On Jun 20, 8:50 am, "Tim BandTech.com" <tttppp...@yahoo.com> wrote: > On Jun 20, 2:00 am, galathaea <galath...@gmail.com> wrote: > > On Jun 19, 5:48 am, "Tim BandTech.com" <tttppp...@yahoo.com> wrote: > > > On Jun 19, 12:44 am, galathaea <galath...@gmail.com> wrote: > > > > > [...] > > > > > there has been a lot of contradictory descriptions given > > > > on formal polynomial rings > > > > and i think it may be obscuring > > > > the fact that a lot of good descriptions were also given > > > > > at one point you were told these weren't vectors > > > > at about the same time another poster > > > > was using that as an illustration > > > > > i think most of the approaches > > > > even the oes that had really good information > > > > probably were not the right approach to take for you > > > > > with the formal interpretations given > > > > you will want to ask questions on meaning > > > > and that types of investigation elicits little patience > > > > > i think a better approach > > > > is to just point out that we can write programs > > > > to deal with these > > > > > using the two structure building primitives > > > > (product and union > > > > aka > > > > tuple and variant) > > > > and the two type classes > > > > (value and function) > > > > i think you know you could write > > > > a c++/java/c#/OCaml/whatever class > > > > that would do everything that has been mentioned > > > > > you might start with something like (c++ here) > > > > > template <typename CoefficientBase> > > > > class FormalPolynomial > > > > { > > > > public: > > > > // construct default 0 > > > > FormalPolynomial(); > > > > > // construct by coefficient list > > > > FormalPolynomial(std::vector<CoefficientBase> const& coefficients); > > > > > // ring operations > > > > FormalPolynomial<CoefficientBase> operator+( > > > > FormalPolynomial<CoefficientBase> const& summand); > > > > FormalPolynomial<CoefficientBase> operator*( > > > > FormalPolynomial<CoefficientBase> const& multiplicand); > > > > > private: > > > > std::vector<CoefficientBase> coefficients_; > > > > > }; > > > > > i am sure you would know how to implement this > > > > and could easily understand how to use this class > > > > > this simple little thing here > > > > is all any of this really is > > > > > if you ask "what is X" here > > > > the answer seems much less exciting > > > > than some of what has been discussed > > > > > it is simply an object constructed something like > > > > > Real coefficientArray[] = > > > > { > > > > 0., > > > > 1. > > > > > }; > > > > > std::vector<Real> coefficients(coefficientArray, coefficientArray + > > > > 2); > > > > > FormalPolynomial<Real> X(coefficients); > > > > > notice that there is no function we've defined here > > > > that takes a FormalPolynomial and returns a Real > > > > and we don't need anything like that > > > > > the operations on X > > > > immediately give answers to "what is X^2?" > > > > and similar questions on the ontology > > > > > this may seem very uninteresting > > > > but that's the whole point: > > > > > the most powerful way of thinking about these things > > > > is as uninteresting and simple things > > > > whose operations you probably already understand > > > > > this is what the adjective "formal" usually means > > > > when applied as "formal polynomials" > > > > or "formal series" > > > > and such > > > > > it means: > > > > keep it simple > > > > no extra requirements > > > > think of them as these objects with the interface given > > > > no other details needed > > > > > in the theory of formal series > > > > for instance > > > > they don't even worry about convergence > > > > > it's just sequences of coefficients > > > > and a sequence algebra defined > > > > > -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- > > > > galathaea: prankster, fablist, magician, liar > > > > Thank you galathaea. > > > > As I look at your code I wonder about having a class with the name > > > 'Ring'. > > > This would provide sum and product operators. > > > Then all of this fuss would evaporate into elemental terms. > > > > There is a ring definition which claims to be at the crux of all of > > > this fuss. It clearly states that its members are in the same set. > > > Whether they be variable or constant has nothing to do with the set > > > which they come from. It would seem that upon entering the polynimial > > > form that three new operators were introduced by insistence of x not > > > being a set element and that has been Fishcake's analysis which I now > > > see as the establishment position. Yet in that form there is no actual > > > use of ring operators, since the type of x is insistently denied to be > > > in the ring. Thus the products and sums in > > > a0 + a1 x + a2 x x + ... > > > are not ring operators. Yet the establishment position thus far has > > > clung to the insistence that this form as elemental does behave like a > > > ring within the polynomial context, and it is nearly true, but not > > > quite. > > > > A slim disproof exists, for upon coming down to a single term in that > > > polynomial > > > a x > > > we should freely replace this value ax with two polynomials > > > (a) (x) > > > and within the the ring context with a third polynomial c > > > a x = c > > > by the closure axiom of the product operator which had been declared > > > on a higher form than A so named 'the polynomial ring A[X]'. If this > > > is not true then the value under scrutiny is not a ring and so the > > > usage of the term ring was not only polluted by the formal > > > interpretation on x, but it was destroyed. The polynomial ring is a > > > misnomer when interpreted the 'formal' way. > > > > Instead witness that upon specifying a set A then all elements are > > > specified. Thus the ring definition alone is suitable to building up > > > polynomials or any other structure you wish of products and sums. > > > Whether elements are variable or constant is about implementation > > > which has not occurred yet since nothing has been done with the > > > construction. > > > creating the class Ring is definitely the right starting point > > but it is often convenient to make Ring an interface > > and implement different rings in different manners > > because there are often "more convenient" > > or "more optimised" > > means of representing the data of certain types of rings > > > still > > the idea is the same > > > so you could make a class RealRing > > > which could even just be an alias for the previous Real > > mentioned as coefficients > > > and you could make a Ring-derived class Complex > > and many other such rings > > > now let's define a simple "adapter class" concept > > > for any Ring r > > where r may be any type derived from Ring > > so we're looking at a polymorphic form > > let's apply the earlier "FormalPolaynomial" template > > > do you see how the class FormalPolynomial<Real> > > has different structure than Real > > > however > > it is clear there is a canonical inclusion > > canon: Real --> FormalPolynomial<Real> > > given by r |-> (r, 0, 0, 0, ...) > > (or the finite rep (r) or whatever is your fancy) > > > the additional structure of FormalPolynomial<> > > over some base ring > > is that we have added this unbounded dimenional vector space > > (but no element of infinite dimension) > > > now > > instead of talking about single elements of the base field > > we can build an algebra of ordered collections > > > or an algebra of finite sequences of symbols > > > still > > i am sure that you can see > > in almost any computer language with a type system > > the construction of a library would make > > Real and FormalPolynomial<Real> two distinct types of values > > > the class doesn't even have an internal representation of X > > at least none with operational meaning > > > (it should just be an index into a structure like a string) > > > in Real > > we have things like r1 . r2 = r3 > > > in FormalPolynomial<Real> > > there are things like (r0, r1) . (s0, s1, s2) = > > (r0 . s0, r1 . s0 + r0 . s1, r0 . s2 + r1 . s1, > > r1 . s2) > > > in several different senses > > this can be seen as "adjoining a transcendent" to a ring > > and similar extension interpretations > > but you can see the added structure > > Well here we go with the magic again in your quotes > adjoining a transcendent > which is merely a fancy term for a product which does not fit the ring > nomenclature which is where product was introduced formally. Thus > while you all couch your terminology on a 'formal' polynomial the > usage of this interpretation is exactly breaking the formality which > the ring created. You've created and immediately destroyed that > principle by this usage.
it's the same thing that happens when a software engineer gets an assignment "hey, we need to add functionality F to class X"
sure it violates the old guarantees but you try to do the addition in such a way that there are no broken promises from before because you don't want to have to change other code loci
usually this means "orthogonal extension"
the mathematicians have formalised this in many ways and one of them is adding transcendent elements to rings which is basically the "polynomialisation operation"
this "formal" element X is the basis for the process of looking up coefficients in the sequence
a software engineer does this by using a dynamic container for the coefficients
> Again, coming from the ring interpretation multiple types are not > accomodated. For instance the complex number as composed of real > numbers does not fit the ring nomenclature due to the seed of its > construction > b i > where b is defined as a real and i is not real. This is a unit vector > product and this product does not fit the ring nomenclature. This will > lead to your usage of a 2D form > ( 0, b ) > in order to remedy the situation. But this form is on two sets, not > one and so it does not fit the ring nomenclature. Upon encapsulating > these two-typed numbers into a singular element form such as > z1 z2 = z3 > z1 + z2 = z4 > then the ring definition becomes compatible, but carrying around the > two-form is a substructure that contaminates the ring requirements if > opened up.
it is a part of the laziness of the language which i think is more of an engineering trait
many writers will skip the formality of the "marking of elements"
as i mentioned in passing earlier something i called the "canonical inclusion"
say a ring class takes r bytes of data to describe a member of the ring (in other words say each instance of some Ring class is r bytes large) graphically: [ r bytes ]
inside FormalPolynomial it stores the data of each element as a dynamic container of r byte locations possibly with some container management data as well graphically: [ container ] [ r bytes ] [ r bytes ] ... [ r bytes ]
the "canonical inclusion" simply "names" elements of FormalPolynomial type using names from the base ring type
the container with only one element [ container ] [ r bytes ]
behaves in the exact same way as the base element sitting in that lone cell
in the vector representation of FormalPolynomial (a1) # (a2) = (a1 + a2) (a1) * (a2) = (a1 . a2) respect the same arithmetic relationships as the base ring
..
so... what all of this means is mathematicians feel lazy writing something like a x instead of (a) x or related notation to indicate "image of naming map" it is because it isn't really necessary
as you mention the parsing would fail and as computer linguists often take advantage parse failures are places where a language may be extended adding anything from notational conventions to complete new behaviors
here it was simple notational convenience adding something like parseFail(condition72B, failedNode): interpret failedNode -> (failedNode)
there is no ambiguity introduced those are encountered by linguistic additions that parse _correctly_ with some other semantics
> > the point about looking at it like this > > as program structures > > is because the dynamics of most mathematical discourse > > often has prediction hindered by loose semantics > > > but as one becomes comfortable with a language > > and many of the modern languages are often good domain fits > > the ontology becomes very predictable > > > you get a better understanding of the evolution of it's types > > > i've seen your web site and resume > > so i know you understand how to implement this > > > can you see how Real works from these loose descriptions? > > how FormalPolynomial works? > > how they are different types? > > Yes, exactly. They are claimed to be different types and so they do > not fit the ring method. > The product in use under this interpretation is outside of the ring > nomenclature since it mimics the unit vector format. Yet it is so easy > to see that this inconsistency need not exist. From the ring we can > freely construct the polynomial and maintain integrity. So long as the > form remains abstract then there is no trouble. Upon specifying a real > valued coefficient however then the conflict ensues. The ring > definition is the core principle here, not the polynomial. The > insistence on respecting the polynomials abstract form while > collapsing its coefficients is inconsistent with the ring terminology > and so you land in terminology like 'transcendent' which is entirely > unmathematical just as Connely's usage of "variable" or > "symbol" (quotes his, not mine). > Are we not fine working with variables as symbols already without the > need to put them in quotes? > Whatever happened here is a sad thing. One of the high priests made a > misstep and decided to never own it. Now that misstep has propagated > and the proof of a religion within mathematics is established. This > causes me to nearly accept religion, except that there is no principal > of truth there. The best means of arriving at truth is not via > mimicry, yet mimicry is exactly what all of you purvey here. Academia > is a lost cause so long as it works like this.
the same proof shows the religiosity of software engineers
it always happens when nomenclatures have to be decided without years of dedicated thought
you see tim everyone is shooting from the hip here and everyone is riding waves of epiphanies trying to get it all out and see what sense it makes
when most people publish proofs they try to capture the way they thought of things but when it doesn't matter what symbolic choice is made often tradition and chance enter into it particularly when hurried to get the new ideas out because it is stalling investigation elsewhere
and when software engineers are racing against the deadline and trying to get a module implemented the litany of derivations requires some simple technique something familiar and easy to access in which to automate the long task
all large intellectual projects accumulate the mud of the many varying arbitrary choices of all the past engineers
refactoring is always an important skill
> Noone here other than > Leland has managed to wander off the path that has already been laid. > Why not consider some other ground? I know that you galathaea are a > master of the polynomial, but I also know you to be openminded. I > guess the simplest question to focus down on is: > Which is more fundamental, the ring or the polynomial?
looking at the computer science model the question is: which is more fundamental BaseRing or FormalPolynomial<BaseRing>
in this particular implementation if fundamental means what i think you may mean here BaseRing can be used on it's own FormalPolynomial<> can be defined without reference to BaseRing but it cannot be instantiated without reference to the base
so BaseRing is more fundamental here?
but as a fellow engineer i know you know we don't have to implement things that way
we could implement our own FormalPolynomialZ2 class that doesn't refer to the generic form or the base and it might even be able to be optimised better and all sorts of other things
and maybe someone programming after a terrible head injury had implemented Z2 as a class where they couldn't get the product to work right so they kept adding and adding to the algorithm until it was 3000 lines long and finally worked for all four cases
is that more fundamental?
> We can build polynomials from rings, but we cannot build rings from > polynomials so long as X is untouchable. This is what you all have > created: a caste of untouchables. Then you will turn around and wipe > them away without a care for what you so carefully layed down as law. > Within a set of laws a form of truth is claimed but whether that form > is accurate must be left open. Here I have provided plenty of evidence > taking the law of the ring as the first law. I do accept this first > law fully, whereas you have brushed it aside yet maintain its > terminology so as to keep up appearances. You do claim to be a > prankster, fablist, magician, and liar and here is a fine low down > instance. Of course you understand I am not slinging an insult at you > emotionally here. I would simply ask that if we are in a mathematical > system that such conflicts ought to work out via logically constructed > arguments. I have pointed directly at the flaw. Noone here has > bothered to admit that the product of the ring transcends the product > of the polynomial. Instead they go over to make a fresh symbol '#' as > a new product operator without any regard for the ring.
think of X like "marks the 1st location in the container"
it's an indeXer that's all
X^2 "marks the 2nd location in the container"
when looking at basis elements in vector spaces we have for arbitrary vector v
dim(v) --- \ v = / v e --- j j j=1
and when looking at an element that is a formal polynomial for arbitrary polynomial p
deg(p) --- \ j p = / p x --- j j=0
ultimately it is just a shorthand for selecting coefficients which is all that happens when this is implemented in code
