Constructive Geometry

Euclidean geometry, as presented by Euclid, consists of straightedge-and-compass constructions and rigorous reasoning about the results of those constructions. A consideration of the relation of the Euclidean “constructions” to “constructive mathematics” leads to the development of a first-order theory ECG of the “Euclidean Constructive Geometry”, which can serve as an axiomatization of Euclid rather close in spirit to the Elements of Euclid. ECG is axiomatized in a quantifier-free, disjunction-free way. Unlike previous intuitionistic geometries, it does not have apartness. Unlike previous algebraic theories of geometric constructions, it does not have a test-for-equality construction. We show that ECG is a good geometric theory, in the sense that with classical logic it is equivalent to textbook theories, and its models are (intuitionistically) planes over Euclidean fields. We then apply the methods of modern metamathematics to this theory, showing that if ECG proves an existential theorem, then the object proved to exist can be constructed from parameters, using the basic constructions of ECG (which correspond to the Euclidean straightedge-and-compass constructions). In particular, objects proved to exist in ECG depend continuously on parameters. We also study the formal relationships between several versions of Euclid’s parallel postulate, and show that each corresponds to a natural axiom system for Euclidean fields.