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

On Jul 10, 6:32 pm, Bill Dubuque <w...@xxxxxxxxxxxxxxxxxxxx> wrote:
"Tim Golden BandTech.com" <tttppp...@xxxxxxxxx> wrote:

Great post Bill.
I have read parts of all of your links.
It is fascinating how much dancing around the terminology of sign has
gone on.
As in your quote Hamilton uses the phrase
"the sign sqrt(-1)"
which says alot about how freely some mathematicians are willing to
speak, and how intepretation of symbollogy does leave some room to
admit that beneath every definition may lay some motive of a less
secure form. We are caught attempting to decode as much as to
construct. The less cryptic the construction then the better it will
be for its simplicity and lack of potential for conflict.

A positive development is the elimination of redundant material for
the sake of achieving the simplest definition, thus yielding more
information as consequential rather than axiomatic. Here is where your
own description of a reliance upon tuples takes on a weakness. The
polynomial algebra is already inherent within the ring definition. In
order to do it consistently with your format I have to accept the ring
A[x] as you define it but treated simply as a set. Thus I might assign
B to be equivalent to this set A[x], whereupon some elements a(n) and
one special element x are chosen from B. From these choices of
elements we can freely construct the polynomial with its algebraic
behaviors using the product and sum of that ring without ever
approaching any tuple notation or definitions. This is merely one
choice of many possible constructions from this set B. The rules
imposed on the tuple sum and product are consistent with the ring
algebra.

I am starting to view set theory as alive rather than dead. Your
Kanamori link shows it to be so. Have you ever troubled over the
plurality of elements? Typically a set is claimed to have only one
uniquely defined element. Yet we witness a plurality of that element
in expressions so that I might have
a0, a1, a2
being in the set of reals and all being the value 2.0. This argument
sounds almost as silly as the generalization of sign, yet if I claim
to remove a0 from the set of reals have I then removed a1 and a2
indirectly? I know that the language I use here can be criticized as
poor, yet this concept remains a simple one so I see no way to confuse
the language. This point is indirectly related to the topic here and
makes itself apparent when considering real coefficients and
attempting to pull the reals from A[X] in order to see what remains in
A[X]. I don't know that this is pivotal to the argument, but it is
ground to be covered. The keyword here is 'indirect'. The free usage
of reversals by mathematicians contorts indirection. There are
occasions when a generalized form is taken as fundamental whereas that
generalized form was constructed via its simpler forms. This is
pertinent to the usage of X and its workspace A[X].

My criticism of the constuction A[X] does not become substantial until
real coefficients are specified. Before this stipulation the set A[X]
was completely abstract. The specification of real coefficients then
requires that this set A[X] has taken in something substantial and
possibly concrete, such as the value 2.0. This is far from abstract.
The question arises
What else is in A[X]?
The answer to this question from the establishment posters is that
whatever else is in A[X] those real values are merely a subset of a
larger set which somehow still remains abstract. By analogy the way
that the reals can be viewed as a subset of the complex numbers has
been applied where a real x can be mapped into a complex value as
x + 0 i .
No such mapping is possible from the reals into A[X] and so this
interpretation carries little meaning by way of analogy. Interestingly
on the interpretation of a real which is a one tuple being converted
to a complex which is a two tuple your Peano axiom will not provide
the mapping since an equivalence requires a two tuple. Especially your
quote of his interpretation is supportive of this beyond definition.

My own interpretation goes back to the ring definition and treats A[X]
as fully specified by substantiating real values, thus leaving A[X] as
merely the set of reals. This would break the value of the
construction. Hankel seems to have thought similarly and it is
interesting that the axiom of choice is nearby, though I don't claim
to understand it. I believe the Hankel criticism was in your Bergman
link. I appreciate Bergman's willingness to speak his mind.

The context of the thread is mostly about A[x] with real coefficients.
What type of product is bx where b is real? Is this a ring product?
And when we add a real to that a + bx is this a ring sum?
Here is more the crux that I would like your opinion on.

This is a matter of _definition_, not a matter of _opinion_.
By the definition of the standard set-theoretic construction
(or reduction) of A[x] that I described at this threads' start,
A[x] is the set of almost-null vectors in A^N with operations
of componentwise addition and with Cauchy product (convolution).

By definition the symbol 'x' denotes (0,1,0...) in A[x].
Further, since a -> (a,0...) is a ring isomorphism of A
onto the constant polys in A[x], by abuse of _notation_,
we may (re)use any existing notation 'a' for elts of A
as names for their constant poly images (a,0...) in A[x].
E.g. we may notate constant polys in Z[x] by some integer
notation, say by radix notation, decimal or binary, etc.

Composing these 'atomic' names with names for ring operations
yields names for all elts of A[x], since A, x generate A[x].

Thus the term 'a + b x' is a _notation_ for an elt of A[x].
By definition this denotes the element of A[x] obtained by
evaluating (a,0...) + (b,0...) * (0,1,0...) in A[x], i.e.
it denotes (a,b,0...). Similarly '32' is decimal notation
for the integer SS0 + SSS0 * (S^10 0) in a Peano algebra.
Don't confuse notation (syntax) with denotation (semantics).

So, to answer your question, b*x indeed denotes the result
of evaluating the product in A[x] of elts denoted by b, x.
Ditto for said sum. We employ the notation 'b' for (b,0....)
merely for convenience. Use of such notation does not imply
that we work simultaneously in the two rings A and A[x],
by 'typecasting' or 'coercing' elts of A to elts of A[x].
In fact one can eliminate such notational abuse and work
exclusively with the fundamental set-theoretic tuple
notation (a,b,c...0...) for elts of A[x]. Doing this
completely eliminates any and all notational subtleties.

Indeed, historically, one of the major accomplishments of the
set-theoretical definition of algebraic structure definitions
was to eliminate imprecise notations. Thus, by eliminating
the syntactic term 'a+bx+cxx' in favor of the semantical elt
(a,b,c,0...) there is no longer any doubt about what 'x' or
'+' denotes, or about when such terms are equal, since, by
set theory definition, tuples are equal iff they have equal
components. Similarly for the field C of complexes a + bi
vs. their pair representation (a,b) discovered by Hamilton.
Before Hamilton gave this semantic reduction of C to pairs
of reals, prior syntactic constructions (e.g. by Cauchy) as
formal terms 'a+bi' were subject to heavy criticism regarding
the precise denotation of their constituent symbols, e.g.
precisely what is the meaning of the symbols 'i', '+', '='?.
In modern language, Cauchy's construction of C is simply the
the quotient ring R[x]/(xx+1), which he described essentially
as real polynomials modulo xx+1. However, in Cauchy's time
mathematics lacked the necessary (set-theory) foundations to
rigorously define the polynomial term algebra R[x], and the
congruence classes modulo xx+1, etc, The best Cauchy could
do was to attempt to describe the construction in terms of
imprecise natural (human) language; e.g, in 1821 he wrote:

In analysis, we call a symbolic expression any combination of
symbols or algebraic signs which means nothing by itself but
which one attributes a value different from the one it should
naturally be [...] Similarly, we call symbolic equations those
that, taken literally and interpreted according to conventions
generally established, are inaccurate or have no meaning, but
from which can be deduced accurate results, by changing and
altering, according to fixed rules, the equations or symbols
within [...] Among the symbolic expressions and equations
whose theory is of considerable importance in analysis, one
distinguishes especially those that have been called imaginary

While nowadays we can rigorously interpret such expressions
in terms of formal languages or term algebras, it was far too
imprecise in Cauchy's time to have any hope of making sense
to his colleagues, e.g. Hankel countered scathingly:

If one were to give a critique of this reasoning, we can not
actually see where to start. There must be something "which
means nothing," or "which is assigned a different value than
it should naturally be" something that has "no sense" or is
"incorrect", coupled with another similar kind, producing
something real. There must be "algebraic signs" - are these
signs for quantities or what? as a sign must designate something
- combined with each other in a way that has "a meaning." I do
not think I'm exaggerating in calling this an unintelligible
play on words, ill-becoming of mathematics, which is proud
and rightly proud of the clarity and evidence of its concepts

Thus it should come as no surprise that Hamilton's elimination
of such "meaningless" symbols - in favor of pairs of reals -
served as a major step forward in placing such numbers on a
foundation more amenable to the mathematicians of the times.
Although there was not yet any theory of sets in which to
rigorously axiomatize the notion of pairs, they were far easier
to accept naively - esp. given the already known closely
associated geometric interpretations of complex numbers.
Hamilton introduced pairs as 'couples' in 1837 [1]:

p.6 The author acknowledges with pleasure that he agrees with
M. Cauchy, in considering every (so-called) Imaginary Equation
as a symbolic representation of two separate Real Equations:
but he differs from that excellent mathematician in his method
generally, and especially in not introducing the sign sqrt(-1)
until he has provided for it, by his Theory of Couples,
a possible and real meaning, as a symbol of the couple (0,1)

p.111 But because Mr. Graves employed, in his reasoning, the
usual principles respecting about Imaginary Quantities, and
was content to prove the symbolical necessity without showing
the interpretation, or inner meaning, of his formulae, the
present Theory of Couples is published to make manifest that
hidden meaning: and to show, by this remarkable instance, that
expressions which seem according to common views to be merely
symbolical, and quite incapable of being interpreted, may pass
into the world of thoughts, and acquire reality and significance,
if Algebra be viewed as not a mere Art or Language, but as the
Science of Pure Time.

It wasn't until the much later development of set-theory that
the fundamental nature of the pairing construction was explicitly
realized. Indeed, as Kanamori wrote on p.289 (17) of [2] in
his interesting article on the history of set theory:

In 1897 Peano explicitly formulated the ordered pair using
"(x;y)" and moreover raised the two main points about the
ordered pair: First, equation 18 of his Definitions stated
the instrumental property which is all that is required of
the ordered pair:

(x,y) = (a,b) iff x = a and y = b

Second, he broached the possibility of reducibility, writing:
"The idea of a pair is fundamental, i.e., we do not know how
to express it using the preceding symbols."

Here is somewhat challenging the reals as a subset of the complex
numbers, presuming that the tuple-centric notation is authentic to
these forms. Let's not forget that geometry itself is nearby along
with the conception of dimension which became a sore point on this
thread back a ways when applied to these tuples. You are awakening
that point here. Polysign can clean house if it is granted fundamental
status. Building polysign from the real numbers is not at all what I
am talking about, though that is how many maintain their senses.
http://bandtechnology.com/polysigned

Great post Bill. Thanks.

- Tim

Once set-theory was fully developed one had the raw materials
(syntax and semantics) to provide rigorous constructions of
algebraic structures and precise languages for term algebras.
Nowadays the polynomial ring A[x] is just a special case of
much more general constructions of free algebras. Such objects
and their genesis via so-called 'universal mapping properties'
are topics normally discussed at length in any course on
Universal Algebra - e.g. see Bergman [3] for a particularly
lucid presentation.

--Bill Dubuque

[1] William Rowan Hamilton. Theory of conjugate functions,
or algebraic couples; with a preliminary and elementary
essay on algebra as the science of pure time
Trans. Royal Irish Academy, v.17, part 1 (1837), pp. 293-422.)http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/PureTime/PureTim...

[2] Akihiro Kanamori. The Empty Set, the Singleton, and the Ordered Pair
The Bulletin of Symbolic Logic, Vol. 9, No. 3. (Sep., 2003), pp. 273-298.http://links.jstor.org/sici?sici=1079-8986%28200309%299%3A3%3C273%3AT...
PS http://www.math.ucla.edu/~asl/bsl/0903/0903-001.ps
PDFhttp://ifile.it/87wb214

[3] George M. Bergman. An Invitation to General Algebra
and Universal Constructions.
PS http://math.berkeley.edu/~gbergman/245/
PDFhttp://ifile.it/hwex61q

.

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