ABSTRACT
This chapter introduces a number of mathematical preliminaries. It considers the nature of concrete embodiments. Concrete embodiments involve a basic set of objects, actions, or relationships and one or more operations and/or relations defined on this basic set, together with possibly one or more distinguished elements of the set. The objects of abstract mathematical study are called algebraic systems. Algebraic systems are like embodiments in the sense that they can be characterized in essentially the same way but they are unlike embodiments in the sense that the basic elements, operations and relations have no meaning. In standard mathematical treatments, mathematical systems are defined first and the basic terms are said to be undefined. Theories may be thought of as sets of properties of embodiments or algebraic systems. Formal systems are constructed to parallel axiomatic theories more typically studied by mathematicians. Meta-mathematics or proof theory is the informal study of formal mathematical systems.
