The mathematical ontology defines the mathematics required to describe physical processes. The EngMath ontology [Gruber & Olsen, 1994], available in the Ontolingua ontology library, is perfectly suited for this job and has therefore been (re)used for this. In this section, only a very short description is given that should be sufficient to understand the projection of the process ontology onto mathematics. For detailed information on EngMath see [Gruber & Olsen, 1994].

The EngMath ontology formalizes mathematical modelling in engineering. The ontology includes conceptual foundations for scalar, vector, and tensor quantities, physical dimensions, units of measure, functions of quantities, and dimensionless quantities.

A *physical quantity* is a measure of some quantifiable aspect of
the modelled world characterized by a *physical dimension* such as
length, mass or time. Quantities in the EngMath ontology can be
expressed in various *units of measure* e.g. meter, inch,
kilogram, pound, etc. The ontology defines all relations between
quantities, units of measure and dimensions. A special class of
physical quantities are *time-dependent quantities*. These are in
fact continuous functions from a quantity with the time dimension to
another physical quantity, and can therefore be interpreted as dynamic
quantities, varying over time.

Another important EngMath class for the PHYSSYS ontology are the
Ontolingua (KIF) expressions that serve as a meta-level description of
mathematical relations between physical quantities. For instance, a
relation `r` between two time-dependent physical quantities
`x` and `y` can be defined as:

Here, the expression `x = 2 * y` is the (infix form of the)
Ontolingua expression used to define the EngMath relation `r`. In
the PHYSSYS ontology, two time-dependent physical quantities are
used to mathematically describe an energy flow and relations similar to
`r` define the mathematical relationships between these
quantities.

Fri Sep 27 13:28:43 MET DST 1996