#boolean algebra
Neo-Boolean - II: Logic Gates
Thinking Inside the Lines
(continued from here)
We have already looked briefly at three of the more important Boolean operators or logic gates: AND, OR,andXOR.NOT just toggles any two Boolean truth values (true/false; on/off; yes/no). Here we introduce two new logic gates which do not occur in Boolean algebra. Both play an important role in mandalic geometry though.
We’ll refer to the first of these new operators or logic gates as INV standing for inversionorinvert. This is similar to Boole’s NOT except that it produces toggling betweeen yang/+ and yin/- instead of 1 and 0. Because it is based on binary arithmetic, Boole’s NOT has been thought of as referring to inversion also (as in ONorOFF). Although both ANDandINV act as toggling logic gates they have very different results in the greater scheme of things, since nature has created a prepotent disparity between a -/+ toggle and a 0/1 toggle in basic parameters of geometry, spacetime, and being itself. This makes Boole’s AND just a statement of logical opposition, notinversion.
Recognition of this important difference is built into mandalic geometry structurally and functionally, as it is also into Cartesian coordinate dynamics and the logic of the I Ching, but lacking in Boole’s symbolic logic. This is necessarily so, as there is no true negative domain in Boolean algebra. The OFF state of electronics and computers, though it may sometimes be thought of in terms of a negative state, is in fact not. It relates to the Western zero (0), not the minus one of the number line. Where Boolean algebra speaks of NOT 1 it refers specifically to zero and only to zero. When mandalic geometry asserts INV 1 it refers specifically to -1 and only to -1 . The inversion of yang then is yin and the inversion of yinisyang.[1] In the I Ching, Taoist thought, and mandalic geometry the two are not opposites but complements and, as such, interdependent.
The second added logic gate that will be introduced now is the REV operator standing for reversionorrevert. This operator produces no change in what it acts upon. It is the multiplicative identity element (also called the neutral elementorunit element), as INV is the inverse element. In ordinary algebra the inverse element is -1, while the identity element is 1. In mandalic geometry and the I Ching the counterparts are yinandyang, respectively. If Boolean algebra lacks a dedicated identity operator, it nonetheless has its Laws of Identity which accomplish much the same in a different way:
- A = A
- NOT A = NOT A
Again, Boolean algebra has no true correlate to the INV operator. There can be no sign inversion formulation as it lacks negatives entirely. Although Boolean algebra may have served analog and digital electronics and digital computers quite well for decades now, it is incapable of doing the same for any quantum logic applications in the future, if only because it lacks a negative domain.[2] It offers up bits readily but qubits only with extreme difficulty and those it does are like tears shed by crocodiles while feeding.
(to be continued)
Image: Boolean Search Operators. [Source]
Notes
[1] Leibniz’s binary number system, on which Boole based his logic, escapes this criticism, as Leibniz uses 0 and 1 simply as notational symbols in a modular arithmetic and not as contrasting functional elements in an algebraic context of either the Boolean or ordinary kind.
In the field of computers and electronics, Boolean refers to a data type that has two possible values representing true and false. It is generally used in context to a deductive logical system known as Boolean Algebra. Binary in mathematics and computers, refers to a base 2 numerical notation. It consists of two values 0 and 1. The digits are combined using a place value structure to generate equivalent numerical values. Thus, both are based on the same underlying concept but used in context to different systems. [Source]
[2] Moreover, I expect physics will soon enough discover that what it now calls antimatter is in some sense and to some degree a necessary constituent of ordinary matter. I can already hear the loudly objecting voices declaring matter and antimatter in contact necessarily annihilate one another, but that need not invalidate the thesis just proposed. My supposition revolves around the meaning of “contact” at Planck scale and the light speed velocity at which subatomic particles are born, interact and decay only to be revived again in an eternal dance of creation and re-creation. Material particles exist in some kind of structural and functional homeostasis, not all that unlike the anabolic and catabolic mechanisms that by means of negative feedback maintain all entities of the biological persuasion in the steady state we understand as life. Physics has yet to get a full grip on this aspect of reality, though moving ever closer with introduction of quarks and gluons to its menagerie of performing particles.
© 2016 Martin Hauser
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-Page 304-
Bootstrapping Neo-Boolean - I
(continued from here)
So yes. This is very much a work in progress. And we have strayed now as it happens into unfamiliar territory. Terra incognita. Therewill be dragons.[1] Dragons are errors. Errors are dangerous, and we must slay them. But all in good time. First, we should scout out the terrain. That would be prudent.
Descartes in constructing his system of coordinates built upon the bedrockofelementary algebra and the number line. We’vepreviously called attention to the important but mostly overlooked issue of the 1:1 congruence between number and geometric/spatial position he incorporated implicitly in the logic of his coordinates and questioned the validity of such correspondence, at least with respect to subatomic scales.
Working two centuries later but very much under the influence of Descartes’ thought, George Boole introduced his own unique brand of algebra. A second major influence on the development of his symbolic logic was the binary number system of Leibniz, himself influenced to a large degree by Descartes. We need to carefully follow and connect the dots here. Great advances in human cognition rarely, if ever, occur in isolation and seclusion. There is a fine line to tread though. If progress requires the shoulders of giants to stand on, it is still difficult at times not to be overly influenced by those who came before.
Boole’s new logic, constructed in the wake of what by his time were firmly entrenched systematizations of thought by two of the most highly regarded philosopher mathematicians, was devised in such a manner as to conform to both of these conventions of system design. Significant to our purposes here are the facts that first, Boolean logic echoes Cartesian convention of attributing to each and every location in geometric space a single unique number, and second, it adheres to Leibniz’s convention of using a modulo-2 number system based on binary elements 1 and 0.[2]
The symbolic logic systems of mandalic geometry and the I Ching do not abide by either of these conventions. Instead they are based on what is best described as composite dimensions with four unique truth values (or vector directions) each, ranging from -1 through two distinctive zeros (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]
(continuedhere)
Image:Here Be Dragons Map. Detail of he Carta marina (Latin “map of the sea” or “sea map”), drawn by Olaus Magnus in 1527-39. This is the earliest map of the Nordic countries that gives details and place names, by Olaus Magnus [Public domain], via Wikimedia Commons. The map was in production for 12 years. The first copies were printed in 1539 in Venice. [Wikipedia]
Notes
[1] Mapmakers during the Age of Exploration sometimes placed the phrase “here be dragons” at the edges of their known world, presumably to warn of the dangers lying in wait for sailing vessels and travelers by land who strayed too far from well-traveled routes. Here is a list of all known historical maps on which these words appear.
[2] Or in Boole’s case, we might say, attributing to each proposition in concept space a single truth value: TRUE or FALSE (var YES or NO; or, in electronics applications, ON or OFF.) What we have here, I believe, is in many instances a false dilemma or the old Aristotelian dichotomy of either/or. Quantum physics demands and deserves better. OK, true enough, Boole gets around to extending possibilities by means of multi-term propositions, which his system can readily handle. The question here, though, is whether nature can or does handle such similarly. I think not. I think it approaches the question at a more fundamental level of reasoning and reality: at the most basic level of spacetime itself.
[3] This echoes the view of cybernetics, a transdisciplinary approach for exploring regulatory systems, their structures, constraints, and possibilities.
Cybernetics is relevant to the study of systems, such as mechanical, physical, biological, cognitive, and social systems. Cybernetics is applicable when a system being analyzed incorporates a closed signaling loop; that is, where action by the system generates some change in its environment and that change is reflected in that system in some manner (feedback) that triggers a system change, originally referred to as a “circular causal” relationship. [Wikipedia]
[4] This entire blog and its predecessor are in some sense the chronology of a journey from the familiar shoreline into largely uncharted waters. Hesitant at first, increasingly more daring as time has gone on and I’ve come to see errors to be stepping stones along the way. And there have beenmanyerrors along the way. Some I am not yet cognizant of. But of those I am aware, I have left most intact in spite of since being superseded by ideas superior, more correct or better formulated. I’ve done this because I think it important to map the course of a conceptual journey, how the ideas evolved from A to B to C to D. It also allows readers to participate, to a degree, in the thrill of an exciting adventure of mind, should they so choose. Happy travels.
© 2016 Martin Hauser
Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)
Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)
-Page 303-
Mandalic geometry, Cartesian coordinates and Boolean algebra: Relationships - I
(continued from here)
In attempting to understand the logic of the I Ching it is important to know the differences between ordinary algebra and Boolean algebra and how Boolean algebra is related to the binary number system.[1]
In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted
1 and 0 respectively. Instead of elementary algebra where the values of the variables are numbers, and the main operations are addition and multiplication, the main
operations of Boolean algebra are the conjunctionand, denoted ∧, the disjunctionor, denoted ∨, and the negationnot, denoted ¬. It is thus a formalism for describing logical relations in the same way that ordinary algebra describes
numeric relations. [Wikipedia]
Whereas in elementary algebra expressions denote mainly numbers, in Boolean algebra they denote the truth values false and true. These values are represented with the bits (or binary digits), namely 0 and 1. They do not behave like the integers 0 and 1, for which
1 + 1 = 2, but may be identified with the elements of the two-element field GF(2), that is, integer arithmetic modulo 2, for which 1 + 1 = 0. Addition and multiplication then play the Boolean roles of XOR (exclusive-or) and AND (conjunction) respectively, with disjunction x∨y (inclusive-or) definable as x + y + xy. [Wikipedia][2]
Mandalic logic already occurs fully in the structure and manner of divinatory practice of the I Ching, if some of it only implicitly. Although mandalic geometry does not originate from either Boolean algebra or the Cartesian coordinate system but from the primal I Ching which predates them by millennia, it does combine and augment aspects of both of these conceptual systems. It extends Boole’s system of symbolic logic to include an additional logic value represented by the number -1. This necessitates modification of some of Boole’s postulates and rules, and increases their total number through introduction of some new ones. The hexagrams or native six-dimensional mandalic coordinates of the I Ching are related to Cartesian triads composed of the numbers -1, 0, and 1, making these two geometric systems commensurate by means of composite dimension, a 6D/3D hybridization or mandalic coordination of structure and function (or space and time).[3]
The introduction of composite dimension produces four distinct dimensional amplitudes and is solely responsible for the mandalic form. For anyone reading this who might be down on sacred geometry, itself a subject which I respect and admire, let it be known that I am talking here about genuine mathematics and symbolic logic, and my suspicion is that there is some genuine physics involved as well.
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
Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)
Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)
-Page 302-
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 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
n = x + 1 - p. :)
-Page 297-
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
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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. :)
-Page 294-
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
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-Page 282-
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 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-
Neo-Boolean - II: Logic Gates
Thinking Inside the Lines
(continued from here)
We have already looked briefly at three of the more important Boolean operators or logic gates: AND, OR,andXOR.NOT just toggles any two Boolean truth values (true/false; on/off; yes/no). Here we introduce two new logic gates which do not occur in Boolean algebra. Both play an important role in mandalic geometry though.
We’ll refer to the first of these new operators or logic gates as INV standing for inversionorinvert. This is similar to Boole’s NOT except that it produces toggling betweeen yang/+ and yin/- instead of 1 and 0. Because it is based on binary arithmetic, Boole’s NOT has been thought of as referring to inversion also (as in ONorOFF). Although both ANDandINV act as toggling logic gates they have very different results in the greater scheme of things, since nature has created a prepotent disparity between a -/+ toggle and a 0/1 toggle in basic parameters of geometry, spacetime, and being itself. This makes Boole’s AND just a statement of logical opposition, notinversion.
Recognition of this important difference is built into mandalic geometry structurally and functionally, as it is also into Cartesian coordinate dynamics and the logic of the I Ching, but lacking in Boole’s symbolic logic. This is necessarily so, as there is no true negative domain in Boolean algebra. The OFF state of electronics and computers, though it may sometimes be thought of in terms of a negative state, is in fact not. It relates to the Western zero (0), not the minus one of the number line. Where Boolean algebra speaks of NOT 1 it refers specifically to zero and only to zero. When mandalic geometry asserts INV 1 it refers specifically to -1 and only to -1 . The inversion of yang then is yin and the inversion of yinisyang.[1] In the I Ching, Taoist thought, and mandalic geometry the two are not opposites but complements and, as such, interdependent.
The second added logic gate that will be introduced now is the REV operator standing for reversionorrevert. This operator produces no change in what it acts upon. It is the multiplicative identity element (also called the neutral elementorunit element), as INV is the inverse element. In ordinary algebra the inverse element is -1, while the identity element is 1. In mandalic geometry and the I Ching the counterparts are yinandyang, respectively. If Boolean algebra lacks a dedicated identity operator, it nonetheless has its Laws of Identity which accomplish much the same in a different way:
- A = A
- NOT A = NOT A
Again, Boolean algebra has no true correlate to the INV operator. There can be no sign inversion formulation as it lacks negatives entirely. Although Boolean algebra may have served analog and digital electronics and digital computers quite well for decades now, it is incapable of doing the same for any quantum logic applications in the future, if only because it lacks a negative domain.[2] It offers up bits readily but qubits only with extreme difficulty and those it does are like tears shed by crocodiles while feeding.
(to be continued)
Image: Boolean Search Operators. [Source]
Notes
[1] Leibniz’s binary number system, on which Boole based his logic, escapes this criticism, as Leibniz uses 0 and 1 simply as notational symbols in a modular arithmetic and not as contrasting functional elements in an algebraic context of either the Boolean or ordinary kind.
In the field of computers and electronics, Boolean refers to a data type that has two possible values representing true and false. It is generally used in context to a deductive logical system known as Boolean Algebra. Binary in mathematics and computers, refers to a base 2 numerical notation. It consists of two values 0 and 1. The digits are combined using a place value structure to generate equivalent numerical values. Thus, both are based on the same underlying concept but used in context to different systems. [Source]
[2] Moreover, I expect physics will soon enough discover that what it now calls antimatter is in some sense and to some degree a necessary constituent of ordinary matter. I can already hear the loudly objecting voices declaring matter and antimatter in contact necessarily annihilate one another, but that need not invalidate the thesis just proposed. My supposition revolves around the meaning of “contact” at Planck scale and the light speed velocity at which subatomic particles are born, interact and decay only to be revived again in an eternal dance of creation and re-creation. Material particles exist in some kind of structural and functional homeostasis, not all that unlike the anabolic and catabolic mechanisms that by means of negative feedback maintain all entities of the biological persuasion in the steady state we understand as life. Physics has yet to get a full grip on this aspect of reality, though moving ever closer with introduction of quarks and gluons to its menagerie of performing particles.
© 2016 Martin Hauser
Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)
Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)
-Page 304-
Mandalic geometry, Cartesian coordinates and Boolean algebra: Relationships - I
(continued from here)
In attempting to understand the logic of the I Ching it is important to know the differences between ordinary algebra and Boolean algebra and how Boolean algebra is related to the binary number system.[1]
In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted
1 and 0 respectively. Instead of elementary algebra where the values of the variables are numbers, and the main operations are addition and multiplication, the main
operations of Boolean algebra are the conjunctionand, denoted ∧, the disjunctionor, denoted ∨, and the negationnot, denoted ¬. It is thus a formalism for describing logical relations in the same way that ordinary algebra describes numeric relations. [Wikipedia]
Whereas in elementary algebra expressions denote mainly numbers, in Boolean algebra they denote the truth values false and true. These values are represented with the bits (or binary digits), namely 0 and 1. They do not behave like the integers 0 and 1, for which
1 + 1 = 2, but may be identified with the elements of the two-element field GF(2), that is, integer arithmetic modulo 2, for which 1 + 1 = 0. Addition and multiplication then play the Boolean roles of XOR (exclusive-or) and AND (conjunction) respectively, with disjunction x∨y (inclusive-or) definable as x + y + xy. [Wikipedia][2]
Mandalic logic already occurs fully in the structure and manner of divinatory practice of the I Ching, if some of it only implicitly. Although mandalic geometry does not originate from either Boolean algebra or the Cartesian coordinate system but from the primal I Ching which predates them by millennia, it does combine and augment aspects of both of these conceptual systems. It extends Boole’s system of symbolic logic to include an additional logic value represented by the number -1. This necessitates modification of some of Boole’s postulates and rules, and increases their total number through introduction of some new ones. The hexagrams or native six-dimensional mandalic coordinates of the I Ching are related to Cartesian triads composed of the numbers -1, 0, and 1, making these two geometric systems commensurate by means of composite dimension, a 6D/3D hybridization or mandalic coordination of structure and function (or space and time).[3]
The introduction of composite dimension produces four distinct dimensional amplitudes and is solely responsible for the mandalic form. For anyone reading this who might be down on sacred geometry, itself a subject which I respect and admire, let it be known that I am talking here about genuine mathematics and symbolic logic, and my suspicion is that there is some genuine physics involved as well.
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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Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
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-Page 302-
November 2, 2015
Today is the 200th anniversary of the birth of George Boole. In our modern digital world, we’re always hearing about “Boolean variables”—1 or 0, true or false. And one might think, “What a trivial idea! Why did someone even explicitly need to invent it?” But as is so often the case, there’s a deeper story—for Boolean variables were really just a side effect of an important intellectual advance that George Boole made.
When George Boole came onto the scene, the disciplines of logic and mathematics had developed quite separately for more than 2000 years. And George Boole’s great achievement was to show how to bring them together, through the concept of what’s now called Boolean algebra. And in doing so he effectively created the field of mathematical logic, and set the stage for the long series of developments that led for example to universal computation.
When George Boole invented Boolean algebra, his basic goal was to find a set of mathematical axioms that could reproduce the classical results of logic. His starting point was ordinary algebra, with variables like x and y, and operations like addition and multiplication.
At first, ordinary algebra seems a lot like logic. After all, p and q is the same as q and p, just as p×q = q×p. But if one looks in more detail, there are differences. Like p×p = p2, but p and p is just p. Somewhat confusingly, Boole used the notation of standard algebra, but added special rules to create an axiom system that he then showed could reproduce all the usual results of logic.
Image: Progressive simplifications of Boolean logic axiom systems over time. Stephen Wolfram
