Mandalic Geometry @mandalicgeometry - Tumblr Blog

Can a number system be both the new kid on the block and older than written history?

The real number system as it exists today has been with us for a few centuries. In foundation it is monovalent, monophasic, and sequential.

The probable number system dates to prehistory but was lost in the mists of time until recently rediscovered and resurrected. In contrast to the real number system it is foundationally bivalent, biphasic, and cyclic.

The probable number system has considerably more structure than the real number system and is therefore more robust. In this sense, it is similar to the complex number system.

In contrast to the complex number system, the probable number system in its foundation presupposes that numbers can assume wavelike forms capable of constructive and destructive interference operationally through the compositing of higher to lower dimension.

By means of compositing of dimension probable numbers are able to distribute throughout the entire mandalic unit vector cube (which is structurally a superposition of the 6-dimensional unit vector hypercube on the 3-dimensional unit vector cube) a function analogous in important ways to that performed in the complex number system by the centralized imaginary unit i.

Another important way in which the probable number system differs from both the real number system and the complex number system is the absence of nothingness and the zero representing it. In its place we find the concepts of balance and equilibrium. Nullification still exists in form of annihilation and its opposite in the form of creation. But the Cartesian coordinate system of ordered pairs and ordered triads is transformed by this approach to handling number and dimension from a ring into a field of hyperdimensional numbers over real numbers in three dimensions.

(to be continued)

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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Toward a new geometry

Definitions and guiding principles

Dimension is a primitive concept which for purposes here can be defined as any linearly independent parameter or variable.

Mathematics is most fundamentally about measurement.

Rules of mathematics are conceptual tools developed by the human mind and are subject to change if and when such should prove necessary or utilitarian.

Points, lines and planes are products of the human mind. Neither they nor the realities they represent have continuous existence in space or time but can recur intermittently at intervals which may be either regular or irregular. As a matter of convenience for purposes of description, however, any or all of these may be represented as existing in some continuous abstract sense.

The whole is greater than the part and has emergent properties not found in any single part or any number of subdivisions of the whole.

The consequences of number structure and interaction cannot be explained entirely on a local basis but emerge from holo-interactive dynamic processs throughout an entire system or any portion thereof in conceptual focus.

It is always possible to express a reflection as a rotation and a rotation as a reflection. The two are isomorphic via properly chosen operations.

Symmetry is of great importance but not always obvious.

Two foundational guiding principles of reality are continuity and change. These then provide the basis and primary focus of mandalic geometry.

Of the two types of change, cyclic and sequential, cyclic change is the more fundamental.

Neither coordinates nor coordinate systems are a feature of nature. They are man-made devices which are pragmatic and utilitarian means by which to grasp a reality which itself has no need of them. Geometry nevertheless requires these crutches to exist and execute its functions.

Geometry frequently also requires conventions of expression to promote widespread understanding of content but should strive to be as convention -free as possible.

Ambiguity is a permissible feature, in fact a necessary feature, of mandalic geometry. This is related to its probabilistic nature and multiple-valued logic which reflect what we, from our limited vantage point, misunderstand to be paradoxes of nature.

Axioms

Numbers can be characterized by dimension of context.

A number may be embedded in multiple dimensions concurrently, existing in variant forms specific to the dimension(s) of context.

The same number may function differently in different dimensional contexts. The dimension of context of a "point" described by a number or subsidary number delimits its expression. In other words, the expression of a number in spatiotemporal terms is determined by its dimensional context. In referring to a number or subsidiary number in any of their variants, therefore, the dimension of context must always be specified.

It may not always be possible to identify the full dimensional context of a number. It is sufficient to determine and elucidate those contexts essential to elaboration of the specific operation(s) under present consideration.

Numbers are not necessarily elemental. They may consist of parts or subsidiary numbers which refer to various different dimensional contexts.

A point is not dimensionless extension in space. It is an emergent feature of the system-as-a-whole which appears intermittently at the common intersection of three or more dimensions. Points, lines and planes are evanescent occurrences in terms of geometry and spacetime. They, and the things they represent, are fleeting events which come and go. Repetition is possible (in a conceptual sense certainly; possibly in a material/energetic sense as well) but not inevitable.

Rejected definitions

The definition of a point found in Euclid’s Elements: A point is that which has no part. [This definition may or may not have been in Euclid’s original Elements.]

The definition of a line found in Euclid’s Elements: A line is breadthless length.

The definition of a surface found in Euclid’s Elements: A surface is that which has length and breadth only.

Rejected axioms

The real numbers are order-isomorphic to the linear continuum of geometry. In other words, the axiom states that there is a one to one correspondence between real numbers and points on a line. This has been described as the Cantor–Dedekind axiom. The Cartesian coordinate system implicitly assumes this axiom which then becomes the cornerstone of analytic geometry.

Rejected notions

Things which coincide with one another equal one another. [Euclid's Common notion 4]

Also important to note:

Euclid’s first postulate states that any two points can be joined by a straight line segment. It does not say that there is only one such line; it merely says that a straight line can be drawn between any two points.

Image credit: James Gyre-Naked Geometry

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#mandalic geometry #geometry #axioms #definitions #guiding principles #algebra #mathematics #toward a new geometry #new geometry #Euclid #Descartes #philosophy of mathematics #physics #quantum logic #interdisciplinary #transdisciplinary

Magic Theatre: For Madmen Only

If we describe a Cartesian ordered triad by x,y,z we can describe an analogous 6-dimensional ordered sextuplet or 6-tuple by xa,ya,za,xb,yb,zb

The definitions that translate a 6-dimensional ordered sextuplet (hexagram in Taoist terminology) into a 3-dimensional ordered triad (trigram in Taoist terminology) are:[1]

(xa + xb) / 2 = x

(ya + yb) / 2 = y

(za + zb) / 2 = z

I think the methodology will work for all scalar quantities. But as currently formulated, mandalic geometry (MG) is a discrete geometry based entirely on unit vectors. We are talking about the line segments between -1 and +1 in the various dimensions and only points -1, 0, and +1 in each line segment in Cartesian terms.

In essence we are not yet particularly concerned with scalars here but only with vectors : -, +, and neutral (0).

Mathematically √−1 is important because by adding it to the real number field, as we have done, we create the algebraically complete field of complex numbers. In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. The real numbers and complex numbers are both complete fields. Cartesian coordinates- - - ordered pairs and ordered triads- - - although based on real numbers, do not form a field. This has important implications, implications which can be ignored only at peril to the particular conceptual system involved..

The definitions above all give three possible results in Cartesian terms: -1, 0, +1. Remember though MG hybridizes six dimensions with three dimensions and represents them superimposed. Wherever one or more zeros occurs in Cartesian coordinates we have also corresponding 6-dimensional forms, composed of just +1s and -1s, of which there are always two for each Cartesian zero. A Cartesian ordered triad with one zero is associated with two such 6-dimensional forms; an ordered triad with two zeros, with four; an ordered triad with three zeros (the origin), with eight. An ordered triad without zeros will have only one associated 6-dimensional form. This constitutes the mandalic pattern, which is an essential feature of the 6D/3D formulation of this geometric system and isomorphism naturally comes into play here as well.

And that is how and why all numbers in this coordinate system based on higher-dimensional extensions of the real numbers "square" to numbers which can be either positive or negative and then reduce or "collapse" to corresponding Cartesian forms that preserve the same sign. This is a necessary result of the fact that a primary "zero form" in 6-dimensional terms is lacking, only +1s and -1s exist. These can then interfere constructively and destructively as number waves, to produce a "secondary zero" by destructive interference whenever linked forms differ in sign in one or more paired dimensions. Since the two linked 6-dimensional numbers are always inverse to one another, any Cartesian zero then can be substituted with two such 6-dimensional forms. This is the process that makes imaginary numbers unnecessary, replacing them with two inversely related probable numbers which behave in most ways like real numbers and are distributed throughout the entire geometric system.

"Hybridization" is probably not the best term here but will be used until I can think of a better descriptor. What I intend is not actual joining and unification, but rather a superposition and conceptual commingling in three-dimensional terms. Such a representational mapping substitutes for all Cartesian forms "equivalent" forms containing only 1s and -1s, no zeros. In so doing, it effectively converts the Cartesian coordinate system from just a ring to a field as well, properly interpreted. Basically then, the probable numbers do for the real numbers much the same as the complex numbers do, but with even greater and more utilitarian results which are also more easily managed.

In operational terms, complex numbers perform two rather simple binary operations: a scaling and a rotation. Scaling capability is clearly inherited through its real number lineage; rotational capacity, from its imaginary number lineage. Together, scaling and rotation combine to augment or diminish an axis of growth and produce vector ambulation in a circular path about a central origin point of reference. The scaling factor could be said to detemine the radius of revolution; the rotation factor, the angle of revolution. And that's pretty much all there is to the "great mystery" of complex numbers. Their importance resides in the great number of fields of endeavor where the combination of these two superpowers is necessary and/or convenient.

Nature uses this combination of scaling and rotation in many of its processes. Atomic and subatomic proceedings are probably not among these. How then did it come about that quantum mechanics arrived at the notion that rotation and scaling could be applicable to modeling of discontinuos states of being? Both refer to changes through continuous space. I think it was an accident of history. In 1925, Erwin Schrödinger, in his search for a way to explain certain mysteries then perplexing the greatest physicists of the day, hit upon his eponymous equation which appeared to do the trick. So well, in fact, that quantum mechanics has been justly considered the single most successful description of reality ever devised. And the equation that basically accomplished this success involves the imaginary number i and complex numbers.[2]

An important aspect of the operation of rotation, one which may have bearing on the Schrödinger equation and its huge success, has been largely overlooked. The result of a rotation can often mimic the result of inversion (reflection through a point), making the two indistinguishable by measurement alone. To someone wearing a blindfold there is no way to tell whether i has by the operations of squaring and rotation changed itself into -1 or -1, the inversion element of multiplication, has simply reflected +1, the identity element of multiplication, through the origin point to -1. Explaining away a 90° rotation with a right angle reflection will no doubt prove more difficult but let's not just yet deny that it might be doable.

Could there be a way to reformulate the Schrödinger equation then so it contains no imaginary or complex numbers? Many have tried to do that very thing and failed. No one has succeeded in nearly a century. Still, we might wonder if the time is ripe now to remove the blindfold. Perhaps we might do well to inquire whether quantum physics is, in some manner we don't quite understand, a victim of its own success.

In theory, circumventing use of complex numbers in a defining equation of quantum mechanics should be possible. On what basis do I say this? The equation we have now relies on complex numbers. These in turn derive an ability to produce rotation from the imaginary number √−1 . But there are other mathematical means to accomplish the same. Trigonometry comes most immediately to mind. The circle and cyclicity it models have a very long and distinguished history. Complex numbers as we've noted can also produce scaling. But so can real numbers. And close examination reveals that complex numbers inherit their ability to scale from the two real numbers they contain. The hard truth ultimately is there is nothing all that special about complex numbers or complex plane. Possibly it is their utilitarian ease of use that positions them as an attractive methodology. Other routes to ease of use exist as well. There is always more than one way to skin the proverbial cat (even a cat residing only in the mind of a physicist named Schrödinger.)

Consider also, how great is the actual need for scaling in quantum mechanics? The distance from centermost part of the atom to the outer reaches of electron orbital space is in fact quite small. Furthermore, the elements of this universe of discourse are quantized, so actual distances involved are moot. In the extreme, the question persists as to whether "distance" is a concept even applicable in this context of quantum logic. Quantum numbers themselves range between 0 and 2. I can count the allowed values on the fingers of one hand.

Regarding rotation, where exactly does that come into play in the quantum realm? Electrons do not orbit the nucleus of the atom. They jump from orbital to orbital by discretized changes in energy involving photon exchange. In the nucleus it seems such discretized instanteous changes take place as well, obviating any need for rotation. Obviously physics misguided here by labeling one of the quantum numbers "spin". Sometimes a rose is best referred to as a rose. The problem here is that we don't really know what it is that "spin" refers to.

The quintessential equation of quantum mechanics was formulated by a physicist, not a mathematician. It is not a simple algebraic equation, but in general a linear partial differential equation, describing the time-evolution of the system's wave function ("state function"). "Derivations” of the Schrödinger equation do generally demonstrate its mathematical plausibility for describing wave-particle duality. To date, however, there are no universally accepted derivations of Schrödinger's equation from appropriate axioms. Nor is there any general agreement as to what the equation actually signifies. Moreover, some authors have demonstrated that certain properties emerging from Schrödinger's equation can even be deduced from symmetry principles alone. This would appear to be a worthwhile direction of investigation to pursue. Quantum mechanics is most fundamentally about symmetry. Let's make Emmy Noether proud by giving her the recognition she deserves.

Finally, it was not without considerable difficulty that Schrödinger developed his equation. In the end, it almost seems he pulled it out of a hat, as a magician might a rabbit.[3] Part of the Zeitgeist of the physics community in the early 1920s revolved around the peculiar notion that particles behaved as waves. Schrödinger decided to follow this direction of thought and find an appropriate 3-dimensional wave equation for the electron. His equation succeeded beyond his wildest dreams. Adopted in the canon of the new physics, it became the cornerstone of that radically different physics, changed forever. Physics has never looked back since.

Still, one startling and haunting fact persists: nowhere else in all of physics has it ever been found necessary to invoke complex numbers.

Once, quite a long time ago, I believed imaginary numbers were wrong. I was the one that was wrong. Later, having grown a little more clever, I came to think that √−1 was a necessary evil- - -correct but not validly applicable to quantum physics. Wrong again. Currently it is my belief that imaginary numbers are guilty of an even worse offense: both true from the mathematical standpoint and partly applicable to physics. The worst of both worlds. Yielding results that are in large part correct, imaginary and complex numbers have managed to lead us all down the garden path for the better part of a century. Have we then gone past the point of no return? My contention is that it is possible to complete the ring that Cartesian coordinates present and transform it to a field over the real numbers, with appeal only to higher-dimensional analogues of the reals and no need for imaginary or complex numbers, an approach which, if actually possible, would offer certain undeniable advantages.[4]

Essentially the method of composite dimension does away with i and complex numbers by distributing an operation analogous to that of i throughout six dimensions or three in Cartesian terms and then working with same by means of reflections (inversions) only. So an algebra based on the system necessitates use of only the real numbers and their higher dimension extensions that I have called probable numbers. Only simple addition and multiplication are required. For those in the audience who are "sufficiently mad", there is the added bonus that a kind of division by zero becomes possible. We'll find out soon enough whether you qualify.

A few additional explanatory remarks are in order here:

Depending on the variant, Cartesian geometry (CG), represents space in two or three dimensions. Points in the former are referenced to two pairwise perpendicular axes; in the latter, to three.

Because Descartes assumes as axiomatic a 1:1 correspondence of number to spatial location each of his three axes becomes a facsimile of the number line, only in different dimensions.

Mandalic geometry (MG) approaches representation of space differently, using a hybrid coordinate system which relates a higher dimension space to a lower dimension space with a 2:1 correlation.

It can be represented entirely commensurate with CG, but in so doing a "glass slipper effect" occurs. Just as Cinderella's stepsisters can manage to force a too fat foot into her glass slipper, the results leave something to be desired. In our context here, the "something to be desired" is a clear and full understanding of six-dimensional reality in its own right. We end up interpreting it in time-sharing terms of probabilities and randomness.

What Descartes refers to as an ordered pair requires two higher dimension ordered pairs to represent in MG; a Cartesian ordered triad requires three higher dimension ordered pairs to represent in MG.

In Taoist terminology the notational equivalent of a Cartesian ordered pair is a "bigram", a two-line symbol, each line of which can take one of two values. As a result there are four types of bigram. Two bigrams make up a tetragram; three, a hexagram.

Descartes views a point as having only two essential characteristics:

It is dimensionless.

It is just a location in space which can be uniquely represented by a single ordered pair or ordered triad.

Mandalic geometry rejects both of these axioms. It regards a point, or a particle so represented, as an evanescent entity emerging from interaction of two higher dimensions expressed in our world of three dimensions in such limited manner.

This can be represented in context of Cartesian space but in making mandalic coordinates commensurate with Cartesian coordinates it is no longer possible to represent every "point" in space uniquely with a single mapping of number to location. What results instead is the probabilistic distribution pattern of the mandala, which we, from our limited vantage in spacetime, misinterpret as something it is not.

MG is a discrete geometry. The result of the mapping formula used is a mandalic configuration in which the 3-dimensional cube composed of unit vectors in Cartesian space becomes a "probability distribution" in combined mandalic space.

I have placed the quotation marks around probability distribution because this is a perspective that arises from our inability to see all that is involved accurately. I suspect this has repercussions pertinent to a full comprehension or grokking of quantum mechanics and possibly of string theory as well.

Since the 64 discrete "points" of the unit vector hypercube of six dimensions represented by the hexagrams cannot "fit" simultaneously in the 27 discrete points of the 3-dimensional unit vector cube by any representational method available to our inherited bio-psychocultural mechanism, a sort of time-sharing process occurs in observations and measurements of reality which we interpret in terms of probability.

What has been described here occurs at enormous velocities close to that of light, and likely refers only to processes in the subatomic quantum realm. For MG, which is also a hybridization of mathematics and physics, context is always of the essence.

There is much more to be said in explanation of mandalic geometry. I see, though, this post has already run rather long, so we will end it here. Enough has already been said in way of introduction of basic material.

Notes

[1] Since the coordinate system is describing a cube with an n-hypercube superimposed, there is an additional constraint placed on all coordinates in the 6-tuples. All scalar values must be identical for x, y and z values. That constraint assures that all vectors though they may differ in sign (direction) maintain equal magnitudes. When the 6-tuples are dimensionally reduced to 3-tuples by the method I've called "compositing of dimension" the resulting geometric figure consists of four different dimensional amplitudes of 6-tuples collapsed. The amplitudes of dimension correspond in spatial terms to the vertices, edge centers, face centers and cube center. The pattern that emerges is that of a mandala. This is a highly symmetric pattern though all symmetries aren't necessarily apparent immediately, even using Taoist notation. The probability distribution of the 6-tuples allots the hexagrams in the following manner: one to each vertex; two to each edge center; four to each face center; one to the cube center. The result is placement of 64 6-tuples in 27 positions of discrete 3-tuples in the specific mandalic distribution pattern described. Think here of the analogy of a hydrogen atom confined within a cubic space of specified side length determined by the nuclear and atomic force fields. The single electron, existing in such quantized energy levels that are possible, can assume various different locations in different orbital shells, but every location in a given orbital must be equidistant from the nuclear proton. Once reduced by dimensional compositing the 6-tuples described here fill four distinct shells that have different radii or distances from the center. From center to periphery these distances can be described as zero; one (or square root one); square root 2; and square root 3. (Pythagorean theorem)

[2] Schrödinger was not entirely comfortable with the implications of quantum theory. About the probability interpretation of quantum mechanics that came out of Solvay '27 he wrote: "I don't like it, and I'm sorry I ever had anything to do with it." ["A Quantum Sampler". The New York Times. 26 December 2005.]

[3] In later years another great physicist, Richard Feynman, would remark, “Where did we get that (equation) from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind of Schrödinger.”

[4] A different approach to avoiding the need for complex numbers from the one I am suggesting is described here. To my mind it offers little of value other than an interesting alternative explanation of what complex numbers are and do. A similar conclusion seems to have been reached by the author.

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A Recap of Some Important Ideas Regarding Mandalic Geometry

Mandalic geometry (MG) is a new kind of mathematical methodology based on a worldview having roots that predate written history.

It is a discrete geometry which currently consists of just a coordinate system but can be extended as Descartes did his to encompass an entire analytic geometry.

Mandalic geometry introduces and is based on a new number system, the probable number system (or probabilistic number system.)

Just as the complex number system combines real numbers and imaginary numbers and is more robust than either, the probable number system combines real numbers and probable numbers and is more robust than either.

The probable number system is also more robust than the complex number system. Complex numbers combine real numbers with imaginary numbers to form the single complex plane. Composite numbers combine real numbers with probable numbers to form six interdependent composite planes.

Axiomatic to the system is the contention that numbers can exist in different dimensions and therefore can be described as being of some particular dimension. Numbers are always viewed and treated within context of a stated dimension.

Probable numbers are an extension of the real numbers to higher dimensions and are independent of imaginary and complex numbers.

Mandalic geometry does not admit the existence of square root of -1 in the real world other than in mathematics invented by the human mind. In place of square root of negative numbers, MG introduces the new concept of contra-square root. In brief this involves substitution of a combination form of interactive two-dimensional analogues of +1 and -1 for -1 as currently used in imaginary number contexts. This is more fully explained elsewhere in the blog.

Put another way, in place of imaginary numbers MG posits the existence of probable numbers. These can be considered the result of what is essentially wavelike interactions of higher dimensional numbers to form the real numbers we know in the 3-dimensional world.

Higher dimensional numbers can interact with one another through wavelike constructive and destructive interference to generate ordinary 3-dimensional numbers. Numbers are not viewed as constants to be acted upon as Descartes so views them but rather as being themselves active and changeable. They participate in process. This feature alone enables composite numbers to mediate between mathematics and physics better than either real or complex numbers can.

The interactions of higher dimensional numbers in the process of dimensional compositing to yield 3-dimensional numbers is a function of time and therefore probabilistic from our limited ordinary point of view. From this perspective, certain probablity distributions are the result of dimensional compositing and the consequent mandalic form. MG considers the probabilistic nature of quantum mechanics likely to be based on such.

The probabilistic nature in three dimensions of what are here called probable numbers is what gives rise to the mandalic form which can in a sense be considered the 3-dimensional evolution of 6-dimensional numbers from protean representations through progressive differentiation of form to the stage of maximal differentiation and back again to the undifferentiated state of greatest probability.

The mandalic form has a geometric progression of its line structures in the three Euclidean/Cartesian dimensions such that series of numbers of the form 1:2:1, 2:4:2, and 4:8:4 occur throughout all of those dimensions when a hybrid 6D/3D coordinate system results from performing 2:1 compositing from six to three dimensions.

Mandalic geometry views points and lines in three dimensions as convenient fictions that exist only as evanescent probabilistic concurrences of analogous entities in higher dimensions.

The probabilistic nature of MG makes it ideal for investigations and descriptions of quantum mechanics.

The exclusion of imaginary and complex numbers and substitution of probable and composite numbers which are easily reducible to ordinary algebraic/arithmetic forms and can be worked with using the same methods as those mathematical disciplines makes MG more utilitarian and appropriate to application to quantum mechanics than are complex numbers. All operations performed are based on simple inversion (reflection through a point) and on real numbers, maintaining all the usual rules and properties of ordinary arithmetic, including commutativity (which quaternions fail to preserve.)

MG is currently based on discrete numbers and is concerned mainly with the positive and negative integers. Fractions and irrational numbers are not excluded from the system but do not currently play a significant role. Future incarnations of MG will extend it outward beyond the unit vector cube to tile the geometric universe and inward to encompass fractional entities and fractals.

It is a hybrid geometry resulting from superposition of 6-dimensional numbers and 3-dimensional numbers and is fully commensurate with 3-dimensional Cartesian geometry.

It describes a linear mapping of two dimensions to one dimension which forms a field of probable numbers over the field of real numbers, analogous to the field of complex numbers but constructed on a different principle and extending to the real numbers in all three Cartesian dimensions rather than just one. The two independent higher dimensions so mapped become dependent variables in the mandalic "line" that results from the compositing of the two. This is expressed, in a sense, as two sine waves 180 degrees out of phase that mutually intersect a common Cartesian axis (x,y or z) at Cartesian +1 and -1 and are maximally separated at Cartesian 0.

This phase difference produces wave interference of both constructive and destructive varieties. So-called "points" or "particles" they represent come into existence only discretely and intermittently at Cartesian -1, +1, and 0, the locations of intersection or confluence (-1 and +1) and maximum separation, the maxima/minima of the two entangled sine waves that occur at Cartesian 0.

As the unit vector cube corresponds to and describes only half of each of the two sine waves, two unit vector cubes are required for a full cycle. Mandalic geometry as currently formulated with a single unit cube then needs to be extended to at least two of these. Extension in both directions of all three Cartesian axes is easily accomplished by repeatedly inverting the current single unit vector cube.

This means that mandalic coordinates alternate positive and negative on both sides of Cartesian 0. The extensions can be continued to infinity in both directions, but not, properly speaking, positive and negative infinity since the manner of extension has created what is essentially a convention-free coordinate system which consists of repeated units of consecutively inverted unit vector cubes in which positive and negative alternate ad infinitum and every Cartesian even-numbered coordinate becomes a "zero equivalent" , or better, a neo-zero in this extended mandalic coordinate system.

The resulting geometry is a dynamic one with "points", "lines", and "planes" coming into and passing out of existence intermittently in a time-sharing of corresponding Cartesian entities. It "persists" in time and space by means of continuous creation, destruction and re-creation and is "held together" by "force fields" produced and maintained by means of tensegrity which is based ultimately on dimension and number, and by a process that.might best be described as a "weaving of reality" with warp and woof.

The 2:1 compositing of dimension involved creates a new number system the members of which are like the real integers in all ways except that they map differently to a Cartesian geometric space. Whereas Decartes assumes that one number maps to one point, MG does not make this assumption which is just an unproved axiom that Descartes makes implicit use of.

The method of dimensional compositing automatically results in a mandalic formation having a geometric progression through three Euclidean/Cartesian dimensions from periphery to center (origin).

Currently MG is limited to a description of unit vectors in a composite hybrid 6D/3D geometry but can be extended to include all scalar values and any even number of dimensions.

The notation system used is borrowed from Taoism and foreign to most Western mathematicians. It is, however, basically equivalent to Cartesian coordinate signs (yin=minus; yang=plus); ordered pairs (=bigrams); and ordered triads (=trigrams); and extends these concepts to include ordered quads (=tetragrams) and ordered sextuplets (=hexagrams).

This notation system is used rather than the usual Cartesian notation because it is much easier for the mind to manipulate dimensional numbers using it. It takes only a little practice to become accustomed to using it. Without its use, understanding of mandalic geometry becomes extremely difficult, if not impossible.

As MG views a point as a concurrence of various different dimensions, it interprets Cartesian ordered pairs and triads, and their extensions to higher dimensions, as tensors and treats them as such. This makes it possible to apply operations of addition and multiplication to these mathematical entities in a manner analogous to the way William Rowan Hamilton applied these operations to complex numbers by way of what he called "algebraic couples".

The probabilistic mandalic form that is the hallmark of MG conveys and necessitates a new interpretation of zero(0). In MG "zero" is not the empty null that it is in Cartesian geometry and Western mathematics generally, but rather a fount of being, so to speak, and a logic gate spanning dimensions. Wherever a zero occurs in Cartesian coordinates two Cartesian-equivalent forms are found in mandalic coordinates. So in the mandalic cube based on unit vectors the twelve edge centers, having a single Cartesian zero, have two Cartesian-equivalent forms (hexagrams); the six face centers, having two Cartesian zeros, have four Cartesian-equivalent forms; and the single cube center, the Cartesian origin point with three zeros, has eight Cartesian-equivalent forms.

This alternative zero and the mandalic structure it inhabits force the creation of four different amplitudes of dimension in the 6-dimensional unit vector cube. These are not independent but all mutually dependent and holo-interactive within the composite 6D/3D coordinate system. All of this occurs in a context reminiscent of the one inhabited by nuclear particles. The mapping proposed by MG may in fact model the elementary force fields, electromagnetism and quantum chromodynamics. It suggests a possible mechanism for formation of the state of matter known as a quark-gluon plasma. Hidden within it may even be the secret of quantum gravity.

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#mandalic geometry #recap #dimensional numbers #dimension #probability distribution #probable numbers #composite numbers #composite dimension #conplex numbers #complex plane #imaginary numbers #real numbers #algebra over a field #tensor #mandalic form #mandala #zero

Beyond the Enlightenment Rationalists: From imaginary to probable numbers - VI

(continued from here)

"O Oysters, come and walk with us!" The Walrus did beseech. "A pleasant walk, a pleasant talk, Along the briny beach: We cannot do with more than four, To give a hand to each."

* * *

“The time has come,” the Walrus said, “To talk of many things: Of shoes–and ships–and sealing-wax– Of cabbages–and kings– And why the sea is boiling hot– And whether pigs have wings.”

-Lewis Carroll, The Walrus and the Carpenter

In this segment, probable numbers will be shown to grow out of a natural context inherently rather than through geometric second thought as transpired in the history of Western thought with imaginary numbers and complex plane. To continue with development of probable numbers it will be necessary to leave behind, for the time being, all preoccupation with imaginary numbers and complex plane. It will also be necessary to depart from our comfort zone of Cartesian spatial coordinate axioms and orientation.

Probable coordinates do not negate validity of Cartesian coordinates but they do relegate them to the status of a special case. In the probable coordinate system the three-dimensional coordinate system of Descartes maps only one eighth of the totality. This means then, that the Cartesian two-dimensional coordinate plane furnishes just one quarter of the total number of corresponding probable coordinate mappings projected to a two-dimensional space.[1] It suggests also that Cartesian localization in 2-space or 3-space is just a small part of the whole story regarding actual spatial and temporal locality and their accompanying physical capacities, say for instance of momentum or mass, but actually encompassing a host of other competencies as well.

Although this might seem strange it is a good thing. Why is it a good thing? First, because nature, as a self-sustaining reality, cannot favor any one coordinate scheme but must encompass all possible - if it is to realize any. Second, because both the Schrödinger equation and Feynman path integral approaches to quantum mechanics say it is so.[2] Third, because Hilbert space demands it. This may leave us disoriented and bewildered, but nature revels in this plan of probable planes. Who are we to argue?

So how do we accomplish this feat? Well, basically by reflections in all dimensions and directions. We extend the Cartesian vectors every way possible. That would give us a 3 x 3 grid or lattice of coordinate systems (the original Cartesian system and eight new grid elements surrounding it), but there are only four different types, so we require only four of the nine to demonstrate. It is best not to show all nine in any case because to do so would place our Cartesian system at direct center of this geometric probable universe and that would be misleading. Why? Because when we tile the two-dimensional universe to infinity in all directions, there is no central coordinate system. Any one of the four could be considered at the center, so none actually is. Overall orientation is nondiscriminative.[3]

LOOKING GLASS CARTESIAN COORDINATE QUARTET

The image seen immediately above shows four Looking House Cartesian coordinate systems, correlated within a mandalic plane. This mandalic plane is one of six faces of a mandalic cube, each of which is constructed to a different plan but composed of similar building blocks, the four bigrams in various positions and orientations. A 2-dimensional geometric universe can be tiled with this image, recursively repeating it in all directions throughout the two dimensions.[4] It should not be very difficult for the reader to determine which of the four mandalic moieties references our particular conventional Cartesian geometric universe.[5]

It remains only to be added here and now that potential dimensions, probable planes, and probable numbers arise immediately and directly from the remarks above. In some ways it's a little like valence in chemical reactions. We'll likely take a look at that combinatory dynamic in context of mandalic geometry at some time down the road. Next though we want to see how the addition of composite dimension impacts and modifies the basic geometry of the probable plane discussed here.[6]

(to be continued)

Top image: The four quadrants of the Cartesian plane. These are numbered in the counterclockwise direction by convention. Architectonically, two number lines are placed together, one going left-right and the other going up-down to provide context for the two-dimensional plane. This image has been modified from one found here.

Notes

[1] To clarify further: There are eight possible Cartesian-like orientation variants in mandalic space arranged around a single point at which they are all tangent to one another. If we consider just the planar aspects of mandalic space, there are four possible Cartesian-like orientation variants which are organized about a central shared point in a manner similar to how quadrants are symmetrically arranged about the Cartesian origin point (0,0) in ordinary 2D space. But here the center point determining symmetries is always one of the points showing greatest rather than least differentiation. That is to say it is formed by Cartesian vertices, ordered pairs having all 1s, no zeros. That may have confused more than clarified, but it seemed important to say. We will be expanding on these thoughts in posts to come. Don't despair. For just now the important takeaway is that the mandalic coordinate system combines two very important elements that optimize it for quantum application: it manages to be both probabilistic and convention-free (in terms of spatial orientation, which surely must relate to quantum states and numbers in some as yet undetermined manner.) At the same time, imaginary numbers and complex plane are neither.

[2] Even if physics doesn't yet (circa 2016) realize this to be true.

[3] It is an easy enough matter to extrapolate this mentally to encompass the Cartesian three-dimensional coordinate system but somewhat difficult to demonstrate in two dimensions. So we'll persevere with a two-dimensional exposition for the time being. It only needs to be clarified here that the three-dimensional realization involves a 3 x 3 x 3 grid but requires just eight cubes to demonstrate because there are only eight different coordinate system types.

[4] I am speaking here in terms of ordinary dimensions but it should be understood that the reality is that the mandalic plane is a composite 4D/2D geometric structure, and the mandalic cube is a composite 6D/3D structure. The image seen here does not fully clarify that because it does not yet take into account composite dimension nor place the bigrams in holistic context within tetragrams and hexagrams. All that is still to come. Greater context will make clear how composite dimension works and why it makes eminent good sense for a self-organizing universe to invoke it. Hint: it has to do with quantum interference phenomena and is what makes all process possible. ADDENDUM (12 APRIL, 2016) The mandalic plane I am referring to here corresponds to the Cartesian 2-dimensional plane and is based on four extraordinary dimensions that are composited to the ordinary two dimensions, hence hybrid 4D/2D. It should be understood though that any number of extra dimensions could potentially be composited to two or three ordinary dimensions. The probable plane described in this post is not such a mandalic plane as no compositing of dimensions has yet been performed. What is illustrated here is an ordinary 2-dimensional plane that has undergone reflections in x- and y-dimensions of first and second order to form a noncomposited probable plane. The distinction is an important one.

[5] This is perhaps a good place to mention that the six planar faces of the mandalic cube fit together seamlessly in 3-space, all mediated by the common shared central point, in Cartesian terms the origin at ordered triad (0.0.0) where eight hexagrams coexist in mandalic space. Moreover the six planes fit together mutually by means of a nuclear particle-and-force equivalent of the mortise and tenon joint but in six dimensions rather than two or three, and both positive and negative directions for each.

[6] It should also be avowed that tessellation of a geometric universe with a nondiscriminative, convention-free coordinate system need not exclude use of Cartesian coordinates entirely in all contextual usages. Where useful they can still be applied in combination with mandalic coordinates since the two can be made commensurate, irrespective of specific Cartesian coordinate orientation locally operative. Whatever the Cartesian orientation might be it can always be overlaid with our conventional version of the same. More concretely, hexagram Lines can be annotated with an ordinal numerical subscript specifying Cartesian location in terms of our local convention should it prove necessary or desirable to do so for whatever reason. On the other hand, before prematurely throwing out the baby with the bath water, we might do well to ask ourselves whether these strange juxtapositions of coordinates might not in fact encode the long sought-after hidden variables that could transform quantum mechanics into a complete theory. In mandalic coordinates of the reflexive nature described, these so-called hidden variables could be hiding in plain sight. Were that to prove the case, David Bohm and Louis de Broglie would be immediately and hugely vindicated in advancing their pilot-wave theory of quantum mechanics. We could finally consign the Copenhagen Interpretation to the scrapheap where it belongs, along with both imaginary numbers and the complex plane. ADDENDUM (24 APRIL, 2016) Since writing this I've learned that de Broglie disavowed Bohm's pilot wave theory upon learning of it in 1952. Bohm had derived his interpretation of QM from de Broglie's original interpretation but de Broglie himself subsequently converted to Niels Bohr's prevailing Copenhagen interpretation.

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Beyond the Enlightenment Rationalists: From imaginary to probable numbers - V

(continued from here)

The four Cartesian quadrants provide the two-dimensional analogue of the number line and its graphic representation in Cartesian coordinate space. This is the true native habitat of the square and, by implication, of square root. Because Enlightenment mathematicians found fit to define square root in a different context inadvertently -that of the number line- we will find it necessary to devise a different name for what ought rightly to have been called square root, but wasn't. I propose that we retain the existent definition of tradition and refer to the new relationship between opposite numbers in the square, that is to say, opposite vertices through two dimensions or antipodal numbers, as contra-square root.[1]

Modified from image found here.

Given this fresh context - one of greater dimension than the number line - it soon becomes clear with little effort that a unit number[2] of any dimension multiplied by itself gives as result the identity element of that express dimension. For the native two-dimensional context of the square the identity element is OLD YANG, the bigram composed of two stacked yang (+) Lines, which corresponds to yang (+1), the identity element in the one-dimensional context of the number line. In a three-dimensional context, the identity element is the trigram HEAVEN which is composed of three stacked yang (+) Lines. The crucial idea here is that the identity element differs for each dimensional context, and whatever that context might be, it produces no change when in the operation of multiplication it acts as operator on any operand within the stated dimension.[3]

As a corollary it can be stated that any number in any dimension n composed of any combination of yang Lines (+1) and yin Lines (-1) if multiplied by itself (i.e., squared) produces the identity element for that dimension. In concrete terms this means, for example, that any bigram multiplied by itself equals the bigram OLD YANG; any of eight trigrams multiplied by itself equals the trigram HEAVEN; and any of the sixty-four hexagrams multiplied by itself equals the hexagram HEAVEN; etc. (valid for any and all dimensions without exception). Consequently, the number of roots the identity element has in any dimension n is equal to the number 2n, these all being real roots in that particular dimension.

Similar contextual analysis would show that the inversion element of any dimension n has 2n roots of the kind we have agreed to refer to as contra-square roots in deference to the Mathematics Establishment.[4]

That leads us to the possibly startling conclusion that in every dimension n there is an inversion element that has the same number of roots as the identity element and all of them are real roots. For two dimensions the two pairs that satisfy the requirement are bigram pairs

For one dimension there is only a single pair that satisfies. That is (surprise, surprise) yin (-1)/yang (+1). What it comes down to is this:

If we are going to continue to insist on referring to square root in terms of the one-dimensional number line, then

+1 has two real roots of the traditional variety, +1 and -1

-1 has two real roots of the newly defined contra variety, +1/-1 and -1/+1

So where do imaginary numbers and quaternions fit in all this? The short answer is they don't. Imaginary numbers entered the annals of human thought through error. There was a pivotal moment[5] in the history of mathematics and science, an opportunity to see that there are in every dimension two different kinds of roots - - - what has been called square root and what we are calling contra-square roots. Enlightenment mathematicians and philosophers essentially allowed the opportunity to slip through their fingers unnoticed.[6]

Descartes at least saw through the veil. He called the whole matter of imaginary numbers 'preposterous'. It seems his venerable opinion was overruled though. Isaac Newton had his say in the matter too. He claimed that roots of imaginary numbers "had to occur in pairs." And yet another great mathematician, philosopher opined. Gottfried Wilhelm Leibniz, in 1702 characterized √−1 as “that amphibian between being and non-being which we call the imaginary root of negative unity.” Had he but preserved such augury conspicuously in mind he might have elaborated the concept of probable numbers in the 18th century. If only he had truly understood the I Ching, instead of dismissing it as a primitive articulation of his own binary number system.

(continued here)

Image: The four quadrants of the Cartesian plane. By convention the quadrants are numbered in a counterclockwise direction. It is as though two number lines were placed together, one going left-right, and the other going up-down to provide context for the two-dimensional plane. Sourced from Math Is Fun.

Notes

[1] My preference might be for square root to be redefined from the bottom up, but I don't see that happening in our lifetimes. Then too this way could be better.

[2] By the term unit number, I intend any number of a given dimension that consists entirely of variant elements of the number one (1) in either its positive or negative manifestation. Stated differently, these are vectors having various different directions within the dimension, but all of scalar value -1 (yin) or +1 (yang). All emblems of I Ching symbolic logic satisfy this requirement. These include the Line, bigram, trigram, tetragram, and hexagram. In any dimension n there exist 2n such emblems. In sum, for our purposes here, a unit number is any of the set of numbers, within any dimension n, which when self-multiplied (squared) produces the multiplicative identity of that dimension which is itself, of course, a member of the set. ADDENDUM (01 MAY 2016): I've since learned that mathematics has a much simpler way of describing this. It calls all these unit vectors. Simple, yes?

[3] I think it fair to presume that this might well have physical correlates in terms of quantum mechanical states or numbers. Here's a thought: why would it be necessary that all subatomic particles exist in the same dimension at all times given that they have a playing field of multiple dimensions, - some of them near certainly beyond the three with which we are familiar? And why would it not be possible for two different particles to be stable and unchanging in their different dimensions, yet become reactive and interact with one another when both enter the same dimension or same amplitude of dimension?

[4] Since in any contra-pair (antipodal opposites) of any dimension, either member of the pair must be regarded once as operator and once as operand. So for the two-dimensional square, for example, there are two antipodal pairs (diagonals) and either vertex of each can be either operator or operand. So in this case, 2 x 2 = 4. For trigrams there are four antipodal pairs, and 2 x 4 = 8. For hexagrams there are thirty-two antipodal pairs and 2 x 32 = 64. In general, for any dimension n there are 2 x 2n/2 = 2n antipodal pairs or contra-roots.

[5] Actually lasting several centuries, from about the 16th to the 19th century. Long enough, assuredly, for the error to have been discovered and corrected. Instead, the 20th century dawned with error still in place, and physicists eager to explain the newly discovered bewildering quantum phenomena compounded the error by latching onto √−1 and quaternions to assuage their confusion and discomfiture. This probably took place in the early days of quantum mechanics when the Bohr model of the atom still featured electrons as traveling in circular orbits around the nucleus or soon thereafter, visions of minuscule solar systems still fresh in the mind. At that time rotations detailed by imaginary numbers and quaternions may have still made some sense. Such are the vagaries of history.

[6] I think an important point to consider is that imaginary and complex numbers were, -to mathematicians and physicists alike,- new toys of a sort that enabled them to accomplish certain things they could not otherwise. They were basically tools of empowerment which allowed manipulation of numbers and points on a graph more easily or conveniently. They provided their controllers a longed for power over symbols, if not over the real world itself. In the modern world ever more of what we humans do and want to do involves manipulation of symbols. Herein, I think, lies the rationale for our continued fascination with and dependence on these tools of the trade. They don't need to actually apply to the world of nature, the noumenal world, so long as they satisfy human desire for domination over the world of symbols it has created for itself and in which it increasingly dwells, to a considerable degree apart from the natural world's sometimes seemingly too harsh laws.

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Beyond the Enlightenment Rationalists: From imaginary to probable numbers - IV

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One of the notable things the Rationalists failed to take into account in their analysis and codification of square roots was the significance of context. In so doing they assured that all related concepts they developed would eventually degenerate into a series of errors of conflation. Do not ever underestimate the importance of context.

Mathematicians, for example, can show that for any 3-dimensional cube there exists a 2-dimensional square, the area of which equals the volume of the cube.[1] And although that is true, something has been lost in translation. This is another of the sleights of hand mathematicians are so fond of. Physicists cannot afford to participate in such parlor tricks as these, however mathematically true they might be.[2]

We will begin now, then, to examine how the mandalic coordinate approach stacks up against that of imaginary numbers and quaternions. The former are holistic and respective of the natural order; the latter are irresponsibly rational, simplistic and, in final analysis, wrong about how nature works.[3] Ambitious endeavor indeed, but let's give it a go.

We've already looked at how the standard geometric interpretation of imaginary numbers in context of the complex plane is based on rotations through continuous Euclidean space. You can brush up on that aspect of the story here if necessary. The mandalic approach to mapping of space is more complicated and far more interesting. It involves multidimensional placement of elements in a discrete space, which is to say a discontinuous space, but one fully commensurate with both Euclidean and Cartesian 3-dimensional space. The holo-interactive manner in which these elements relate to one another leads to a probabilistic mathematical design which preserves commutative multiplication, unlike quaternions which forsake it.

Transformations between these elements are based on inversion (reflection through a point) rather than rotation which cannot in any case reasonably apply to discrete spaces. The spaces that quantum mechanics inhabits are decidedly discrete. They cannot be accurately detailed using imaginary and complex numbers or quaternions. To discern the various, myriad transitions which can occur among mandalic coordinates requires some patience. I think it cannot be accomplished overnight but at least in the post next up we can make a start.[4]

(continued here)

Image: A drawing of the first four dimensions. On the left is zero dimensions (a point) and on the right is four dimensions (A tesseract). There is an axis and labels on the right and which level of dimensions it is on the bottom. The arrows alongside the shapes indicate the direction of extrusion. By NerdBoy1392 (Own work) [CC BY-SA 3.0 or GFDL], via Wikimedia Commons

Notes

[1] If only in terms of scalar magnitude. Lost in translation are all the details relating to vectors and dimensions in the original. Conflation does not itself in every case involve what might be termed 'error' but because it always involves loss or distortion of information, it is nearly always guaranteed to eventuate in error somewhere down the line of argument. The point of all this in our context here is that, in the history of mathematics, something of this sort occurred when the Rationalists of the Enlightenment invented imaginary and complex numbers and again when quaternions were invented in 1843. These involved a disruption of vectors and dimensions as treated by nature. The loss of information involved goes a long way in explaining why no one has been able to explain why and how quantum mechanics works in a century or more. These misconstrued theses of mathematics behave like a demon or ghost in the machine that misdirects, albeit unintentionally, all related thought processes. What we end up with is a plethora of confusion. The fault is not in quantum mechanics but in ourselves, that we are such unrelentingly rational creatures, that so persistently pursue an unsound path that leads to reiterative error.

[2] Because physicists actually care about the real world; mathematicians, not so much.

[3] It must be admitted though that it was not the mathematicians who ever claimed imaginary numbers had anything to do with nature and the real world. Why would they? Reality is not their concern or interest. No, it was physicists themselves who made the mistake. The lesson to be learned by physicists here I expect is to be careful whose petticoat they latch onto. Not all are fabricated substantially enough to sustain their thoughts about reality, though deceptively appearing to do just that for protracted periods of time.

[4] My apologies for not continuing with this here as originally intended. To do so would make this post too long and complicated. Not that transformations among mandalic coordinates are difficult to understand, just that they are very convoluted. This is not a one-point-encodes-one-resident-number plan like that of Descartes we're talking about here. This is mandala country.

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Beyond the Enlightenment Rationalists: From imaginary to probable numbers - III

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My objection to the imaginary dimension is not that we cannot see it. Our senses cannot identify probable dimensions either, at least not in the visually compelling manner they can the three Cartesian dimensions. The question here is not whether imaginary numbers are mathematically true. How could they not be? The cards were stacked in their favor. They were defined in such a manner, -- consistently and based on axioms long accepted valid, -- that they are necessarily mathematically true. There's a word for that sort of thing. --The word is tautological.-- No, the decisive question is whether imaginary numbers apply to the real world; whether they are scientifically true, and whether physicists can truly rely on them to give empirically verifiable results with maps that accurately reproduce mechanisms actually used in nature.[1]

The geometric interpretation of imaginary numbers was established as a belief system using the Cartesian line extending from -1,0,0 through the origin 0,0,0 to 1,0,0 as the sole real axis left standing in the complex plane. In 1843, William Rowan Hamilton introduced two additional axes in a quaternion coordinate system. The new j and k axes, similar to the i axis, encode coordinates of imaginary dimensions. So the complex plane has one real axis, one imaginary; the quaternion system, three imaginary axes, one real, to accomplish which though involved loss of commutative multiplication. The mandalic coordinate system has three real axes upon which are superimposed six probable axes. It is both fully commensurate with the Cartesian system of real numbers and fully commutative for all operations throughout all dimensions as well.[2]

All of these coordinate systems have a central origin point which all other points use as a locus of reference to allow clarity and consistency in determination of location. The mandalic coordinate system is unique in that this point of origin is not a null point of emptiness as in all the other locative systems, but a point of effulgence. In that location where occur Descartes' triple zero triad (0.0.0) and the complex plane's real zero plus imaginary zero (ax=0, bi=0), we find eight related hexagrams, all having neutral charge density, each of these consisting of inverse trigrams with corresponding Lines of opposite charge, canceling one another out. These eight hexagrams are the only hexagrams out of sixty-four total possessing both of these characteristics.[3]

So let's begin now to plot the points of the mandalic coordinate system with the view of comparing its dimensions and points with those of the complex plane.[4] The eight centrally located hexagrams all refer to and are commensurate with the Cartesian triad (0,0,0). In a sense they can be considered eight alternative possible states which can exist in this locale at different times. These are hybrid forms of the four complementary pair of hexagrams found at antipodal vertices of the mandalic cube. The eight vertex hexagrams are those with upper and lower trigrams identical. This can occur nowhere else in the mandalic cube because there are only eight trigrams.[5]

From the origin multiple probability waves of dimension radiate out toward the central points of the faces of the cube, where these divergent force fields rendezvous and interact with reciprocal forces returning from the eight vertices at the periphery. converging toward the origin. Each of these points at the six face centers are common intersections of another eight particulate states or force fields analogous to the origin point except that four originate within this basic mandalic module and four without in an adjacent tangential module. Each of the six face centers then is host to four internal resident hexagrams which share the point in some manner, time-sharing or other. The end result is the same regardless, probabilistic expression of characteristic form and function. There is a possibility that this distribution of points and vectors could be or give rise to a geometric interpretation of the Schrödinger equation, the fundamental equation of physics for describing quantum mechanical behavior. Okay, that's clearly a wild claim, but in the event you were dozing off you should now be fully awake and paying attention.

The vectors connecting centers of opposite faces of an ordinary cube through the cube center or origin of the Cartesian coordinate system are at 180° to each other forming the three axes of the system corresponding to the number of dimensions. The mandalic cube has 24 such axes, eight of which accompany each Cartesian axis thereby shaping a hybrid 6D/3D coordinate system. Each face center then hosts internally four hexagrams formed by hybridization of trigrams in opposite vertices of diagonals of that cube face, taking one trigram (upper or lower) from one vertex and the other trigram (lower or upper) from the other vertex. This means that a face of the mandalic cube has eight diagonals, all intersecting at the face center, whereas a face of the ordinary cube has only two.[6]

The circle in the center of this figure is intended to indicate that the two pairs of antipodal hexagrams at this central point of the cube face rotate through 90° four times consecutively to complete a 360° revolution. But I am describing the situation here in terms of revolution only to show an analogy to imaginary numbers. The actual mechanisms involved can be better characterized as inversions (reflections through a point), and the bottom line here is that for each diagonal of a square, the corresponding mandalic square has a possibility of 4 diagonals; for each diagonal of a cube, the corresponding mandalic cube has a possibility of 8 diagonals. For computer science, such a multiplicity of possibilities offers a greater number of logic gates in the same computing space and the prospect of achieving quantum computing sooner than would be otherwise likely.[7]

Similarly, the twelve edge centers of the ordinary cube host a single Cartesian point, but the superposed mandalic cube hosts two hexagrams at the same point. These two hexagrams are always inverse hybrids of the two vertex hexagrams of the particular edge. For example, the edge with vertices WIND over WIND and HEAVEN over HEAVEN has as the two hybrid hexagrams at the center point of the edge WIND over HEAVEN and HEAVEN over WIND. Since the two vertices of concern here connect with one another via the horizontal x-dimension, the two hybrids differ from the parents and one another only in Lines 1 and 4 which correspond to this dimension. The other four Lines encode the y- amd z-dimensions, therefore remain unchanged during all transformations undergone in the case illustrated here.[8]

This post began as a description of the structure of the mandalic coordinate system and how it differs from those of the complex plane and quaternions. In the composition, it became also a passable introduction to the method of composite dimension. Additional references to the way composite dimension works can be found scattered throughout this blog and Hexagramium Organum. Basically the resulting construction can be thought of as a tensegrity structure, the integrity of which is maintained by opposing forces in equilibrium throughout, which operate continually and never fail, a feat only nature is capable of. We are though permitted to map the process if we can manage to get past our obsession with and addiction to the imaginary and complex numbers and quaternions.[9]

In our next session we'll flesh out probable dimension a bit more with some illustrative examples. And possibly try putting some lipstick on that PIG (Presumably Imaginary Garbage) to see if it helps any.

(continued here)

Notes

[1] For more on this theme, regarding quaternions, see Footnote [1] here. My own view is that imaginary numbers, complex plane and quaternions are artificial devices, invented by rational man, and not found in nature. Though having limited practical use in representation of rotations in ordinary space they have no legitimate application to quantum spaces, nor do they have any substantive or requisite relation to square root, beyond their fortuitous origin in the Rationalists' dissection and codification of square root historically, but that part of the saga was thoroughly misguided. We wuz bamboozled. Why persist in this folly? Look carefully without preconception and you'll see this emperor's finery is wanting. It is not imperative to use imaginary numbers to represent rotation in a plane. There are other, better ways to achieve the same. One would be to use sin and cos functions of trigonometry which periodically repeat every 360°. (Read more about trigonometric functions here.) Another approach would be to use polar coordinates.

[SOURCE]

A quaternion, on the other hand, is a four-element vector composed of a single real element and three complex elements. It can be used to encode any rotation in a 3D coordinate system. There are other ways to accomplish the same, but the quaternion approach offers some advantages over these. For our purposes here what needs to be understood is that mandalic coordinates encode a hybrid 6D/3D discretized space. Quaternions are applicable only to continuous three-dimensional space. Ultimately, the two reside in different worlds and can't be validly compared. The important point here is that each has its own appropriate domain of judicious application. Quaternions can be usefully and appropriately applied to rotations in ordinary three-dimensional space, but not to locations or changes of location in quantum space. For description of such discrete spaces, mandalic coordinates are more appropriate, and their mechanism of action isn't rotation but inversion (reflection through a point.) Only we're not speaking here about inversion in Euclidean space, which is continuous, but in discrete space, a kind of quasi-Boolean space, a higher-dimensional digital space (grid or lattice space). In the case of an electron this would involve an instantaneous jump from one electron orbital to another.

[2] I think another laudatory feature of mandalic coordinates is the fact that they are based on a thought system that originated in human prehistory, the logic of the primal I Ching. The earliest strata of this monumental work are actually a compendium of combinatorics and a treatise on transformations, unrivaled until modern times, one of the greatest intellectual achievements of humankind of any Age. Yet its true significance is overlooked by most scholars, sinologists among them. One of the very few intellectuals in the West who knew its true worth and spoke openly to the fact, likely at no small risk to his professional standing, was Carl Jung, the great 20th century psychologist and philosopher.

It is of relevance to note here that all the coordinate systems mentioned are, significantly, belief systems of a sort. The mandalic coordinate system goes beyond the others though, in that it is based on a still more extensive thought system, as the primal I Ching encompasses an entire cultural worldview. The question of which, if any, of these coordinate systems actually applies to the natural order is one for science, particularly physics and chemistry, to resolve.

Meanwhile, it should be noted that neither the complex plane nor quaternions refer to any dimensions beyond the ordinary three, at least not in the manner of their current common usage. They are simply alternative ways of viewing and manipulating the two- and three-dimensions described by Euclid and Descartes. In this sense they are little different from polar coordinates or trigonometry in what they are attempting to depict. Yes, quaternions apply to three dimensions, while polar coordinates and trigonometry deal with only two. But then there is the method of Euler angles which describes orientation of a rigid body in three dimensions and can substitute for quaternions in practical applications.

A mandalic coordinate system, on the other hand, uniquely introduces entirely new features in its composite potential dimensions and probable numbers which I think have not been encountered heretofore. These innovations do in fact bring with them true extra dimensions beyond the customary three and also the novel concept of dimensional amplitudes. Of additional importance is the fact that the mandalic method relates not to rotation of rigid bodies, but to interchangeability and holomalleability of parts by means of inversions through all the dimensions encompassed, a feature likely to make it useful for explorations and descriptions of particle interactions of quantum mechanics. Because the six extra dimensions of mandalic geometry may, in some manner, relate to the six extra dimensions of the 6-dimensional Calabi–Yau manifold, mandalic geometry might equally be of value in string theory and superstring theory.

It is possible to use mandalic coordinates to describe rotations of rigid bodies in three dimensions, certainly, as inversions can mimic rotations, but this is not their most appropriate usage. It is overkill of a sort. They are capable of so much more and this particular use is a degenerate one in the larger scheme of things.

[3] This can be likened to a quark/gluon soup. It is a unique and very special state of affairs that occurs here. Physicists take note. Don't let any small-minded pure mathematicians dissuade you from the truth. They will likely write all this off as "sacred geometry." Which it is, of course, but also much more. Hexagram superpositions and stepwise dimensional transitions of the mandalic coordinate system could hold critical clues to quantum entanglement and quantum gravity. My apologies to those mathematicians able to see beyond the tip of their noses. I was not at all referring to you here.

[4] Hopefully also with dimensions and points of the quaternion coordinate system once I understand the concepts involved better than I do currently. It should meanwhile be underscored that full comprehension of quaternions is not required to be able to identify some of their more glaring inadequacies.

[5] In speaking of "existing at the same locale at different times" I need to remind the reader and myself as well that we are talking here about particles or other subatomic entities that are moving at or near the speed of light,- - -so very fast indeed. If we possessed an instrument that allowed us direct observation of these events, our biologic visual equipment would not permit us to distinguish the various changes taking place. Remember that thirty frames a second of film produces the illusion of motion. Now consider what thirty thousand frames a second of repetitive action would do. I think it would produce the illusion of continuity or standing still with no changes apparent to our antediluvian senses.

[6] Each antipodal pair has four different possible ways of traversing the face center. Similarly, the mandalic cube has thirty-two diagonals because there are eight alternative paths by which an antipodal pair might traverse the cube center. This just begins to hint at the tremendous number of transformational paths the mandalic cube is able to represent, and it also explains why I refer to dimensions involved as potential or probable dimensions and planes so formed as probable planes. All of this is related to quantum field theory (QFT), but that is a topic of considerable complexity which we will reserve for another day.

[7] One advantageous way of looking at this is to see that the probabilistic nature of the mandalic coordinate system in a sense exchanges bits for qubits and super-qubits through creation of different levels of logic gates that I have referred to elsewhere as different amplitudes of dimension.

[8] Recall that the Lines of a hexagram are numbered 1 to 6, bottom to top. Lines 1 and 4 correspond to, and together encode, the Cartesian x-dimension. When both are yang (+), application of the method of composite dimension results in the Cartesian value +1; when both yin (-), the Cartesian value -1. When either Line 1 or Line 4 is yang (+) but not both (Boole's exclusive OR) the result is one of two possible zero formations by destructive interference. Both of these correspond to (and either encodes) the single Cartesian zero (0). Similarly hexagram Lines 2 and 5 correspond to and encode the Cartesian y-dimension; Lines 3 ane 6, the Cartesian z-dimension. This outline includes all 9 dimensions of the hybrid 6D/3D coordinate system: 3 real dimensions and the 6 corresponding probable dimensions. No imaginary dimensions are used; no complex plane; no quaternions. And no rotations. This coordinate system is based entirely on inversion (reflection through a point) and on constructive or destructive interference. Those are the two principal mechanisms of composite dimension.

[9] The process as mapped here is an ideal one. In the real world errors do occur from time to time. Such errors are an essential and necessary aspect of evolutionary process. Without error, no change. And by implication, likely no continuity for long either, due to external damaging and incapacitating factors that a natural world devoid of error never learned to overcome. Errors are the stepping stones of evolution, of both biological and physical varieties.

-Page 308-

Beyond the Enlightenment Rationalists: From imaginary to probable numbers - II

(continued from here)

When a geometric interpretation of imaginary numbers was at last proposed, long after they were invented, it was as though accomplished by central committee. The upshot was easily enough understood but also simplistic. In broad brushstroke here is what seems to have gone down.

The 3 dimensions of Descartes' coordinate system-a number already deficient from the perspective of mandalic geometry-were reduced to just one. Of the real number axes then only the x-axis remained. This meant from the get-go that any geometric figure that ensued could encompass only linearity in terms of real numbers and dimensions. It was applicable only to a line segment, so the complex plane that resulted could describe just one real dimension and one imaginary dimension. It consecrated the number line in a single dimension, to exclusion of its proper habitation in two others besides. Strike one for imaginary numbers.[1]

With that as background let's look now at the rotations described by this geometric interpretation of imaginary numbers in the context of the complex plane.

i in the complex or cartesian plane. Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axis By Loadmaster (David R. Tribble) (Own work) [CC BY-SA 3.0 or GFDL], via Wikimedia Commons

The number 1 is the multiplicative identity element for real numbers and the number -1 is the reflection inversion element for real numbers. Put another way, the number one times any number equals that number; the number -1 times any number is a negative of that number or the inverse number through a reference point, usually taken as zero. Multiplying by 1 then leaves 1, -1, i and -i all unchanged. Multiplying by -1 changes -1 to 1, 1 to -1, i to -i, and -i to i. In terms of rotations in the complex plane, these changes all involve a rotation through 180 degrees. Multiplication of the number 1 by i changes it to i; i by i changes it to -1; -1 by i to -i; and -i by i to 1. These changes all involve rotations through 90 degrees. And finally, multiplication of 1 by -i changes it to -i; -i by -i changes it to -1; -1 by -i to i; and i by -i to 1: all changes involving rotations through -90 degrees.

The figure below shows another way to interpret these rotations that amounts to the same tbing: i1 = i; i2 =-1; i3 = -i; i4 = 1. Click to enlarge.

Four numbers on the real line multiplied by integer powers of the imaginary unit, which corresponds to rotations by multiples of the right angle. By Keφr [CC0], via Wikimedia Commons

I think a committee of some sort must have come up with this resplendent plan. For certain it was an Academy of Mathematics and Sciences that endorsed and enthroned it. All bow to central authority.

I had planned to include a comparison of imaginary numbers and probable numbers in this post as well but because that is a long discussion itself, it will have to wait till the next post. I might add it should prove well worth the wait.

(continued here)

Notes

[1] Mathematician William Rowan Hamilton subsequently addressed this deficiency in 1843 with his quaternions, a number system that extends the complex numbers to three-dimensional space. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or, in other words, as the quotient of two vectors. This complicated matters even more by introducing a non-commutative multiplication operation to the system, though to be fair the quaternion coordinate system has found some useful applications mainly for calculations involving 3-dimensional rotations, as in 3-dimensional computer graphics, computer vision, and crystallographic texture analysis. Still it becomes problematic when theoretical physics attempts use of quaternions in calculations pertaining to atomic and subatomic spaces where rotations do not actually take place. The conclusion to be drawn here is that quaternions can be usefully, if somewhat clumsily, applied to 3-dimensional macro-spaces but are inapproriate for accurate description of higher dimensional spaces. What is here unfortunate and misleading is that quaternions apparently do describe outcomes of events in the quantum realm to some partial degree, if not the mechanisms of the events themselves. Physicists would not long tolerate them were that not so. [ADDENDUM (24 APRIL, 2016) Since writing this I've learned that quaternions are not currently used in quantum physics nor were they ever, to any great degree, in the past.] In other words, sometimes the right answer can be reached by a wrong method. In the case under discussion here, we should note that it is possible for a rotation to mimic inversion (reflection through a point). A 90° rotation in two dimensions can mimic a single inversion in a single plane through an edge of a square, and a 180° rotation in two dimensions can mimic a single inversion through a diagonal of a square or two successive inversions through two perpendicular edges of a square. A 180° rotation in three dimensions can mimic three inversions through three mutually perpendicular edges of a square; a combination of one inversion through a diagonal of a square and another through an edge perpendicular to the plane of the first inversion; or a single inversion through a diagonal of the cube. Subatomic paricles exist as discrete or quantized entities and would follow such methods of transformation rather than rotations through a continuous space. Of course, transformations involving a diagonal would require more transformative energy than one involving a single edge. Such patterns of relationship and transformation could no doubt be described in terms of quantum states and quantum numbers without too much difficulty by a knowledgeable theoretical physicist. Surely doing so could be no more difficult than using quaternions, which may give a correct answer while also misleading and limiting knowledge of the the true workings of the quantum realm by using an incorrect mechanism, one non-commutative to boot. Nature doesn't approve of hat tricks like that.

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Beyond the Enlightenment Rationalists: From imaginary to probable numbers - I

Imaginary numbers arose in the history of mathematics as a result of misunderstanding the dimensional character of numbers. There was a failure to acknowledge that numbers exist in a context of dimension. This has earlier been addressed at length.[1] Simply put, numbers exist always in a particular dimensional context. Square numbers pertain to a context of two dimensions and therefore to a plane, not a line. Square roots then ought justly reference a two-dimensional geometrical context rather than the linear one mathematics has maintained ever since mathematicians of the Age of Enlightenment decreed it so. Square roots contrary to the way mathematics would have it can neither exist in nor be found in any single line segment, because they do not originate in the number line but in the two-dimensional square.

Algebra, not geometry, provided the breeding ground for imaginary numbers. They were given a geometric interpretation as an afterthought only, long after the fact of their invention. Rationalist algebraists, feeling compelled to give meaning to equations of the form b2 = -4 came up with the fantastic notion of imaginary numbers. Only indirectly did these grow out of nature, by way of minds of men obsessed with reason.[2]

Descartes knew of the recently introduced square roots of negative numbers. He thought them preposterous and was first to refer to the new numbers by the mocking name imaginary, a label which stuck and which continues to inform posterity of the exact manner in which he viewed the oddities. It is one of the ironies of history that when at last a geometrical interpretation of square root of negative numbers was offered it involved swallowing up Descartes' own y-axis. Poetic justice? Or ultimate folly?

Had the essential dimensional nature of numbers been recognized there would have been no need to inquire what the square root of -1 was. It would have been clear that there was no square root of -1 nor any need for such as +1 also has no square root. As linear numbers, neither -1 nor +1 can legitimately be said to have a square root. Both, though, have two-dimensional analogues and these do have square roots, not recognized as such unfortunately by the mathematics hegemony.[3]

In the next post we will look at a comparison between imaginary numbers, which were formulated in accordance with this misconstrual about how numbers relate to dimensions, and probable numbers which grow organically out of a consideration of how numbers and dimensions actually relate to one another in nature.[4] The first of these approaches can be thought of as rational planning by a central authority; the second, as the holistic manner in which nature attends to everything, all at once, and without rational forethought.

(continued here)

Notes

[1] See the series of about nine posts that begins here.

[2] The Rationalists missed here a golden opportunity to relate number and dimension by defining square root much too narrowly. They seem to have been so mesmerized by their algebraic equations that they failed to pursue the search into deeper significance pertaining to essential linkages between dimension and number that intuition and imagination might have bestowed.

[3] As Shakespeare correctly pointed out, a rose by any name would smell as sweet. Plus one times plus one certainly equals plus one but that has nothing to do with actual square root really, just with algebraic linear multiplication. Note has often been made in these pages of the difference between mathematical truth and scientific truth. Whereas mathematics demands only adherence to its axioms and consistency, science requires empirical proof. Mathematics defined square root in a certain manner centuries ago, and has since been devoutly consistent in its adherence to that definition. In so doing it has preserved a cherished doctrine of mathematical truth, as though in formaldehyde. It has also for many centuries contrived to be consistently scientifically incorrect. The problem lies in the fact it has converted physicists and near everyone else to its own insular worldview.

[4] For an early discussion of the probable plane, potential dimensions, and probable numbers see here.

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A Short Philosophical Aside

The scrupulous 3-dimension world we humans inhabit is in fact biological, not physical, in origin. Its limitations are determined by our specific sensory, motor and mental apparatus and abilities. It only hints at the real world, and while doing so it combines some highly erroneous observations as well. Molluscs and insects and arachnids all have a very different perspective of their environment. We would find discomfort in the world view of an octopus, as we do in the quantum world view.[1][2]

Dimension is a term laymen toss about haphazardly. Mathematicians and physicists have a more precise interpretation concerning dimension. For them, any independent parameter constitutes a separate dimension. But when it comes down to the nitty-gritty, what if anything can truly be separate and independent? Those are both relative terms. Nothing that exists is really fully isolate and independent. That is one of the substratal premises from which mandalic geometry evolves: relationships invariably exist. And relationships can always change. Mandalic geometry therefore is a geometry of process - a spacetime geometry, not one of space alone.

For those who created the primal I Ching relationship was considered a fundamental aspect of reality. When they thought of dimension - - - and they did, in their own way - - - relationships were always involved. Flash-forward a few thousand years - quantum mechanics accomplishes much the same with its view of interacting particles in continual motion, ever-changing, and incessantly forging transient effective links with numerous other particles of similar and different type under the influence of various fields of force.

Kant thought that human concepts and categories determine our view of the world and its laws. He held that inborn features of our minds structure our experiences. Since, in his view, mind shapes and structures experience, at some level of representation all human experience shares certain essential operational features. Among these according to Kant are our concepts relating to space and time, integral to all human experience. The same might be said about our concepts of cause and effect.

Kant further asserts that we never have direct experience of things, referred to in his writings as the noumenal world. All we experience is the phenomenal world that is relayed to us by our senses. Kant views noumena as the thing-in-itself or true reality and phenomena as our experience or perception of that thing, filtered through our senses and reasoning. According to Kant science can be applied only to things that can be observed and studied. The entire world of noumena is beyond the scope and reach of science. As an heir to Enlightenment philosophy Kant respects the value of reason but believes the noumenal world to be beyond its scope and reach. So are we fated then never to experience the noumena directly? Not by a long shot. Kant claims the noumena to be accessible but only by intellectual intuition without the aid of reason.[3]

In the world of phenomena nothing is self-existent. Everything exists by virtue of dependence on something else. Point to something, anything at all, that refutes that view and I'll tell you you're out of your mind - and in the noumenal world. What, pray tell, are you doing there and how did you get there anyway? If you can clearly communicate the how I may give it a try myself.[4]

Image:

One of a set of illustrations by Emma V. Moore titled Noumena - Collages © Emma V Moore 2013 courtesy of the artist. More of her exceptional art can be found at http://www.emmavmoore.co.uk. Follow also on Bēhance Please do not remove credits.

Notes

[1] The world view granted us by our inherited biologic capacities has been millions of years in the making. Indeed. But that makes it still not a whit truer than had we groped it only yesterday. Evolution seems to have sacrificed a full immersive sense of reality to grant a greater degree of interoperability essential to dealing with vicissitudes of a material world and confer durability within that domain. The quest after true apprehension we feel impelled to pursue is a siren not without danger.

"The search for reality is the most dangerous of all undertakings, for it destroys the world in which you live." -Nisargadatta Maharaj

[2] Regarding the origin and transformations of the word "scrupulous":

Scrupulous and its close relative "scruple" ("an ethical consideration") come from the Latin noun scrupulus, the diminutive of "scrupus." "Scrupus" refers to a sharp stone, so scrupulus means "small sharp stone." "Scrupus" retained its literal meaning but eventually also came to be used with the metaphorical meaning "a source of anxiety or uneasiness," the way a sharp pebble in one's shoe would be a source of pain. When the adjective "scrupulous" entered the language in the 15th century, it meant "principled." Now it also commonly means "painstaking" or "careful." [Source]

Sad to say, this fascinating word that so successfully wended its way through several related incarnations in a number of different Indo-European languages prior to its appearance in English, c.15th century, appears to be passing out of usage among English speakers in modern times. We will likely be left with the occasional utterance of "scruples" but "scrupulous" itself seems destined for oblivion.

Curiously, my election of the word here was not rationally motivated. As I was framing the thought expressed in the paragraph in my mind, the word just appeared out of nowhere and seemed to insist, "I belong here though you may not yet understand why. You really need a word with my complex heritage of multiple meanings here." And so I went with it, not fully knowing why. Funny thing about it, my rational mind is quite unable now to come up with any other single word that suits as well.

[3] Kant's epistemology recognizes three different sources of knowlege: sensory experience, reason, and intuition. He views intuition as independent of the other two and the only one of the three with direct access to the world of noumena. This may present as suspect at first, but then how do we explain things like what Einstein did a century ago? Einstein himself has hinted in his writings at the essential role of intuition and imagination in his thinking.

Slide 25 of 48

Click here for more slides on Kant's philosophy by William Parkhurst from Introduction to Philosophy Lecture 13, source of the above slide reproduction.

[4] Our human penchant for categorization inevitably leads to dismemberment of holistic reality into an endless number of manifest objects, many of which we no longer recognize as essentially related.

"People normally cut reality into compartments, and so are unable to see the interdependence of all phenomena. To see one in all and all in one is to break through the great barrier which narrows one's perception of reality." -Thích Nhất Hạnh

-Page 305-

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, and XOR. 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 inversion or invert. 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 ON or OFF). Although both AND and INV 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, not inversion.

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 yin is yang.[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 reversion or revert. This operator produces no change in what it acts upon. It is the multiplicative identity element (also called the neutral element or unit 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 yin and yang, 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.

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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. There will 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 bedrock of elementary algebra and the number line. We've previously 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]

(continued here)

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 been many errors 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.

-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 conjunction and, denoted ∧, the disjunction or, denoted ∨, and the negation not, 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.

Kalachakra Mandala

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, yin and yang, 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.

The I Ching uses 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.

(continued here)

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 3.0 or GFDL], via Wikimedia Commons

Notes

[1] Boole's algebra predated the modern developments in abstract algebra and mathematical logic but is seen as connected to the origins of both fields. Similarly to elementary algebra, the pure equational part of the theory can be formulated without regard to explicit values for the variables.

[2] If you are new to Boolean algebra these definitions may be confusing because in some ways they seem to fly in the face of ordinary algebra. I'll admit, I find them somewhat daunting. Let me see if I can clarify the three examples given in this quote. Those of you more familiar with the language of Boolean algebra might kindly correct me in the event I err. I'm growing more comfortable with being wrong at times. And this is after all a work in progress.

Boolean XOR (exclusive-or) allows that a statement of the form (x XOR y) is TRUE if either x or y is TRUE but FALSE if both are TRUE or if both are FALSE. Since Boolean algebra uses binary numbers and represents TRUE by 1, FALSE by 0, then for x = TRUE, y = TRUE x + y = 1 + 1 = 0 , so FALSE for x = FALSE, y = FALSE x + y = 0 + 0 = 0 , so FALSE for x = TRUE, y = FALSE x + y = 1 + 0 = 1 , so TRUE for x = FALSE, y = TRUE x + y = 0 + 1 = 1 , so TRUE

Boolean AND (conjunction) allows that a statement of the form (x AND y) is TRUE only if both x is TRUE and y is TRUE. If either x or y is FALSE or both are FALSE then x AND y is FALSE. Here algebraic multiplication of binary 1s and 0s plays the role of Boolean AND. (Incidentally, binary multiplication works exactly the same way as algebraic multiplication. There's a gift!) for x = TRUE, y = TRUE xy = 1(1) = 1, so TRUE for x = FALSE, y = FALSE xy = 0(0) = 0, so FALSE for x = TRUE, y = FALSE xy = 1(0) = 0 , so FALSE for x = FALSE, y = TRUE xy = 0(1) = 0 , so FALSE

Boolean OR (inclusive-or) is the truth-functional operator of (inclusive) disjunction, also known as alternation. The OR of a set of operands is true if and only if one or more of its operands is true. The logical connective that represents this operator is generally written as ∨ or +. As stated in the Wikipedia article logical disjunction x∨y (inclusive-or) is definable as x + y + xy [(x OR y) OR (x AND y)] as shown below. [Note: x AND y is often written xy in Boolean algebra. So watch out which algebra 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.

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Mandalic Line Segments, Entanglement and Quantum Gravity Part I

(continued from here)

We are going to consider once again now geometric line segments of mandalic geometry and their relation to Cartesian line segments and the Western number line. Yes, this is sort of a detour from what I stated we would look at next. But this is not unrelated and lies at the very heart of mandalic geometry, and I’m not yet ready to address what I projected in the last remark of my previous post.

I keep returning to this subject because of its extreme importance. Beyond its significance to understanding the logic encoded in mandalic geometry and the I Ching, I believe it may also hold the key to quantum entanglement and quantum gravity. Despite the fact that mandalic line segments are really fundamentally mental constructs, a fiction of sorts, it is still important to understand how they are composed and how their components interact. Though they may themselves be fictions, the line segments and the points that compose them do in fact map a number of physical entities, realities that may be related to quantum numbers and quantum particles and states.

When Descartes invented his coordinate system, with its points and line segments, he based his system on the number line extended to two or three dimensions. In modeling it on the number line the space he described was imagined to bear a necessary one to one correspondence to the real numbers.[1] However this 1:1 mapping of geometric space to the real numbers was a premise implicitly assumed by Descartes. It was in fact axiomatic,[2] but apparently not stated as such.[3] As a result, the presumed relation has become a blind spot[4] in Western thought, never proved nor disproved, at least not at subatomic scales.[5]

Neither mandalic geometry nor the primal I Ching make such an assumption. In place of Descartes' 1:1 correspondence of geometric space and the numbers on the number line, we find a mandalic arrangement in which there are different categories of spatial location which can host one or more discrete numbers in a probabilistic manner. This creates various dimensional amplitudes and a multidimensional waveform of component entities.[6]

To my mind these characteristics of the mandalic coordinate system in combination with others described elsewhere make it more relevant to investigation and interpretation of many quantum phenomena which are as yet poorly understood than Cartesian coordinate dynamics may be and without need for recourse to imaginary numbers and complex plane.

(continued here)

Image: 6 steps of the Sierpinski carpet, animated. By KarocksOrkav (Own work) [CC BY-SA 3.0], via Wikimedia Commons

Notes

[1] Real numbers are numbers that can be found on the number line. This includes both the rational and irrational numbers.

[2] That is to say, taken for granted as self-evident.

[3] See Note [4] here.

[4] We have lived with this unproved premise so long that we no longer even question it, or imagine that there might be an alternative which conforms better to reality at certain scales, notably subatomic scales. The I Ching also seems to suggest that a complete true description of complex relationships that involve a large number of dimensions, including complex societal relationships, requires more than a simple 1:1 correspondence between the notational symbols involved and the realities they represent.

[5] And from what I can see, no one seems to have much interest in proving or disproving this assumption.

[6] When speaking about hexagrams the number of dimensions involved is six as each Line of the hexagram encodes a value for a single distinct dimension in a 6-dimensional space. In a hybrid 6D/3D compositing of dimensions though, two such Lines in relation reference a single Cartesian dimension in 2- or 3-space. A concept not to be missed here is that interactions of quantum particles may well involve such integration of dimension, of dimensions we are not even aware of beyond the unsettling fact they upset the neat applecart of customary conceptual categories.

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Beyond Taoism - Part 5 A Vector-based Probabilistic Number System Part II

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Taoism and the primordial I Ching are in agreement that temporal changes have two different aspects: sequent and cyclic. Western thought in general follows suit. The I Ching differs from the other two in asserting that the direction of change - for both sequent and cyclic change - is fully reversible, with the proviso that sufficiently small units of measurement are involved.[1] The probability that reversal can be achieved diminishes proportionately to the magnitude of change that has taken place.[2]

Taoist appropriation of bigrams and trigrams of the I Ching to model such phenomena as change of seasons and phases of the moon is plausible if not quite legitimate. The natural phenomena so modeled are macroscopic and vary continuously and inexorably throughout an ever-repeating cyclic spectrum. And there's the rub.

As they occur and function in the I Ching bigrams and trigrams are dicontinuous discrete elements, formed by other similarly discontinuous discretized entities, and they follow evolutionary courses which are most often nonrepetitive. So the Taoist usage is misleading at best, annihilative at worst. Unfortunately, as the I Ching itself evolved through centuries of commentaries and reinterpretations, it became ever more contaminated and tainted by these Taoist corruptions of meaning, at the same time that it was being inundated by Confucian sociological and ethical reworkings. What we have today is an amalgam, the various parts of which do not sit well with one another.[3]

Though it may in part be hyperbole to prove a point, the stark difference between the two approaches, that of Taoism and that of the I Ching, is epitomized by comparison of the Taoist diagram of the cycle of seasons with diagrams at the top and bottom of the page, which are based on the number, logic, and coordinate systems of The Book of Changes.[4] The increased complexity of the latter diagrams should not prove a stumbling block, as they can be readily understood in time with focus and attention to detail. The important take-away for now is that in the I Ching bigrams exist within a larger dimensional context than the Taoist diagram avows, and this context makes all their interactions more variable, conditional, and complex. As well, the same can be said of trigrams and hexagrams.

One of the more important aspects of these differences has to do with the notion of equipotentiality. As bigrams and trigrams function within higher dimensional contexts in the I Ching, this introduces a possibility of multiple alternative paths of movement and directions of change. Put another way, primordial I Ching logic encompasses many more degrees of freedom than does the logic of Taoism.[5] There is no one direction or path invariably decreed or favored. An all-important element of conditionality prevails. And that might be the origin of what quantum mechanics has interpreted as indeterminism or chance.

Next up, a closer look at equipotentiality and its further implications.

Section FH(n)[6]

(continued here)

Notes

[1] There are exceptions. Taoist alchemy describes existence of certain changes that admit reversibility under special circumstances. Other than the Second Law of Thermodynamics (which is macroscopic in origin, not result of any internally irreversible microscopic properties of the bodies), the laws of physics neglect all distinction between forward-moving time and backward-moving time. Chemistry recognizes existence of certain states of equilibrium in which the rates of change in both directions are equal. Other exceptions likely occur as well.

[2] Since change is quantized in the I Ching, which is to say, it is divided into small discretized units, which Line changes model, the magnitude of change is determined by the number of Line changes that have occurred between Point A and Point B in spacetime. Reversal is far easier to achieve if only a single Line change has occurred than if three or four Lines have changed for example.

[3] Ironically, Taoism itself has pointed out the perils of popularity. Had the I Ching been less popular, less appealing to members of all strata of society, it would have traveled through time more intact. Unless, of course, it ended up buried or burned. What is fortunate here is that much of the primordial logic of the I Ching can be reconstructed by focusing our attention on the diagrammatic figures and ignoring most of the attached commentary.

[4] These diagrams do not occur explicitly in the I Ching. The logic they are based on, though, is fully present implicitly in the diagramatic structural forms of hexagrams, trigrams, and bigrams and the manner of their usage in I Ching divinatory practices.

[5] Or, for that matter, than does the logic of Cartesian coordinate space if we take into account the degrees of freedom of six dimensional hexagrams mapped by composite dimensional methodology to model mandalic space. (See Note [4] here for important related remarks.)

[6] This is the closest frontal section to the viewer through the 3-dimensional cube using Taoist notation. See here for further explanation. Keep in mind this graph barely hints at the complexity of relationships found in the 6-dimensional hypercube which has in total 4096 distinct changing and unchanging hexagrams in contrast to the 16 changing and unchanging trigrams we see here. Though this model may be simple by comparison, it will nevertheless serve us well as a key to deciphering the number system on which I Ching logic is based as well as the structure and context of the geometric line that can be derived by application of reductionist thought to the associated mandalic coordinate system of the I Ching hexagrams. We will refer back to this figure for that purpose in the near future.

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Beyond Taoism - Part 4 A Vector-based Probabilistic Number System Introduction

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Leibniz erred in concluding the hexagrams of the I Ching were based on a number system related to his own binary number system. He had a brilliant mind but was just as fallible as the rest of us. He interpreted the I Ching in terms of his own thought forms, and he saw the hexagrams as a foreshadowing of his own binary arithmetic.[1]

So in considering the hexagram Receptive, Leibniz understood the number 0; in the hexagram Return, the number 1; in the hexagram Army, the number 2; in the hexagram Approach, the number 3; in the hexagram Modesty, the number 4; in the hexagram Darkening of the Light, the number 5; and so on, up to the hexagram Creative, in which he saw the number 63.[2] His error is perhaps excusable in light of the fact that the Taoists, though much closer to the origin of the I Ching in time, themselves misinterpreted the number system it was based on.[3]

From our Western perspective I Ching hexagrams are composed of trigrams, tetragrams, bigrams, and ultimately yin and yang Lines. From the native perspective of the I Ching this order of arrangement is putting the cart before the horse. Dimensions and their interactions are, in the view of I Ching philosophy and mandalic geometry, antecedent logically and materially to any cognitive parts we may abstract from them. Taoism in certain contexts has abstracted the parts and caused them to appear as if primary. It has the right to do so if creating its own philosophy, but not as interpretation of the logic of the I Ching. It is a fallacy if so intended.[4]

The Taoists borrowed from the I Ching two-dimensional numbers, treated them as one-dimensional and based their quasi-modular number system on the dimension-deficient result. This is the way they arrived at their seasonal cycle consisting of bigrams: old yin (Winter), young yang (Spring), old yang (Summer), young yin (Autumn), old yin (Winter), and so forth. This represents a very much impoverished and impaired version of the original configuration in the primal strata of the I Ching.[5]

The number system of the I Ching is not a linear one-dimensional number system like the positional decimal number system of the West; nor is it like the positional binary number system invented by Leibniz. It is not even like the quasi-modular number system of Taoism. The key to the number system of the hexagrams is located not in the 64 unchanging explicit hexagrams, but rather in the changing implicit hexagrams found only in the divination practice associated with the I Ching. These number 4032.[6] The manner in which these operate, however, is actually fairly simple and is uniform throughout the system. So once understood, they can be safely relegated to the implicit background, coming into play only during procedures involving divination or in attempts to understand the system fully, logically and materially. When dealing with more ordinary circumstances just the 64 more stable hexagrams need be attended to in a direct and explicit manner.

The Taoist sequence of bigrams is in fact a corruption of the far richer asequential multidimensional arrangement of bigrams that occurs in I Ching hexagrams and divination. There we see that change can occur from any one of the four stable bigrams to any other. If this is so then no single sequence can do justice to the total number possible. The ordering of bigrams presented by Taoism is just one of many that make up the real worlds of nature and humankind. Taoism imparts special significance to this sequence; the primal I Ching does not. It views all possible pathways of change as equally likely.[7]

Next time around we will look further into the implications of this equipotentiality and see how it plays out in regard to the number system of the I Ching.

Section FH(n)[8]

(continued here)

Notes

[1] By equating yang with 1 and yin with 0 it is possible to sequence the 64 I Ching hexagrams according to binary numbers 0 through 63. The mere fact that this is possible does not, however, mean that this was intended at the time the hexagrams were originally formulated. Unfortunately, this arrangement of hexagrams seems to have been the only one of which Leibniz had knowledge. This sequence was, in fact, the creation of the Chinese philosopher Shao Yong (1011–1077). It did not exist in human mentation prior to the 11th century CE.

This arrangement was set down by the Song dynasty philosopher Shao Yong (1011–1077 CE), six centuries before Wilhelm Leibniz described binary notation. Leibniz published 'De progressione dyadica' in 1679. In 1701 the Jesuit Joachim Bouvet wrote to him enclosing a copy of Shao Yong's 'Xiantian cixu' (Before Heaven sequence). [Source]

Note also that the author of Calling crane in the shade, the source quoted above, calls attention to confusion that exists about whether the "true binary sequence of hexagrams" should begin with the lowest line as the least significant bit (LSB) or the highest line. He points out that the Fuxi sequence as transmitted by Shao Yong in both circular and square diagrams takes the highest line as the LSB, although in fact it would make more sense in consideration of how the hexagram form is interpreted to take the lowest line as the LSB. My thinking is that either Shao Yong misinterpreted the usage of hexagram form or, more likely, the conventional interpretation of the Shao Yong diagrams is incorrect. Here I have chosen to use the lowest line of the hexagram as the LSB, and I think it possible Leibniz may have done the same.

If one considers the circular Shao Yong diagram, the easier of the two to follow, one can reconstruct the binary sequence, with the lowest line as LSB, by beginning with the hexagram EARTH at the center lower right half of the circle, reading all hexagrams from outside line (bottom) to inside line (top), progressing counterclockwise to MOUNTAIN over WIND at top center, then jumping to hexagram MOUNTAIN over EARTH bottom center of left half of the circle, and progressing clockwise to hexagram HEAVEN at top center. Of the two, this is the interpretation that makes the more sense to me and the one I have followed here, despite the fact that it is not the received traditional interpretation of the Shao Yong sequence. Historical transmissions have not infrequently erred. Admittedly it is difficult to decipher all Lines of some of the hexagrams in the copy Leibniz received due to passage of time and its effects on paper and ink. Time is not kind to ink and paper, nor for that matter to flesh and products of intellect.

In the final analysis, which of the two described interpretations is the better is moot because neither conforms to the logic of the I Ching which is not binary to begin with. Moreover, there is a third interpretation of the Shao Yong sequence that is superior to either described here. It is not binary-based. And why should it be? After all the Fuxi trigram sequence which Shao Yong took as model for his hexagram sequence is itself not binary-based. Perhaps we'll consider that interpretation somewhere down the road. For now, the main take-away is that Leibniz, in his biased interpretation of the I Ching hexagrams made one huge mistake. Ironically, had he not some 22 years prior already invented binary arithmetic, this error likely would have led him to invent it. It was "in the cards" as they say. At least in certain probable worlds.

[2] Receptive and Creative are alternative names for the hexagrams Earth and Heaven, respectively. The sequence detailed can be continued ad infinitum using yin-yang notation, though of course this takes us beyond the realm of hexagrams into what would be, for mandalic geometry and logistics of the I Ching, domains of dimensions numbering more than six. Keep in mind here though that Leibniz was not thinking in terms of dimension but an alternative method of expressing the prevalent base 10 positional number system notation of the West. He held in his grasp the key to unlocking an even greater treasure but apparently never once saw that was so. This seems strange considering his broadly diversified interests and pursuits in the fields of mathematics, physics, symbolic logic, information science, combinatorics, and in the nature of space. Moreover, his concern with these was not just as separate subjects of investigation. He envisaged uniting all of them in a universal language capable of expressing mathematical, scientific, and metaphysical concepts.

[3] Earlier in this blog I have too often confused Taoism with pre-Taoism. The earliest strata of the I Ching belong to an age that preceded Taoism by centuries, if not millennia. Though Taoism was largely based on the philosophy and logic of the I Ching, it didn't always interpret source materials correctly, or possibly at times it intentionally used source materials in new ways largely foreign to the originals. The number system of the I Ching is a case in point. In the interest of full disclosure, I am not an expert in the history or philosophy of Taoism. Taoist philosophies are diverse and extensive. No one has a complete set or grasp of all the thoughts, practices and techniques of Taoism. The two core Taoist texts, the Tao Te Ching and Chuang-tzu, provide the philosophical basis of Taoism which derives from the eight trigrams (bagua) of Fu Xi, c. 2700 BCE, the various combinations of which created the 64 hexagrams documented in the I Ching. The Daozang, also referred to as the Taoist canon, consists of around 1,400 texts that were collected c. 400, long after the two classic texts mentioned. What I describe as Taoist thought then is abstracted in some manner from a huge compilation, parts of which may well differ from what is presented here. Similar effects of time and history can be discerned in Buddhism, Christianity, Islam and secular schools of thought like Platonism, Aristotelianism, Humanism, etc.

[4] Recent advances in the sciences have begun to raise new ideas regarding the structure of reality. Many of these have parallels in Eastern thought. There has been a shift away from the reductionist view in which things are explained by breaking them down then looking at their component parts, towards a more holistic view. Quantum physics notably has changed the way reality is viewed. There are no certainties at a quantum level, and the experimenter is necessarily part of the experiment. In this new view of nature everything is linked and man is himself one of the linkages.

[5] It is not so much that this is incorrect as that it is extremely limiting with respect to the capacities of the I Ching hexagrams. A special case has here been turned into a generalization that purports to cover all bases. This may serve well enough within the confines of Taoism but it comes nowhere near elaborating the number system native to the I Ching. We would be generous in describing it as a watered down version of a far more complex whole. Through the centuries both Confucianism and Taoism restructured the I Ching to make it conducive to their own purposes. They edited it and revised it repeatedly, generating commentary after commentary, which were admixed with the original, so that the I Ching as we have it today, the I Ching of tradition, is a hodgepodge of many convictions and many opinions. This makes the quest for the original features of the I Ching somewhat akin to an archaeological dig. I find it not all that surprising that the oracular methodology of consulting the I Ching holds possibly greater promise in this endeavor than the written text. The early oral traditions were preserved better, I think, by the uneducated masses who used the I Ching as their tool for divination than by philosophers and scholars who, in their writings, played too often a game of one-upmanship with the original.

[6] A Line can be either yin or yang, changing or unchanging. Then there are four possible Line types and six Lines to a hexagram. This gives a total of 4096 changing and unchanging hexagrams (46 = 4096). Since there are 64 unchanging hexagrams (26 = 64) there must be 4032 changing hexagrams (4096-64 = 4032).

[7] This calls to mind the path integral formulation of quantum mechanics which was developed in its complete form by Richard Feynman in 1948. See, for example, this description of the path integral formulation in context of the double-slit experiment, the quintessential experiment of quantum mechanics.

[8] This is the closest frontal section to the viewer through the 3-dimensional cube using Taoist notation. See here for further explanation. Keep in mind this graph barely hints at the complexity of relationships found in the 6-dimensional hypercube which has in total 4096 distinct changing and unchanging hexagrams in contrast to the 16 changing and unchanging trigrams we see here. Though this model may be simple by comparison, it will nevertheless serve us well as a key to deciphering the number system on which I Ching logic is based as well as the structure and context of the geometric line that can be derived by application of reductionist thought to the associated mandalic coordinate system of the I Ching hexagrams. We will refer back to this figure for that purpose in the near future.

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