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.

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. 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 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. 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.

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. 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. 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. 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.


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.

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) all that needs to be done is to build out the real
number more generally. Then, rather than worry about the need for
infinite length sequences 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. 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. 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.

- Tim
.

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