Tentatively, I plan to close the course by connecting the old with the new, but this might be a quixotic dream. While I know quite a bit about hyperbolic space, it is from this modern viewpoint rather than the axiomatic viewpoint of Lobachevsky. By bringing the foundations of the subject in line with analysis, he brought Calculus to bear on the study of geometry allowing differential and algebraic geometry, together with Klein’s Erlangen program, to subsume the classical subject. The course design is a learning experience for me because, although my mathematical expertise is in geometry, that expertise is in the subject geometry blossomed into during the 20th century after Hilbert’s revolution. We also have a course specifically on Euclidean geometry that many of these students also take, which is why the mathematical focus is on noneuclidean geometry. one part a course in hyperbolic geometry. one part a course in logic: discussing axiomatic systems and their properties and Hilbert’s triumphant axiomatization of geometry bringing it in line with set theory and analysis.one part history: examining the critiques of Euclid’s Elements and the attempts to eliminate the parallel postulate.It’s unlike any other course I’ve taught before because it has a bit of a split personality: This quarter I’m teaching “Foundations of Geometry,” a senior level course aimed primarily at future secondary mathematics teachers.
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