Re: Is magnitude more fundamental than the real numbers?


Gene Ward Smith wrote:
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.

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

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