**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)**

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** (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

**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. :)

**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)**

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**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

**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. :)

**Thich Nhat Hanh** (born 1926)

Image: Thich Nhat Hanh, a Vietnamese

Buddhist monk, a poet, a scholar, and

a peace activist. [Source]

**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.

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**(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

**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. :)

**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.

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© 2014 Martin Hauser