In the history of western thought, mathematics has had a central place. Its own history has “internal” aspects, characterised by a cumulative development of great and greater complexity and a proliferation of powerful methods for solving problems which mathematics generated from within itself, and which thereby gradually transformed its very nature. However, this class will seek to regard such “internal” aspects in the context of their place within and their contribution to broader philosophical inquiries and to conceptions of how the natural world is know. Central “moments” in the history will be featured, beginning with ancient Greek mathematics through to the development of non-Euclidean geometry in the 19th Century and beyond.