#change in the i ching

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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