#directional locatives

Beyond Descartes - Part 5

Reciprocation, Alternation, Decussation
Imaginary and Complex Numbers

(continued from here)

Previously in this blog a number of attempts have been made to explicate the Taoist number line and contrast it with the Western version of the same.  It is essential to do this and to do it flawlessly,  first because different systems of arithmetic result from the two, and secondly because the mandalic coordinate system is based on the former perspective while the Cartesian coordinate system is based on the latter.^[1]

What has been offered earlier has been accurate to a degree, a good first approximation. Here we intend to present a more definitive account of the Taoist number line,  describing both how it is similar to and how it differs from the  Western number line  used by Descartes in formation of his coordinate system.  This will inevitably transport us  well beyond that comfort zone offered by the more accessible three-dimensional cubic box that has heretofore engaged us.

Both Taoist and Western number lines observe directional locative division of their single dimension into two major partitions:  positive and negative for the West;  yinandyang for Taoism.^[2]  There the similarities essentially end.  From its earliest beginnings Taoism recognized a second directional divisioning in its number line, that of manifest/unmanifestorbeingandbecoming.^[3]  The West never did such.  As a result, some time later the West found it necessary to invent imaginary numbers.^[4][5]

Animaginary number is a complex number that can be written as a real numbermultiplied by theimaginary uniti, which is defined by its property i² = −1. [Wikipedia]

Descartes knew of these numbers but was not particularly fond of them.  It was he, in fact, who first used the term “imaginary” describing them in a derogatory sense. [Wikipedia]  The term “imaginary number” now just denotes a complex number with a real part equal to 0,  that is, a number of the form bi. A complex number where the real part is other than 0 is represented by the form a + bi.

In place of the complex plane, Taoism has (and always has had from time immemorial)  a plane of potentiality.  An explanation of this alternative plane was attempted earlier in this blog,  but it can likely be improved. This post has simply been a broad brushstrokes overview. In the following posts we will look more closely at the specifics involved.^[6]

(continuedhere)

Image (lower): A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram representing the complex plane. “Re” is the real axis, “Im” is the imaginary axis, and i is the imaginary unit which satisfies i² = −1. Wolfkeeper at English Wikipedia [GFDL (http://www.gnu.org/copyleft/fdl.html) or CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons

Notes

[1] The arithmetic system derived from the Taoist number line can perhaps best be understood as a  noumenal  one. It applies to the world of ideas rather than to our phenomenal world of the physical senses, but it may also apply to the real world, that is, the real real world which we can never fully access.

Much of modern philosophy has generally been skeptical of the possibility of knowledge independent of the physical senses, and Immanuel Kant gave this point of view its canonical expression: that the noumenal world may exist, but it is completely unknowable to humans. In Kantian philosophy, the unknowable noumenon is often linked to the unknowable “thing-in-itself” (Ding an sich, which could also be rendered as “thing as such” or “thing per se”), although how to characterize the nature of the relationship is a question yet open to some controversy. [Wikipedia]

[2] From the perspective of physics this involves a division into two major quanta of charge, negative and positive, which like yinandyang can be either complementary or opposing.  Like forces repel one another and unlike attract. This is the basis of electromagnetism, one of four forces of nature recognized by modern physics. But it is likely also the basis, though not fully recognized as such, of the strong and weak nuclear forces, possibly of the force of gravity as well. I would suspect that to be the case. The significant differences among the forces  (or force fields, the term physics now prefers to use)  lie mainly, as we shall see, in intricate twistings and turnings through various dimensions or directions that negative and positive charges undergo in particle interactions.

[3] It is this additional axis of probabilistic directional location, along with composite dimensioning, both of which are unique to mandalic geometry, that make it a geometry of spacetime,  in contrast to Descartes’ geometry which, in and of itself, is one of space alone. The inherent spatiotemporal dynamism that is characteristic of  mandalic coordinates  makes them altogether more relevant for descriptions of particle interactions than Cartesian coordinates, which often demand complicated external mathematical mechanisms to sufficiently enliven them to play even a partial descriptive role, however inadequate.

[4] In addition to their use in mathematics, complex numbers, once thought to be  "fictitious" and useless,  have found practical applications in many fields, including chemistry, biology, electrical engineering, statistics, economics,  and, most importantly perhaps, physics..

[5] The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers. He called them “fictitious” during his attempts to find solutions to cubic equations in the 16th century.  At the time, such numbers were poorly understood,  consequently regarded by many as fictitious or useless as negative numbers and zero once were. Many other mathematicians were slow to adopt use of imaginary numbers, including Descartes, who referred to them in his La Géométrie, in which he introduced the term imaginary,  that was intended to be derogatory. Imaginary numbers were not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855).  Geometric interpretation of  complex numbers as points in a complex plane  was first stated by mathematician and cartographer Caspar Wessel in 1799. [Wikipedia]

[6] What I have called here the plane of potentiality occurs only implicitly in the Taoist I Ching but is fully developed in mandalic geometry. It may be related to  bicomplex numbers  or tessarines in abstract algebra, the existence of which I only just discovered. Unlike the quaternions first described by Hamilton in 1843, which extended the complex plane to three dimensions, but unfortunately are not commutative,  tesserines or bicomplex numbers  are hypercomplex numbers in a commutative,  associative  algebra over real numbers,  with two imaginary units (designated i and k). Reading further, I find the following fascinating remark,

The tessarines are now best known for their subalgebra of real tessarines t = w + y j, also called split-complex numbers, which express the parametrization of the unit hyperbola. [Wikipedia]

The rectangular hyperbola x2-y2 and its conjugate, having the same asymptotes. The Unit Hyperbola is blue, its conjugate is green, and the asymptotes are red. By Own work (Based on File:Drini-conjugatehyperbolas.png) [CC BY-SA 2.5],via Wikimedia Commons

Note to self:  Also investigate Cayley–Dickson constructionandzero divisor. Remember,  this is a work still in progress,  and if a  bona fide mathematician  believes division by zero is possible in some circumstances,  (as is avowed by mandalic geometry), I want to find out more about it.

© 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 281-

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-