Joel David Hamkins, Professor of Logic, Oxford University
This lecture is based on chapter 4 of my book, Lectures on the Philosophy of Mathematics, published with MIT Press, https://mitpress.mit.edu/books/lectur....

Lecture 4. Geometry

Classical Euclidean geometry is the archetype of a mathematical deductive process. Yet the impossibility of certain constructions by straightedge and compass, such as doubling the cube, trisecting the angle, or squaring the circle, hints at geometric realms beyond Euclid. The rise of non-Euclidean geometry, especially in light of scientific theories and observations suggesting that physical reality is not Euclidean, challenges previous accounts of what geometry is about. New formalizations, such as those of David Hilbert and Alfred Tarski, replace the old axiomatizations, augmenting and correcting Euclid with axioms on completeness and betweenness. Ultimately, Tarski’s decision procedure points to a tantalizing possibility of automation in geometrical reasoning.

See lecture course information, including the schedule of topics, at http://jdh.hamkins.org/lectures-on-th....