The Philosophical and Pedagogical Implications of a Computerized Diagrammatic System for Euclidean Geometry

This chapter briefly describes the computer proof system CDEG, version 2.0, a computerized formal system for giving diagrammatic proofs in Euclidean Geometry. The existence of such a computer proof system shows that geometric arguments that rely on diagrams can be made rigorous, and this fact has important philosophical implications for how we understand and teach geometry. The full details of the computer system and its underlying logic are complicated and are discussed elsewhere; this chapter instead gives a brief example of how the computer system can be used to prove a theorem of Euclidean geometry, and then discusses some of the philosophical implications of such a system. The chapter starts with a brief history of the use of diagrams in geometry, and explains why we might be interested in such a formal system. It then gives an introduction to what constitutes a geometric diagram in this context, and how such diagrams are manipulated by this computer system. As an example, it shows how Euclid’s first proposition from Book I of the Elements [1] can be derived in this system. CDEG is meant to formalize the proof methods found in Euclid’s Elements, and the following sections of this chapter discuss the correspondence between these two proof systems, the question of why we would want to develop a computer system that mimics Euclid’s proof practice, and other related philosophical issues. Finally, the last section of this chapter discusses some ways these philosophical issues might bear on the way we teach geometry courses. CDEG stands for “Computerized Diagrammatic Euclidean Geometry.” This computer proof system implements a diagrammatic formal system for giving diagram-based proofs of theorems of Euclidean geometry that are similar to the informal proofs found in Euclid’s Elements. It is based on the diagrammatic formal system FG, which is described in detail in my book, Euclid and his Twentieth Century Rivals: Diagrams in the Logic of Euclidean Geometry [7]. The book also describes an earlier version of CDEG; however, CDEG has evolved significantly since its publication, and is now publicly available in a beta version. Euclid’s Elements was written around 300 B.C. and was considered the gold standard of careful reasoning and mathematical rigor for roughly the next 2000 years. However, this changed with the development of the idea of a formal proof system in the late 1800s. A formal proof system is one in which all the allowable rules are set out carefully in advance in a way that can be followed mechanically, so as to leave no room for human error. Most formal proof