#real number system
Can a number system be both the new kid on the block and older than written history?
The real number system as it exists today has been with us for a few centuries. In foundation it is monovalent, monophasic, and sequential.
The probable number system dates to prehistory but was lost in the mists of time until recently rediscovered and resurrected. In contrast to the real number system it is foundationally bivalent, biphasic, and cyclic.
The probable number system has considerably more structure than the real number system and is therefore more robust. In this sense, it is similar to the complex number system.
In contrast to the complex number system, the probable number system in its foundation presupposes that numbers can assume wavelike forms capable of constructive and destructive interference operationally through the compositing of higher to lower dimension.
By means of compositing of dimension probable numbers are able to distribute throughout the entire mandalic unit vector cube (which is structurally a superposition of the 6-dimensional unit vector hypercube on the 3-dimensional unit vector cube) a function analogous in important ways to that performed in the complex number system by the centralized imaginary unit i.
Another important way in which the probable number system differs from both the real number system and the complex number system is the absence of nothingness and the zero representing it. In its place we find the concepts of balance and equilibrium. Nullification still exists in form of annihilation and its opposite in the form of creation. But the Cartesian coordinate system of ordered pairs and ordered triads is transformed by this approach to handling number and dimension from a ring into a field of hyperdimensional numbers over real numbers in three dimensions.
(to be continued)
© 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 315-
Beyond Taoism - Part 3
A Multidimensional Number System
(continued from here)
Descartes modeled his coordinate system on the Western number line, itself an extension of the decimal number system to include the new negative numbers, and upon the Euclidean notion of a three-dimensional geometry. All these events took place in historical times. In approaching the I Ching and its number system we are dealing mostly with events that took place before recorded history so it is impossible to say with certainty how anything involved came about. We can’t so much as be sure whether the I Ching was based on an antecedent number system, or predated and foreshadowed a subsequent number system of Chinese antiquity possibly contingent on it. We view all such things as through a glass, darkly.[1]
It is clear, though, that the number system of the I Ching is one far more complex than that of Western mathematics. The number system of the West is unidimensional (linear). Descartes, in his coordinate system, extends it for use in three dimensions. The number system of the I Ching, on the other hand, is in origin multidimensional. It is mandalic as well, which is to say it consists of multiple dimensions interwoven in a specific manner which can best be characterized as mandalic in form, possessing a number of interlaced and interlinked concentric shells or orbitals about a unifying center.
At the important origin of Descartes’ coordinate system is found his triple zero ordered triad (0,0,0). Descartes views this point,[2] asall his points, primarily in terms of location, not relationship. The matter of relationship is left to analytic geometry, the geometry Descartes codified based on his coordinate system.[3] The coordinate system itself seems not to care how points are formed or related beyond the most elementary and trivial operations of addition and subtraction throughout what essentially remain predominantly isolated dimensions.[4] In the end this becomes an effective and prodigious mind snare.[5]
In contrast to the Cartesian approach,theI Ching offers a unified coordinate system and geometry in a single entity which emphasizes the relationship of “points” and other “parts” (e.g., lines, faces) as much, if not more, than location, beginning with wholeness and ending with the same. In between, all sorts of complex and interesting interactions and changes take place. In analyzing these, it is best to begin at the origin of the coordinate system of the I Ching, the unceasing wellspring of being that supplants the triple vacuity of Descartes and Western mathematics.
Section FH(n)[6]
(continuedhere)
Notes
[1] My thinking is that the I Ching was originally primarily a non-numerical relationship system that subsequently was repurposed to include, as one of its more important contextual capacities, numerical relationships. That said, from a contemporary perspective, rooted in a comprehensive awareness that spans combinatorics, Boolean algebra, particle physics, and the elusive but alluring logic of quantum mechanics and the Standard Model, it would seem that this relationship system is an exemplary candidate for an altogether natural number system, one that a self-organizing reality could readily manage.
[2] As do most geometers who follow after Descartes.
[3] Strictly speaking, this approach is not in error, though it does seem a somniferous misdirection. Due to the specific focus and emphasis enfolded in Descartes’ system, certain essential aspects of mathematical and physical reality tend to be overlooked. These are important relational aspects, highly significant to particle physicists among others. These remarks are in no way intended to denigrate Cartesian coordinates and geometry, but to motivate physicists and all freethinkers to investigate further in their explorations of reality.
[4] The Cartesian system neglects, for instance, to express anywhere that the fact the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom, which in mathematical logic
has been used to describe the thesis that the real numbers are order-isomorphic to the linear continuum of geometry. In other words, the axiom states that there is a one to one correspondence between real numbers and points on a line.
This axiom is the cornerstone of analytic geometry. The Cartesian coordinate system developed by René Descartes explicitly assumes this axiom by blending the distinct concepts of real number system with the geometric line or plane into a conceptual metaphor. This is sometimes referred to as the real number line blend. [Wikipedia]
Neither mandalic geometry nor the I Ching, upon which it is based, accept this axiom as true in circumstances other than those restrictive settings, such as Cartesian geometry, where it is explicitly demanded as axiomatic to the system. In other words, they do not recognize the described one to one correspondence between number and geometric space as something that reality is contingent on. The assumption contained in this axiom, however, has been with us so long that we tend to see it as a necessary part of nature. Use of the stated correspondence may indeed be expedient in everyday macro-circumstances but continued use in other situations, particularly to describe subatomic spatial relations, is illogical and counterproductive, to paraphrase a certain Vulcan science officer.
[5] For an interesting take on the grounding metaphors at the basis of the real number line and neurological conflation see The Importance of Deconstructing the Real Number Line. Also on my reading list regarding this subject matter is Where Mathematics Comes From:How the Embodied Mind Brings Mathematics into Being(1,2,3) by George Lakoff and Rafael Nuñez. Neither of the authors is a mathematician, but sometimes it is good to get an outside perspective on what is in the box.
[6] This is the closest frontal section to the viewer through the 3-dimensional cube using Taoist notation. See here for further explanation. Keep in mind this graph barely hints at the complexity of relationships found in the 6-dimensional hypercube which has in total 4096 distinct changing and unchanging hexagrams in contrast to the 16 changing and unchanging trigrams we see here. Simple by comparison though this model may be it will nevertheless serve us well as a key to deciphering the line derived from the mandala of I Ching hexagrams, and we will be referring back to this figure for that purpose in the near future.
© 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 298-
