Re: Math as Religion


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.
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. "

- http://groups.google.com/group/sci.math/msg/a340f8254714b780

The important distinction that allows this conflict is in how we
dissect the number system.
Because the polysign construction imposes the identity law
Sum for s = 1 to n ( s x ) = 0
we need not include this piece of information in the constituent
components of this law. The law builds sign and so to claim that sign
is a component of its constituents is inconsistent. This is a new
dissection and inverts a small piece of branching of the old tree. If
we adopt the new way then the old way (Gene's context) is suspect.
Though the incompatibility is minor the stricture of mathematics does
not allow even the most minor conflict. For n=2 the polysign numbers
are the real numbers and identity law above expands out to
- x + x = 0 .
This is recognizable as accurate to even a grade school child, but this
law is not a standard part of the definition of the real numbers. Yet
it is this form which allows the generalization of sign. Since this
information has been stated at this point there is no need to repeat it
anywhere else. This information principle is what makes a tight
construction. Certainly what is left of the continuous portion x is
merely a magnitude.

The benefit of this approach is that the complex numbers are barely
different than the real numbers. The reals are P2 and the complex
numbers are P3. Simply changing n by 1 gets the complex numbers from
the same laws that define the real numbers. This statement alone is
enough reason to answer why any mathematician should take interest in
the polysign construction. This is a primitive and productive
construction that poses and answers many questions:

Are the field criteria accurate?
Must a linear system obey the magnitudinal law
| A B | = | A | | B | ?
Does time correspond to P1?
Do improper transformations model electron spin?
Do n-poles exist?
Why spacetime?

That magnitude is fundamental and can be married to sign is the
foundation which allows the polysign concept to thrive. Previously in
debating this I have provided the gorilla conjecture, which poses that
since we can teach a gorilla the principle of magnitude that magnitude
is fundamental. Here we may enter into a psychological examination of
mathematical learning. Is it considered sufficient by the teaching
mathematician that a student be capable of repeatable error free
results to demonstrate understanding? Under this criteria the magnitude
is a primitive feature and the reals a sincere failing point. How many
sign errors have been generated by the human race? Yet how many
children unschooled in mathematics at all can pick out the larger of
two objects? This is how brutally simple magnitude is. The uneducated
mind is capable of percieving it, yet the highly educated mathematician
refutes the principle.
And so a farce is made of mathematicians. I do not respect religion and
I do not respect mathematicians who practice their subject as one.

-Tim

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