Euclidean and Non-Euclidean Geometry in the History and Philosophy of Mathematical Practice

In the history of mathematical practice, geometry serves as an important focal point through which numerous changes in practice and associated (mathematical) worldviews can be studied. For instance, eighteenth century Kantian views on the synthetic a priori nature of geometry obfuscate practices of the likes of Menelaus of Alexandria who lived 16 centuries before him and are rarely mentioned. One could argue that Kant’s philosophy on the nature of space (and geometry) was relational to human perception and cognition, as opposed to specific mathematical discoveries per se. Tracing a curated selection of advances in Euclidean and non-Euclidean geometry is useful to understand the mathematical practices of the ancient Greeks, and the remarkable contributions of Arab mathematicians leading to post-Renaissance advances that ushered in the modern era with the work of Grothendieck, and Busemann in the “return to Euclid.”