David Leal wrote: (01)
> The brief exchange on decimal numerals prompts a tenuously related thought 
> not all of life is easily tackled using logic, least of all mathematics.
> MathML is a practical notation which covers a lot of mathematics, and is
> simple because it has no ambitions to be anything more than a practical
> notation.
>
> We could gain benefits from defining good interfaces between notations which
> can represent some logic (such as OWL) and MathML. We could also think about
> how MathML takes advantage of ontologies.
> ... (02)
This is an interesting view of the world, which I find all too common in
the data exchange communities. Thinking in languages like XML Schema
and MathML is the 21st century version of "thinking in Fortran". (And
yes, language does affect your ability to think.) (03)
The issue is _not at all_ about CLIF meets MathML; it is rather, as John
points out, about how the underlying concept models for firstorder
logic and other established mathematical disciplines are related. The
whole idea of CL was to produce a fundamental reference model for FOL
that would serve as a basis for exchange among engines with diverse
languages and somewhat varied capabilities. It is all about the
concepts (the deep structure, if you will); it is not about the surface
syntax. (04)
In knowledge engineering, there is very definitely an advantage to
marrying a generalpurpose reasoner, whether of the FOL kind or of the
DL kind or of the LP kind, with an engine that has builtin mathematical
knowledge. (05)
It is certainly common for reasoners to have builtin knowledge of
arithmetic. But there are also hybrid reasoners with builtin knowledge
of plane geometry (I'm not so sure about solid geometry) and/or about
vector spaces and linear functions. And there may well be other
examples. But it is only useful to talk about marrying exchange
languages when you have some idea what tools can interpret the
permissible utterances. (06)
> Consider the thing that is 5 metres. In MathML we would happily
> define this thing as follow:
>
> <apply>
> <times/>
> <cn>5</cn>
> <ci definitionURL="&unit;Metre"/>
> </apply)
>
> From an ontology for quantities and units, it should be possible
> to learn that:
> a) there is a "timeslike" operator on (real, length) pairs
> which evaluates to give a real,
> b) there is a length (called the Metre) which everybody knows,
> and hence that the MathML is valid. (07)
Actually the mathematics required here is more than arithmetic; it
requires a basic understanding of vector spaces, so that the idea of
multiplying a scalar by a pair is welldefined. Secondly, it requires a
specific model of 'quantity' (or properly 'quantity value') as a set of
vector spaces in which a given measurement unit is the unitvector of
each space, and in which one of the spaces is 'length' and 'Metre' is
the _chosen_ unit vector for that space. Whether "an [unspecified]
ontology for quantities and units" will choose that explicit model is
hardly certain. (08)
In short, thinking of this problem in MathML gives you less than half
the insight needed to understand the knowledge engineering problem.
So I strongly recommend you concentrate first on the mathematical
knowledge concepts you need, and the ontologies that assign those
mathematical models to the underlying engineering concepts, and then ask
whether MathML, as defined, provides a representation for those concepts
that can be conveniently combined with CLXML or OWL or your favorite
RDF language. (09)
Put another way, choosing a particular mathematical model as the basis
for an ontology for a given scientific or engineering phenomenon is
itself a knowledge engineering decision. It is easier to make that
decision when you have an engine that can reason effectively with such a
model. So I come back to the idea that you need to define a hybrid
exchange language only when you know what tools can understand it. (010)
Ed (011)

Edward J. Barkmeyer Email: edbark@xxxxxxxx
National Institute of Standards & Technology
Manufacturing Systems Integration Division
100 Bureau Drive, Stop 8263 Tel: +1 3019753528
Gaithersburg, MD 208998263 FAX: +1 3019754694 (012)
"The opinions expressed above do not reflect consensus of NIST,
and have not been reviewed by any Government authority." (013)
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