Re: Is magnitude more fundamental than the real numbers?


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.

Right, and we can go on to define magnitude for R^n as well.
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

.

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