**Back to Basics: space, time and dimension**

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**Throughout most of the history of Western thought** space and time were regarded as independent aspects of reality. **(1)** Dimension was considered an attribute of both. Space was viewed as consisting of three independent linear dimensions pictured as being mutually perpendicular. Each of these spatial dimensions consisted of two opposite directions and movement in either direction was possible. Different spatial dimensions were independent of one another and space could be traversed in one, two or three dimensions at once. Time was viewed as having a single dimension progressing in a forward direction only. Most often sequent time was emphasized preferentially to cyclic time. **(2)** Material objects were viewed as distinct entities occupying space and time but independent of them. **(3)**

**In the worldview of Taoism space, time and dimension** were never viewed as existing apart from one another but as all intimately related. Furthermore, dimensions are viewed as interrelated, not independent of one another. In general neither space nor time is conceived in terms of single linear dimensions but as interrelated composites of two or more dimensions. Direction in Taoism has to do not exclusively with opposing pairs but also with interdependent polarities. Time like space is considered to be bidirectional. Cyclic change plays a role of at least equal importance to sequent change. Time in the cyclic sense develops in directions of both expansion and contraction. Both evolution and involution, activation and deactivation are all ever-present possibilities. All possible combinations of relationship are explored and the probable eventual future outcomes of each occurrence are always taken into consideration for purposes of understanding events and planning actions. **(4)**

**Image: **A simple cycle. Author: Jerome Tremblay, writeLaTex. (This is used here to illustrate in the most elementary manner possible the basis of cyclic change and cyclic time. The more complex nature of these will be elaborated more fully in future posts.)

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**(1) It was not until the early 20th century** when Einstein introduced his Special theory of relativity, that space and time were fully integrated in a single concept, spacetime.

**(2) Historically the "cyclic" view of time** was of great importance in ancient thought and religions in the West as well as in the East. Attention was certainly paid to periodic recurring cycles related to the lunar month and, with the rise of agriculture, to the solar year. With the subsequent ascendance of the historically based religions however and in more modern times as technological achievements have taken center stage this acute awareness of periodicity and cyclic time has largely declined in the West.

** (3) Leibniz believed that space and time,** far from having independent existence, were determined by these material objects which he supposed were not contained

*in*space and time but rather

*created*them through their positioning relative to one another.

**(1,2,3)**Leibniz, however, was familiar with the I Ching and it is unlikely that his thought processes would not have been influenced to some degree by the relational and relativistic Taoist worldview he found therein.

**(4) In fact, the Taoist I Ching** can be considered first and foremost an exhaustive compendium of the combinatorial probabilities of spacetime relationships in six dimensions. Its alternate title The Book of Changes attests to this. The fact that it has also been used over the centuries as a method of divination should not in the least detract from its more comprehensive value to human knowledge and epistemology.

© 2014 Martin Hauser

## 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.

Image: Turquoise Eyes by thepeachpeddler CC BY 2.0

**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)**

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** (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.

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**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)**

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**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

**David Mumford** (born 1937)

**David Bryant Mumford** is an American mathematician known for distinguished work in algebraic geometry. [Wikipedia]

The statement was apparently made in 1975. Some thoughts on this remark can be found here.

Photo: David Mumford in 2010. By George M. Bergman (http://owpdb.mfo.de/detail?photo_id=13740) [GFDL], via Wikimedia Commons

(via 2radical3)