A new quantum logic - Bigram multiplication
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We are ready now to look more closely at 2-dimensional multiplication. In terms of Taoist notation and philosophy this involves multiplication of the bigrams. The easiest and most direct way to do this is in simple chart form. With the previous few posts as preparation the risk of mathematical conflation(1) here has been largely nullified.
The important thing to keep in mind when doing these simple bigram multiplications is that first and second dimensions are kept separate.(2) The lower line of the multiplier bigram multiplied by the lower line of the multiplicand bigram gives the lower line of the product bigram. The upper line of the multiplier bigram times the upper line of the multiplicand bigram gives the upper line of the product bigram. Nothing else is required here. Bigram multiplication reduces to two easy linear yin/yang multiplications.
In the multiplication charts below the bigram in the pink rectangle at the left is the multiplier and the bigrams in the blue rectangles across the top the multiplicands. The product of each individual multiplication is found in the white rectangle where the two intersect. Incidentally, these charts work for division as well. I’ll let you figure out how that is done.(3)
The top chart of these four shows bigram multiplication with the identity bigram of Quadrant I as the multiplier.
The second shows multiplication with the bigram of Quadrant II (which produces inversion of the horizontal dimension only) as the multiplier.
The third shows multiplication with the inversion bigram of Quadrant III (which produces inversion of both vertical and horizontal dimensions) as the multiplier.
The fourth shows multiplication with the bigram of Quadrant IV (which produces inversion of the vertical dimension only) as the multiplier.
Now that we fully understand where we’re coming from we can restate these results in a manner that involves some mathematical conflation.(4):
Any bigram multiplied by itself gives the identity bigram as product.
Any bigram multiplied by its inversion (polar opposite) gives the inversion bigram as product.
The identity bigram multiplied by any other gives back that same other as product.
The inversion bigram multiplied by any other gives back the inversion (polar opposite) of that other as product.
(to be continued)
(1) See footnote (1) here for a definition of conflation.
(2) Recall here that in the bigram the lower line specifies the first dimension or x-axis coordinate and the upper line the second dimension or y-axis coordinate.
(3) Hint: 2-dimensional bigram division reduces to two simple linear divisions and you already know how to deal with the vector forms of the number one (1). Okay, I’m not actually being condescending here, just slightly facetious.
(4) Not a very good thing to do as has been pointed out previously. But this will hopefully begin to make clear how and why mathematics goes bad. In this particular case no actual information has been lost. It has merely been twisted and distorted into a form which shifts focus from what is actually taking place to a specialized description which forces the reader to deal with issues of translation rather than matters of true importance. Welcome to the world of mathematical conflation. May you all be spared further exposure to it. (Not very likely though. All of us are occasional innocent perpetrators of conflation though I sometimes wonder whether at least some mathematical language isn’t intended mainly as a barrier to dissuade the peasants and other commoners from entry. That may just be my paranoid self talking though.)
© 2014 Martin Hauser
A new quantum logic - Numbers and dimension
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We need to understand that numbers possess dimension. Or let me rephrase that - numbers exist in all dimensions and take on the special character of the particular dimension they are inhabiting. Descartes uses both 2-dimensional numbers, his ordered pairs, and 3-dimensional numbers, the ordered triads, to uniquely identify all resident points of the plane, the 2-dimensional version of his coordinate system, and of the 3-dimensional version respectively.
Mandalic geometry does something similar using Taoist notation. But it is more interested in how different dimensions relate to one another than in merely locating points. For mandalic geometry it is dimensions that are real while points are just the evanescent phantasmagorical embodiments of their interactions. In other words, points are the temporary expression of the intersection of various dimensions.(1) Points are entities existing in spacetime in a manner similar to how fictional characters exist in books and films. As such they are subject to frequent whims and alterations of various sorts and degrees.(2)
The bigrams of the I Ching can be viewed in a manner similar to the ordered pairs of Descartes and treated similarly though with a few specific exceptions.(3) Fundamentally they refer to systems of two dimensions or to 2-dimensional parts of higher-dimensional systems. With all systems, but particularly with nonlinear systems (those having dimension greater than one), consideration must be given to units of measurement, point of reference and orientation. These three specifications are all required to completely and uniquely identify a situation or event.(4) Those wishing to know more about dimensional context can find a good article dealing with it in Wikipedia here.
The first or lower line of a bigram corresponds to the x-axis coordinate of Cartesian geometry; the upper line to the y-axis coordinate. Like the Cartesian ordered pair, the bigram is a 2-dimensional or planar number. Multiplication of bigrams, however, can be treated as multiplication of the individual linear parts. There is a multiplication of the first dimensional part to consider and a multiplication of the second dimensional part. The end result is another bigram which may or may not differ from the original two. This depends solely upon which of the four bigrams are used as multiplier and multiplicand. When both or one of these is the identity operator of bigram multiplication (i.e., +1 or yang in both dimensions of the bigram) the result will be the other initial bigram. As is so with linear numbers, multiplication of like vectors yields a positive vector while multiplication of unlike vectors yields a negative vector.(5)
This multiplication of planar numbers just represents a dimensional evolution of the multiplication of linear numbers. In fact, as described above, it involves two separate linear or one-dimensional multiplications. Mandalic geometry, concerned as it is with Planck-scale units, deals almost exclusively with the scalar number 1 and its natural derivatives(6) unlike Cartesian geometry which deals with scalars ranging to infinity. It is the sign portion (+/-; yang/yin) of vector numbers that is of most interest to mandalic geometry. Both Cartesian and mandalic geometries recognize two fundamental signs which represent the positive and negative polarities of nature. Descartes uses signs only for the purpose of locating points in space. Mandalic geometry uses signs also as a jumping off place for an investigation into the nature of dimension and how dimensions relate to one another both spatially and temporally.
Image: Aid to defining fractal dimensions. By Brendan Ryan.Nazlfrag at en.wikipedia [Public domain], from Wikimedia Commons
(1) Euclid proclaimed points to be dimensionless. It is the view of mandalic geometry that they participate in all dimensions. This may be a matter only of differing perspective. But small differences in perspective snowball eventually into varying worldviews which may differ considerably. Physics shares many of the perspectives held by Western mathematics as historically this was where it sourced them from. Though all of these viewpoints may be true in the mathematical sense they are likely not uniformly valid in the manner required by physics to satisfy the stricter definition of scientific truth. Physics would do well, I think, to do some of its sourcing in the vast storehouses of biology and chemistry. See here for an excellent short discussion of the differences between three categories of truth - subjective, deductive, and inductive (or scientific truth).
(2) This may be true of material particles as well. Like points, different particles possess a dynamic and energy specific to themselves and as with points are interchangeable one to another. The interchangeability of points is one of the thought forms expressed in both analytic geometry and the Taoist I Ching. The one graphs all of the varied changes decreed by the equations of mathematics; the other is a compendium or microcosm of all possible changes and combinatoric variants in a universe of six dimensions.
(3) In the context of higher dimension systems found in the I Ching and mandalic geometry the bigrams are not restricted to a single plane as are the Cartesian ordered pairs but are capable of entering and dwelling in any of the planar structures which may occur within the context of those disciplines. This will be explained fully in posts to follow. For now it is only necessary to note that whereas Cartesian ordered pairs are confined to and in a sense define a particular plane, Taoist bigrams are transposable among the many different planes of any number of dimensions and are themselves defined by the dimensional context in which they exist. At some level this is related to the subject of fractals but in the interest of clarity and parsimony we’ll save that discussion for another time.
(4) For Descartes a point is an object with no properties other than location; a space is a collection of locations; spaces can be characterized by their degrees of freedom. For mandalic geometry none of these three is separate from the others. That being so, none exists in itself or by itself alone. The most general of the three is degrees of freedom which amounts to the same thing as dimension. Then dimension determines both location of points and the spaces in which those points are embedded. None of this is static. All is fluid, constantly in process of change. A point that changes cannot be determined by location in space alone. Not even by location in spacetime except as a moving, changing point of reference. That is what the graphs of analytic geometry and the changes of The Book of Changes actually describe - changing location and relationships through spacetime.
(5) This is to be understood as a general principle which will be found to hold true throughout all dimensional contexts. In a sense it is a simple mathematical tautology. If +1 times +1 equals +1 and -1 times -1 equals +1 in a single linear dimension then the same holds true in a set of any number of linear dimensions. Also if +1 times -1 equals -1 and -1 times +1 equals -1 in a single linear dimension then the same holds true in a set of any number of linear dimensions. The qualifier here, if any exists, is that mandalic geometry views every dimension as a set of linear dimensions in relationships specific to that particular dimension.
(6) As used here the term natural derivatives refers neither to natural numbers specifically nor to the derivatives of differential calculus. It refers rather to those numbers, such as pi, square root 2, and square root 3, which occur in nature. Some of the natural numbers do qualify here as well - - - the integers 2, 3, 4, 8 for example as well as certain others.
© 2014 Martin Hauser
A new quantum logic - Introduction
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In this module we begin exploration of a new kind of quantum logic, one based in part on the identity and inversion functions of multiplication introduced in the preceding post. It would be easy enough to present the material in summary chart form but we will hold off doing that till the very end of the exposition because it would only lead to confusion and loss of information through conflation(1). Instead we will look graphically first at the way this logic works in two dimensions then proceed, still with graphic representation, to the 3-dimensional and higher dimensional categories.(2)
As it turns out this will also provide us an excellent opportunity to demonstrate how much easier this all goes down with Taoist notation as opposed to Cartesian notation. With the former we just go with the flow while with the latter we get mired down with details and directions - - -literally. Using Cartesian notation for this is sort of like going for a swim in quicksand. With a whole lot of luck and perseverance you might make it to the other side. But what’s the point of setting up such an obstacle course for something that can be so much simpler and a lot more enjoyable. We’re going Taoist here.
The image above shows four unit squares mutually tangent at the origin of the coordinate system(O). The four vertices are labeled A, B, C, D. The extreme points of the horizontal x-axis are labeled L, N and the extreme points of the vertical y-axis are labeled K, M. The side of each unit square equals 1 unit in length. The diagonal of each unit square is equal to square root 2 units in length. The Cartesian coordinates of all the major points are shown. The Taoist coordinates of the four vertices are shown as well.(3) Everything is clearly and accurately labeled. I assure you this is not going to be a hat trick of any sort. Still you might want to hold on to your hat if you happen to be wearing one. Ready? And awaaay we go! . . .
(1) See Footnote (1) here for a short expose of conflation.
(2) SPOILER: Would I be spilling the beans if I told you that the way this works is exactly the same across all dimensions?
(3) The two axes are not labeled with Taoist coordinates because there is no symbol corresponding to Cartesian zero (0) in Taoist notation. There is a really good reason for that though. It wasn’t just an oversight. We could easily substitute the star symbol (*) for the Cartesian zero (0) with the understanding that rather than being used to signify zero it would stand for a wild card (i.e., signify either yin or yang.) This solution is not native to Taoism, however, but of my own invention. When we discuss composite dimension more fully in the future we will present the Taoist solution to this problem. That is a bit more complicated and there is no real need to get involved with it at this point when dealing with only two dimensions.
© 2014 Martin Hauser
The identity and inversion operators of multiplication
We are all familiar with the number 1 and its remarkable powers. It is the basis of all arithmetic. Its absolute (scalar) value (1,2) combined with the signs + or - produce the vectors +1 and -1. Here we look at the roles of +1 and -1 as the identity function and inversion function respectively of multiplication. If multiplication by 1 sounds a bit too basic for you, read on. You may be in store for some surprises. Maybe not in this module but in others that build upon the fundamental concepts introduced here.(1)
The first thing to note is the dual aspects of magnitude and direction. As a scalar, the number 1 without a sign is the sole existing number which produces no change in scale or magnitude when applied as a multiplicative operator. As a vector the absolute value 1 takes on two distinct natures. In the positive version +1 (or the equivalent yang unity of Taoism) this number becomes the identity function of multiplication. In the negative form -1 (or the equivalent yin unity of Taoism) it serves as the inversion function of multiplication.
Both these forms of unity are vectors which acting as multiplicative operators preserve magnitude and scale while yielding differing effects upon direction.(2) The negative (yin) form causes inversion of direction through a central point.(3) The positive (yang) form has no effect upon direction, leaving it unchanged.
(1) Certain remarkable events occurring in the subatomic world that quantum mechanics studies, for example, are predicated upon the vectors +1 and -1 as the identity and inversion functions of multiplication. Throw the notion of scalar quantites into the mix and we have combined scalars and vectors in the operation of scalar multiplication which encompasses both determining direction of movement and scaling of magnitude.
(2) It is important to distinguish the scalar (magnitudinal) and vector (magnitudinal plus directional) aspects of numbers. The numbers of major concern in mandalic geometry are the vector forms of the number one as the unidimensional polarities +1 (yang) and -1 (yin); the corresponding composite bi- and tri-dimensional vector forms (Cartesian ordered pairs of the unit square and ordered triads of the unit cube and the bigrams and trigrams of the Taoist I Ching); and the polar zeros (composites of yin and yang).
(3) In some contexts this could also be viewed as a rotation of 180 degrees though it is probably best not to do so. It is not truly a rotation operator as such an entity can exist only in two or more dimensions while we are still concerned here only with the real number line and a single dimension. Also, when we consider higher dimensional numbers and their properties we will see that all results of multiplication of real numbers of any order of dimension can be treated exclusively in terms of occurrence or non-occurrence of inversions through one or more dimensions. Recourse to considerations of rotation will not be necessary.
© 2014 Martin Hauser
Mathematical conflation: A historical example
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Above is an example of the type of equation that led mathematicians of the European Age of Enlightenment to invent imaginary numbers. Recall that these were individuals who were still not entirely comfortable with negative numbers. Yet within a few short generations they had concocted square roots of negative numbers and the complex plane. In the words of a man of the Enlightenment, “Fools rush in where angels fear to tread.”(1)
Let’s look briefly at where, in my opinion, they went wrong:
If we produce the square algebraically in each of the four quadrants of the Cartesian plane the way mathematics instructs, we do indeed get x^2 = a x a. But there is something very fishy here. Something strange is going on. I think we’re being played for fools by the old hat trick.
In Quadrant I:
x times y = x times x = x^2 because both x and y are equal to a which is positive (a positive vector).
In Quadrant III:
x times y = x times x = x^2 because both x and y are equal to a which is negative (a negative vector).
So yes, in both these cases the resulting square (algebraic and geometric) is positive. Any two vectors having the same sign when multiplied together yield a positive result.
But here’s the catch:
In Quadrants II and IV:
x times y does not equal x times x because x and y differ in sign. One is a positive vector, the other a negative vector. Any two vectors differing in sign when multiplied together always produce a negative result.
In Quadrant II: x times y = -a times +a = -(a^2)
In Quadrant IV: x times y = +a times -a = -(a^2)
The algebraic square and the geometric square differ from one another in Quadrants II and IV. Rationalists of the Enlightenment overlooked this detail, dismissed it or discredited it. Any way you care to look at it they disavowed the fact. The hat trick staged was one of conflation and the rationalists having themselves perpetrated it were the first to fall for it.
Something important has been lost in the rationalists’ translation through the error of conflation committed here.(2) The concept of square root that emerged from this unfounded merging of algebra and geometry is a flagrant error, one that has been turned into orthodoxy by four centuries of consistent misuse. Tell a lie often enough and it becomes truth - - - or at very least academic gospel.
What has been lost through conflation is the truth about squares and square roots and also the momentous realization that negative squares (negative 2-dimensional numbers) are every bit as real as are negative numbers (negative 1-dimensional or linear numbers.) Unfortunately, that realization never followed as one might expect it would have. Now at last the rabbit is out of the hat and a reassessment of square roots is in order.
Rooted as their thought was in pictorial geometry and in practical engineering the ancients never would have committed an error such as this. Nor would have the Romantics who followed the rationalist thinkers of the Enlightenment.(3) But the die had been cast, the cards dealt, and we moderns left with the choice to play the dishonest hand received or call for a reshuffle on grounds we now feel shaky about with Academia crying, “Foul!” should we muster courage to make the attempt.
Mathematicians cannot validly treat a given line segment as negative in one context and positive in another when it suits their purpose. What they’ve done here is mathematical conflation. The result is misleading and in my opinion untrue, a mathematical error - or would be if mathematics didn’t have its own very special definition of mathematical truth which is capable of glossing over cases like this. Physics can not afford a similar nonchalance due to the very different nature of scientific truth which is grounded in empirical observation and experience.
(1) See here for some interesting comments about the meaning and origin of this remark.
(2) Just as the essential reality and meaning of the square root of 2 is obvious in geometry but obscured in decimal arithmetic and algebra, so the true meaning of squaring and square roots is lost in translation from living, breathing geometry into crystalline algebra.
(3) The Age of Enlightenment was not entirely the villain I’ve painted it here with poetic license but it certainly had its shortcomings.
The Scientific Revolution is closely tied to the Enlightenment, as its discoveries overturned many traditional concepts and introduced new perspectives on nature and man’s place within it. The Enlightenment flourished until about 1790–1800, at which point the Enlightenment, with its emphasis on reason, gave way to Romanticism, which placed a new emphasis on emotion; a Counter-Enlightenment began to increase in prominence. The Romantics argued that the Enlightenment was reductionistic insofar as it had largely ignored the forces of imagination, mystery, and sentiment. [Wikipedia]
© 2014 Martin Hauser