Metaphysics and the Mathematical Diagram

Accounts of geometry are caught between the demands of history and philosophy, and are difficult to reduce to either. In a profoundly influential move, Plato used geometrical proof as one means of bootstrapping his Theory of Forms and what came to be called metaphysics, and the emergence of ontological modes of thinking. This has led to a style of thinking still common today that gets called ‘mathematical Platonism’. By contrast, the sheer diversity of mathematical practices across cultures and time has been adduced to claim their historical contingency, which has recently prompted Ian Hacking to question why there is philosophy of mathematics at all. The different roles assigned to geometrical diagrams in these debates form the focus of this chapter, which analyses in detail the contrasting discussions of diagrams, and of the linearization and spatialization of thinking, by Plato (especially Meno and the Republic), by the cognitive historian Reviel Netz, the media theorist Sybille Krämer, and the anthropologist Tim Ingold.