The discussions in the OMG forum raised an old puzzle about two
different ways of defining circles and ellipses. Euclid's way
makes circles a special case of ellipses, but some programmers
might define them as two independent classes. (01)
Examples like this arise in any ontology. What happens when
different people define two types with specifications that
might sometimes, but not always coincide? (02)
John (03)
 Original Message 
Subject: Re: [architecturestrategy] Relationship between types, classes
and sets
Date: Thu, 20 Jan 2011 10:08:43 0500
From: John F. Sowa
To: architecturestrategy@xxxxxxxxxxxxxxx (04)
> It would help things along is you could you explain what you see the
> predicates for the OO uses of Circle and Ellipse in
> http://en.wikipedia.org/wiki/Circleellipse_problem are.
> And what you see the interpretation of OO inheritance may be, where this
> shows circles inheriting from ellipses  where mathematicians would expect
> ellipses to be subtypes of circles. (05)
The puzzle about circles and ellipses does not violate the following
criterion: (06)
Every class C has a predicate isinC(x), which is true iff object x
is in class C. (07)
As for the circleellipse issue, it is misleading to say that one
definition is more "mathematical" than the other. Both are equally
mathematical (in the sense that they are defined by mathematical
specifications). (08)
But mathematicians since Euclid have preferred to say that a circle
is a special case of an ellipse because that assumption simplifies
and generalizes the theorems and proofs. But the other definition
could be assumed if anybody found it useful for some purpose. (09)
Solution: There is a trivial solution to the socalled problem
that is not mentioned in the Wikipedia: drop the requirement that
two isomorphic figures must belong to the same classes. That would
imply that no stretched ellipse could ever *be* a circle  the
specification would take priority over the appearance in determining
class membership. (010)
Re mutators: If you define a circle as having one center, and an
ellipse as having two foci, you don't get any problems with mutators
or inheritance. By that definition, a stretched ellipse with both
foci at the same point would not be a circle, even though it looked
like a circle. (011)
Re inheritance: Since no circle would ever be an ellipse, no circle
could inherit two foci from the definition of ellipse. (012)
Euclid's definition is more "elegant" and "general", but elegance and
generality are not prerequisites for being mathematical. (013)
John (014)
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