Re: Datatype disjointness implemented

We had a version of this discussion before wherein I pointed to  
several standard papers on issues with the ontology of mathematics.  
It's very sad that you did not bother to read or even glance at them.

On 12 Mar 2009, at 19:55, Alan Ruttenberg wrote:

> The change includes:
>
> xsd:double
> xsd:float
> Value Spaces. The value spaces of xsd:double and xsd:float are
> specified in XML Schema Datatypes [XML Schema Datatypes]. Furthermore,
> the value spaces of owl:real, xsd:double, and xsd:float are pairwise
> disjoint.
>
> This is inconsistent, as it is inconsistent in XML Schema.

Nope.

> The value
> spaces of both decimal and float and doubles are all "numbers" in XML
> Schema, notwithstanding their later protestation that the sets are
> disjoint.

The value spaces of both negative integer and positive integer are all  
"numbers" in XML Schema, notwithstanding their actual disjointness.  
(IOWs, it takes a lot more than what you provided to get an  
inconsistency.)

There is no univocal set of "numbers". For example, one can interpret  
the integers (all of them) into the positive integers into the even  
integers into the odd integers. You can use various, non identical set  
theoretic constructs (e.g, (), (()), ((()))... vs. (), (()), ((),  
(())), i.e., Zermelo vs. von Neumann reductions)  for the integers,  
and they will answer to all the purposes of arithmetic (i.e., meet the  
Peano Axioms) and yet, trivially and obviously) they are mostly non- 
disjoint (0 and 1 are the same in each, but we could fix that easily).

So, in principle, we cannot rely on the denotation of "numbers" to be  
determinate, thus we cannot reason from the use of "numbers" in a  
document which is *not* being particularly precise about various  
things to *anything*. We *can* reason from the formal constraints they  
put in, i.e., the axiomitization they use. There, what's central is  
precisely what you dismiss, the "further protestations".

> Specifically, it says, for floats:
>
> "Its value space is a subset of the rational numbers."
>
> This is inconsistent with floats being disjoint with owl:real, which
> has as a subset, owl:rational.

Not at all. There's no presumption that owl:rational has to map into  
"the" rationals (since there is no such set). xsd:float can be a  
subset of pairs of Zermelo natural numbers and owl:rational can be the  
von Neumann ratios. Both have both mathematical and historical claim  
to being "the rationals" and yet, ontologically (and set  
theoretically) they are disjoint (pace 0 and 1).

Further more, we can, without any change in anything, map xsd:float  
into a subset of the rationals between 0 and 1 and owl:rational into  
all the others. This exactly models everything (including the  
disjointness).

> We can't repeat this inconsistency in OWL.

Since it isn't inconsistent by any standard, we can repeat it.  
Furthermore, there are simpler readings (e.g., "the floats are a  
subset of a identity-distinct copy of the rationals") which also  
perfectly makes things work out..

> If the value spaces are to
> be disjoint, it needs to be explicitly stated how this comes about.

No, we don't. There are many ways that it could come about and they're  
all perfectly fine. We just need the axiom...the models can take care  
of themselves.

If we map them into anything it should be bit patterns and point out  
that there is an associated axiomization over that bit patterns which  
yields the same entailments as if you mapped them into the rationals.

> However I see no such theory other than the informal:
>
> "Thus, the value spaces of xsd:double and xsd:float can be understood
> as containing "fresh copies" of the appropriate subsets of the value
> space of owl:real."

This isn't informal at all. Or at least, it's as formal as the use of  
"numbers" was. We don't *need* to formalize it further by reducing it  
to some other "more primitive" notion, as the axiomization more than  
sufficiently characterizes it, i.e., floats are a set of entities  
disjoint with owl:real but with an axiomization which makes them  
indiscernable wrt facet axioms from a certain subset of owl:real.

That's absolutely formal.

You confuse formalization with reduction, that is, with a *preferred*  
formalization.

> An appropriate way to do this would be to have the value spaces of OWL
> numbers not be numbers, but instead be pairs of numbers and tag.

Since this way furthers your agenda against disjointness, I don't  
think it's a very appropriate way.

> I
> interpret this to mean that we are *not* adopting the value spaces as
> specified by XML Schema,

Straw man arguments are not especially compelling.

> somewhat nullifying the motivation for the
> change.

This is why it's inappropriate.

> Note that I do not, in pointing this out, consider such a repair as
> removing our overruled objection to the overall change.

  I hope it is now clear that this is just a blind ally. Oppose  
disjointness if you want against history, implementor wishes, and  
(arguably) usability. There's no accounting for taste and there's no  
*technical* incoherence, per se, in adopting disjointness or non- 
disjointness. It is, truly, a judgement call. However wrongheaded I  
might think your judgement is in this case, it is yours and that's  
fair enough. But please cease these specious arguments.

> However, if
> the change is to be adopted, it needs to done in a careful and
> consistent manner.

As it has been.

Cheers,
Bijan.

Received on Thursday, 12 March 2009 22:14:29 UTC