Le mardi 15 mars 2022 à 18:17:18 UTC+1, timba...@gmail.com a écrit :
> Another quick thought and it is just my opinion, but you are roaming back and forth between (h,i,j) in R and R1 + R2i + R3j, and this you may be consistent at, but may be annoying particularly to people of the quaternion variety, which is likely a large part of your readership future or present. I do see the amount of work you have into your document and I wouldn't rush it in, but rather think it over and try to optimize notation. Possibly you are correct to stay where you are. In polysign the ordered form does work though the values are not signed at all. So a value like (1.2,2.3,3.4) is unambiguously a value in P3 which is equivalent to -1.2+2.3*3.4, or a value *1.2-2.3+3.4, depending on where the identity product goes. These sorts of notational difficulties should be repeatedly addressed consistently so that confusion cannot arise. I don't need the ordered form very often and there is actually a need of a zero sign '@' in polysign so that the identity sign does not shift around, at which point the identity position is (1,0,0,...) as in:
> (1,0,0,0,...) z = z
> or:
> ( @ 1 ) z = z
> In P2 this zero sign '@' is equal to '+' with two strokes to draw the sign and modulo two these are the same thing yet in P3 the '*' sign with three strokes to draw it is the same as the zero sign. If polysign were ever seriously accepted this zero sign would need to be introduced, yet it can coexist with the traditional real value's usage of '+', which is really merely coincidence. And of course our usage of '+' to mean summation has to be carefully dealt with in polysign; the '@' symbol is unambiguously the sum in Pn using what is left of traditional notation.
>
> This idea that notation matters: to some they will cast it off, yet our usage of notation with a sum requiring a symbol yet the product requiring mere juxtaposition is a matter of convention only. At some level that we communicate here in a singular series of characters could be an impediment to such things as general dimensional notions as we are working on. The Cartesian product as well proves to be problematic. Functional analysis relies heavily on a Cartesian product to define sum and product within the modern math curriculum. Also these are defined as binary operators whereas the n-ary form would be more general. Strangely enough in the n-ary form we see the singleton show its face as well. Translating the usual binary operator into n-ary would suggest that a sum of three values is a three dimensional problem in say RxRxR for real values. I assure you that this is not the case, and that even the binary form is a fraud. Closure as a mathematical concept requires just one set to work in. For instance if I offer you a sum of a and b yielding c where is the need of a Cartesian product?
> a+b=c ; a in R, b in R, c in R.
> It's one set only.
>
> Ooh, it seems somebody has been at this page:
> https://en.wikipedia.org/wiki/Ring_(mathematics)
> fixing it up to hide this problem.
>
> Still, here we see the old problem:
> https://en.wikipedia.org/wiki/Binary_operation
>
> and the earlier page does reference the binary operator still. I know these concepts are subtle, but if there is room for reinterpretation at some fundamental level then possibly the upset will unshuffle the deck, so to speak. The accumulation in mathematics is extremely annoying and I certainly prefer to distance myself from it as you do, though we can't help but bump our heads straight back into it. Possibly the upside is that it is a tolerant place, and academia needs room enough for so many PhDs now that this tolerance could be obscenely permissive. On the downside claims of perfection within mathematics are definitely false. Escapism is more the truth of the matter. Escape into your specialty mostly with no regard for physical correspondence. For those who want physics to be born this cannot be good. Still as individuals room for everybody is established.
>
> I don't think this tension will go away until we have a TOE or some unification and even then the idea that such a theory will carry parallel interpretations is entirely plausible. So a necessary tension between mimicry and freedom somehow will lead us into a progression, but if an invalid assumption lays beneath our works than all are flawed. I do see the Cartesian product as a candidate for this analysis. Since when are you allowed two copies of the real value? since the Cartesian product, really. So then what are two independent systems doing tied together at zero and perfectly perpendicular to each other? Is this really a natural occurrence? And here polysign do form an entirely different means of achieving the general dimensional situation. Indeed, the word 'dimension' as tied to the real line has to be carefully noted as old language. We can still use it, but the nonorthogonal coordinate systems that polysign natively develops are based on a balance amongst signs.

“Another quick thought and it is just my opinion, but you are roaming back and forth between (h,i,j) in R and R1 + R2i + R3j, and this you may be consistent at, but may be annoying particularly to people of the quaternion variety, which is likely a large part of your readership future or present. I do see the amount of work you have into your document and I wouldn't rush it in, but rather think it over and try to optimize notation. Possibly you are correct to stay where you are. ”

You are always sharp in your thinking. Indeed, it is bad to be annoying and go back and forth when presenting my work. This means that I have not well adapted my thinking to the way the readers think. After reading your comment, I have written an explanation for you and the others to make clear the main idea and concept of my multidimensional complex system, which is in fact based on multiplication rather than on addition. The text is here for download.
https://drive.google.com/file/d/15ATCrPo8c9PCXVbWZDOCDcdPCITByhuq/view?usp=sharing

The main idea is to represent a 3D vector as the product of two 2D complexes. For example, A is a 2D complex in the horizontal plane, B a complex in the vertical plane. Then, the 3D complex that is defined in the 3D space with the horizontal and vertical planes is the product AB.
A=e^(i*theta), B=e^(j*psi), then, AB=e^(i*theta+j*psi)

This way, 3D complex can be multiplied. In the same way, nD complex is the product
e^(a1) e^(a2) e^(a3)… e^(an).
Such construction makes the nD complex number apt to be multiplied.

As for the vector in space like R1 + R2i + R3j, it is an expression for AB=e^(i*theta+j*psi). Because the angular form is suitable for multiplication and the vector form for addition, we have no choice when doing multiplication and addition, but to use the appropriate form.

The angular form and the vector form represent the same and unique point in space. I have put the clear explanation in the link above.

“I understand that your math comes out of the angular form and I've sort of been pushing on the vector form. At some level the ultimate form is the single variable form z so that we can simply discuss z1z2 for instance as the product and z1+z2 as the sum. We don't generally expect any surprises in the sum but the product always seems to be of interest. For instance division as the reverse operator of the product and its mechanics can get difficult.”

In fact, a geometrical point in space is one thing and a mathematical representation or expression is another. e^(i*theta) represents a point say A, and cos(theta)+ sin(theta)i represents a point B. It happens that A=B, but not necessarily always the case. This is a construction by human, not a natural object like rock or tree.

As e^(i*theta) and cos(theta)+ sin(theta)i are not the same algebraic thing, they do not necessarily bear the same operation, such as z1z2 or z1+z2. So, z3(z1+z2) is not the same thing as z3z1+ z3z2. However, in geometry, they could represent the same point in space because human has defined so.

“This certainly is true of polysign. Only just recently did division get an algorithm, and it is not as if it is easy. The old familiar behavior:
| z1 z2 | = | z1 | | z2 |
does not hold in polysign and there are nonzero products yielding zero such as:
( - 1 + 1 )( + 1 # 1 ) = 0 [P4]”

My system of multidimensional complex deals well division because
A=e^(i*theta), B=e^(j*psi), A/B= e^(i*theta) e^(-j*psi)

“Supposedly Dedekind proved that associative ( z1(z2z3) = (z1z2)z3 ) systems will always portray images of R and C as RxC (Cartesian product), CxCxR, CxCxC, etc., these being higher dimensional forms.”
My system of multidimensional complex is also well associative for multiplication because
e^(a1) e^(a2) e^(a3)=[ e^(a1) e^(a2) ] e^(a3)= e^(a1) [e^(a2) e^(a3) ]

“It turns out that the interpretation of the continuum as if resolved by the rational value is problematic according to my own analysis.”
It seems that you are into research in continuum. Does this mean that you are interested in the definition that the line of real is a continuum and that the set of all the real numbers is the continuum? From my research, I have got the conclusion that the set of all the real numbers is not a continuum but a discrete set.

“…who cares to question a value such as
2.01
and the mathematician's version of this value versus the engineer's version..”
Because the set of all the real numbers is discrete, mathematician will agree with engineer about 2.01

“This then reflects back onto Dedekind and the interpretation of the continuum. Chasing digits is epsilon/delta theory.”
Because real numbers is discrete, chasing digits with epsilon/delta theory is a discrete operation which will never reach the continuum.