I am reading the paper of Jeremy Avigad, Edward Dean and John Mumma, entitled "A formal system for Euclid's Elements".
Could this approach be extended to the books of Apollonius of Perga dealing with conics ?
I am interested in complexity questions and the completeness/incompleteness of axiomatic systems for Greek geometry.
Maybe there is more to the late ideas of Frege about basing Arithmetic on Geometry than is generally believed...
Also, inspired by the well-known correspondence between the elementary theory of field extensions and the classical constructions with ruler and compass, one can ask: what kind of field extensions correspond to constructions in which we can draw conics as well as circles and lines ?
You may focus on the relationship between Euclidean Geometry vs Riemannian Geometry or Hilbert or Banach Spaces. See e.g., https://arxiv.org/pdf/1807.11290.pdf
Peter Simons is doing some wonderful work on a mereological approach to classical spherical geometry.
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