#sinology

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 3
A Multidimensional Number System

(continued from here)

Descartes modeled his coordinate system on the Western number line,  itself an extension of the decimal number system to include the new negative numbers, and upon the Euclidean notion of a three-dimensional geometry.  All these events took place in historical times.  In approaching the I Ching and its number system we are dealing mostly with events that took place before recorded history so it is impossible to say with certainty how anything involved came about.  We can’t so much as be sure whether the I Ching was based on an antecedent number system,  or predated and foreshadowed a subsequent number system of Chinese antiquity possibly contingent on it. We view all such things as through a glass, darkly.^[1]

It is clear, though, that the number system of the I Ching is one far more complex than that of Western mathematics.  The number system of the West is unidimensional (linear).  Descartes,  in his coordinate system, extends it for use in three dimensions. The number system of the I Ching, on the other hand,  is  in origin  multidimensional.  It is mandalic as well, which is to say it consists of multiple dimensions interwoven in a specific manner which can best be characterized as mandalic in form,  possessing a number of interlaced and interlinked concentric shells or orbitals about a unifying center.

At the important origin of Descartes’ coordinate system is found his triple zero ordered triad (0,0,0).  Descartes  views this point,^[2] asall his points, primarily in terms of location, not relationship.  The matter of relationship is left to analytic geometry,  the geometry Descartes codified based on his coordinate system.^[3] The coordinate system itself seems not to care how points are formed or related beyond the most elementary and trivial operations of addition and subtraction throughout what essentially remain predominantly isolated dimensions.^[4] In the end this becomes an effective and prodigious mind snare.^[5]

In contrast to the Cartesian approach,theI Ching offers a unified coordinate system and geometry in a single entity which emphasizes the relationship of “points” and other “parts” (e.g.,  lines,  faces) as much,  if not more,  than location,  beginning with wholeness and ending with the same.  In between,  all sorts of  complex and interesting interactions and changes take place.  In analyzing these,  it is best to begin at the origin of the coordinate system of the I Ching,  the unceasing wellspring  of  being that supplants the triple vacuity of Descartes and Western mathematics.

Section F_H(n)^[6]

(continuedhere)

Notes

[1] My thinking is that the I Ching was originally primarily a non-numerical relationship system that subsequently was repurposed to include,  as one of its more important contextual capacities, numerical relationships. That said, from a contemporary perspective,  rooted in  a comprehensive awareness that spans combinatorics,  Boolean algebra,  particle physics, and the elusive but alluring logic  of  quantum mechanics and the Standard Model,  it would seem that this relationship system is an exemplary candidate for an altogether natural number system, one that a self-organizing reality could readily manage.

[2] As do most geometers who follow after Descartes.

[3] Strictly speaking, this approach is not in error, though it does seem a somniferous misdirection.  Due to the specific focus and emphasis enfolded in Descartes’ system, certain essential aspects of mathematical and physical reality tend to be overlooked. These are important relational aspects,  highly significant to particle physicists among others. These remarks are in no way intended to denigrate  Cartesian  coordinates and geometry,  but to motivate physicists and all freethinkers  to investigate further in their explorations of reality.

[4] The Cartesian system neglects, for instance, to express anywhere that the fact  the algebra of the real numbers  can be employed to yield results about the linear continuum of geometry relies on  the Cantor–Dedekind axiom,  which in mathematical logic

has been used to describe the thesis that the real numbers are order-isomorphic to the linear continuum of geometry. In other words, the axiom states that there is a one to one correspondence between real numbers and points on a line.

This axiom is the cornerstone of analytic geometry. The Cartesian coordinate system developed by René Descartes explicitly assumes this axiom by blending the distinct concepts of real number system with the geometric line or plane into a conceptual metaphor. This is sometimes referred to as the real number line blend. [Wikipedia]

Neither mandalic geometry nor the I Ching,  upon which it is based,  accept this axiom as true in circumstances other than those restrictive settings, such as Cartesian geometry, where it is explicitly demanded as axiomatic to the system. In other words,  they do not recognize the described one to one correspondence between number and geometric space as something that reality is contingent on. The assumption contained in this axiom, however, has been with us so long that we tend to see it as a necessary part of nature.  Use of the stated correspondence may indeed be expedient in everyday macro-circumstances but continued use in other situations,  particularly to describe subatomic spatial relations,  is illogical and counterproductive, to paraphrase a certain Vulcan science officer.

[5] For an interesting take on the grounding metaphors at the basis of the real number line and neurological conflation see  The Importance of Deconstructing the Real Number Line.  Also on my reading list regarding this subject matter  is Where Mathematics Comes From:How the Embodied Mind Brings Mathematics into Being(1,2,3) by George Lakoff and Rafael Nuñez. Neither of the authors is a mathematician, but sometimes it is good to get an outside perspective on what is in the box.

[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.  Simple by comparison though this model may be it will nevertheless serve us well as a key to deciphering the line derived from the mandala of I Ching hexagrams, and we will be referring back to this figure for that purpose in the near future.

© 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 298-

Beyond Taoism - Part 2
Number System of the I Ching

(continued from here)

Many different number systems exist in the world today. Others have existed in times past. The number system we are most familiar with is base 10 or radix 10,  which makes use of ten digits,  numbered  0  to  9. Beyond the number 9, the numbers recapitulate, beginning again with 0 and shifting a new “1” to the 10s position, in a positional number system. Using this conventional technique all integers and decimals can be easily and uniquely expressed.  This familiar  numeral system  is also known as the decimal system.^[1]

Another number system we are familiar with and use every day is the modular numeral system, particularly in its manisfestation of modulo 12, better known as clock arithmetic.  This is a system of arithmetic in which integers “wrap around” and begin again upon reaching a set value, called the modulus. For clock arithmetic, the modulus used is 12. On the typical 12-hour clock,  the day is divided into two equal periods of 12 hours each. The 24 hour / day cycle starts at 12 midnight  (often indicated as 12 a.m.), runs through 12 noon  (often indicated as 12 p.m.),  and  continues  to the midnight at the end of the day. The numbers used are 1 through 11 and 12 (the modulus,  acting as zero).  Military time is similar,  only is based on a 24-hour clock with modulus-24 rather than modulus-12. The modulus-24 system is the most commonly used time notation in the world today.

Binary arithmetic is similar to clock arithmetic, but is modulo-2 instead of modulo-12.  The only integers used in this system are  0 and 1, with the “wrap around” back to zero occurring each time the number 1 is reached.  Computers, in particular, handle this arithmetic system,  which we owe to Leibniz, with remarkable acumen. George Boole also based his true/false logic on binary arithmetic.  This, in itself, accounts for some of its strange, counterintuitive aspects,  like the fact that in Boolean algebra the sum of 1 + 1 equals 0.  Not your father’s arithmetic.  But both Leibniz and Boole found profound uses for it. As did the entire digital revolution.

When we come to consideration of the number system and arithmetic used in the I Ching we can anticipate encountering equal difficulty in comprehension, possibly more. The system employed is a modular one - sort of.  However,  it uses negative 1 (yin) as well as positive 1 (yang) whereas zero (0) is nowhere to be seen, at least not in guise of  an explicit dedicated symbol  earmarked for the purpose. The "wrap around" appears to occur at both -1 (yin)  and  +1 (yang). Something different and quite extraordinary is going on here. This is no simple modular numeral system, though it may be masquerading as one.

Thus far the number system of the I Ching sounds much like that of Taoism. It is not, though. We have some big surprises in store for us.

Section F_H(n)^[2]

(continuedhere)

Notes

[1] See here for a list/description of numeral systems having other bases. A more comprehensive list of numeral systems can be found here.

[2] For explanation of this diagram see here.

© 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 297-

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 Descartes - Part 6

The Fiction Formerly Known As the Line

(continued from here)

Rereading the last post a moment ago I see I fell into the same old trap, namely describing a concept arising from an alternative worldview in terms of our Western worldview.  It is so astonishingly easy to do this. So it is important always to be on guard against this error of mind.

In saying that the Taoist number line is the basis of its coordinate system I was phrasing the subject in Western terminology,  which doesn’t just do an injustice to the truth of the matter,  it does violence to it,  in the process destroying the reality:  that within Taoism, the coordinate system is primary.  It precedes the line,  which follows from it.  What may be the most important difference between the Taoist apprehension of space and that of Descartes lies encoded within that single thought.

Descartes continues the fiction fomented in the Western mind by Euclid that  the point and the line  have independent reality. Taking that to be true,  Descartes constructs his coordinate system using  pointsandlines  as the elemental building blocks. But to be true to the content and spirit of Taoism, this fabrication must be surrendered.  For Taoism,  the coordinate system, which models space, or spacetime rather, is primary. Therefore to understand the fictional Taoist line we must begin there, in the holism and the complexity of its coordinate system where dimension, whatever it may be, reigns supreme.^[1]

And that means we can no longer disregard composite dimension, postponing discussion of it for a later time,  because it is the logical basis on which the I Ching is predicated. It is related to what we today know as combinatorics,Boolean algebra, and probability,  and is what gives rise to what I have called the plane of potentiality. It is the very pith of mandalic geometry, what makes it a representation of mandalic spacetime.^[2]

(continuedhere)

Notes

[1] In my mind, dimension is a category of physical energetic description before it is a category of geometrical description.  When particle physicists speak about “quantum numbers” I think they are actually, whether intended or not, referring to dimensions. If this is true, then our geometries should be constructed to reflect that primordial reality, not arbitrarily as we choose.

[2] In speaking of logic and the I Ching in the same breath I am using the term in its broadest sense as any formal system in which are defined axioms and rules of inference. In reference to the I Ching,  the logic involved is far removed from the rationalism bequeathed to Descartes by his times.  It is a pre-rationalist logic that prevailed in human history for a very long time before the eventual splitting off of the irrational from the rational.  This means also that the I Ching is among other things a viable instrument to access strata of human minds long dormant in historical times,  other than possibly,  at times,  in poetry and art and the work of those select scientists who make extensive use of intuition in the development of their theories.

Note to self:  Two contrasting systems of thought based on very different worldviews can never be adequately explained in terms of one another. At times though, for lack of anything better, we necessarily fall back on just such a strategy, however limited, and make the best of it we can.

© 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 282-

Beyond Descartes - Part 5

Reciprocation, Alternation, Decussation
Imaginary and Complex Numbers

(continued from here)

Previously in this blog a number of attempts have been made to explicate the Taoist number line and contrast it with the Western version of the same.  It is essential to do this and to do it flawlessly,  first because different systems of arithmetic result from the two, and secondly because the mandalic coordinate system is based on the former perspective while the Cartesian coordinate system is based on the latter.^[1]

What has been offered earlier has been accurate to a degree, a good first approximation. Here we intend to present a more definitive account of the Taoist number line,  describing both how it is similar to and how it differs from the  Western number line  used by Descartes in formation of his coordinate system.  This will inevitably transport us  well beyond that comfort zone offered by the more accessible three-dimensional cubic box that has heretofore engaged us.

Both Taoist and Western number lines observe directional locative division of their single dimension into two major partitions:  positive and negative for the West;  yinandyang for Taoism.^[2]  There the similarities essentially end.  From its earliest beginnings Taoism recognized a second directional divisioning in its number line, that of manifest/unmanifestorbeingandbecoming.^[3]  The West never did such.  As a result, some time later the West found it necessary to invent imaginary numbers.^[4][5]

Animaginary number is a complex number that can be written as a real numbermultiplied by theimaginary uniti, which is defined by its property i² = −1. [Wikipedia]

Descartes knew of these numbers but was not particularly fond of them.  It was he, in fact, who first used the term “imaginary” describing them in a derogatory sense. [Wikipedia]  The term “imaginary number” now just denotes a complex number with a real part equal to 0,  that is, a number of the form bi. A complex number where the real part is other than 0 is represented by the form a + bi.

In place of the complex plane, Taoism has (and always has had from time immemorial)  a plane of potentiality.  An explanation of this alternative plane was attempted earlier in this blog,  but it can likely be improved. This post has simply been a broad brushstrokes overview. In the following posts we will look more closely at the specifics involved.^[6]

(continuedhere)

Image (lower): A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram representing the complex plane. “Re” is the real axis, “Im” is the imaginary axis, and i is the imaginary unit which satisfies i² = −1. Wolfkeeper at English Wikipedia [GFDL (http://www.gnu.org/copyleft/fdl.html) or CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons

Notes

[1] The arithmetic system derived from the Taoist number line can perhaps best be understood as a  noumenal  one. It applies to the world of ideas rather than to our phenomenal world of the physical senses, but it may also apply to the real world, that is, the real real world which we can never fully access.

Much of modern philosophy has generally been skeptical of the possibility of knowledge independent of the physical senses, and Immanuel Kant gave this point of view its canonical expression: that the noumenal world may exist, but it is completely unknowable to humans. In Kantian philosophy, the unknowable noumenon is often linked to the unknowable “thing-in-itself” (Ding an sich, which could also be rendered as “thing as such” or “thing per se”), although how to characterize the nature of the relationship is a question yet open to some controversy. [Wikipedia]

[2] From the perspective of physics this involves a division into two major quanta of charge, negative and positive, which like yinandyang can be either complementary or opposing.  Like forces repel one another and unlike attract. This is the basis of electromagnetism, one of four forces of nature recognized by modern physics. But it is likely also the basis, though not fully recognized as such, of the strong and weak nuclear forces, possibly of the force of gravity as well. I would suspect that to be the case. The significant differences among the forces  (or force fields, the term physics now prefers to use)  lie mainly, as we shall see, in intricate twistings and turnings through various dimensions or directions that negative and positive charges undergo in particle interactions.

[3] It is this additional axis of probabilistic directional location, along with composite dimensioning, both of which are unique to mandalic geometry, that make it a geometry of spacetime,  in contrast to Descartes’ geometry which, in and of itself, is one of space alone. The inherent spatiotemporal dynamism that is characteristic of  mandalic coordinates  makes them altogether more relevant for descriptions of particle interactions than Cartesian coordinates, which often demand complicated external mathematical mechanisms to sufficiently enliven them to play even a partial descriptive role, however inadequate.

[4] In addition to their use in mathematics, complex numbers, once thought to be  "fictitious" and useless,  have found practical applications in many fields, including chemistry, biology, electrical engineering, statistics, economics,  and, most importantly perhaps, physics..

[5] The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers. He called them “fictitious” during his attempts to find solutions to cubic equations in the 16th century.  At the time, such numbers were poorly understood,  consequently regarded by many as fictitious or useless as negative numbers and zero once were. Many other mathematicians were slow to adopt use of imaginary numbers, including Descartes, who referred to them in his La Géométrie, in which he introduced the term imaginary,  that was intended to be derogatory. Imaginary numbers were not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855).  Geometric interpretation of  complex numbers as points in a complex plane  was first stated by mathematician and cartographer Caspar Wessel in 1799. [Wikipedia]

[6] What I have called here the plane of potentiality occurs only implicitly in the Taoist I Ching but is fully developed in mandalic geometry. It may be related to  bicomplex numbers  or tessarines in abstract algebra, the existence of which I only just discovered. Unlike the quaternions first described by Hamilton in 1843, which extended the complex plane to three dimensions, but unfortunately are not commutative,  tesserines or bicomplex numbers  are hypercomplex numbers in a commutative,  associative  algebra over real numbers,  with two imaginary units (designated i and k). Reading further, I find the following fascinating remark,

The tessarines are now best known for their subalgebra of real tessarines t = w + y j, also called split-complex numbers, which express the parametrization of the unit hyperbola. [Wikipedia]

The rectangular hyperbola x2-y2 and its conjugate, having the same asymptotes. The Unit Hyperbola is blue, its conjugate is green, and the asymptotes are red. By Own work (Based on File:Drini-conjugatehyperbolas.png) [CC BY-SA 2.5],via Wikimedia Commons

Note to self:  Also investigate Cayley–Dickson constructionandzero divisor. Remember,  this is a work still in progress,  and if a  bona fide mathematician  believes division by zero is possible in some circumstances,  (as is avowed by mandalic geometry), I want to find out more about it.

© 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 281-

Beyond Descartes - Part 4
Directional Locatives

(continued from here)

Descartes derives his directional locatives from considerations of human anatomy, as does most of Western culture. The descriptive terms generally used for orientation purposes include left/right;up/down; and forward/backward.^[1] The first two sets have been extended also to refer to the cardinal directions, North/South and East/West.

To the degree that they conform to Cartesian coordinates, mandalic coordinates adhere to this schema as well.  However, mandalic geometry and the Taoist I Ching upon which it is largely based constitute a system of combinatorial relationships that is rooted mainly in  radial symmetry rather than bilateral symmetry. For mandalic coordinates, the principal directional locatives can be characterized as  divergentandconvergent, and the principal movements or changes in position, as centrifugalandcentripetal.^[2]

One of the important consequences of this alternative geometric perspective is that the frame of reference as well as the complex pattern produced are more integrative than in the method of Descartes. Looked at another way, Descartes is most enamored by specification of location of individual points whereas mandalic geometry is more concerned with relationships of parts - and the overall unification of the entire complex holistic system.^[3]

From this one seemingly small difference an enormous disparity grows in a manner reminiscent of chaos theory.^[4] Cartesian coordinates and mandalic coordinates can be made commensurate, but remain after all two exclusive systems of spatial awareness,  leading to very disparate results arising out of what seem small initial differences.^[5]

(continuedhere)

Image (bottom): Animation of a double compound pendulum showing chaotic behaviour. By Catslash (Own work). [Public domain], via Wikimedia Commons.^[6]

Notes

[1] Such terminology is of little use, despite its biological origins, to an amoeba or octopus,  not to mention those  extraterrestrials  who have been blessed with a second set of eyes at the back of their heads. (We wuz cheated.)

[2] To be more correct, the radial symmetry involved is of a special type. It is not simple planar radial symmetry, nor even the three-dimensional symmetry of a cube and its circumscribed and inscribed spheres. It is all of those but also the symmetry involved in all the different faces of a six-dimensional hypercube and the many relationships among them.

[3] To be fair, Descartes eventually gets around to relating his points in a systematic whole we now know as analytic geometry (1,2).  But as great an achievement though it might be,  Cartesian geometry  lacks the overarching cosmographical implications which characterize mandalic geometry and the I Ching. Descartes’ system is purposed differently, arising as it does out of a very different world view. To paraphrase George Orwell,

“All geometries are sacred, but some geometries are more sacred than others.”

[4] Chaos theory was summarizedbyEdward Lorenzas:

“When the present determines the future, but the approximate present does not approximately determine the future.”

[5] An example of one unique result of mandalic coordination of space is the generation of a geometric/logical probability wave of all combinatorial elements that occur in the 6D/3D hybrid composite dimension specification of the system. I envision this as offering a possible model at least,  if not an actual explanation, of the  probabilistic nature  of quantum mechanics.  Extrapolating this thought to its uttermost conclusion, it is not entirely inconceivable, to my mind at least, that probability itself might be the result of composite dimensioning. (And for such a brash remark I would almost surely be excommunicated from the fold were I but a member.)

[6] Starting the pendulum from a slightly different initial condition would result in a completely different trajectory.  The double rod pendulum is one of the simplest dynamical systems that has chaotic solutions. [Wikipedia]

© 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 280-

Beyond Descartes - Part 3
Logic Gates and Switches: Introduction

(continued from here)

It has been often noted throughout this work that mandalic geometry does not view points as fundamental geometrical elements in the manner Descartes and Euclid do. It considers them to be evanescent communions of two or more dimensions.  This  alternative perspective  conveys further the insight  that such conjoint formative interface locations both separate and connect. They are both boundaries and tipping points between all the participating dimensions,  what I have whimsically referred to  previously as dimension interchange lanes.  This is a far cry from the way Descartes regards and handles hispoints.

Descartes’points are locations, pure and simple, defining occupants of a uniform geometrical space. They don’t really doorattempt anything; they simply are.  They do not act,  but are acted upon by the equations of Cartesian geometry.  The  points themselves,  for all the reality Descartes attempts to imbue them with, turn out,  when the curtain is drawn,  to be no more capable of mustering an original thought  than is  the Scarecrow in  L. Frank Baum’s  The Wonderful Wizard of Oz.  Being of feeble mind themselves,  they just sit there awaiting brainy algebra to act upon them. In and of themselves,  beyond determining location,  they are essentially impotent.^[1]

A useful way to apprehendpoint locations of mandalic coordinates is to  interpret them  as  logic gates  which can handle  transition operations in a variety of different ways  depending upon the  dimension amplitudes verged on.  Passage through such locations is potentially bidirectional,  in theory if not always in actuality at a given moment, so they accommodate both  convergent and divergent flows  throughout varied amplitude levels of the mandalic structure.  To wit,  they can promote both  differentiationandpotentialization  phases of an evolving process.  Because these points arise through confluence of dimensions,  they bear within their transitory being information imparted by the participating dimensions.  Contrary to Descartes’ simpleminded points, these points have the capacity to encode an intelligence derived from their parent dimensions.^[2]

In electrical engineering,aswitch is an electrical component that can control an electrical circuit  by initiating or interrupting the current  or by diverting it from one conductor to another.  The most usual configuration consists of  a manually operated electromechanical device  having  one or more sets of electrical contacts.  These contacts are connected to external circuits. Each set of contacts can be in either of two states: either “closed” meaning the contacts are touching and electricity can flow between them, or “open”, meaning the contacts are separated in which case the switch is nonconducting. The mechanism that brings about the transition between these two states - openorclosed - can be either a “toggle”  (flip switch for continuous “on” or “off”)  or  “momentary”  (depress and hold for “on” or “off”) type.

Understand that logic gates don’t apply only to electronic devices nor are they controlled only by such devices. The concepts and methodologies involved go far beyond simple electronics.

• Logic gates are primarily implemented using diodes or transistors acting as electronic switches, but can also be constructed using vacuum tubes, electromagnetic relays (relay logic), fluidic logic, pneumatic logic, optics, molecules, or even mechanical elements. With amplification, logic gates can be cascaded in the same way that Boolean functions can be composed, allowing construction of a physical model of all of Boolean logic, and therefore, all of the algorithms and mathematics that can be described with Boolean logic. Wikipedia

For our purposes here and now, we need only mention that scalar numbers and vectors can be implemented in the context of Boolean logic as well.  Indeed, the incessant complex cotillion performed by subatomic particles can likely be subjected to such an analysis or one similar.^[3] And, of course, also digital circuits and computer architecture.

This has been just an introductory teaser to the topic of logic gates in mandalic geometry.  I’m getting my feet wet now myself. This is all still quite new to me so we’ve barely scratched the surface here.  An upcoming post will survey the logic gates and switches identifiable among groups of transliteration Cartesian coordinates and mandalic coordinates. This may take a while to materialize, but I think will be worth the wait.  And in case I forget to bring up the subject of how fractals fit into all this sometime in the next month or two, remind me please that I intended to.

(continuedhere)

Notes

[1] This could be a mathematician’s beautiful dream, but a physicist’s abhorrent nightmare.

[2] Although this statement pertains especially to composite dimension points, it is true, to a degree, of ordinary three-dimensional points as well when viewed in a manner similar to that using trigram tranliterations of Cartesian triads.  This means then that Cartesian coordinates could do the same and to the same degree, if  they were handled in the same manner as trigram coordinates are. The point is they are not and presumably never were.

[3] With that last remark I likely committed quantum mechanical heresy. If I in fact did, so be it. If it doesn’t quite hit the intended mark we can refer to it as steampunk mechanics.

Image (lower): Boolean lattice of subsets. KSmrq. Licensed under CC BY-SA 3.0viaCommons.

© 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 279-

Earlier to Later Heaven: Fugue VII Beyond Descartes - Part 2
A Different Zero

(continued from here)

Mandalic geometry has been formulated in such a manner as to be fully commensurate with Descartes’ coordinate system. Firstly, because it can be.  Beyond that,  because Descartes’ system is known throughout the world, and is endorsed by all conversant in disparate fields of science and mathematics. Moreover, the Cartesian coordinate system is a special case of the mandalic coordinate system,  bearing a relationship to it analogous to that which Newtonian mechanics does to quantum mechanics.

One of the fundamental differences lies in the way the two regard zero locations. Descartes, taking his cue from the Western number line, constructs a coordinate system which envisages a single common origin to all three dimensions, while maintaining between those dimensions a rigid uncompromising distinction. Mandalic geometry views dimension as primary rather than points, lines, or two or three dimensional figures. It does not regard dimensions as intrinsically separate in the manner in which they  exist and relate  to one another.  This allows for a far greater degree of flexibility of what we view as parts of the system, including the possibility of folding each into another,  through different dimensions as well as the same dimension.

For Descartes, zero is the empty location, the no man’s land where positive and negative vectors of each dimension invert or fail to invert.  A negative vector acting on a positive vector or another negative vector will cause inversion.  A positive vector, acting on a negative vector or another positive vector, will not. For mandalic geometry, zeros are that, but more. They are dimension interchange lanes,  and also locations of dimensional amplitudetransition.^[1]

Descartes, influenced still by the number line, proceeds to build a geometric universe based largely on scale. It is an imposing edifice nearly purely divergent,  constructed from three largely independent linear axes of evolutionary zeal.  Taoist cosmology and mandalic coordinates equally eschew an impressive but mundane number line in pursuance of complex twisting and intertwining of parts evolved on the underlying principles of modularity, repetition, reflection, relationship and recursion.^[2]

These are two very different universes of logic.  Descartes’ approach leads to a description of space as being homogenous, isotropic, and fixed while that of mandalic geometry leads alternatively to a spacetime which is inhomogeneous, anisotropic and dynamically variable.^[3] For Descartes space is a background arena,  the theater in which all events transpire.^[4] For mandalic geometry,  space-time is foreground and background both. It is the sole ground which defines the nature of reality.

(continuedhere)

Notes

[1] The first,  dimensional interchanges,  occur in the Cartesian coordinate system but are generally neither recognized nor treated as such. Dimensional amplitude transition locations do not occur in Cartesian coordinates,  nor are they found in the simple 3D trigram Cartesian equivalent,  reproduced in the upper diagram above, as they are a manifestation only of compositing of two or more dimensions. They are attributes of all hybrid composite dimensional systems,  for our purposes here, either the 6D/3D hybrid mandalic system of hexagrams,  the 4D/2D hybrid mandalic system of tetragrams,  or the 2D/1D hybrid mandalic system of bigrams.

[2] An important consequence here is that Descartes’ number line-based axes each contain a single zero. When mandalic coordinates are scaled up beyond the basic modular unit, every even number maintains all characteristics of the initial zero, including, most significantly, its multipotentiality. This is a basic axiomatic result of the intermingling, sharing nature of mandalic structure.

[3] It is this variability and dynamism of mandalic coordinates that make the method potentially suitable to mappings of subatomic particles as these are similarly variable and dynamic,  sharing importantly also the ability of exchanges / interchanges among their diverse numbers.

[4] Witness for example how Descartes exploits his newly formed coordinate system to stage, what was then, a cutting-edge geometric exposition of algebra, now referred to as analytic geometry. Mandalic geometry employs coordinates which are pre-invested with the ability to directly impart information regarding spatial transmutations themselves, without requirement of any intermediary.

© 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 278-

Earlier to Later Heaven: Fugue VI Beyond Descartes - Part 1

(continued from here)

In this post we take a short detour within our current central topic, that of relationship of Earlier Heaven and Later Heaven arrangements of the trigrams. The new material included here grew out of ruminations on the aforesaid primary topic though,  and is actually not so much a detour as a preparing the way for what I hope will be the eventual solution of our problem at hand.

Mandalic geometry, as we’ve seen, is fully commensurate with the coordinate system of Descartes, but its principal forebears lie elsewhere. It is derived largely from Taoist and pre-Taoist thought structures, most importantly the I Ching,  the earliest strata of which were formed before the separation of rational and irrational thought in the history of human cognition. As a result it is capable of far exceeding the possibilities of the Cartesian coordinate system, a product of the Enlightenment and Age of Rationalism. It offers geometry the possibility of a structural fluidity and a functional variability that Cartesian geometry lacks.^[1]

From the very beginning of this project I’ve been much puzzled by the lack in traditional Chinese thought  of a symbol corresponding to the zero of the Western number line and number theory.^[2] Traditional Asian thought does not uniformly lack a zero symbol.^[3] And yet the I Ching and Taoism manage well enough without one, electing to base their numerical relationships instead entirely on combinatorics involving permutations of yinandyang – what we in the West call  negativeandpositive – through multiple dimensions. It is an entirely different perspective arising out of a very different worldview.^[4]

What Taoism invented in the process was a unique,  thoroughly self-consistent brilliant system of logic/geometry/combinatorics which has been masquerading, all these many centuries,  as “just a method of divination.”^[5]  In essence, Chinese thought invented a discrete number system and geometry, one based on vectors rather than scalars, a vector geometry that can be extrapolated to any desired number of dimensions. The I Ching settles for just six,  the first whole number multiple of three. That is complicated enough.^[6][7]

(continuedhere)

Notes

[1] For one example of the advantages such variability and fluidity offer, in this particular case in creating  dynamic,  phase-shifting forms of nanomaterials,  see here.

[2] For a short history of the concept of zeroseeWho Invented Zero?

[3] The West, after all, derived its zero symbol ultimately via India.

[4] One might well speculate whether the significant root difference in world view between traditional Indian and Chinese thought lay in the fact that Indian mathematicians could have created a Zero out of nothingness (Śūnyatā),  a key term in Mahayana Buddhism and also some schools of Hindu philosophy while Taoist thought did not include a concept of nothingness. Instead it conceived of a formlessness prior to manifestation. In Taoist cosmology Taiji is a term for the “Supreme Ultimate” state of undifferentiated absolute and infinite potential,  the oneness before duality,  from which  yinandyang  originate.  So it might be that lacking a concept of nothingness forestalled invention of a zero symbol.  Still, it also allowed creation of an original,  unique holistic philosophy of reality, found perhaps nowhere else.

[5] The Russian philosopher, mathematician and authorPeter D. Ouspensky (1878-1947)  relates an apocryphal legend regarding the origin of the Tarot,  the moral of which has significance also to the history of the I Ching.

[6] In its emphasis on vector analysis and primacy of dimension the philosophy which underlies the I Ching and mandalic geometry  shares some characteristics of Clifford algebra.

[7] One of the important things with respect to physics I hope to show with mandalic geometry is that it is possible to construct an integrated geometrical / logical system which is self-sufficient and self-consistent, capable of modeling interactions of subatomic particles of the Standard Model and then some.  This goal is,  I believe,  approximated in mandalic geometry by meticulous coupling of the methodologies of composite dimension and trigram toggling,  although it quickly becomes apparent that a system based upon what is after all a relatively small number of dimensions - six in the case in point - becomes vastly complex and difficult to follow, at least initially.  One can’t help wondering how physics will be able to correlate all the intricate data resulting from its countless particle accelerator collisions and combine it into a consistent whole without some very fancy mental acrobatics on the part of theoretical physicists.  Without a suitable logical scaffold that might take an inordinately long time to achieve.

© 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 277-

Earlier to Later Heaven: Fugue V Alchemy Is Not a Dirty Word

(continued from here)

As is the case with all great mysteries, many different explanations for the origins of both the EARLIER and LATER HEAVEN arrangements of the trigrams have been proposed. In seeking plausible explanations,  it is well to keep in mind what the worldview contextual origin of these two different arrangements might have been.  The dates of their origins seem to be separated by thousands of years.^[1]   They grew out of very different worldviews.  Still there was a clear attempt to correlate LATER HEAVEN with EARLIER HEAVEN.  At the time LATER HEAVEN appeared on the scene,  there was a longstanding tradition in place which was not entirely discarded. A kind of conceptual amalgamation took place.^[2] The mystery lies in how and why that was accomplished.

Alchemy in its most fundamental aspect has to do with all kinds of relationships and from such a perspective both Earlier Heaven and Later Heaven are alchemical in nature. But differently so.  They are reflections of different ways of seeing reality.  Both,  however,  are debilitated, in the sense that they are missing a third dimension.  The trigram, having three Lines, is a combination form which maps three dimensions.  It cannot be adequately represented by figures of one or two dimensions. Any attempt to do so inevitably provokes injustice to reality by misrepresenting all the combinatoric relationships possible among the eight trigrams.

Placing Earlier Heaven in context of three dimensions enables it to express the full range of relationships and changes that can occur among the eight trigrams,  thereby creating a combinatoric system resembling a Boolean lattice.  This is very likely the form in which Earlier Heaven was originally understood,  whether explicitly or implicitly.  At some stage of the development of Chinese philosophy, it lost its clear connection to the third dimension,  possibly as a result of the new method of encoding and storing ideas in writing, which demanded linear text, displacing the older oral tradition. And that suggests a possible clue to the mystery of the two trigram arrangements.

(continuedhere)

Notes

[1] Both arrangements of the trigrams are of great antiquity, Earlier Heaven possibly dating to around 3000 BCE and Later Heaven to the 8th century BCE. The point here is that the two arose in widely different cultural contexts having differing worldviews with all that entails. Yet they were related one to the other in the Chinese mind.  Imagine,  if you will,  say,  the unlikely amalgamation of the phlogiston theory  with the scientific oxygen theory of combustion  or with modern thermodynamics in the West. How likely is it that such amalgamations could have occurred though the ideas involved were separated not by millennia but only a few centuries.  Something very strange and unexplained happened in the history of Chinese thought when these two widely different arrangements of trigrams were entangled with one another.  Things are not as they appear on the surface here. Something hermetic and profound is going on at a deeper level.

[2] Something similar did occur in the West when chemistry took over from alchemy. Though chemistry owes a debt to alchemy for its very existence it is somewhat in denial regarding its origins or at least about this particular aspect of its origins. Having alchemy in its family tree, however, is not something for chemistry to be ashamed about.  Alchemy had been practiced in many parts of the world for several millennia before chemistry appeared and can boast many important accomplishments in the history of human cognition. It could be said chemistry  threw out the baby with the bath water  when it conclusively broke with alchemy, except that it never did do that, not completely.  Alchemy in the broadest and best sense has to do with relationships of objects,  one to another. Much the same can be said about chemistry.

Modern astronomy also pays homage to an earlier form of ideation when it invokes the notion of a constellation, a term first used in astrology which was considered a scholarly tradition throughout most of its history.  Constellations in most cases are composed of stars which, though visible in the same general area, are often located at very different distances from Earth.  Nonetheless the tradition of referring to constellations is still in use by astronomers today.  Its expediency is considerable as any given point in a celestial coordinate system can easily and unambiguously be assigned to a constellation,  88 of which are officially recognized and used in modern astronomy.  The past lives on, in the history of ideas, but changed.

© 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 276-

Chinese history and culture (click for more)

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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