Re: Math as Religion

In article <1163162640.035005.250540@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> "Timothy Golden BandTechnology.com" <tttpppggg@xxxxxxxxx> writes:
Gene Ward Smith wrote:
stephen@xxxxxxxxxx wrote:

Who has said that mathematics is THE truth? The "mainstream"
folks disagreeing with Timothy Golden have made no claims
of truth. Timothy Golden is the one who seems to think that
it is THE truth that magnitude is more fundamental than the reals.
If he wishes to define magnitude rigourously and then define
the reals based on that, he is free to.

As has been pointed out on numerous times, this is in fact an old idea,
going back to the Greeks. Landau, for a modern example, develops
positive reals from second order arithmetic of
positive integers, and goes on from there. This has some advantages, in
particular that the positive rationals, as the ratios of positive
integers, may be constructed without worrying about division by zero,
and then the positive reals (or magnitude) can be constructed next.

One can use polysigned numbers, if one so chooses, for constructions.
But Tim seems unable to say why we should.

Gene, like ***, represents the establishment.

Oh.

This is Gene on a prior thread:
"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. "

And Gene was right.

The important distinction that allows this conflict is in how we
dissect the number system.

No. The important distinction is that you use terms that you define
in existing terms (the reals). So you assume the existence of the
reals. Otherwise the definition is void.

In the quote on top Gene *did* show how you could define magnitude
without any reference to the reals. But for some reason you do not
want to follow that road.

Because the polysign construction imposes the identity law
Sum for s = 1 to n ( s x ) = 0

Again that basic notational flaw. Proper notation would be
sum for i = 1 to n ( s_i x) = 0

This is a primitive and productive
construction that poses and answers many questions:

Are the field criteria accurate?

Eh? What can be inaccurate in criteria?

Must a linear system obey the magnitudinal law
| A B | = | A | | B | ?

To me this makes no sense. In a linear system we have a linear operator
that transforms input to output. I think that with A and B you mean the
linear operators. But in that case you have to define the meaning of |.|.
In linear algebra, when we define the norm of a vector as the Euclidean
norm, and the norm of a matrix as
sup |A.x|/|x|
the above certainly does not hold. In that case we can only show:
| A B | <= | A | | B |.

Does time correspond to P1?
Do improper transformations model electron spin?
Do n-poles exist?
Why spacetime?

These are not mathematical problems.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.

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