Dear John, (01)
Thank you for your comments. I agree with them. (02)
Perhaps I am groping towards a more subtle point - often a mathematical
understanding is deemed useful because it justifies a useful notation.
MathML is a notation which is limited to thet 6th form (en-uk) or 12th grade
(en-us) mathematics, which is all that engineers usually use. This can
define the level of mathematical understanding we are looking for. My example: (03)
> <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. (04)
is naive (but not so naive that it is indifferent to the nature of operators
and operands). At a slightly less naive level, we are postulating an
"algebraic structure" which contains the set of length values, the
"times-like" operator, and some other things. (05)
We are not going to create an ontology for all of mathematics. Instead we
need an ontology that is pitched at the right level to support our requirements. (06)
Best regards,
David (07)
At 08:43 03/06/2009 -0400, you wrote:
>Dear David,
>
>DL> 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.
>
>Many people become passionate about notations because any change
>in notation can make their programs incompatible or obsolete.
>
>That is why Common Logic was defined with an abstract syntax,
>examples of several different syntaxes, and the option for letting
>anyone define whatever new syntaxes they might find useful.
>
>For CL, no notation has a privileged status, and any notation
>that supports the standard semantics is as acceptable as any
>other -- provided that the notation has a formally defined
>mapping to the abstract syntax.
>
>The Wolfram Mathematica system is an example of a very large
>library of modules (called 'notebooks'), which support all the
>major versions of mathematics. The Mathematica software uses
>its own internal notation (which most users never see), and
>it supports an open-ended variety of external notations:
>a printed form for the traditional mathematical publications,
>a simplified form for typed input, MathML, and a variety of
>notations specialized for particular languages such as FORTRAN
>or LISP.
>
>I would recommend an approach along the lines of Mathematica,
>Common Logic, and many other systems that are based on an
>abstract syntax but an open-ended number of concrete syntaxes.
>
>John Sowa
>
>
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> (08)
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