Variations on a Theme of Makowski

TL;DR

This paper explores the axiomatic foundations of various geometries, clarifies terminology, and investigates the connection between affine geometry axioms and the stability of related field theories, with historical reflections.

Contribution

It disambiguates key geometric terms, axiomatizes affine geometry over complex fields, and links geometric axioms to model-theoretic stability classifications.

Findings

Axiomatization of affine geometry over complex fields.

Clarification of metric, orthogonal, isotropic, and hyperbolic terms.

Connection between geometric axioms and stability of field theories.

Abstract

We distinguish the axiomatic study of proofs in geometry from 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 20th 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 [Wu94] or [Szm78], 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 Jan\'{o}s.