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Re: the "set paradigm" is harmful



 


On 9 Feb 2009 at 9:19, Michel Hack wrote:

> Earlier I wrote:
>   ... when the radius exceeds either an absolute or a relative threshold.
> 
> Change "either" to "both".  One applies for small absolute values, the
> other for the rest.
> 
> Btw, I realise that one can also have an approximate number arithmetic
> with containment properties, and that this is probably what Svetoslav
> had in mind -- but my comments about restriction to narrow intervals
> remain applicable.
> 
> Michel.


 Surely, MR-presented intervals, resp. approximate numbers, 
admit probabilistic interpretation, e.g. midpoint can 
be mean value, in Gaussian distributions radius can be 
standard deviation or half-width of confidence interval, etc. 
(Note that the guaranteed containment property can be still there,
say within the 95% of probability). Such interpretations depend on the
particular models/problems. MR-presentation supports various
interpretations as summarized in Arnolds classification. Note that
MR-form intervals incorporate the set paradigm --- up to the 
special cases (oo, a], [b, oo). That is why MR-form is even more
important than sup-inf form. MR-form is more convenient than the
sup-inf form in many applications (possibly for Taylor polynomials
as well). Some authors often use this form as an intrinsic form in 
computations/proofs and then translate results in sup-inf form
(notably J. Rohn). This form incorporates and enhances  several
suggestions made by U. Kulisch.  

Conclusion is: MR-presentation of intervals deserves (at least) equal
consideration in the standard as the sup-inf form.

Remark 1. Narrow intervals indeed have certain important properties.

Remark 2. Centered multiplication may be optionally implemented,
whereas the "true" interval multiplication should be (of course) obligatory
in the standard. Centered multiplication is only an approximation of
the true one, but is faster.

 Svetoslav