Re: Math as Religion


*** T. Winter wrote:
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/~***/

I am willing to be flexible with notation and I don't believe that
there is a singular way. Above in this thread I have criticized your
underscore notation as lacking generality and I would suggest on that
issue if you want to take it up to take it up there:
http://groups.google.com/group/sci.math/msg/db97b7686861d8c0

Your refutation of the polysign construction as relying on the reals is
exactly what this thread is trying to address. I am looking squarely at
this from one side and you from the other.
So this is fine. This is the debate that I wish to have. And so this
debate leads to the question of whether a fine mathematician such as
yourself is capable of breaking such a rule in the interest of
improving mathematics. Hindsight tells me that breaking this rule is
not a conflict ad so this rule is no law. It is merely a tradition
which has cost mathematics the deficit of the polysign numbers.

Your ability to go here is the difference between scientific
mathematics and religious mathematics. Currently you are practicing
religious mathematics. Breaking rules is a lot of fun; especially false
rules, for these shroud the truth and should be broken. It is a
scientific mathematicians duty to tread here.

-Tim

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