Axiomatizing changing conceptions of the geometric continuuum II: Archimedes – Descartes –Tarski – Hilbert

In Part I of this paper we argued that the first-order systems HP5 and EG are modest complete descriptive axiomatization of most of Euclidean geometry. In this paper we discuss two further modest complete descriptive axiomatizations: Tarksi’s for Cartesian geometry and new systems for adding π. In contrast we find Hilbert’s full second order system immodest for geometrical purposes but appropriate as a foundation for mathematical analysis. Part I [Baldwin 2017b] dealt primarily with Hilbert’s first order axioms for geometry; Part II deals with his ‘continuity axioms’ – the Archimedean and completeness axioms. Part I argued that the first-order systems HP5 and EG (defined below) are ‘modest’ complete descriptive axiomatization of most (described more precisely below) of Euclidean geometry. In this paper we consider some extensions of Tarski’s axioms for elementary geometry to deal with circles and contend: 1) that Tarski’s first-order axiom set E is a modest complete descriptive axiomatization of Cartesian geometry; 2) that the theories EGπ,C,A and E π,C,A are modest complete descriptive axiomatizations of extensions of these geometries designed to describe area and circumference of the circle; and 3) that, in contrast, Hilbert’s full second-order system in the Grundlagen is an immodest axiomatization of any of these geometries but a modest descriptive axiomatization the late 19th century conception of the real plane. We elaborate and place this study in a more general context in [Baldwin 2017a]. We recall some of the key material and notation from Part I. That paper involved two key elements. The first was the following quasi-historical description. Eu∗Research partially supported by Simons travel grant G5402.