Re: Possible flaw in the polynomial ring A[X] construction

In article
<160c5496-7d33-4f12-aa78-56d79ef7a0a8@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Tim Golden BandTech.com" <tttpppggg@xxxxxxxxx> wrote:

On Aug 3, 4:44 pm, Virgil <Vir...@xxxxxxxx> wrote:
In article
<3353112e-c949-4607-a6a7-4040507f8...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Tim Golden BandTech.com" <tttppp...@xxxxxxxxx> wrote:



On Aug 2, 3:01 pm, Virgil <Vir...@xxxxxxxx> wrote:
In article
<25aaa9e2-7fd5-466d-aeef-7b08e0c39...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Tim Golden BandTech.com" <tttppp...@xxxxxxxxx> wrote:


Those who wish to understand mathematics must learn how not to be caught.
mathematics requires that those who wish to understand her be most
horribly literally minded.

Yes, and skeptical too. Particularly mistakes are not to be made. What
we see in the polynomial ring construction are two interpretations of
x: one as a variable and one as a constant. This explains the
literature which waters x down to a 'symbol'. I am well within reason
to criticize this math. Just because it has been written and has
proliferated does not mean that it is correct. I feel even more
comfortable if I consider that the modern work is a shuffling of the
deck; an attempt to bring in multiple interpretations under one cover.
A brief summary of the problems with the polynomial ring construction:
a x
is not a ring product if a and x are not in the same set.

But they are in the same set as you have been told many times. The set
is not the set of real numbers, since x is not in the set of real
numbers. However, they are in the same set (the set of polynomials over
the reals): e.g., 37.51 and x both belong to that set.

This product goes unstudied in the literature even though the ring
definition went to the trouble of granting a product operator to work
with just pages before. Instead to push further away from the ordinary
algebraic form of a sum of products the polynomial is pushed into a
tuple representation, where the strictures of math require both an
infinite length format

An infinite length format is not required to define the ring of
polynomials over the reals. I could define the set of real polynomials
to be the set of finite tuples with real numbers as entries of the
tuples. I could then define addition and multiplication on these finite
tuples that would yield (an isomorphic copy of) the ring of real
polynomials. This is not done to my knowledge, but math would "allow" it.
For example, (0, 1) * (0, 1) = (0, 0, 1). Nothing prevents me from
defining such a product.

and a finite restriction. This is the only way
to resolve the ballooning indices

This is not the only way to resolve the ballooning indices: see above.

and keep everything looking
coherent. The ability to work in N-space is not workable due to
products resulting in 2N, which is not a ring behavior.

It is workable.

In other words
if we were to accept the finite tuple form
a0 + a1 x + a2 x x = (a0,a1,a2)
and consider products we see that
(a0,a1,a2) (b0,b1,b2) = (c0,c1,c2,c3,c4)
while a true and coherent representation, does not meet the ring
requirement since the tuple form is growing,

Yes.

and this conflicts with
the usage of the tuple from a set theoretic format of a cartesian
product, whose influence has been dodged by the users of this format.

There is no conflict. With tensors and differential forms, similar
mixture of different dimensional forms are done all the time.

The dimensional interpretation of the tuple is too much freedom to
grant the polynomial. Taking this into account some comfort can be had
in building out the infinite length tuple. It is merely a standard
algebraic expression walking around in chains, which stretch back a
very long length away in the infinite form. How one comes to require
this infinite length form is something which the tuple people here
have not cared to share. Here I believe that I've exposed the
reasoning clearly as to how the infinite length form arises, something
which the robotic approaches leave out.

The tuple form takes on primary reliance for those who wish to protect
the subject. Yet the motivations of that form clearly reflect the
algebraic format. If one were to consider the motivation then we see
that the algebraic form actually carries the ring definition closer to
it since it is an expression of sums of products. The cartesian sense
that the tuple format of the polynomial mimics is false. This is
apparent since
(0,a)(0,b) = (0,0,c) .
There is no independence for these components. The very meaning of
this operation as a behavior of a variable with a coefficient
(a x)(b x) = c x x
is easy enough to see, yet there is some denial of the equivalence of
the form in defense of the polynomial ring. This defensive stance is
only necessary because of the weakness which I attack.

The opposition to your thesis is that your notion of a polynomial ring
over some base ring is merely that base ring itself, which
mathematically is pointless. Unless a polynomial ring differs in some
way from its base ring, there is no mathematical purpose in having one.

Well, maybe you are not such a mime. Are you able to take this thought
that you have expressed above here seriously? All that you need to
assimilate this is the ring definition, which is a solid and stable
footing to work from.

But that definition is apparently is something that you do not
comprehend.

I have a direct argument for why I believe what I believe.
The polynomial as built in abstract algebra is a poor construction.

Your notion of it certainly is poor enough.

Yes, I do see it as a poor notion.

Then try to follow better notions, like those of Arturo Magidin.

I likewise view Arturo's tuple reliance as a poor notion as argued
above.4


It is obvious from the notation that A[x] is a relative of A.
Should it be so surprising that A[x] is A given that x is a variable?

In proper mathematics, variable have domains.
What is the domain of your x in A[x]?

x is in the set A[x].

That does not answer my question.

This is precisely the point. The literature does no better. Connell
defines x to be a "symbol" or a "variable". You prefer "fixed
element". These divergent attempts to define x indicate a problem.


Please let's not slip on the term domain, which
I will take as the set theoretic meaning above, rather the the field
sensetive form within abstract algebra; another terminological cf.

In proper mathematics, when one designates something as a variable, one
automatically imbues it with a domain, even when one does not bother to
make that specification explicit.

So that if "Tim Golden BandTech.com" cannot answer my question about
the domain of x, it is because he dose not understand the mathematical
meaning of "variable."

Here also your language is in conflict with your usage of x above as a
'fixed element'.

I am not the one calling 'x' a variable, you are.
My calling x a fixed element not in the base ring is quite compatible
with my, and Magidin's, model of a polynomial ring.

I am not the only one calling 'x' a variable. I simply am pointing out
a conflict in this math and you are swearing up and down that there is
no conflict. The treatment of x as a variable is precisely what drives
the tuple form. Yet in it x lands as a 'fixed element'. The wisest
among us will admit that we are not discussing two different things.
They will see that we are merely discussing two representations of the
same thing. The simpler representation is back in the algebraic form,
where the product and sum behaviors are already covered by the ring
definition. This is merely a matter of choosing the simpler of two
alternatives. I am left guessing that the tuple form is so strongly
stressed in order to head off or rather to confuse the student away
from simplistic thinking. This you and Arturo may preach sincerely,
but I am free to voice my own opinion.

- Tim
.

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