Re: Understanding the quotient ring nomenclature

On Jun 15, 5:59 pm, "Tim BandTech.com" <tttppp...@xxxxxxxxx> wrote:
On Jun 15, 3:14 pm, Leland McInnes <leland.mcin...@xxxxxxxxx> wrote:

Now, as to your complaint about "continuum modulo math": no one has
tried to explain what an ideal is yet, let alone try and explain (in
anything but the most vague handwaving way) what a quotient ring is
and how it works. Your assumptions about what must be going on are
wrong, and that's why it isn't very convincing to you. If you actually
understood how it did work it would be convincing. I won't bother
trying to explain, however, as I'll simply get accused of trying to
explain the second thing while pretending the first is crystalline. We
don't even have polynomial rings halfway sorted, so there's no point
in discussing the rest.

Yes, I'd like to get to the ideal. That seems crucial.

Sadly I really don't believe we are in any position to do that.

Let me just ask a small question here: was the issue I picked on here
a misunderstanding on your part, or an unfortunate use of language on
your part. You haven't made that clear at all, and it would help if I
actually knew.

A ring is defined as containing elements. Such elements have clear
elemental instances in the reals, the integers, or the complex
numbers, whereas for R[X] there are no such elemental forms.

Okay, so are you implying that a ring must contain numbers as
elements? That a ring cannot have abstract or non numeric (or even non
vector) elements? Or are you saying that a ring as defined can have
that, but to do so is bad?

Instead
we see a form
   (a0,a1,a2,...,an,...)
which is not elemental.

Define "elemental" please. It isn't at all clear to me exactly what
you mean. Do you mean it has to be a single number? What counts as a
number? How do you even define "number"? Why does your definition of
"elemental" apply to the concept of rings which don't use your
definitions?

Already this form carries a variable of
undefined domain and an infinite length of terms.

I'm not sure what you mean by undefined domain. The domain is pretty
clear as far as I can tell -- it is the space of all such tuples. Now
I admit there is some potential confusion here (that I see crops up
later), so let's be more exacting: we can define the domain as the set
of all functions f from N (the natural numbers) to R (the real
numbers) such that f(n) is 0 for all but finitely many n in N. This is
clearly equivalent to the set of all nearly null infinite tuples
where, specifically, a tuple is ordered sequence of real number values
with no infinite ordinal positions. Alternatively we can define the
domain as the set of all polynomials with real coefficients that have
finitely many terms. That's three very specific definitions of the
domain, so I'm not at all clear why you think it is "undefined".

This is far from
elemental. In that the form
   (a1, a2, a3 )
is simpler than the infinite length above then there exists a more
elemental form than the infinite series. In fact, the elements in the
infinite series are reals. Thus there is a hint of an elemental form
in the representation and such an elemental form is instantiable such
as
   - 2.35
which then can be generalized to a symbolic form
   a
which is then developed up to a very complicated form at the
polynomial level especially since no complete instance can be
specified due to the infinite length. Consider whether a derived form
is more elemental rather than less elemental than the starting
element. This would be a wise consideration: if a choice of form
exists then a choice which is more elemental would always choose the
simpler form. Like atomic theory we would make a mistake by labelling
a molecule an atom.

Your rambling here, and arguing about "what is atomic" or some such,
when that doesn't really have any meaning in this context. I could
argue that a digit is clearly more atomic than a real number (since it
is, after all, composed of digits) and thus real numbers are nonsense
and we should only have number 0-9. The "complexity" of the objects
exists so as to allow for a degree of generality. Nothing simpler will
suffice for the needs to which these objects will be put. For you to
claim otherwise is simply a case of ignorance on your part of the
needs to which these objects will be applied. Que sera sera. Try
having a little patience and let these ideas actually be developed
instead of complaining at each and every step.

Still, here we will not resolve our difference so I announce Uncle in
the interest of moving onward toward the ideal.

This is a pointless gesture. You clearly don't understand what I'm
talking about, so there's no point in me going on, regardless of what
you choose to announce. We will simply end up with even greater
misunderstandings on the later material. I'm snipping the rest of you
"Uncle" comments because they are equally pointless.


Apparently I need to ask you a lot more questions because it is less
and less clear to me what on earth you think is going on. Can you
explain to me clearly and concisely how you think these "dense
structures" are "collapsed down to one informational unit"?

You seemed happy enough with Arturo's presentation where he takes
   (a, 0, 0, ... )
to be simply a. It's all the same point isn't it?

No, no it isn't. We can "identify" (a,0,0,...) with a. That is, we can
think of (a,0,0,...) as analogous in the new ring to a in the real
numbers. However analogous to does not mean "is the same object as".
They are different things.

So I must bow and
accept your insistence that the infinite length series is as
fundamental as a singular element of that series in order to move on.
My own perspective really is quite defensible here.

Only in as much as it is a misinterpretation and misunderstanding of
what was said. Without your misinterpretations you don't have a leg to
stand on.

Alright, here you have contradicted yourself within this one
paragraph. First you say
   "For each and every polynomial we care about every non-zero term."
Then you say
   "For certain purposes these terms aren't important"
Again we are on the same silly point of contention and we have fully
amplified it ad nauseum.

I was a little sloppy with my language (it happens to all of us). What
I meant was that, for *general* purposes we care about each and every
non-zero term. For *specific* purposes we can sometimes forgo having
to worry about each and every term, but it matters greatly what that
specific purpose is. *generally* speaking, and unless you know exactly
what you are doing, all the non-zero terms of the polynomial will
matter.

Now, the question is: why is this necessarily a problem? At a glance
it might be, or it might not be. You can't know until you try it out.
I have tried it out, and it isn't a problem. Can you explain very very
clearly and very specifically why this apparently "breaks" anything?

Because of the insistence on infinite length sequences the product of
two such sequences whose terms compute out to length 2n will not be
computable.

Really? Why? I think you simply have a misunderstanding here, because
I'm quite certain they are computable and you are just
misunderstanding something.

If the multiple of any two polynomials has only finitely many terms as
long as the two polynomials being multiplied have finitely many terms,
is it possible to ever get a polynomial that doesn't have finitely
many terms by multiplying two polynomials with finitely many terms?

Polynomials are defined to have an infinite number of terms,

Really, that's news to me. s this your definition, or a textbook one?
anyway, let's press on and see what the misunderstanding lies.

however
the number of nonzero terms obeys the m+n principle only for the
simplest product form that you have selected. Since rings are
concerned with these operators generally then the consideration of the
quantity of product operations ought to be considered within this
specific study. Thus for instance a product series form would fill
this study out nicely, rather than repeating the same old single
product operation. Also it should be noted that my own usage of
'nonzero terms' is weak since the expression
   (0,0,0,0,4.5,0,0,0,5.6,0,0,0,...)
contains two nonzero terms. The degree then matters more than the
quantity of terms. Polynomials in two nonzero terms may exist where
the polynomial is infinite in length. Any position may be occupied by
these two nonzero terms by symmetry and so the expressibility of the
form is not general. We can have a two nonzero term polynomial
represented by
   (0,0,0,...,0, 4.5,0,0,0,0,...,0,5.6,0,0,0,...) .

Right, I believe this is where the misunderstanding has crept in. It
was implicit in the notation (and the use of the word "tuple"), but
apparently this wasn't clear to you: each position in the infinite
sequence is a finite ordinal. That is we cannot have infinitely many
zeros followed by a non-zero term (which is I presume what you intend
with that notation). Indeed, you cannot have infinitely many anything
followed by any term -- there is no omega'th nor (omega+1)th position
-- we can always talk about the n'th component where n is some
(finite) natural number. Now, there are infinitely many (finite)
natural numbers, and hence there are infinitely many positions/
components, but each position/component is describable by *some*
finite natural number. That rules out what you give above (presuming
you intend to have infinitely many zeros preceeding 4.5) since we
cannot give any natural number position to 4.5. I'm sorry for the
misunderstanding -- perhaps this should have been made clearer. It
would have helped if you had actually explained specifically what you
thought the problem might be at the outset however, instead of simply
jumping to the conclusion that we were all horribly wrong.

It should also have become clear had you considered the other
presentations given: polynomials can't have infinite degree; functions
from N to R (as given by Virgil) should also make this point obvious.
Anyway, sorry for the confusion, hopefully you can see what we mean
now?

Do the previous questions give a valid proof by induction that any
finite product of polynomials with finitely many terms gives a
polynomial with only finitely many terms?

No. This subject is an invalid construction as the ring definition
provides the superposition operator and thus to provide a
superposition operator within as a declared ring is not consistent,
especially one which refuses to be placed in the same domain as its
realvalued elements dictate.

Could you please clearly and concise define exactly what you mean by
"superposition operator" and what "domain" you think any of this takes
place in (and even what you think constitutes a "domain"). As it
stands that sentence means nothing to me.
Because the ring itself is providing the quality of superposition and
multiplication the declaration of a form in X which does not allow the
superposition to be taken consistent with the ring operator's quality
yields an invalid construction.

I'll have to ask for the definition of "superposition operator" again,
and preferably the source where you are getting the fact that a ring
is defined to have one. Could you also define what is means for
something to "provide the quality of superposition" and clear
statement as to what quality superposition is supposed to be provided
with would also be nice. While we're at it, sources for "quality of
superposition" relating to ring definitions would be useful too. I
might actually be able to understand where you are coming from at
last.

These entanglements that I've laid out are sort of second order. I'm
sure I've really got you riled up now.

No, mostly just a combination of disappointed (that we had yet more
fundamental misunderstandings) and confused (since you use terms that
I've never heard of in connection with rings before, claim definitions
I'm unaware of, and don't explain or source any of this non-standard
stuff).

My freestanding argument now goes that the ring R[X] is an inherently
invalid construction by the very usage of superposition within the
polynomial, for that superposition does inherently allow the
substitution of real valued X. Thus the insistence on X as
unevaluatable is invalid. By superposing a real value with X as in the
form
   (ax+b)(cx+d)
the ring behavior dictates that X be real. Thus the polynomial form
collapses to a single real value.

I really have no idea what any of this even means. Can you define and
source your terms.


I do get the
polynomial form that you purvey. I've gotten it from the beginning,
but admittedly the stricture of language as communication is
frustrating.

I'm not at all convinced that you. In fact I'm pretty solidly
convinced that you don't -- that there are still deep
misunderstandings and we are a long way from being on the same page.
Certainly I have no idea what you are talking about when you start
talking about qualities of superpositions and collapsing X to real
values: If you want to discuss this field it really helps if you use
the same terms and definitions as everyone else. Please source your
definitions of rings and polynomials that use the terms you do in the
way you do, and provide clear definitions for all of these terms.
Otherwise you're just talking right past me and there is little point.

I'll have to ask what you mean by R[X]^m. Are we talking about the
Cartesian product of m copies of R[X], or are we talking about
polynomials that are m-fold multiples of polynomials in R[X] or are we
talking about something else? Neither construction presents any
difficulties. I might (reasonably) question your desire to bother with
the former, but given any halfway decent argument as to why one might
want to consider it, I would be happy to do so. I'm not sure I see the
problem.

It's just the mth power of an existing polynomial.

Then you have some serious notational abuse going on here since R[X]
is not a polynomial, but a set of infinitely many polynomials. Taking
the mth power of any given element of R[X] (the individual element
being a polynomial) certainly presents no problems, so clearly you are
still not talking about the same objects as I am.

Well, I have abstracted and generalized here a bit. I suppose there is
a crux in claiming that
    (0,0,0,0,...,0,2.2,0,0,0,...)
is symmetrical to
    (2.2,0,0,0,0,...).
The informational content is the same.

The information content is *not* the same since the position is part
of the information. You may as well say that 23 and 32 are the same --
I mean its just the positions of the digits that have changed -- the
information content is the same.

Right, so let's move on to the ideal. I follow you perfectly.

You very clearly do not. I'm not moving on till I know we're on the
same page.

.

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