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Re: [ontolog-forum] Common Logic and decimal numerals

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Ed Barkmeyer <edbark@xxxxxxxx>
Date: Wed, 03 Jun 2009 13:48:47 -0400
Message-id: <4A26B77F.1070303@xxxxxxxx>
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 first-order 
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 general-purpose reasoner, whether of the FOL kind or of the 
DL kind or of the LP kind, with an engine that has built-in mathematical 
knowledge.    (05)

It is certainly common for reasoners to have built-in knowledge of 
arithmetic.  But there are also hybrid reasoners with built-in 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 "times-like" 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 well-defined.  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 unit-vector 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 CL-XML 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: +1 301-975-3528
Gaithersburg, MD 20899-8263                FAX: +1 301-975-4694    (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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