Re: CRITICAL TORUS !!!


Spoonfed wrote:
Timothy Golden BandTechnology.com wrote:
Gene Ward Smith wrote:
Timothy Golden BandTechnology.com wrote:

I disagree with 1) .
The n-signed numbers are n-1 dimensional.
This is embodied by the identity law. It requires that equal amounts in
each sign direction yields zero. The general law can be stated as:
Sum over s from 1 to n of ( s x ) = 0
where s is a sign and x is a magnitude. In P3 this yields:
- x + x * x = 0.

OK; I was taking my ideas from your postings, which gave a
multiplication law. This adds another wrinkle to the definition, which
removes the value from the idempotent
(r[0] + r[1] + ... +r[n-1])/n from consideration. You now end up with
(n-1)/2 copies of the complex numbers when n is odd, and (n-2)/2
copies, plus one copy of the reals, when n is even. You can describe
this as R[Cn]/(e0), where e0 = r[0] + ... + r[n-1].

It's general and begets the geometry of the polysigned numbers. It is
an application of symmetry that allows the generalization of sign.

I'm not sure what you are saying, but this allows you to write every
polysign in terms of positive reals only.

The
simplex coordinate system and its nonorthogonal geometry is directly
implied by this simple equation, as is the n-1 dimensional consequence.
You have either obfuscated or obliterated this law in your rendition.
I wish you would comment on this since it has come up twice now with
you and has still not been addressed. The other time was a P3 product
such as:
( 1, 1, 1 )( 2, 3, 1 ) = 0
which is indeed zero but it is consistently zero because the first term
is zero by definition.

If you are hoping to have rid yourself of zero divisors by this means,
in general you haven't. The 3-polysigns are now the complex numbers,
and have no zero divisors, but the higher dimensions do have zero
divisors.

The arithmetic nature of the system is no different than the real
numbers which are the same as P2. P3 is 2D and is exactly equivalent to
the complex numbers under product and sum.

Exactly. P2 is R, P3 is C, but for n>3 Pn is not a field.

I would like to understand why the information is not passing clearly.

Mathematicians speak mathematise.

I'm afraid that It's the old real number mantra creeping in. Still even
if you insist on using the reals the system will work out. But the
ambiguity of such a construction is questionable. You are unwilling to
grant the notion of three signs aren't you?

I'm willing to grant it if you can define it in a way which makes
sense, but so far I haven't seen a definition.
So, in order to convince you I'll need to understand the terms under
which the definition will be acceptable. Do you accept:
http://en.wikipedia.org/wiki/Magnitude_%28mathematics%29
and
http://en.wikipedia.org/wiki/Quantity
as legitimate and fundamental references?

If there is the possibility that a little effort might make the article
make sense to you, use Wikipedia. If 25% or more of the words on the
page are unfamiliar to you, you might try looking elsewhere.

As I look up scalar it is a word with many definitions and I should
avoid its use in deference to magnitude, which according to the above
definition is not dependent upon the real numbers.

If you are talking about an end quantity that you could easily slap a
real-valued unit on (force, distance, momentum, length, time (?),
etc...) you can call it a magnitude. You can call it a scalar whether
you can slap a unit on it or not.

Aren't the real numbers simply elements with a choice of two signs and
one magnitude? If not then the mnemonic sign should be optional and
another representation more fundamental should be available that would
be preferable.

The word "should" is not a mathematical term. You have "givens" and
you have "therefores." You can question the givens but generally not
the therefores. If you're representing anything in reference to zero,
I don't see any "choice" except which way to define as positive, and
"mnemonic" means a method to help you remember something, so I think
you may be misusing the term. Perhaps you mean convention.

<Snipping some dubious points you've made more convincingly elsewhere.>

I suppose another way of looking at the quagmire of the polysign
definition is that the current definitions may indeed make it very
difficult to build three-signed numbers and that helps explain why
noone has done it before. Should mathematics be open to new
constructions? Must all work be derived from past work?

The book is written. One must preserve the book. Math as religion.

-Tim

Ehhh, no, I don't think that's what's going on. Mathematics is more of
a language. As such you can say just about anything you want with it,
except something ambiguous. The reason ambiguities come up is usually
because people have created different notations for the same thing, or
the same notation for something completely different. Mathematics is
open to new constructions, but whenever possible, you should maintain
the traditional conventions, terminology, and notation. This isn't
because one notation is more "true" than the other, but because the two
notations are like foreign languages.

I think with some modification, Gene's mathematical description could
serve as a template for a very tight definition and summary of
properties of the polysigned numbers. On your website, something like
this should go right up front, where you say what you KNOW about them.
Then applications and possible applications can be covered.

I'm trying to be patient and play by the rules, but when I look at real
number definitions and see six of them and not one treats magnitude as
a fundamental component then I am still stuck until this aspect gets
addressed. I have built a system that allows n-signed numbers. By
definition it should not be built from the two-signed numbers, which
are a member of the family. Now, when I look at the wiki on magnitude
mentioned above I realize that magnitude is not tied to the real
numbers by definition. That is just the most common usage of them.
Let's face it, distance comes before the real numbers come. That's
mathematical evolution. You can define a distance for the real numbers,
but that does not make the distance concept restricted to the real
numbers. Anyone with a piece of string will be able to demonstrate
distance without the use of the real numbers. This is embodied by that
wiki page. So I construct the polysign numbers from magnitude and sign
as two entities bonded together.
The real numbers are a member and therefor they can be defined this
way. Big deal. Now there are seven definitions instead of six. Who
knows how many definitions there actually are out there. Do they really
matter? Don't we understand what they are? Where is the conflict? If we
make a mistake that mistake is detectable. Where is my mistake? Two
signs and a magnitude compose the elementary real number. What is the
problem?

Perhaps the problem is that people accept that phrase 'real' as
literal. The 'real' numbers have broken symmetry. That doesn't seem
very real, yet we still represent space with them. Even though the
space does not have broken symmetry. I'm just pushing it along further
and out comes complex numbers and spacetime. So I am not creating any
more conflict than is already present. Perhaps the solution is to
rename reality. Since we have such a dim view of it currently perhaps
we can leave the mathematicians with their two pronged nipple and move
on to singality. It should be a reserved name which disallows naming of
mathematical spaces after it so that the rule makers cannot contaminate
it. Whatever it is we are subject to it regardless of our beliefs. That
should not be mistaken for a simple little number system that people
have gotten carried away with to the point of mistaking it for reality.

Wow, that's a rant!
I put a link to the P4 product analysis over on:
http://groups.google.com/group/sci.math/msg/1972adafbe1158b9

-Tim

.

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