Re: Understanding the quotient ring nomenclature

On Jun 15, 12:39 pm, Leland McInnes
<leland.mcin...@xxxxxxxxx> wrote:
On Jun 15, 9:37 am, "Tim BandTech.com"
<tttppp...@xxxxxxxxx> wrote:


Thanks for the simple advice Virgil. On the one
hand I see the polynomial form as a farily simplistic thing, but on the other hand I
see the requirement of not evaluating X as very
challenging.


Because a polynomial is different from a polynomial
_function_. If you mix these two, you will surely be
confused.

If you can just get your mind around it, trust me,
it won't be challenging at all.

Treating
this expression as elemental when it cannot be
evaluated is a hypocritical step.

In what sense is it hypocritical exactly?

On the one hand you've insisted that X will never be
evaluated.
On the other hand you're going to wind up evaluating
X.

It is not evaluated when X is an element of R[X],
but it will be evaluated when we consider a polynomial
function. You treat things differently in different
contexts/situations. An abstract, general function
f(x) cannot be evaluated, but once you have a specific
function, say, f(x)=x^2 -- a poly. function -- then you
can evaluate it. You may define an algebra of functions,
with f(x)+g(x)=(f+g)(x) , without evaluating.
What's wrong with that?.

For instance
in the buildout of the complex numbers from your
infinite dimensional
X-form.

I'm pretty sure your problem
is that you don't understand it. I'm going to have
another go at
explaining it in a different way in another post,
but please believe
me when I say that my understanding of it is good,
and I've tried in
vain to see how any of the complaints you make
could apply to my
understanding (really, I have tried) and they just
don't. That means,
to me, that you don't understand it properly.

Entering the quotient ring stage with this basis
is
a fraudulent construction, especially if claiming
to have constructed
an instantiable basis such as the complex
numbers.

What, to you, makes something "instantiable"
exactly? What is more
instantiable about the symbol "i" than the symbol
"X"?

Nice question. I'm not so sure that there is much
better about the
symbol i.
The math which I spoke of that replaces the X format
has no need of
the i format either.
The motivation of deriving i from a real basis seems
pretty far
fetched. From the perspective of ring definition the
sum and product
operations are fundamental and their good behavior is
enough. We need
to address
X X = - 1 .
Here we see an assignment of a real value to one of
these mysterious
domains of unassignables in X^n.

This is false. This operation is taking place at a
_coset_ level. XX and -1 are _cosets_ . Cosets
satisfy the property:

[A+B]=[B]+[B]

[-A]=-[A] (in the Abelian case)

[0] is the id. element in the quotient ring, i.e.,

[C]+[0]=[C]

The coset [XX+1] is equivalent, under this relation,
to the coset, 0, to [0]. Then:

[XX+1]=[XX]+[1]=[0] so

[XX]=[-1]+[0]=[-1]

This is the solution to the mystery of X^2=-1

So this is an identity at a _coset_ level.
Under this construction of the complexes,
the numbers are actually cosets. We don't usually
use coset notation because it is cumbersome.
But in this construction, each term a+ib is
a coset representative.

i is a representative for the coset [X] , the
equiv. class of polys. in R[X] , with remainder
x , when dividing by x^2+1

This statement above
I believe to be
invalid by your own usage of X. You've insisted that
these X do not
perform this way. A real value is being assigned here
to XX.

See above.

You've
already insisted that this need never happen. These X
are somehow
sacred cows, or scare crows, all in a line forming
some fence that
need never be touched in a field of virtual hay which
can somehow
still contain real values.

Of course, you are here excluding the possibility that
you have not fully understood things, and you have
refused to take a middle-of-the-road approach of
reading a book, doing some of the exercises, and then
getting back to us with questions after being informed.
Your misunderstanding of the meaning of XX=-1 shows
that you could benefit from hitting the books.


Complex analysis tags along with real analysis fairly
well. Also
complex math is involved in a fair amount of physics
and engineering
computation. I do accept the utility of this
rotational form. That
rotational form is also existent within the real
value, though it's
apparency in the real form is diminished beneath what
most will
perceive.

In math, you need to define things more clearly.
I don't think anyone can help you unless you clarify
for us what you mean with expressions like:
"existent within the real value" , or "its apparency..."


I accept the junction of these two maths;
the real numbers
and the complex numbers. There need not be so much
magic in getting to the next dimension as is going on in the ring
quotient.

Again, you are using a term 'dimension'that has different (but specific) meanings in different contexts
loosely. Math is an inherently technical area, and
does not lend itself well to informal talk, nor
informal talk by practitioners that are not aware
of the precise meaning assigned in precise contexts.

So: what do you mean by dimension?. I assume you
are considering some vector space dimension. Why
are you doing this when working with rings?

You are being very loose with your terminology
and phrasing. There is no way around understanding
the precise meaning of each term and definition.
And I agree with others that you have not done
your part of the deal in trying to understand.

Again, why not meet us halfway and read a chapter
on this, do a few exercises, and then get back to us?


Rather, the
fundamental definitions of the real numbers (as I
recall there are already six or seven versions of them) can be generalized.

It would be great to see your quotient ring go to
work. I want to
understand how this happens. Just how you can breach
the hump of
claiming X to not carry any instantiable value and
then turn around
and substitute real values in is puzzling to me.

It is the difference between a formal expression and
a function. You can analyze (formal) properties of
an object like R[X] without instantiating. And
you can talk about an element of R[X]

I
will try to keep
open minded. I am ready for your presentation. Still,
it amazes me and amuses me how much you all are willing to eat that hay from the field
and obey the bizzarre fence that has been laid.

Of course, you are completely dismissing off-hand
the possibility that you just don't get it.


I do find this
circumstance understandable as mathematicians in
their plurality and over generations cover new ground and fill out any voids in the
possibility space. This is where 'new' work lays.
Yet here I suspect that eventually this area will be withdrawn.

Seriously?. You have been dealing with quotients
for what, a month?. And you think your opinion
is well-informed?.

I'm very confident that it won't be. Look,
polynomial rings are used
regularly in all manner of modern mathematics, and
are particularly
foundational in modern algebraic geometry and
algebraic number theory.
Not only have these fields made significant
theoretical progress,
they've produced very solid practical results.
Polynomial rings are
quite rigorous, well defined, and lead to solid
practical results.
They aren't going anywhere.

The rotational aspects of
the polynomial are already inherently within the
real number. So
rather than build a confusing structure up above
the polynomial (the
quotient ring)

You are being loose and imprecise. This is built
above the polynomial _ring_

all that needs to be done

For whom?. Why?. We cannot read your mind. You
seem to be making assumptions that are clear only
to you.

is to
build out the real
number more generally. Then, rather than worry
about the need for
infinite length sequences

why do you worry about them?. Many have made
an effort to explain to you. Why not respond
in kind and explain clearly what your problem is,
assuming as little as possible?. I think many
here are not clear on what your doubt is about.

Specifically pinpoint the contradictions
that you see.


the n-length sequence
can hold its own since
products will remain at length n. This math
relies upon a modulo
effect which is already present within the real
valued number.

Right, well please demonstrate how all of this
building up can be used practically in algebraic number theory and algebraic geometry. I
assure you, you'll simply end up reconstructing the
same thing with
different terminology.

If this is true then all the better. There will be no
'X' at the bottom of the construction which has no substitutable quality.

So when you describe something axiomatically, what
should you do?. You can define an object that satisfies
, say, the group axioms, and derive some properties of
them in a purely formal way. Should we abandon
axiomatic descriptions?



Yes, I
do believe that this is what I'm talking about. There
will be no need of infinite length progressions.

Further the complex numbers take
their place as a natural extension from the real
numbers inherently within the construction.

??

At the basis of the
terminology that I
suggest is merely a generalization of the real number
and its own
discrete symbollic signage. Thus the chasm between
continous and
discrete terminology is layed out on a different map,
yet this map
does match up to much of existing mathematics and
especially the
development of what might be called multidimensional
geometry.

??. I have no idea of what you mean by all this.

The
naming of all these genera of mathematics is a
disgusting thing and it has taken me a long time to arrive at the subject of abstract algebra.
It does seem to be the correct spot to focus on.

I think that more could be done, like, say the
way the IEEE goes about setting standards.



- Tim
.

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