Variations on a Theme of Makowsky
摘要
We distinguish the axiomatic study of proofs in geometry from the study about geometry from general axioms for mathematics. We briefly report on an abuse of that distinction and its unfortunate effect on US high school education. We review a number of twentieth century approaches to synthetic geometry. In doing so, we disambiguate (in the Wikipedia sense) the terms: metric, orthogonal, isotropic and hyperbolic. With some of these systems we are able to axiomatize ‘affine geometry’ over the complex field. The argument is trivial from Wu (Mechanical Theorem Proving in Geometry. Texts and Monographs in Symbolic Computation. Springer-Verlag, New York, 1994) or Szmielew (From Affine to Euclidean Geometry: An Axiomatic Approach. D. Reidel, Dordrecht, 1978), but not remarked by either of them.). We examine the general question of the connections between axioms for Affine geometries and the stability classification of associated complete first order theories of fields. We conclude with reminiscences of a half-century friendship with Janós.