Re: Understanding the quotient ring nomenclature

On Jun 15, 11:31 am, "Tim BandTech.com" <tttppp...@xxxxxxxxx> wrote:
On Jun 14, 10:44 pm, Leland McInnes <leland.mcin...@xxxxxxxxx> wrote:

On Jun 14, 12:58 pm, "Tim BandTech.com" <tttppp...@xxxxxxxxx> wrote:

On Jun 13, 3:28 pm, Arturo Magidin <magi...@xxxxxxxxxxxxxx> wrote:

On Jun 13, 9:02 am, "Tim BandTech.com" <tttppp...@xxxxxxxxx> wrote:

You've presented a polynomial in real coefficients but insist that X
is an indeterminate variable.

People have merely been insisting that the X doesn't represent some
element of R, but is another ring element entirely, here represented
as a formal symbol.

Here I am engaged in a game of whack-a-mole.

Trust me, I feel the same way: each time I try and explain something
you've apparently misunderstood, you come back with a reply that shows
misunderstandings of 5 or 6 new things. Your background really is
lacking, and that makes the explanations I and others give often seem
weird to you because we are often using language very specifically,
and you don't seem to know what the various specifics mean,
misinterpret them, and than take those misinterpretations and build a
whole new edifice of misunderstanding out of them.

You have just stated that X is another ring element entirely.

Yes, and it seems you've misunderstood what I meant entirely, so let's
nip this in the bud and try and explain it all carefully because it
might provide us with another way to explain to you what the X
actually means. Bear with me as I'm about to go over something that
you already know, but pay attention because it is going to provide the
groundwork for an analogy later.

Let's start with a concrete example you do understand: the real
numbers. Now, suppose we want to consider the complex numbers. How do
we do that? Well we take the ring of real numbers, and add to it an
element "i". Now, "i" is not a real number, it's an object from
outside the ring of real numbers. It is so elusive that, in fact, we
can't use a number to denote it, instead we have to just use a symbol
(usually "i"). Now, to make the real numbers plus this extra symbol
"i" into a ring we need to have addition and multiplication defined.
Addition and multiplication of real numbers is fine (since it was
already a ring) but what happens if we try and do addition and
multiplication that involve "i"? We want to have a+i be meaningful for
any real number a. Now we can't actually evaluate a+i to any real
number, so we'll just have to say that a+i is another element of the
ring (thus, we started by adding just i, but to make sure we get a
ring we've now added a+i for real number a). We also want b*i to be
meaningful, so we'll just have to say all of those are elements of the
ring too. But then once we have all those b*i we'll have to allow
addition with each of them too so we'll have to add a+b*i for each
pair of real number a and i to to thing ring. But wait, we only
considered what happens when we add or multiply real numbers with
"i"; what happens when we add or multiply "i" with itself? Well we
want i+i to be meaningful, so we better add that to the ting as
well ... except that we have the distributive axiom for rings so i+i =
i(1+1) = 2*i, and we already added 2*i to the ring, so we're okay. You
can verify that the rest of the possibilities for adding "i" to itself
work out similarly. Now what of multiplying "i" with itself? Well to
get the complex numbers we have a special requirement for this extra
element "i" that we are adding: specifically we require that i*i = -1.
That means we don't need to add any new elements for multiples of "i"
with itself since -1 is already in the ring, and further multiples
will either be a real number again, or a real number times "i", which
we already added to the ring. You may want to check that the
distributivity axioms for rings (along with knowledge that the real
number are already a ring) ensure that these cases mean that there is
nothing else we need to add to make the real numbers, extended by this
extra (non real number) element "i" into a ring. The result is that we
need all objects of the form a +b*i where a and b are real numbers and
"i" is an extra (non real number!) element we added to the ring of
real numbers.

We could go through the same sort of process to construct the
quaternions by adding extra elements non real number elements "i",
"j", and "k" to the real numbers, and then adding all the extra
elements we'll need to make this larger set into a ring (with the
appropriate properties applied to "i", "j", and "k".

Note that in both these cases "i" is nothing more than a symbol used
to denote a new ring element we wish to add (so as to create a new
ring) to the real numbers. That is, to build a bigger ring we add a
new element to the ring, assume that element has particular properties
with regard to how it multiplies with itself (and with any other new
elements we wish to add), and then add all the extra elements we need
to make sure the resulting new ring is closed under addition and
multiplication.

Now, what say we want to add a new element a bit like "i" to the ring
of real numbers, but we don't want to presume it has specific
properties as to how it multiplies with itself? This is reasonable,
since by not assuming special properties we get the "fully general"
case. So, all we a re going to do is pick a symbol to denote this new
element we want to create, add it to the ring of real numbers, and
then add in all the extra elements we'll need to ensure that the real
numbers along with this new element form a ring. I'm going to pick the
symbol "X" to denote this new element. We then go through the process
just as we did with "i" for the complex numbers: we'll need a + X, and
also b*X, and then a + b*X. Now what about adding "X" to itself? Just
as with the complex number example, we'll have X + X = X(1+1) = 2*X
and so on, meaning we have already added all the possible sums of
"X"'s with those b*X. What about multiplying X with itself? Well we
don't have any requirement as to what happens when we multiply X with
itself, because we wanted to maintain the greatest generality we
could, so X*X is just X*X -- we'll have to add that to the ring. But
then we'll have to go back and add the elements a + b*X*X for each
pair of real numbers a and b; but wait -- we can also form the sum a
+b*X + c+d*X*X = (a+c) + b*X + d*X*X. Now (a+c) is just another real
number, so we're okay there, but the rest... well, we'll just have to
add elements of the form a + b*X + c*X*X for each triple of real
numbers a,b,c as well. But then what about X*X*X? We'll have to add
that to the ring as well. And then we'll have to allow elements of
them form a + b*X + c*X*X + d*X*X*X as well ... and we'll just keep
going on. Now, before you panic of infinities introduced by this "keep
going on", we'll deal with that issue later in this post; forget about
it for now, because it will only distract you from my point.

Note that all we've done is exactly as we would do to create the
complex numbers, or the quaternions, except we didn't place any
restrictions on how the new element we are introducing is to behave
with respect to multiplication with itself. If you want to look at it
as a more general version of the complex numbers that might be a start
-- it is merely an analogy, but it might help you get your head around
it. When I say "X" is another ring element entirely I mean it in the
same sense that "i" is another ring element entirely when we go about
constructing the complex numbers from the real numbers. It is
introduced from outside the ring of real numbers and thus can't be
written as any real number, doesn't stand for any real number, and
can't be evaluated as any real number. It is just "something else".
The "X" is just like that: it is somethign from outside the ring of
real numbers, doesn't stand for any real number, and can't be
evaluated as any real number. It is an extra element for which we have
no other name than "X". Does this make it clearer?

Yet in
the quotient ring construction on the one hand we might take a ring
instance such as Z which does have fully instantiable elements such as
+ 3, + 5
yet when I ask what the elements of X are there is no answer.

I'm not sure whether I'm nipicking your language, or picking a
misunderstanding here, but "X" doesn't have "elements", it *is* an
element. The ring Z[X] of all polynomials with *integer* coefficients
has elements. "X" is one of those elements. So is 2+3*X+7*X*X*X*X.

Thus
expressions in X are not elemental. I simply give you Z as a fine
counterexample.

I'm still not sure what you mean by "instantiable". If you mean: can I
give you a set of things with "X" many elements in them ... well no.
But then you can't give me a set of things with "-7" things in it, nor
can you show me a line that is exactly "sqrt(2)" long, not can you
point out to me an example of "2+3*i" apples. Thus if I am taking your
meaning of "instantiable" right, then neither the integers, nor the
real numbers, nor the complex numbers are "instantiable". You'll have
to define "instantiable" for me before I can answer this properly
though.

Thus in specifying a ring quotient usage of R[X] in
the construction takes on dubious meaning.

It has an explicit meaning, but it is an abstract one. Then again, so
is the notion of "number" to begin with -- try reading some philosophy
of mathematics. Certainly limits in calculus are equally abstract. So
are Dedekind cuts, or Cauchy sequences. So are irrational numbers. If
you don't like taking an abstract generalisation and seeing what you
can get from it (and as I said in another post, the results gained
from polynomial rings in various fields of mathematics are significant
and remarkable) then you don't really want to do mathematics;
mathematics really is the art of abstraction. That's fine, but stop
trying to pretend that all of this isn't rigorous when it is -- it's
just abstract.

There is nothing elemental
about it. Especially when enforced to be infinite in length there is
something rather anti-elemental about R[X].

On the contrary, when viewed the right way it is indeed quite
fundamental -- it's the most general way we can add a single "new"
element to R and close it up under multiplication and addition to get
a ring. The complex numbers are a special case of this. All your
different polysigned numbers can be thought of as special cases of
this (vary the requirements you put on the element you add and you can
get all your polysigned numbers). A whole host of other constructions
are equally special cases -- we opt for the most general since we can
build all these other special cases out of it in terms of quotients:
the complex numbers, your polysigned numbers, and an infinite variety
of other cases which you haven't considered.

Yet upon choosing a real
value or a complex value for X, which I believe is where practical
usage of the polynomial form lays,

No, it really doesn't.

then all of these concerns
evaporate and the entire construction simply collapses to a real or a
complex value, which then exposes the fact that the entire polynomial,
while capable of introducing complexity of raw values, does nothing
for dimensional structure of the domain X, which is completely
counterintuitive to the information enclosed in its general form.

Well yes, that would be true if the only way to do anything was to
consider X to be a some real number. That isn't the only way to do
things, however, so this criticism is quite moot.

This
is just one of several contradictory contexts of the ring quotient
construction
R[ X ] / ( X X + 1 )

I'll repeat again: there are no contradictions if you actually bother
to understand the construction. We have even gotten as far as getting
you to fully understand R[X], let alone quotient rings, so can you
leave the speculative criticism aside until you actually understand
what you're talking about.

Right, except for the requirement that all but finitely many of the
components are zero. I shied away from explaining the reason for this
since the only good explanations I know (that don't involve explaining
it in terms of polynomials of finite but unbounded degree, which were
apparently not a good way to explain it to you) wade us into rather
deeper waters.

I think this is more like a game of trickery. You get a student to eat
a fist step that doesn't quite jibe with the next step. Then you cover
that first over, tunnel over to the next thing and present it as if it
were a crystalline form.

No one is trying to trick you Tim. There are libraries with lots of
books on these topics. There are a whole host of free online resources
for these topics. Feel free to actually read any of those instead of
listening to us. Perhaps read some of those and ask about anything
that doesn't quite match with what we say. You'll likely get
explanations for why the different things do jibe, it just isn't
necessarily obvious at first.

As to the issue of moving to the second thing before the first is
clear: that's your doing, not ours. You are the one who jumps ahead
and brings up issues about things that have not yet been addressed in
our explanations and demands answers. As I said, it feels like Whack-a-
mole to me too, since as soon as I explain one point, you are
demanding answers on something else down at the third of fourth step
and getting way ahead of yourself. Am I to simply ignore your demands
for explanation, or do things somewhat out of order out of necessity
of keeping up with your demands?

I understand that series which converge to zero are nice things to
work with. I am fine with your relaince upon these.

Language use nitpick time: series that converge to zero are something
quite different from nearly null infinite tuples. A series is a sum
(usually infinite) and to say it converges to zero is to say that the
sequence of partial sums has limit zero. That has nothing to do with
nearly null infinite tuples. Either you are misunderstanding something
here, or you are being very sloppy with language. It's this sort of
thing that I believe infuriates Arturo because he will take you at
your word on "series which converge to zero", see that it has nothing
to do with what he was talking about at all, and call your comment
nonsense. I'm going to give you the benefit of the doubt and assume
you meant something else but were sloppy with your language.

But this does not
alter my criticism of the quotient ring as a constructor of the
complex numbers. Especially the quotient ring as a definitional stage
which leaps from the simplicity of modulo discrete math over onto
continuum modulo math is not convincing. I already know you'll say
these words are meaningless.

Polynomial rings and quotient rings as constructors of the complex
numbers are not necessarily the most natural way to go about it. If
all you ever wanted to do was construct the complex numbers then
polynomial rings and quotient rings thereof really aren't worth the
trouble. The key is that they aren't just used to construct the
complex numbers. You seem worried about how complicated this
construction is (although it is simpler than you are making it for
yourself, honest). You seem to want to build up to the specific things
you are interested in -- and that's okay if you're only interested in
those specific things. Polynomial rings are far more general (and are
abstract so as to allow that). Thus instead of building up to the few
specific cases you happen to think of, you can at once encompass all
the possible cases and narrow to whatever specific case you care to.
Importantly you can narrow to specific cases you might not have
otherwise thought of. Equally, given the powerful general framework,
you can prove things for the general cases and have those results
immediately apply to all the specific cases rather than having to
prove the same thing over and over again. So yes it is a little
complicated, and yes it is abstract, but the complication comes from
the abstraction, and the abstraction provides vast sweeping generality
that allows you to do far more, and see further, than special cases
allow. This is why you struggled to see that P4 was just RxC, but
others using polynomial rings and quotient rings can make the
connection simply and easily: the greater generality allows for much
greater flexibility and deeper insight.

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.

Well yes, the issue arises, but then you stop and think about it
carefully for half a minute and realise that it isn't a problem.
Before we get to why, let me pick a nit in your use of language: you
say "the polynomials are not rings" which is fundamentally wrong. The
*set* of *all* polynomials is a ring (note the singular). Saying "the
polynomials are not rings" is saying that the individual polynomials
are not each rings (by the failure to denote the space/set/collection
of polynomials, and by the use of the plural "rings") which is just
nonsense. Either you were sloppy with your words (which happens, but
is probably why people call what you say nonsense -- it probably often
is if you read what you say (as others are forced to) and not what you
mean) or you have truly fundamental misunderstandings.

Yes, I pushed my criticism too far in my last post. You see the change
in responses too. Here Arturo now says that he did concede the
dimensional interpretation whereas before he obfuscates that. I am
open to having truly fundamental misunderstandings. I am also open to
mathematics having constructed truly fundamental misunderstandings.
This latter stage is one which good mathematicians need to open to.
Yet it is just where the treachery lays. By stepping over the edge
you'll make yourself incredible. You are willing to dance closer to
this edge than the others. I respect that. I respect that I will not
convince you to walk the tightrope over to the edge of another
precipice. I would merely like to cleanly understand the holy ground
that you've decided to inhabit, plastic hammers holstered but ready
for action.

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.

Now, back to why we don't have to worry about
"superinfinite" (whatever that means) issues. Remember when I pointed
out that the nearly null requirement on tuples is equivalent to the
requirement that the polynomials have finite degree? Well given a
polynomial of finite degree n and a polynomial of finite degree m,
then their product will have *finite* degree m+n. Thus, as long as our
polynomials have finite degree, we know more have to worry about
multiplication causing problems than we have to worry about addition
of finite (but arbitrarily large) natural numbers causing problems.
You don't believe there is some fundamental issue with arithmetic
where it has to concern itself with "superfinite" numbers do you?

When all practical usage of polynomials collapse these informationally
dense structures down to one informational unit of their claimed
content then yes, I do see a problem here.

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"? For each
and every polynomial we care about every non-zero term. Yes we talk
about the degree of the polynomial, but that is simply extracting one
property out so as to prove that in general certain properties are
preserved. For practical purposes we usually care about all the
individual terms. For certain purposes these terms aren't important --
the proof above just happens to be one of them.

You have carefully stated
that you will work in null terminated sequences. These taken under
product will propagate upward in length. I don't see that an honest
answer can dismiss this awareness either way.

And no one did, except for technical issues with your sloppy use of
language again. You say the "length" of the sequence will increase --
the length of the sequences are always the same: countably infinite.
Thus Arturo simply complains you are talking nonsense, because you are
if read literally. Being more generous in interpretation: the position
of the last non-zero term will, indeed, appear later if you multiply
two sequences together; you actually have to say that though (or
define beforehand specifically what you mean by length, and explain
why you are using it in a non-standard way).

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?

Let me instead ask you a series of questions, and hopefully we can pin
down where you think the problems arise.

How many terms does the result of multiplying the 3 term polynomial 1
+ X + 2*X^2 with the 2 term polynomial 2 + 4*X^6 have?

Is the answer to the previous question a finite number?

How many terms (at most!) do we expect the result of multiplying a n
term polynomial with an m term polynomial to have (where n and m are
finite numbers)?

Is the answer to the previous question a finite number?

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?

If the answer to the previous question is yes, please give an example
of two polynomials with finitely many terms whose product doesn't have
finitely many terms.

Suppose I want to find the result of multiplying together n different
polynomials where n is a finite number and all the polynomials have
only finitely many terms. For purposes of induction let's assume that
the result of multiplying n-1 polynomials that all have finitely many
terms is a polynomial with finitely many terms. Why can I not multiply
together the first n-1 polynomials to get a single polynomial with
finitely many terms, and be left with finding the multiple of two
polynomials, each with only finitely many terms?

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?

If we take the set of polynomials with finitely many terms, and *all*
finite products thereof, are there any polynomials in that set that
don't have finitely many terms?

If the answer to the above question is no, in what way does the ring
formed by taking all polynomials with finitely many terms and all
finite products thereof have any problem with multiplication?

Like working with
infinity within calculus only its limit can be taken. I believe that
infinity is something to be careful with.

Yes, so do I. The thing is, I have been careful with it, and I've
convinced myself that no entanglements arise when using infinity in
the way that I've described.

It can be used, but should
be used carefully.

See above.

I think that dismissal of this feature especially
to a concerned student is not wise. Simply owning this feature is
enough and then moving on from there rather than being in denial is
appropriate.

But that's what I'm doing. The only point I'm arguing here is your
claim that this somehow falsifies, or otherwise shatters the
consistency of polynomial rings. It doesn't. Can we move on now?

The future usages of the math being constructed may cause
conflict here. Aren't you open to this? Especially within the concept
of ring whereby the operators are abstracted we should be capable of
building some continuous iterate product or some such form where the
result remains within the original domain, otherwise we are not in a
ring system. You have chosen to work in an unistantiated form R[X] and
if we raise this form to
R[X] ^ m
then we have constructed the same format but at a level of greater
complexity.

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.

Since we are engaged in generalities to the fullest you
can take this one as a sneering example. You happily limit the length
of the series whilst you insist that the series be infinite in length.

No. The requirement is that there be infinitely many components to the
vector, but that all but finitely many of those components must be
zero. At this point it is much easier to describe in terms of
polynomials: we simply require that the polynomial have finitely many
terms. Exactly how many terms that is is not bounded -- any natural
number is possible -- but it must be a *finite* number.

Please do not take this criticism personally. This is the
establishment mathematics you are upholding. I seriously doubt if one
man alone would ever arrive at this format of construction. This is
the work of one man standing on another man's shoulders. Yes, we'd not
get very far without this behavior. Yet where this behavior can get us
is still dubious. Twisting a zero into an infinity as a child plays
with a loop of yarn should we assume the child's deep understanding?
This situation is far more complicated yet the claim of mathematics as
fundamental construction should be upheld. Scrutiny on this area does
not resolve to the simplistic underpinning that some believe. Instead
we have waivering nuances.

No, what we have a a disparate range of attempts to explain the
situation to you, given at different levels of formality, by different
people, all in the space of a few USENET posts, and with you lacking
the background to make the explanations relatively easy. The result is
that you don't get a single clear consistent development from first
principles up. If you add to that the fact that you are rarely if ever
willing to stick it out with the first principles (and instead want to
leap ahead before you've made sense of the beginning), and the result
is, yes, a little all over the map. Certainly my explanations, trying
to hit the different misunderstandings (which occur all over the map)
on the head, has rambled across the place. But then if you want a
clear and consistent development from first principles you should do
as several people here have suggested and actually invest in a good
textbook, or take a course.

You aren't making sense anymore. Take a moment and actually read what

Sorry, but this is still making sense to me. If in calculus we were to
specify a large n and then this large n lead to an even larger m
within a computation we would not arrive at a computable result.

Okay, let's try with the questions again:

How many terms must a product of two polynomials have for it to become
uncomputable?

If a product of polynomials has only finitely many terms, can we
compute each of those terms?

If we compute all of the terms appearing in a product of polynomials,
have we computed the product of the polynomials?

Given that any finite product of polynomials with finitely many terms
must itself have only finitely many terms, in what way is it not
computable?

Finally:

Given that the real numbers contain uncomputable numbers, how do you
reconcile your need for computability with your desire to work with
real numbers?

Yet
the math which you purvey does contain this behavior. Products reach
ever greater complexity, hence the requirement of not just a length n
sequence, but an infinite length sequence. It's so simple to see this.

Well yes, we have infinite length sequences. I believe this was stated
at the very outset. Why is this a problem? You can complain that it is
unnecessarily complex, but see my earlier point about abstraction and
generality. Certainly it doesn't lead to any inconsistency or
contradiction. If it does, then please provide a nice complete proof
thereof.

No, this point does not prevent you from going onward, but it is a
point of valid awareness.

Sure, whatever. No one objected to the notion that we will require
infinite length tuples (or polynomials of arbitrarily high degree).
People objected to your claim that this somehow made the mathematics
inconsistent, or false, or wrong. It doesn't. So: it happens, it
doesn't cause any problems, can we move on now?
.

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