For the field of manufacturing to become a science, it is necessary to develop general mathematical descriptions for the analysis and synthesis of manufacturing systems. Standard analytic models, as used extensively in the past, are ineffective for describing the general manufacturing situation due to their inability to deal with discontinuous and nonlinear phenomena. These limitations are transcended by algebraic models based on set structures. Set-theoretic and algebraic structures may be used to (1) express with precision a variety of important qualitative concepts such as hierarchies, (2) provide a uniform framework for more specialized theories such as automata theory and control theory, and (3) provide the groundwork for quantitative theories. By building on the results of other fields such as automata theory and computability theory, algebraic structures may be used as a general mathematical tool for studying the nature and limits of manufacturing systems. This paper shows how manufacturing systems may be modeled as automatons, and demonstrates the utility of this approach by discussing a number of theorems concerning the nature of manufacturing systems. In addition symbolic logic is used to formalize the Design Axioms, a set of generalized decision rules for design. The application of symbolic logic allows for the precise formulation of the Axioms and facilitates their interpretation in a logical programming language such as Prolog. Consequently, it is now possible to develop a consultive expert system for axiomatic design.