the "set paradigm" is harmful
Dear colleagues,
The phase:
(S) "intervals are sets of numbers",
is repeatedly stated in the documents
discussed by now.
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. (S) imposes the
priority of the inf-sup presentations of intervals and thus
prohibits the equally important view that intervals
are approximate numbers (having a "main" or "mean"
value and error bound). It has been shown that intervals
presented in midpoint-radius notation satisfy the same
algebraic structures as intervals using inf-sup presentation.
The argument of some supporters of the set-paradigm
that an interval of the form [a, oo) cannot be presented in
midpoint-raduis form seems to me ridiculous.
The set-paradigm implies the consideration of only
the "standard" interval arithmetic structures and prohibits
their natural algebraic extensions. The supporters of (S)
argue that every statement using improper intervals
can be presented by proper ones. Imagine that in
the IEEE 754 standard one adopts the thesis that negative
numbers are excluded of consideration. Indeed, every
proposition using negative numbers can be modelled by using
just positive numbers! To those who know the embedding theorem
(for the embedding of a semigroup into a group) thesis (S) is
equivalent to prohibiting negative numbers.
The set-paradigm leads to unnecessary limitations in the
application of the standard.
Interval arithmetic structures should be considered in the way
we consider real arithmetic structures. Interval arithmetic structures
are rich structures arising from just two basic operations and one
relation. These abstract structures can be used for various
purposes depending on the semantic contents which one
attaches to the models/problems of consideration. It is true that
the idea of a set is underlying in the semantic contents but with various
nuances. Algebraically complete intervals structure are needed
when using advanced techniques of interval analysis.
The extraordinary diversity of the extended interval arithmetic structures
allow various uses and interpretations, which incorporate
the set paradigm as a special case. This paradigm in its narrow sense
is harmful for the future development of interval analysis.
The implementation of complete interval arithmetic does not require
additional costs. Apart from standard applications it can serve for many other
purposes in extended applications such as modal interval analysis, range
interval analysis using inner interval arithmetic operations etc. Up to
certain extent Kahan intervals can be also modelled by generalized
intervals.
S. Markov