Re: Is magnitude more fundamental than the real numbers?


*** T. Winter wrote:
In article <1154979553.224582.269590@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> "Timothy Golden BandTechnology.com" <tttpppggg@xxxxxxxxx> writes:
> *** T. Winter wrote:
...
> > Well, that is already covered in algebra. If a is a zero-divisor in
> > a ring, so is p.a for all p in that ring. However, the set of zero-
> > divisors is not necessarily 1-dimensional. For instance in P4, all of
> > (1, 0, 1, 0), (1, 1, 0, 0) and (0, 0, 1, 1) are zero-divisors, and they
> > do not span a one-dimensional subset.
...
> OK, now I'm catching on. Are you willing to exclude the identity axis?
> I'm fairly sure that I can get this arbitrary division to work in P4
> for anything other than points on the identity axis, which is a line
> passing from +1#1 through the origin and onward through -1*1.

Nope. It will not work with either #1+1, #1*1 and +1-1. Because they
are all three zero-divisors.

Could you please explain the #1*1 instance?
I don't understand why you have that value in this list.
I understand that
( # 1 + 1 ) ( # 1 * 1 ) = 0 .
But this is an artifact of the general behavior that any value
multiplied by #1+1 will land on the axis. It is equally lacking
sensibility whether the resultant is is zero or nonzero. So this in
effect begs that we take the axis as representing something like zero.
It essentially destroys a value, leaving the resultant always upon the
axis just as a standard zero multiplication yields zero thereby
destriying any original image. It is true that there is a remnant
two-signed value that remains but informationally the notion of the
higher sign space is destroyed by multiplying any object by any axis
value. Other values retain some semblance of their original image.
Their resultant is a rotation and scaling that is directionally
dependent, but which retains information (unlike the axis values). If
anything the number of results of the quotient rises. If we look at
z ( # 1 + 1 ) = c
we can automatically state the the constant c will be somewhere on the
axis. Now we seek values z that satisfy this relationship so that
z = c / ( # 1 + 1 ) .
We find many results. For instance if c is zero z can be any scaled
version of
( 1, 1, 0, 0 ), ( 0, 1, 1, 0 ), ( 0, 0, 1, 1 ), ( 1, 0, 0, 1 )
When c is nonzero we see that it must still be on the axis.
The restriction on c seems to be the sticky point. You'd like to put
anything over there, but the product already states that this is not
possible. So we can sit on different sides and point our fingers at the
other side, but progress will not be made until some convincing new
argument arises. To impose the old math rules on this new construction
may not be appropriate. The identity axis is a new beast. The old cage
will not necessarily hold it in.

-Tim

--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/

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