RE: the "set paradigm" is harmful
I think what Svetoslav has in mind is that, e.g., Taylor arithmetic is
automatically excluded by this definition, since in Taylor arithmetic,
the main object is not an interval but rather a tuple consisting of real
numbers (coefficients of the Taylor polynomial) + the Taylor bounds for
the remainder.
My understanding is that the gist of his email is that we should keep in
mind these important ways of estimating interval uncertainty related to
use of affine expressions, Taylor polynomials, etc.
I am not so sure how to modify the proposal along these lines, I think a
constructive suggestions may be perceived more positively.
-----Original Message-----
From: stds-1788@xxxxxxxx [mailto:stds-1788@xxxxxxxx] On Behalf Of Arnold
Neumaier
Sent: Monday, February 09, 2009 1:43 AM
To: Svetoslav Markov
Cc: stds-1788@xxxxxxxxxxxxxxxxx
Subject: Re: the "set paradigm" is harmful
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