Re: Motion on interval flavors
On 2012-07-09 12:35:25 +0300, Svetoslav Markov wrote:
> Dear Alexandre,
>
> I agree with you. I also beleive that, in principle,
> Kaucher/modal intervals can be avoided.
>
> However, the analogy with negative numbers
> comes again into play. Everything that is formulated
> using negative numbers can be reformulated without
> using negative numbers. Your arguments are similar
> to the arguments of those prominent mathematicians
> who up to the 18th century refused to work with negative
> numbers.
[...]
There is an important difference with Kaucher/modal intervals:
nowadays, everyone needs negative numbers for their problems, while
Kaucher/modal intervals are useful only for very specific problems.
> Note also that the applications of negative numbers increased
> only when mathematicians started to use them systematically and got
> used to the (group-like) properties of real numbers. The history
> teaches us that soon or later the same will happen with Kaucher intervals.
Even if applications would increase, this doesn't imply that
Kaucher intervals should be in the main interval arithmetic
standard. Perhaps in another standard.
> Now my conclusion and my question:
>
> Conclusion: Kaucher (improper) intervals can be avoided,
> real (negative) numbers also.
>
> Question: what for are then real (negative) numbers?
>
> Those who can answer this question can understand what for
> are Kaucher intervals.
Well, I think that a better analogy would be complex numbers.
Even complex numbers, which appear in more problems than Kaucher
intervals (and are the right way to define some functions on a
theoretical point of view), are not in the IEEE 754 standard.
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
100% accessible validated (X)HTML - Blog: <http://www.vinc17.net/blog/>
Work: CR INRIA - computer arithmetic / AriC project (LIP, ENS-Lyon)