seen as connected to the origins of both fields. Similarly to elementary algebra, the pure equational part of the theory can be formulated without regard to explicit values for the variables.

[2] If you are new to Boolean algebra these definitions may be confusing because in some ways they seem to fly in the face of ordinary algebra.  I’ll admit, I find them somewhat daunting.  Let me see if I can clarify the three examples given in this quote. Those of you more familiar with the language of Boolean algebra might kindly correct me in the event I err.  I’m growing more comfortable with being wrong at times.  And this is after all a work in progress.

• Boolean XOR (exclusive-or) allows that a statement of the form (x XOR y) is TRUE
  if either x or y is TRUE but FALSE if both are TRUE or if both are FALSE.  Since Boolean algebra uses binary numbers and represents  TRUE by 1,  FALSE by 0,  then
            for  x = TRUE,   y = TRUE    x + y = 1 + 1 = 0 ,    so FALSE
            for  x = FALSE,  y = FALSE   x + y = 0 + 0 = 0 ,  so FALSE
            for  x = TRUE,    y = FALSE   x + y = 1 + 0 = 1 ,   so TRUE
            for  x = FALSE,   y = TRUE    x + y = 0 + 1 = 1 ,   so TRUE

• Boolean AND (conjunction) allows that a statement of the form (x AND y) is TRUE
  only if both x is TRUE and y is TRUE. If either x or y is FALSE or both are FALSE
  then x AND y is FALSE. Here algebraic multiplication of binary 1s and 0s plays the
  role of Boolean AND. (Incidentally, binary multiplication works exactly the same
  way as algebraic multiplication. There’s a gift!)
            for  x = TRUE,    y = TRUE      xy  =  1(1) = 1,    so TRUE
            for  x = FALSE,   y = FALSE     xy = 0(0) = 0,   so FALSE
            for  x = TRUE,    y = FALSE      xy = 1(0) = 0 ,  so FALSE
            for  x = FALSE,    y = TRUE      xy = 0(1) = 0 ,  so FALSE

• Boolean OR (inclusive-or) is the truth-functional operator of (inclusive) disjunction,
  also known as alternation. The OR of a set of operands is true if and only if one or
  more of its operands is true. The logical connective that represents this operator is
  generally written as ∨ or +. As stated in the Wikipedia article logical disjunction x∨y
  (inclusive-or) is definable as x + y + xy [(x OR y) OR (x AND y)] as shown below.
  [Note: x AND y is often written xy in Boolean algebra. So watch out whichalgebra
  is being referred to, ordinary or Boolean. Are we confused yet?]
            for  x = TRUE,    y = TRUE      x + y = 1 , xy = 1 ,    so TRUE
            for  x = FALSE,   y = FALSE     x + y = 0 , xy = 0 ,   so FALSE
            for  x = TRUE,     y = FALSE     x + y = 1 , xy = 0 ,   so TRUE
            for  x = FALSE,    y = TRUE      x + y = 1 , xy = 0 ,   so TRUE
  

[3] Fundamentally, though,  the  coordinates of mandalic geometry  refer to vector directions alone, rather than to both vectors and scalars (or direction and magnitude) as do Cartesian coordinates. Yin specifies actually the entire domain of negative numbers rather than just the scalar value -1. Yang similarly refers to the entire domain of positive numbers rather than the scalar value 1 alone. Their conjunction  through the compositing of dimensions,  though represented by the symbol zero (0)  in the format commensurate with Cartesian coordinates,  refers actually to a  state or condition  not found in Western thought  outside of certain forms of mysticism  and other outsider philosophies like alchemy;  equilibration of forces in physics; equilibrium reactions in chemistry; and the kindred concept of homeostasis mechanisms of living organisms found in biology.

[4] This is to Westerners counterintuitive. Our customary logic and arithmetic allows for but a single zero. That two different zeros might exist concurrently or consecutively is - to our minds - irrational and we wrestle mightily with the idea. To complicate matters still more,  neither of these zeros is  conveniently  like our familiar Western zero.  So which should win out here?  Rationality or reality?  In fact,  the decision is not ours.  In the end nature decides.  Nature always decides. It stuffs the ballot box  and  casts the deciding vote much to our chagrin,  leaving us powerless to contradict what we may interpret as a whim. Our votes count for bupkis.

[5] This calls to mind also the Möbius strip which involves a twist that looks very much like a decussation to me.  The decussation or  twist in space  we are talking about here though has a sort of wormhole at its center that connects two contiguous dimensional amplitudes. I can’t say more about this just now. I need to think on it still. It seems a promising subject for reflection. (1,2,3)

[6] It needs to be pointed out here that in mandalic geometry, and similarly in the primal I Ching as well,  a bigram can be formed from any two related Lines of  hexagrams,  trigrams,  and tetragrams. The two Lines need not be (and often are not) adjacent to one another. I would think such versatility might well prove useful for modeling and mapping quantum states and interactions.

[7] Note that yin and yang in composite dimension can each take the absolute values 0, 1, and 2  but when yin has absolute value 2, yang has absolute value 0; when yang has absolute value 2,  yin has absolute value 0.  This inverse relation in fact is what makes the arrangement here a superimposed, actually interwoven, dual modulo-3 number system. It also makes the center points of mandalic lines,squares,  and cubes  more protean and less differentiated  than their vertices and elicits the different amplitudes of dimension.

The composite dimension value at the origin points(centers) of all of these geometric figures is  always  zero  in  Cartesian  terms  since the values of the differing Lines  in  the  two entangled 6-dimensional hexagrams  located here add to zero. But neither of these 6-dimensional entities is in its ground state at the center.  Both  have absolute value 1  at Cartesian 0.  Let me say that again: composite dimension values at the center or origin are zero in Cartesian terms but the values of both individual constituents are non-zero.Yin is in its ground state when yang is at its maximum and vice versa. At the center, since the two are equal and opposite they interfere destructively. This results in a composite zero ground state.

So from the perspective of  Cartesian coordinate dynamics, which is after all the customary perspective in our subjective lives,  we encounter only emptiness. But it is this very emptiness that opens to a new dimension. In the hybrid 6D/3D mandalic cube  only line centers and the cube center  have direct access through change of one dimension to face centers and only the face centers have a similar direct access through a single dimension to the cube center and edge centers. All coexist in an ongoing harmony of tensegrity. There is method to all this madness then.

© 2016 Martin Hauser

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 302-

Mandalic Line Segments,
Entanglement and Quantum Gravity
Part I

(continued from here)

We are going to consider once again now geometric line segments of mandalic geometry  and  their relation to Cartesian line segments and the Western number line. Yes,  this is sort of a detour from what I stated we would look at next. But this is not unrelated and lies at the very heart of mandalic geometry, and I’m not yet ready to address what I projected in the last remark of my previous post.

I keep returning to this subject because of its extreme importance. Beyond its significance to understanding the logic encoded in mandalic geometry and the I Ching, I believe it may also hold the key to quantum entanglement and quantum gravity.  Despite the fact that mandalic line segments are really fundamentally mental constructs,  a fiction of sorts, it is still important to understand how they are composed and how their components interact.  Though they may themselves be fictions,  the line segments and the points that compose them do in fact map a number of physical entities,  realities that may be related to quantum numbers and quantum particles and states.

When Descartes invented his coordinate system, with its points and line segments,  he based his system on the number line extended to two or three dimensions. In modeling it on the number line the space he described was imagined to bear a  necessary  one to one correspondence to the real numbers.^[1]  However this  1:1 mapping  of geometric space to the real numbers was a premise implicitly assumed by Descartes.  It was in fact axiomatic,^[2]  but apparently not stated as such.^[3]  As a result, the presumed relation has become a blind spot^[4] in Western thought,  never proved nor disproved, at least not at subatomic scales.^[5]

Neither mandalic geometry nor the primal I Ching make such an assumption. In place of Descartes’ 1:1 correspondence of geometric space and the numbers on the number line, we find a mandalic arrangement in which there are different categories of spatial location which can host one or more discrete numbers in a probabilistic manner.  This creates various dimensional amplitudes and a multidimensional waveform of component entities.^[6]

To my mind these characteristics of the mandalic coordinate system in combination with others described elsewhere make it more relevant to investigation and interpretation of many quantum phenomena which are as yet poorly understood than Cartesian coordinate dynamics may be and without need for recourse to imaginary numbers and complex plane.

(continuedhere)

Image: 6 steps of the Sierpinski carpet, animated. By KarocksOrkav (Own work) [CC BY-SA 3.0],via Wikimedia Commons

Notes

[1] Real numbers are numbers that can be found on the number line. This includes both the rational and irrational numbers.

[2] That is to say, taken for granted as self-evident.

[3] See Note [4] here.

[4] We have lived with this unproved premise so long that we no longer even question it,  or imagine that there might be an alternative which conforms better to reality at certain scales, notably subatomic scales.  The I Ching also seems to suggest  that a complete true description of complex relationships that involve a large number of dimensions,  including complex societal relationships,  requires more than a simple 1:1 correspondence between the notational symbols involved and the realities they represent.

[5] And from what I can see, no one seems to have much interest in proving or disproving this assumption.

[6] When speaking about hexagrams the number of dimensions involved is six as each Line of the hexagram encodes a value for a single distinct dimension in a 6-dimensional space.  In a hybrid 6D/3D compositing of dimensions though, two such Lines in relation reference a single Cartesian dimension in 2- or 3-space.  A concept not to be missed here is that  interactions of quantum particles  may well involve such  integration of dimension,  of dimensions  we are not even aware of beyond the unsettling fact  they upset the neat applecart of customary conceptual categories.

© 2016 Martin Hauser

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 301-

Beyond Taoism - Part 5
A Vector-based Probabilistic
Number System
Part II

(continued from here)

Taoism and the primordial I Chingare in agreement that temporal changes have two different aspects: sequent and cyclic.  Western thought in general follows suit. The I Ching differs from the other two in asserting that  the direction of change - for both sequent and cyclic change - is fully reversible,  with the proviso  that sufficiently small units of measurement are involved.^[1]  The probability that reversal can be achieved  diminishes proportionately to the magnitude of change that has taken place.^[2]

Taoist appropriation of bigrams and trigrams of the I Ching to model such phenomena as change of seasons and phases of the moon  is plausible if not quite legitimate. The natural phenomena so modeled are macroscopic and vary continuouslyandinexorably throughout an ever-repeating cyclic spectrum. And there’s the rub.

As they occur and function in the I Ching bigrams and trigrams are dicontinuous discrete elements,  formed by other similarly discontinuous discretized entities,  and they follow evolutionary courses which are most often nonrepetitive. So the Taoist usage is misleading at best, annihilative at worst. Unfortunately, as the I Ching itself evolved through centuries of commentaries and reinterpretations,  it became  ever more contaminated and tainted by these Taoist corruptions of meaning, at the same time that it was being inundated by  Confucian sociological and ethical reworkings.  What we have today is an amalgam, the various parts of which do not sit well with one another.^[3]

Though it may in part be hyperbole to prove a point,  the stark difference between the two approaches,  that of Taoism and that of the I Ching, is epitomized by comparison of the Taoist diagram of the cycle of seasons with diagrams at the top and bottom of the page,  which are based on  the  number,  logic,  and coordinate systems of   The Book of Changes.^[4] The increased complexity of the latter diagrams should not prove a stumbling block, as they can be readily understood in time with focus and attention to detail.  The  important take-away  for now is that in the I Ching bigrams  exist within a larger dimensional context  than the Taoist diagram avows,  and this context makes all their interactions more variable,  conditional,  and complex. As well, the same can be said of trigrams and hexagrams.

One of the more important aspects of these differences has to do with the notion of equipotentiality.  As bigrams and trigrams function within  higher dimensional contexts  in the  I Ching,  this introduces a possibility of multiple alternative paths of movement and directions of change.  Put another way,  primordial I Ching logic encompasses many more  degrees of freedom  than does the logic of Taoism.^[5]  There is no one direction or path  invariably decreed or favored.  An all-important element of conditionality prevails.  And that might be the origin of what quantum mechanics has interpreted as indeterminism or chance.

Next up, a closer look at equipotentiality and its further implications.

Section F_H(n)^[6]

(continuedhere)

Notes

[1] There are exceptions. Taoist alchemy describes existence of certain changes that admit reversibility under special circumstances.  Other than the Second Law of Thermodynamics (which is macroscopic in origin,  not result of any internally irreversible microscopic properties of the bodies), the laws of physics neglect all distinction between forward-moving timeandbackward-moving time. Chemistry recognizes existence of certain states of equilibrium in which the rates of change in both directions are equal. Other exceptions likely occur as well.

[2] Since change is quantized in the I Ching, which is to say, it is divided into small discretized units,  which Line changes model,  the magnitude of change is determined by the number of Line changes that have occurred  between Point A and Point B in spacetime.  Reversal is far easier to achieve if only a single Line change has occurred than if three or four Lines have changed for example.

[3] Ironically, Taoism itself has pointed out the perils of popularity. Had the I Ching been less popular, less appealing to members of all strata of society, it would have traveled through time more intact.  Unless,  of course,  it ended up buried or burned. What is fortunate here is that much of the primordial logic of the I Ching can be reconstructed by focusing our attention on the diagrammatic figures and ignoring most of the attached commentary.

[4] These diagrams do not occur explicitly in the I Ching. The logic they are based on, though, is fully present implicitly in the diagramatic structural forms of hexagrams, trigrams, and bigrams and the manner of their usage in  I Ching divinatory practices.

[5] Or, for that matter, than does the logic of Cartesian coordinate space if we take into account the degrees of freedom of six dimensional hexagrams mapped by composite dimensional methodology to model mandalic space. (See Note [4] here for important related remarks.)

[6] This is the closest frontal section to the viewer through the 3-dimensional cube using Taoist notation.  See here for further explanation.  Keep in mind this graph barely hints at the complexity of relationships found in the 6-dimensional hypercube which has in total 4096 distinct changing and unchanging hexagrams in contrast to the 16 changing and unchanging trigrams we see here. Though this model may be simple by comparison,  it will nevertheless serve us well as a key to deciphering the number system on which I Ching logic is based as well as the structure and context of the geometric line that can be derived by application of reductionist thought to the associated mandalic coordinate system of the I Ching hexagrams. We will refer back to this figure for that purpose in the near future.

© 2016 Martin Hauser

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 300-

Beyond Taoism - Part 4
A Vector-based Probabilistic
Number System
Introduction

(continued from here)

Leibniz erred in concluding the hexagrams of the I Ching were based on a number system related to his own  binary number system.  He had a brilliant mind but was just as fallible as the rest of us.  He interpreted the I Ching in terms of his own thought forms,  and he saw the hexagrams as a foreshadowing of his own binary arithmetic.^[1]

So in considering the hexagram Receptive,  Leibniz understood the number 0; in the hexagram Return, the number 1; in the hexagram Army, the number 2; in the hexagram Approach, the number 3;  in the hexagram Modesty,  the number 4;  in the hexagram  Darkening of the Light, the number 5;  and so on, up to the hexagram Creative, in which he saw the number 63.^[2]  His error is perhaps excusable in light of the fact that the Taoists, though much closer to the origin of the I Ching in time, themselves misinterpreted the number system it was based on.^[3]

From our Western perspectiveI Ching hexagrams are composed of trigrams, tetragrams, bigrams, and ultimately yinandyang Lines. From the native perspective of the I Ching this order of arrangement is putting the cart before the horse.  Dimensions  and their interactions  are,  in the view of I Ching philosophy and mandalic geometry,  antecedent logically and materially to any cognitive parts we may abstract from them. Taoism in certain contexts has abstracted the parts and caused them to appear as if primary. It has the right to do so if creating its own philosophy,  but not as interpretation of the logic of the I Ching. It is a fallacy if so intended.^[4]

The Taoists borrowed from the I Ching two-dimensional numbers, treated them as one-dimensional and based their quasi-modular number system on  the dimension-deficient result.  This is the way they arrived at their seasonal cycle consisting of bigrams:   old yin (Winter),  young yang (Spring), old yang (Summer), young yin (Autumn), old yin (Winter),  and so forth. This represents a very much impoverished and impaired version of the original configuration in the primal strata of the I Ching.^[5]

The number system of the I Ching is not a linear one-dimensional number system like  the positional decimal number system  of the West; nor is it like the positional binary number system invented by Leibniz. It is not even like the quasi-modular number system of Taoism.  The key to the number system of the hexagrams is located not in the 64 unchanging explicit hexagrams,  but rather in the changing implicit hexagrams found only in the divination practice associated with the I Ching. These number 4032.^[6]  The manner in which these operate,  however,  is actually  fairly simple and is uniform throughout the system.  So once understood,  they can be safely relegated to the implicit background, coming into play only during procedures involving divination or in attempts to understand the system fully, logically and materially.  When dealing with more ordinary circumstances just the 64 more stable hexagrams need be attended to in a direct and explicit manner.

The Taoist sequence of bigrams is in fact a corruption of the far richer asequential multidimensional arrangement of bigrams that occurs in I Ching hexagrams and divination. There we see that change can occur from any one of the four stable bigrams to any other.  If this is so then no single sequence can do justice to the total number possible. The ordering of bigrams presented by Taoism is just one of many that make up the real worlds of nature and humankind.  Taoism imparts special significance to this sequence; the primal I Ching does not. It views all possible pathways of change as equally likely.^[7]

Next time around we will look further into the implications of this equipotentiality and see how it plays out in regard to the number system of the I Ching.

Section F_H(n)^[8]

(continuedhere)

Notes

[1] By equating yang with 1 and yin with 0 it is possibletosequence the 64 I Ching hexagrams according to binary numbers 0 through 63.  The mere fact that this is possible does not, however, mean that this was intended at the time the hexagrams were originally formulated. Unfortunately, this arrangement of hexagrams seems to have been the only one of which Leibniz had knowledge. This sequence was, in fact, the creation of the Chinese philosopher Shao Yong (1011–1077). It did not exist in human mentation prior to the 11th century CE.

This arrangement was set down by the Song dynasty philosopher Shao Yong (1011–1077 CE), six
centuries before Wilhelm Leibniz described binary notation. Leibniz published ‘De progressione
dyadica’ in 1679. In 1701 the Jesuit Joachim Bouvet wrote to him enclosing a copy of Shao Yong’s 'Xiantian cixu’ (Before Heaven sequence). [Source]

Note also that the author of Calling crane in the shade, the source quoted above, calls attention to confusion that exists about whether the “true binary sequence of hexagrams” should begin with the lowest line as the least significant bit (LSB) or the highest line. He points out that the Fuxi sequence as transmitted by Shao Yong in both circular and square diagrams takes the highest line as the LSB, although in fact it would make more sense in consideration of how the hexagram form is interpreted to take the lowest line as the LSB. My thinking is that either Shao Yong misinterpreted the usage of hexagram form or, more likely, the conventional interpretation of the Shao Yong diagrams is incorrect. Here I have chosen to use the lowest line of the hexagram as the LSB,  and I think it possible  Leibniz may have done the same.

If one considers the circular Shao Yong diagram,  the easier of the two to follow,  one can reconstruct the binary sequence,  with the lowest line as LSB,  by beginning with the hexagram EARTH at the center lower right half of the circle, reading all hexagrams from outside line (bottom) to inside line (top),  progressing counterclockwise to  MOUNTAIN over WIND at top center, then jumping to hexagram  MOUNTAIN over EARTH  bottom center of left half of the circle,  and progressing clockwise to hexagram  HEAVEN  at top center.  Of the two,  this is the interpretation that makes the more sense to me and the one I have followed here, despite the fact that it is not the received traditional interpretation of the Shao Yong sequence. Historical transmissions have not infrequently erred. Admittedly it is difficult to decipher all Lines of some of the hexagrams  in the copy Leibniz received due to passage of time and its effects on paper and ink.  Time is not kind to ink and paper, nor for that matter to flesh and products of intellect.

In the final analysis, which of the two described interpretations is the better is moot because neither conforms to the logic of the I Ching which is not binary to begin with. Moreover,  there is a third interpretation of the Shao Yong sequence that is superior to either described here.  It is not binary-based.  And why should it be? After all the Fuxi trigram sequence  which Shao Yong took as model for his hexagram sequence  is itself not binary-based. Perhaps we’ll consider that interpretation somewhere down the road. For now, the main take-away is that Leibniz, in his biased interpretation of the I Ching hexagrams made one huge mistake.  Ironically,  had he not some 22 years prior already invented  binary arithmetic, this error likely would have led him to invent it.  It was “in the cards” as they say. At least in certain probable worlds.

[2]ReceptiveandCreative are alternative names for the hexagrams EarthandHeaven, respectively. The sequence detailed can be continued ad infinitum using yin-yang notation, though of course this takes us beyond the realm of hexagrams into what would be, for mandalic geometry and logistics of the I Ching, domains of dimensions numbering more than six.  Keep in mind here though that Leibniz was not thinking in terms of dimension but an  alternative method  of expressing the prevalent base 10 positional number system notation of the West.  He held in his grasp the key to unlocking an even greater treasure but apparently never once saw that was so.  This seems strange considering his broadly diversified interests and pursuits in the fields of  mathematics,  physics,  symbolic logic,  information science,  combinatorics,  and in the nature of space.  Moreover,  his concern with these was not just as separate subjects of investigation.  He envisaged uniting all of them in a  universal language  capable of expressing mathematical,  scientific, and metaphysical concepts.

[3] Earlier in this blog I have too often confused Taoism with pre-Taoism. The earliest strata of the I Ching belong to an age that preceded Taoism by centuries, if not millennia.  Though Taoism was largely based on the philosophy and logic of the I Ching,  it didn’t always interpret source materials correctly,  or possibly at times it intentionally used source materials in new ways largely foreign to the originals. The number system of the I Ching is a case in point.

In the interest of full disclosure, I am not an expert in the history or philosophy of Taoism.  Taoist philosophies are diverse and extensive. No one has a complete set or grasp of all the thoughts, practices and techniques of Taoism. The two core Taoist texts, the  Tao Te ChingandChuang-tzu,   provide the philosophical basis of Taoism which derives from the eight trigrams (bagua) of Fu Xi, c. 2700 BCE, the various combinations of which created the 64 hexagrams documented in the I Ching.  The Daozang,  also referred to as  the Taoist canon,  consists of around 1,400 texts that were collected c. 400, long after the two classic texts mentioned. What I describe as Taoist thought then is abstracted in some manner from a huge compilation, parts of which may well differ from what is presented here. Similar effects of time and history can be discerned in Buddhism, Christianity, Islam and secular schools of thought like Platonism,Aristotelianism,Humanism, etc.

[4] Recent advances in the sciences have begun to raise new ideas regarding the structure of reality. Many of these have parallels in Eastern thought.  There has been a shift away from the reductionist view in which things are explained by breaking them down then looking at their component parts, towards a more holistic view. Quantum physics notably has changed the way reality is viewed. There are no certainties at a quantum level, and the experimenter is necessarily part of the experiment. In this new view of nature everything is linked and man is himself one of the linkages.

[5] It is not so much that this is incorrect as that it isextremelylimiting with respect to the capacities of the I Ching hexagrams.  A special case has here been turned into a generalization that purports to cover all bases. This may serve well enough within the confines of Taoism but it comes nowhere near elaborating the number system native to the I Ching. We would be generous in describing it as a watered down version of a far more complex whole.  Through the centuries both Confucianism and Taoism  restructured the I Ching to make it conducive to their own purposes.  They edited it and revised it repeatedly,  generating commentary after commentary,  which were admixed with the original,  so that the I Ching as we have it today,  the I Ching of tradition,  is a hodgepodge of many convictions and many opinions. This makes the quest for the original features of the I Ching somewhat akin to an archaeological dig.  I find it not all that surprising  that the oracular methodology of consulting the I Ching  holds possibly greater promise in this endeavor than the written text.  The  early oral traditions  were preserved better,  I think,  by the uneducated masses who used the I Ching as their tool for divination than by philosophers and scholars who,  in their writings,  played too often a game of one-upmanship with the original.

[6] A Line can be either yin or yang, changing or unchanging. Then there are four possible Line types and six Lines to a hexagram.  This gives a total of 4096 changing and unchanging hexagrams (4⁶ = 4096). Since there are 64 unchanging hexagrams (2⁶ = 64) there must be 4032 changing hexagrams (4096-64 = 4032).

[7] This calls to mind the path integral formulation of quantum mechanics which was developed in its complete form by Richard Feynman in 1948. See, for example, this description of the path integral formulation in context of the double-slit experiment, the quintessential experiment of quantum mechanics.

[8] This is the closest frontal section to the viewer through the 3-dimensional cube using Taoist notation.  See here for further explanation.  Keep in mind this graph barely hints at the complexity of relationships found in the 6-dimensional hypercube which has in total 4096 distinct changing and unchanging hexagrams in contrast to the 16 changing and unchanging trigrams we see here. Though this model may be simple by comparison,  it will nevertheless serve us well as a key to deciphering the number system on which I Ching logic is based as well as the structure and context of the geometric line that can be derived by application of reductionist thought to the associated mandalic coordinate system of the I Ching hexagrams. We will refer back to this figure for that purpose in the near future.

© 2016 Martin Hauser

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-Page 299-

“…Moreover, even if verbal economics could be successfully translated into mathematical symbols and then translated into English so as to explain the conclusions, the process makes no sense and violates the great scientific principle of Occam’s Razor: avoiding unnecessary multiplication of entities…

More recently, Boris Ischboldin has emphasized the difference between verbal, or “language,” logic (“the actual analysis of thought stated in language expressive of reality as grasped in common experience”) and “construct” logic, which is “the application of quantitative (economic) data of the constructs of mathematics and symbolic logic which constructs may or may not have real equivalents.”10

Although himself a mathematical economist, the mathematician son of Carl Menger wrote a trenchant critique of the idea that mathematical presentation in economics is necessarily more precise than ordinary language:

Consider, for example, the statements (2) To a higher price of a good, there corresponds a lower (or at any rate not a higher) demand.

(2’) If p denotes the price of, and q the demand for, a good, then

q = f(p) and dq/dp = f’ (p) ≤ 0

Those who regard the formula (2’) as more precise or “more mathematical” than the sentence (2) are under a complete misapprehension … the only difference between (2) and (2’) is this: since (2’) is limited to functions which are differentiable and whose graphs, therefore, have tangents (which from an economic point of view are not more plausible than curvature), the sentence (2) is more general, but it is by no means less precise: it is of the same mathematical precision as (2’).11…

— Murray Rothbard, Praxeology: the Method of Austrian Economics