[om] Standard .. Ontologies for Math

J H Davenport jhd at maths.bath.ac.uk
Fri Jun 8 16:30:02 CEST 2001

On Thu, 7 Jun 2001, Richard Fateman wrote:
> Andrew Solomon wrote:
> > As always, I applied the "semigroup" test. It fails.
I think Andrew is referring to the distinction between a semigroup and a
monoid, and indeed this ontology is confused: what it calls a semigroup is
what I (and probably Andrew) would call a monoid. Hence t is not clear
that OpenMath would wish to follow this ontology in detail.
I also don't understand its definition of an Intengral Domain, but this is
probably stupidity on my part.
> I don't know what the semigroup test is, but I think the motivation
> for OM and knowledge systems overlap considerably.

> I looked at STS (small type system) and found it to be much less detailed than
> the Stanford site for the kinds of things I might like to see
> for statements of axioms and properties. But maybe I didn't
> look long enough or in the right place.
It is less detailed. The problem is that one rapdily has to complicate the
language to get any further. It would be possible (and this might be
interesting) to build more semantics for OpenMath in terms of the
Stanford basic ontology.
> >  At random I poked at "NumericalValue" and found the description
> "Denotes an OpenMath object that is to be thought of as a
>  numerical value (possibly symbolic)."
Yes, this is not the world's best piece of prose - I'm open to suggestions.
The problem is simple: what would one quote as the type of the argument to
arctan? The NAG libraries would expect an explicit number (real or complex
depending on the precise route, but that's a problem for the phrasebook,
whereas Macsyma will accept a symbolic value. However, a typed system
ought not to accept a value that is typed to be, say, a set.
> This consists not only of a circular definition, but vaguely
It's not actually circular, but defining a formal OM concept in terms of
an informal one. However, it is indeed relying on an informal concept: one
has to start somewhere.
> suggests the possibility of a thought-crime if one disagrees,
> and then suggests that a numerical value could possibly be
> symbolic. (Which it is not, in my view.  But since symbolic
> is not defined, who knows?)
> There is no author indicated for STS.  Was this really reviewed
> on 2000-09-01? By whom?
My copy (from www.opnemath.org) has a review date of 2003-04-01.


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