From Geometry to Algebra

Our aim is to see which practices of Greek geometry can be expressed in various logics. Thus we refine Detlefsen’s notion of descriptive complexity by providing a scheme of increasing more descriptive formalizations of geometry Following Hilbert we argue that defining a field structure on a line in ‘Euclidean geometry’ provides a foundation for both geometry and algebra. In particular we prove from first principles: √ 2 · √ 3 = √ 6, similar triangles have proportional sides, Euclid’s 3rd axiom: circle intersection, the area of every triangle is measured by a segment. For these as Hilbert showed, no theory of limits is needed. Thus, the first order theory as described by Hilbert or Tarski is adequate for proportion and polygonal area. We further consider the role of π and determining the area and circumference of a circle. and the area of a circle of radius r is πr. Here we extend the first order geometry by adding a constant for the length π. Here we will rely on the axiom of Archimedes but only in the metatheory and not at all on Dedekind completeness. The natural numbers are not definable in these geometries. Finally Dedekind completeness add a second order axiom to give the modern basics of modern analysis. Very preliminary -not yet for general release. I expect to give a major reorganization to this material. But the present version explains the role of multiplication as similarity vrs as repeated addition and sketches the study of circles. This requires adding formal notions of area and arc length. This is partially carried out in Section 7. This version just clarifies a few points from the presentation at the Urbana meeting.