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.