Geometry and Proof

We discuss the relation between the specific axiomatizations, specifically Hilbert’s [Hil71] reformulation of geometry at the beginning of the last century, and the way elementary geometry has been expounded in high schools in the United States. Further we discuss the connections among formal logic, the teaching of logic and the preparation of high school teachers. In part our goal is to describe how high school geometry instruction developed in the United States during the 20th century in hopes of learning of the development in other countries. We conclude with some recommendations concerning teaching reasoning to high school students and preparing future teachers for this task. We view Hilbert’s geometry as a critique of Euclid and focus on three aspects of it: a) the need for undefined terms, b) continuity axioms, c) the mobility postulate. (We are at the moment being historically cavalier and using ‘Hilbert’ as a surrogate for an analysis by many contributors including in particular Pasch and Dedekind. The problems that Hilbert addressed had been raised since classical times; Hilbert’s simultaneous solution to many of them in the wake of the discovery of non-Euclidean geometry made this book seminal.) a) Hilbert’s recognition that Euclid’s definition of e.g a point was meaningless and that the appropriate procedure is to consider a set of axioms that represents objects which ”might as well be chairs, tables and beer mugs ” plays a foundational role in the modern approach to mathematics and in particular to model theory. Two technical notions arise from the common notion of ‘definition’. The basic notions are not defined; rather the system (geometry) is ‘defined’ by the axioms relating them; auxiliary notions are ‘defined’ as abbreviation for relations among the basic notions. This modern conception was influenced by Frege [Ste] as well as Dedekind. b) Hilbert’s introduction of continuity axioms meant that he was studying ‘geometry over the reals’. This is a notion that was likely meaningless to Euclid. The distinction between the Greek conception of numbers and the modern view of number systems is another variant of insight a). More technically, Hilbert is pointing out are geometries over various fields that have to be considered and founding the modern understanding of the relation between the algebraic