#descartes

Toward a new geometry

Definitions and guiding principles

• Dimension is a primitive concept which for purposes here can be defined as any linearly independent parameter or variable.
• Mathematics is most fundamentally about measurement.
• Rules of mathematics are conceptual tools developed by the human mind and are subject to change if and when such should prove necessary or utilitarian.
• Points, lines and planes are products of the human mind. Neither they nor the realities they represent have continuous existence in space or time but can recur intermittently at intervals which may be either regular or irregular. As a matter of convenience for purposes of description, however, any or all of these may be represented as existing in some continuous abstract sense.
• The whole is greater than the part and has emergent properties not found in any single part or any number of subdivisions of the whole.
• The consequences of number structure and interaction cannot be explained entirely on a local basis but emerge from holo-interactive dynamic processs throughout an entire system or any portion thereof in conceptual focus.
• It is always possible to express a reflection as a rotation and a rotation as a reflection. The two are isomorphic via properly chosen operations.
• Symmetry is of great importance but not always obvious.
• Two foundational guiding principles of reality are continuity and change. These then provide the basis and primary focus of mandalic geometry.
• Of the two types of change, cyclic and sequential, cyclic change is the more fundamental.
• Neither coordinates nor coordinate systems are a feature of nature. They are man-made devices which are pragmatic and utilitarian means by which to grasp a reality which itself has no need of them. Geometry nevertheless requires these crutches to exist and execute its functions.
• Geometry frequently also requires conventions of expression to promote widespread understanding of content but should strive to be as convention -free as possible.
• Ambiguity is a permissible feature, in fact a necessary feature, of mandalic geometry. This is related to its probabilistic nature and multiple-valued logic which reflect what we, from our limited vantage point, misunderstand to be paradoxes of nature.

Axioms

• Numbers can be characterized by dimension of context.
• A number may be embedded in multiple dimensions concurrently, existing in variant forms specific to the dimension(s) of context.
• The same number may function differently in different dimensional contexts. The dimension of context of a “point” described by a number or subsidary number delimits its expression. In other words, the expression of a number in spatiotemporal terms is determined by its dimensional context. In referring to a number or subsidiary number in any of their variants, therefore, the dimension of context must always be specified.
• It may not always be possible to identify the full dimensional context of a number. It is sufficient to determine and elucidate those contexts essential to elaboration of the specific operation(s) under present consideration.
• Numbers are not necessarily elemental. They may consist of parts or subsidiary numbers which refer to various different dimensional contexts.
• A point is not dimensionless extension in space. It is an emergent feature of the system-as-a-whole which appears intermittently at the common intersection of three or more dimensions. Points, lines and planes are evanescent occurrences in terms of geometry and spacetime. They, and the things they represent, are fleeting events which come and go. Repetition is possible (in a conceptual sense certainly; possibly in a material/energetic sense as well) but not inevitable.

Rejected definitions

• The definition of a point found in Euclid’s Elements: A point is that which has no part. [This definition may or may not have been in Euclid’s original Elements.]
• The definition of a line found in Euclid’s Elements: A line is breadthless length.
• The definition of a surface found in Euclid’s Elements: A surface is that which has length and breadth only.

Rejected axioms

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

Rejected notions

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

Also important to note:

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

Image credit: James Gyre-Naked Geometry

© 2016 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 314-

Magic Theatre: For Madmen Only

If we describe a Cartesian ordered triad by x,y,z we can describe an analogous 6-dimensional ordered sextuplet  or 6-tuple  by x_a,y_a,z_a,x_b,y_b,z_b

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

• (x_a + x_b) / 2 = x
• (y_a + y_b) / 2 = y
• (z_a + z_b) / 2 = z

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A few additional explanatory remarks are in order here:

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

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

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

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

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

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

Descartes views a point as having only two essential characteristics:

• It is dimensionless.
• It is just a location in space which can be uniquely represented
  by a single ordered pairorordered triad.

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

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

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

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

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

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

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

Notes

[1] Since the coordinate system is describing a cube with an n-hypercube superimposed,  there is an additional constraint placed on all coordinates in
the 6-tuples.  All scalar values must be identical for x, y and z values.  That constraint assures that all vectors though they may differ in sign (direction) maintain equal magnitudes.

When the 6-tuples are dimensionally reduced to 3-tuples by the method I’ve called “compositing of dimension”  the resulting geometric figure consists of four different dimensional amplitudes of 6-tuples collapsed.  The amplitudes of dimension correspond  in spatial terms  to the vertices,  edge centers,  face centers and cube center. The pattern that emerges is that of a mandala. This is a highly symmetric pattern though all symmetries aren’t necessarily apparent immediately, even using Taoist notation. The probability distribution of the 6-tuples allots the hexagrams in the following manner:  one to each vertex;  two to each edge center; four to each face center; one to the cube center. The result is  placement of 64 6-tuples  in 27 positions of discrete 3-tuples  in the specific mandalic distribution pattern described.

Think here of the analogy of a hydrogen atom confined within a cubic space of specified side length determined by the nuclear and atomic force fields. The single electron,  existing in such quantized energy levels that are possible,  can assume various different locations in different orbital shells,  but every location in a given orbital must be equidistant from the nuclear proton. Once reduced by dimensional compositing the 6-tuples described here fill four distinct shells that have different radii or distances from the center.  From center to periphery these distances can be described as zero;  one (or square root one);  square root 2; and square root 3. (Pythagorean theorem)

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

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

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

© 2016 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 313-

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

(continued from here)

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

Modified from image found here.

Given this fresh context - one of greater dimension than the number line - it soon becomes clear with little effort that a unit number^[2]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.

(continuedhere)

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.

© 2016 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 310-

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

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

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

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

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

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

(continuedhere)

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

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

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

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

Bootstrapping Neo-Boolean - I

(continued from here)

So yes. This is very much a work in progress. And we have strayed now as it happens  into  unfamiliar territory.  Terra incognita.  Therewill be dragons.^[1]  Dragons  are  errors.  Errors  are  dangerous,  and we must slay them.  But  all  in  good  time.  First,  we should scout out the terrain. That would be prudent.

Descartes in constructing his system of coordinates built upon the bedrockofelementary algebra and the number line. We’vepreviously called attention to the important  but mostly overlooked issue of the 1:1 congruence between number and geometric/spatial position he incorporated implicitly in the logic of his coordinates and questioned the validity of such correspondence, at least with respect to subatomic scales.

Working two centuries later but very much under the influence of Descartes’ thought,  George Boole introduced his own unique brand of algebra.  A second major influence on the development of his symbolic logic was the binary number system of Leibniz, himself influenced to a large degree by Descartes. We need to carefully follow and connect the dots here. Great advances in human cognition rarely,  if ever,  occur in isolation and seclusion. There is a fine line to tread though. If progress requires the shoulders of giants to stand on,  it is still difficult at times not to be overly influenced by those who came before.

Boole’s new logic, constructed in the wake of what by his time were firmly entrenched systematizations of thought by two of the most highly regarded philosopher mathematicians, was devised in such a manner as to conform to both of these conventions of system design.  Significant to our purposes here are the facts that first, Boolean logic echoes Cartesian convention of attributing to each and every location in geometric space a single unique number,  and second, it adheres to Leibniz’s convention of using a modulo-2 number system based on binary elements 1 and 0.^[2]

The symbolic logic systems of mandalic geometry and the I Ching do not abide by either of these conventions.  Instead they are based on what is best described as  composite dimensions with four unique truth values (or vector directions) each, ranging from -1 through two distinctive zeros (0_a; 0_b) to +1, and assignment of numbers to spatial locations through all dimensions by means of probability distributions in place of a simple and simplistic 1:1 distribution.  To accommodate these alternative conceptual concepts, we will need to expand and modify traditional Boolean logic as we have already done as regards Cartesian coordinate theory.

For starters here we should doubtless add, the mandalic form is the probability distribution through all dimensions, and the probability distributions are the mandalas.  And movement through either or both can only be accomplished by  discretized stepwise maneuvers  between different amplitudes of dimension separated by obscure quantum leaps of endless being and becoming and being and unbecoming, toward and away from  the centers and subcenters of holistic systems,  the parts of which are always aiming towards some kind of equilibrium never quite within reach. Which then makes error also a necessary aspect of reality and not simply the fearful monster we imagined.  It is error that makes achievement possible.^[3][4]

(continuedhere)

Image:Here Be Dragons Map. Detail of he Carta marina (Latin “map of the sea” or “sea map”), drawn by  Olaus Magnus  in 1527-39.  This is the earliest map of the Nordic countries that gives details and place names, by Olaus Magnus [Public domain], via Wikimedia Commons. The map was in production for 12 years.  The first copies were printed in 1539 in Venice. [Wikipedia]

Notes

[1] Mapmakers during the Age of Exploration sometimes placed the phrase “here be dragons” at the edges of their known world,  presumably to warn of the dangers lying in wait for sailing vessels  and  travelers by land who strayed too far from well-traveled routes.  Here is a list of all known historical maps on which these words appear.

[2] Or in Boole’s case, we might say,  attributing to each proposition in concept space a single truth value:  TRUE or FALSE (var YES or NO;  or, in electronics applications,  ON or OFF.)  What we have here, I believe, is in many instances a false dilemma  or the old Aristotelian dichotomy of  either/or.  Quantum physics demands and deserves better.  OK, true enough,  Boole gets around to extending possibilities  by means of multi-term propositions,  which his system can readily handle.  The question here, though,  is whether  nature  can or does  handle such similarly.  I think not.  I think it approaches the question  at a more fundamental level of reasoning and reality: at the most basic level of spacetime itself.

[3] This echoes the view of cybernetics,  a transdisciplinary approach for exploring regulatory systems, their structures, constraints, and possibilities.

Cybernetics is relevant to the study of systems, such as mechanical, physical, biological, cognitive, and social systems. Cybernetics is applicable when a system being analyzed incorporates a closed signaling loop; that is, where action by the system generates some change in its environment and that change is reflected in that system in some manner (feedback) that triggers a system change, originally referred to as a “circular causal” relationship. [Wikipedia]

[4] This entire blog and its predecessor are in some sense the chronology of a journey from the familiar shoreline into largely uncharted waters.  Hesitant at first, increasingly more daring as time has gone on and I’ve come to see  errors  to be stepping stones along the way. And there have beenmanyerrors along the way. Some I am not yet cognizant of.  But of those I am aware,  I have left most intact in spite of since being superseded by ideas superior, more correct or better formulated.  I’ve done this  because I think it  important  to  map the course  of a conceptual journey,  how the ideas evolved from A to B to C to D.  It also allows readers to participate,  to a degree,  in the thrill of an exciting adventure of mind, should they so choose. Happy travels.

© 2016 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 303-

Mandalic Line Segments,
Entanglement and Quantum Gravity
Part I

(continued from here)

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

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

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

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

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

(continuedhere)

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

Notes

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

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

[3] See Note [4] here.

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

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

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

© 2016 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form.  Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 301-

Beyond Boole - Part 1
Symbolic Logic for the 21st Century

Boolean Algebra:
Fundamental Operations

(continued from here)

Looking back on how we arrived at this stage of reconstruction of Western thought,  I see the difficulty arose in attempting to explain the “missing zero” of Taoism. Blame our troubles on Leibniz. It was he who introduced binary numbers to the West,  and made the fateful choice of using zero(0) instead of -1 to counter with +1.  Leibniz knew full well of the I Ching, but did not understand it well. He missed the point, seeing in it only a resemblance to his own newly devised system of numbers.

By Leibniz’s time negative numbers were firmly entrenched in the European mind.  Why did  Leibniz  ignore them completely?  In doing so he blazed a new trail that led eventually to the digital revolution of recent times. It also led to a dead end in the history of Western thought, one the West has not yet come fully face to face with. It will, though. Give it a few more years.^[1]

George Boole, the inventor of what we know today as Boolean logic or Boolean algebra, was one of the thinkers who followed in the footsteps of Leibniz, building on the trail he blazed.^[2]  When he came to devise his truth tables,  he also chose zero(0) as the counterpart to one(1).  This led to certain resounding successes.  And ultimately,  to certain failures  that introduced yet another layer to the  blind spot  of Western symbolic logic. Here we are, almost two centuries later,^[3] saddled with and hampered by the unfortunate fallout of that eventful decision still.^[4]

Most arguments in elementary algebra denote numbers. However, in Boolean algebra, they denote  truth values  falseandtrue.  Convention has decreed these values are represented with the  bits (or binary digits), namely 0 and 1.  They do not behave like the integers 0 and 1 though, for which 1 + 1 = 2,  but are identified with the elements of the  two-element field GF(2), that is, integer arithmetic modulo 2, for which 1 + 1 = 0. (1,2) This causes a substantial problem when we attempt correlation of Taoist logic and Boolean logic. As we will soon discover, Taoist logic is a hybrid logic that is based on both vector inversion and arithmetic modulo 2.  As such,  it ought prove relatable to both Cartesian coordinates and Boolean algebra, though it may necessitate “forcing a larger foot in a smaller glass slipper.”

Taoism chose ages ago to use ‘yin’ and 'yang’ as its logical symbols. Although this appears, at first, to be a binary system, like those of Leibniz and Boole, on closer inspection it proves not to be.  It is one of far greater logical complexity, alternatively binary or ternary with intermediate third element understood. This implied third element is able to bestow balance and equilibrium throughout all of the Taoist logical system.  This is where the 'missing zero’ of Taoism went.  Only it is a very different zero than the 'zero’ of Western thought.  It is a zero of infinite potential rather than one of absolute emptiness.  It is a  zero  of  continual beginnings and endings, not of finality. It is one of the things that make the I Ching totally unique in the history of human cognition.  All these hidden zeros are wormholes between dimensions and between different amplitudes of dimension.

So where does this all lead to, then? We’ve seen that the Taoist 'yin’ can readily be made commensurate with 'minus 1’ of Western arithmetic, the number line,  and  Cartesian coordinates.^[5]  But if it is to remain true to Taoist logic,  it cannot be made commensurate with the Western 'zero’. We’ve found the Taoist number system and geometry to be Cartesian-like but not Cartesian. Now we discover them to be Boolean-like, not Boolean. Sorry, Leibniz,  they are not so much as remotely like your binary system. You were far too quick to disesteem the unique qualities of the I Ching.^[6]

This all has far-reaching consequences for Western thought in general. Especially though, for symbolic logic, mathematics, and physics. More specifically for our purposes here it means that when we create our Taoist notation transliteration of Cartesian coordinates, we will need also to translate Boolean logic into terms compatible with Taoist thought, that is to say, from a two-value system based on '1s’ and '0s’ into a three-value system based on '1s’, ’-1s’, and the ever-elusive invisible balancing-act '0s’ of Taoism.^[7] We turn to that undertaking next.

(continuedhere)

Image: Fundamental operations of Boolean algebra.  Symbolic Logic, Boolean Algebra and the Design of Digital Systems. By the Technical Staff of Computer Control Company, Inc.  Other logical operations exist and are found useful by non-engineer logicians.  However, these can always be derived from the three shown. These three are most readily implementable by electronic means. The digital engineer, therefore,  is usually concerned only with these fundamental operations of conjunction, disjunction, and negation.

Notes

[1] It is at times like this that I am thankful I am not a member of Academia. Were I so, I could not afford, from a practical standpoint, to make claims such as this. Tenure notwithstanding.

[2] A knowledge of the binary number system is an important adjunct to an understanding of the fundamentals of Symbolic Logic.

[3] If we look back far enough in time, it was the introduction of “zero” as a number and a philosophical concept that led us down this tangled garden path, though the history of human thought is nothing if not interesting.

[4] Far out speculative thought here:  Were binary numbers and Boolean logic based on +1s and -1s instead of +1s and 0s,  might it not be possible to construct today a software-based quantum computer requiring no fancy juxtapositions and superpositions of subatomic particles?  Think on it for a while before dismissing the thought as irrational folly.

[5] More correctly expressed, it can be made commensurate with the domain of negative numbers, since it is a vector symbol, properly speaking, concerned only with direction, not magnitude.

[6] Unfortunately there is still little understanding of the true nature of the symbolic logic encoded in the I Ching, as exemplified by this quote:

The I Ching dates from the 9th century BC in China. The binary notation in the
I Ching is used to interpret its quaternary divination technique.

It is based on taoistic duality of yin and yang.Eight trigrams (Bagua) and a set of 64 hexagrams (“sixty-four” gua), analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China.

The contemporary scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, although he did not intend his arrangement to be used mathematically. Viewing the least significant bit on top of single hexagrams in Shao Yong’s square and reading along rows either from bottom right to top left with solid lines as 0 and broken lines as 1 or from top left to bottom right with solid lines
as 1 and broken lines as 0 hexagrams can be interpreted as sequence from 0 to 63.

[Wikipedia]

It was this Shao Yong sequence of hexagrams (Before Heaven sequence) that Leibniz viewed six centuries after the Chinese scholar created it, so maybe he can be forgiven his error after all.

The more significant point here might be that an important  Neo-Confucian philosopher, cosmologist, poet, and historian of the 11th century either was no longer able to access the original logic and meaning of the I Ching or, at the very least, was hellbent on reinterpreting it in a manner contradictory to its original intent.  The latter is a distinct possibility,  as Neo-Confucianism was an attempt to create a more rationalist secular form of Confucianism by rejecting superstitious and mystical elements of  Taoism and Buddhism that had influenced Confucianism since the Han Dynasty (206 BC–220 AD).

[7] Taoist logic and mandalic geometry share some of the characteristics of both Cartesian coordinates and Boolean logic,  but not all of either.  Descartes’ system is indeed a ternary one when viewed in terms of vector direction rather than scalar magnitude. That fits with the requirements of Taoist logic.  It is, on the other hand, dimension-poor,  as Taoist logic and geometry require a full six independent dimensions for execution.  Boolean logic lacks the necessary third logical element -1, which causes inversion through a central point of mediation. But we shall see, it does bestow the ability to enter and exit a greater number of dimensional levels by means of its logical gates. Used together in an appropriate manner, these two can provide a key to understanding Taoist logic and geometry. Speculating even further, Taoist thought might provide a key to interpretation of quantum mechanics, the same quantum mechanics devised in the early twentieth century that no one can yet explain. Well,  I mean, actually,  Taoist thought in the formulation given it by mandalic geometry.  Why feign modesty, when this work will likely linger in near-total obscurity for the next hundred years gathering dust or whatever it is that pixels gather in darkness undisturbed.

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 294-

Beyond Descartes - Part 10
Taoism Meets Boolean Logic: Introduction

Logic gate symbols

(continued from here)

Before we can hope to comprehend Taoist arithmetic and geometry we need to take a short detour through Boolean logic. First and foremost, we need to see how Boolean logic^[1] relates to Cartesian coordinates. That will provide what may be the best foundation available for understanding the Taoist approach to mapping of spacetime and the methodology which mandalic geometry derived from it.^[2]

For Descartes, his coordinate system is one thing,  his coordinate geometry another.  For Taoism, the coordinate system is the geometry.^[3] Boolean logic helps to explain how the two perspectives are similar,  how different. Cartesian coordinates are static and passive. Taoist coordinates and the derivative mandalic coordinates are active and dynamic.  In brief, the latter are changeable and self-changeable, a feat carried out by means of a brand of Boolean logic intrinsic to the system. Although it is true that Descartes’ coordinates do encode much the same information,  that is not where their focus of interest lies. Accordingly they turn our own attention elsewhere and we overlook those inherent possibilities.^[4]

Descartes’ geometric system is one based on vectors, that is, on both  magnitude and direction.  But in the scheme of things,  the former has somehow eclipsed the preeminence of the latter in the Western hive mind.  The opposite is true of Taoist thought and of mandalic geometry. Direction is uniformly revered as primary and prepotent. Magnitude, or scale,  is viewed as secondary and subordinate.  This mindset allows the Boolean nuances inherent in the system to come to the fore, where they are more easily recognized and deployed.

From such small and seemingly insignificant differences ensue entirely disparate worldviews.

(continuedhere)

Notes

[1] George Boole’s monumental contribution to symbolic logic was published in 1854 but was viewed as only an interesting academic novelty until the second decade of the twentieth century,  when it was at last exhumed as a mathematical masterpiece by Whitehead and Russell in their Principia Mathematica.

[2] In Boolean logic (Boolean algebra) logical propositions are represented by algebraic equations in which  multiplication  and  addition  (and negation) are replaced with ‘and’ and 'or’ (and 'not’),  and where the numbers  '0’ and '1’ represent 'false’ and 'true’ respectively. Boolean logic has played a significant role in the development of computer programming and continues to do so.

[3] This is true also of mandalic geometry in its current formulation.

[4] This might be a proper place to proclaim that nature has little use for Descartes’ breed of coordinates,  finding them far too stagnant and limiting for her purposes. Fortuitously, she devised her own choice coordinate stock long before Descartes thought to invent his.

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 293-

Beyond Descartes - Part 8:
A Good Convention Gone Bad,
An Opportunity Missed

Composite Dimension and
Amplitudes of Potentiality
Episode 2

(continued from here)

We cannot blame Descartes for imaginary numbers. It was he, after all,  who christened these numbers “imaginary” due to his disdain for them.  We can,  however,  fault him  for his lack of insight  into how his coordinate system could be extended to create a viable substitute to show that imaginary numbers and the complex plane were nonsensical and make them unnecessary. Alas, that was not to be. Certain powerful forces of history decreed that imaginary numbers were here to stay and we seem stuck with them still, nearly five centuries later.

Not all would agree that imaginary numbers are a bad convention. We should all,  however,  be able to agree that they are  a convention and nothing more. They were invented by humanity.^[1]  Mathematics may not have taken to them at first - but did eventually welcome them into its fold for better or worse. The real damage was done when physics did the same without first subjecting the mathematical concepts involved to the kind of scrutiny and empirical review it demands of its own theories.

Where is the proof that imaginary numbers and complex plane in fact apply to the real world and particularly to the subatomic realm?  It is lacking in the main, and though the geometric concepts have indeed been successfully applied to a number of branches of physics  and explanations of  a variety of physical phenomena,  the reconciliation is incomplete,  the fit an uncomfortable one, and too many mysteries remain unexplained.

The term imaginary unit refers to a solution to the equation  x² = -1. By convention, the solution is usually denoted i. As no real number exists with this property,  the imaginary number i extends the real numbers and creates an entirely new and different category of numbers.  And crucially, at this point an assumption is made,  a rather sweeping assumption.  It is assumed that the properties of addition and multiplication we’re familiar with - (closure, associativity, commutativity and distributivity) - continue to hold true for this new species of number, or I should say, for this newly derived artificial species of number.  That may fly in the ivory tower^[2]  of pure mathematics,  but it lacks the wings and propelling force required to maneuver effectively in the real world that physics investigates.  Still,  the complex plane,  generated by mathematically motivated minds,  was soon adopted by physicists the world over.^[3]

Mandalic geometry offers an alternative solution in the effective combination of  dimensional numbers,  composite dimension,  and plane of potentiality. We’ll take a close look at potential numbers first. Let’s see how they stack up against  the imaginary numbers,  how  and where  they differ. Distinctions between complex plane and potential plane are subtle but they make for a world - a universe, actually - of difference. When next we meet, kindly check all preconceptions at the door.  Entirely untrodden paths await.

(continuedhere)

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] Let those who suppose differently, who believe them to be an indelible part of nature itself, prove their case. Until they do, I will see fit to call such numbers manmade inventions.

[2] I use the term ivory tower without malice of any kind in this context, rather judiciously, because mathematics demands no more than internal consistency for its particular brand of truth. It is not much interested in examining its definitions and axioms to determine how they shape up against hard reality. Mathematicians leave that  "sordid work"  to physicists and philosophers, both of whom are more willing to dig in  the mire of nature,  seeking its actual relics.  Enthusiastically to persist in such a real world-oblivious manner as pure mathematicians do, I think, requires a very special type of mind, one I don’t fully understand myself.

[3] In some circles this would be considered no less than a monumental leap of faith, particularly in view of the many unproved assumptions made in creation of imaginary and complex numbers. This was, in fact,  the New Faith  promulgated by Descartes’ contemporaries, the rationalists of the Age of Reason,  to supplant the Old Faiths of Religion and Scholasticism.

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 284-

Beyond Descartes - Part 6

The Fiction Formerly Known As the Line

(continued from here)

Rereading the last post a moment ago I see I fell into the same old trap, namely describing a concept arising from an alternative worldview in terms of our Western worldview.  It is so astonishingly easy to do this. So it is important always to be on guard against this error of mind.

In saying that the Taoist number line is the basis of its coordinate system I was phrasing the subject in Western terminology,  which doesn’t just do an injustice to the truth of the matter,  it does violence to it,  in the process destroying the reality:  that within Taoism, the coordinate system is primary.  It precedes the line,  which follows from it.  What may be the most important difference between the Taoist apprehension of space and that of Descartes lies encoded within that single thought.

Descartes continues the fiction fomented in the Western mind by Euclid that  the point and the line  have independent reality. Taking that to be true,  Descartes constructs his coordinate system using  pointsandlines  as the elemental building blocks. But to be true to the content and spirit of Taoism, this fabrication must be surrendered.  For Taoism,  the coordinate system, which models space, or spacetime rather, is primary. Therefore to understand the fictional Taoist line we must begin there, in the holism and the complexity of its coordinate system where dimension, whatever it may be, reigns supreme.^[1]

And that means we can no longer disregard composite dimension, postponing discussion of it for a later time,  because it is the logical basis on which the I Ching is predicated. It is related to what we today know as combinatorics,Boolean algebra, and probability,  and is what gives rise to what I have called the plane of potentiality. It is the very pith of mandalic geometry, what makes it a representation of mandalic spacetime.^[2]

(continuedhere)

Notes

[1] In my mind, dimension is a category of physical energetic description before it is a category of geometrical description.  When particle physicists speak about “quantum numbers” I think they are actually, whether intended or not, referring to dimensions. If this is true, then our geometries should be constructed to reflect that primordial reality, not arbitrarily as we choose.

[2] In speaking of logic and the I Ching in the same breath I am using the term in its broadest sense as any formal system in which are defined axioms and rules of inference. In reference to the I Ching,  the logic involved is far removed from the rationalism bequeathed to Descartes by his times.  It is a pre-rationalist logic that prevailed in human history for a very long time before the eventual splitting off of the irrational from the rational.  This means also that the I Ching is among other things a viable instrument to access strata of human minds long dormant in historical times,  other than possibly,  at times,  in poetry and art and the work of those select scientists who make extensive use of intuition in the development of their theories.

Note to self:  Two contrasting systems of thought based on very different worldviews can never be adequately explained in terms of one another. At times though, for lack of anything better, we necessarily fall back on just such a strategy, however limited, and make the best of it we can.

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 282-

Beyond Descartes - Part 4
Directional Locatives

(continued from here)

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]

(continuedhere)

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.”

[4] Chaos theory was summarizedbyEdward Lorenzas:

“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]

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering.  To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 280-

Beyond Descartes - Part 3
Logic Gates and Switches: Introduction

(continued from here)

It has been often noted throughout this work that mandalic geometry does not view points as fundamental geometrical elements in the manner Descartes and Euclid do. It considers them to be evanescent communions of two or more dimensions.  This  alternative perspective  conveys further the insight  that such conjoint formative interface locations both separate and connect. They are both boundaries and tipping points between all the participating dimensions,  what I have whimsically referred to  previously as dimension interchange lanes.  This is a far cry from the way Descartes regards and handles hispoints.

Descartes’points are locations, pure and simple, defining occupants of a uniform geometrical space. They don’t really doorattempt anything; they simply are.  They do not act,  but are acted upon by the equations of Cartesian geometry.  The  points themselves,  for all the reality Descartes attempts to imbue them with, turn out,  when the curtain is drawn,  to be no more capable of mustering an original thought  than is  the Scarecrow in  L. Frank Baum’s  The Wonderful Wizard of Oz.  Being of feeble mind themselves,  they just sit there awaiting brainy algebra to act upon them. In and of themselves,  beyond determining location,  they are essentially impotent.^[1]

A useful way to apprehendpoint locations of mandalic coordinates is to  interpret them  as  logic gates  which can handle  transition operations in a variety of different ways  depending upon the  dimension amplitudes verged on.  Passage through such locations is potentially bidirectional,  in theory if not always in actuality at a given moment, so they accommodate both  convergent and divergent flows  throughout varied amplitude levels of the mandalic structure.  To wit,  they can promote both  differentiationandpotentialization  phases of an evolving process.  Because these points arise through confluence of dimensions,  they bear within their transitory being information imparted by the participating dimensions.  Contrary to Descartes’ simpleminded points, these points have the capacity to encode an intelligence derived from their parent dimensions.^[2]

In electrical engineering,aswitch is an electrical component that can control an electrical circuit  by initiating or interrupting the current  or by diverting it from one conductor to another.  The most usual configuration consists of  a manually operated electromechanical device  having  one or more sets of electrical contacts.  These contacts are connected to external circuits. Each set of contacts can be in either of two states: either “closed” meaning the contacts are touching and electricity can flow between them, or “open”, meaning the contacts are separated in which case the switch is nonconducting. The mechanism that brings about the transition between these two states - openorclosed - can be either a “toggle”  (flip switch for continuous “on” or “off”)  or  “momentary”  (depress and hold for “on” or “off”) type.

Understand that logic gates don’t apply only to electronic devices nor are they controlled only by such devices. The concepts and methodologies involved go far beyond simple electronics.

• Logic gates are primarily implemented using diodes or transistors acting as electronic switches, but can also be constructed using vacuum tubes, electromagnetic relays (relay logic), fluidic logic, pneumatic logic, optics, molecules, or even mechanical elements. With amplification, logic gates can be cascaded in the same way that Boolean functions can be composed, allowing construction of a physical model of all of Boolean logic, and therefore, all of the algorithms and mathematics that can be described with Boolean logic. Wikipedia

For our purposes here and now, we need only mention that scalar numbers and vectors can be implemented in the context of Boolean logic as well.  Indeed, the incessant complex cotillion performed by subatomic particles can likely be subjected to such an analysis or one similar.^[3] And, of course, also digital circuits and computer architecture.

This has been just an introductory teaser to the topic of logic gates in mandalic geometry.  I’m getting my feet wet now myself. This is all still quite new to me so we’ve barely scratched the surface here.  An upcoming post will survey the logic gates and switches identifiable among groups of transliteration Cartesian coordinates and mandalic coordinates. This may take a while to materialize, but I think will be worth the wait.  And in case I forget to bring up the subject of how fractals fit into all this sometime in the next month or two, remind me please that I intended to.

(continuedhere)

Notes

[1] This could be a mathematician’s beautiful dream, but a physicist’s abhorrent nightmare.

[2] Although this statement pertains especially to composite dimension points, it is true, to a degree, of ordinary three-dimensional points as well when viewed in a manner similar to that using trigram tranliterations of Cartesian triads.  This means then that Cartesian coordinates could do the same and to the same degree, if  they were handled in the same manner as trigram coordinates are. The point is they are not and presumably never were.

[3] With that last remark I likely committed quantum mechanical heresy. If I in fact did, so be it. If it doesn’t quite hit the intended mark we can refer to it as steampunk mechanics.

Image (lower): Boolean lattice of subsets. KSmrq. Licensed under CC BY-SA 3.0viaCommons.

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x + 1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 279-

Earlier to Later Heaven: Fugue VI Beyond Descartes - Part 1

(continued from here)

In this post we take a short detour within our current central topic, that of relationship of Earlier Heaven and Later Heaven arrangements of the trigrams. The new material included here grew out of ruminations on the aforesaid primary topic though,  and is actually not so much a detour as a preparing the way for what I hope will be the eventual solution of our problem at hand.

Mandalic geometry, as we’ve seen, is fully commensurate with the coordinate system of Descartes, but its principal forebears lie elsewhere. It is derived largely from Taoist and pre-Taoist thought structures, most importantly the I Ching,  the earliest strata of which were formed before the separation of rational and irrational thought in the history of human cognition. As a result it is capable of far exceeding the possibilities of the Cartesian coordinate system, a product of the Enlightenment and Age of Rationalism. It offers geometry the possibility of a structural fluidity and a functional variability that Cartesian geometry lacks.^[1]

From the very beginning of this project I’ve been much puzzled by the lack in traditional Chinese thought  of a symbol corresponding to the zero of the Western number line and number theory.^[2] Traditional Asian thought does not uniformly lack a zero symbol.^[3] And yet the I Ching and Taoism manage well enough without one, electing to base their numerical relationships instead entirely on combinatorics involving permutations of yinandyang – what we in the West call  negativeandpositive – through multiple dimensions. It is an entirely different perspective arising out of a very different worldview.^[4]

What Taoism invented in the process was a unique,  thoroughly self-consistent brilliant system of logic/geometry/combinatorics which has been masquerading, all these many centuries,  as “just a method of divination.”^[5]  In essence, Chinese thought invented a discrete number system and geometry, one based on vectors rather than scalars, a vector geometry that can be extrapolated to any desired number of dimensions. The I Ching settles for just six,  the first whole number multiple of three. That is complicated enough.^[6][7]

(continuedhere)

Notes

[1] For one example of the advantages such variability and fluidity offer, in this particular case in creating  dynamic,  phase-shifting forms of nanomaterials,  see here.

[2] For a short history of the concept of zeroseeWho Invented Zero?

[3] The West, after all, derived its zero symbol ultimately via India.

[4] One might well speculate whether the significant root difference in world view between traditional Indian and Chinese thought lay in the fact that Indian mathematicians could have created a Zero out of nothingness (Śūnyatā),  a key term in Mahayana Buddhism and also some schools of Hindu philosophy while Taoist thought did not include a concept of nothingness. Instead it conceived of a formlessness prior to manifestation. In Taoist cosmology Taiji is a term for the “Supreme Ultimate” state of undifferentiated absolute and infinite potential,  the oneness before duality,  from which  yinandyang  originate.  So it might be that lacking a concept of nothingness forestalled invention of a zero symbol.  Still, it also allowed creation of an original,  unique holistic philosophy of reality, found perhaps nowhere else.

[5] The Russian philosopher, mathematician and authorPeter D. Ouspensky (1878-1947)  relates an apocryphal legend regarding the origin of the Tarot,  the moral of which has significance also to the history of the I Ching.

[6] In its emphasis on vector analysis and primacy of dimension the philosophy which underlies the I Ching and mandalic geometry  shares some characteristics of Clifford algebra.

[7] One of the important things with respect to physics I hope to show with mandalic geometry is that it is possible to construct an integrated geometrical / logical system which is self-sufficient and self-consistent, capable of modeling interactions of subatomic particles of the Standard Model and then some.  This goal is,  I believe,  approximated in mandalic geometry by meticulous coupling of the methodologies of composite dimension and trigram toggling,  although it quickly becomes apparent that a system based upon what is after all a relatively small number of dimensions - six in the case in point - becomes vastly complex and difficult to follow, at least initially.  One can’t help wondering how physics will be able to correlate all the intricate data resulting from its countless particle accelerator collisions and combine it into a consistent whole without some very fancy mental acrobatics on the part of theoretical physicists.  Without a suitable logical scaffold that might take an inordinately long time to achieve.

© 2015 Martin Hauser

Please note:  The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent).  This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added.  To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x + 1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)

-Page 277-

i do not think, therefore i ain’t

Details of “The resolution of Cartesian dualism in a nutshell”

The resolution of Cartesian dualism in a nutshell (ft. Spinoza)

“At the end all of Decartes’ problems has been solved by Spinoza. First of all, his dualism” (cit. Philosophy professor)

Hello guys! Yes, I did what you’ve just seen above. And I think that Sailor Moon’s meme is extremely accurate to explain Spinoza’s reaction in the history of philosophy.