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