Axiomatizing Changing Conceptions of the Geometric Continuum I: Euclid-Hilbert†

Abstract

We begin with a general account of the goals of axiomatization, introducing a variant (modest) on Detlefsen’s notion of ‘complete descriptive axiomatization’. We examine the distinctions between the Greek and modern view of number, magnitude and proportion and consider how this impacts the interpretation of Hilbert’s axiomatization of geometry. We argue, as indeed did Hilbert, that Euclid’s propositions concerning polygons, area, and similar triangles are derivable (in their modern interpretation in terms of number) from Hilbert’s first order axioms. We argue that Hilbert’s axioms including continuity show much more than Euclid’s theorems on polygons and basic results in geometry and thus are an immodest complete descriptive axiomatization of that subject.