I’m Not Afraid of AI Overlords— I’m Afraid of Whoever’s Training Them To Think That Way
by Damien P. Williams
I want to let you in on a secret: According to Silicon Valley’s AI’s, I’m not human.
Well, maybe they think I’m human, but they don’t think I’m me. Or, if they think I’m me and that I’m human, they think I don’t deserve expensive medical care. Or that I pose a higher risk of criminal recidivism. Or that my fidgeting behaviours or culturally-perpetuated shame about my living situation or my race mean I’m more likely to be cheating on a test. Or that I want to see morally repugnant posts that my friends have commented on to call morally repugnant. Or that I shouldn’t be given a home loan or a job interview or the benefits I need to stay alive.
Now, to be clear, “AI” is a misnomer, for several reasons, but we don’t have time, here, to really dig into all the thorny discussion of values and beliefs about what it means to think, or to be a mind— especially because we need to take our time talking about why values and beliefs matter to conversations about “AI,” at all. So instead of “AI,” let’s talk specifically about algorithms, and machine learning.
Machine Learning (ML) is the name for a set of techniques for systematically reinforcing patterns, expectations, and desired outcomes in various computer systems. These techniques allow those systems to make sought after predictions based on the datasets they’re trained on. ML systems learn the patterns in these datasets and then extrapolate them to model a range of statistical likelihoods of future outcomes.
Algorithms are sets of instructions which, when run, perform functions such as searching, matching, sorting, and feeding the outputs of any of those processes back in on themselves, so that a system can learn from and refine itself. This feedback loop is what allows algorithmic machine learning systems to provide carefully curated search responses or newsfeed arrangements or facial recognition results to consumers like me and you and your friends and family and the police and the military. And while there are many different types of algorithms which can be used for the above purposes, they all remain sets of encoded instructions to perform a function.
And so, in these systems’ defense, it’s no surprise that they think the way they do: That’s exactly how we’ve told them to think.
[Image of Michael Emerson as Harold Finch, in season 2, episode 1 of the show Person of Interest, “The Contingency.” His face is framed by a box of dashed yellow lines, the words “Admin” to the top right, and “Day 1” in the lower right corner.]
“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.”
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 equationandFeynman 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 andLouis 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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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]
Given this fresh context - one of greater dimension than the number line - it soon becomes clear with little effort that a unit number[2]ofany 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 elementandall 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.
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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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]
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 3.0orGFDL],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 whyandhow 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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Taoism and the primordial I Chingare 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 continuouslyandinexorably 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.
[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 timeandbackward-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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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. :)
Leibniz erred in concluding the hexagrams of the I Ching were based on a number system related to his own binary number system. He had a brilliant mind but was just as fallible as the rest of us. He interpreted the I Ching in terms of his own thought forms, and he saw the hexagrams as a foreshadowing of his own binary arithmetic.[1]
So in considering the hexagram Receptive, Leibniz understood the number 0; in the hexagram Return, the number 1; in the hexagram Army, the number 2; in the hexagram Approach, the number 3; in the hexagram Modesty, the number 4; in the hexagram Darkening of the Light, the number 5; and so on, up to the hexagram Creative, in which he saw the number 63.[2] His error is perhaps excusable in light of the fact that the Taoists, though much closer to the origin of the I Ching in time, themselves misinterpreted the number system it was based on.[3]
From our Western perspectiveI Ching hexagrams are composed of trigrams, tetragrams, bigrams, and ultimately yinandyang Lines. From the native perspective of the I Ching this order of arrangement is putting the cart before the horse. Dimensions and their interactions are, in the view of I Ching philosophy and mandalic geometry, antecedent logically and materially to any cognitive parts we may abstract from them. Taoism in certain contexts has abstracted the parts and caused them to appear as if primary. It has the right to do so if creating its own philosophy, but not as interpretation of the logic of the I Ching. It is a fallacy if so intended.[4]
The Taoists borrowed from the I Ching two-dimensional numbers, treated them as one-dimensional and based their quasi-modular number system on the dimension-deficient result. This is the way they arrived at their seasonal cycle consisting of bigrams: old yin (Winter), young yang (Spring), old yang (Summer), young yin (Autumn), old yin (Winter), and so forth. This represents a very much impoverished and impaired version of the original configuration in the primal strata of the I Ching.[5]
The number system of the I Ching is not a linear one-dimensional number system like the positional decimal number system of the West; nor is it like the positional binary number system invented by Leibniz. It is not even like the quasi-modular number system of Taoism. The key to the number system of the hexagrams is located not in the 64 unchanging explicit hexagrams, but rather in the changing implicit hexagrams found only in the divination practice associated with the I Ching. These number 4032.[6] The manner in which these operate, however, is actually fairly simple and is uniform throughout the system. So once understood, they can be safely relegated to the implicit background, coming into play only during procedures involving divination or in attempts to understand the system fully, logically and materially. When dealing with more ordinary circumstances just the 64 more stable hexagrams need be attended to in a direct and explicit manner.
The Taoist sequence of bigrams is in fact a corruption of the far richer asequential multidimensional arrangement of bigrams that occurs in I Ching hexagrams and divination. There we see that change can occur from any one of the four stable bigrams to any other. If this is so then no single sequence can do justice to the total number possible. The ordering of bigrams presented by Taoism is just one of many that make up the real worlds of nature and humankind. Taoism imparts special significance to this sequence; the primal I Ching does not. It views all possible pathways of change as equally likely.[7]
Next time around we will look further into the implications of this equipotentiality and see how it plays out in regard to the number system of the I Ching.
[1] By equating yang with 1 and yin with 0 it is possibletosequence the 64 I Ching hexagrams according to binary numbers 0 through 63. The mere fact that this is possible does not, however, mean that this was intended at the time the hexagrams were originally formulated. Unfortunately, this arrangement of hexagrams seems to have been the only one of which Leibniz had knowledge. This sequence was, in fact, the creation of the Chinese philosopher Shao Yong (1011–1077). It did not exist in human mentation prior to the 11th century CE.
This arrangement was set down by the Song dynasty philosopher Shao Yong (1011–1077 CE), six centuries before Wilhelm Leibniz described binary notation. Leibniz published ‘De progressione dyadica’ in 1679. In 1701 the Jesuit Joachim Bouvet wrote to him enclosing a copy of Shao Yong’s 'Xiantian cixu’ (Before Heaven sequence). [Source]
Note also that the author of Calling crane in the shade, the source quoted above, calls attention to confusion that exists about whether the “true binary sequence of hexagrams” should begin with the lowest line as the least significant bit (LSB) or the highest line. He points out that the Fuxi sequence as transmitted by Shao Yong in both circular and square diagrams takes the highest line as the LSB, although in fact it would make more sense in consideration of how the hexagram form is interpreted to take the lowest line as the LSB. My thinking is that either Shao Yong misinterpreted the usage of hexagram form or, more likely, the conventional interpretation of the Shao Yong diagrams is incorrect. Here I have chosen to use the lowest line of the hexagram as the LSB, and I think it possible Leibniz may have done the same.
If one considers the circular Shao Yong diagram, the easier of the two to follow, one can reconstruct the binary sequence, with the lowest line as LSB, by beginning with the hexagram EARTH at the center lower right half of the circle, reading all hexagrams from outside line (bottom) to inside line (top), progressing counterclockwise to MOUNTAIN over WIND at top center, then jumping to hexagram MOUNTAIN over EARTH bottom center of left half of the circle, and progressing clockwise to hexagram HEAVEN at top center. Of the two, this is the interpretation that makes the more sense to me and the one I have followed here, despite the fact that it is not the received traditional interpretation of the Shao Yong sequence. Historical transmissions have not infrequently erred. Admittedly it is difficult to decipher all Lines of some of the hexagrams in the copy Leibniz received due to passage of time and its effects on paper and ink. Time is not kind to ink and paper, nor for that matter to flesh and products of intellect.
In the final analysis, which of the two described interpretations is the better is moot because neither conforms to the logic of the I Ching which is not binary to begin with. Moreover, there is a third interpretation of the Shao Yong sequence that is superior to either described here. It is not binary-based. And why should it be? After all the Fuxi trigram sequence which Shao Yong took as model for his hexagram sequence is itself not binary-based. Perhaps we’ll consider that interpretation somewhere down the road. For now, the main take-away is that Leibniz, in his biased interpretation of the I Ching hexagrams made one huge mistake. Ironically, had he not some 22 years prior already invented binary arithmetic, this error likely would have led him to invent it. It was “in the cards” as they say. At least in certain probable worlds.
[2]ReceptiveandCreative are alternative names for the hexagrams EarthandHeaven, respectively. The sequence detailed can be continued ad infinitum using yin-yang notation, though of course this takes us beyond the realm of hexagrams into what would be, for mandalic geometry and logistics of the I Ching, domains of dimensions numbering more than six. Keep in mind here though that Leibniz was not thinking in terms of dimension but an alternative method of expressing the prevalent base 10 positional number system notation of the West. He held in his grasp the key to unlocking an even greater treasure but apparently never once saw that was so. This seems strange considering his broadly diversified interests and pursuits in the fields of mathematics, physics, symbolic logic, information science, combinatorics, and in the nature of space. Moreover, his concern with these was not just as separate subjects of investigation. He envisaged uniting all of them in a universal language capable of expressing mathematical, scientific, and metaphysical concepts.
[3] Earlier in this blog I have too often confused Taoism with pre-Taoism. The earliest strata of the I Ching belong to an age that preceded Taoism by centuries, if not millennia. Though Taoism was largely based on the philosophy and logic of the I Ching, it didn’t always interpret source materials correctly, or possibly at times it intentionally used source materials in new ways largely foreign to the originals. The number system of the I Ching is a case in point.
In the interest of full disclosure, I am not an expert in the history or philosophy of Taoism. Taoist philosophies are diverse and extensive. No one has a complete set or grasp of all the thoughts, practices and techniques of Taoism. The two core Taoist texts, the Tao Te ChingandChuang-tzu, provide the philosophical basis of Taoism which derives from the eight trigrams (bagua) of Fu Xi, c. 2700 BCE, the various combinations of which created the 64 hexagrams documented in the I Ching. The Daozang, also referred to as the Taoist canon, consists of around 1,400 texts that were collected c. 400, long after the two classic texts mentioned. What I describe as Taoist thought then is abstracted in some manner from a huge compilation, parts of which may well differ from what is presented here. Similar effects of time and history can be discerned in Buddhism, Christianity, Islam and secular schools of thought like Platonism,Aristotelianism,Humanism, etc.
[4] Recent advances in the sciences have begun to raise new ideas regarding the structure of reality. Many of these have parallels in Eastern thought. There has been a shift away from the reductionist view in which things are explained by breaking them down then looking at their component parts, towards a more holistic view. Quantum physics notably has changed the way reality is viewed. There are no certainties at a quantum level, and the experimenter is necessarily part of the experiment. In this new view of nature everything is linked and man is himself one of the linkages.
[5] It is not so much that this is incorrect as that it isextremelylimiting with respect to the capacities of the I Ching hexagrams. A special case has here been turned into a generalization that purports to cover all bases. This may serve well enough within the confines of Taoism but it comes nowhere near elaborating the number system native to the I Ching. We would be generous in describing it as a watered down version of a far more complex whole. Through the centuries both Confucianism and Taoism restructured the I Ching to make it conducive to their own purposes. They edited it and revised it repeatedly, generating commentary after commentary, which were admixed with the original, so that the I Ching as we have it today, the I Ching of tradition, is a hodgepodge of many convictions and many opinions. This makes the quest for the original features of the I Ching somewhat akin to an archaeological dig. I find it not all that surprising that the oracular methodology of consulting the I Ching holds possibly greater promise in this endeavor than the written text. The early oral traditions were preserved better, I think, by the uneducated masses who used the I Ching as their tool for divination than by philosophers and scholars who, in their writings, played too often a game of one-upmanship with the original.
[6] A Line can be either yin or yang, changing or unchanging. Then there are four possible Line types and six Lines to a hexagram. This gives a total of 4096 changing and unchanging hexagrams (46 = 4096). Since there are 64 unchanging hexagrams (26 = 64) there must be 4032 changing hexagrams (4096-64 = 4032).
[7] This calls to mind the path integral formulation of quantum mechanics which was developed in its complete form by Richard Feynman in 1948. See, for example, this description of the path integral formulation in context of the double-slit experiment, the quintessential experiment of quantum mechanics.
[8] This is the closest frontal section to the viewer through the 3-dimensional cube using Taoist notation. See here for further explanation. Keep in mind this graph barely hints at the complexity of relationships found in the 6-dimensional hypercube which has in total 4096 distinct changing and unchanging hexagrams in contrast to the 16 changing and unchanging trigrams we see here. Though this model may be simple by comparison, it will nevertheless serve us well as a key to deciphering the number system on which I Ching logic is based as well as the structure and context of the geometric line that can be derived by application of reductionist thought to the associated mandalic coordinate system of the I Ching hexagrams. We will refer back to this figure for that purpose in the near future.
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Descartes modeled his coordinate system on the Western number line, itself an extension of the decimal number system to include the new negative numbers, and upon the Euclidean notion of a three-dimensional geometry. All these events took place in historical times. In approaching the I Ching and its number system we are dealing mostly with events that took place before recorded history so it is impossible to say with certainty how anything involved came about. We can’t so much as be sure whether the I Ching was based on an antecedent number system, or predated and foreshadowed a subsequent number system of Chinese antiquity possibly contingent on it. We view all such things as through a glass, darkly.[1]
It is clear, though, that the number system of the I Ching is one far more complex than that of Western mathematics. The number system of the West is unidimensional (linear). Descartes, in his coordinate system, extends it for use in three dimensions. The number system of the I Ching, on the other hand, is in origin multidimensional. It is mandalic as well, which is to say it consists of multiple dimensions interwoven in a specific manner which can best be characterized as mandalic in form, possessing a number of interlaced and interlinked concentric shells or orbitals about a unifying center.
At the important origin of Descartes’ coordinate system is found his triple zero ordered triad (0,0,0). Descartes views this point,[2] asall his points, primarily in terms of location, not relationship. The matter of relationship is left to analytic geometry, the geometry Descartes codified based on his coordinate system.[3] The coordinate system itself seems not to care how points are formed or related beyond the most elementary and trivial operations of addition and subtraction throughout what essentially remain predominantly isolated dimensions.[4] In the end this becomes an effective and prodigious mind snare.[5]
In contrast to the Cartesian approach,theI Ching offers a unified coordinate system and geometry in a single entity which emphasizes the relationship of “points” and other “parts” (e.g., lines, faces) as much, if not more, than location, beginning with wholeness and ending with the same. In between, all sorts of complex and interesting interactions and changes take place. In analyzing these, it is best to begin at the origin of the coordinate system of the I Ching, the unceasing wellspring of being that supplants the triple vacuity of Descartes and Western mathematics.
[1] My thinking is that the I Ching was originally primarily a non-numerical relationship system that subsequently was repurposed to include, as one of its more important contextual capacities, numerical relationships. That said, from a contemporary perspective, rooted in a comprehensive awareness that spans combinatorics, Boolean algebra, particle physics, and the elusive but alluring logic of quantum mechanics and the Standard Model, it would seem that this relationship system is an exemplary candidate for an altogether natural number system, one that a self-organizing reality could readily manage.
[2] As do most geometers who follow after Descartes.
[3] Strictly speaking, this approach is not in error, though it does seem a somniferous misdirection. Due to the specific focus and emphasis enfolded in Descartes’ system, certain essential aspects of mathematical and physical reality tend to be overlooked. These are important relational aspects, highly significant to particle physicists among others. These remarks are in no way intended to denigrate Cartesian coordinates and geometry, but to motivate physicists and all freethinkers to investigate further in their explorations of reality.
[4] The Cartesian system neglects, for instance, to express anywhere that the fact the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom, which in mathematical logic
has been used to describe the thesis that 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 axiom is the cornerstone of analytic geometry. The Cartesian coordinate system developed by René Descartes explicitly assumes this axiom by blending the distinct concepts of real number system with the geometric line or plane into a conceptual metaphor. This is sometimes referred to as the real number line blend. [Wikipedia]
Neither mandalic geometry nor the I Ching, upon which it is based, accept this axiom as true in circumstances other than those restrictive settings, such as Cartesian geometry, where it is explicitly demanded as axiomatic to the system. In other words, they do not recognize the described one to one correspondence between number and geometric space as something that reality is contingent on. The assumption contained in this axiom, however, has been with us so long that we tend to see it as a necessary part of nature. Use of the stated correspondence may indeed be expedient in everyday macro-circumstances but continued use in other situations, particularly to describe subatomic spatial relations, is illogical and counterproductive, to paraphrase a certain Vulcan science officer.
[5] For an interesting take on the grounding metaphors at the basis of the real number line and neurological conflation see The Importance of Deconstructing the Real Number Line. Also on my reading list regarding this subject matter is Where Mathematics Comes From:How the Embodied Mind Brings Mathematics into Being(1,2,3) by George Lakoff and Rafael Nuñez. Neither of the authors is a mathematician, but sometimes it is good to get an outside perspective on what is in the box.
[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. Simple by comparison though this model may be it will nevertheless serve us well as a key to deciphering the line derived from the mandala of I Ching hexagrams, and we will be referring back to this figure for that purpose in the near future.
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Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where n = x + 1 - p. :)
Many different number systems exist in the world today. Others have existed in times past. The number system we are most familiar with is base 10 or radix 10, which makes use of ten digits, numbered 0 to 9. Beyond the number 9, the numbers recapitulate, beginning again with 0 and shifting a new “1” to the 10s position, in a positional number system. Using this conventional technique all integers and decimals can be easily and uniquely expressed. This familiar numeral system is also known as the decimal system.[1]
Another number system we are familiar with and use every day is the modular numeral system, particularly in its manisfestation of modulo 12, better known as clock arithmetic. This is a system of arithmetic in which integers “wrap around” and begin again upon reaching a set value, called the modulus. For clock arithmetic, the modulus used is 12. On the typical 12-hour clock, the day is divided into two equal periods of 12 hours each. The 24 hour / day cycle starts at 12 midnight (often indicated as 12 a.m.), runs through 12 noon (often indicated as 12 p.m.), and continues to the midnight at the end of the day. The numbers used are 1 through 11 and 12 (the modulus, acting as zero). Military time is similar, only is based on a 24-hour clock with modulus-24 rather than modulus-12. The modulus-24 system is the most commonly used time notation in the world today.
Binary arithmetic is similar to clock arithmetic, but is modulo-2 instead of modulo-12. The only integers used in this system are 0 and 1, with the “wrap around” back to zero occurring each time the number 1 is reached. Computers, in particular, handle this arithmetic system, which we owe to Leibniz, with remarkable acumen. George Boole also based his true/false logic on binary arithmetic. This, in itself, accounts for some of its strange, counterintuitive aspects, like the fact that in Boolean algebra the sum of 1 + 1 equals 0. Not your father’s arithmetic. But both Leibniz and Boole found profound uses for it. As did the entire digital revolution.
When we come to consideration of the number system and arithmetic used in the I Ching we can anticipate encountering equal difficulty in comprehension, possibly more. The system employed is a modular one - sort of. However, it uses negative 1 (yin) as well as positive 1 (yang) whereas zero (0) is nowhere to be seen, at least not in guise of an explicit dedicated symbol earmarked for the purpose. The "wrap around" appears to occur at both -1 (yin) and +1 (yang). Something different and quite extraordinary is going on here. This is no simple modular numeral system, though it may be masquerading as one.
Thus far the number system of the I Ching sounds much like that of Taoism. It is not, though. We have some big surprises in store for us.
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. :)
We come now to the Taoist/Cartesian transliteration sections of the three-dimensional cube.[1] The frontal FH section seen below is the Cartesian xy-plane we’re all familiar with from the 2-dimensional version of the Cartesian coordinate system with the third Cartesian dimension (z) added to the labeling of points. This gives us nine distinct Cartesian triad points: four vertices, four edge centers, and one face center. For all of the points, the third Cartesian dimension (z) is constant in this slice, and the vector value is positive (located toward the viewer with respect to the z=0 value of the z-axis or FHE plane which we’ll be viewing in a future post.)
The diagram shown here relates changing and unchanging trigrams of the I Ching to corresponding Cartesian ordered triads. Descartes views each of his ordered triads as referring to a single point having substantive reality in Cartesian geometric space. The I Ching and mandalic geometry, on the other hand, regard the trigrams as evanescent composite states of being in a spacetime which is ever-changing. They are relational elements in some ways analagous to the subatomic entities of particle physics.
Accordingly, it should be further understood each “point” here, though shown as a flat “square”, has a third dimension implied, and is therefore actually a “cube”, only one face of which is seen.[2] Mandalic geometry considers the point a fictional device which actually refers to a common intersection of three or more planes in a three-dimensional context, or two or more lines in a two-dimensional context. Moreover, mandalic geometry is a discretized geometry, and the trigram must be considered as having a distributed domain of action. This is illustrated in all the Cartesian transliteration points by distributing eight copies of trigrams with appropriate changing and unchanging lines among eight vertex-analogues of each Cartesian point.
The key to labeling of points in this section[3] and all those to follow can be found here. Additionally, by tradition, adding an “x” to a yin line indicates it is a changing line and adding an “o” to a yang line indicates it is a changing line. A changing yin line is considered an old yin line which is changing to a yang line; a changing yang line, an old yang line that is changing to a yinline.
Vector addition of two or more yinlines yields a yin line as result. Vector addition of two or more yang lines gives a yang line as the result. Vector addition of an unequal number of yin lines and yang lines yields as result that vector (yinoryang) in excess. Vector addition of an equal number of yin lines and yang lines gives as result Cartesian zero which, in mandalic systematics is to be considered a vector (direction) rather than a scalar (magnitude). This goes far in explaining how the I Ching and Taoism managed without an explicit zero.
Thezero was implicit or understood without using a special symbol of designation. Moreover, it was conceived as representative of an order of reality entirely different from that distinguished by the Western zero. It is, however, fully commensurate with Cartesian coordinate dynamics. It is this alternative zero, with its extraordinary capacities, that provides access to potential dimensions and to different amplitudes of dimension. This will be further elaborated in a future post where we will address how Boolean logic impacts what we’ve covered here.
For now simply note that the changing yin Line and changing yang Line in the horizontal first dimension (x) in each “point” shown in the middle column add to zero, not the zero of scalar magnitude though, but the zero of vector equilibrium.
Section FH(n)
In this section of the cube, as in all frontal sections, the third Line/dimension (z) never changes; the second Line/dimension (y) changes only in columns, as one progresses up or down; the first Line/dimension changes only in the rows, progressing left or right. This is just a consequence of viewing a two-dimensional Cartesian xy-plane in context of a section of the three-dimensional Cartesian xyz-cube. Although not the manner in which we are accustomed to viewing the plane, it is nonetheless fully compatible with ordinary Cartesian coordinates. It is simply an alternative perspective, one more suited for analysis/demonstration of trigram relationships in a Cartesian setting.
[1] This should be viewed as a work in progress. I’m still feeling my way with this so the content and/or format may change in the future. What is demonstrated here does not yet take into account the manner in which Boolean logic relates to the distribution of changing and unchanging trigrams nor does this series of cube sections include the all-important geometric method of composite dimension. As described, this is simply a Taoist notation transliteration of Cartesian coordinate structure. The meat and potatoes of the matter is yet to come. Of particular note here, though, is the fact that even at this early stage of translation to a version of mandalic geometry that can be considered comprehensive, what is possibly best described as a decussationbetweenyinandyang lines is already evident at every Cartesian triad point containing a “Cartesian zero”. Worth mentioning here, this will be a key feature addressed in future posts.
[2]Point, square, and cube, have all been placed in quotation marks to indicate that what is being referred to here is actually a different category of objects or elements which should in some sense be understood as relating to fractals or fractal structure and of a different dimensionality entirely than are those ordinary geometric objects. The admittedly deficient terminology used here is necessitated by the fact that sufficiently appropriate vocabulary terms to describe the reality intended do not currently exist, or if they do are not as yet known to me. Since we are representing a Cartesian point (ordered triad) as a quasi-cubic structure here, it must have a near face (n) and a far face (f) in each section with respect to the viewer. The chart displayed details the near face (n) of Section FH.
[3] This is the frontal section through the cube nearest a viewer. It is Descartes’ xy-plane with label of the third dimension (z) added so each point label shown is a Cartesian ordered triad rather than an ordered pair as textbooks generally show the plane. Why the difference? Because the geometry texts are interested only in demonstrating the two-dimensional plane in isolation, whereas we want to see it as it exists in the context of three or more dimensions. Cartesian triads are shown by convention as (x,y,z), so the xy-plane emerges from the first two coordinates of the points in this section, and all the z-coordinates seen here are positive (+1). The FE plane has all of its x and y coordinates identical to those seen here but its z-coordinates are all negative (-1). The FHE plane has all the x and y coordinates identical to those seen here but its z-coordinates are all zero (0).
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. :)
Looking back on how we arrived at this stage of reconstruction of Western thought, I see the difficulty arose in attempting to explain the “missing zero” of Taoism. Blame our troubles on Leibniz. It was he who introduced binary numbers to the West, and made the fateful choice of using zero(0) instead of -1 to counter with +1. Leibniz knew full well of the I Ching, but did not understand it well. He missed the point, seeing in it only a resemblance to his own newly devised system of numbers.
By Leibniz’s time negative numbers were firmly entrenched in the European mind. Why did Leibniz ignore them completely? In doing so he blazed a new trail that led eventually to the digital revolution of recent times. It also led to a dead end in the history of Western thought, one the West has not yet come fully face to face with. It will, though. Give it a few more years.[1]
George Boole, the inventor of what we know today as Boolean logic or Boolean algebra, was one of the thinkers who followed in the footsteps of Leibniz, building on the trail he blazed.[2] When he came to devise his truth tables, he also chose zero(0) as the counterpart to one(1). This led to certain resounding successes. And ultimately, to certain failures that introduced yet another layer to the blind spot of Western symbolic logic. Here we are, almost two centuries later,[3] saddled with and hampered by the unfortunate fallout of that eventful decision still.[4]
Most arguments in elementary algebra denote numbers. However, in Boolean algebra, they denote truth values falseandtrue. Convention has decreed these values are represented with the bits (or binary digits), namely 0 and 1. They do not behave like the integers 0 and 1 though, for which 1 + 1 = 2, but are identified with the elements of the two-element field GF(2), that is, integer arithmetic modulo 2, for which 1 + 1 = 0. (1,2) This causes a substantial problem when we attempt correlation of Taoist logic and Boolean logic. As we will soon discover, Taoist logic is a hybrid logic that is based on both vector inversion and arithmetic modulo 2. As such, it ought prove relatable to both Cartesian coordinates and Boolean algebra, though it may necessitate “forcing a larger foot in a smaller glass slipper.”
Taoism chose ages ago to use ‘yin’ and 'yang’ as its logical symbols. Although this appears, at first, to be a binary system, like those of Leibniz and Boole, on closer inspection it proves not to be. It is one of far greater logical complexity, alternatively binary or ternary with intermediate third element understood. This implied third element is able to bestow balance and equilibrium throughout all of the Taoist logical system. This is where the 'missing zero’ of Taoism went. Only it is a very different zero than the 'zero’ of Western thought. It is a zero of infinite potential rather than one of absolute emptiness. It is a zero of continual beginnings and endings, not of finality. It is one of the things that make the I Ching totally unique in the history of human cognition. All these hidden zeros are wormholes between dimensions and between different amplitudes of dimension.
So where does this all lead to, then? We’ve seen that the Taoist 'yin’ can readily be made commensurate with 'minus 1’ of Western arithmetic, the number line, and Cartesian coordinates.[5] But if it is to remain true to Taoist logic, it cannot be made commensurate with the Western 'zero’. We’ve found the Taoist number system and geometry to be Cartesian-like but not Cartesian. Now we discover them to be Boolean-like, not Boolean. Sorry, Leibniz, they are not so much as remotely like your binary system. You were far too quick to disesteem the unique qualities of the I Ching.[6]
This all has far-reaching consequences for Western thought in general. Especially though, for symbolic logic, mathematics, and physics. More specifically for our purposes here it means that when we create our Taoist notation transliteration of Cartesian coordinates, we will need also to translate Boolean logic into terms compatible with Taoist thought, that is to say, from a two-value system based on '1s’ and '0s’ into a three-value system based on '1s’, ’-1s’, and the ever-elusive invisible balancing-act '0s’ of Taoism.[7] We turn to that undertaking next.
Image: Fundamental operations of Boolean algebra. Symbolic Logic, Boolean Algebra and the Design of Digital Systems. By the Technical Staff of Computer Control Company, Inc. Other logical operations exist and are found useful by non-engineer logicians. However, these can always be derived from the three shown. These three are most readily implementable by electronic means. The digital engineer, therefore, is usually concerned only with these fundamental operations of conjunction, disjunction, and negation.
Notes
[1] It is at times like this that I am thankful I am not a member of Academia. Were I so, I could not afford, from a practical standpoint, to make claims such as this. Tenure notwithstanding.
[2] A knowledge of the binary number system is an important adjunct to an understanding of the fundamentals of Symbolic Logic.
[3] If we look back far enough in time, it was the introduction of “zero” as a number and a philosophical concept that led us down this tangled garden path, though the history of human thought is nothing if not interesting.
[4] Far out speculative thought here: Were binary numbers and Boolean logic based on +1s and -1s instead of +1s and 0s, might it not be possible to construct today a software-based quantum computer requiring no fancy juxtapositions and superpositions of subatomic particles? Think on it for a while before dismissing the thought as irrational folly.
[5] More correctly expressed, it can be made commensurate with the domain of negative numbers, since it is a vector symbol, properly speaking, concerned only with direction, not magnitude.
[6] Unfortunately there is still little understanding of the true nature of the symbolic logic encoded in the I Ching, as exemplified by this quote:
The I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique.
The contemporary scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, although he did not intend his arrangement to be used mathematically. Viewing the least significant bit on top of single hexagrams in Shao Yong’s square and reading along rows either from bottom right to top left with solid lines as 0 and broken lines as 1 or from top left to bottom right with solid lines as 1 and broken lines as 0 hexagrams can be interpreted as sequence from 0 to 63.
It was this Shao Yong sequence of hexagrams (Before Heaven sequence) that Leibniz viewed six centuries after the Chinese scholar created it, so maybe he can be forgiven his error after all.
The more significant point here might be that an important Neo-Confucian philosopher, cosmologist, poet, and historian of the 11th century either was no longer able to access the original logic and meaning of the I Ching or, at the very least, was hellbent on reinterpreting it in a manner contradictory to its original intent. The latter is a distinct possibility, as Neo-Confucianism was an attempt to create a more rationalist secular form of Confucianism by rejecting superstitious and mystical elements of Taoism and Buddhism that had influenced Confucianism since the Han Dynasty (206 BC–220 AD).
[7] Taoist logic and mandalic geometry share some of the characteristics of both Cartesian coordinates and Boolean logic, but not all of either. Descartes’ system is indeed a ternary one when viewed in terms of vector direction rather than scalar magnitude. That fits with the requirements of Taoist logic. It is, on the other hand, dimension-poor, as Taoist logic and geometry require a full six independent dimensions for execution. Boolean logic lacks the necessary third logical element -1, which causes inversion through a central point of mediation. But we shall see, it does bestow the ability to enter and exit a greater number of dimensional levels by means of its logical gates. Used together in an appropriate manner, these two can provide a key to understanding Taoist logic and geometry. Speculating even further, Taoist thought might provide a key to interpretation of quantum mechanics, the same quantum mechanics devised in the early twentieth century that no one can yet explain. Well, I mean, actually, Taoist thought in the formulation given it by mandalic geometry. Why feign modesty, when this work will likely linger in near-total obscurity for the next hundred years gathering dust or whatever it is that pixels gather in darkness undisturbed.
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Below are the three sagittal sections of the Cartesian 3-cube. All three have nine distinct ordered triads, one located at each discretized Cartesian spatial locus. SH and SE sections have four vertices, four edge centers and one face center as did the FH, FE, TH, and TE sections seen earlier. The SHE section contains four edge centers and four face centers and also, as its central point, the single cube center, as did the FHE and THEsections.[1]
These are all Cartesian yz-planes, seen in three-dimensional context at different x-values. For SH, x = +1. For SHE, x = 0. For SE, x = -1.
The key to labeling of points in these sections and all those to follow can be found here.
Section SH
Section SHE
Section SE
Having completed our survey of sections of the Cartesian 3- cube, we are now ready to view the Taoist transliteration equivalents. Almost. But first …
[1] See here for comment regarding the spurious appearance of edge and face centers in section FHE. That comment applies to the SHE and the THE sections as well. The deceptive appearance can be described as being an initially misleading artifact of the sectioning methodology.
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This post builds on orientational material offered in the previous post. An explanation of the procedural method of graphic demonstration used in this post and those following can be found there, and it would be helpful to review that earlier post, if not already done, before proceeding further.
Due in part to the challenging subject matter, in part to arduous graphic demonstration, we’ll approach this investigation in three stages of progressive difficulty. In the first stage we’ll just dangle our feet in the water by looking at how the "slicing methodology" works with ordinary three-dimensional Cartesian coordinates. In the second stage, we’ll go waist-deep, and consider the same Cartesian coordinates in their Taoist notation transliteration equivalents. And in the final stage, we’ll go for full immersion, with graphic representation of true mandalic geometry, that is, plotting all 64 hexagrams in a hybrid 6D/3D coordinate system using the methodology of composite dimension which, of course, has no analogue in purely Cartesian terms.
At each stage - Cartesian, Taoist transliteration, and mandalic - we’ll look at the respective cube in frontal,transverse, and sagittal slices, always in that order and always progressing from identity face containing Cartesian (1,1,1), trigram HEAVEN, or hexagram HEAVEN to inversion face, containing Cartesian (-1,-1,-1), trigram EARTH, or hexagram Earth, as the case may be.
To accomplish our purpose we will require an effective, consistent way to refer to the individual “slices” and each of the 27 Cartesian points. There are three “slices” for each type of sectioning of the “cube”, so a total of nine. I propose that we uniquely identify each “slice” by labeling it with the first letter of the section type (frontal, transverse, or sagittal) and the subscript letters “H” for planes containing trigram or hexagram HEAVEN but not Earth, “E” for planes containing trigram or hexagram EARTH but not HEAVEN, and “HE” for planes containing both trigram forms.[1]
The labels of the sections, then, will be:
FH frontal section containing HEAVEN but not EARTH
FHE frontal section containing both HEAVEN and EARTH
FE frontal section containing EARTH but not HEAVEN
TH transverse section containing HEAVEN but not EARTH
THE transverse section containing both HEAVEN and EARTH
TE transverse section containing EARTH but not HEAVEN
SH sagittal section containing HEAVEN but not EARTH
SHE sagittal section containing both HEAVEN and EARTH
SE sagittal section containing EARTH but not HEAVEN
For the 27 individual discretized Cartesian points, I propose the following labeling convention:
Each point is to be first identified as to type. There are four point types: vertex(V), edge center(E), face center(F), and cube center(O). The cube center corresponds to the Cartesian triad (0,0,0), the origin point of the Cartesian coordinate system. In the Cartesian/Euclidean cube there are 8 vertices, 12 edge centers, 6 face centers, and a single cube center. The higher dimensional mandalic cube has many more of each of these.
Vertices
Having identified the point type, each point is then further identified by a subscript consisting of the first letter of the name of trigram or hexagram that is resident at the point. The single exception to this will be WATER. To differentiate between WATER and WIND, I propose using the letter “A” (first letter of “aqua”, Latin for “water”) to specify WATER. This plan allows us, then, to discriminate among the various vertex points, and also to distinguish them from the other point types. Accordingly, we arrive at these labels for the 8 vertex points:
VH HEAVEN
VE EARTH
VT THUNDER
VW WIND
VA WATER
VF FIRE
VM MOUNTAIN
VL LAKE
Edge centers
Edge centers will be labeled “E” along with a subscript consisting of the first letter of its two vertices, “A” being used instead of “W” for WATER. Though this may initially seem excessively complicated, the reasons for setting things up this way will soon be made clear, and it will all become second nature. The 12 edge centers will be labeled as below:
EHW
EHF
EHL
EET
EEA
EEM
ETF
ETL
EAW
EAL
EMW
EMF
Face centers
There are six face centers. Three occur in identity faces of the cube that contain the trigram or hexagram HEAVEN; three, in inversion faces that contain the trigram or hexagram EARTH. Labeling will be with the letter “F” and a subscript consisting of either “E” for EARTH along with one of its companion diagonal vertices, “W” for WIND, “F”, FIRE, “L”, LAKE or “H” for HEAVEN, along with one of its companion diagonal vertices, “T” for THUNDER, “A”, WATER, “M”, MOUNTAIN. So these six face center labels are:
FEW
FEF
FEL
FHT
FHA
FHM
Cube center
The cube center, which is singular in Cartesian terms but a multiple composite in terms of mandalic geometry, will be labeled as:
O
identifying it as the origin of the coordinate system, that is to say, of both the Cartesian coordinate system and the mandalic coordinate system.
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Because nature is ever playful, grokking mandalic geometry is much like a game. We view it as a largely serious one, though, one that involves combinatorics, Boolean logic, and magic squares and cubes. Groundwork for what appears in this post, and several to follow, was laid in May, 2014 in a series titled “Quantum Naughts and Crosses” which began here.
The game is played on a board or field made of three-dimensional coordinates of the Cartesian variety upon which are superimposed the six additional extraordinary dimensions unique to mandalic coordinates. For convenience and ease of representation, the board will be displayed here in two dimensional sections abstracted from the Cartesian cube and from the superimposed mandalic hypercube in a manner analogous to the way computed tomography renders sections of the human body.
Planes perpendicular to the z-axis viewed from front to back of cube
Planes perpendicular to the y-axis viewed from top to bottom of cube
Planes perpendicular to the x-axis viewed from side right to left of cube
These “cuts” will produce square sections through xy-, xz-, and yz-planes, respectively, of the Cartesian cube and, in the case of the mandalic cube, analogous sections of higher dimension.
These choices of sections are made largely for convenience and ease of communicability. They are mainly of a conventional nature.[2] On the other hand, there is special significance in the fact that all three section types progress from identity faces of the cube, containing the trigram or hexagram HEAVEN, to inversion faces, containing trigram or hexagram EARTH. Some manner of consistency of this sort is necessary. The one chosen here will make things easier as we progress.
Ourgameboard has 27 discretized Cartesian points, centered in 3 amplitude levels about the Cartesian origin (0,0,0).[3] Each point in the figure on the right above is represented by a single small cube, but in the two-dimensional sections we’ll be using for elaboration, they will appear as small squares. So the gameboard is “composed of” 27 cubes arranged in a 3x3x3 pattern. But in descriptions of sections, we will view 9 squares in a 3x3 pattern. This configuration will appear as
But keep in mind each small square in this figure is actually a small cube representing one of the 27 discretized Cartesian points we’ve described.
[1] The origin of the word "tomography" is from the Greek word “tomos” meaning “slice” or “section” and “graphe” meaning “drawing.” A CT imaging system produces cross-sectional images or “slices” of anatomy, like the slices in a loaf of bread. The “slices” made are transverse (cross-sections from head to toes or, more often, a portion thereof), but reconstructions of the other types of sections described above are sometimes made, and MRI generates all three types natively.
[2] Admittedly, I’ve chosen the convention here myself and to date it is shared by no one else. Perhaps at some future time it will be a shared convention. One can only hope.
[3] These three discrete amplitude levels of potentiality in the mandalic 9-cube correspond geometrically to face centers, edge centers and vertices of the 3-cube of Cartesian coordinates. They are encoded by the six new potential dimensions interacting with the three ordinary Cartesian dimensions in context of the hybrid 6D/3D mandalic cube. They are a feature of the manner of interaction of all nine temporospatial dimensions acting together in holistic fashion. This should begin to give an idea why there is no Taoist line that can generate a 9-cube in a fashion analogous to the way the Western number line is used to generate the Cartesian / Euclidean 3-cube. The 9-dimensional entity is primeval and a variety of different types of "line" can be derived from it. Similarly, the mandala of the I Ching hexagrams cannot be derived from the logic encoded in any linear structure. An overarching perspective is required to derive first the mandala of hexagrams and then from it, a variety of Taoist line types. Nature may be playful, but it is not nearly as simplistic as our Western science, mathematics, and philosophy would have it.
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We have seen that an imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property ixi =−1. The square of an imaginary number bi is −b2. For example, 6i is an imaginary number, and its square is −36.[1] Other than 0, imaginary numbers yield negative real numbers when they are squared.[2]
Turning now to potential numbers, we can similarly define a unit of potentiality p by the property p x -p = -1. [Long pause here waiting for the other shoe to drop.] Just a minute, you say, that’s just like 1 x -1 = -1. Yes, it is. And that is just the point. All real numbers. Nothing to imagine. And Descartes finally vindicated after all these years - imaginary numbers just imaginary after all. But how does this work? Or does it even work? What exactly is the point? Is this a joke? It’s no joke, I assure you. It’s an easier and better way to achieve the same ends - - - and more. Muchmore.
The secret is in the sauce, I say slyly. Really? Well, yes - in a way. Though imaginaries use a sauce with nearly identical ingredients. The recipe is p + (-p) = 0. And, of course, i + (-i) = 0 as well. The trick is in how - - - and where - - - the sauce is applied. In the potential plane the sauce is applied more liberally in more locations for greater lubrication.
Levity aside. (This is after all a TST[3].) The complex plane uses a single axis. This axis represents a new dimension, wholly distinct from the x, y and z dimensions. Strangely, we’re never informed where this axis/dimension might be located, just that it is somewhere other than where x, y and z are located. Stranger still, the complex plane allocates the y-axis of the Cartesian plane for its own use in location of its points. Although never specifically mentioned, to my knowledge, I surmise the imaginary dimension exists in what mathematics and physics both call phase space.[4]
The mandalic or potential plane uses no such underhanded plan. It openly posits the existence of six new dimensions, allocated equally with two accompanying each of the Cartesian dimensions, all overtly evident. (All nine spatial dimensions in plain sight together, that is.) Nothing left to the imagination. As the new dimensions are made commensurate with the old in a hybrid geometric display, no imaginary dimension is needed. Coordinates of all potential dimensions are readily communicable with the real number system through all of the ordinary Cartesian dimensions concurrently along with the Cartesian coordinates. Moreover, mandalic geometry conjectures that the ordinary Cartesian dimensions may in fact originate in interactions among number species of potential dimensions filtered through impacts on inherited biological sensory mechanisms.[5] This raises yet another interesting possibility.[6]
In the long convoluted history of mathematics, the imaginary numbers were introduced as a correlative to the number line with its real numbers. That meant, among other things, that they were linear, consisting of a single dimension. The complex plane related the two in a kind of hybrid geometry that consisted of one real dimension and one imaginary dimension. Mathematician William Rowan Hamilton in 1843 proffered the quaternions, a number system that extends the complex numbers to three dimensions, whereupon things went, to my mind, from bad, to very much worse.
Quaternions came with certain dysfunctional characteristics, among them, the fact that multiplication of two quaternions is noncommutative. This is problematic. The imaginary and complex numbers, at least, had both been commutative. Nevertheless, physics endorsed the quaternions as it earlier had imaginary and complex numbers.
Why? Because the quaternions do in fact give partly correct results, and when investigating a dimly illuminated region of reality, such as the subatomic world still is today, even partial results are heartily welcomed if that is all that can be had. The sad consequence of this, is that physics has been led astray in its quest for truth for over a century now, because partial truths can be much more misleading than complete errors. Total error is often uncovered much sooner than partial truth, which can pass undiscovered, depending upon circumstances, for a very long time.
Mandalic geometry will be shown to be free of the difficulty posed by noncommutative multiplication. It is fully commutative throughout its nine dimensions (three ordinary, six extraordinary). It was not composed that way from a number line, with elements that could be commutatively multiplied with one another. It came that way fully formed from the start, in its primeval embodiment as a multidimensional structure, expressing behavior intrinsic to holistic nature.
Next time around, we’ll begin to look under the hood of the mandalic approach to geometry and see if we can grokit.
Image: (lower left) Imaginary unit i in the complex or Cartesian plane. Real numbers lie on horizontal axis, imaginary numbers on the vertical axis. By Loadmaster (David R. Tribble), (Own work) [CC BY-SA 3.0orGFDL], via Wikimedia Commons; (lower right) A diagram of the complex plane. The imaginary numbers are on the vertical axis, the real numbers on the horizontal axis. By Oleg Alexandrov [GFDLorCC-BY-SA-3.0],via Wikimedia Commons
Notes
[1] 62xi2 = 36 x (-1) = -36.
[2] Zero (0) is considered both real and imaginary, and both the real part and the imaginary part are defined as real numbers. (If that makes little sense to you, don’t blame me. I’m just the messenger here, reporting what the mathematicians have stated to be the case.) This seems to me to be purely an arbitrary definition, and it confuses me as much as it probably does you. Could it be they did this to avoid the situation where 02 x (-1) = -0? I think I would find that definition less disturbing, welcome even.
[3] Newly coined Internet acronym for Truly Serious Topic. (Not to be confused with TSR Totally Stupid Rules.)
Speaking about “greater lubrication”(wewere a moment ago, remember?), I use the phrase not simply as a figure of speech, or a simile, but rather, as a metaphor. "Spicing" of mandalic geometry with all those zeros of potentiality makes for a very “fluidic dish” which, I believe, reflects the changeable nature of reality far better than the stricter, strait-laced coordinates of Descartes or the complex plane are able to do. And it’s not just a matter of fluidity involved here. The mandalic form so begotten is, in fact, a probability distribution through the three Cartesian dimensions concurrently, which feature alone makes mandalic geometry an ideal candidate for application to quantum physics.
[4] A phase space of a dynamical system is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. In a phase space every degree of freedom or parameter of the system is represented as an axis of a multidimensional space; a one-dimensional system is called a phase line, while a two-dimensional system is called a phase plane. For every possible state of the system (that is to say, any allowed combination of values of the system’s parameters) a point is included in the multidimensional space. [Wikipedia]
[5] I am speaking here of the hybrid 6D/3D formulation of mandalic geometry which combines the features of dimensional numbers, potential numbers, and composite dimension, this being a fully open access geometric system that has nothing hidden, nothing held back. What you see is what you get. (WYSIWYG)
[6] It is tempting to wonder whether there might be a close connection between the composite dimensions/potential coordinates proposed by mandalic geometry and the pilot wave theoryorde Broglie–Bohm theory of quantum mechanics. At least there seems to be a correlation between David Bohm’s implicate/explicate order and the manifest/unmanifest (potential) coordinates of mandalic geometry.
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Rereading the last post a moment ago I see I fell into the same old trap, namely describing a concept arising from an alternative worldview in terms of our Western worldview. It is so astonishingly easy to do this. So it is important always to be on guard against this error of mind.
In saying that the Taoist number line is the basis of its coordinate system I was phrasing the subject in Western terminology, which doesn’t just do an injustice to the truth of the matter, it does violence to it, in the process destroying the reality: that within Taoism, the coordinate system is primary. It precedes the line, which follows from it. What may be the most important difference between the Taoist apprehension of space and that of Descartes lies encoded within that single thought.
Descartes continues the fiction fomented in the Western mind by Euclid that the point and the line have independent reality. Taking that to be true, Descartes constructs his coordinate system using pointsandlines as the elemental building blocks. But to be true to the content and spirit of Taoism, this fabrication must be surrendered. For Taoism, the coordinate system, which models space, or spacetime rather, is primary. Therefore to understand the fictional Taoist line we must begin there, in the holism and the complexity of its coordinate system where dimension, whatever it may be, reigns supreme.[1]
And that means we can no longer disregard composite dimension, postponing discussion of it for a later time, because it is the logical basis on which the I Ching is predicated. It is related to what we today know as combinatorics,Boolean algebra, and probability, and is what gives rise to what I have called the plane of potentiality. It is the very pith of mandalic geometry, what makes it a representation of mandalic spacetime.[2]
[1] In my mind, dimension is a category of physical energetic description before it is a category of geometrical description. When particle physicists speak about “quantum numbers” I think they are actually, whether intended or not, referring to dimensions. If this is true, then our geometries should be constructed to reflect that primordial reality, not arbitrarily as we choose.
[2] In speaking of logic and the I Ching in the same breath I am using the term in its broadest sense as any formal system in which are defined axioms and rules of inference. In reference to the I Ching, the logic involved is far removed from the rationalism bequeathed to Descartes by his times. It is a pre-rationalist logic that prevailed in human history for a very long time before the eventual splitting off of the irrational from the rational. This means also that the I Ching is among other things a viable instrument to access strata of human minds long dormant in historical times, other than possibly, at times, in poetry and art and the work of those select scientists who make extensive use of intuition in the development of their theories.
Note to self: Two contrasting systems of thought based on very different worldviews can never be adequately explained in terms of one another. At times though, for lack of anything better, we necessarily fall back on just such a strategy, however limited, and make the best of it we can.
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Descartes derives his directional locatives from considerations of human anatomy, as does most of Western culture. The descriptive terms generally used for orientation purposes include left/right;up/down; and forward/backward.[1] The first two sets have been extended also to refer to the cardinal directions, North/South and East/West.
To the degree that they conform to Cartesian coordinates, mandalic coordinates adhere to this schema as well. However, mandalic geometry and the Taoist I Ching upon which it is largely based constitute a system of combinatorial relationships that is rooted mainly in radial symmetry rather than bilateral symmetry. For mandalic coordinates, the principal directional locatives can be characterized as divergentandconvergent, and the principal movements or changes in position, as centrifugalandcentripetal.[2]
One of the important consequences of this alternative geometric perspective is that the frame of reference as well as the complex pattern produced are more integrative than in the method of Descartes. Looked at another way, Descartes is most enamored by specification of location of individual points whereas mandalic geometry is more concerned with relationships of parts - and the overall unification of the entire complex holistic system.[3]
From this one seemingly small difference an enormous disparity grows in a manner reminiscent of chaos theory.[4] Cartesian coordinates and mandalic coordinates can be made commensurate, but remain after all two exclusive systems of spatial awareness, leading to very disparate results arising out of what seem small initial differences.[5]
Image (bottom): Animation of a double compound pendulum showing chaotic behaviour. By Catslash (Own work). [Public domain], via Wikimedia Commons.[6]
Notes
[1] Such terminology is of little use, despite its biological origins, to an amoeba or octopus, not to mention those extraterrestrials who have been blessed with a second set of eyes at the back of their heads. (We wuz cheated.)
[2] To be more correct, the radial symmetry involved is of a special type. It is not simple planar radial symmetry, nor even the three-dimensional symmetry of a cube and its circumscribed and inscribed spheres. It is all of those but also the symmetry involved in all the different faces of a six-dimensional hypercube and the many relationships among them.
[3] To be fair, Descartes eventually gets around to relating his points in a systematic whole we now know as analytic geometry (1,2). But as great an achievement though it might be, Cartesian geometry lacks the overarching cosmographical implications which characterize mandalic geometry and the I Ching. Descartes’ system is purposed differently, arising as it does out of a very different world view. To paraphrase George Orwell,
“All geometries are sacred, but some geometries are more sacred than others.”
“When the present determines the future, but the approximate present does not approximately determine the future.”
[5] An example of one unique result of mandalic coordination of space is the generation of a geometric/logical probability wave of all combinatorial elements that occur in the 6D/3D hybrid composite dimension specification of the system. I envision this as offering a possible model at least, if not an actual explanation, of the probabilistic nature of quantum mechanics. Extrapolating this thought to its uttermost conclusion, it is not entirely inconceivable, to my mind at least, that probability itself might be the result of composite dimensioning. (And for such a brash remark I would almost surely be excommunicated from the fold were I but a member.)
[6] Starting the pendulum from a slightly different initial condition would result in a completely different trajectory. The double rod pendulum is one of the simplest dynamical systems that has chaotic solutions. [Wikipedia]
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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]
Given this fresh context - one of greater dimension than the number line - it soon becomes clear with little effort that a unit number[2]ofany 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 elementandall 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.
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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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. :)
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 3.0orGFDL],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.
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 3.0orGFDL],via Wikimedia Commons
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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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. :)
“Hands off: neither the whole truth nor the whole good is revealed to any single observer, although each observer gains a partial superiority of insight from the peculiar position in which he stands. Even prisons and sick-rooms have their special revelations. It is enough to ask each of us that he should be faithful to his own opportunities and make the most of his own blessings, without presuming to regulate the rest of the vast field.”
The Atomists were a fascinating group of philosophers. We have seen the eminence of Greek scientists in various scientific discoveries in the past several chapters, but the Atomists are possibly the most surprising. Pertaining to their namesake, the Atomists built a science and philosophy around the positing of atoms to build up the world. The world was made from two things, atoms and void.
Sophism is when one reasons fallaciously in hopes of deceiving someone. In other words, it is when someone employs fallacious, yet convincing, reasoning to sway someone. In some cases, people with these tendencies will project by accusing their opponent of sophistry or they will employ a No True Scotsman in saying that their opponent cannot possible be a “real” philosopher. I do not take kindly to such ad hominem and that is why I discontinued the discussion. Some of you may have seen this in my opponent’s response yesterday. These issues are minor. The major issue is in how he defines words.
Sophists tend to define words by omitting the use their opponent is using. When I say voting rights infringe upon other more integral, unalienable rights like the right to life or healthcare, I am not at all talking about a negative right, as he defined, in where one can hypothetically defend their right using force. While this isa definition of a right, it is not the only definition on offer. A right is also a principle of entitlement, a positive right, and so, when I say someone has the right to life, what I am saying is that they are entitled to live, irrespective of what the Constitution says; the phrase right to life in The Declaration of Independence is described as unalienable, god-given if you prefer. While there are clauses attached to this entitlement, such as they are entitled to live given that they do not murder someone, my definition is just as valid as the one my opponent employed. The difference is that my opponent dismissed my definition in order to deceive his readers. That is to say nothing of the validity of the distinction of negative and positive rights; plenty of philosophers (e.g. Eric Nelson, Ian Carter, Henry Shue) do not think the distinction is valid or even necessary.
He, for instance, continued to accuse me of not knowing what rights are, as though definitions themselves do not describe words in a self-evident fashion. A right is sometimes synonymous with a certain entitlement, but not all entitlements, real or imagined, are rights. A man may feel that after dating his girlfriend for five years, he is entitled to have sex with her. Consent is still at play no matter how long a couple has dated and so, he is not entitled to have sex with his girlfriend; she is not entitled to sex with her boyfriend either. These are matters of consent and as such, it is a privilege that they grant one another. The right to life is self-evident as even the Declaration of Independence attests. I do not need to go any further on that.
In that same vein, he mentions consent of the governed and people providing healthcare and bizarrely asserts that taxation is a violation of bodily autonomy; he does nothing at all to ground this claim, but, ironically enough, begs the question. Under the current government, 100 million or so people forgo their voting rights every election and many more forgo their rights as it pertains to electing state and local officials on a year-to-year basis. This implies that the right to vote is not as integral as some argue and definitely not as integral as my right to life. I may willingly surrender my right to vote given that I’m not particularly drawn to any of the candidates; I will not willingly surrender my right to live, assuming I am not terminally ill or mentally incapacitated. I am entitled to live and that is an integral entitlement; I am also entitled to vote, but that is not an integral entitlement as I can willingly choose not to.
What I have proposed, as Plato and others before me have, is an Epistocracy. Also of note is that he flat-out asserts Plato was wrong without justifying it; that is more more evidence that he has presupposed his conclusions. It is not a soft tyranny as he claimed. It is rule of the knowledgeable. What I am basically arguing is that if a third of the population is not going to vote anyway, we should decide on which one-third that is. The one-third that I temporally want to exclude are the least informed and that is assuming that such people even comprise one-third of the population; they might comprise a smaller portion than one-third and as such, I can say that at least I am not excluding as many people as are currently excluded and who have been excluded, at times, with malicious intent. The least informed are individuals who have not learned to or do not care to think critically. Since they do not think critically, they are prone to ignoring crucial issues and engaging in cult-like, conspiracy-based reasoning. A White Supremacist, on paper, is entitled to vote, but since he votes to harm minorities, he should not retain that entitlement.
Felons are largely excluded from the political process because they surrendered that entitlement in breaking the law. So it is up to my hypothetical government to decide at which point someone has committed to all that is required prior to breaking the law. What separates the average White Supremacist from Kyle Rittenhouse? The question boils down to who is armed and who is not and who is willing to harm or murder minorities versus who is not. Who then is the ideologue and who is willing to act on erred convictions? Since there is no sound reasoning to justify racism, discrimination, and prejudice, then White Supremacists should not be entitled to vote. Since there is no way of predicting which White Supremacist will act on their erred convictions, they should not be entitled to vote. Full stop!
Theconsent of the governed does not reduce to mere voting rights. In being a citizen or legal immigrant in the United States, you have de facto consented to be governed whether you vote or not, whether you are entitled to vote or not. Our current government already excludes a large portion of the population due to criminal records, gerrymandering, and other forms of voter suppression. So there is no material difference in my saying that we should exclude certain people for reasons separate from the ones the government uses to justify their exclusion and disenfranchisement of certain voters. As I have shown, however, I think my reasons for excluding the woefully ignorant are far better than the reasons given to exclude an entire demographic in a certain district or most felons without distinction. The primary reason is that voting rights cannot be prioritized over unalienable rights, so if a person votes with the intent to harm minorities, the minority’s right to live supersedes the White Supremacist’s right to vote. If I have to ground an entire moral framework to prove that conclusion, then my opponent is basically arguing that the right to life is not unalienable and is therefore, a privilege reserved for some and not others.
All felons are not created equal. Sure, a murderer on death row has long surrendered his entitlement to vote. Someone wrongfully accused of a crime or someone serving a marijuana-related sentence should not be excluded. Yet, in most cases, no distinction is made between the former felon and the latter. Then there is the real crux: my exclusion is notpermanent. You can be a White Supremacist today and not be one tomorrow. That means that you can learn why you are wrong about non-Whites and come to see common humanity in minorities. Any and all kinds of ignorance can be rectified given time, so it is entirely possible to justify a vote for any candidate in an informed manner. What my hypothetical government would guarantee is an informed voter who does not vote along party lines, who does not double-down on a quasi-fascist like Trump, who does not ignore science and the urgency of Climate Change, and so on. A more informed electorate is absolutely a good thing and the exclusion stemming from my hypothetical government is preferable to the extant exclusion in the current U.S. government.
In any case, this is why I refused to exchange further. Sophists define words by omitting definitions they dislike. They accuse, commit fallacies, and project their errors onto you. Ultimately, sophists tend to be disingenuous because they have predilections and surmises they think are self-evident and so they do not commit to the philosophical work of reasoning to their conclusion; this was observed in my opponent’s bizarre claim that taxation violates bodily autonomy and that the provision of healthcare, in where one is paid by the government, is also a violation of bodily autonomy. These conclusions are not argued for or justified in any way and entirely ignore state-provided healthcare in other countries in where people have consented to pay their taxes for sake of receiving free healthcare and tuition-free college educations.
I have reasoned to my conclusion. I have seen the real harm in letting ignorant people vote year after year; these people have been given no (dis)incentive to rectify that ignorance. So basically what I am saying is that if we disincentivize ignorance, people will want to become more informed. They would not call every disagreeable story about their favored candidate “fake news.” They would not go down the rabbit-hole of conspiracy theories. They would have good reason to change. I see nothing at all wrong with telling people this: if you want to vote, demonstrate that you are informed enough and empatheticenough to participate in this process because your vote has palpable effects on other lives. After nearly four years of suffering through the lack of empathy, apathy, hatred, and incompetence of the Trump Administration, I am more resolute now than I was two years ago: everyone should not be entitled to vote; only the demonstrably informed in the U.S. population should do so and as such, I propose Epistocracy, the rule of the knowledgeable as that incentivizes everyone to become more knowledgeable before casting a vote.
I will conclude by saying that the false equivalence he made between Epistocracy and tyranny can be dismissed very easily: Epistocracy does not permanently exclude anyone, so if anyone has an issue with being governed by the knowledgeable, then it is incumbent on them to demonstrate the aptitude to join the ranks of the knowledgeable; tyranny, on the other hand, excludes the governed and subjects them to any number of abuses. Epistocracy is not about abuse, but rather about preventing the abuse suffered by the more empathetic and knowledgeable at the hands of the cruel, apathetic, and ignorant. Perhaps we should want to exclude malignant Psychopaths, Narcissists, Sadists, and Machiavellians, most especially when they have dehumanizing and degrading views of people they do not agree with. This is beyond, “I do not like your voice” or “I do not like these people.” This is about people who speak harm and carry out actions consistent with dangerous and potentially fatal beliefs.
The United States cannot continue to tolerate such ignorance and it is clear that the entitlement to vote has fallen into the wrong hands. In the least, I can say what a lot of other people cannot say: I have proposed a viable solution. I also happen to think it is among the better solutions, especially in light of my opponent’s tacit anarchism and admiration for Capitalism. I will not challenge a sophist on such erred points of view, as they have already presupposed the conclusion; this is also painfully obvious in his ego-stroking as it pertains to Marxism. He has claimed to debase all of Marxism and this should not surprise anyone given that my opponent’s love for Capitalism entails feeling threatened by an anti-Capitalist like Marx. There is no argument to be had with such people. In any event, be mindful of the tendency to define words by omitting key definitions. Such an individual does not want a genuine dialogue; they just want to win. Nothing productive comes from that.
