Loose ends: Routes, results and commutation

 

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As an example of a context in which commutation fails consider the transforms of cube inversions. To clarify, we have already demonstrated that commutation in the narrowly focused mathematical sense does hold true in this context. But here we are considering the larger more inclusive case of objective reality that physics and the sciences in general are concerned with.

Referring to the diagram above and focusing here on the forward facing plane we can see that WIND times MOUNTAIN equals FIRE and also that MOUNTAIN time WIND equals FIRE as well. So the purely mathematical definition of commutativity is satisfied. But hold on a moment. Consider how we got from here to there and whether that makes a difference in the real world. The world of physics, chemistry, biology, literature, music and so on. Of the very lives we live even.

When WIND as operator acts upon MOUNTAIN as operand in the operation of dimensional multiplication it causes MOUNTAIN to move horizontally one step to the right to FIRE. When MOUNTAIN as operator acts upon WIND as operand, however, it causes it either to move first horizontally one step to the right to HEAVEN then down one step to FIRE or down first one step to MOUNTAIN then horizontally one step to FIRE. Another alternative, given sufficient force, would be to cause WIND to move along the diagonal down and to the right simultaneously to FIRE.

Pure mathematics would have us believe that none of this makes a difference. In other words, all these real world differences make no difference. Really? How is that? Solely because pure mathematics has decreed it so in its rule book which it is ever so careful to maintain as internally self-consistent.

Mandalic geometry, however, is concerned with that real world that pure mathematics would have us ignore because it is a hybrid discipline the allegiance of which is to the way things actually work rather than to the preconceived notions of a self-consistent book of tautological rules. For that reason it insists that the route taken in arriving at a destination can be as important or more so than the destination itself. Whether or not this is true in a particular case can only be determined by experiment and experience. There is no rule book that will reveal real world truth for every occurrence or all parameters of exploration.

Science often marvels at how mathematics so successfully applies to the real world. It would do well to take note of the many situations where this platitudinal truism falls far short of the truth that science seeks.

(to be continued)

 © 2014 Martin Hauser

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Loose ends: Commutation’s dirty little secret

 

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If we accept as definition of the commutative property the misleading simplistic statement that binary operation is commutative if changing the order of the operands does not change the result then we too must number ourselves among the many sheep of the world who do not object to having the wool pulled over their eyes. Though the statement may well be true in a narrow mathematical sense it fails the more inclusive test of what in fact occurs in physical reality. That is the condition we must insist on if we wish all our prized philosophies not to degenerate into tautological redundancy.

There is another definition sometimes given describing commutation which at first seems to be saying the same as the definition above. It states that being commutative means having the property that one term operating on a second is equal to the second operating on the first, as a × b = b × a. The difference between the two definitions resides in the nuance of different significance between the terms “is equal to” and “the result”. This second definition is not even true in the narrow mathematical sense if by “is equal to” it intends “is the same as”. But in erring it does us the favor of revealing the conflationary sleight of hand performed by mathematics in its manner of usage of the concept of commutation.(1)

We have already agreed that all scalar numbers do indeed commute in the operation of multiplication. The two points mandalic geometry takes issue with are

  1. that vectors or dimensional numbers necessarily commute in the same manner either in multiplication or addition
  2. that the result of an operation is the only factor of importance to take into consideration in the real world

To clarify the second objection, in physical processes the route(s) taken in arriving at a result are often as important as and at times more important than the actual result itself. Pure mathematics, unconcerned as it is with the world of objective reality, would have us deny this experiential truth. Physics and all the sciences in general must remain constantly vigilant and be wary about what particular aspects of mathematics creep into their intellectual disciplines.

In the next post we will demonstrate how route and result are related in inversion transforms of the cube and show how the mathematical definition of commutation fails in that context.

(to be continued)

 Image:   Illustration from “SLEIGHTS: A Number of Incidental Effects, Tricks, Sleights, Moves, and Passes (1914)” by Burling Hull (found here)

 

(1) Somewhat ironically one of the definitions of commutation offered outside the field of mathematics is 

  1. (n.) A substitution, as of a less thing for a greater, esp. a substitution of one form of payment for another, or one payment for many, or a specific sum of money for conditional payments or allowances; as, commutation of tithes; commutation of fares; commutation of copyright; commutation of rations. [Source]

In other words commutation sometimes signifies the return of that which is inferior in place of the original. An interesting definition, one not specifically directed toward usage of the same term within mathematics but tantalizingly apropos nonetheless.

 © 2014 Martin Hauser

Please note - This is a beta post meaning that the content and/or format may not yet be in finalized form. A reblog as a TEXT post now will contain this caveat to warn readers to refer to the current version which appears in the source blog. This note will not however appear in a LINK post which itself accomplishes the same. Thank you for not removing this statement and for your understanding. :)

Loose ends: Notation, notation, notation

 

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Before we begin to consider the transforms of cube inversions there are a few loose ends we still need to address.

First, we need to make perfectly clear that with appropriate maneuvers everything that can be done with Taoist-notation mandalic coordinates can also be accomplished with Indo-Arabic-notation Cartesian coordinates. Just the difficulty involved is such that it seems hardly worth the while. Use of the Taoist notation not only makes the required mental manipulations much easier, it also brings to light a number of important relationships which otherwise pass unnoticed.(1)

Both Cartesian and Taoist coordinate systems are positional notational systems. They both assign different dimensions to different positions in the respective notational structure. Cartesian ordered triads, for example, order dimensions from left to right; Taoist trigrams, from below to above. If we view the horizontal x dimension as dimension one, the vertical y dimension as dimension two and the forward/backward z dimension as dimension three then Cartesian notation represents this arrangement of dimensions as x,y,z (e.g., 1,-1,1) by convention while Taoist notation by convention represents the same by image, the trigram FIRE. All eight correspondences are shown in the diagram above.

If one learns to associate the notational form of a trigram with both its English name assignation and its family association in the group of trigrams mental manipulation becomes quite easy. The same cannot be said about the Cartesian ordered triads no matter what one does with them to attempt easily performed mental jugglery. Once the mechanism which deals with composite dimension is introduced and we begin to consider the hexagrams, the six dimensional figures of Taoist notation, the I Ching and mandalic geometry, this critical difference in difficulty of usage between Cartesian coordinates and mandalic coordinates becomes all the more apparent and significant as it increases, one might say, almost exponentially.

(to be continued)

 

(1) The caveat here is that some minds work very differently from my own. Of these there may well be a few that could manipulate the Cartesian notation as easily as the Taoist. The point is that I believe most minds will find the Taoist notation far easier in application than the Cartesian notation for investigation of groups of the 3-dimensional cube and symmetries and asymmetries of the 6-dimensional hypercube or mandalic cube.

 © 2014 Martin Hauser

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A new quantum logic: Inversion group of the cube

 

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This is an important topic which involves the permutation group of the inversions of the vertices of the cube. This is not so much complicated as it is involved. Several posts will be aimed at introduction and development of the basics of the subject. This is fundamental material which is prerequisite for understanding the structure of the 6-dimensional mandalic cube and how its parts interact in the whole functionally.

This subject is similar to the rotation group of the cube but involves inversions or reflections through a point rather than rotations. The trigram vertices of the cube are multiples having split personalities. This is the case because, as has been stated many times before, the eight points which they specify and distinguish are fictions which result from intersections of various different dimensions. The inversions they engage in can therefore take place in a finite number of different ways, all of which must be taken into account in any consideration of the cube that aspires to a complete description and full understanding.(1)

We have already met the list of characters. The eight trigrams can be compared not so much to characters in a book, film or play as to the actors playing those characters. As sometimes occurs in theatrical works the actors here can play multiple roles. They must be conceived as having more than a single potential in terms of alternate character development, a requirement which issues from the fact that the lines and planes that intersect to form the unique vertex points do so functionally in such a manner that there are multiple ways the whole can be constituted from the parts and the parts from the whole.(2) We will look at all the inversion transforms of a cube that map it to itself.

(to be continued)

Image: Cubic crystal system, modified to demonstrate positioning of trigrams and Cartesian triads. Generalic, Eni. “Cubic crystal system.” Croatian-English Chemistry Dictionary & Glossary. 31 July 2014. KTF-Split.19 Aug. 2014. <http://glossary.periodni.com>.

 

(1) This should call to mind the effects of measurement on quantum systems. In the quantum world objects seem to exist natively in a state of totipotentiality. Until measured they exist in many places at the same time. Only when measured do they appear to us to have a definite location. The Copenhagen interpretation asserts that quantum mechanics rather than describing an objective reality deals only with probabilities of observing, or measuring, various aspects of energy quanta.

The act of measurement causes the set of probabilities to immediately and randomly assume only one of the possible values. This feature of mathematics is known as wavefunction collapse. The essential concepts of the interpretation were devised by Niels Bohr, Werner Heisenberg and others in the years 1924–27. [Wikipedia]

Mandalic geometry holds an alternative view. It professes that the set of probabilities described by the Schrödinger equation and quantum mechanics does in fact refer to an objective reality, one having a logic of higher dimension and which coordinates space and time together though not in quite the manner Einstein envisaged. Within this higher logical patterning multiple outcomes are always possible and these we interpret as probabilities. Measurement does not collapse the wave function of the system but simply allows us to see some particular aspect of it uniquely as it happens to exist at a given moment in space and time. It asserts also that were we able to revisit that precise moment in space and time and reproduce all the conditions and the measurement made exactly as before, the result could well be other than what was found initially. Philosophically speaking, mandalic geometry adheres to a combined belief in both free will and determinism, but understood at a different dimensional level of geometrical reality than we generally interpret things.

(2) This hints at the pluripotentiality of subatomic particles which can take alternate forms and interact in various different manners depending upon context and with what specifically the interaction occurs. The same came be said about the chemical elements and other chemical entities in chemical reactions. Context is always of importance in determination of outcome. On the printed page this actuality is too often lost either in the confusion of breadth or in the overly narrow focus employed.

 © 2014 Martin Hauser

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A new quantum logic: Continuing investigation - 3

 

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Suppose for sake of argument we have 24 apples to be divided equally among a specified number of recipients. If the number of recipients is, for example, six, each will receive four apples. On the other hand if there are four recipients the take of each will be six apples. We saw previously that the multiplication of dimensionless numbers is commutative and it does not matter whether we multiply 6 times 4 or 4 times 6 since the product of multiplication in both cases is 24.

Here the situation is less clear. Surely it makes some difference whether we give 4 apples to each of 6 children or 6 apples to each of 4. Although the product of apples total is the same in both cases, the number of sets differs, as does the distribution of apples within the sets. Elementary mathematics tells us we can multiply sets and, like multiplying numbers, the operation is commutative. What are we missing here? I’m confused. Frankly what I think we’re missing is the purity of thought of pure mathematicians who need not be concerned with the egalitarianism required of socially competent parents distributing party favors who are not looking to deal with real-life quarreling and demands.(1)

One more important point we’ve overlooked is that we’ve now entered the realm of dimensional numbers where everything that quacks is no longer a duck and there are situations where commutativity of multiplication is no longer universally true in the way we’ve heretofore been led to believe.(2)

(to be continued)

 

(1) What to do, for instance, if there are 25 apples and 6 children or, worse still, only 23 apples and 6 irrationally greedy screaming children. For further elaboration of the purity of thought of pure mathematicians check out this quote of one of the most celebrated mathematicians of the 20th century.

(2) Despite that little mathematical devil with its overly insistent pitchfork on my left shoulder. If any of what has been stated above is incorrect would some of the math experts out there explain in simple terms we can all understand where the error is and why it is wrong. I apologize if I’ve seemed to denigrate pure mathematics in any way. Such is nowhere near my intention. Though I do believe that some mathematics is at times misappropriated by other disciplines, most importantly physics. Also it seems to me that mathematicians as a group are too little concerned with communication of mathematical content to those outside their field. On those occasions when they do attempt to explain a mathematical theorem it too often comes across more as an elaboration and reiteration than an actual explanation. Yes, I know they have a very specialized language. But so do scientists, artists, and composers yet somehow they all manage to explain their respective discipline pretty darn well and they usually welcome the opportunity to do so. The caveat here is that there are always exceptions to any observation.

© 2014 Martin Hauser

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A new quantum logic: Continuing investigation - 2

 

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We recently embarked on an expedition of mathematical discovery. To be honest I’m not entirely sure where this will lead. We are currently exploring the full nature of the commutative property of the operation of multiplication which involves a lot more than our teachers let on in school. Or perhaps they didn’t know themselves, just acted as though they did. Perhaps they were themselves confused by the conflation lurking in the traditional presentation of the subject in Western mathematics. That is a dragon worthy of our most valiant confrontation.(1)

We’ve agreed that dimensionless “pure” numbers in a binary multiplication operation are fully commutative. We are now addressing the crucial question whether dimensional numbers and sets are also. The concept of dimensional numbers can be approached in a variety of different ways. We will be looking first at the way sets invoke this powerful aspect of numbers. The sets we will examine initially introduce 2-dimensional numbers which can be represented easily in any geometric coordinate system with mutually perpendicular axes such as those used in Cartesian geometry and mandalic geometry.

We will allow in the examples to follow that the value along the vertical y-axis represents number of sets while the value along the horizontal x-axis the number of members in each set. The first of these will be considered the multiplier, the second multiplicand, in the operation of binary multiplication. The product of the two specifies the total number of members in all the sets combined.(2)

(continued here)

 Image: St. George and the Dragon [detail] (public domain) [Source]

 

(1) Implicit uses of the commutative property go back to ancient times. Formal uses arose in the late 18th and early 19th centuries, when mathematicians began to work on a theory of functions. For an all too brief description of the history and etymology of the commutative property see here.

(2) Note that before we even begin, we are assigning distinctive names to the elements that participate in the binary operation of multiplication as mathematics instructs we do. This surely hints at the gnawing thought that we do in fact recognize an abiding difference residing within the two and in their relationships to one another. This in spite of anything that traditional mathematics whispers in our other ear about commutativity of elements of multiplication. If either of the two can be first in the operation why is it so important to bestow upon them specific labels by which to distinguish them. We can all tell which is first and which second in any interaction. Why insist on names? We can anticipate that a well-designed search might reveal a concealed conflationary deception lurking in the background here trying its best to not reveal itself in the darkness where it resides. Here be dragons.

© 2014 Martin Hauser 

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A new quantum logic: The investigation continues

 

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In the previous post we showed that multiplication is commutative and we gave the example seen above as our poster child for the commutative property of multiplication. Now we will attempt to disprove the universality of the commutative property of multiplication.

Multiplication is not always commutative. Whether it is or not depends on certain specifications. The 1-dimensional multiplication that we described before is always commutative. But what about the multiplication of higher dimensional numbers and the multiplication of sets. Let’s delve into these matters further.

Say we have 6 apples. Taken four times that would be 24 apples as we have seen. But  suppose we wish for some reason to place those 6 apples in sets of 2, or 3, or 6. And suppose we are as much concerned about the sets and the number of sets as about the apples themselves, either individually or in total. Now we have a completely different circumstance to deal with, a different specification of numbers and their possible arrangements. We can anticipate this will have some effect on the operation of multiplication and possibly on the property of commutation as well.

Now 6 sets times 4 equals 24 sets, and to be sure 4 sets times 6 still equals 24 sets. But do they really and without exception equal one another? Is there then no difference between these two arrangements of sets? Think on it. The two sets belong to different categories or species of pattern. They may contain the same number of apples in toto but they are not identical because the patterning of apples differs between the two sets. This truth is covered up by an error of conflation and the job of concealment isn’t even done all that well. It’s like a hasty and very poor paint job. Still it tends to confound us.(1)

In the next post we will begin to look more closely at the commutativity of sets. Geometry knows the truth and will reliably lead the way. Forward and onward. Now we are tracking dimensional numbers. We are hot on the trail.

(continued here)

 

(1) What unfortunately tends at times to get lost here is the pattern of distribution, a rather important aspect of the entirety under consideration.

© 2014 Martin Hauser 

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A new quantum logic: The investigation begins

 

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As a logical plan of attack we will begin with what appears to be the easiest route of access and progress sequentially from there to the more difficult. We begin then with garden variety grade school multiplication. I don’t know how core curriculum develops this but back in the day we were taught multiplication as a kind of variation of addition. The most important thing to note here is that this form of multiplication deals exclusively with scalar magnitudes. No signs, directions, or vectors. And it involves only dimensionless quantities. These are quantities lacking physical dimension. They are therefore “pure” numbers, and as such always have a dimension of 1. What could be easier?

So multiplying, say, 4 by 6 is no different than multiplying 6 by 4 since both equations yield 24 as a result. This example is like the poster child for commutability. If you take 6 and add 6 more to it 3 times you get 24. And if you take 4 and add 4 more to it 5 times you get 24. I can see a pattern forming here. Ordinary multiplication is commutative just because ordinary addition is commutative. Case closed. But not quite yet. We still need to address how vectors and dimensions relate to multiplication and also what happens when you throw geometric reflection through a point into the mix. We have a way to go still before we can rest our case.

Next we will look at how set theory alters the picture and introduce the concepts of dimensional numbers and dimensional multiplication. This stew needs more spices in the mix to make it an honest dish.

(continued here)

© 2014 Martin Hauser