The way I see it
One of the things contemporary physics crucially needs if it is to progress beyond the impasse it has reached is a “zero alternative”, one more robust and fertile than the nothingness it has been shackled with for far too long.
This is basically a matter of epistemology involving the respective philosophies of mathematics and science. Mathematics, like physics, is very much a work in progress. It changes from time to time. Those who have neglected the study of the history of mathematics sometimes miss this essential point.
Significantly, mathematics and science do not share the same kind of truth. Much of what mathematics views as “true” is not applicable to physics. None of mathematics is apropos to science out of the box so to speak. A different criterion of proof prevails for the one than for the other. Mathematics seeks only consistency based upon rules of logic but science pursues a more pragmatic course. It stalks reality-based truth, truth fully supported by results of experience and experimentation.
Were physics to discard the notion of an empty, impotent “zero” it would not be the first time in its history that it chose a conflicting form of mathematics in preference to one more popular. Einstein, for example, famously scuttled certain prevailing aspects of mathematics in favor of new concepts recently formulated by Hendrik Lorentz and Henri Poincaré which he found more appropriate to his determined purpose.
His conscious selection early in the 20th century of an alternative mathematics resulted in the development of special relativity which soon received a new geometric interpretation by Hermann Minkowski. Although Einstein initially viewed this innovative four-dimensional spacetime as just a mathematical sleight of hand he soon realized that a geometrical view of space-time would be prerequisite for his later work in general relativity.
Physicists already know that the vacuum of space is not empty but filled with energy of an elusive neutral sort. It is time they admitted also that the reigning “zero” of mathematics is not applicable to their needs at all times and in all situations. Neutrality is not emptiness. Emptiness does not give rise to the fertile universe; fullness does.
This is not religion, nor pure mathematics, nor sacred geometry. It is rather a question of the philosophy of mathematics, the philosophy of science and how the two interact harmoniously and productively or fail to do so. The wrong mathematics in combination with the quest of physics can have results akin to destructive interference of wave propagation.
The “zero” of mathematics does well enough under artificial lighting in the private halls of finance and industry. But in the full-spectrum natural daylight of physics and the other sciences it can be seen for what it really is: a pale fiction masquerading as truth, prancing like the emperor proudly displaying his newly fashioned finery. Only in this case the finery is already antique, long overdue for requisite alterations if not a complete overhaul.
Back to Basics: polarity and the number lines - II
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Taoism natively is more inclined to think in terms of 2-, 3-, and higher-dimensions than in 1-dimensional linear terms. Taoism has a number line analogue but an implicit one which is treated as an abstraction, more a distant consequence of real processes in the universe than a fundamental building block. Reference to higher dimensions is not fully relinquished even in this 1-dimensional abstraction. It is little used in isolation but features more prominently in Taoist diagrams of the analogue of the 2-dimensional Cartesian plane. In a sense this makes the Taoist number line much more robust than the number line of Western mathematics. Whereas from the narrowly focused perspective of Western mathematics the “number line” of Taoism might be viewed as “hyperdimensional” from the perspective of Taoism itself it is “dimension poor” and therefore degenerate.(1)
Regarding two distinct kinds of change, sequent and cyclic, Western thought is, in general, more concerned with sequent and Eastern thought with cyclic change.(2) Whereas the Western number line stretches out to infinity in both directions as in an orgiastic celebration of sequent change, the Taoist number line, exhibiting more restraint confines itself to some more realistic terminus of magnitude. It does so first because the taijitu (infinity analogue) of Taoism is non-polarized and exists at the center where Western thought places its “zero”. But also because it envisions change mainly in terms of cycles and invariably selects more realistic points of maximum and minimum extension than infinity.(3)
From the point of view of Taoism infinity though unbounded is also undifferentiated, existing in a non-polarized state of pure potential and potency whereas all differentiated states having polarity are limited in degree of potential and subject ultimately to constraint of extension.(4)
(to be continued)
(1) Taoism is a worldview based largely on relationships. From its very beginnings it likely considered a single dimension insufficient to express the full complexity of relationships possible. The I Ching, based largely upon the Taoist worldview, is a treatise which makes use of 64 hexagrams to correlate six dimensions of relationship. It may be the world’s earliest text on combinatorics and dimensionality. The true significance of this seminal work of humankind has sadly been too frequently overlooked.
(2) This is entirely a matter of degree and of preferred focus but has nevertheless profound consequences reflected in the resulting respective worldview of the different cultures. From an oversimplified bird’s eye view, Western thought regards significance best revealed by way of historical development through time experienced sequentially; Eastern thought, by way of recursive phenomena of nature expressed through cyclic time.
(3) This means also that there can be no single representative number line as there is in Western mathematics. Not at least if distances along the line are marked off in customary units of consecutive digits. For each specific Taoist number line unique complementary terminal maximum and minimum values must be selected. In the case illustrated above the value was chosen to be 20 so as to conform in terms of number of intervals to the Western number line segment shown (ten negative and ten positive intervals.) Had the value been chosen as 10 instead, the Taoist line would extend only from yin = 10; yang = 0 to yin = 0; yang = 10 and the number of intervals encompassed would have been a total of ten rather than the required twenty.
One way to surmount this difficulty in labeling described would be to number the intervals along the Taoist number line in terms of percentages rather than specific sequent intervals. Were this procedure followed every Taoist number line would extend from yin = 100%; yang = 0% to yin = 0%; yang = 100% with the central point of origin (corresponding to “zero” in the Western number line) labeled as yin = 50%; yang = 50%.
The two “zeros” that occur at the extreme ends of the Taoist line (yang = 0 to the left; yin = 0 to the right) should not be viewed as numbers but rather in a sense similar to that in which “zeros” are used as unit ten placeholders in our decimal number system.
(4) In any case, labeling of the central origin point with either specific sequent intervals having identical absolute values or equal percentages (yin 50%/yang 50%) signifies the potential of the non-polarized and unbounded taijitu (infinity) to change by means of polarization into its polarized, bounded aspect. This process can be viewed also in terms of pair production (as understood by both Taoism and particle physics.)
© 2014 Martin Hauser
Back to Basics: polarity and the number lines - I
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The number line is represented as a straight line on which every point is assumed to correspond to a single real number and every real number to a single point in a 1:1 mapping. It can be graphed along either a horizontal axis or a vertical axis. In the former case, negative is shown to the left of zero and positive to the right by convention. In the latter case, negative is shown below zero and positive above.
The number lines of Western mathematics and Taoism are similar in that both make constant use of the concept of polarity. That is where the similarities end. Even the manner in which this concept is used in the two worldviews differs.
For mathematics the basic polarity entrenched in its number line(1,2) is that between positive and negative. These two polarities are thought of as opposite and mutually exclusive. They are mediated by the concept of “zero”, a sort of no-man’s-land, the boundary between the two which belongs rightfully to neither. It is thought of as being in a sense an empty buffer zone between the polarities of positive and negative and functions not so much to balance or unify the two as to keep them apart from one another, or failing that to nullify both. Hence zero generally is treated as having no sign. Additionally, zero in the number line of mathematics and in the one-dimensional line of Cartesian geometry has neither magnitude nor preferred direction.(1)
It should come as no surprise that division by zero is not possible in Western mathematics when “zero” has itself been conceived as a kind of singularity.(2) This fact of itself should have indicated something amiss with the way “zero” has been conceptualized.(3) The “zero” of Western thought works well in the field of finance and most everyday practical fields of endeavor in general. Where the concept falls short is in the attempt to apply it indiscriminately in modern physics and certain other fields of science.
A close corollary here is the misapprehension of the actual manner in which mathematical signs (and hence all polarities) operate in the real world. In place of division by a non-existent “zero” Taoism advances the concepts of “polarization”, “depolarization”, and “repolarization”, all of which its “zero” alternative is fully capable of accomplishing. In a very real sense this “zero” alternative represents pure potential, both in the mathematical and physical senses. In the physical world it corresponds to the taijitu which may legitimately be considered pure potential energy as opposed to differentiated matter. Taoism in its way realized long before Einstein that the two are interchangeable.
With the preceding as background we are ready to consider next what the Taoist number line equivalent might look like.
Image: The Number Line. art: Zach Sterba/mC. writer: Kevin Gallagher/mC. [Source]
(1) This combination of parameters, comprising as it were a particular worldview, contrasts markedly with the worldview of Taoism. Unlike the demilitarized buffer zone that “zero” represents in Western mathematics, the taijitu of Taoism which occurs in its place both mediates between the polarities of negative and positive and gives rise to those same polarities repeatedly by the process of polarization. In place of the additive inverse negation operation of “zero” we have the operations of “depolarization” and “repolarization”. Having more in common with the worldview of Taoism than that embraced by Western mathematics, mandalic geometry treats the central buffer zone as a point of equilibrium, balance and potentiality which lacks neither content nor direction. It has in fact two directions, one centripetal, one centrifugal, which over the course of time can be alternatively chosen as a preferred direction repeatedly in cyclic fashion.
(2) This naturally brings to mind the informal rule of Computer Science holding that integrity of output is dependent upon the integrity of input. [GIGO (1960-1965): acronym for garbage in, garbage out.]
(3) This has far-reaching consequences, more importantly in the fields of modern physics and epistemology than that of mathematics. It is my contention that physics since the introduction of quantum mechanics and the theory of relativity (or “invariance” as Einstein preferred to refer to it) has been thwarted in its development by (among other things) too strict adherence to a limited concept of “zero” unsuited to its purposes. For Taoism and mandalic geometry “zero” is not a number but a polarizing function, itself without magnitude as commonly understood or permanent sign. Modern physics already partially leans toward this point of view but has not yet entirely shed the old outgrown “skin” that the number “zero” represents.
© 2014 Martin Hauser
Back to Basics: the fundamental polarity
For Taoism the fundamental polarity is that between yin and yang. This is usually presented along a vertical axis when considered in a single dimension. Yang is always above and associated with the South compass direction. Whenever two dimensions are treated simultaneously the yang polarity of the second dimension is presented along a horizontal axis to the left by convention and is associated with the East compass direction. (1)
For Western mathematics the fundamental polarity is that between negative and positive. When considered in context of one dimension this may be presented either along a horizontal axis (positive to the right) or vertical axis (positive up). When two dimensions are under consideration the horizontal axis is generally referred to as the x-axis and the vertical axis, the y-axis, both with directions labeled as noted above.
The two thought systems can be made commensurate in terms of mathematics. For instructional purposes here the Western conventions of direction are followed. (2) Also used here exclusively is the right-hand rule convention of three-dimensional vector geometry. Since the letter “x” is used to refer to the horizontal dimension and the letter “y” to the vertical dimension, the third dimension or “z” dimension must then necessarily have its positive direction toward the viewer. (3)
In physics, polarity is an attribute with two possible values. An electric charge, for example, can have either a positive or negative polarity. A voltage or potential difference between two points of an electric circuit has a polarity determined by which of the two points has the higher electric potential. A magnet has a polarity with the two poles distinguished as “north” and “south” pole. More generally, the polarity of an electric or magnetic field can be viewed as the sign of the vectors describing the field. (4)
Image: Yin yang. Public domain.
(1) Early Chinese cartography traditionally placed South above, North below, East to the left and West to the right. Though all reversed from Western presentations these are clearly conventional choices rather than matters of necessity. Many other ancient conventional associations of “yin” and “yang” have been preserved in Taoism. Most of the ancient traditional associations of “positive” and “negative” have long since been lost to Western thought.
(2) It should be noted that blindmen6.tumblr.com, this blog’s predecessor, presented instead the conventions used in the I Ching. That choice of convention has been abandoned here in favor of the Western convention in order to avoid unnecessary confusion.
(3) “Necessarily” only because the die has already been cast by choice of the directions of the horizontal and vertical axes and choice of adherence to the generally accepted right-hand rule. These, though, are all matters of convention. That should be kept in mind, if only because foresight suggests at a certain stage of development mandalic geometry may find it necessary to give the boot to some conventions and possibly as well to the use of any convention at all. Indeed the ultimate goal is a convention-free geometry though we are very far from that at this point in time.
(4) Although the text of this blog often equates the “yang” polarity with “+1” and the “yin” polarity with “-1” that is to be taken as a shorthand of sorts used instead of referring to “the positive sign of the vector +1” and “the negative sign of the vector -1” each time. Although doing so is most decidedly a convenience it is not strictly correct as these Taoist concepts actually refer to the entire poles of positivity and negativity. It is possible to use this shorthand only because to this point and for the foreseeable future the discretized number system of mandalic geometry requires only +1, -1 and 0 in terms of Western mathematics. It can be extrapolated to higher scalar values but will not be in the near time frame.
© 2014 Martin Hauser
A few words would seem in order at this point to remind readers, new and old, about the origins of mandalic geometry. The system represents an amalgamation of various thought streams deriving from the disciplines of mathematics, physics, chemistry, biology, philosophy (both Western and Eastern) and sinology. Although it has many diverse forebears, there are just two principal progenitors. These are the Cartesian coordinate system which is the foundation of analytic geometry and the Taoist worldview inherent to the I Ching.
The first of these two antecedents was invented by René Descartes in the 17th century and revolutionized mathematics by supplying the first systematic link between Euclidean geometry and algebra. The second dates back to prehistory in its original form and its inventor or inventors are unknown other than in legend. The I Ching has played a central role in traditional Chinese culture for thousands of years as an organizer and choreographer of modes of thought among other differing roles. Both of these tools of mind are intimately bound up with epistemology, the branch of philosophy concerned with the nature and scope of knowledge and which is also referred to as “theory of knowledge”.
Descartes’ innovation is concerned mainly with location in space. It is a coordinate system that specifies every point in a given plane (in its 2-dimensional version) uniquely by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular lines or axes, measured in the same unit of length. The space described may also be a 3-dimensional one if three mutually perpendicular axes are cited and Cartesian ordered triples used to specify a single given point in space with respect to all three axes of reference.
The I Ching, on the other hand, is concerned mainly with relationships, their possible and probable complex interwoven combinations, and their development or evolution through time. It is fundamentally an ancient treatise on combinatorics, relationship, changes evolving in spacetime, probability, and the mechanisms of (ostensibly) human power structures. Its human focus can be readily and successfully translated though into other relational constructs of a mathematical and/or physical nature.
Mandalic geometry is the result of crossbreeding these two distinct currents of thought or worldviews which arose in very different eras of human history and different geographical settings. As such it is a relativistic spatiotemporal discipline which unifies and extends all the individual components inherited from both its conceptual parents. It focuses on spacetime from a higher dimensional perspective than does either of its principal progenitors. Of particular interest is the particular manner in which it unifies space, time, change and probability in a single over-arching cognitive function of mind and mental processing. It is this coordination of characteristics that makes mandalic geometry uniquely suited for investigation and understanding of quantum processes in what we label the “real world”.
Image: Interlude musical, oil on canvas, 1874. Alexander Louis Leloir [Public domain], via Wikimedia Commons
© 2014 Martin Hauser
Quantum Naughts and Crosses 13
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This is the yz-plane with x = +1. It is the face of the mandalic cube that would be presented were the cube rotated 90 degrees clockwise.(1) Its resident tetragrams are formed from lines 2, 3, 5 and 6. Lines 1 and 4 are not included in the resident tetragrams because the x-dimension value is unchanging (+1) throughout this face of the mandalic cube. The z-axis (lines 3 and 6) is presented horizontally here, positive toward the viewer’s left. The y-axis (lines 2 and 5) is presented vertically, positive toward the top. The corresponding Cartesian triples are shown directly beneath the hexagram(s) they relate to.
The complementary(2) face of the mandalic cube, the yz-plane with x = -1, can be generated by changing lines 1 and 4 in every hexagram above from yang(+1) to yin(-1). Were we to do that and also view the resulting plane from a vantage point inside the cube we would then see a patterning of resident tetragrams identical to that in the plane above. The only difference apparent would be the substitution of yin lines for yang lines at positions 1 and 4.
We might have justifiably started out here by viewing the yz-plane with x = -1 from without the cube and followed with viewing the yz-plane with x = +1 from inside the cube had the die not already been cast. The problem with that attack given the present circumstances is that we have previously begun our consideration of the members of the the other two face pairs with the positive member from outside the cube. By preserving that consistency we end up with a jigsaw puzzle the parts of which can readily be fitted together to recreate the whole. Any inconsistency at this point can only result in failure.(3)
(1) This assumes that we begin with the reference face we have been using (xy-plane, z = +1) toward the observer seated at the bridge table.
(2) Mandalic geometry views opposite faces of the mandalic cube as being complementary rather than antagonistic or adversarial. This seems almost unnecessary to point out when the six planes that constitute the Cartesian cube are viewed as a single complex whole. There is a synergy of action simultaneously involving all component parts of the whole and there is an even greater degree of complex interactivity involving the component parts of the higher dimension mandalic cube.
The parts may indeed at times be in conflict or opposition with one another but at other times work together to create an effect. For a possible analogy think here of the constructive and destructive interference in which two or more wave fronts may participate. The I Ching, although it does not explicitly view the hexagrams and their component trigrams and tetragrams in the context of a geometric cube, nonetheless attributes these alternative and alternating reciprocal capacities to yin and yang and to all the line figures formed from them.
(3) This is much more than a simple matter of human convention. In this case we really are dealing with actual laws of nature, however cryptic and concealed they might be. This is not the right time to elaborate fully on what is involved here. Suffice it for now to point out that the approach we have chosen allows the three Cartesian and six additional mandalic dimensions to conform together with one another to certain combinatorial principles that nature demands they do.
For example, the three faces of the cube in which the hexagram consisting of six yang lines is found must fit together at a single point which forms one of the eight vertices of the cube by superimposing the three occurrences of this hexagram in the three different Cartesian planes at that single point. A similar requirement exists for all the other vertices of the cube as well. When all these various requirements are met all six faces can fit together snugly to form the cube. Were even just one of the requirements not satisfied the cube as a structural and functional whole would be unable to form.
We are talking here not simply about geometric shapes but about energetic physical phenomena as well. Ultimately this is not just a matter of composing a cube but of confronting the reality that dimensions fit together and force fields interact only in specific predetermined ways which we have no power to change. Moreover, this is just one indication that mandalic geometry describes more than literal locations existing in a topological space. It also corresponds in some sense to a state space, an abstract space in which different “positions” represent states of some physical system.
© 2014 Martin Hauser