Shaping Plato: interpreting Plato’s philosophy through his geometry

‘Let no one who is not a geometer enter’ was allegedly displayed outside Plato’s Academy and his love of geometry is revealed by the many geometrical references occurring throughout his work. They are often ignored but I demonstrate their importance by showing that Plato’s use of geometry can help to shape our understanding of his philosophy. Although I mention other examples I focus on two geometrical references. I examine Plato’s exploration of apparently diminutive distant objects, whose real size can be found using geometrical calculations, and his employment of diagramma, which implies a diagram associated with theorems. Both are repeated and I track them across several dialogues to analyse, from Plato’s perspective, the validity of the associated philosophical arguments, and the significance of the repetitions. I also use my findings to critique, from a modern viewpoint, some theories of Platonic interpretation. I found that Plato associates distant objects with pleasures, which can also appear deceptively small, and diagramma with the acquisition of eternal and unchanging knowledge. Examining each individual account leaves unresolved issues, but I show that Plato’s repetition of the geometrical examples allows him to continue discussions aimed at philosophers across several dialogues, rectifying omissions and demonstrating the value of writing as a reminder. Analysis of his use of geometry also supports various interpretations of the dialogues, relating to their reception, the role of Socrates, implied references to the Forms, and the spuriousness of the Epinomis. From my examination of Plato’s repetitive use of geometry I deduce that it is incorrect to read each dialogue as an autonomous text. I conclude that the extended discussions have a value for Plato, as he advances his views for the edification of philosophers, and also for us, as we use them to gain insights into aspects of the dialogues beyond those envisaged by Plato.