Learning and Teaching Axiomatic Geometry

Some years ago I prepared for an ICMI conference at Bologna a discussion on “Le Choix d’Axiomes pour la Géométrie”, and later published it in L’Enseignement Mathématique [1]. In that paper I examined some of the reasons for continuing to teach axiomatic geometry in the secondary school and went on to explain my own preference for choosing axioms closely related to Artin’s treatment of affine geometry [2]. It was pointed out that in this approach one deals separately with the axioms of incidence, the axioms of order, and the axioms of orthogonality, thus dividing and isolating the mathematical difficulties to be overcome. This treatment closely parallels the one proposed by Choquet [3], except that the coordinatization of an affine plane or space has to be carried out abstractly in Artin’s manner without assuming the real numbers as already available. The effect of invoking the axioms of order or orthogonality is to impose specific limitations upon the coordinate field. In particular, one sees very clearly how the axioms of incidence and order alone lead to real coordinates, a fact of geometry that, historically speaking, lies behind the invention of the real number system itself.