So yes. This is very much a work in progress. And we have strayed now as it happens into unfamiliar territory. Terra incognita. Therewill be dragons.[1] Dragons are errors. Errors are dangerous, and we must slay them. But all in good time. First, we should scout out the terrain. That would be prudent.
Descartes in constructing his system of coordinates built upon the bedrockofelementary algebra and the number line. We’vepreviously called attention to the important but mostly overlooked issue of the 1:1 congruence between number and geometric/spatial position he incorporated implicitly in the logic of his coordinates and questioned the validity of such correspondence, at least with respect to subatomic scales.
Working two centuries later but very much under the influence of Descartes’ thought, George Boole introduced his own unique brand of algebra. A second major influence on the development of his symbolic logic was the binary number system of Leibniz, himself influenced to a large degree by Descartes. We need to carefully follow and connect the dots here. Great advances in human cognition rarely, if ever, occur in isolation and seclusion. There is a fine line to tread though. If progress requires the shoulders of giants to stand on, it is still difficult at times not to be overly influenced by those who came before.
Boole’s new logic, constructed in the wake of what by his time were firmly entrenched systematizations of thought by two of the most highly regarded philosopher mathematicians, was devised in such a manner as to conform to both of these conventions of system design. Significant to our purposes here are the facts that first, Boolean logic echoes Cartesian convention of attributing to each and every location in geometric space a single unique number, and second, it adheres to Leibniz’s convention of using a modulo-2 number system based on binary elements 1 and 0.[2]
The symbolic logic systems of mandalic geometry and the I Ching do not abide by either of these conventions. Instead they are based on what is best described as composite dimensions with four unique truth values (or vector directions) each, ranging from -1 through two distinctive zeros (0a; 0b) to +1, and assignment of numbers to spatial locations through all dimensions by means of probability distributions in place of a simple and simplistic 1:1 distribution. To accommodate these alternative conceptual concepts, we will need to expand and modify traditional Boolean logic as we have already done as regards Cartesian coordinate theory.
For starters here we should doubtless add, the mandalic form is the probability distribution through all dimensions, and the probability distributions are the mandalas. And movement through either or both can only be accomplished by discretized stepwise maneuvers between different amplitudes of dimension separated by obscure quantum leaps of endless being and becoming and being and unbecoming, toward and away from the centers and subcenters of holistic systems, the parts of which are always aiming towards some kind of equilibrium never quite within reach. Which then makes error also a necessary aspect of reality and not simply the fearful monster we imagined. It is error that makes achievement possible.[3][4]
Image:Here Be Dragons Map. Detail of he Carta marina (Latin “map of the sea” or “sea map”), drawn by Olaus Magnus in 1527-39. This is the earliest map of the Nordic countries that gives details and place names, by Olaus Magnus [Public domain], via Wikimedia Commons. The map was in production for 12 years. The first copies were printed in 1539 in Venice. [Wikipedia]
Notes
[1] Mapmakers during the Age of Exploration sometimes placed the phrase “here be dragons” at the edges of their known world, presumably to warn of the dangers lying in wait for sailing vessels and travelers by land who strayed too far from well-traveled routes. Here is a list of all known historical maps on which these words appear.
[2] Or in Boole’s case, we might say, attributing to each proposition in concept space a single truth value: TRUE or FALSE (var YES or NO; or, in electronics applications, ON or OFF.) What we have here, I believe, is in many instances a false dilemma or the old Aristotelian dichotomy of either/or. Quantum physics demands and deserves better. OK, true enough, Boole gets around to extending possibilities by means of multi-term propositions, which his system can readily handle. The question here, though, is whether nature can or does handle such similarly. I think not. I think it approaches the question at a more fundamental level of reasoning and reality: at the most basic level of spacetime itself.
[3] This echoes the view of cybernetics, a transdisciplinary approach for exploring regulatory systems, their structures, constraints, and possibilities.
Cybernetics is relevant to the study of systems, such as mechanical, physical, biological, cognitive, and social systems. Cybernetics is applicable when a system being analyzed incorporates a closed signaling loop; that is, where action by the system generates some change in its environment and that change is reflected in that system in some manner (feedback) that triggers a system change, originally referred to as a “circular causal” relationship. [Wikipedia]
[4] This entire blog and its predecessor are in some sense the chronology of a journey from the familiar shoreline into largely uncharted waters. Hesitant at first, increasingly more daring as time has gone on and I’ve come to see errors to be stepping stones along the way. And there have beenmanyerrors along the way. Some I am not yet cognizant of. But of those I am aware, I have left most intact in spite of since being superseded by ideas superior, more correct or better formulated. I’ve done this because I think it important to map the course of a conceptual journey, how the ideas evolved from A to B to C to D. It also allows readers to participate, to a degree, in the thrill of an exciting adventure of mind, should they so choose. Happy travels.
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Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where n = x + 1 - p. :)
In attempting to understand the logic of the I Ching it is important to know the differences between ordinary algebra and Boolean algebra and how Boolean algebra is related to the binary number system.[1]
In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively. Instead of elementary algebra where the values of the variables are numbers, and the main operations are addition and multiplication, the main operations of Boolean algebra are the conjunctionand, denoted ∧, the disjunctionor, denoted ∨, and the negationnot, denoted ¬. It is thus a formalism for describing logical relations in the same way that ordinary algebra describes numeric relations. [Wikipedia]
Whereas in elementary algebra expressions denote mainly numbers, in Boolean algebra they denote the truth values false and true. These values are represented with the bits (or binary digits), namely 0 and 1. They do not behave like the integers 0 and 1, for which 1 + 1 = 2, but may be identified with the elements of the two-element field GF(2), that is, integer arithmetic modulo 2, for which 1 + 1 = 0. Addition and multiplication then play the Boolean roles of XOR (exclusive-or) and AND (conjunction) respectively, with disjunction x∨y (inclusive-or) definable as x + y + xy. [Wikipedia][2]
Mandalic logic already occurs fully in the structure and manner of divinatory practice of the I Ching, if some of it only implicitly. Although mandalic geometry does not originate from either Boolean algebra or the Cartesian coordinate system but from the primal I Ching which predates them by millennia, it does combine and augment aspects of both of these conceptual systems. It extends Boole’s system of symbolic logic to include an additional logic value represented by the number -1. This necessitates modification of some of Boole’s postulates and rules, and increases their total number through introduction of some new ones. The hexagrams or native six-dimensional mandalic coordinates of the I Ching are related to Cartesian triads composed of the numbers -1, 0, and 1, making these two geometric systems commensurate by means of composite dimension, a 6D/3D hybridization or mandalic coordination of structure and function (or space and time).[3]
The introduction of composite dimension produces four distinct dimensional amplitudes and is solely responsible for the mandalic form. For anyone reading this who might be down on sacred geometry, itself a subject which I respect and admire, let it be known that I am talking here about genuine mathematics and symbolic logic, and my suspicion is that there is some genuine physics involved as well.
The mandalic number system, then, is a quasi-modular number system, different from Leibniz’s binary number system which is fully modular. Boole’s rule 1 AND 1 = 1 still holds true in mandalic logic. However we must add to this the new logic rule that -1 AND -1 = -1. Individually the two rules are modular, based on a clock arithmetic using a modulo-3 number system rather than Leibniz’s modulo-2 or binary number system, but with yet another added twist.
Together the two rules prescribe a compound system, one which is not singly modular but doubly modular. The two components, yinandyang, are complementary and are inversely related to one another in this unified system. This logic organization appears based on the figure 8 or sine wave and its negative, allowing for periodicity, for recursive periods of interminably repeating duration, and, perhaps most importantly, for wave interference, of constructive and destructive varieties. These two geometric figures also engender an unexpected decussation of dimension not recognized by Western mathematics. This is so because 1 AND -1 = 0 and -1 AND 1 = 0. The surprise here is that there are two distinct zeros: 0a and 0b.[4] In two- or three-dimensional Cartesian terms there exists no difference between these two zeros. However, in terms of 6-dimensional aspects of mandalic geometry and the hexagrams of the I Ching, the two are clearly distinct structurally and functionally.[5]
This arithmetic system is the basis of the logic encoded in the hexagrams of the I Ching. Each hexagram uniquely references a single 6- dimensional discretized point, of which there are 64 total. These 64 6- dimensional points of the mandalic cube are distributed among the 27 discretized points of the ordinary 3-dimensional cube through the compositing of dimensions in such manner that a mandala is formed which positions 1, 2, 4 or 8 hexagrams at each 3-dimensional point according to the dimensional amplitude of the particular point. This necessarily creates a concurrent probability distribution of hexagrams through each of the three Cartesian dimensions.
TheI Chinguses a dual or composite three-valued logic system. In place of truth values, the variables used are yin, yang and the two in conjunction. These fundamentally represent vector directions. Yin is represented by -1, yang by 1, and their conjunction, using Cartesian or Western number terminology, by zero (0). This symbol does not occur natively in the I Ching though where the representation used is simply a combination of yin and yang symbols, most often in form of a bigram containing both and regarded as representing a composite dimension, namely 0[1] or 0[2].[6]
The two bigrams that satisfy the requirement are
young yang
for 0[1]
and
young yin
for 0[2].
Although mandalic logic is in Cartesian terms a 3-valued system, in native terms it is 4-valued. It is not a simple modulo-3 or modulo-4 number system, but two interrelated modulo-3 systems combined. The best way to think about this geometric arrangement is possibly to view it as a single composite dimension having four distinct vector directions: a negative direction represented by mandalic composite yin (Cartesian -1); positive direction represented by mandalic composite yang (Cartesian 1); and two decussating relatively undifferentiated directions in some sort of equilibrium, represented by mandalic 0[1] (composite yin/yang) and 0[2] (composite yang/yin). both of which devolve to Cartesian 0 (balanced vector direction of the origin or center).[7]
So we’ve seen that the number system used in the I Ching is not binary as Leibniz believed but instead doubly trinary with the two halves, in simplest terms, inversely related and intertwined. Still, it was an easy mistake to make because the notation used is binary. We’ve seen too that all trigrams and hexagrams in the system can be rendered commensurate with the Cartesian coordinate system: trigrams by simple transliteration, hexagrams by dimensional compositing. What, then, of George Boole and his eponymous logic? How do they fit in the logic scheme of the I Ching? I’m glad you asked. Stay tuned to find out.
Images: Upper: TRANSFORMATION OF THE SYMBOL OF YIN (LINE split in two) AND YANG (STRAIGHT-LINE). BLEND: 4 bigrams, THEN 8 trigrams. (MORAN, E. ET AL. 2002: 77). Found here. Lower: Modified from an animation showing how the taijitu (yin-yang diagram) may be drawn using circles, then erasing half of each of the smaller circles. O'Dea at WikiCommons [CC BY-SA 3.0orGFDL],via Wikimedia Commons
Notes
[1] Boole’s algebra predated the modern developmentsinabstract algebra and mathematical logic but is seen as connected to the origins of both fields. Similarly to elementary algebra, the pure equational part of the theory can be formulated without regard to explicit values for the variables.
[2] If you are new to Boolean algebra these definitions may be confusing because in some ways they seem to fly in the face of ordinary algebra. I’ll admit, I find them somewhat daunting. Let me see if I can clarify the three examples given in this quote. Those of you more familiar with the language of Boolean algebra might kindly correct me in the event I err. I’m growing more comfortable with being wrong at times. And this is after all a work in progress.
Boolean XOR (exclusive-or) allows that a statement of the form (x XOR y) is TRUE if either x or y is TRUE but FALSE if both are TRUE or if both are FALSE. Since Boolean algebra uses binary numbers and represents TRUE by 1, FALSE by 0, then for x = TRUE, y = TRUE x + y = 1 + 1 = 0 , so FALSE for x = FALSE, y = FALSE x + y = 0 + 0 = 0 , so FALSE for x = TRUE, y = FALSE x + y = 1 + 0 = 1 , so TRUE for x = FALSE, y = TRUE x + y = 0 + 1 = 1 , so TRUE
Boolean AND (conjunction) allows that a statement of the form (x AND y) is TRUE only if both x is TRUE and y is TRUE. If either x or y is FALSE or both are FALSE then x AND y is FALSE. Here algebraic multiplication of binary 1s and 0s plays the role of Boolean AND. (Incidentally, binary multiplication works exactly the same way as algebraic multiplication. There’s a gift!) for x = TRUE, y = TRUE xy = 1(1) = 1, so TRUE for x = FALSE, y = FALSE xy = 0(0) = 0, so FALSE for x = TRUE, y = FALSE xy = 1(0) = 0 , so FALSE for x = FALSE, y = TRUE xy = 0(1) = 0 , so FALSE
Boolean OR (inclusive-or) is the truth-functional operator of (inclusive) disjunction, also known as alternation. The OR of a set of operands is true if and only if one or more of its operands is true. The logical connective that represents this operator is generally written as ∨ or +. As stated in the Wikipedia article logical disjunction x∨y (inclusive-or) is definable as x + y + xy [(x OR y) OR (x AND y)] as shown below. [Note: x AND y is often written xy in Boolean algebra. So watch out whichalgebra is being referred to, ordinary or Boolean. Are we confused yet?] for x = TRUE, y = TRUE x + y = 1 , xy = 1 , so TRUE for x = FALSE, y = FALSE x + y = 0 , xy = 0 , so FALSE for x = TRUE, y = FALSE x + y = 1 , xy = 0 , so TRUE for x = FALSE, y = TRUE x + y = 1 , xy = 0 , so TRUE
[3] Fundamentally, though, the coordinates of mandalic geometry refer to vector directions alone, rather than to both vectors and scalars (or direction and magnitude) as do Cartesian coordinates. Yin specifies actually the entire domain of negative numbers rather than just the scalar value -1. Yang similarly refers to the entire domain of positive numbers rather than the scalar value 1 alone. Their conjunction through the compositing of dimensions, though represented by the symbol zero (0) in the format commensurate with Cartesian coordinates, refers actually to a state or condition not found in Western thought outside of certain forms of mysticism and other outsider philosophies like alchemy; equilibration of forces in physics; equilibrium reactions in chemistry; and the kindred concept of homeostasis mechanisms of living organisms found in biology.
[4] This is to Westerners counterintuitive. Our customary logic and arithmetic allows for but a single zero. That two different zeros might exist concurrently or consecutively is - to our minds - irrational and we wrestle mightily with the idea. To complicate matters still more, neither of these zeros is conveniently like our familiar Western zero. So which should win out here? Rationality or reality? In fact, the decision is not ours. In the end nature decides. Nature always decides. It stuffs the ballot box and casts the deciding vote much to our chagrin, leaving us powerless to contradict what we may interpret as a whim. Our votes count for bupkis.
[5] This calls to mind also the Möbius strip which involves a twist that looks very much like a decussation to me. The decussation or twist in space we are talking about here though has a sort of wormhole at its center that connects two contiguous dimensional amplitudes. I can’t say more about this just now. I need to think on it still. It seems a promising subject for reflection. (1,2,3)
[6] It needs to be pointed out here that in mandalic geometry, and similarly in the primal I Ching as well, a bigram can be formed from any two related Lines of hexagrams, trigrams, and tetragrams. The two Lines need not be (and often are not) adjacent to one another. I would think such versatility might well prove useful for modeling and mapping quantum states and interactions.
[7] Note that yin and yang in composite dimension can each take the absolute values 0, 1, and 2 but when yin has absolute value 2, yang has absolute value 0; when yang has absolute value 2, yin has absolute value 0. This inverse relation in fact is what makes the arrangement here a superimposed, actually interwoven, dual modulo-3 number system. It also makes the center points of mandalic lines,squares, and cubes more protean and less differentiated than their vertices and elicits the different amplitudes of dimension.
The composite dimension value at the origin points(centers) of all of these geometric figures is always zero in Cartesian terms since the values of the differing Lines in the two entangled 6-dimensional hexagrams located here add to zero. But neither of these 6-dimensional entities is in its ground state at the center. Both have absolute value 1 at Cartesian 0. Let me say that again: composite dimension values at the center or origin are zero in Cartesian terms but the values of both individual constituents are non-zero.Yin is in its ground state when yang is at its maximum and vice versa. At the center, since the two are equal and opposite they interfere destructively. This results in a composite zero ground state.
So from the perspective of Cartesian coordinate dynamics, which is after all the customary perspective in our subjective lives, we encounter only emptiness. But it is this very emptiness that opens to a new dimension. In the hybrid 6D/3D mandalic cube only line centers and the cube center have direct access through change of one dimension to face centers and only the face centers have a similar direct access through a single dimension to the cube center and edge centers. All coexist in an ongoing harmony of tensegrity. There is method to all this madness then.
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. :)
