On 3 Jun 2013, at 17:01, Richard Fateman <email@example.com> wrote:
> On 6/3/2013 3:05 AM, Andrzej Kozlowski wrote: > > > > > On 3 Jun 2013, at 09:34, Richard Fateman > > <firstname.lastname@example.org> > > > wrote: > > > > > > > >> On 6/1/2013 9:23 PM, Andrzej Kozlowski wrote: > > > >>> This is one of many examples of "cross purpose" > arguing. I was > > > >>> not discussing implementing non-standard analysis at > all. My > > > >>> point was that that there is nothing logically more > dubious about > > > >>> a finite "number" x such that x+1=1 than there is > about a > > > >>> positive "number" x such that nx< 1 for every > positive integer n, > > > >>> or, alternatively, finite "number" x such that x/n > >1 for every > > > >>> positive integer n. Mathematicians often use the word > "number" > > > >>> when referring to objects belonging to some > "extension" of the > > > >>> real line. > > > >> > > > >> Oh where to start. Here's one place. > > > >> > > > >> You say it is ok to have a number x such that x+1=1. I > agree. I > > > >> say it is NOT ok to have a number x such that x+1=x. You > respond It > > > >> is ok to have a number x such that x+1=1. Can you see the > > > >> difference? > > > > > > > > Yes, I was careless about this, but that was because it does > not > > > > really matter. It is O.K. to extend the real line to an > algebraic and > > > > topological sturcture which contains objects such x such that > x+1=x. > > So such systems have the AK seal of approval. Argument settled?? > > Since such objects do not satisfy the necessary axioms for real numbers, they cannot be part of the real line.
Surely, you must be joking Mr. Fateman. Really. Whoever suggested that they are elements of the real line? When I mentioned non-standard analysis you reacted as if you knew what it was, but now it seems you have no idea. Infinitely small and infinitely large quantities that appear in non-standard analysis also do not belong to the real line, because the reals satisfy the Archimedean Property and infinitesimals do not. But they belong to a non-Archimedean field which contains the reals. It turns out that you can translate any argument that uses infinitesimals (elements of the non-Archimedean extension) into a valid one which uses only reals. Thus using non-standard analysis is only a convenience - it shortens proofs and computations and will never lead to incorrect answers.
In a quite different context, the same is true of interval arithmetic. Everything done with intervals can be proved by arguments that do not refer to real numbers. Intervals simply provide a tool that is much more convenient in certain situations.
Significance arithmetic is a first order approximation to interval arithmetic - in a completely precise sense (essentially the same as approximating the taylor series by its linear part). One can determine mathematically the situation in which taking the linear part of a Taylor expansion is the right thing to do; doing this sort of things goes back to Newton. The fact that by using zero precision numbers you can get strange looking answers (stange looking only to someone who does not understand the above) does not prove anything.
> > Computer algebra systems sometimes have to adopt models of an extension to the real line to accommodate > certain calculations typically involving limits and division by zero or perhaps complex numbers. This > extension may include +inf, -inf, and und (these may be 1/0, -1/0, and 0/0). There are other ways of > doing this. Where to read about a design for such a system ? See the IEEE 754 binary float explanations. > > > Of course such objects do not > have inverses, so you can't conclude > > > > that 0=1. > > That is true, there is no number x such that inf + x == 0. inf+FiniteNumber is inf. > It is usually somewhat discomforting to add to a system that used to be bound by the axioms of the reals, > these exceptional numbers. But there are only a few.
You are simply proving again that you have the habit of discussing things you do not understand. In non-standard analysis you "add" to the real numbers uncountably many quantities that are "infinitesimally small" and "infinitely large". That's "quite a few".
> > > >> Also is the Grobner basis program in Mathematica the > fastest? I > > > >> suspect it is not, though I have not compared it to > Faugere's work, > > > >> or other unnamed systems. Is it the only one using > significance > > > >> arithmetic? I suspect it is. What would that prove? > What would > > > >> that prove about use as a default? > > > > > > > > This is nonsense. Faugere does not work on numerical analysis > and has > > > > not implemented a numerical Groebner basis. Other people have > worked > > > > on approximate Grobener bases using fixed precision > arithmetic but > > > > (as far as I know) there are no working implementations > available. > > > I think the goal of the numerical Groebner basis algorithms is to provide a fast computation > of the exact results by using approximate numerics. If so it is not an entirely different subject. > The numerical solution of systems of algebraic equations to find (one) solution IS a > different task from finding them all. > And of those researchers doing Groebner basis computations with numerical > approximation, how does Mathematica compare?
It is hard to tell since I don't think there exists any implementation that could be used for comparison. This may be out of date, but Daniel should know. In his book on numerical polynomial algebra Stetter describes how this sort of thing might be done with fixed precision. He writes:
"Naturally, for large problems, or for small problems with long decimals, the computation may get drowned in the swell of the integers. Anyway, it appears unreasonable to perform a computation in integer arithmetic only to round the result when it is obtained. Therefore, the development and the efficient and safe implementation of true floating-point GB-algorithms is still a central research topic for numerical polynomial algebra. Kondratyev's Ph.D. thesis [10.3] represents a seminal first result in this direction."
However, a thesis is a thesis and a step from a thesis to a working implementation is not a trivial one. I rather doubt that it has been made.