#history of mathematics

Beyond Taoism - Part 4
A Vector-based Probabilistic
Number System
Introduction

(continued from here)

Leibniz erred in concluding the hexagrams of the I Ching were based on a number system related to his own  binary number system.  He had a brilliant mind but was just as fallible as the rest of us.  He interpreted the I Ching in terms of his own thought forms,  and he saw the hexagrams as a foreshadowing of his own binary arithmetic.^[1]

So in considering the hexagram Receptive,  Leibniz understood the number 0; in the hexagram Return, the number 1; in the hexagram Army, the number 2; in the hexagram Approach, the number 3;  in the hexagram Modesty,  the number 4;  in the hexagram  Darkening of the Light, the number 5;  and so on, up to the hexagram Creative, in which he saw the number 63.^[2]  His error is perhaps excusable in light of the fact that the Taoists, though much closer to the origin of the I Ching in time, themselves misinterpreted the number system it was based on.^[3]

From our Western perspectiveI Ching hexagrams are composed of trigrams, tetragrams, bigrams, and ultimately yinandyang Lines. From the native perspective of the I Ching this order of arrangement is putting the cart before the horse.  Dimensions  and their interactions  are,  in the view of I Ching philosophy and mandalic geometry,  antecedent logically and materially to any cognitive parts we may abstract from them. Taoism in certain contexts has abstracted the parts and caused them to appear as if primary. It has the right to do so if creating its own philosophy,  but not as interpretation of the logic of the I Ching. It is a fallacy if so intended.^[4]

The Taoists borrowed from the I Ching two-dimensional numbers, treated them as one-dimensional and based their quasi-modular number system on  the dimension-deficient result.  This is the way they arrived at their seasonal cycle consisting of bigrams:   old yin (Winter),  young yang (Spring), old yang (Summer), young yin (Autumn), old yin (Winter),  and so forth. This represents a very much impoverished and impaired version of the original configuration in the primal strata of the I Ching.^[5]

The number system of the I Ching is not a linear one-dimensional number system like  the positional decimal number system  of the West; nor is it like the positional binary number system invented by Leibniz. It is not even like the quasi-modular number system of Taoism.  The key to the number system of the hexagrams is located not in the 64 unchanging explicit hexagrams,  but rather in the changing implicit hexagrams found only in the divination practice associated with the I Ching. These number 4032.^[6]  The manner in which these operate,  however,  is actually  fairly simple and is uniform throughout the system.  So once understood,  they can be safely relegated to the implicit background, coming into play only during procedures involving divination or in attempts to understand the system fully, logically and materially.  When dealing with more ordinary circumstances just the 64 more stable hexagrams need be attended to in a direct and explicit manner.

The Taoist sequence of bigrams is in fact a corruption of the far richer asequential multidimensional arrangement of bigrams that occurs in I Ching hexagrams and divination. There we see that change can occur from any one of the four stable bigrams to any other.  If this is so then no single sequence can do justice to the total number possible. The ordering of bigrams presented by Taoism is just one of many that make up the real worlds of nature and humankind.  Taoism imparts special significance to this sequence; the primal I Ching does not. It views all possible pathways of change as equally likely.^[7]

Next time around we will look further into the implications of this equipotentiality and see how it plays out in regard to the number system of the I Ching.

Section F_H(n)^[8]

(continuedhere)

Notes

[1] By equating yang with 1 and yin with 0 it is possibletosequence the 64 I Ching hexagrams according to binary numbers 0 through 63.  The mere fact that this is possible does not, however, mean that this was intended at the time the hexagrams were originally formulated. Unfortunately, this arrangement of hexagrams seems to have been the only one of which Leibniz had knowledge. This sequence was, in fact, the creation of the Chinese philosopher Shao Yong (1011–1077). It did not exist in human mentation prior to the 11th century CE.

This arrangement was set down by the Song dynasty philosopher Shao Yong (1011–1077 CE), six
centuries before Wilhelm Leibniz described binary notation. Leibniz published ‘De progressione
dyadica’ in 1679. In 1701 the Jesuit Joachim Bouvet wrote to him enclosing a copy of Shao Yong’s 'Xiantian cixu’ (Before Heaven sequence). [Source]

Note also that the author of Calling crane in the shade, the source quoted above, calls attention to confusion that exists about whether the “true binary sequence of hexagrams” should begin with the lowest line as the least significant bit (LSB) or the highest line. He points out that the Fuxi sequence as transmitted by Shao Yong in both circular and square diagrams takes the highest line as the LSB, although in fact it would make more sense in consideration of how the hexagram form is interpreted to take the lowest line as the LSB. My thinking is that either Shao Yong misinterpreted the usage of hexagram form or, more likely, the conventional interpretation of the Shao Yong diagrams is incorrect. Here I have chosen to use the lowest line of the hexagram as the LSB,  and I think it possible  Leibniz may have done the same.

If one considers the circular Shao Yong diagram,  the easier of the two to follow,  one can reconstruct the binary sequence,  with the lowest line as LSB,  by beginning with the hexagram EARTH at the center lower right half of the circle, reading all hexagrams from outside line (bottom) to inside line (top),  progressing counterclockwise to  MOUNTAIN over WIND at top center, then jumping to hexagram  MOUNTAIN over EARTH  bottom center of left half of the circle,  and progressing clockwise to hexagram  HEAVEN  at top center.  Of the two,  this is the interpretation that makes the more sense to me and the one I have followed here, despite the fact that it is not the received traditional interpretation of the Shao Yong sequence. Historical transmissions have not infrequently erred. Admittedly it is difficult to decipher all Lines of some of the hexagrams  in the copy Leibniz received due to passage of time and its effects on paper and ink.  Time is not kind to ink and paper, nor for that matter to flesh and products of intellect.

In the final analysis, which of the two described interpretations is the better is moot because neither conforms to the logic of the I Ching which is not binary to begin with. Moreover,  there is a third interpretation of the Shao Yong sequence that is superior to either described here.  It is not binary-based.  And why should it be? After all the Fuxi trigram sequence  which Shao Yong took as model for his hexagram sequence  is itself not binary-based. Perhaps we’ll consider that interpretation somewhere down the road. For now, the main take-away is that Leibniz, in his biased interpretation of the I Ching hexagrams made one huge mistake.  Ironically,  had he not some 22 years prior already invented  binary arithmetic, this error likely would have led him to invent it.  It was “in the cards” as they say. At least in certain probable worlds.

[2]ReceptiveandCreative are alternative names for the hexagrams EarthandHeaven, respectively. The sequence detailed can be continued ad infinitum using yin-yang notation, though of course this takes us beyond the realm of hexagrams into what would be, for mandalic geometry and logistics of the I Ching, domains of dimensions numbering more than six.  Keep in mind here though that Leibniz was not thinking in terms of dimension but an  alternative method  of expressing the prevalent base 10 positional number system notation of the West.  He held in his grasp the key to unlocking an even greater treasure but apparently never once saw that was so.  This seems strange considering his broadly diversified interests and pursuits in the fields of  mathematics,  physics,  symbolic logic,  information science,  combinatorics,  and in the nature of space.  Moreover,  his concern with these was not just as separate subjects of investigation.  He envisaged uniting all of them in a  universal language  capable of expressing mathematical,  scientific, and metaphysical concepts.

[3] Earlier in this blog I have too often confused Taoism with pre-Taoism. The earliest strata of the I Ching belong to an age that preceded Taoism by centuries, if not millennia.  Though Taoism was largely based on the philosophy and logic of the I Ching,  it didn’t always interpret source materials correctly,  or possibly at times it intentionally used source materials in new ways largely foreign to the originals. The number system of the I Ching is a case in point.

In the interest of full disclosure, I am not an expert in the history or philosophy of Taoism.  Taoist philosophies are diverse and extensive. No one has a complete set or grasp of all the thoughts, practices and techniques of Taoism. The two core Taoist texts, the  Tao Te ChingandChuang-tzu,   provide the philosophical basis of Taoism which derives from the eight trigrams (bagua) of Fu Xi, c. 2700 BCE, the various combinations of which created the 64 hexagrams documented in the I Ching.  The Daozang,  also referred to as  the Taoist canon,  consists of around 1,400 texts that were collected c. 400, long after the two classic texts mentioned. What I describe as Taoist thought then is abstracted in some manner from a huge compilation, parts of which may well differ from what is presented here. Similar effects of time and history can be discerned in Buddhism, Christianity, Islam and secular schools of thought like Platonism,Aristotelianism,Humanism, etc.

[4] Recent advances in the sciences have begun to raise new ideas regarding the structure of reality. Many of these have parallels in Eastern thought.  There has been a shift away from the reductionist view in which things are explained by breaking them down then looking at their component parts, towards a more holistic view. Quantum physics notably has changed the way reality is viewed. There are no certainties at a quantum level, and the experimenter is necessarily part of the experiment. In this new view of nature everything is linked and man is himself one of the linkages.

[5] It is not so much that this is incorrect as that it isextremelylimiting with respect to the capacities of the I Ching hexagrams.  A special case has here been turned into a generalization that purports to cover all bases. This may serve well enough within the confines of Taoism but it comes nowhere near elaborating the number system native to the I Ching. We would be generous in describing it as a watered down version of a far more complex whole.  Through the centuries both Confucianism and Taoism  restructured the I Ching to make it conducive to their own purposes.  They edited it and revised it repeatedly,  generating commentary after commentary,  which were admixed with the original,  so that the I Ching as we have it today,  the I Ching of tradition,  is a hodgepodge of many convictions and many opinions. This makes the quest for the original features of the I Ching somewhat akin to an archaeological dig.  I find it not all that surprising  that the oracular methodology of consulting the I Ching  holds possibly greater promise in this endeavor than the written text.  The  early oral traditions  were preserved better,  I think,  by the uneducated masses who used the I Ching as their tool for divination than by philosophers and scholars who,  in their writings,  played too often a game of one-upmanship with the original.

[6] A Line can be either yin or yang, changing or unchanging. Then there are four possible Line types and six Lines to a hexagram.  This gives a total of 4096 changing and unchanging hexagrams (4⁶ = 4096). Since there are 64 unchanging hexagrams (2⁶ = 64) there must be 4032 changing hexagrams (4096-64 = 4032).

[7] This calls to mind the path integral formulation of quantum mechanics which was developed in its complete form by Richard Feynman in 1948. See, for example, this description of the path integral formulation in context of the double-slit experiment, the quintessential experiment of quantum mechanics.

[8] 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. Though this model may be simple by comparison,  it will nevertheless serve us well as a key to deciphering the number system on which I Ching logic is based as well as the structure and context of the geometric line that can be derived by application of reductionist thought to the associated mandalic coordinate system of the I Ching hexagrams. We will refer back to this figure for that purpose in the near future.

© 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
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-Page 299-

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 9:
The Potential Plane
and Probable States of Change

Composite Dimension and
Amplitudes of Potentiality
Episode 3

(continued from here)

We have seen that an imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i,  which is defined by its property ixi =−1.  The square of an imaginary number bi is −b².  For example,  6i is an imaginary number,  and its square is −36.^[1] Other than 0,  imaginary numbers yield negative real numbers when they are squared.^[2]

Turning now to potential numbers, we can similarly define a unit of potentiality p by the property p x -p = -1. [Long pause here waiting for the other shoe to drop.] Just a minute, you say, that’s just like 1 x -1 = -1.  Yes, it is. And that is just the point. All real numbers. Nothing to imagine. And Descartes finally vindicated after all these years - imaginary numbers just imaginary after all.  But how does this work? Or does it even work?  What exactly is the point? Is this a joke? It’s no joke, I assure you.  It’s an easier and better way to achieve the same ends - - - and more. Muchmore.

The secret is in the sauce, I say slyly. Really? Well, yes - in a way. Though imaginaries use a sauce with nearly identical ingredients.  The recipe is p + (-p) = 0. And, of course, i + (-i) = 0 as well.  The trick is in how - - - and where - - - the sauce is applied.  In the potential plane the sauce is applied more liberally in more locations for greater lubrication.

Levity aside. (This is after all a TST^[3].) The complex plane uses a single axis.  This axis represents a new dimension, wholly distinct from the x, y and z dimensions.  Strangely,  we’re never informed where this axis/dimension might be located,  just that it is somewhere other than where x, y and z are located. Stranger still, the complex plane allocates the y-axis of the Cartesian plane for its own use in location of its points. Although never specifically mentioned, to my knowledge, I surmise the imaginary dimension exists in what mathematics and physics both call phase space.^[4]

The mandalic or potential plane uses no such underhanded plan. It openly posits the existence of six new dimensions, allocated equally with two accompanying each of the Cartesian dimensions,  all overtly evident. (All nine spatial dimensions in plain sight together, that is.)  Nothing left to the imagination. As the new dimensions are made commensurate with the old in a hybrid geometric display,  no imaginary dimension is needed. Coordinates of  all potential dimensions  are  readily communicable  with the real number system through all of the ordinary Cartesian dimensions concurrently along with the Cartesian coordinates.  Moreover,  mandalic geometry conjectures that the ordinary Cartesian dimensions may in fact originate in  interactions among number species  of potential dimensions filtered through impacts on inherited biological sensory mechanisms.^[5] This raises yet another interesting possibility.^[6]

In the long convoluted history of mathematics, the imaginary numbers were introduced as a correlative to the number line with its real numbers. That meant, among other things, that they were linear, consisting of a single dimension.  The  complex plane  related the two
in a kind of hybrid geometry that consisted of one real dimension and one imaginary dimension.  Mathematician  William Rowan Hamilton in 1843 proffered the  quaternions,  a number system that extends the complex numbers to three dimensions, whereupon things went, to my mind, from bad, to very much worse.

Quaternions came with certain dysfunctional characteristics, among them,  the fact that multiplication of two quaternions is noncommutative. This is problematic.  The imaginary and complex numbers,  at least,  had both been commutative.  Nevertheless, physics endorsed the quaternions as it earlier had imaginary and complex numbers.

Why? Because the quaternions do in fact give partly correct results, and when investigating a dimly illuminated region of reality, such as the subatomic world still is today, even partial results are heartily welcomed if that is all that can be had.  The sad consequence of this, is that physics has been led astray in its quest for truth for over a century now,  because partial truths can be much more misleading than complete errors. Total error is often uncovered much sooner than partial truth, which can pass undiscovered, depending upon circumstances, for a very long time.

Mandalic geometry will be shown to be free of the difficulty posed by noncommutative multiplication. It is fully commutative throughout its nine dimensions (three ordinary, six extraordinary). It was not composed that way from a number line,  with elements that could be commutatively multiplied with one another. It came that way fully formed from the start, in its primeval embodiment  as a multidimensional structure,  expressing behavior intrinsic to holistic nature.

Next time around, we’ll begin to look under the hood of the mandalic approach to geometry and see if we can grokit.

(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] 6²xi² = 36 x (-1) = -36.

[2] Zero (0) is considered both real and imaginary, and both the real part and the imaginary part are defined as real numbers. (If that makes little sense to you, don’t blame me. I’m just the messenger here, reporting what the mathematicians have stated to be the case.) This seems to me to be purely an arbitrary definition, and it confuses me as much as it probably does you.  Could it be they did this to avoid the situation where 0² x (-1) = -0?  I think I would find that definition less disturbing, welcome even.

[3] Newly coined Internet acronym for Truly Serious Topic. (Not to be confused with TSR Totally Stupid Rules.)

Speaking about “greater lubrication”(wewere a moment ago, remember?), I use the phrase not simply as  a figure of speech,  or a simile,  but rather,  as a metaphor.  "Spicing" of mandalic geometry with all those zeros of potentiality makes for a very “fluidic dish” which, I believe, reflects the changeable nature of reality far better than the stricter, strait-laced coordinates of Descartes or the complex plane are able to do. And it’s not just a matter of fluidity involved here. The mandalic form so begotten is, in fact, a probability distribution through the three Cartesian dimensions concurrently,  which feature alone  makes mandalic geometry an ideal candidate for application to quantum physics.

[4] A phase space of a dynamical system is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. In a phase space every degree of freedom or parameter of the system is represented as an axis of a multidimensional space; a one-dimensional system is called a phase line, while a two-dimensional system is called a phase plane.  For every possible state of the system (that is to say, any allowed combination of values of the system’s parameters) a point is included in the multidimensional space. [Wikipedia]

[5] I am speaking here of the hybrid 6D/3D formulation of mandalic geometry which combines the features of  dimensional numbers,  potential numbers,  and composite dimension,  this being a fully open access geometric system that has nothing hidden, nothing held back. What you see is what you get. (WYSIWYG)

[6] It is tempting to wonder whether there might be a close connection between the composite dimensions/potential coordinates  proposed by mandalic geometry and the pilot wave theoryorde Broglie–Bohm theory of quantum mechanics. At least there seems to be a correlation  between  David Bohm’s implicate/explicate order and the manifest/unmanifest (potential) coordinates of mandalic geometry.

© 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 285-

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-

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-

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-