Is magnitude more fundamental than the real numbers?
Timothy Golden BandTechnology.com
There seems to be a stumbling point in the base of mathematics here.
Many are schooled so heavily in the reals that their definition of
magnitude relies upon the reals. If magnitude is more fundamental then
it should not be defined by the reals. I have a mathematical
construction reliant upon magnitude that defines the reals (P2) and
complex numbers(P3) . They are defined via the same rules but with a
value n = 3 rather than n = 2. This is the only difference between the
reals and the complex numbers by this construction and so it is
superior to the usual definitions of either. More sets of interesting
number come out since n can be any natural number.
http://bandtechnology.com/PolySigned/PolySigned.html
This explains my motivation for attempting this discussion.
If you could please just answer the simple question:
Is magnitude more fundamental than the real numbers?
All responses will be appreciated.
-Tim
Gene Ward Smith
Timothy Golden BandTechnology.com wrote:
> Hi. I answer the above in the affirmative. I'm curious what you think.
I think magnitude basically *is* the real numbers, since the absolute
value takes values in the non-negative reals for not only the reals,
but many other fields besides.
> I have a mathematical
> construction reliant upon magnitude that defines the reals (P2) and
> complex numbers(P3) .
I've pointed out several times that you do not have such a
construction. I'll repeat it: you have NOT constructed the reals. This
is because your definition requires that the reals have already been
constructed.
Timothy Golden BandTechnology.com
Gene Ward Smith wrote:
> Timothy Golden BandTechnology.com wrote:
>
> > Hi. I answer the above in the affirmative. I'm curious what you think.
>
> I think magnitude basically *is* the real numbers, since the absolute
> value takes values in the non-negative reals for not only the reals,
> but many other fields besides.
Magnitudes certainly are not real numbers. They have no operators.
They have no sign.
-Tim
Gene Ward Smith
Timothy Golden BandTechnology.com wrote:
> Magnitudes certainly are not real numbers. They have no operators.
> They have no sign.
Magnitudes can be equated with he nonnegative real numbers, closed
under addition and multiplication, which certainly are operators. The
positive reals are closed under inverses and square roots also, and are
a group under multiplication.
Timothy Golden BandTechnology.com
Magnitudes are not real numbers. They are much simpler than real
numbers.
from http://en.wikipedia.org/wiki/Magnitude_%28mathematics%29 :
The magnitude of a mathematical object is its size:
a property by which it can be larger or smaller than other objects
of the same kind; in technical terms, an ordering of the class of
objects
to which it belongs.
We can define the reals in terms of magnitude and we can define
magnitude in terms of the reals. Which construction is more appropriate
is a matter of putting the simpler concept beneath the more complicated
concept. Magnitude is the less complicated of the two.
Therefor defining magnitude from the reals is less meaningful than
defining the reals from magnitude. The ambiguity is similar to defining
the real numbers in terms of the real numbers. It is ambiguous to
define a simpler concept using a more complicated concept or even an
equally complicated concept. Definitions require that the simpler
concepts be contained by the more complicated, as a branched structure.
Down at the bottom of these definitions are axioms. Can magnitude be
axiomatic? The continuum concept is in there. Magnitude need not go
into all of the number theory. It is primitive. Whether multiplication
and summation can be done with them I feel flexible on. These
operations are well behaved.
Really to do the general product
( s1 x1 )( s2 x2 ) = ( s1 + s2 )( x1 x2 )
where s are natural signs and x are magnitudes the x1x2 does boil down
to multiplying two magnitudes, though the operation as a whole relies
on signs as well.
-Tim
Gene Ward Smith
Timothy Golden BandTechnology.com wrote:
> Magnitudes are not real numbers. They are much simpler than real
> numbers.
>
> from http://en.wikipedia.org/wiki/Magnitude_%28mathematics%29 :
>
> The magnitude of a mathematical object is its size:
> a property by which it can be larger or smaller than other objects
> of the same kind; in technical terms, an ordering of the class of
> objects
> to which it belongs.
Which doesn't contradict anything I said.
More quotes from the same article:
The magnitude of a real number is usually called the absolute value or
modulus. It is written | x |, and is defined by:
| x | = x, if x = 0
| x | = -x, if x < 0
and
Similarly, the magnitude of a complex number, called the modulus, gives
the distance from zero in the Argand diagram. The formula for the
modulus is the same as that for Pythagoras' theorem.
\left| z \right| = \sqrt{\Re(z)^2 + \Im(z)^2 }
where \Re \mbox{ and } \Im are the Real part and Imaginary part of z.
What your Wikipedia article is saying is that in these cases, which are
typical, a magnitude is a non-negative real number.
> We can define the reals in terms of magnitude and we can define
> magnitude in terms of the reals.
Indeed we can, and both constructions are so closely related it makes
no sense to claim that magnitude is easier. They are pretty much the
same thing.
If we were in ancient Alexandria, it would have made a lot of sense to
define the real numbers in terms of magnitude, because people had
already defined magnitude. These days, we are not in that position, and
people normally prefer to proced more directly to the real numbers,
which are a field.
However, people do sometimes do things your way--notably, Edmund
Landau's famous "Foundations of Analysis" starts from positive
integers, defines positive rationals, and then positive reals, or
magnitudes, *before* introducing zero or negative numbers. This has the
advantage that you can't divide by zero in your definitions because you
haven't defined zero yet.
Which construction is more appropriate
> is a matter of putting the simpler concept beneath the more complicated
> concept. Magnitude is the less complicated of the two.
To you. I think they are more or less the same, and reals in some
respects are less complicated, magnitude in other respects.
> Therefor defining magnitude from the reals is less meaningful than
> defining the reals from magnitude.
OK. But then you run into a problem: you haven't defined magnitude
either. Since you haven't done that, you can't very well claim to have
constructed the reals.
> Down at the bottom of these definitions are axioms. Can magnitude be
> axiomatic?
Absolutely.
The continuum concept is in there. Magnitude need not go
> into all of the number theory. It is primitive.
I'd be careful making that claim. Do you regard it as dispensible to
the concept of magnitude that any positive integer defines a magnitude?
What about the claim that for any given magnitude, there is always some
integer whose magnitude is greater?
Timothy Golden BandTechnology.com
Gene Ward Smith wrote:
> Timothy Golden BandTechnology.com wrote:
>
> > Magnitudes are not real numbers. They are much simpler than real
> > numbers.
> >
> > from http://en.wikipedia.org/wiki/Magnitude_%28mathematics%29 :
> >
> > The magnitude of a mathematical object is its size:
> > a property by which it can be larger or smaller than other objects
> > of the same kind; in technical terms, an ordering of the class of
> > objects
> > to which it belongs.
>
> Which doesn't contradict anything I said.
>
> More quotes from the same article:
>
> The magnitude of a real number is usually called the absolute value or
> modulus. It is written | x |, and is defined by:
>
> | x | = x, if x = 0
> | x | = -x, if x < 0
>
> and
>
> Similarly, the magnitude of a complex number, called the modulus, gives
> the distance from zero in the Argand diagram. The formula for the
> modulus is the same as that for Pythagoras' theorem.
>
> \left| z \right| = \sqrt{\Re(z)^2 + \Im(z)^2 }
>
> where \Re \mbox{ and } \Im are the Real part and Imaginary part of z.
>
The fact that magnitude exists in all of these systems is evidence that
it is fundamental and lies beneath them all.
It's too bad nobody else will state an opinion.
> What your Wikipedia article is saying is that in these cases, which are
> typical, a magnitude is a non-negative real number.
>
> > We can define the reals in terms of magnitude and we can define
> > magnitude in terms of the reals.
>
> Indeed we can, and both constructions are so closely related it makes
> no sense to claim that magnitude is easier. They are pretty much the
> same thing.
>
> If we were in ancient Alexandria, it would have made a lot of sense to
> define the real numbers in terms of magnitude, because people had
> already defined magnitude. These days, we are not in that position, and
> people normally prefer to proced more directly to the real numbers,
> which are a field.
That's a fine point. I can demonstrate magnitude with a hair in my
hands to a caveman, whereas the real numbers will bewilder the caveman.
The notion of a negative value is far removed from the senses, and for
that matter a positive value as well, since without a negative concept
there is no need for a positive concept and hence the phrase
'non-negative reals' is a complexity that is unnecessary.
>
> However, people do sometimes do things your way--notably, Edmund
> Landau's famous "Foundations of Analysis" starts from positive
> integers, defines positive rationals, and then positive reals, or
> magnitudes, *before* introducing zero or negative numbers. This has the
> advantage that you can't divide by zero in your definitions because you
> haven't defined zero yet.
>
Interesting. I'd like to dodge the integers for magnitude. When
actually on a continuum the probability of landing on an exact integer
is nill. They should not play a central role in the development of
magnitude.
You bring to light another interesting aspect of the magnitude that the
polysigned numbers require. Zero is inherently mentioned by the
polysign system as a cancellation of equal magnitudes. In this regard
they define zero and so you've got me wondering if this magnitude needs
to even have a lower bound. What the ramifications of that are I do not
know. You would probably know better than I. But at that point I
suppose I'm defining magnitude for the polysign system, rather than
using an existing definition.
> Which construction is more appropriate
> > is a matter of putting the simpler concept beneath the more complicated
> > concept. Magnitude is the less complicated of the two.
>
> To you. I think they are more or less the same, and reals in some
> respects are less complicated, magnitude in other respects.
So you are letting mathematics be a matter of personal opinion?
Within the realm of a given construction a contradiction should arise
if the construction is false. At this level these are open problems.
Could magnitude be so simple that it does not deserve the scrutiny that
we are giving it?
Anyhow you must grant that I contruct the polysign numbers from
magnitude, which is not the real numbers and I have made that choice.
So while you see little difference I suppose that I should state the
differences in the context of the polysign construction.
The identity law states that for P2 (the reals)
- x + x = 0
where x is a magnitude. Now, any properties of the reals that are built
from this law do not need to apply to x since they are defined atop x.
Next we have
The general P2 product says that for magnitudes a, b, c, d
( - a + b )( - c + d ) = + ac - ad - bc + bd .
So any property of the reals that is based on these sign mechanics need
not be included in the behavior of magnitude.
Now, having deleted these two concepts from what you believe to be
interchangeable concepts can't you see the conlict? The magnitude basis
does not contain these properties. They can built on top of it by these
laws and extended to three signs to yield the complex numbers. To claim
that the polysign system is built from the real numbers is false. They
build the real numbers via the above instance P2. It may be that I need
to define this thing that I am calling magnitude, so that it is devoid
of these properties. I thought it already was.
> > Therefor defining magnitude from the reals is less meaningful than
> > defining the reals from magnitude.
>
> OK. But then you run into a problem: you haven't defined magnitude
> either. Since you haven't done that, you can't very well claim to have
> constructed the reals.
Right. I'm starting to understand that. As the natural numbers produce
sign so magnitude produces the continuum in its most primitive form.
How much will it take for a true mathematician to be happy with my
construction?
>
> > Down at the bottom of these definitions are axioms. Can magnitude be
> > axiomatic?
>
> Absolutely.
>
> The continuum concept is in there. Magnitude need not go
> > into all of the number theory. It is primitive.
>
> I'd be careful making that claim. Do you regard it as dispensible to
> the concept of magnitude that any positive integer defines a magnitude?
> What about the claim that for any given magnitude, there is always some
> integer whose magnitude is greater?
Certainly the notion of integer is a completely different notion than
magnitude in my thinking. To superpose the two should not be necessary.
I suppose that this leaves an interesting problem about unity and what
composes a unity multiplication. If I say
There exists a unique magnitude U
such that any other magnitude A
when multiplied by U
gives A.
There is still no notion of integer. The notion of twice the value
perhaps need not arise, therefor no integer concept. This is awfully
abstract. Now you've got me wondering if linearity even needs to be
imposed. When adding a magnitude to itself A+A must we arrive at 2A ?
Certainly seems like the most consistent approach. So when we add U + U
we'd get twice unity, but that isn't much different than twice A
either. In effect the value would remain 2U without ever being resolved
to 2. The notion of choosing Unity is a fascinating one, for in the
physical world we do not observe integer values inherently in the
continuum. We make them up. A choice of unity leads to a scalar in many
equations. At this type of divide you and I will part ways. Nature is
what I study, not math. But still, I want to make a convincing argument
to you and I think this integer concept within a continuum is a false
precept. Instead we see that given a continuum we can derive the
integers!
-Tim
Tom
I agree that +ve reals are simpler than reals, and further, that natural numbers are simpler than +ve reals.
I guess simplicity could be defined as proximity to the real world.
you can have 2, 3 or 20 beans but not -e beans.
Anyway the interesting thought is that perhaps we have all considered the real numbers to be the fundamental numbers rather than magnitudes.
If you consider reals as a unit then this leads to the idea of orthogonal dimensions and cartesian coordinates.
If you consider magnitudes as a unit, this seems to lead to Timothy's ideas on polysigned numbers.
Problem to me with this is that the real world doesn't seem to exist in a multiple of real numbers (we have 3 space dimensions and half a time dimension as it only seems to go one way)... It could be thought of as a multiple of magnitudes though... 5 magnitudes with direction.
Whether this amounts to anything more than an interesting re-interpretation of numbers and dimensions would come down to applications I guess.
Timothy Golden BandTechnology.com
But what you are seeing as half of a dimension (time) I see as a
zero-dimensional entity. It's not that 'I' see it this way. The
polysign construction sees it this way. I just try to understand it.
Taking the construction literally dictates that one-signed numbers are
zero-dimensional. So we are still in whole dimensions if you follow
polysign strictly. It's a tough one I'll admit. But the arrow of time
is still included in P1 (the one-signed numbers), however it is not
measurable and this is what really is convincing about it. Time sits
geometrically parallel with the other dimensions without a measure.
Arithmetic can still be done, but it will always render out to zero.
You'll have to spend some time on
http://bandtechnology.com/PolySigned/OneSigned.html
To be truly convinced one has to accept the polysign construction as
fundamental.
Even without conviction it follows from the definitions and so the
construction is consistent with spacetime wether you believe it to be
spacetime or not.
Thanks so much for weighing in on the topic.
It's probably not humane to keep score but so far it's two magnitudes
to one real.
(chuckle)
-Tim
Tom
However I should say that I don't think the time being uni-directional thing is a very strong argument for polysigned numbers being more suitable than reals. Its quite a loose arguement isn't it....
for example these points:
1. can you really travel forward in time anymore than you can travel backward?
2. making spacetime as a 5-signed number would mean you could go backward in time through a combination of the other signs.. doesn't it?
3. does the nature of time really matter to a number system.. I mean you wouldn't argue a number system based on gravity attracting, so why should you based on time going one way (if it even does).
A couple of random questions about polysigned numbers...
I didn't read in detail but how have you defined your product operators? I mean, do they spring out of the equations for summation.. or have you defined products using your own rules?
Also I'm surprised that the 3-signed numbers equate to complex numbers however the 5-signed numbers don't equate to quaternions (since you say the product doesn't preserve length). Doesn't this seem a bit weird?
Tom.
Timothy Golden BandTechnology.com
Certainly they are weird in the context of ordinary math. Yet they are
not weird since they define the real numbers and the complex numbers
and others that are well behaved algebraically. The product algorithm
straight from my source code is:
for( i = 0; i < n; i++ )
{
for( j = 0; j < n; j++ )
{
k = (i+j)%n;
x[ k ] += s1.x[i] * s2.x[j];
}
}
s1 and s2 are n-signed numbers. x[i] are their magnitude components.
When n = 2 (P2) this generates the real numbers.
When n = 3 (P3) it is the complex numbers.
The sum in code is just:
for( i = 0; i < n; i++ )
{
x[i] = s1.x[i] + s2.x[i];
}
In effect it is the usual vector sum but on magnitudes.
These two operations are the polysign numbers in a nutshell.
Now about time. I don't see much degree of freedom in terms of time.
About the best that I can do is to choose when I make an event occur.
But upon making the event occur I cannot undo it as I might in the
spatial dimensions. So for instance the act of pressing a button I
believe accurately defines the degree of freedom that time permits.
Abstractly this links back to a zero-dimensional point concept.
Attempts at accessing a continuum will invariably put us using the
spatial degrees of freedom. For instance I can ride my bike slowly or
pedal harder and travel faster. Now we have contaminated the problem
with space being intertwined with time. To isolate time I think the
best I can do is this button concept which is again spatial but the
structure is simple and can be abstracted.
Doing anything convincing with time is challenging. But the one-signed
numbers don't require these intuitive troubles. They operate in the
product and sum defined above. But they do not render to anything. They
are subject to the following reduction:
- x = 0 .
This is the identity law of polysign numbers in P1. It used to be I
made exception for P1, but no longer though I still note the paradox.
This paradox exactly matches that of time.
P5 is 4D so I understand the temptation of implementing spacetime under
it. But I don't believe that will work out. The natural progression
P1, P2, P3, P4, ...
is broken beyond P3 by the law
| A || B | = | A B |
and this fundamental distance behavior is a very pretty explanation for
spacetime.
It suggests that the geometrical product is inherent and we see this
somewhat in the classical force equations for charge and mass. A bit of
massaging gets the geometric product by transforming distance:
Y = 1 / ( X + 1 )
So that a discrete charge whose geometry is
q
allows a Force relationship of
F = q1 q2 .
That's tangential and esoteric but in a nutshell that's what I'm
working on.
P5 may be worth a try but I don't think it'll produce much on its own.
Quaternions (Q's) are an entirely different construction.
P5 and Q's exhibit rotational behavior.
P5 and Q's are four-dimensional though some people claim the Q's are
3D.
General n-signed numbers obey the associative, commutative, and
distributive properties of the field requirements, including P5.
Quaternions do not; the commutative property does not work.
Algebraically the polysign numbers are very well behaved.
Yet in higher dimension spaces are ill behaved in terms of ordinary
distance.
There is a deep divide.
The polysign construction does not rely on a Cartesian product to get
dimensionality.
In this regard it is simpler than standard R^n.
That it transforms forward and backward is fortunate.
But P4 and above are effectively anisotropic. They will never succumb
to tensor manipulations.
All of this is highly suggestive for physics models.
If P1 thru P3 are spacetime does matter start at P4 and go up?
Maybe, but if the polysign numbers are true then the break will exist
naturally and should not require human intervention. So following them
around is the best hope for a clean theory built from them. That's
mostly what I do. A point particle model is looking fairly good so far.
The topology
0D + 1D + 2D ...
is the proper substrate. The product of elements in this substrate
should yield the reality we experience under the point particle model
that I am attempting. That puts us a level removed from the substrate
so that we exist in a particle product space, rather than a Cartesian
product space. As you know when a product is taken the result is not
generally in the same category as its operands. Force is a different
category than charge. That the two are linked by a second derivative is
fascinating. Down in these depths hopefully a simple and natural model
will arise. When coupled with the complexities of the high signs (P4+ )
it may find congruence with modern theory. That's quite a gamble, but
the cost of being wrong is nill.
-Tim
Tom
> straight from my source code is:
>
> for( i = 0; i < n; i++ )
> {
> for( j = 0; j < n; j++ )
> {
> k = (i+j)%n;
> x[ k ] += s1.x[i] * s2.x[j];
> }
> }
> General n-signed numbers obey the associative,
> commutative, and
> distributive properties
I'm almost convinced that the product formula is a direct result of requiring the cummulative and distributive properties to hold true.
In other words the number system may define itself by
1. multiple signs (the first representing +ve reals)
2. an add operator that obeys commutivity and associativity
3. a product operator that obeys commutivity and distributivity
I wonder if a set of constraints like this is enough to uniquely define polysign numbers...
Anyway this is getting out of my depth so I'll leave the discussion here. Interesting stuff.
Timothy Golden BandTechnology.com
There seems to be a stumbling point in the base of mathematics here.
Many are schooled so heavily in the reals that their definition of
magnitude relies upon the reals. If magnitude is more fundamental then
it should not be defined by the reals. I have a mathematical
construction reliant upon magnitude that defines the reals (P2) and
complex numbers(P3) . They are defined via the same rules but with a
between the
reals and the complex numbers by this construction and so it is
superior to the usual definitions of either. More sets of interesting
number come out since n can be any natural number.
http://bandtechnology.com/PolySigned/PolySigned.html
This explains my motivation for attempting this discussion.
If you could please just answer the simple question:
Is magnitude more fundamental than the real numbers?
All responses will be appreciated.
This is a repost in the hopes of getting some new responses.
-Tim
Timothy Golden BandTechnology.com
Timothy Golden BandTechnology.com
I answer this question in the affirmative. I'm curious what you think.
There seems to be a stumbling point in the base of mathematics here.
Many are schooled so heavily in the reals that their definition of
magnitude relies upon the reals.
If magnitude is more fundamental then it should not be defined by the
reals.
I have a mathematical construction reliant upon magnitude that defines
the reals (P2) and
complex numbers(P3) .
They are defined via the same rules but with a value n = 3 for P3
whereas n = 2 for P2.
This is the only difference between the real and complex numbers by
this construction and so it is superior to the usual definitions of
either.
More sets of interesting number come out since n can be any natural
number.
http://bandtechnology.com/PolySigned/PolySigned.html
The arithmetic product and sum obey the associative, commutative, and
distributive laws as the real and complex numbers do in any dimension.
Dave L. Renfro
> Hi. I answer this question in the affirmative. I'm curious
> what you think. There seems to be [...]
Why are you posting the same comments/questions?
As far as I can tell, post #1 (made June 30),
post #13 (made July 17), and post #14 (made July 25)
are identical. Unless you delete a post, it continues
to stay in the thread you posted it in.
-------------------------------
http://groups.google.com/group/sci.math/browse_frm/thread/20a7021eaa7f7074
Is magnitude more fundamental than the real numbers?
1 Timothy Golden BandTechnology.com Jun 30
2 Gene Ward Smith Jun 30
3 Timothy Golden BandTechnology.com Jun 30
4 Gene Ward Smith Jun 30
5 Timothy Golden BandTechnology.com Jun 30
6 Gene Ward Smith Jun 30
7 Timothy Golden BandTechnology.com Jul 2
8 Tom Jul 3
9 Timothy Golden BandTechnology.com Jul 3
10 Tom Jul 3
11 Timothy Golden BandTechnology.com Jul 3
12 Tom Jul 3
13 Timothy Golden BandTechnology.com Jul 17
14 Timothy Golden BandTechnology.com Jul 25
-------------------------------
Dave L. Renfro
Ioannis
news:1154376220.4...@b28g2000cwb.googlegroups.com...
>
> Timothy Golden wrote (in part):
>
> > Hi. I answer this question in the affirmative. I'm curious
> > what you think. There seems to be [...]
>
> Why are you posting the same comments/questions?
> As far as I can tell, post #1 (made June 30),
> post #13 (made July 17), and post #14 (made July 25)
> are identical. Unless you delete a post, it continues
> to stay in the thread you posted it in.
Dave, in addition to the above, at least one server (amongst the thousands)
that carries sci.math is misbehaving. Might be mine. I am getting articles
dated back to 21st of July.
Apparently a bad server has regenerated its articles for some reason,
consequently all the other servers are picking them up and showing them as
current. The culprits can be identified by their older dates from their
headers.
> Dave L. Renfro
--
Ioannis
Dave L. Renfro
> Dave, in addition to the above, at least one server
> (amongst the thousands) that carries sci.math is
> misbehaving. Might be mine. I am getting articles
> dated back to 21st of July.
>
> Apparently a bad server has regenerated its articles
> for some reason, consequently all the other servers
> are picking them up and showing them as current.
> The culprits can be identified by their older dates
> from their headers.
I seem to be having trouble with google, too.
In the past few minutes I've noticed that some thread
listings haven't been consistent. There will be newer
posts, posts which I've already seen, that don't show
up in the threads, and then after I "back-button"
a few times and try again, the posts will reappear
in the threads.
I thought the original poster was doing some "hit and
run" posting, and since nothing much of interest to me
has been posted today, I thought I'd play net nanny.
But maybe I targeted an innocent civilian this time.
(It wouldn't be the first time for me, either.)
Dave L. Renfro
Timothy Golden BandTechnology.com
So you critique the thread and yield no response?
I would like more responses.
On Google by reposting the thread comes up chronologically again so
that a fresh viewer may read it.
It is highly unlikely that a thread one week old will yield any fresh
responses.
I don't see this as spamming or misuse.
Misuse would be creating a fresh thread with the same topic.
Here I am just trying to keep the thread alive.
Furthermore the last line of the repost states that it is a repost.
Unfortunately I get no responses so I keep fishing.
I could see that if I was trying to mislead someone that it would be
misuse, but to seek communication is what this is about. When you look
at the thread it is completely apparent.
How else should I procede?
I am fishing for yes's and no's and until I get some I have only worms
to eat.
-Tim
Dave L. Renfro
> So you critique the thread and yield no response?
> I would like more responses.
http://bandtechnology.com/PolySigned/PolySigned.html
I thought Gene Ward Smith made some useful comments
earlier in this thread, but I'm not sure that you
understood the points he was making. At least, if
you did understand his points, your follow-up posts
didn't convince me.
My view is that you shouldn't concern yourself with
deep philosophical and foundational issues. Thousands
of extremely capable people during the last 2000+ years
have thought very hard about these issues, so if you
insist on going this route, there's quite a bit of
prior work that you need to acquaint yourself with,
unless your intellectual capability is an order of
magnitude greater than anyone else who has worked
on these issues. Even the best people study the past
masters before seriously attempting to surpass them,
if for no other reason than to learn what "surpass them"
actually means.
What you should do is investigate the mathematical
properties of the system you've developed. I see some
suggestive pictures and purported properties. Can
you give an arithmetic formulation of your number
system? This was a major concern for 19'th century
mathematicians with the real numbers, and an excellent
non-historical study of the central results of their
work can be found in Landau's book "Foundations of
Analysis". A similar "arithmetic formulation" of a
previously successful mathematical tool is surveyed in
http://www-personal.umich.edu/~jtropp/papers/Tro99-Infinitesimals.pdf
However, something I think you should realize is
that these mathematical areas first arose because
they _served_a_useful_purpose, one that no one could
deny, and only later were they put on a firm logical
and arithmetic grounding. With this in mind, the question
mathematicians are going to ask you is in what way
can your numbers be used to solve problems that can't
be solved with traditional methods, or to solve problems
that are much more difficult to solve with traditional
methods? Until you show the usefulness of your numbers,
they have little more interest to mathematicians than
linguists would have for someone making up a bunch of
meanings for random looking letter sequences such as
'xxuiepep' and 'eqqueo'. I think what you're doing is
less trivial than this (making up meanings for letter
sequences), but you still need to come up with *reasons*
why someone would be interested in your work.
Finally, I think the comments (below) I wrote to someone
recently, a non-mathematician, might be of some use
in connection with my remarks above. These comments
have to do with publishing mathematical papers, and
they might also be of use to other "amateur mathematicians"
in sci.math. I'm probably half-way one of these "amateur
mathematicians" myself, for what it's worth.
These comments pertain to nearly completed manuscripts,
by the way. The issues below often pale in significance
to being able to come up with an appropriate research
topic to investigate or a suitable problem to solve.
-------------------------------------------------------
First, are the results correct? Are my proofs sound and
the various mathematical statements correctly stated?
This part is almost always the easiest, unless one is
working on something extremely deep and difficult. Of
course, there's always the possibility that you could
have overlooked something, such as neglect all the
cases. For example, suppose we're proving something
for all real numbers by proving the result for all
positive numbers and then proving the result for all
negative numbers. [Why break it down like this? Well,
sometimes an argument that works for positive numbers
doesn't work for negative numbers, and you have to
modify it or use an altogether different approach
for negative numbers.] Such a breakdown of cases omits
the number zero. Things are rarely this simple, but
this is a good analogy for what often happens. O-K,
so you omitted zero, and either you or the referee
notice this. It's often easy to handle a special
case, so you just see if the result is true for zero.
If it is, then you include this case in your proof.
If the result is false for zero, then you restate
the theorem as holding for all *nonzero* numbers,
and after the proof you include the comment that the
result fails for zero (with some details, if it's not
easy to verify that it fails for zero). And, if you
can't prove whether it's true for zero, then state
the theorem for nonzero numbers and mention that
you've been unable to resolve the case for zero.
Second, do the results merit publication? If they're
trivial or have almost no conceivable interest, it's
unlikely to get the referee and/or editor's approval.
A couple of my papers fell into this category, although
in each case I'm sure a change to a different journal
will work (and would be a better "home" for the paper),
but I haven't gotten around to doing this yet (something
I should probably dig up sometime and look into). The
result you're publishing should have some significance
and not be something "everyone knows".
Third, how do the results fit in with the rest of
the literature on the subject? Have others made
partial attempts at solving the problem you're
addressing? This is something I'm pretty good at,
but I think for some mathematicians, especially
those who are fairly new (those who have only had
their Ph.D.'s for a few years), this can be more
difficult than either of the other issues. This is
one of the main functions of a referee, at least in
math. Sure, the referee looks over the paper to see
if the results are correct, but this rarely means
going over it with a fine-tooth comb and checking
every detail. Sure, the referee may offer some
suggestions on how the result can be improved
(maybe the referee knows how to prove the result
for zero) or how the proofs can be simplified
(i.e. maybe the referee sees a different way of
proving the result for nonzero numbers that doesn't
require two separate proofs, one for positive numbers
and one for negative numbers). But, along with these
obviously useful suggestions from a referee to the
author, the referee is also likely to point out
related results in the literature that the author
hadn't been aware of. Of course, if the referee
finds that the result has already been published
somewhere (e.g. a paper in some obscure 1913 Czech
journal that the author hadn't known about), it's very
unlikely that the referee would recommend publication.
Still, the referee might recommend publication if the
original published result is virtually unknown and
the present author's proof is much nicer in one way
or another than the original. In short, probably the
primary function of the referee is to make a judgment
about the significance of the result, supporting this
judgement, as appropriate, with how it fits in with
the rest of the literature relating to it.
-------------------------------------------------------
Dave L. Renfro
Timothy Golden BandTechnology.com
Thanks Dave.
I agree that the higher sign math needs more development.
I'm working a bit on quotients.
There is no other math that will follow the algebraic properties in any
dimension for general product and sum so the interest to mathematicians
is obvious from the start. The fact that they generalize between the
real and complex numbers and provide congruency with spacetime via a
natural progression should also be of interest.
But if magnitude is refuted as built from the reals one of these claims
has to go away. It's obvious to me that magnitude is more fundamental.
For modern mathematicians is this a problem? That is what I am trying
to address and the only way I can see is a yes/no answer. We build
mathematics from simple constructs upward to more complicated ones, not
the other way around. Yet this is problematic isn't it? Otherwise I'd
get more solid answers. The vacancy of input from others is meaningful.
How much energy to spend here is not clear to me. This is the bowels of
math, where the real numbers have six unique definitions on Wikepedia
alone. That others who have done the reading you suggest are not able
to answer clearly suggests that I am free to proceed without too much
worry. Still, this means that I am making another definition of the
real numbers based on magnitude. The value of this definition is that
the complex numbers can be arrived at very shortly thereafter, as well
as other interesting higher dimensional systems, as well as a
definition of zero-dimensional numbers. Dimension is generated via a
new approach.
I appreciate your detailed response.
-Tim
Hero
> There seems to be a stumbling point in the base of mathematics here.
> Many are schooled so heavily in the reals that their definition of
> magnitude relies upon the reals. If magnitude is more fundamental then
> it should not be defined by the reals.
>
> Is magnitude more fundamental than the real numbers?
Yes.
Is direction more fundamental than the real numbers?
Direction in space, direction of counting, opposites like growing and
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
Tim, You are working so hard on this and You have some results, as far
as my level allows me to judge, but..
Your notation makes it extremely difficult to understand. Why don't You
use four different signs for the four-signed, please?!
( My proposal: ' ( or ° , or . ), | , Y, N
with the background: one-point sign, two-point sign ( | ), three point
sign (Y ) and four points (N) ).
Friendly greetings to all of You
Hero
Aluminium Holocene Holodeck Zoroaster
"squaring the cardioid," although it must be said that
the M-set is *entirely* an artifact of the floatingpoint spec, and
its many, many implimentations in hardware & software
(it's IEEE-755, or some thing, with a more recent update);
this was confirmed when monsieur M. begged my question
about this, at a rather dull "general audience" talk
that he gave at Royce Hall, UCLA, some time ago.
others have tried this, called Quadray for a spatial/tetrahedral one,
and
it doesn't seem to offer any utility, although it's possible that
some interesting numbertheory could lurk therein (whereas,
the quadray folks were rather more utilitatrian, seeking only
to establish "Synergetics" as better than "cartesianism,"
with no really interesting result -- and
when they both largely suck).
"spacetime" is a hopelessly useless abstraction,
since it was really already highly formalized
as phase-spatialization, using hamiltonians & lagrangians;
unfortunately, it tends to give the "time travel" crowd a funny
platform,
like time is "going to go" some where, some how. well,
"there's no where, therein," thanks to a momentary lapse
by AE's teacher, Minkowski -- good N-d numbertheorist,
as far as you could go with it!
> > http://bandtechnology.com/PolySigned/PolySigned.html
> There is no other math that will follow the algebraic properties in any
> dimension for general product and sum so the interest to mathematicians
> is obvious from the start. The fact that they generalize between the
> real and complex numbers and provide congruency with spacetime via a
> natural progression should also be of interest.
>
> But if magnitude is refuted as built from the reals one of these claims
> has to go away. It's obvious to me that magnitude is more fundamental.
> For modern mathematicians is this a problem? That is what I am trying
> to address and the only way I can see is a yes/no answer. We build
> mathematics from simple constructs upward to more complicated ones, not
> the other way around. Yet this is problematic isn't it? Otherwise I'd
> get more solid answers. The vacancy of input from others is meaningful.
> How much energy to spend here is not clear to me. This is the bowels of
> math, where the real numbers have six unique definitions on Wikepedia
> alone. That others who have done the reading you suggest are not able
> to answer clearly suggests that I am free to proceed without too much
> worry. Still, this means that I am making another definition of the
> real numbers based on magnitude. The value of this definition is that
> the complex numbers can be arrived at very shortly thereafter, as well
> as other interesting higher dimensional systems, as well as a
> definition of zero-dimensional numbers. Dimension is generated via a
> new approach.
thus:
speaking of Kyoto,
there is an almost historical article on the emmissions trading schemes
of yore & today, in yesterday's NYTMagazine,
via its billionaire proponent from the junkbond biz. only thing
missing:
how many billions of dollars per year is CCX hedging, and
has it ever had any effect on, like, the price of oil?
> apt phraseology, "in sympathy," compared
> to the typical Muslim Fisikist hypothesis that
> there is no essential connection between the buildings
> on the site -- when there are very many.
>
> the simplest one is that there is a *huge* concourse
> running under the site, containing a subway, parking,
> malls, utilities etc., into which the towers collapsed.
>
> > WTC7 seems to be forgotten here.
> thus:
> what Miss Manners or Wikipedia Authority enshrines that,
> that one should not stick one's reply at the beginning?...
> assuming adequate referential skills, on both sides of the screen,
> doesn't it save one from some repeatative strange injury?
> thus:
> Pierre Duh, that's what our Muslim Fisikist, Schonfeld,
> is trying to say:
> that the biggest bombs ever to have hit the biggest buildings,
> could not have resulted in such a "free fall" collapse, although
> that's just a fisikal hypothesis.
> thus:
> I may be extrememly socially retarded but I'm not stupid --
> this is a *classic* analysis of Muslim Fisiks,
> which also tend to be hardcore examples of it....
> the real question is,
> Why should Earth's tallest, rather highly tensile structure not
> collapse
> at the speed of freefall?...
> I said, Why not?...
> anyway, there's a good analysis of the comparison
> between a surreptitious bombing, and
> an inside controlled demo, in the current issue of that MIT mag
> -- *Technology & Innovation*, or some thing --
> using the Murrah Building for the example....
> still, it is high time to impeach Trickier Dick Cheeny,
> who did *what* in the Nixon Administration with Don Rumfseld?... oh,
> you were convinced by that braindead guy, that he did it?
--it takes some to jitterbug!
http://members.tripod.com/~american_almanac
http://www.21stcenturysciencetech.com/2006_articles/Amplitude.W05.pdf
http://www.rwgrayprojects.com/synergetics/plates/figs/plate01.html
http://larouchepub.com/other/2006/3322_ethanol_no_science.html
http://www.wlym.com/pdf/iclc/howthenation.pdf
Dik T. Winter
...
> But if magnitude is refuted as built from the reals one of these claims
> has to go away. It's obvious to me that magnitude is more fundamental.
But nowhere do you give a mathematical definition of "magnitude". What
is it? How would you define it? What axioms are you using in that
definition? I see only an amount of text that mentions magnitude
frequently, but there is no basis there.
> That is what I am trying
> to address and the only way I can see is a yes/no answer. We build
> mathematics from simple constructs upward to more complicated ones, not
> the other way around.
Yup. We build the natural numbers using the peano axioms. Next we
create all integers. Followed by the rationals. This is again followed
by the reals, and finally we come at the complex numbers. (The order
varies only slightly, that is, negative numbers can be added after the
reals have been defined, but 0 is needed earlier.)
> This is the bowels of
> math, where the real numbers have six unique definitions on Wikepedia
> alone.
There is no problem with that as long as it can be shown that the
definitions are equivalent (and that can be done).
> Still, this means that I am making another definition of the
> real numbers based on magnitude.
You should first provide a definition of "magnitude".
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Aluminium Holocene Holodeck Zoroaster
nothing, googol, googplex -- the hundreds [*] !-)
seruiously, the "bug/feature" of his system is laid-out,
graphically on his site, although it wasn't apparent
from any of his dyscussion, that I scanned:
any point on the plane only has two of the three coordinates
of the presumed "triple" -- the write-up is so quickie as to ignore
that
he's giving each point a triple, which would actually
include "the opposite" for one of the numbers, but
ithe negative one is simply left out, as with "quadrays" in space.
hard to "tetragon that" with the endless say-so
about the arithmetic properties; eh?
* my new googolified numbers,
whis is at least as dumb as the original googolplex:
just say "hundred" after every two places, and
be sure to keep a tally; like,
"the speed of light is about 18,60,00 miles per hour;"
said, "18-hundred 60-hundred hundred mph."
> But nowhere do you give a mathematical definition of "magnitude". What
> is it? How would you define it? What axioms are you using in that
> definition? I see only an amount of text that mentions magnitude
> frequently, but there is no basis there.
thus:
see, you're begging the question;
what is the "normal" rate of collapse
of a building, controlled or surreptitious?
you probably believe in "global" warming,
just because Al Gore invented it ... and
it might get the DNC to run the creep,
sine they gutted the Voting Rights Act on March 27, 2000,
thanks to the God-am Supreme Court's refusal
to hear an appeal.... ah; what case was that?
> The kind of free-fall suggested by the US Govnerment Physicist
thus:
speaking of Kyoto,
there is an almost historical article on the emmissions trading schemes
of yore & today, in yesterday's NYTMagazine,
via its billionaire proponent from the junkbond biz. only thing
missing:
how many billions of dollars per year is CCX hedging, and
has it ever had any effect on, like, the price of oil?
--it takes some to jitterbug!
Gene Ward Smith
Dik T. Winter wrote:
> Yup. We build the natural numbers using the peano axioms. Next we
> create all integers. Followed by the rationals. This is again followed
> by the reals, and finally we come at the complex numbers. (The order
> varies only slightly, that is, negative numbers can be added after the
> reals have been defined, but 0 is needed earlier.)
The second order seems more relevant to this magnitude discussion.
> You should first provide a definition of "magnitude".
That would be nice. It does not seem to be forthcoming.
Timothy Golden BandTechnology.com
Dik T. Winter wrote:
> In article <1154462758....@i3g2000cwc.googlegroups.com> "Timothy Golden BandTechnology.com" <tttp...@yahoo.com> writes:
> ...
> > But if magnitude is refuted as built from the reals one of these claims
> > has to go away. It's obvious to me that magnitude is more fundamental.
>
> But nowhere do you give a mathematical definition of "magnitude". What
> is it? How would you define it? What axioms are you using in that
> definition? I see only an amount of text that mentions magnitude
> frequently, but there is no basis there.
Yes. A definition of magnitude is needed beneath the polysign
definition.
But is it my burden to define it? Is it also my burden to define the
multiplication of two magnitudes? These things are in their usual
context. I am not asking for anything more of them than their usual
properties. How about
http://en.wikipedia.org/wiki/Magnitude_%28mathematics%29
Is this sufficient? The concept is so elementary as to be at the base.
Somewhere the definitions come to an end. If they don't they'll chase
themselves in circles and become invalid. A magnitude is an unsigned
number. I then attach signs to these numbers and allow arithmetic
product and sum to take place via the signs. That three-signed numbers
are equivalent to the complex numbers can only strengthen the validity.
The real number definitions speak of a product but do they really
define it?
Or is that a mechanical operation that we have been trained to do on a
series of symbols?
What is
ab
takes on quite a challenge via the definitions of real numbers. Must
they define it?
At some low level this will break down into symbollic mechanics that
mimic summation. So where is this summation expression in terms of a
and b? Is this a valid concern? Or are we discussing matters so
fundamental that we should dismiss this route?
If the reals can't tell you what ab is until instantiated as concrete
values then they cannot define those values and define a product. The
values must come first. They are magnitudes.
>
> > That is what I am trying
> > to address and the only way I can see is a yes/no answer. We build
> > mathematics from simple constructs upward to more complicated ones, not
> > the other way around.
>
> Yup. We build the natural numbers using the peano axioms. Next we
> create all integers. Followed by the rationals. This is again followed
> by the reals, and finally we come at the complex numbers. (The order
> varies only slightly, that is, negative numbers can be added after the
> reals have been defined, but 0 is needed earlier.)
I do have a criticism of the usual ways. The choice of unity is
arbitrary along a real number line. There is no inherent integral
property of the line as claimed by this construction. That we can
generate an integral construction on the line (a+a) is true, and that
is a better way to approach the analogy between natural numbers and the
line. Instantiating unity even poses it's arbitrary nature in the
physical world where we impose inches and meters without conflict.
For more please see
http://groups.google.com/group/sci.math/msg/e77dac4322485e10
-Tim
Gene Ward Smith
Timothy Golden BandTechnology.com wrote:
> Yes. A definition of magnitude is needed beneath the polysign
> definition.
> But is it my burden to define it?
Only if you want to back up your claim to have a new approach to
defining the reals.
Timothy Golden BandTechnology.com
Aluminium Holocene Holodeck Zoroaster wrote:
> feh, Peano axioms; magnitude is from zero to more (one, too;
> nothing, googol, googplex -- the hundreds [*] !-)
>
> seruiously, the "bug/feature" of his system is laid-out,
> graphically on his site, although it wasn't apparent
> from any of his dyscussion, that I scanned:
> any point on the plane only has two of the three coordinates
> of the presumed "triple" -- the write-up is so quickie as to ignore
I must defend myself here.
If there is any way to improve my website then I am interested in
learning it.
It has been carefully worded and may be a bit terse, but all the
principles are there and should be clear. Many have a difficult time
understanding the construction but I will have to argue that the bug is
the real number around whose context we have been trained.
It is true that allowing negative floating point values will not break
the system, but the point is that the construction is of magnitudes and
that signed numbers will be developed from them. So you can't choose a
negative number. That would be building sign from sign.
> that
> he's giving each point a triple, which would actually
> include "the opposite" for one of the numbers, but
> ithe negative one is simply left out, as with "quadrays" in space.
>
> hard to "tetragon that" with the endless say-so
> about the arithmetic properties; eh?
Not at all. You must be misunderstanding something.
The dimensionality argument is critical.
For P3 we can have a value like:
- 2.3 + 1.2 * 3.3 .
It appears as though it would be 3D from first glance but because
- x + x * x = 0
we will be able to cancel out 1.2 from each component yielding
- 1.1 * 3.1 .
This is called the reduced form.
Because every instance can be reduced this way the n-signed numbers are
n-1 dimensional. For the reals the reduced form yields a solitary
value, but in higher signs if generally yields more values. The
geometry is already defined by the identity relation. Unit vectors in
each sign direction yield zero. The simplex is the only geometry that
matches.
>
> * my new googolified numbers,
> whis is at least as dumb as the original googolplex:
> just say "hundred" after every two places, and
> be sure to keep a tally; like,
> "the speed of light is about 18,60,00 miles per hour;"
> said, "18-hundred 60-hundred hundred mph."
>
> > But nowhere do you give a mathematical definition of "magnitude". What
> > is it? How would you define it? What axioms are you using in that
> > definition? I see only an amount of text that mentions magnitude
> > frequently, but there is no basis there.
>
> thus:
> see, you're begging the question;
> what is the "normal" rate of collapse
> of a building, controlled or surreptitious?
>
> you probably believe in "global" warming,
> just because Al Gore invented it ... and
> it might get the DNC to run the creep,
> sine they gutted the Voting Rights Act on March 27, 2000,
> thanks to the God-am Supreme Court's refusal
> to hear an appeal.... ah; what case was that?
Chuckle...
I didn't vote for Gore.
It's a shame really. The parties could have had McCain and Bradley and
then I wouldn't know who to vote for. But instead they chose Gush and
Bore.
Who's God do you work for?
-Tim
Timothy Golden BandTechnology.com
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
Dik T. Winter
> Dik T. Winter wrote:
> > In article <1154462758....@i3g2000cwc.googlegroups.com> "Timothy Golden BandTechnology.com" <tttp...@yahoo.com> writes:
> > ...
> > > But if magnitude is refuted as built from the reals one of these claims
> > > has to go away. It's obvious to me that magnitude is more fundamental.
> >
> > But nowhere do you give a mathematical definition of "magnitude". What
> > is it? How would you define it? What axioms are you using in that
> > definition? I see only an amount of text that mentions magnitude
> > frequently, but there is no basis there.
>
> Yes. A definition of magnitude is needed beneath the polysign
> definition.
> But is it my burden to define it?
Yes.
> Is it also my burden to define the
> multiplication of two magnitudes?
Yes.
> These things are in their usual
> context. I am not asking for anything more of them than their usual
> properties.
What is their usual context in this context? What are there usual
properties?
> How about
> http://en.wikipedia.org/wiki/Magnitude_%28mathematics%29
> Is this sufficient?
No, there magnitude is defined for some kind of objects, in terms of
those objects (real numbers, complex numbers, vectors in Euclidean
space). As complex numbers and Euclidean space rely on the real
numbers for their definition, the only concept of magnitude defined
there is based on real numbers. Clearly you can not define real
numbers based on such a definition.
> The concept is so elementary as to be at the base.
Perhaps, but mathematicians need definitions.
> Somewhere the definitions come to an end. If they don't they'll chase
> themselves in circles and become invalid.
You clearly do not know how mathematics works.
> A magnitude is an unsigned
> number.
What is a number in this context?
> I then attach signs to these numbers and allow arithmetic
> product and sum to take place via the signs. That three-signed numbers
> are equivalent to the complex numbers can only strengthen the validity.
Oh.
> The real number definitions speak of a product but do they really
> define it?
Yes. It takes some work, but in a good textbook that is really done.
> Or is that a mechanical operation that we have been trained to do on a
> series of symbols?
Not in mathematics.
> What is
> ab
> takes on quite a challenge via the definitions of real numbers. Must
> they define it?
Yes. If a is a real, there is a Cauchy sequence that converges to a.
If b is a real, there is a Cauchy sequence that converges to b.
Call the first sequence: a_0, a_1, ... and the second: b_0, b_1, ...;
where the a_i and b_i are rational numbers. Now form the product
sequence: a_0 * b_0, a_1 * b_1, ..., if the first sequences are
Cauchy sequences, the last sequence is also one (that can be proven,
and has been proven),so it converges to a real number: a * b.
Now you may ask, how do you define the product on rational numbers?
Well, conventionally a rational numbers is (an equivalence class of)
a pair of integers, say a = (a_n, a_d) and b = (b_n, b_d), then
a * b = (a_n * b_n, a_d * b_d), where a_n, a_d, b_n and b_d are
integers. And so on, back down to the definition of multiplication
on the non-negative numbers. How is it defined there? Using the
Peano axioms.
> At some low level this will break down into symbollic mechanics that
> mimic summation.
Only in the definition of multiplication of the non-negative numbers
by the Peano axioms.
> If the reals can't tell you what ab is until instantiated as concrete
> values then they cannot define those values and define a product.
I have no idea what this means. The product is not defined in terms of
the values.
...
> > Yup. We build the natural numbers using the peano axioms. Next we
> > create all integers. Followed by the rationals. This is again followed
> > by the reals, and finally we come at the complex numbers. (The order
> > varies only slightly, that is, negative numbers can be added after the
> > reals have been defined, but 0 is needed earlier.)
>
> I do have a criticism of the usual ways. The choice of unity is
> arbitrary along a real number line. There is no inherent integral
> property of the line as claimed by this construction. That we can
> generate an integral construction on the line (a+a) is true, and that
> is a better way to approach the analogy between natural numbers and the
> line. Instantiating unity even poses it's arbitrary nature in the
> physical world where we impose inches and meters without conflict.
But unity is not arbitrary at all. Unity is *that* real number u such
that for *all* real numbers r, u * r = r. There is only one such real
number, and it is not arbitrary. And I am *not* talking about a number
line, or whatever realisation you have for the real numbers.
Timothy Golden BandTechnology.com
Dik T. Winter wrote:
> In article <1154538787.5...@i42g2000cwa.googlegroups.com> "Timothy Golden BandTechnology.com" <tttp...@yahoo.com> writes:
> > Dik T. Winter wrote:
> > > In article <1154462758....@i3g2000cwc.googlegroups.com> "Timothy Golden BandTechnology.com" <tttp...@yahoo.com> writes:
> > > ...
> > > > But if magnitude is refuted as built from the reals one of these claims
> > > > has to go away. It's obvious to me that magnitude is more fundamental.
> > >
> > > But nowhere do you give a mathematical definition of "magnitude". What
> > > is it? How would you define it? What axioms are you using in that
> > > definition? I see only an amount of text that mentions magnitude
> > > frequently, but there is no basis there.
> >
> > Yes. A definition of magnitude is needed beneath the polysign
> > definition.
> > But is it my burden to define it?
>
> Yes.
>
> > Is it also my burden to define the
> > multiplication of two magnitudes?
>
> Yes.
>
> > These things are in their usual
> > context. I am not asking for anything more of them than their usual
> > properties.
>
> What is their usual context in this context? What are there usual
> properties?
magnitude is an unsigned value.
They are an abstract measure that is probably demonstrable to a
gorilla.
We could train the gorilla with a stick to see if other stickes are
exactly as long. When he seperates those sticks into a pile and puts
shorter ones in another pile and longer ones in a third pile he will
have mastered magnitude. He can then seperate each pile using the same
method. Eventually all of the sticks will be sorted in order of length.
So if a gorilla can understand magnitude nothing prevents it from being
a fundamental property.
This I would call the gorilla conjecture.
Now that's solid information. Wow. I really appreciate you taking the
time to explain all of that. So in effect if all this can be done in
the context of unsigned numbers then it will legitimate the polysign
numbers. I did once take a real analysis course that covered cauchy
sequences but it has been a long time. I'll try to dig into this a bit.
I don't see why there would be a problem mimicing all of this for a
value without any sign, so long as naturals are enough to go up to the
continuum.
>
> > At some low level this will break down into symbollic mechanics that
> > mimic summation.
>
> Only in the definition of multiplication of the non-negative numbers
> by the Peano axioms.
>
> > If the reals can't tell you what ab is until instantiated as concrete
> > values then they cannot define those values and define a product.
>
> I have no idea what this means. The product is not defined in terms of
> the values.
Right. My statement is invalid. The definition of the reals is really a
long long trail.
Any one page def. is just a summary. The product is so complicated that
it takes some pages to define it.
>
> ...
> > > Yup. We build the natural numbers using the peano axioms. Next we
> > > create all integers. Followed by the rationals. This is again followed
> > > by the reals, and finally we come at the complex numbers. (The order
> > > varies only slightly, that is, negative numbers can be added after the
> > > reals have been defined, but 0 is needed earlier.)
> >
> > I do have a criticism of the usual ways. The choice of unity is
> > arbitrary along a real number line. There is no inherent integral
> > property of the line as claimed by this construction. That we can
> > generate an integral construction on the line (a+a) is true, and that
> > is a better way to approach the analogy between natural numbers and the
> > line. Instantiating unity even poses it's arbitrary nature in the
> > physical world where we impose inches and meters without conflict.
>
> But unity is not arbitrary at all. Unity is *that* real number u such
> that for *all* real numbers r, u * r = r. There is only one such real
> number, and it is not arbitrary. And I am *not* talking about a number
> line, or whatever realisation you have for the real numbers.
Right. I am talking about practical usage of the real number line, and
particularly the geometrical usage of it. In this case a choice of
unity is arbitrary. This would then pose the problem that this is not
an application of the real numbers since as you say there is only one
unity. And indeed the product seems to play no sensible role
geometrically other than scaling, which is the problem with choosing
unity all over again. Perhaps relativity theory has to offer these
scales straight from the reference frame inherently.
Granting magnitude as fundamental would certainly simplify things.
Allowing a continuous entity to be inherent rather than built from a
discrete one would cut out quite a lot of complication. So perhaps in
conjunction with the Peano axioms there is a congruent set of axioms on
a continuum. The successor notion is certainly there, though in a
continuous sense. It would be a sort of precalculus on magnitudes. This
would alleviate the conflict of relying upon a discrete geometry to get
a continuous one.
Thanks again for your detailed and helpful response.
-Tim
Mariano Suárez-Alvarez
Maybe before refounding the theory of real numbers
on magnitude (whatever that might possibly mean) you could
first study the theory of real numbers as we have it founded
today?
Anyhow...
Your construction of `polysigned' numbers can be cast
in the language of algebra (and thereby make it precise)
quite simply. Pick n >= 1, let Z_n be the group of integers
modulo n, let P = R_0^+ be the semiring of non-negative
real numbers with operation given by the usual sum
and product. With this data one can construct a
semi-group ring A = P[Z_n], the elements of which are
formal linear combinations of elements of Z_n with
coefficients taken from P; this is completely analogous
to the construction of the usual group ring, only with
`semi' coefficients.
Now let S be the subset of A of elements which are
scalar multiples of the sum of the elements of Z_n; in order
to implement your idea that `the sum of signs is zero', we
need to take a quotient, but since A is only a semi-ring,
we have to do this with care: I use a sort of `stabilization'
with respect to the set S, as follows. One can define an
equivalence relation ~ in A as follows: if x, y are in A,
then x~y iff there exist s, t in S such that x+s = y+t.
This is indeed an equivalence relation, and is compatible
with the operations of A, so we can form the quotient
B = A/~ and it becomes a semiring. In fact, this is a ring,
as one can show without much work.
Now, if n = 2, the map
a_0 s_0 + a_1 s_1 in A ---> a_0 - a_1 in R
(R is the real numbers) induces a map B -> R, which is
a map of rings. It is an isomorphism. So yes, you have
`reconstructed' the real numbers.
Likewise, for n = 3, one can do something in the same
style mapping the s_i into cubic roots of 1 in the complex
numbers. I have not looked at what happens for higher
n, but I honestly doubt there is much there.
As for the value of this as a foundation for the theory of
real numbers, I am quite sure it is not tha great, as
the only hing one has gained is `reducing' the problem
to the foundation of the non-negative reals (or magnitudes,
if you prefer)
Cheers,
-- m
Dik T. Winter
> Dik T. Winter wrote:
> > Yes. If a is a real, there is a Cauchy sequence that converges to a.
> > If b is a real, there is a Cauchy sequence that converges to b.
> > Call the first sequence: a_0, a_1, ... and the second: b_0, b_1, ...;
> > where the a_i and b_i are rational numbers. Now form the product
> > sequence: a_0 * b_0, a_1 * b_1, ..., if the first sequences are
> > Cauchy sequences, the last sequence is also one (that can be proven,
> > and has been proven),so it converges to a real number: a * b.
> > Now you may ask, how do you define the product on rational numbers?
> > Well, conventionally a rational numbers is (an equivalence class of)
> > a pair of integers, say a = (a_n, a_d) and b = (b_n, b_d), then
> > a * b = (a_n * b_n, a_d * b_d), where a_n, a_d, b_n and b_d are
> > integers. And so on, back down to the definition of multiplication
> > on the non-negative numbers. How is it defined there? Using the
> > Peano axioms.
>
> Now that's solid information. Wow. I really appreciate you taking the
> time to explain all of that. So in effect if all this can be done in
> the context of unsigned numbers then it will legitimate the polysign
> numbers.
Yes. But doing that is the same as stating upfront that the magnitudes
you are talking about as the non-negative reals (as Gene Ward Smith
wrote in response). In that case you do not need all those definitions,
but you get all the operations you want for free (as they are defined
on the reals).
> I did once take a real analysis course that covered cauchy
> sequences but it has been a long time. I'll try to dig into this a bit.
> I don't see why there would be a problem mimicing all of this for a
> value without any sign, so long as naturals are enough to go up to the
> continuum.
But mimicing it leaves your magnitudes as isomorphic the the non-negative
reals, so why mimic it and not say upfront that they are the non-negative
reals?
> Granting magnitude as fundamental would certainly simplify things.
> Allowing a continuous entity to be inherent rather than built from a
> discrete one would cut out quite a lot of complication. So perhaps in
> conjunction with the Peano axioms there is a congruent set of axioms on
> a continuum. The successor notion is certainly there, though in a
> continuous sense. It would be a sort of precalculus on magnitudes. This
> would alleviate the conflict of relying upon a discrete geometry to get
> a continuous one.
But I think that would become horribly complicated.
But back to your paper, now I know what magnitude is (just a non-negative
real). Back to doubly signed numbers. Apparently a doubly signed number
is a pair (p1, p2) (where the first component carries the plus sign and
the second component the minus sign). I think (although you do not state
that) that for arbitrary magnitude, (p1+a, p2+a) = (p1, p2), as I think
(although you do not state that either), (a, a) = 0. This means that
the doubly signed numbers form equivalence classes, and those equivalence
classes (with the induced operations) are isomorphic to the reals through
the mappings (p1, p2) => p1-p2, and r => (r, 0) if r is non-negative and
(0, -r) if r is negative. (Here (p1, p2), (r, 0) and (0, r) are just
some representatives of the equivalence classes.) Also there is always
one representative in an equivalence class that has one of the components
equal to 0. So far so good.
With triply signed numbers something similar can be done. Define
w = (1 + sqrt(-3))/2. Look at the triple (p1, p2, p3) (I use the
sign order *, +, - here), again we have here equivalence classes,
and again in each class there is a representative with one component
equal to 0. Now it is easy to establish an isomorphism with the
complex numbers through the relation: (p1, p2, p3) => p1 + w.p2 + w^2.p3.
The reverse relation is a bit complicated, but can be done. Again, so
far so good.
I have to look further what the multiply signed numbers are for higher
orders.
Timothy Golden BandTechnology.com
similar assumption about P3 (the three-signed numbers). There is
absolutely no need to artificially assign them the cube roots of one
in the complex plane. That relation is derived, not imposed.
As Mariano uses a notation
a_0 s_0
where a is a magnitude and s is a sign I have grown closer to this
representation though it is not used on my website, which favors an
approach that a child might learn from.
I use
s x
where s is natural numbered sign and x is a magnitude. This then is an
elemental polysigned value which gets declared in a domain of say the
three-signed (P3) numbers.
At this point we can say two things.
Sum over s of ( s x ) = 0 .
( s1 x1 ) ( s2 x2 ) = ( s1 + s2 ) x1x2 .
The latter is the elemental product and this will yield the
correspondenct to the complex numbers. The sum s1 + s2 is in mod form
as Mariano properly sees. There is nothing more but to clear up any
discrepancies or holes in the above two lines. That these two lines may
be applied in P2 to yield the real numbers and in P3 to yield the
complex numbers and in P4 to yield a true 3D product and onward up
through any dimension is why I bother with a website at all. There is
no artifice in the construction. Everything is strict and simple, yet
yields the usual math and then some.
The s portion of
s x
is traditionally a symbol that is either '+' or '-'. Since we need to
use a third sign to continue the progression I choose '*' which when I
write it on a piece of paper has three lines through it. So now rather
than the abstract form we also have a concrete arithmetic form just as
the real numbers have:
- 1.1 + 2.3 * 4.5 .
The treatment of these values is like the reals; the field properties
(probably with the exception of quotient property) work on them so the
notation is entirely appropriate.
For generic values of
- a + b * c
and
- d + e * f
where the letters are magnitudes we get the generic P3 product:
+ ad * ae - af
* bd - be + bf
- cd + ce * cf .
And this result graphed on the geometry imposed by
- x + x * x = 0
will generate the complex numbers. The proof is here:
http://bandtechnology.com/PolySigned/ThreeSignedComplexProof.html
There is no usage of square roots of minus one in this development of
the complex numbers.
This construction declares that the complex numbers may also be gotten
by generalizing sign and that they are next in the progression after
the two-signed real numbers.
-Tim
Dik T. Winter
...
> - x + x * x = 0
> will generate the complex numbers. The proof is here:
> http://bandtechnology.com/PolySigned/ThreeSignedComplexProof.html
> There is no usage of square roots of minus one in this development of
> the complex numbers.
From that page, line 6: 'i(sqrt(3)'.
So it is not clear to me what you are meaning here. To establish
an isomorphism with the complex numbers you need sqrt(-3). (And
actually cbrt(1), but that is a bit hidden in that presentation.)
Timothy Golden BandTechnology.com
Dik T. Winter wrote:
> In article <1154695623.0...@p79g2000cwp.googlegroups.com> "Timothy Golden BandTechnology.com" <tttp...@yahoo.com> writes:
> ...
> > And this result graphed on the geometry imposed by
> > - x + x * x = 0
> > will generate the complex numbers. The proof is here:
> > http://bandtechnology.com/PolySigned/ThreeSignedComplexProof.html
> > There is no usage of square roots of minus one in this development of
> > the complex numbers.
>
> From that page, line 6: 'i(sqrt(3)'.
> So it is not clear to me what you are meaning here. To establish
> an isomorphism with the complex numbers you need sqrt(-3). (And
> actually cbrt(1), but that is a bit hidden in that presentation.)
The structure of the proof requires a transform between the two spaces.
So one of the spaces is P3(three-signed numbers) and the other is
C(complex numbers).
The transform from P3 to C is just a little bit of pythagorean theorem
in the plane.
So we have:
c = T ( p )
where c is in C and p is in P3 and T is a transform.
Instantiate two values p1 and p2 in P3.
Transform them to C so that c1=T(p1), c2=T(p2).
Take these products which are each well defined in their own domains.
So we have c3=c1c2 and p3=p1p2.
Now perform the transform T(p3) and it is found equal to c3, quod erat
faciendum.
I think you are just looking at the application of pythagorean theorem
which does not involve any square roots of minus one. The sqrt(3) is
just coming out because of the 2pi/3 angles. I have no idea what
cbrt(1) means.
The complex form is just the (a+bi) form and stays in that form, so I
never substitute a sqrt(-1); it just remains i.
The transform is just pure geometry in terms of distance. The only
thing that matters is that we line up the * pole of P3 with the
positive real axis and the - pole close to the positive imaginary axis.
This is the proper orientation to get the math to work out and the
standard complex axes superposed on the graph of the P3 plane is
accurate. I suppose that would be a superior graphic if it had both
coordinate systems explicitly laid out. I'll see what I can do to
improve the graphic.
Perhaps I should do the sum out first which I don't even bother with.
That would then demonstrate the form for the product proof. Would this
be an improvement?
Anything you suggest I will take seriously. The proof is solid. Making
it legible to you is of utmost importance.
-Tim
Dik T. Winter
> Dik T. Winter wrote:
> > In article <1154695623.0...@p79g2000cwp.googlegroups.com> "Timothy Golden BandTechnology.com" <tttp...@yahoo.com> writes:
> > ...
> > > And this result graphed on the geometry imposed by
> > > - x + x * x = 0
> > > will generate the complex numbers. The proof is here:
> > > http://bandtechnology.com/PolySigned/ThreeSignedComplexProof.html
> > > There is no usage of square roots of minus one in this development of
> > > the complex numbers.
> >
> > From that page, line 6: 'i(sqrt(3)'.
> > So it is not clear to me what you are meaning here. To establish
> > an isomorphism with the complex numbers you need sqrt(-3). (And
> > actually cbrt(1), but that is a bit hidden in that presentation.)
> I think you are just looking at the application of pythagorean theorem
> which does not involve any square roots of minus one. The sqrt(3) is
> just coming out because of the 2pi/3 angles. I have no idea what
> cbrt(1) means.
cbrt means cube root, but you are missing the i. In that formula you
have cbrt(1) in disguise. It can be done simpler, take a triple
(p1, p2, p3) [order: * , +, -]. Now investigate the behaviour of
a = (1, 0, 0), b = (0, 1, 0) and c = (0, 0, 1); we see immediately
that a is the unit, so it maps to 1, and that b^3 = a, so b maps to
the cube root of 1.
> The complex form is just the (a+bi) form and stays in that form, so I
> never substitute a sqrt(-1); it just remains i.
That is just form; there is no difference.
Aluminium Holocene Holodeck Zoroaster
whether you describe it or not, that
you could use just two of the "signed" coordinates,
leaving the other as "unused," although
it would have a negative value in any homogenous coordinates.
that is, it's a kind of homogenous coordinates
-- which are generally better to work with --
such as "quadrays."
> The dimensionality argument is critical.
> For P3 we can have a value like:
> - 2.3 + 1.2 * 3.3 .
> It appears as though it would be 3D from first glance but because
> - x + x * x = 0
> we will be able to cancel out 1.2 from each component yielding
> - 1.1 * 3.1 .
> > * my new googolified numbers,
> > whis is at least as dumb as the original googolplex:
> > just say "hundred" after every two places, and
> > be sure to keep a tally; like,
> > "the speed of light is about 18,60,00 miles per hour;"
> > said, "18-hundred 60-hundred hundred mph."
--it takes some to jitterbug!
Dik T. Winter
> In article <1154708298....@i3g2000cwc.googlegroups.com> "Timothy Golden BandTechnology.com" <tttp...@yahoo.com> writes:
> > The complex form is just the (a+bi) form and stays in that form, so I
> > never substitute a sqrt(-1); it just remains i.
>
> That is just form; there is no difference.
BTW, I just have looked at the quadruply signed numbers. They have a
property that makes them less nice than the complex numbers. There
exist two non-zero quadruply signed numbers with a product that is
equal to 0 (actually, there are many pairs of such numbers). Defining
division on them is, eh, problematical.
Timothy Golden BandTechnology.com
Dik T. Winter wrote:
> In article <J3Ht4...@cwi.nl> "Dik T. Winter" <Dik.W...@cwi.nl> writes:
> > In article <1154708298....@i3g2000cwc.googlegroups.com> "Timothy Golden BandTechnology.com" <tttp...@yahoo.com> writes:
> ...
> > > The complex form is just the (a+bi) form and stays in that form, so I
> > > never substitute a sqrt(-1); it just remains i.
> >
> > That is just form; there is no difference.
>
> BTW, I just have looked at the quadruply signed numbers. They have a
> property that makes them less nice than the complex numbers. There
> exist two non-zero quadruply signed numbers with a product that is
> equal to 0 (actually, there are many pairs of such numbers). Defining
> division on them is, eh, problematical.
Right, and so the field axioms do not work out in higher signs.
Still, with this trouble comes the claim for a derivation of spacetime.
You are close to discovering what I call the identity axis, though that
name is possibly a misnomer and might better be called the null axis.
Your zero products include points that are in the plane passing through
the origin that is normal to +1#1 and points on this(+1#1) axis.
Still, you will readily see that the associative, commutative, and
distributive properties over the product and the sum work for any sign
level. I'm still not clear on whether or not division will ever be had
for the higher signs. I can't find a conjugate. I've done some plodding
through linear equations but some have been too challenging. Anyhow my
skills in that department are weak. Eventually I'll have a finder in my
code that will hunt down quotients.
Did you have a look at
http://bandtechnology.com/PolySigned/Deformation/DeformationUnitSphereP4.html
yet? That's the P4 product survey. If you have a slow computer or a
slow connection you might try just getting the animation one at a time,
particularly:
http://bandtechnology.com/PolySigned/Deformation/AxisDualDeformStudy.gif
That's the most coherent view of the P4 product.
-Tim
Dik T. Winter
> Dik T. Winter wrote:
> > In article <J3Ht4...@cwi.nl> "Dik T. Winter" <Dik.W...@cwi.nl> writes:
> > > In article <1154708298....@i3g2000cwc.googlegroups.com> "Timothy Golden BandTechnology.com" <tttp...@yahoo.com> writes:
> > ...
> > > > The complex form is just the (a+bi) form and stays in that form, so I
> > > > never substitute a sqrt(-1); it just remains i.
> > >
> > > That is just form; there is no difference.
> >
> > BTW, I just have looked at the quadruply signed numbers. They have a
> > property that makes them less nice than the complex numbers. There
> > exist two non-zero quadruply signed numbers with a product that is
> > equal to 0 (actually, there are many pairs of such numbers). Defining
> > division on them is, eh, problematical.
>
> Right, and so the field axioms do not work out in higher signs.
> Still, with this trouble comes the claim for a derivation of spacetime.
Sorry, this is outside mathematics. Mathematics has no knowledge about
time, and the notion space in mathematics is not what (I think) you would
call space. It is possible though, that the quintuply signed numbers do
not have zero divisors.
> Still, you will readily see that the associative, commutative, and
> distributive properties over the product and the sum work for any sign
> level. I'm still not clear on whether or not division will ever be had
> for the higher signs.
Not for the quadruply signed numbers. I suspect only for those n-signed
numbers where n is prime.
> Did you have a look at
> http://bandtechnology.com/PolySigned/Deformation/DeformationUnitSphereP4.html
> yet? That's the P4 product survey.
Well, I have no use for pictures. Sorry.
Timothy Golden BandTechnology.com
Dik T. Winter wrote:
> In article <1154737699....@i42g2000cwa.googlegroups.com> "Timothy Golden BandTechnology.com" <tttp...@yahoo.com> writes:
> > Dik T. Winter wrote:
> > > In article <J3Ht4...@cwi.nl> "Dik T. Winter" <Dik.W...@cwi.nl> writes:
> > > > In article <1154708298....@i3g2000cwc.googlegroups.com> "Timothy Golden BandTechnology.com" <tttp...@yahoo.com> writes:
> > > ...
> > > > > The complex form is just the (a+bi) form and stays in that form, so I
> > > > > never substitute a sqrt(-1); it just remains i.
> > > >
> > > > That is just form; there is no difference.
> > >
> > > BTW, I just have looked at the quadruply signed numbers. They have a
> > > property that makes them less nice than the complex numbers. There
> > > exist two non-zero quadruply signed numbers with a product that is
> > > equal to 0 (actually, there are many pairs of such numbers). Defining
> > > division on them is, eh, problematical.
> >
> > Right, and so the field axioms do not work out in higher signs.
> > Still, with this trouble comes the claim for a derivation of spacetime.
>
> Sorry, this is outside mathematics. Mathematics has no knowledge about
> time, and the notion space in mathematics is not what (I think) you would
> call space. It is possible though, that the quintuply signed numbers do
> not have zero divisors.
Right. Now there is one. The one-signed numbers define a
zero-dimensional construction. Superpostion will only yield larger
values in one-signed math and they always render to zero. This allows
them to sit alongside the others:
P1 P2 P3 P4 ...
with the breakpoint beneath P4 due to the ill behaved properties of the
high signs. This is a natural progression and the well behaved parts
yield spacetime in a form that is far more congruent than the 4D
version.
>
> > Still, you will readily see that the associative, commutative, and
> > distributive properties over the product and the sum work for any sign
> > level. I'm still not clear on whether or not division will ever be had
> > for the higher signs.
>
> Not for the quadruply signed numbers. I suspect only for those n-signed
> numbers where n is prime.
You don't have it yet. You could try to give me a counterexample if you
are so sure. I'll give you a positive one and I assure you it is not
preengineered:
z1 = - 2 + 3
z2 = # 1 * 2
z3 = -1 * 3
now we have
z1( z2 + z3) [where the + is summation]
= ( - 2 + 3 ) ( # 1 * 2 - 1 * 3 )
= ( - 2 + 3 ) ( -1 * 5 # 1 )
= + 2 # 10 - 2 * 3 - 15 + 3
= - 17 + 5 * 3 # 10
= - 14 + 2 # 7
and z1 z2 + z1 z3
= ( - 2 + 3 )( #1 * 2) # (- 2 + 3 )( -1 * 3 ) [ where # is four-signed
concrete summation]
= ( - 2 # 4 + 3 - 6 ) # ( + 2 # 6 * 3 - 9 )
= ( - 8 + 3 # 4 ) # ( - 9 + 2 * 3 # 6 )
= - 17 + 5 * 3 # 10
= - 14 + 2 # 7
To prove that
z1 ( z2 + z3 ) = z1 z2 + z1 z3
in P4 in general is trivial but long since each product has sixteen
terms.
I don't yet have the proof in general sign which would be nice since
then it could be done just once. I'm pretty sure that using the
elemental form
s x
will be sufficient but I haven't tried to do it out. There are larger
fish to fry.
Anyhow, after you do enough of the math it is clear that the same
principles that work for real and complex numbers with product and sum
work for any dimension under the polysign construction.
-Tim
>
> > Did you have a look at
> > http://bandtechnology.com/PolySigned/Deformation/DeformationUnitSphereP4.html
> > yet? That's the P4 product survey.
>
> Well, I have no use for pictures. Sorry.
Oh. That's alright. It is of interest that the product's unique
features are fully exposed on the unit shell (unit sphere surface) All
other products are just scaled versions. I suppose that you can see
better than I that the product is symmetrical and well-behaved. It's
true that the even signs will generate zero results for non zero
operands and that sounds bad, but it relates to the identity axis which
is a fairly interesting feature.
Thanks for taking a look at the polysign numbers.
The magnitude definition is necessary to define the real numbers via
the polysign construction, which it does with the two basic rules. In
effect any information declared there need not be defined beneath in
the definition of magnitude, and so I believe that magnitude is
fundamental. Thanks very much for your help.
-Tim
cbr...@cbrownsystems.com
> Maybe before refounding the theory of real numbers
> on magnitude (whatever that might possibly mean) you could
> first study the theory of real numbers as we have it founded
> today?
>
Wow! If I understand you correctly, you suggest you will refund the
real numbers, because they have not lived up to the claims that were
made of them.
'Cause I paid $40 (US, mind you) for the privilege, and now I find out
that, basically, they suck because they woefully lack in the magnitude
department.
Do you send me your Paypal info now, or what? Or is there someone else
I should contact?
(Fortunately, all financial transactions can be carried out in Q, which
is public domain).
Eagerly awaiting your reply - Chas
Mariano Suárez-Alvarez
A classical theorem of Milnor and Kervaire asserts that the
dimension of a (finite dimensional) division algebra over the
real numbers must be 1, 2, 4 and 8 (and these dimensions
are realized b the field of real numbers itself, the complex
numbers, the quaternions and the (non-associative) Cayley
numbers (a.k.a. octonions))
(The proof of this is quite sophisticated (it involves
algebraic topology and K-theory; see Wikipedia for
an idea of what this subjects are...) There are a number
of simpler results, for which purely algebraic proofs
are known. Hurwitz's theorem, for example, states
that if a real composition division algebra is necessarily of
dimension a power of two (an algebra is a composition
algebra if there is a `norm' quadratic function which
behaves as the square of the absolute value.))
This leaves very few possibilities for algebras of
`polysigned numbers' to be division algebras: just
for n = 2, 3, 5 and 9. One can show that no
real associative algebra of dimension 8 isa division
algebra, so that rules n = 9 out. Next, one can see
quite simply that the only two real division algebras
which are commutative are the field of real numbers
itself and the complex numbers, so---since polysigned
numbers with n=5 are commutative--this rules out
n=5 too.
Thus, if division is wanted among polysigned numbers,
you are stuck with n=2 and n=3.
-- m
Dik T. Winter
> Dik T. Winter wrote:
> > Sorry, this is outside mathematics. Mathematics has no knowledge about
> > time, and the notion space in mathematics is not what (I think) you would
> > call space. It is possible though, that the quintuply signed numbers do
> > not have zero divisors.
>
> Right. Now there is one.
Makes no sense. What was right about what I wrote? That possibly the
quintuply signed numbers do not have zero divisors? What does the
"Now there is one" mean?
> > > I'm still not clear on whether or not division will ever be had
> > > for the higher signs.
> >
> > Not for the quadruply signed numbers. I suspect only for those n-signed
> > numbers where n is prime.
>
> You don't have it yet. You could try to give me a counterexample if you
> are so sure.
Try (I am using my notation of quaduples with sign order #, *, +, -):
(1, 0, 1, 0)/(0, 0, 1, 1) = (p1, p2, p3, p4).
Solving the equations you will find p1 = p3 + 1 and p3 = p1 + 1. There
is no solution. If there are zero divisors in a ring you can not divide
by them in general.
> z1 = - 2 + 3
> z2 = # 1 * 2
> z3 = -1 * 3
What does your example show about division? It only shows that the
operations are distributive, but I never claimed otherwise. Yes,
indeed, z1(z2 + z3) = z1 z2 + z1 z3. That is the distributive law.
But we were talking about division.
> To prove that
> z1 ( z2 + z3 ) = z1 z2 + z1 z3
> in P4 in general is trivial but long since each product has sixteen
> terms.
Indeed, trivial.
> I don't yet have the proof in general sign which would be nice since
> then it could be done just once. I'm pretty sure that using the
> elemental form
> s x
> will be sufficient but I haven't tried to do it out. There are larger
> fish to fry.
But using that notation, it is indeed fairly trivial. Rest assured, the
operations are distributive, commutative and associative. But I was
talking about division, that is something completely different.
> Anyhow, after you do enough of the math it is clear that the same
> principles that work for real and complex numbers with product and sum
> work for any dimension under the polysign construction.
Yes, they work, and they form a ring. But you will not be able to
define division on the quadruply signed numbers. It may be possible
with the quintuply signed numbers (and for all cases where the number
of signs is a prime).
Hero
> Try (I am using my notation of quaduples with sign order #, *, +, -):
> (1, 0, 1, 0)/(0, 0, 1, 1) = (p1, p2, p3, p4).
> Solving the equations you will find p1 = p3 + 1 and p3 = p1 + 1. There
> is no solution. If there are zero divisors in a ring you can not divide
> by them in general.
> .......
> Yes, they work, and they form a ring. But you will not be able to
> define division on the quadruply signed numbers.
And Mariano wrote:
> > A classical theorem of Milnor and Kervaire asserts that the
> > dimension of a (finite dimensional) division algebra over the
> > real numbers must be 1, 2, 4 and 8 (and these dimensions
> > are realized b the field of real numbers itself, the complex
> > numbers, the quaternions and the (non-associative) Cayley
> > numbers (a.k.a. octonions))
Now You've got the main analytic features of the four-signed
multiplication. But the tetra is always good for a surprise, as we are
geometrically educated with a dominant right angle.
And there is one interesting geometric property i wonder how this
reflects analytically, with the four-signed multiplication or
otherwise.
The tips of the four unit-vectors, that is the four corners of a tetra
with edges of equal length brought into a succesion by four signs, like
-, +, *, # , are situated on a cylindrical spiral (helix) with equal
angels of about 130 degrees and equal distances, when projected onto
the axis.
Friendly greetings
Hero
Timothy Golden BandTechnology.com
Nice analysis. P4 mimics R x C very closely and it has been suggested
elsewhere(by Gene Ward Smith) that images of R and C will be embedded
in the higher sign spaces. This seems to be adjacent to the Clifford
algebra and that region of math. Yet, while there is some congruence of
P4 to P3XP2 a difference is observed:
http://bandtechnology.com/PolySigned/Deformation/P4T3Comparison.html
This difference appears to be a scalar error, but upon trying to
normalize the difference I find that as it is squeezed out of the
embedded real portion it is still exhibited in the complex portion. If
you'd like graphics I can publish them.
I have not read the theorem by Milnor and Kervaire, but to claim access
to a division algebra necessarily requires a defined product doesn't
it? Here the product is something different than the quaternions and
others in that family so I'd take this into consideration before
accepting this analysis. It would be quite an informational statement
to claim that a 3D product process is not 'reversible' in general., but
that a 4D is.
In the polysign domain these zero resultant products are in the even
signs. For instance in P6 we can see the equivalent of the P4 as:
( 1, 0, 1, 0, 1, 0 )( 2, 2, 0, 0, 0, 0 )
= ( 0, 2, 0, 2, 0, 2 )
+ ( 2, 0, 2, 0, 2, 0 )
= 0 .
You probably already understand that, but this is the artifact that
indicates general division will fail. The most obvious perspective is
that what was excepted in the field axioms ( namely zero ) for division
needs to be expanded to include this 'null' axis ( 1, 0, 1, 0, ... )
for indeed any value that is multiplied by it winds up on it. For
instance:
( 1, 0, 1, 0 )( 1, 2, 3, 0 )
= ( 1, 0, 1, 0 )
+ ( 3, 0, 3, 0 )
+ ( 0, 2, 0, 2 )
= ( 4, 2, 4, 2 )
= ( 2, 0, 2, 0 ).
In effect the definition of zero could be viewed as expanded for the
polysign numbers to include the identity axis for even sign systems.
Since any value multiplied by a value on the identity axis(possibly
misnamed) yields a value on the identity axis the information contained
in the result is nearly null.
The magnitude behavior has not been discussed by you all. Quaternions
preserve magnitude under their product. P4+ do not. I don't know if
that makes you question the theorem more, but it is an important
feature of the polysign system. But are the division concept and this
magnitude concept related?
In that division is merely the reverse of product we already know that
if
a = b c
then we can define the quotient
a / b = c .
At this level there is no prefered sign level.
The product works across all dimensions for all values.
Therefore the quotient necessarily exists.
Primes, evens, powers of two are all misnomered restrictions.
Whether the algorithm exists and how many results it can return for c
in the above is the simplest form of the problem.
Even without the algorithm the existence of the quotient is automatic
isn't it?
-Tim
Timothy Golden BandTechnology.com
I apologize. As I reread the original I see that I had placed two
disparate statements in one paragraph. You expressed denial without
addressing which and I took that as a denial of the first. You clearly
do agree with the algebraic properties so I must recant the statement
that you don't understand the polysign numbers. You do!!! I am grateful
that you spend time pondering them and look forward to any
contributions that you make, including hard criticism.
I've already responded in part to your prime conjecture in the response
to Mariano above here. I am sorry about the misunderstanding.
-Tim
Dik T. Winter
> Mariano Su=E1rez-Alvarez wrote:
...
> > A classical theorem of Milnor and Kervaire asserts that the
> > dimension of a (finite dimensional) division algebra over the
> > real numbers must be 1, 2, 4 and 8 (and these dimensions
> > are realized b the field of real numbers itself, the complex
> > numbers, the quaternions and the (non-associative) Cayley
> > numbers (a.k.a. octonions))
> I have not read the theorem by Milnor and Kervaire, but to claim access
> to a division algebra necessarily requires a defined product doesn't
> it? Here the product is something different than the quaternions and
> others in that family so I'd take this into consideration before
> accepting this analysis.
Does not matter. As you can add and subtract the same number from all
of the coordinates, and the result is the same, you can subtract the
last coordinate from all of them (obtaining reals rather than magnitudes),
the result is an algebra over the reals. For division to be possible,
the dimension of this algebra must be 1, 2, 4 or 8, as has been proven
(and that is regardless on how the multiplication is defined). It can be
shown that with dimension 8, the multiplication is not assiociative and
with dimension 4 multiplcation is not commutative. As your multiplications
ar commutative and associative, dimension 4 and 8 fall out, so the only
division algebras possible in your case are P2 and P3.
> The most obvious perspective is
> that what was excepted in the field axioms ( namely zero ) for division
> needs to be expanded to include this 'null' axis ( 1, 0, 1, 0, ... )
> for indeed any value that is multiplied by it winds up on it.
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.
> In that division is merely the reverse of product we already know that
> if
> a = b c
> then we can define the quotient
> a / b = c .
> At this level there is no prefered sign level.
> The product works across all dimensions for all values.
> Therefore the quotient necessarily exists.
Not for each pair of numbers. Indeed, if a = b c, a/b and a/c both
do exist. But given arbitrary p and q then p/q does not necessarily
exist. For instance, if q is a zero-divisor and p is not, the quotient
does not exist.
Timothy Golden BandTechnology.com
In your model the identity axis can match the axis of your cylinder and
give you the arm pairing.
In other regards this is merely a remnant of their perfect symmetry.
While the unit rays fit into a cylinder they'll also fit into a sphere.
Seeking details in general dimension may be more relevant, since those
results will extend. I understand the predisposition with 3D space and
the nicety of the coordinate system.
Vector addition is still vector addition in its usual Cartesian sense,
with the exception that there is a notion of accumulation that allows
ever growing coordinates to remain local so long as those coordinates
are balanced.
Informationally the act of reduction is the act of rendering or
graphing and in that regard every time it is performed arithmetically
it is optional. Though it makes the results comparable it is not a
necessary operation, but it is an inherent operation of graphing.
This 'render' concept is almost an operator and is the identity law
Sum(sa)=0.
The four directions of 3D space can be regarded as unidirectional. That
their inverse is composed of the other directions makes this apparent.
This unidirectionality is a universal feature of the polysign numbers.
That then leads to the signon which I think is an even more significant
feature than the simplex:
http://bandtechnology.com/PolySigned/Lattice/Lattice.html
This construction still does not use the product. It also does not need
the continuum.
Following the real number construction (composed from the natural
numbers) what happens if we make the composition polysigned rather than
two signed at the stage of developing integers? There seems to be
several avenues there. I don't like this route yet but someone else
might. The Cauchy sequence and all of that in polysign.
At what level is sign really appropriate? Could it be that what we call
sign has some more apt term? The combinatorics of modulo math take on
their own dynamic patterns which pose categories of symmetry. These are
the sign products and finessing to the formal term 'signature' seems
sensible; it connotes many unique forms. Still, the notion that our
existent spacetime has a signed representation is false until the
product takes on a geometrical significance. Simply squaring some
values on the real line will tell you how far off the real numbers are
from a true geometrical representation of space as we observe it. The
intuitive sense of sign that we take from the real numbers is in
conflict.
The truer visible remnant is magnitude, not sign.
-Tim
>
> Friendly greetings
> Hero
Timothy Golden BandTechnology.com
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. The
product images warrant this claim. This claim is for a computer
algorithm to get within an epsilon error that is practical (i.e. it
doesn't break the floating point math operations). This will get more
and more difficult as one approaches the identity axis, but for points
away from it there should be no problem getting the arbitrary quotient
a/b to generate the value c within a distance of arbitrary epsilon. The
only place that we should see multiple solutions is for values whose b
portion are perpendicular to the identity axis through the origin where
we will get two results. I think that this will work for higher signs
as well. I don't know that this will help generate a way for a human to
do quotient math.
So if you want to give me some values I'll try to generate the quotient
for you.
-Tim
Sean Holman
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
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.
Dik T. Winter
> Dik T. Winter wrote:
...
> > 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.
Timothy Golden BandTechnology.com
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
Timothy Golden BandTechnology.com
Dik T. Winter wrote:
> In article <1154979553.2...@i42g2000cwa.googlegroups.com> "Timothy Golden BandTechnology.com" <tttp...@yahoo.com> writes:
> > Dik 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
Dik T. Winter
> Dik T. Winter wrote:
...
> > 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 .
And that means that *both* are zero divisors. It also means the inverse
of #1*1 does not exist, so you can not divide by it in general. It also
mean that when you take an arbitrary number, the product with #1*1 is
a zero divisor. And so on.
> But this is an artifact of the general behavior that any value
> multiplied by #1+1 will land on the axis.
Any value multiplied by #1*1 will land on another axis.
And BTW, all numbers that have exactly two 1's in their representation
and 0's otherwise, are zero divisors.
> 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.
The "identity axis" is nothing new at all. It is well known that when
you multiply a number by a zero divisor that the result is a zero
divisor. And that is what your "identity axis" states, the only point
is that there is not a single "identity axis", there are multiple.
Timothy Golden BandTechnology.com
There is only one of these axes. It acts like a real line embedded in
the even signed P4+ domains. It always has the notation ( 1, 0, 1, 0,
... ).
The term 'zero divisor' is one that I am not yet comfortable with, but
it is fairly easy to see that in the reals
( 0 ) ( +5 ) = 0 .
This does not make any strong statement about the number +5, just as
the example in P4
( # 1 + 1 ) ( # 1 * 1 ) = 0 .
does not make a strong statement about ( # 1 * 1 ). It is one of many
values that will provide this zero result just as +5 can be replaced
for the example in the reals.
Also, you've neglected the tail portion of my statement that makes a
fairly strong argument, particularly the analysis of
z ( # 1 + 1 ) = c
for constant c. There is perhaps a misunderstanding.
The identity axis is a dominant feature of the even signs. as is
demonstrated by
http://bandtechnology.com/PolySigned/Deformation/DeformationUnitSphereP4.html
When the sphere turns into a line that is when the red dot has taken a
value of the identity axis. Information is destroyed. The only similar
behavior downward in sign is multiplication by zero. Here there is a
slight difference due to the fact that a real value remainder exists so
it is not exactly a zero concept, but very similar. The phrase
'identity axis' perhaps should be rephrased 'null axis'. The existing
field definition for the reciprocal uses a strictly zero exception. The
polysign system suggests that there ought to be more excepted;
particularly all values along the identity axis provide multiple
solutions.
Is this perhaps a counterexample to your claims above?:
( # 1 + 1 )( # 1 + 1 ) = # 2 + 2 ;
Therefor
( # 2 + 2 ) / ( # 1 + 1 ) = # 1 + 1 .
Do you see the real line embedded behavior?
-Tim
Dik T. Winter
> Dik T. Winter wrote:
...
> > you multiply a number by a zero divisor that the result is a zero
> > divisor. And that is what your "identity axis" states, the only point
> > is that there is not a single "identity axis", there are multiple.
> There is only one of these axes. It acts like a real line embedded in
> the even signed P4+ domains. It always has the notation ( 1, 0, 1, 0,
> ... ).
No, there are more of them.
> The term 'zero divisor' is one that I am not yet comfortable with, but
> it is fairly easy to see that in the reals
> ( 0 ) ( +5 ) = 0 .
Yup. But a zero-divisor is a *non-zero* number that can be multiplied by
another non-zero number to give 0. So when a and b are non-zero and
a.b = 0, a and b are both zero divisors.
> This does not make any strong statement about the number +5, just as
> the example in P4
> ( # 1 + 1 ) ( # 1 * 1 ) = 0 .
> does not make a strong statement about ( # 1 * 1 ).
But ( #1 + 1) is non-zero, so it makes a strong statement. Anyhow, you
will not find a solution to:
( #1 *1 +1 )/( #1 *1)
because the first is not a zero-divisor and the second is.
In P4 there are numerous zero-divisors, some basic ones are the following
six (coordinates in order #, *, +, -):
(1, 1, 0, 0) (1, 0, 1, 0) (1, 0, 0, 1) (0, 1, 1, 0) (0, 1, 0, 1) (0, 0, 1, 1)
for *all* six of them you can not find a quotient if you try to divide
(1, 1, 1, 0) by any one of them.
> Also, you've neglected the tail portion of my statement that makes a
> fairly strong argument, particularly the analysis of
> z ( # 1 + 1 ) = c
> for constant c. There is perhaps a misunderstanding.
But z (1, 1, 0, 0) is also on some axis (a different one).
Some linear algebra: to solve
(a1, a2, a3, a4)/(b1, b2, b3, b4) = (p1, p2, p3, p4)
you get at the matrix
( b1 b2 b3 b4 )
( b4 b1 b2 b3 )
( b3 b4 b1 b2 )
( b2 b3 b4 b1 )
we are interested in the determinant, because if it is zero there is in
general no solution. The determinant is zero if either b1+b2+b3+b4 = 0
or b1-b2+b3-b4 = 0. It is however a bit more tricky as we are in
equivalence classes where incrementing each coordinate by the same value
does not change the number: (1, 1, 1, 1) = 0. But whatever increment
we use, the determinant remains zero if b1-b2+b3-b4 = 0. So we can never
find a generic solution for (1, 1, 0, 0), (0, 1, 1, 0), (0, 0, 1, 1) and
(1, 0, 0, 1) (or in general for (b1, b2, b3, b4) if b1-b2+b3-b4 = 0).
And, indeed, if b1-b2+b3=b4 = 0, (b1, b2, b3, b4) is a zero-divisor.
In that case: (b1,b2,b3,b4)(1,0,1,0) = (b1,b2,b3,b4)(0,1,0,1) = 0.
You are only seeing that (1, 0, 1, 0) and (0, 1, 0, 1) can only divide
(a1, a2, a3, a4) if a1 = a3 and a2 = a4.
> The identity axis is a dominant feature of the even signs. as is
> demonstrated by
Sorry those pictures do not tell me anything.
> Is this perhaps a counterexample to your claims above?:
> ( # 1 + 1 )( # 1 + 1 ) = # 2 + 2 ;
I see no counterexamples. Try to divide (1, 1, 1, 0) by (5, 3, 1, 3) and
see that it fails. More knowledge about zero-divisors would help you.
Timothy Golden BandTechnology.com
Hi Dik.
I believe that I have resolved our apparent disagreement. I've
scrambled together a low grade reciprocal finder (results with large
error) and sure enough the divisor counterexamples that you provide
have no decent reciprocal. I've read the definition of zero divisor and
it is fine though it does not discuss anything about the underlying
structure of the system it is applied to. I suppose that is up to us
and I am now curious what we'll see above p4 in this context.
The 'other axes' are actually a plane. It is the plane that is
perpendicular to the identity axis that passes through the origin.
We've discussed it a bit before but now I can see it in the context of
division. Just as a multiplication by a point on the identity axis
removes information so does a multiplication by any point in this
plane. Instead of squashing the information into a line it yields two
dimensions of result, but still provides an irreversible effect on any
object. In effect the object has been squashed into a plane and can
never be 'unsquashed' informationally. This is exposed in
http://bandtechnology.com/PolySigned/Deformation/AxisDualDeformStudy.gif
When the sphere crosses over itself (flipping handedness) this is the
place.
This is a product view which explains why the division will not yield
results since the quotient is the reversal of the product. We cannot
get three dimensions back from two. There must be a theorem on
dimensional reduction and zero divisors. It's clear in the
informational context.
Because of this dimensional behavior the only meaningful division on
the zero divisor parts are more values from those parts. The zero
product is composed of one value from each of these parts, those parts
being the identity axis and the plane orthogonal to it on the origin.
There should be a theorem there as well. Upon excluding this plane as
well as the line that I call the identity axis we can have division.
This is quite a structure built in isn't it? The line literally
combined with the plane. Very pretty.
Physics could be here, and in a dimensionally consequential way.
I suppose you are nearing an R x C style equivalent definition of the
P4 product. As I scramble to follow along now I am wondering what you
will find next. And what of P6? Is the zero divisor exclusion the
identity axis and the points orthogonal to it through the origin? What
is a plane in P4 would be a 4D space in P6. This would be a ( 1, n-1 )
type of relation. That is consistent with the zero products, whose
possibility space rises dimensionally.
If the Hopf, Milnor and Kervaire dimensional theorems apply then we
should be able to get a zero divisor structure in P7 as well. But they
suggest that P5 will work perfectly. I'll have a better reciprocal
finder eventually. For now it is slow but could find reciprocals up in
these spaces. If you want me to look for something specific I'll be
happy to try.
I'm not very good at linear algebra but I'll try to follow along. So
far I don't see how you got the matrix in b. Is this a conjugation
style of division? It should be possible to specify a conjugate in P4
so long as we are off of the zero divisor structure but I don't have
it.
-Tim
Dik T. Winter
...
> I believe that I have resolved our apparent disagreement. I've
> scrambled together a low grade reciprocal finder (results with large
> error) and sure enough the divisor counterexamples that you provide
> have no decent reciprocal. I've read the definition of zero divisor and
> it is fine though it does not discuss anything about the underlying
> structure of the system it is applied to. I suppose that is up to us
> and I am now curious what we'll see above p4 in this context.
Zero-divisors are indeed pat of the structure, but much can be said
about zero-divisors without even investigating the underlying
structure (when that is a ring, of course).
> The 'other axes' are actually a plane. It is the plane that is
> perpendicular to the identity axis that passes through the origin.
Yup, the set of obvious zero-divisors are, using coordinates
(x1, x2, x3, x4), the plane x1 + x3 = x2 + x4 and the line
x1 = x3 & x2 = x4.
> We've discussed it a bit before but now I can see it in the context of
> division. Just as a multiplication by a point on the identity axis
> removes information so does a multiplication by any point in this
> plane. Instead of squashing the information into a line it yields two
> dimensions of result, but still provides an irreversible effect on any
> object. In effect the object has been squashed into a plane and can
> never be 'unsquashed' informationally.
Indeed, the product of a zero-divisor and an arbitrary element is a
zero-divisor. That is easy to prove when the product is commutative.
> We cannot
> get three dimensions back from two. There must be a theorem on
> dimensional reduction and zero divisors. It's clear in the
> informational context.
There is not necessarily dimensional reduction. There are rings where
every element is a zero-divisor. But in general the zero-divisors form
subspaces of lower dimensionality (assuming we have some dimensional
algebra over the reals, which you have). But it is all quite tricky,
because if a and b are zero-divisors, a + b is not necessarily a
zero-divisor.
> The zero
> product is composed of one value from each of these parts, those parts
> being the identity axis and the plane orthogonal to it on the origin.
> There should be a theorem there as well. Upon excluding this plane as
> well as the line that I call the identity axis we can have division.
Indeed, you can have division with the elements that are not zero-divisors,
because they all have multiplicative inverses. It is just like in
linear algebra, where a matrix has an inverse if and only if the
determinant is not equal to 0. But if you look at it, you will see
that the structure of the set of matrices with determinant 0 is quite
tricky.
> I suppose you are nearing an R x C style equivalent definition of the
> P4 product.
Pretty close, but not there yet. The line of zero-divisors should
correspondent with R, and the plane with C. What remains to be done
is finding the exact mapping. (That that is true follows because
in R x C we have also two sets of zero-divisors, one 1D and one 2D,
corresponding to R and C respectively. I think an exact mapping can
be found.)
> As I scramble to follow along now I am wondering what you
> will find next. And what of P6? Is the zero divisor exclusion the
> identity axis and the points orthogonal to it through the origin?
There are many more zero-divisors in P6. I will explain, and that shows
also the reason that I originally thought that Pn might have no zero-
divisors if n is prime. Consider in P6:
(1, 1, 1, 0, 0, 0) * (1, 0, 0, 1, 0, 0) = 0
and
(1, 1, 0, 0, 0, 0) * (1, 0, 1, 0, 1, 0) = 0
These are obvious zero-divisors (you get the other obvious zero-divisors
by circulating the coordinates). That we have two pairs of clearly
different kinds is because 6 = 2 * 3. I do not know whether there are
non-obvious zero-divisors (I am pretty sure, there are none in P4).
> What
> is a plane in P4 would be a 4D space in P6. This would be a ( 1, n-1 )
> type of relation. That is consistent with the zero products, whose
> possibility space rises dimensionally.
What you will find (I think) in P6 is that there are four spaces of
zero-divisors.
> If the Hopf, Milnor and Kervaire dimensional theorems apply then we
> should be able to get a zero divisor structure in P7 as well. But they
> suggest that P5 will work perfectly.
In P5 and P7 there are no obvious zero-divisors. So looking for them
can be problematical. But starting at Pn, we can get an (n-1)-
dimensional algebra over R by subtracting the last coordinate from
all coordinates. The last coordinate now is 0. It is now an algebra
over the reals of dimension n-1. So by Hopf, if it is a division
algebra, n-1 should be a power of 2. We excluded already all non-prime
n, now we can also exclude all primes that are not of the form 2^k + 1.
But there is a stronger theorem by Hopf: every commutative division
algebra over k has dimension 1 or 2. (H. Hopf, Ein topologischer
Beitrag zur reellen Algebra, Comment. Math. Helv. 13 (1940), 219-239.)
So we do not even need the theorems by Milnor and Kervaire. What they
did show was that *any* division algebra over R has at most dimension
8, and that those algebras are precisely R, C, Q and O. (But Q is
non-commitative and O is non-associative. If we get further we get
the sedonians, but now there are zero-divisors.)
What remains is looking at P5 and see whether zero-divisors can be found
(there must be some, otherwise it would be a division algebra, and that
can not be the case by the theorem of Hopf.)
and so by the theorem they can only be a division algebra (that is,
division does exist) if n-1 = 1, 2, 4 or 8.
> I'm not very good at linear algebra but I'll try to follow along. So
> far I don't see how you got the matrix in b. Is this a conjugation
> style of division?
No, it is just plain linear algebra notation to denote a set of linear
equations, which division in your systems is: solving sets of linear
equations.
Timothy Golden BandTechnology.com
Also it can be said that if a and b are zero divisors then their
product is not necesarily zero.
For instance -1*1 and #1+1 are both zero divisors. But their product is
-2*2.
This is where the structure matters, for it can be said that iff a is
on the 1D part and b is on the 2D part that their product will be zero.
This is all in the P4 context. The 1D part forces a result on its axis
and the 2D part forces a result on its plane, yielding the origin. The
two are inherently tied together. I think that this dimensional level
is more appropriate a focus than the phrase 'zero divisor' for this
phrase connotes a general phenomenon but without the structure it is
too abstract. Each zero divisor loses its meaning without being
attributed to its place in the structure. For P4 this is an axis and
its orthogonal plane at the origin.
In P6 the value (0,0,1,1,0,0) does not intuit to be normal to
(1,0,1,0,1,0) but it is according to my computer a right angle. So thus
far the zero divisor concept laid out here maintains an orthogonal
relation in P6 as well. In P6 the space orthogonal to the identity axis
is a 4D object. But this is just pinning down a known behavior. The P5
challenge sounds much more interesting since there is a sliver of hope
of proving an exception to Hopf.
>
> > The zero
> > product is composed of one value from each of these parts, those parts
> > being the identity axis and the plane orthogonal to it on the origin.
> > There should be a theorem there as well. Upon excluding this plane as
> > well as the line that I call the identity axis we can have division.
>
> Indeed, you can have division with the elements that are not zero-divisors,
> because they all have multiplicative inverses. It is just like in
> linear algebra, where a matrix has an inverse if and only if the
> determinant is not equal to 0. But if you look at it, you will see
> that the structure of the set of matrices with determinant 0 is quite
> tricky.
Yes, and you can have division with the elements that are zero
divisors.
>From the example above
(-2*2) / (-1*1) = #1+1 .
>
> > I suppose you are nearing an R x C style equivalent definition of the
> > P4 product.
>
> Pretty close, but not there yet. The line of zero-divisors should
> correspondent with R, and the plane with C. What remains to be done
> is finding the exact mapping. (That that is true follows because
> in R x C we have also two sets of zero-divisors, one 1D and one 2D,
> corresponding to R and C respectively. I think an exact mapping can
> be found.)
Yeah. It sure looks that way. I have gotten very close with the
independent products. That this product generates a self-similar error
that could be removed with more products suggests that there is an
exact answer lurking. Would mean is that there is an alternate way to
develop the polysign system that will look a lot more like the Clifford
algebra? That would be too big of a leap I suppose. Understanding
division on P5 should help.
>
> > As I scramble to follow along now I am wondering what you
> > will find next. And what of P6? Is the zero divisor exclusion the
> > identity axis and the points orthogonal to it through the origin?
>
> There are many more zero-divisors in P6. I will explain, and that shows
> also the reason that I originally thought that Pn might have no zero-
> divisors if n is prime. Consider in P6:
> (1, 1, 1, 0, 0, 0) * (1, 0, 0, 1, 0, 0) = 0
> and
> (1, 1, 0, 0, 0, 0) * (1, 0, 1, 0, 1, 0) = 0
Above I've verified that this is an orthogonal set of vectors.
Yes, I've already acknowledged you above here on this. I think this is
worth focusing on.
I still have not found the Hopf proof. Since you have carefully noted
its publication I'll see what can be gotten in english. In twenty pages
is it possible that this assumption has been made without evaluation?:
" The assumption that the square of a unit vector
is positive unity leads to an algebra whose characteristic
quantities
are non-associative. " - Cargill Gilston Knott
This is a quote of a quote from
http://en.wikipedia.org/wiki/Cargill_Gilston_Knott
I have not read any of his work but it may be that the polysign
construction livens such fundamental debates. Dimensionality is
produced differently under the polysign construction.
Squaring unit vectors in P4 yields a cone. Squaring its components will
not yield a sensible distance. That requires use of cross terms as
well.
>
> and so by the theorem they can only be a division algebra (that is,
> division does exist) if n-1 = 1, 2, 4 or 8.
>
> > I'm not very good at linear algebra but I'll try to follow along. So
> > far I don't see how you got the matrix in b. Is this a conjugation
> > style of division?
>
> No, it is just plain linear algebra notation to denote a set of linear
> equations, which division in your systems is: solving sets of linear
> equations.
> --
> dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
> home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
One would hope that there is a means of doing the division out
shorthand and this would imply a conjugate that yields the reciprocal
indirectly. That this would be tied to the structure of the zero
divisors is likely since they are what will break the division process.
I still don't see how to do this yet the product is simply defined.
Does the existence of zero divisors necessarily conflict with the
notion of a shorthand division process?
-Tim
Dik T. Winter
> Dik T. Winter wrote:
...
> > every element is a zero-divisor. But in general the zero-divisors form
> > subspaces of lower dimensionality (assuming we have some dimensional
> > algebra over the reals, which you have). But it is all quite tricky,
> > because if a and b are zero-divisors, a + b is not necessarily a
> > zero-divisor.
>
> Also it can be said that if a and b are zero divisors then their
> product is not necesarily zero.
Indeed. Take matrices and their determinants. A zero divisor is a matrix
with a determinant equal to 0. But the product of two matrices with
determinant equal to 0 does not necessarily result in the 0 matrix.
> This is where the structure matters, for it can be said that iff a is
> on the 1D part and b is on the 2D part that their product will be zero.
That is because in P4 the zero-divisors split in two different, separated,
sets. In P6 it is already more difficult. And in matrices it can be
told how it works. If in some sense "orthogonal" (you have to define it, but
in general it works out nicely) zero-divisors are multiplied, the result
is 0.
> In P6 the value (0,0,1,1,0,0) does not intuit to be normal to
> (1,0,1,0,1,0) but it is according to my computer a right angle. So thus
> far the zero divisor concept laid out here maintains an orthogonal
> relation in P6 as well.
Yes, and indeed in every ring with zero-divisors there is orthogonality
involved when the product is actually 0.
> In P6 the space orthogonal to the identity axis
> is a 4D object.
There are four classes of zero divisors: (1, 1, 0, 0, 0, 0),
(1, 0, 1, 0, 1, 0), (1, 1, 1, 0, 0, 0) and (1, 0, 0, 1, 0, 0) (where
circulants of the given elements belong to the same class). They
yield (as far as I can see) in order, 4D, 1D, 3D and 2D spaces.
> The P5
> challenge sounds much more interesting since there is a sliver of hope
> of proving an exception to Hopf.
Only a sliver; throw it away. On the other hand, finding zero-divisors
in P5 can be challenging, as not all of the coordinates are integer (as
far as I have been able to ascertain).
> > Indeed, you can have division with the elements that are not zero-divisors,
> > because they all have multiplicative inverses. It is just like in
> > linear algebra, where a matrix has an inverse if and only if the
> > determinant is not equal to 0. But if you look at it, you will see
> > that the structure of the set of matrices with determinant 0 is quite
> > tricky.
>
> Yes, and you can have division with the elements that are zero
> divisors.
Yes, you can divide some zero-divisors by some other zero-divisors, but
there it stops.
> Understanding
> division on P5 should help.
Understandign the zero-divisors of P5 would help a bit more.
>
> >
> > > As I scramble to follow along now I am wondering what you
> > > will find next. And what of P6? Is the zero divisor exclusion the
> > > identity axis and the points orthogonal to it through the origin?
> >
> > There are many more zero-divisors in P6. I will explain, and that shows
> > also the reason that I originally thought that Pn might have no zero-
> > divisors if n is prime. Consider in P6:
> > (1, 1, 1, 0, 0, 0) * (1, 0, 0, 1, 0, 0) = 0
> > and
> > (1, 1, 0, 0, 0, 0) * (1, 0, 1, 0, 1, 0) = 0
>
> Above I've verified that this is an orthogonal set of vectors.
There are two sets involved. 1D, 2D, 3D and 4D.
> I still have not found the Hopf proof. Since you have carefully noted
> its publication I'll see what can be gotten in english. In twenty pages
> is it possible that this assumption has been made without evaluation?:
> " The assumption that the square of a unit vector
> is positive unity leads to an algebra whose characteristic
> quantities are non-associative. " - Cargill Gilston Knott
> This is a quote of a quote from
> http://en.wikipedia.org/wiki/Cargill_Gilston_Knott
I think there is no relevance to this dicussion. But whatever the
case, if he wrote that it was certainly wrong with the current
terminology in mind.
> > No, it is just plain linear algebra notation to denote a set of linear
> > equations, which division in your systems is: solving sets of linear
> > equations.
>
> One would hope that there is a means of doing the division out
> shorthand and this would imply a conjugate that yields the reciprocal
> indirectly. That this would be tied to the structure of the zero
> divisors is likely since they are what will break the division process.
> I still don't see how to do this yet the product is simply defined.
> Does the existence of zero divisors necessarily conflict with the
> notion of a shorthand division process?
Linear algebra provides a shorthand notation.
Mariano Suárez-Alvarez
You will not find an exception to a theorem.
> [snip]
> I still have not found the Hopf proof. Since you have carefully noted
> its publication I'll see what can be gotten in english.
You can look for similar arguments in Husemoller's "Fiber Bundles",
if I recall c0orrectly.
Hopf's theorem, as well as Milnor-Kervaire's and others in this
direction are *not* proved using algebra, but algebraic topology.
The topological arguments stem from a rather deep study of
what spheres can be given the structure of H-spaces (see
wikipedia por a definition) and similar ideas.
-- m
Timothy Golden BandTechnology.com
The polysign numbers are a new construction so there is a sliver of a
chance that they could defy some of existing mathematics. I'm hunting
for P5 zero divisors now. It looks like they are there, but I have yet
to get all the way to zero. For instance
(1.077, 0.889, 0.228, 0, 0.519 ) * ( 1.004, 0.420, 0.371, 1.043, 0 )
= ( 0.005, 0.005, 0.011, 0.016, 0 )
and whose magnitude is 0.014 . (These values are chopped, not rounded)
A perturbative algorithm on these unity magnitude operands should get
a zero value, and expose the structure that these zero divisors are on.
The zero divisors are still a coherent part of a continuous mapping.
I'm not sure in strict math what the proper teminology is, but right
nearby the zero divisors are points just off of zero, so in this way
the whole 'division algebra' problem makes me ask the question 'so
what?' What is it that we are gaining by studying the division problem?
We can express general dimensional relations such as
0 = a0 + a1 z + a2 z z + a3 z z z ...
but where will the division algebra become consequential? I understand
that it helps characterize a construction, but I'm wondering where it
goes from there?
-Tim
Mariano Suárez-Alvarez
Again, you will not defy existing mathematics.
> I'm hunting
> for P5 zero divisors now. It looks like they are there, but I have yet
> to get all the way to zero. For instance
>
> (1.077, 0.889, 0.228, 0, 0.519 ) * ( 1.004, 0.420, 0.371, 1.043, 0 )
> = ( 0.005, 0.005, 0.011, 0.016, 0 )
>
> and whose magnitude is 0.014 . (These values are chopped, not rounded)
> A perturbative algorithm on these unity magnitude operands should get
> a zero value, and expose the structure that these zero divisors are on.
Pick n >= 4.
Write e_i for the i-th "sign" in your n-polysigned numbers. Let
x = sum_{k=0..n-1} cos(2 pi k / n) e_i
and
y = sum_{k=0..n-1} cos(4 pi k / n) e_i
(If you do not like negative coefficients, just add a large
enough multiple of
sum_{i=0..n-1} e_i
so that all coefficients become positive; this does not
change the end value...)
Then a little trigonometry should convince you that
x y = 0.
NB: You may enjoy computing the product for 1 <= n < 4, and
what is different in those three cases.
Note that I did not come up with the this example out of
thin air: this is just a particular instance of a rather more
general phenomenon, studied (essentially completely and
exhaustively) by Frobenius, Schur, Wedderburn and others
nearly two centuries ago.
> The zero divisors are still a coherent part of a continuous mapping.
> I'm not sure in strict math what the proper teminology is, but right
> nearby the zero divisors are points just off of zero, so in this way
> the whole 'division algebra' problem makes me ask the question 'so
> what?' What is it that we are gaining by studying the division problem?
Well, for one thing, you are studying the division problem.
> We can express general dimensional relations such as
> 0 = a0 + a1 z + a2 z z + a3 z z z ...
> but where will the division algebra become consequential?
It really depends on what you want to do. You have to
decide what you want to use your polysigned numbers
for, and then evaluate if for *that* purpose the fact that
divisors of zero exist for almost all cases is consequential
or not.
>I understand
> that it helps characterize a construction, but I'm wondering where it
> goes from there?
I would heartily recomend that you try to go through an
introduction to the theory of algebras, such as Pierce's
`Associatve Algebras'.
-- m
Timothy Golden BandTechnology.com
> Timothy Golden BandTechnology.com wrote:
snip
Neat. I've verified it for P4 thru P9. I don't understand it.
> > The zero divisors are still a coherent part of a continuous mapping.
> > I'm not sure in strict math what the proper teminology is, but right
> > nearby the zero divisors are points just off of zero, so in this way
> > the whole 'division algebra' problem makes me ask the question 'so
> > what?' What is it that we are gaining by studying the division problem?
>
> Well, for one thing, you are studying the division problem.
>
> > We can express general dimensional relations such as
> > 0 = a0 + a1 z + a2 z z + a3 z z z ...
> > but where will the division algebra become consequential?
>
> It really depends on what you want to do. You have to
> decide what you want to use your polysigned numbers
> for, and then evaluate if for *that* purpose the fact that
> divisors of zero exist for almost all cases is consequential
> or not.
>
> >I understand
> > that it helps characterize a construction, but I'm wondering where it
> > goes from there?
>
> I would heartily recomend that you try to go through an
> introduction to the theory of algebras, such as Pierce's
> `Associatve Algebras'.
>
> -- m
Alright. Thanks for the reference. I'll try to get to it.
What do you think of P1?
The polysign construction does not require the usage of
Sum( sx ) = 0
except for purposes of graphing or 'rendering' a result.
In this sense the components are accumulators.
Zero-dimensional(P1) operations work but always render to zero.
There is congruence with time.
In some regard this 'disappearance' is what we are seeing in P4+
through zero divisors.
Dimensions are disappearing. Physics has an accepted 'tunneling' effect
modelled by energy wells. Perhaps there is a linkage to the
arithmetical behavior of polysign numbers in the higher signs.
Do you believe there is a way to express the equivalent of an
exponential for Pn?
Even in P3 the closest that I have gotten yet is just to take a value
near unity and take the series z^n. A clean value will iterate a
circle. I've tried this for values in higher signs but the results
become degenerate. Now I am wondering if I simply haven't chosen the
correct initial value; For instance in P4 the correct choice may be
just off the unity vector along +1#1.
Can I trace out a unit shell in any dimension via z^n? I'm fairly
certain that the answer is no simply due to magnitude nonconservation
in P4+ products. Whether that would break an exponential definition I
am unsure.
-Tim
Aluminium Holocene Holodeck Zoroaster
of the math, to justify your own polemics. anyway,
I may've misspoken on the third (planar) coordinate being negative;
it depends on the homogenous coordinates; the simplest
are trilinears, but you should try others. the study of the various
points and
lines and circles and on & on of the "general trigon" is much eaiser,
using homogenous ones.
only two coordinates are *neccesary*, but three gets rid
of all sorts of degeneracies & special cases.
(I'd say, it's obvious, that tunnelling depends upon defects
in the semiconductor, beyond just the use of dopants, themselves.)
> In some regard this 'disappearance' is what we are seeing in P4+
> through zero divisors.
> Dimensions are disappearing. Physics has an accepted 'tunneling' effect
> modelled by energy wells. Perhaps there is a linkage to the
> arithmetical behavior of polysign numbers in the higher signs.
thus:
Polymath / Scholar Extraordinaire / Inventor / Architect / Poet /
Ecologist / Futurist / Philosopher
http://www.channer.tv/Friday.htm
thus:
it seems like the frequency shift would
be manageable, going 600mph on a plane;
you/pilots know the trajectory thereby....
sometimies a cellphone is just, "Hi, Mom;
--it takes some to jitterbug!
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