Re: Csets, modals and Motion 3
> Let me add something about the "K.I.S.S. principle".
>
> From mathematical point of view a definition of a mathematical object
> (in our case "interval") is done by means of axioms. A definition is
> simple if the corresponding system of axioms is simple and applicable.
>
> Thus the definition of a real number is based on the algebraic systems of
> a ring (when addition and multiplication are involved) and a vector space
> (when addition and multiplication by scalar are involved). These two
> systems proved to be the simplest and eventually were accepted worldwide.
> Consequently zero and negative numbers were accepted (although
> they are not easy to be understood).
>
> Similarly, the simplest definition of an interval is based on the Kaucher
> system of axioms (when addition, multiplication and inclusion are
> involved)
> and the quasivector system (when addition, multiplication by scalar and
> inclusion are involved). These systems involve improper intervals --
> similarly to real number systems, which involve negative numbers. The
> reason behind is that computing in a group is simpler than computing in
> a semigroup.
>
> However, the concept of a group derived from a semigroup by means of
> embedding leads to improper elements. It is not easy for the human mind
> to understand this. That is why it took more than 20 centuries for the
> humanity before the concept of a real number was finally coined.
>
> Most colleagues have specialized in techniques, using just proper
> intervals. Certainly, they do not like Kaucher/modal intervals. It is not
> wondering that these colleagues will vote against an implementation of
> Kaucher/modal intervals in the standard. For them Kaucher intervals are
> too "complex" and do not conform to "K.I.S.S. principle".
There must be a formal proposal so that people can make an informed
decision. So this is partly the responsibility of the modal subgroup.
Whatever is the fate of Kaucher intervals in the standard, a reason and
rationale must be provided.
> However, the truth is that Kaucher IA is simpler than the "standard" one.
> Just because a group is simpler than a semigroup.
I am convinced even from an electrical engineering perspective it is the
simplest design.... even simpler than the classical arithmetic.
Nate