Re: the "set paradigm" is harmful
Arnold,
from practical point of view the mid-rad
presentation differs from the sup-inf one in that
midpoint and radius do not need to be presented
uniformly in the computer, e g radius may need
a very short mantissa. From abstract algebraic point
of view both components belong to spaces of
different structure.
The standard should consider both presentations
as equally possible. The set-paradigm as stated
in both discussed documents imposes the
priority of the sup-inf presentation.
Svetoslav
On 9 Feb 2009 at 9:42, Arnold Neumaier wrote:
> Svetoslav Markov schrieb:
> >
> > The phrase:
> >
> > (S) "intervals are sets of numbers",
> >
> > is repeatedly stated in the documents
> > discussed by now.
>
> This is the consensus in the wider community of all
> mathematicians. No amount of lobbying will change this.
>
>
> > Thereby (S) is understood in the sense,
> > that intervals are boxes of the form [a, b].
> > I shall further call this theoretical framework
> > the "set paradigm".
> >
> > In my opinion a standard based on the set paradigm will
> > be able to serve only a limited number of applications and
> > is harmful for the future development of interval analysis.
> >
> > The set paradigm excludes the view the intervals can be
> > considered as approximate numbers.
>
>
> What are meaningful definitions of approximate numbers?
>
> 1. A number known to lie between to known numbers.
> This gives traditional interval arithmetic.
>
> 2. A number known to deviate from a given number by at most
> a given amount. This is equivalent to a special case of 1.,
> special since it does not cater for unbounded intervals.
>
> 3. Probabilistic versions of 1. or 2.
>
> It is clear that we should not consider option 3. in the
> present standard discussion. This leaves 1. as the more
> general (and in practice almost exclusively used) option.
>
>
> Arnold Neumaier