Abstract

In this dissertation we show, by developing a formal system for geometry that consists only of diagrams, that diagrams can be used in mathematics in essential ways. A grammar for the diagrams that determines what diagrams are well-formed and that distinguishes essential features from accidental ones is described. Also, the different objects of the diagrams are given a meaning that relates the diagrams to the abstract realm of geometry. The system has some transformation rules that allow the user to obtain new diagrams from given ones. A rigorous proof that the system is sound is given. That means that, by using the transformation rules on the diagrams of the system, it is impossible to prove a false statement--in other words, it is proven that no fallacies can arise from the system. We also describe some limitations of the system and show how they can be overcome by extending the system in different ways