Re: Is magnitude more fundamental than the real numbers?


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?

.

Relevant Pages

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