Re: Is magnitude more fundamental than the real numbers?
- From: "Timothy Golden BandTechnology.com" <tttpppggg@xxxxxxxxx>
- Date: 8 Aug 2006 06:41:36 -0700
Sean Holman wrote:
Hopefully I haven't misunderstood any of the stuff you wrote on the polysigned numbers, and I must admit I did not read the entire site, but here are my thoughts:
Hero wrote:
Timothy Golden BandTechnology.com schrieb:of mathematics here.
There seems to be a stumbling point in the base
their definition ofMany are schooled so heavily in the reals that
more fundamental thenmagnitude relies upon the reals. If magnitude is
question:it should not be defined by the reals.
If you could please just answer the simple
numbers?
Is magnitude more fundamental than the real
numbers?
Yes.
Is direction more fundamental than the real
opposites like growing and
Direction in space, direction of counting,
shrinking ( size times three versus size times(three to the power of
(- 1) ) ),...
Is it?
I have a mathematical construction ....
http://bandtechnology.com/PolySigned/PolySigned.html
some results, as far
Tim, You are working so hard on this and You have
as my level allows me to judge, but..understand. Why don't You
Your notation makes it extremely difficult to
use four different signs for the four-signed,please?!
( My proposal: ' ( or ° , or . ), | , Y, N( | ), three point
with the background: one-point sign, two-point sign
sign (Y ) and four points (N) ).
Friendly greetings to all of You
Hero
Hi Hero.
Nice to hear from you. I'm wondering also if another
form would be more
accessible. The most obvious to me is to get away
from sign symbols and
go to i,j,k,l, etc. which is what Spoonfed likes to
do. It does make
some sense in that the product operation will work
out with the usual
cyclic behavior but actually dong arithmetic in these
and switching
from P3 to P4 let's say will require quite a lot of
mental adaptation
whereas the numerically connected sign symbols allow
an immediate
computation. I am wondering if meeting half way
between these two would
make sense so that in P4 I normally would write:
- 1.2 + 2.3 * 3.4 # 4.5
whose equivalent in unit vector notation would be
1.2 i + 2.3 j + 3.4 k + 4.5 l
but in a half-way would be:
1s1.2 + 2s2.3 + 3s3.4 + 4s4.5 .
But in either of these two forms the freedom to throw
a minus sign in
does not gel with the notion of defining sign. I
agree that there is
notational conflict but I cannot see any way out of
it. And then the
identity sign also needs to be considered as a zero
sign. We talked
about using an '@' symbol for the zero sign a long
time ago but I've
not engaged it. I see your notational mnemonic. And
then what of the
real numbers? The current notation is consistent with
them. That I
don't yet have as fifth sign doesn't seem to be a
problem yet since P4
is hard enough to grapple with. But to look at the
reals as
` 2.3 | 3.4 = | 1.1
is going to gross people out in another way. I see
conflicts but no
resolution.
Also for general values z in any n-signed domain it
is possible to
write things like:
z1 z2 + z3 = z4 z1 .
where '+' means vector summation.
It is completely valid then to write
z1 ( z2 - z4 ) + z3 = 0.
where - z4 is the inverse of z4. This all works fine.
But it should not
be confused with concrete instances where for
instance in P4
- 1.2 + 2.3
is not reducible.
Incidentally the inverse of the above expression is
+ 1.2 * 1.2 # 1.2 - 2.3 * 2.3 # 2.3
= - 2.3 + 1.2 * 3.5 # 3.5
= - 1.1 * 2.3 # 2.3 .
-Tim
You're poly-signed numbers seem to me to be an interesting algebraic construction. Perhaps you've already considered it, but the notation that jumps to my mind is the following, for example in P4:
Let: #1 = 1, -1 = z, +1 = z^2, *1 = z^3.
Indicate addition simply by +, and multiplication by juxtaposition. Then the multiplication rule simple matches up with exponent addition mod 4:
(#1)(#1) = (1)(1) = 1, (-1)(-1) = (z)(z) = z^2 = +1
This makes good sense. There will then be the confusion that since you
are using an
a + bz + czz
form that the factors need to be unsigned. Then the real numbers will
be represented by
a + bz
which someone will still complain about. To stay in that form for
concrete instances may not be as sensible, but I do agree that this
vector form is fine for the mechanics. In effect we are carrying a
natural number around with a magnitude and any notation that connotes
this will work. We also have to specify the domain somewhere (e.g.
announce P4 in your construction above). I don't see any one perfect
notation. Making a break with the real signs may help. I'm happy that
the sign symbollism will allow a less experienced person to do this
math. The higher forms take more education don't they? Even the vector
form
(P4: 2.3, 4.1, 3.5, 0 )
works fine so in effect all of these methods are available and work
fine. They are just means of organizing and nothing more. The sign
mnemonic -+*# will be the most accessible to the least educated. Others
should have no problem seeing these other forms. Abstracting to that
level may allow you the comfort of not seeing the construction as sign.
That may be a good thing, but for instance the development of the
complex numbers from P2 in the traditional way would take on a form of
( a + bz) + (c+ dz)i
and that would be a weak abstraction.
Another comment, and maybe you've addressed this as well, but if you want to rigorously define the polysigned numbers, it is not sufficient to simply give the multiplication rule and addition. For example again with P4, if you simply take the definition as the multiplication rule you give, and the relation +1-1*1#1 = 0, it is possible that either #1+1 = 0, or #1+1 does not equal zero. Note that if #1+1 = 0, then you have the complex numbers (#1 = 1, -1 = i, etc).
I imagine someone may have pointed this out, but there is more to the reals than the +/- structure, which makes your question about whether the reals or magnitudes are more fundamental a little strange. You cannot simply say magnitudes are thought of as "size", and then the real numbers are P2. There's more to it then that. You need to construct "magnitudes" to have appropriate algebraic and analytic properties first if you want to do it that way, and in so doing, you're essentially just reconstructing the non-negative reals.
Yeah, this seems to be the standard mathematician perspective. My
simplest complaint is that it would be a contradiction to define the
reals from the non-negative reals wouldn't it?
Since the polysign rules
Sum(sx)=0
(s1 x1)(s2 x2) = (s1+s2)x1x2
cover some of the properties of the reals what is left?
I see magnitude, you see 'non-negative reals'
But I now complain that it is a conflict to define the reals from the
non-negative reals.
At some level we are quibbling over such a fundamental concept that I
am happy to dismiss it. Those who are steeped in a sort of law that
allows a fundamental concept to be defined by a more complicated one
cause me to defend the polysign construction this way. Suppose we
consider definition to be a breakdown of a construct into independent
parts? These parts should not be redundant. Otherwise they should have
been broken out a different way. The parts should be independent
concepts that when joined together form a consequential structure. Now,
knowing of the polysign construction when we go back down the ladder of
the construction of the reals to the point that two signs are engaged
to yield the integers one might argue that the polysign is the general
choice here. This effectively reiterates the natural numbers onto
themselves in a sort of
n m
format, where the first portion takes an entirely different meaning
than the second part.
What would this complicated animal be? The polysign lattice has some
interesting consequences:
http://bandtechnology.com/PolySigned/Lattice/Lattice.html
In effect
d( a, b ) = (n-1) d( b, a )
for 'adjacent' positions when the distance is taken by walking the
lattice.
Yet on a continuum this effect seems to disappear.
Does this invalidate anything? If a definition's parts reiterate a
basic concept(here the natural numbers) should it care that it takes
those parts as orthogonal when they are equivalent?
I don't really know. I am beginning to see the definition a continuum
via the natural numbers as a false construct. All of the quibbling that
goes on between the two to traverse up to the continuum is a strong
sign of the weak connection. Natural numbers can be formed from a
continuum almost immediately, and without the unity problem that is
mentioned elsewhere in this thread. This then becomes a more
satisfactory notion of physical space. But still the existence of a
product that makes no physical sense remains.
-Tim
.
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