(Fwd) Re: Motion on interval flavors

["Svetoslav Markov" <smarkov@xxxxxxxxxx>]

Dear George,

> What are Kaucher/modal IA useful for?

This is a strange question, but let me take it seriously.

To answer this question the analogy with negative numbers
comes again into work.

Negative numbers open a whole new world of problems 
to be solved. They also open a  world of new basic theories,
like linear algebra, rings etc. From practical point of
view these tools induce new semantic. Do I have to give
examples? All sciences give examples of the interpretation
of negative numbers.

Similarly, Kaucher/modal intervals open a new world of problems
and new basic tools. Also, new semantic comes into play, namely  
 the so-called modal sematic. The latter allows to formulate
 problems involving inclusions of sets, by means of logical quantors
(exists-for all). These problems are then  translated into algebraic
 problems using kaucher intervals. Then kaucher arithmetic is 
applied to solve the problems.

There exists already a vast literature on the application of
Kaucher/modal arithmetic.  Sergei Shary and Alexandre Goldsztejn 
are just two names to be mentioned. 

Best regards,

Svetoslav



On 8 Jul 2012 at 11:31, Corliss, George wrote:

From:	"Corliss, George" <george.corliss@xxxxxxxxxxxxx>
To:	Svetoslav Markov <smarkov@xxxxxxxxxx>
Copies to:	"Corliss, George" <george.corliss@xxxxxxxxxxxxx>,
       	stds-1788 <stds-1788@xxxxxxxxxxxxxxxxx>
Subject:	Re: Motion on interval flavors
Date sent:	Sun, 8 Jul 2012 11:31:49 +0000

> Negative numbers may be useful for describing the state of my bank account.  Also in 18th century England, it was common to throw debtors into prison, so the authorities clearly understood the concept of negative net worth.
> 
> What are Kaucher/modal IA useful for?
> 
> George Corliss
> George.Corliss@xxxxxxxxxxxxx
> 
> 
> 
> On Jul 8, 2012, at 5:52 AM, Svetoslav Markov wrote:
> 
> > Dear prof Kulisch,
> > 
> > I am quite puzzled by your attitude  to Kaucher/modal IA.
> > 
> > It seems to me that you, and probably other
> > participants in this forum,  are not aware
> > with the history of the acceptance of negative numbers!
> > 
> > I make the analogy with negative numbers because 
> > improper (Kaucher/modal) intervals are just
> > like negative numbers (as their radii are negative).
> > 
> > There is a lot in internet about the history of the acceptance
> > of negative numbers, see e. g.:
> > 
> > http://en.wikipedia.org/wiki/Negative_number
> > (section "history")
> > 
> > This section finishes like this:
> > 
> > In A.D. 1759, Francis Maseres, an English mathematician, wrote that negative 
> > numbers "darken the very whole doctrines of the equations and make dark of the 
> > things which are in their nature excessively obvious and simple". He came to 
> > the conclusion that negative numbers were nonsensical.[14]
> > 
> > In the 18th century it was common practice to ignore any negative results 
> > derived from equations, on the assumption that they were meaningless.[15]
> > 
> > Are we living in the 18th century? Indeed, negative numbers were not
> > real numbers before mathematicians  accustomed  to use them. 
> > 
> > Regards,
> > 
> > Svetoslav
> > PS. Here are some other links to "The History of Negative Numbers":
> > 
> > http://nrich.maths.org/5961
> > 
> > http://logica.ugent.be/albrecht/thesis/HPM2008.pdf
> > 
> > http://www.ma.utexas.edu/users/mks/326K/Negnos.html
> > 
> > http://www.basic-mathematics.com/history-of-negative-numbers.html
> > 
> > http://logica.ugent.be/centrum/preprints/Numberline-ScienceAndEducation-rev.pdf
> > 
> > 
> > 
> > A BBC film:
> > 
> > http://www.bbc.co.uk/iplayer/episode/p003hyd9/In_Our_Time_Negative_Numbers/
> > 
> > 
> > 
> > On 6 Jul 2012 at 15:56, Ulrich Kulisch wrote:
> > 
> > Date sent:      	Fri, 6 Jul 2012 15:56:50 +0200
> > From:           	Ulrich Kulisch <ulrich.kulisch@xxxxxxx>
> > To:             	John Pryce <prycejd1@xxxxxxxxxxxxx>
> > Copies to:      	stds-1788 <stds-1788@xxxxxxxxxxxxxxxxx>
> > Subject:        	Re: Motion on interval flavors
> > 
> >> Am 20.06.2012 07:49, schrieb John Pryce:
> >>> I'm not very happy with the name "set-based" intervals, but calling them
> >>> "standard" intervals is no longer appropriate, as well as clashing with the
> >>> main meaning we give to "standard" (as a noun and also an adjective). Any
> >>> better name welcomed.
> >>> 
> >>> Regards
> >>> 
> >>> John Pryce
> >>> 
> >> 
> >> I see two kinds of set-based intervals, the closed and bounded intervals 
> >> of IR and the closed intervals of \overline{IR}. What about calling the 
> >> first cb-intervals and the second c-intervals.
> >> 
> >> Kaucher or modal intervals are not really intervals. They are abstract 
> >> entities. Why not calling them i-intervals for inverse or improper 
> >> intervals or m-intervals.
> >> 
> >> Best regards
> >> Ulrich
> >> 
> >> -- 
> >> Karlsruher Institut f"ur Technologie (KIT)
> >> Institut f"ur Angewandte und Numerische Mathematik
> >> D-76128 Karlsruhe, Germany
> >> Prof. Ulrich Kulisch
> >> 
> >> Telefon: +49 721 608-42680
> >> Fax: +49 721 608-46679
> >> E-Mail: ulrich.kulisch@xxxxxxx
> >> www.kit.edu
> >> www.math.kit.edu/ianm2/~kulisch/
> >> 
> >> KIT - Universit"at des Landes Baden-W"urttemberg
> >> und nationales Grossforschungszentrum in der
> >> Helmholtz-Gesellschaft
> > 
> >

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