Reviving the philosophy of geometry

In the Anglophone world, the philosophical treatment of geometry has fallen on hard times. While in Germany as late as the 1920s there were vibrant discussions concerning the nature of geometry— especially in relation to the direction of its development, the role of intuition and the perception of physical space—the rise of logical empiricism largely brought these to a close. Accounts of mathematics in its totality as uniformly reducible to a language such as set theory have led the recipients of logical empiricist doctrines to ignore the thematic contours of modern mathematics. I argue, however, that geometry in all of its many guises flourishes in contemporary mathematics, and that the term ‘geometry’ itself continues to have an important meaning. One way then for philosophy to come to a better understanding of mathematics is to provide an account of modern geometry, including its development of new forms of space, both for mathematical physics and for arithmetic. I argue that we can find a good starting point for this work by returning to the discussions of Weyl and Cassirer on geometry. With the help of modern interpreters, we see that issues salient to these thinkers are very much relevant to us today. I also propose that an effective way to encompass a great part of modern geometry is via (1, 1)-toposes, also known as homotopy toposes, an important development of category theory. Recently an internal language for such categories has been devised, known as ‘homotopy type theory’, a variant of which captures the ‘cohesiveness’ of geometric spaces. With these tools in place, we can now start to see what an adequate philosophical account of current geometry might look like.