#binary numbers
Mandalic geometry, Cartesian coordinates and Boolean algebra: Relationships - I
(continued from here)
In attempting to understand the logic of the I Ching it is important to know the differences between ordinary algebra and Boolean algebra and how Boolean algebra is related to the binary number system.[1]
In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted
1 and 0 respectively. Instead of elementary algebra where the values of the variables are numbers, and the main operations are addition and multiplication, the main
operations of Boolean algebra are the conjunctionand, denoted ∧, the disjunctionor, denoted ∨, and the negationnot, denoted ¬. It is thus a formalism for describing logical relations in the same way that ordinary algebra describes
numeric relations. [Wikipedia]
Whereas in elementary algebra expressions denote mainly numbers, in Boolean algebra they denote the truth values false and true. These values are represented with the bits (or binary digits), namely 0 and 1. They do not behave like the integers 0 and 1, for which
1 + 1 = 2, but may be identified with the elements of the two-element field GF(2), that is, integer arithmetic modulo 2, for which 1 + 1 = 0. Addition and multiplication then play the Boolean roles of XOR (exclusive-or) and AND (conjunction) respectively, with disjunction x∨y (inclusive-or) definable as x + y + xy. [Wikipedia][2]
Mandalic logic already occurs fully in the structure and manner of divinatory practice of the I Ching, if some of it only implicitly. Although mandalic geometry does not originate from either Boolean algebra or the Cartesian coordinate system but from the primal I Ching which predates them by millennia, it does combine and augment aspects of both of these conceptual systems. It extends Boole’s system of symbolic logic to include an additional logic value represented by the number -1. This necessitates modification of some of Boole’s postulates and rules, and increases their total number through introduction of some new ones. The hexagrams or native six-dimensional mandalic coordinates of the I Ching are related to Cartesian triads composed of the numbers -1, 0, and 1, making these two geometric systems commensurate by means of composite dimension, a 6D/3D hybridization or mandalic coordination of structure and function (or space and time).[3]
The introduction of composite dimension produces four distinct dimensional amplitudes and is solely responsible for the mandalic form. For anyone reading this who might be down on sacred geometry, itself a subject which I respect and admire, let it be known that I am talking here about genuine mathematics and symbolic logic, and my suspicion is that there is some genuine physics involved as well.
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.
(continuedhere)
Images: Upper: TRANSFORMATION OF THE SYMBOL OF YIN (LINE split in two) AND YANG (STRAIGHT-LINE). BLEND: 4 bigrams, THEN 8 trigrams. (MORAN, E. ET AL. 2002: 77). Found here. Lower: Modified from an animation showing how the taijitu (yin-yang diagram) may be drawn using circles, then erasing half of each of the smaller circles. O'Dea at WikiCommons [CC BY-SA 3.0orGFDL],via Wikimedia Commons
Notes
[1] Boole’s algebra predated the modern developmentsinabstract algebra and mathematical logic but is seen as connected to the origins of both fields. Similarly to elementary algebra, the pure equational part of the theory can be formulated without regard to explicit values for the variables.
[2] If you are new to Boolean algebra these definitions may be confusing because in some ways they seem to fly in the face of ordinary algebra. I’ll admit, I find them somewhat daunting. Let me see if I can clarify the three examples given in this quote. Those of you more familiar with the language of Boolean algebra might kindly correct me in the event I err. I’m growing more comfortable with being wrong at times. And this is after all a work in progress.
Boolean XOR (exclusive-or) allows that a statement of the form (x XOR y) is TRUE
if either x or y is TRUE but FALSE if both are TRUE or if both are FALSE. Since Boolean algebra uses binary numbers and represents TRUE by 1, FALSE by 0, then
for x = TRUE, y = TRUE x + y = 1 + 1 = 0 , so FALSE
for x = FALSE, y = FALSE x + y = 0 + 0 = 0 , so FALSE
for x = TRUE, y = FALSE x + y = 1 + 0 = 1 , so TRUE
for x = FALSE, y = TRUE x + y = 0 + 1 = 1 , so TRUEBoolean 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 FALSEBoolean OR (inclusive-or) is the truth-functional operator of (inclusive) disjunction,
also known as alternation. The OR of a set of operands is true if and only if one or
more of its operands is true. The logical connective that represents this operator is
generally written as ∨ or +. As stated in the Wikipedia article logical disjunction x∨y
(inclusive-or) is definable as x + y + xy [(x OR y) OR (x AND y)] as shown below.
[Note: x AND y is often written xy in Boolean algebra. So watch out whichalgebra
is being referred to, ordinary or Boolean. Are we confused yet?]
for x = TRUE, y = TRUE x + y = 1 , xy = 1 , so TRUE
for x = FALSE, y = FALSE x + y = 0 , xy = 0 , so FALSE
for x = TRUE, y = FALSE x + y = 1 , xy = 0 , so TRUE
for x = FALSE, y = TRUE x + y = 1 , xy = 0 , so TRUE
[3] Fundamentally, though, the coordinates of mandalic geometry refer to vector directions alone, rather than to both vectors and scalars (or direction and magnitude) as do Cartesian coordinates. Yin specifies actually the entire domain of negative numbers rather than just the scalar value -1. Yang similarly refers to the entire domain of positive numbers rather than the scalar value 1 alone. Their conjunction through the compositing of dimensions, though represented by the symbol zero (0) in the format commensurate with Cartesian coordinates, refers actually to a state or condition not found in Western thought outside of certain forms of mysticism and other outsider philosophies like alchemy; equilibration of forces in physics; equilibrium reactions in chemistry; and the kindred concept of homeostasis mechanisms of living organisms found in biology.
[4] This is to Westerners counterintuitive. Our customary logic and arithmetic allows for but a single zero. That two different zeros might exist concurrently or consecutively is - to our minds - irrational and we wrestle mightily with the idea. To complicate matters still more, neither of these zeros is conveniently like our familiar Western zero. So which should win out here? Rationality or reality? In fact, the decision is not ours. In the end nature decides. Nature always decides. It stuffs the ballot box and casts the deciding vote much to our chagrin, leaving us powerless to contradict what we may interpret as a whim. Our votes count for bupkis.
[5] This calls to mind also the Möbius strip which involves a twist that looks very much like a decussation to me. The decussation or twist in space we are talking about here though has a sort of wormhole at its center that connects two contiguous dimensional amplitudes. I can’t say more about this just now. I need to think on it still. It seems a promising subject for reflection. (1,2,3)
[6] It needs to be pointed out here that in mandalic geometry, and similarly in the primal I Ching as well, a bigram can be formed from any two related Lines of hexagrams, trigrams, and tetragrams. The two Lines need not be (and often are not) adjacent to one another. I would think such versatility might well prove useful for modeling and mapping quantum states and interactions.
[7] Note that yin and yang in composite dimension can each take the absolute values 0, 1, and 2 but when yin has absolute value 2, yang has absolute value 0; when yang has absolute value 2, yin has absolute value 0. This inverse relation in fact is what makes the arrangement here a superimposed, actually interwoven, dual modulo-3 number system. It also makes the center points of mandalic lines,squares, and cubes more protean and less differentiated than their vertices and elicits the different amplitudes of dimension.
The composite dimension value at the origin points(centers) of all of these geometric figures is always zero in Cartesian terms since the values of the differing Lines in the two entangled 6-dimensional hexagrams located here add to zero. But neither of these 6-dimensional entities is in its ground state at the center. Both have absolute value 1 at Cartesian 0. Let me say that again: composite dimension values at the center or origin are zero in Cartesian terms but the values of both individual constituents are non-zero.Yin is in its ground state when yang is at its maximum and vice versa. At the center, since the two are equal and opposite they interfere destructively. This results in a composite zero ground state.
So from the perspective of Cartesian coordinate dynamics, which is after all the customary perspective in our subjective lives, we encounter only emptiness. But it is this very emptiness that opens to a new dimension. In the hybrid 6D/3D mandalic cube only line centers and the cube center have direct access through change of one dimension to face centers and only the face centers have a similar direct access through a single dimension to the cube center and edge centers. All coexist in an ongoing harmony of tensegrity. There is method to all this madness then.
© 2016 Martin Hauser
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Beyond Taoism - Part 2
Number System of the I Ching
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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 FH(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
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Beyond Boole - Part 1
Symbolic Logic for the 21st Century
Boolean Algebra:
Fundamental Operations
(continued from here)
Looking back on how we arrived at this stage of reconstruction of Western thought, I see the difficulty arose in attempting to explain the “missing zero” of Taoism. Blame our troubles on Leibniz. It was he who introduced binary numbers to the West, and made the fateful choice of using zero(0) instead of -1 to counter with +1. Leibniz knew full well of the I Ching, but did not understand it well. He missed the point, seeing in it only a resemblance to his own newly devised system of numbers.
By Leibniz’s time negative numbers were firmly entrenched in the European mind. Why did Leibniz ignore them completely? In doing so he blazed a new trail that led eventually to the digital revolution of recent times. It also led to a dead end in the history of Western thought, one the West has not yet come fully face to face with. It will, though. Give it a few more years.[1]
George Boole, the inventor of what we know today as Boolean logic or Boolean algebra, was one of the thinkers who followed in the footsteps of Leibniz, building on the trail he blazed.[2] When he came to devise his truth tables, he also chose zero(0) as the counterpart to one(1). This led to certain resounding successes. And ultimately, to certain failures that introduced yet another layer to the blind spot of Western symbolic logic. Here we are, almost two centuries later,[3] saddled with and hampered by the unfortunate fallout of that eventful decision still.[4]
Most arguments in elementary algebra denote numbers. However, in Boolean algebra, they denote truth values falseandtrue. Convention has decreed these values are represented with the bits (or binary digits), namely 0 and 1. They do not behave like the integers 0 and 1 though, for which 1 + 1 = 2, but are identified with the elements of the two-element field GF(2), that is, integer arithmetic modulo 2, for which 1 + 1 = 0. (1,2) This causes a substantial problem when we attempt correlation of Taoist logic and Boolean logic. As we will soon discover, Taoist logic is a hybrid logic that is based on both vector inversion and arithmetic modulo 2. As such, it ought prove relatable to both Cartesian coordinates and Boolean algebra, though it may necessitate “forcing a larger foot in a smaller glass slipper.”
Taoism chose ages ago to use ‘yin’ and 'yang’ as its logical symbols. Although this appears, at first, to be a binary system, like those of Leibniz and Boole, on closer inspection it proves not to be. It is one of far greater logical complexity, alternatively binary or ternary with intermediate third element understood. This implied third element is able to bestow balance and equilibrium throughout all of the Taoist logical system. This is where the 'missing zero’ of Taoism went. Only it is a very different zero than the 'zero’ of Western thought. It is a zero of infinite potential rather than one of absolute emptiness. It is a zero of continual beginnings and endings, not of finality. It is one of the things that make the I Ching totally unique in the history of human cognition. All these hidden zeros are wormholes between dimensions and between different amplitudes of dimension.
So where does this all lead to, then? We’ve seen that the Taoist 'yin’ can readily be made commensurate with 'minus 1’ of Western arithmetic, the number line, and Cartesian coordinates.[5] But if it is to remain true to Taoist logic, it cannot be made commensurate with the Western 'zero’. We’ve found the Taoist number system and geometry to be Cartesian-like but not Cartesian. Now we discover them to be Boolean-like, not Boolean. Sorry, Leibniz, they are not so much as remotely like your binary system. You were far too quick to disesteem the unique qualities of the I Ching.[6]
This all has far-reaching consequences for Western thought in general. Especially though, for symbolic logic, mathematics, and physics. More specifically for our purposes here it means that when we create our Taoist notation transliteration of Cartesian coordinates, we will need also to translate Boolean logic into terms compatible with Taoist thought, that is to say, from a two-value system based on '1s’ and '0s’ into a three-value system based on '1s’, ’-1s’, and the ever-elusive invisible balancing-act '0s’ of Taoism.[7] We turn to that undertaking next.
(continuedhere)
Image: Fundamental operations of Boolean algebra. Symbolic Logic, Boolean Algebra and the Design of Digital Systems. By the Technical Staff of Computer Control Company, Inc. Other logical operations exist and are found useful by non-engineer logicians. However, these can always be derived from the three shown. These three are most readily implementable by electronic means. The digital engineer, therefore, is usually concerned only with these fundamental operations of conjunction, disjunction, and negation.
Notes
[1] It is at times like this that I am thankful I am not a member of Academia. Were I so, I could not afford, from a practical standpoint, to make claims such as this. Tenure notwithstanding.
[2] A knowledge of the binary number system is an important adjunct to an understanding of the fundamentals of Symbolic Logic.
[3] If we look back far enough in time, it was the introduction of “zero” as a number and a philosophical concept that led us down this tangled garden path, though the history of human thought is nothing if not interesting.
[4] Far out speculative thought here: Were binary numbers and Boolean logic based on +1s and -1s instead of +1s and 0s, might it not be possible to construct today a software-based quantum computer requiring no fancy juxtapositions and superpositions of subatomic particles? Think on it for a while before dismissing the thought as irrational folly.
[5] More correctly expressed, it can be made commensurate with the domain of negative numbers, since it is a vector symbol, properly speaking, concerned only with direction, not magnitude.
[6] Unfortunately there is still little understanding of the true nature of the symbolic logic encoded in the I Ching, as exemplified by this quote:
The I Ching dates from the 9th century BC in China. The binary notation in the
I Ching is used to interpret its quaternary divination technique.
It is based on taoistic duality of yin and yang.Eight trigrams (Bagua) and a set of 64 hexagrams (“sixty-four” gua), analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China.
The contemporary scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, although he did not intend his arrangement to be used mathematically. Viewing the least significant bit on top of single hexagrams in Shao Yong’s square and reading along rows either from bottom right to top left with solid lines as 0 and broken lines as 1 or from top left to bottom right with solid lines
as 1 and broken lines as 0 hexagrams can be interpreted as sequence from 0 to 63.
[Wikipedia]
It was this Shao Yong sequence of hexagrams (Before Heaven sequence) that Leibniz viewed six centuries after the Chinese scholar created it, so maybe he can be forgiven his error after all.
The more significant point here might be that an important Neo-Confucian philosopher, cosmologist, poet, and historian of the 11th century either was no longer able to access the original logic and meaning of the I Ching or, at the very least, was hellbent on reinterpreting it in a manner contradictory to its original intent. The latter is a distinct possibility, as Neo-Confucianism was an attempt to create a more rationalist secular form of Confucianism by rejecting superstitious and mystical elements of Taoism and Buddhism that had influenced Confucianism since the Han Dynasty (206 BC–220 AD).
[7] Taoist logic and mandalic geometry share some of the characteristics of both Cartesian coordinates and Boolean logic, but not all of either. Descartes’ system is indeed a ternary one when viewed in terms of vector direction rather than scalar magnitude. That fits with the requirements of Taoist logic. It is, on the other hand, dimension-poor, as Taoist logic and geometry require a full six independent dimensions for execution. Boolean logic lacks the necessary third logical element -1, which causes inversion through a central point of mediation. But we shall see, it does bestow the ability to enter and exit a greater number of dimensional levels by means of its logical gates. Used together in an appropriate manner, these two can provide a key to understanding Taoist logic and geometry. Speculating even further, Taoist thought might provide a key to interpretation of quantum mechanics, the same quantum mechanics devised in the early twentieth century that no one can yet explain. Well, I mean, actually, Taoist thought in the formulation given it by mandalic geometry. Why feign modesty, when this work will likely linger in near-total obscurity for the next hundred years gathering dust or whatever it is that pixels gather in darkness undisturbed.
© 2015 Martin Hauser
Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)
Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)
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